Path Integrals in Quantum Mechanics, Statistics and Polymer Physics

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Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets Hagen Kleinert Professor of Physics Freie Universität Berlin

To Annemarie and Hagen II

Nature alone knows what she wants.

Goethe

Preface The third edition of this book appeared in 2004 and was reprinted in the same year without improvements. The present fourth edition contains several extensions. Chapter 4 includes now semiclassical expansions of higher order. Chapter 8 offers an additional path integral formulation of spinning particles whose action contains a vector field and a Wess-Zumino term. From this, the Landau-Lifshitz equation for spin precession is derived which governs the behavior of quantum spin liquids. The path integral demonstrates that fermions can be described by Bose fields—the basis of Skyrmion theories. A further new section introduces the Berry phase, a useful tool to explain many interesting physical phenomena. Chapter 10 gives more details on magnetic monopoles and multivalued fields. Another feature is new in this edition: sections of a more technical nature are printed in smaller font size. They can well be omitted in a first reading of the book. Among the many people who spotted printing errors and helped me improve various text passages are Dr. A. Chervyakov, Dr. A. Pelster, Dr. F. Nogueira, Dr. M. Weyrauch, Dr. H. Baur, Dr. T. Iguchi, V. Bezerra, D. Jahn, S. Overesch, and especially Dr. Annemarie Kleinert.

H. Kleinert Berlin, June 2006

vii

viii

H. Kleinert, PATH INTEGRALS

Preface to Third Edition This third edition of the book improves and extends considerably the second edition of 1995: • Chapter 2 now contains a path integral representation of the scattering amplitude and new methods of calculating functional determinants for timedependent second-order differential operators. Most importantly, it introduces the quantum field-theoretic definition of path integrals, based on perturbation expansions around the trivial harmonic theory. • Chapter 3 presents more exactly solvable path integrals than in the previous editions. It also extends the Bender-Wu recursion relations for calculating perturbation expansions to more general types of potentials. • Chapter 4 discusses now in detail the quasiclassical approximation to the scattering amplitude and Thomas-Fermi approximation to atoms. • Chapter 5 proves the convergence of variational perturbation theory. It also discusses atoms in strong magnetic fields and the polaron problem. • Chapter 6 shows how to obtain the spectrum of systems with infinitely high walls from perturbation expansions. • Chapter 7 offers a many-path treatment of Bose-Einstein condensation and degenerate Fermi gases. • Chapter 10 develops the quantum theory of a particle in curved space, treated before only in the time-sliced formalism, to perturbatively defined path integrals. Their reparametrization invariance imposes severe constraints upon integrals over products of distributions. We derive unique rules for evaluating these integrals, thus extending the linear space of distributions to a semigroup. • Chapter 15 offers a closed expression for the end-to-end distribution of stiff polymers valid for all persistence lengths. • Chapter 18 derives the operator Langevin equation and the Fokker-Planck equation from the forward–backward path integral. The derivation in the literature was incomplete, and the gap was closed only recently by an elegant calculation of the Jacobian functional determinant of a second-order differential operator with dissipation. ix

x • Chapter 20 is completely new. It introduces the reader into the applications of path integrals to the fascinating new field of econophysics. For a few years, the third edition has been freely available on the internet, and several readers have sent useful comments, for instance E. Babaev, H. Baur, B. Budnyj, Chen Li-ming, A.A. Dr˘agulescu, K. Glaum, I. Grigorenko, T.S. Hatamian, P. Hollister, P. Jizba, B. Kastening, M. Krämer, W.-F. Lu, S. Mukhin, A. Pelster, ¨ C. Ocalır, M.B. Pinto, C. Schubert, S. Schmidt, R. Scalettar, C. Tangui, and M. van Vugt. Reported errors are corrected in the internet edition. When writing the new part of Chapter 2 on the path integral representation of the scattering amplitude I profited from discussions with R. Rosenfelder. In the new parts of Chapter 5 on polarons, many useful comments came from J.T. Devreese, F.M. Peeters, and F. Brosens. In the new Chapter 20, I profited from discussions with F. Nogueira, A.A. Dr˘agulescu, E. Eberlein, J. Kallsen, M. Schweizer, P. Bank, M. Tenney, and E.C. Chang. As in all my books, many printing errors were detected by my secretary S. Endrias and many improvements are due to my wife Annemarie without whose permanent encouragement this book would never have been finished.

H. Kleinert Berlin, August 2003


Preface to Second Edition Since this book first appeared three years ago, a number of important developments have taken place calling for various extensions to the text. Chapter 4 now contains a discussion of the features of the semiclassical quantization which are relevant for multidimensional chaotic systems. Chapter 3 derives perturbation expansions in terms of Feynman graphs, whose use is customary in quantum field theory. Correspondence is established with Rayleigh-Schrödinger perturbation theory. Graphical expansions are used in Chapter 5 to extend the Feynman-Kleinert variational approach into a systematic variational perturbation theory. Analytically inaccessible path integrals can now be evaluated with arbitrary accuracy. In contrast to ordinary perturbation expansions which always diverge, the new expansions are convergent for all coupling strengths, including the strong-coupling limit. Chapter 10 contains now a new action principle which is necessary to derive the correct classical equations of motion in spaces with curvature and a certain class of torsion (gradient torsion). Chapter 19 is new. It deals with relativistic path integrals, which were previously discussed only briefly in two sections at the end of Chapter 15. As an application, the path integral of the relativistic hydrogen atom is solved. Chapter 16 is extended by a theory of particles with fractional statistics (anyons), from which I develop a theory of polymer entanglement. For this I introduce nonabelian Chern-Simons fields and show their relationship with various knot polynomials (Jones, HOMFLY). The successful explanation of the fractional quantum Hall effect by anyon theory is discussed — also the failure to explain high-temperature superconductivity via a Chern-Simons interaction. Chapter 17 offers a novel variational approach to tunneling amplitudes. It extends the semiclassical range of validity from high to low barriers. As an application, I increase the range of validity of the currently used large-order perturbation theory far into the regime of low orders. This suggests a possibility of greatly improving existing resummation procedures for divergent perturbation series of quantum field theories. The Index now also contains the names of authors cited in the text. This may help the reader searching for topics associated with these names. Due to their great number, it was impossible to cite all the authors who have made important contributions. I apologize to all those who vainly search for their names. xi

xii In writing the new sections in Chapters 4 and 16, discussions with Dr. D. Wintgen and, in particular, Dr. A. Schakel have been extremely useful. I also thank Professors G. Gerlich, P. Hänggi, H. Grabert, M. Roncadelli, as well as Dr. A. Pelster, and Mr. R. Karrlein for many relevant comments. Printing errors were corrected by my secretary Ms. S. Endrias and by my editor Ms. Lim Feng Nee of World Scientific. Many improvements are due to my wife Annemarie.

H. Kleinert Berlin, December 1994


Preface to First Edition These are extended lecture notes of a course on path integrals which I delivered at the Freie Universität Berlin during winter 1989/1990. My interest in this subject dates back to 1972 when the late R. P. Feynman drew my attention to the unsolved path integral of the hydrogen atom. I was then spending my sabbatical year at Caltech, where Feynman told me during a discussion how embarrassed he was, not being able to solve the path integral of this most fundamental quantum system. In fact, this had made him quit teaching this subject in his course on quantum mechanics as he had initially done.1 Feynman challenged me: “Kleinert, you figured out all that grouptheoretic stuff of the hydrogen atom, why don’t you solve the path integral!” He was referring to my 1967 Ph.D. thesis2 where I had demonstrated that all dynamical questions on the hydrogen atom could be answered using only operations within a dynamical group O(4, 2). Indeed, in that work, the four-dimensional oscillator played a crucial role and the missing steps to the solution of the path integral were later found to be very few. After returning to Berlin, I forgot about the problem since I was busy applying path integrals in another context, developing a field-theoretic passage from quark theories to a collective field theory of hadrons.3 Later, I carried these techniques over into condensed matter (superconductors, superfluid 3 He) and nuclear physics. Path integrals have made it possible to build a unified field theory of collective phenomena in quite different physical systems.4 The hydrogen problem came up again in 1978 as I was teaching a course on quantum mechanics. To explain the concept of quantum fluctuations, I gave an introduction to path integrals. At the same time, a postdoc from Turkey, I. H. Duru, joined my group as a Humboldt fellow. Since he was familiar with quantum mechanics, I suggested that we should try solving the path integral of the hydrogen atom. He quickly acquired the basic techniques, and soon we found the most important ingredient to the solution: The transformation of time in the path integral to a new path-dependent pseudotime, combined with a transformation of the coordinates to 1

Quoting from the preface of the textbook by R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965: “Over the succeeding years, ... Dr. Feynman’s approach to teaching the subject of quantum mechanics evolved somewhat away from the initial path integral approach.” 2 H. Kleinert, Fortschr. Phys. 6 , 1, (1968), and Group Dynamics of the Hydrogen Atom, Lectures presented at the 1967 Boulder Summer School, published in Lectures in Theoretical Physics, Vol. X B, pp. 427–482, ed. by A.O. Barut and W.E. Brittin, Gordon and Breach, New York, 1968. 3 See my 1976 Erice lectures, Hadronization of Quark Theories, published in Understanding the Fundamental Constituents of Matter , Plenum press, New York, 1978, p. 289, ed. by A. Zichichi. 4 H. Kleinert, Phys. Lett. B 69 , 9 (1977); Fortschr. Phys. 26 , 565 (1978); 30 , 187, 351 (1982).

xiii

xiv “square root coordinates” (to be explained in Chapters 13 and 14).5 These transformations led to the correct result, however, only due to good fortune. In fact, our procedure was immediately criticized for its sloppy treatment of the time slicing.6 A proper treatment could, in principle, have rendered unwanted extra terms which our treatment would have missed. Other authors went through the detailed timeslicing procedure,7 but the correct result emerged only by transforming the measure of path integration inconsistently. When I calculated the extra terms according to the standard rules I found them to be zero only in two space dimensions.8 The same treatment in three dimensions gave nonzero “corrections” which spoiled the beautiful result, leaving me puzzled. Only recently I happened to locate the place where the three-dimensional treatment went wrong. I had just finished a book on the use of gauge fields in condensed matter physics.9 The second volume deals with ensembles of defects which are defined and classified by means of operational cutting and pasting procedures on an ideal crystal. Mathematically, these procedures correspond to nonholonomic mappings. Geometrically, they lead from a flat space to a space with curvature and torsion. While proofreading that book, I realized that the transformation by which the path integral of the hydrogen atom is solved also produces a certain type of torsion (gradient torsion). Moreover, this happens only in three dimensions. In two dimensions, where the time-sliced path integral had been solved without problems, torsion is absent. Thus I realized that the transformation of the time-sliced measure had a hitherto unknown sensitivity to torsion. It was therefore essential to find a correct path integral for a particle in a space with curvature and gradient torsion. This was a nontrivial task since the literature was ambiguous already for a purely curved space, offering several prescriptions to choose from. The corresponding equivalent Schrödinger equations differ by multiples of the curvature scalar.10 The ambiguities are path integral analogs of the so-called operator-ordering problem in quantum mechanics. When trying to apply the existing prescriptions to spaces with torsion, I always ran into a disaster, some even yielding noncovariant answers. So, something had to be wrong with all of them. Guided by the idea that in spaces with constant curvature the path integral should produce the same result as an operator quantum mechanics based on a quantization of angular momenta, I was eventually able to find a consistent quantum equivalence principle 5

I.H. Duru and H. Kleinert, Phys. Lett. B 84 , 30 (1979), Fortschr. Phys. 30 , 401 (1982). G.A. Ringwood and J.T. Devreese, J. Math. Phys. 21 , 1390 (1980). 7 R. Ho and A. Inomata, Phys. Rev. Lett. 48 , 231 (1982); A. Inomata, Phys. Lett. A 87 , 387 (1981). 8 H. Kleinert, Phys. Lett. B 189 , 187 (1987); contains also a criticism of Ref. 7. 9 H. Kleinert, Gauge Fields in Condensed Matter , World Scientific, Singapore, 1989, Vol. I, pp. 1–744, Superflow and Vortex Lines, and Vol. II, pp. 745–1456, Stresses and Defects. 10 B.S. DeWitt, Rev. Mod. Phys. 29 , 377 (1957); K.S. Cheng, J. Math. Phys. 13 , 1723 (1972), H. Kamo and T. Kawai, Prog. Theor. Phys. 50 , 680, (1973); T. Kawai, Found. Phys. 5 , 143 (1975), H. Dekker, Physica A 103 , 586 (1980), G.M. Gavazzi, Nuovo Cimento 101 A, 241 (1981); M.S. Marinov, Physics Reports 60 , 1 (1980). 6


xv for path integrals in spaces with curvature and gradient torsion,11 thus offering also a unique solution to the operator-ordering problem. This was the key to the leftover problem in the Coulomb path integral in three dimensions — the proof of the absence of the extra time slicing contributions presented in Chapter 13. Chapter 14 solves a variety of one-dimensional systems by the new techniques. Special emphasis is given in Chapter 8 to instability (path collapse) problems in the Euclidean version of Feynman’s time-sliced path integral. These arise for actions containing bottomless potentials. A general stabilization procedure is developed in Chapter 12. It must be applied whenever centrifugal barriers, angular barriers, or Coulomb potentials are present.12 Another project suggested to me by Feynman, the improvement of a variational approach to path integrals explained in his book on statistical mechanics13 , found a faster solution. We started work during my sabbatical stay at the University of California at Santa Barbara in 1982. After a few meetings and discussions, the problem was solved and the preprint drafted. Unfortunately, Feynman’s illness prevented him from reading the final proof of the paper. He was able to do this only three years later when I came to the University of California at San Diego for another sabbatical leave. Only then could the paper be submitted.14 Due to recent interest in lattice theories, I have found it useful to exhibit the solution of several path integrals for a finite number of time slices, without going immediately to the continuum limit. This should help identify typical lattice effects seen in the Monte Carlo simulation data of various systems. The path integral description of polymers is introduced in Chapter 15 where stiffness as well as the famous excluded-volume problem are discussed. Parallels are drawn to path integrals of relativistic particle orbits. This chapter is a preparation for ongoing research in the theory of fluctuating surfaces with extrinsic curvature stiffness, and their application to world sheets of strings in particle physics.15 I have also introduced the field-theoretic description of a polymer to account for its increasing relevance to the understanding of various phase transitions driven by fluctuating line-like excitations (vortex lines in superfluids and superconductors, defect lines in crystals and liquid crystals).16 Special attention has been devoted in Chapter 16 to simple topological questions of polymers and particle orbits, the latter arising by the presence of magnetic flux tubes (Aharonov-Bohm effect). Their relationship to Bose and Fermi statistics of particles is pointed out and the recently popular topic of fractional statistics is introduced. A survey of entanglement phenomena of single orbits and pairs of them (ribbons) is given and their application to biophysics is indicated. 11

H. Kleinert, Mod. Phys. Lett. A 4 , 2329 (1989); Phys. Lett. B 236 , 315 (1990). H. Kleinert, Phys. Lett. B 224 , 313 (1989). 13 R.P. Feynman, Statistical Mechanics, Benjamin, Reading, 1972, Section 3.5. 14 R.P. Feynman and H. Kleinert, Phys. Rev. A 34 , 5080, (1986). 15 A.M. Polyakov, Nucl. Phys. B 268 , 406 (1986), H. Kleinert, Phys. Lett. B 174 , 335 (1986). 16 See Ref. 9. 12

xvi Finally, Chapter 18 contains a brief introduction to the path integral approach of nonequilibrium quantum-statistical mechanics, deriving from it the standard Langevin and Fokker-Planck equations. I want to thank several students in my class, my graduate students, and my postdocs for many useful discussions. In particular, T. Eris, F. Langhammer, B. Meller, I. Mustapic, T. Sauer, L. Semig, J. Zaun, and Drs. G. Germán, C. Holm, D. Johnston, and P. Kornilovitch have all contributed with constructive criticism. Dr. U. Eckern from Karlsruhe University clarified some points in the path integral derivation of the Fokker-Planck equation in Chapter 18. Useful comments are due to Dr. P.A. Horvathy, Dr. J. Whitenton, and to my colleague Prof. W. Theis. Their careful reading uncovered many shortcomings in the first draft of the manuscript. Special thanks go to Dr. W. Janke with whom I had a fertile collaboration over the years and many discussions on various aspects of path integration. Thanks go also to my secretary S. Endrias for her help in preparing the manuscript in LATEX, thus making it readable at an early stage, and to U. Grimm for drawing the figures. Finally, and most importantly, I am grateful to my wife Dr. Annemarie Kleinert for her inexhaustible patience and constant encouragement.

H. Kleinert Berlin, January 1990


Contents

1 Fundamentals 1.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . 1.2 Relativistic Mechanics in Curved Spacetime . . . . . . . . 1.3 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 1.3.1 Bragg Reflections and Interference . . . . . . . . . . 1.3.2 Matter Waves . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Schrödinger Equation . . . . . . . . . . . . . . . . . 1.3.4 Particle Current Conservation . . . . . . . . . . . . . 1.4 Dirac’s Bra-Ket Formalism . . . . . . . . . . . . . . . . . . 1.4.1 Basis Transformations . . . . . . . . . . . . . . . . . 1.4.2 Bracket Notation . . . . . . . . . . . . . . . . . . . . 1.4.3 Continuum Limit . . . . . . . . . . . . . . . . . . . . 1.4.4 Generalized Functions . . . . . . . . . . . . . . . . . 1.4.5 Schrödinger Equation in Dirac Notation . . . . . . . 1.4.6 Momentum States . . . . . . . . . . . . . . . . . . . 1.4.7 Incompleteness and Poisson’s Summation Formula . 1.5 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Uncertainty Relation . . . . . . . . . . . . . . . . . . 1.5.2 Density Matrix and Wigner Function . . . . . . . . . 1.5.3 Generalization to Many Particles . . . . . . . . . . . 1.6 Time Evolution Operator . . . . . . . . . . . . . . . . . . . 1.7 Properties of Time Evolution Operator . . . . . . . . . . . 1.8 Heisenberg Picture of Quantum Mechanics . . . . . . . . . 1.9 Interaction Picture and Perturbation Expansion . . . . . . 1.10 Time Evolution Amplitude . . . . . . . . . . . . . . . . . . 1.11 Fixed-Energy Amplitude . . . . . . . . . . . . . . . . . . . 1.12 Free-Particle Amplitudes . . . . . . . . . . . . . . . . . . . 1.13 Quantum Mechanics of General Lagrangian Systems . . . . 1.14 Particle on the Surface of a Sphere . . . . . . . . . . . . . 1.15 Spinning Top . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16.1 Scattering Matrix . . . . . . . . . . . . . . . . . . . 1.16.2 Cross Section . . . . . . . . . . . . . . . . . . . . . . 1.16.3 Born Approximation . . . . . . . . . . . . . . . . . . 1.16.4 Partial Wave Expansion and Eikonal Approximation xvii

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1 1 10 11 12 13 15 17 18 18 20 22 23 25 26 28 31 32 33 34 34 37 39 42 43 45 47 51 57 59 67 67 68 70 70

xviii 1.16.5 Scattering Amplitude from Time Evolution Amplitude 1.16.6 Lippmann-Schwinger Equation . . . . . . . . . . . . . 1.17 Classical and Quantum Statistics . . . . . . . . . . . . . . . 1.17.1 Canonical Ensemble . . . . . . . . . . . . . . . . . . . 1.17.2 Grand-Canonical Ensemble . . . . . . . . . . . . . . . 1.18 Density of States and Tracelog . . . . . . . . . . . . . . . . . Appendix 1A Simple Time Evolution Operator . . . . . . . . . . . Appendix 1B Convergence of Fresnel Integral . . . . . . . . . . . . Appendix 1C The Asymmetric Top . . . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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72 72 76 77 77 81 83 84 85 87

2 Path Integrals — Elementary Properties and Simple Solutions 2.1 Path Integral Representation of Time Evolution Amplitudes . 2.1.1 Sliced Time Evolution Amplitude . . . . . . . . . . . . . 2.1.2 Zero-Hamiltonian Path Integral . . . . . . . . . . . . . . 2.1.3 Schrödinger Equation for Time Evolution Amplitude . . 2.1.4 Convergence of Sliced Time Evolution Amplitude . . . . 2.1.5 Time Evolution Amplitude in Momentum Space . . . . . 2.1.6 Quantum-Mechanical Partition Function . . . . . . . . . 2.1.7 Feynman’s Configuration Space Path Integral . . . . . . 2.2 Exact Solution for Free Particle . . . . . . . . . . . . . . . . . 2.2.1 Direct Solution . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Fluctuations around Classical Path . . . . . . . . . . . . 2.2.3 Fluctuation Factor . . . . . . . . . . . . . . . . . . . . . 2.2.4 Finite Slicing Properties of Free-Particle Amplitude . . . 2.3 Exact Solution for Harmonic Oscillator . . . . . . . . . . . . . 2.3.1 Fluctuations around Classical Path . . . . . . . . . . . . 2.3.2 Fluctuation Factor . . . . . . . . . . . . . . . . . . . . . 2.3.3 The iη-Prescription and Maslov-Morse Index . . . . . . 2.3.4 Continuum Limit . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Useful Fluctuation Formulas . . . . . . . . . . . . . . . . 2.3.6 Oscillator Amplitude on Finite Time Lattice . . . . . . . 2.4 Gelfand-Yaglom Formula . . . . . . . . . . . . . . . . . . . . . 2.4.1 Recursive Calculation of Fluctuation Determinant . . . . 2.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Calculation on Unsliced Time Axis . . . . . . . . . . . . 2.4.4 D’Alembert’s Construction . . . . . . . . . . . . . . . . 2.4.5 Another Simple Formula . . . . . . . . . . . . . . . . . . 2.4.6 Generalization to D Dimensions . . . . . . . . . . . . . 2.5 Harmonic Oscillator with Time-Dependent Frequency . . . . . 2.5.1 Coordinate Space . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Momentum Space . . . . . . . . . . . . . . . . . . . . . 2.6 Free-Particle and Oscillator Wave Functions . . . . . . . . . . 2.7 General Time-Dependent Harmonic Action . . . . . . . . . . .

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89 89 89 91 92 92 94 96 97 101 101 102 104 110 111 111 113 114 115 116 118 119 120 120 122 123 124 126 126 127 129 131 133


xix 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15

Path Integrals and Quantum Statistics . . . . . . . . . . . . . . Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Statistics of Harmonic Oscillator . . . . . . . . . . . . Time-Dependent Harmonic Potential . . . . . . . . . . . . . . . Functional Measure in Fourier Space . . . . . . . . . . . . . . . Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation Techniques on Sliced Time Axis via Poisson Formula Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15.1 Zero-Temperature Evaluation of Frequency Sum . . . . . . 2.15.2 Finite-Temperature Evaluation of Frequency Sum . . . . . 2.15.3 Quantum-Mechanical Harmonic Oscillator . . . . . . . . . 2.15.4 Tracelog of First-Order Differential Operator . . . . . . . 2.15.5 Gradient Expansion of One-Dimensional Tracelog . . . . . 2.15.6 Duality Transformation and Low-Temperature Expansion 2.16 Finite-N Behavior of Thermodynamic Quantities . . . . . . . . 2.17 Time Evolution Amplitude of Freely Falling Particle . . . . . . . 2.18 Charged Particle in Magnetic Field . . . . . . . . . . . . . . . . 2.18.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18.2 Gauge Properties . . . . . . . . . . . . . . . . . . . . . . . 2.18.3 Time-Sliced Path Integration . . . . . . . . . . . . . . . . 2.18.4 Classical Action . . . . . . . . . . . . . . . . . . . . . . . 2.18.5 Translational Invariance . . . . . . . . . . . . . . . . . . . 2.19 Charged Particle in Magnetic Field plus Harmonic Potential . . 2.20 Gauge Invariance and Alternative Path Integral Representation 2.21 Velocity Path Integral . . . . . . . . . . . . . . . . . . . . . . . . 2.22 Path Integral Representation of Scattering Matrix . . . . . . . . 2.22.1 General Development . . . . . . . . . . . . . . . . . . . . 2.22.2 Improved Formulation . . . . . . . . . . . . . . . . . . . . 2.22.3 Eikonal Approximation to Scattering Amplitude . . . . . 2.23 Heisenberg Operator Approach to Time Evolution Amplitude . . 2.23.1 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . 2.23.2 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 2.23.3 Charged Particle in Magnetic Field . . . . . . . . . . . . . Appendix 2A Baker-Campbell-Hausdorff Formula and Magnus Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2B Direct Calculation of Time-Sliced Oscillator Amplitude Appendix 2C Derivation of Mehler Formula . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134 136 142 146 150 153 154 157 158 161 163 164 166 167 174 176 178 178 181 181 183 184 185 187 188 189 189 192 193 193 194 196 196 200 203 204 205

3 External Sources, Correlations, and Perturbation Theory 208 3.1 External Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 208 3.2 Green Function of Harmonic Oscillator . . . . . . . . . . . . . . 212 3.2.1 Wronski Construction . . . . . . . . . . . . . . . . . . . . 212

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3.3

3.4 3.5 3.6 3.7 3.8

3.9 3.10 3.11 3.12

3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22

3.23

3.2.2 Spectral Representation . . . . . . . . . . . . . . . . . . . 216 Green Functions of First-Order Differential Equation . . . . . . 218 3.3.1 Time-Independent Frequency . . . . . . . . . . . . . . . . 218 3.3.2 Time-Dependent Frequency . . . . . . . . . . . . . . . . . 225 Summing Spectral Representation of Green Function . . . . . . 228 Wronski Construction for Periodic and Antiperiodic Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Time Evolution Amplitude in Presence of Source Term . . . . . 231 Time Evolution Amplitude at Fixed Path Average . . . . . . . 235 External Source in Quantum-Statistical Path Integral . . . . . . 236 3.8.1 Continuation of Real-Time Result . . . . . . . . . . . . . 237 3.8.2 Calculation at Imaginary Time . . . . . . . . . . . . . . . 241 Lattice Green Function . . . . . . . . . . . . . . . . . . . . . . . 248 Correlation Functions, Generating Functional, and Wick Expansion 248 3.10.1 Real-Time Correlation Functions . . . . . . . . . . . . . . 251 Correlation Functions of Charged Particle in Magnetic Field . . . 253 Correlation Functions in Canonical Path Integral . . . . . . . . . 254 3.12.1 Harmonic Correlation Functions . . . . . . . . . . . . . . 255 3.12.2 Relations between Various Amplitudes . . . . . . . . . . . 257 3.12.3 Harmonic Generating Functionals . . . . . . . . . . . . . . 258 Particle in Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . 261 Heat Bath of Photons . . . . . . . . . . . . . . . . . . . . . . . . 265 Harmonic Oscillator in Ohmic Heat Bath . . . . . . . . . . . . . 267 Harmonic Oscillator in Photon Heat Bath . . . . . . . . . . . . 270 Perturbation Expansion of Anharmonic Systems . . . . . . . . . 271 Rayleigh-Schrödinger and Brillouin-Wigner Perturbation Expansion 275 Level-Shifts and Perturbed Wave Functions from Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Calculation of Perturbation Series via Feynman Diagrams . . . . 281 Perturbative Definition of Interacting Path Integrals . . . . . . . 286 Generating Functional of Connected Correlation Functions . . . 287 3.22.1 Connectedness Structure of Correlation Functions . . . . . 288 3.22.2 Correlation Functions versus Connected Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 3.22.3 Functional Generation of Vacuum Diagrams . . . . . . . . 293 3.22.4 Correlation Functions from Vacuum Diagrams . . . . . . . 297 3.22.5 Generating Functional for Vertex Functions. Effective Action 299 3.22.6 Ginzburg-Landau Approximation to Generating Functional 304 3.22.7 Composite Fields . . . . . . . . . . . . . . . . . . . . . . . 305 Path Integral Calculation of Effective Action by Loop Expansion 306 3.23.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . 306 3.23.2 Mean-Field Approximation . . . . . . . . . . . . . . . . . 307 3.23.3 Corrections from Quadratic Fluctuations . . . . . . . . . . 311 3.23.4 Effective Action to Second Order in h ¯ . . . . . . . . . . . 314 H. Kleinert, PATH INTEGRALS

xxi 3.23.5 Finite-Temperature Two-Loop Effective Action . . . . . 3.23.6 Background Field Method for Effective Action . . . . . 3.24 Nambu-Goldstone Theorem . . . . . . . . . . . . . . . . . . . 3.25 Effective Classical Potential . . . . . . . . . . . . . . . . . . . 3.25.1 Effective Classical Boltzmann Factor . . . . . . . . . . . 3.25.2 Effective Classical Hamiltonian . . . . . . . . . . . . . . 3.25.3 High- and Low-Temperature Behavior . . . . . . . . . . 3.25.4 Alternative Candidate for Effective Classical Potential . 3.25.5 Harmonic Correlation Function without Zero Mode . . . 3.25.6 Perturbation Expansion . . . . . . . . . . . . . . . . . . 3.25.7 Effective Potential and Magnetization Curves . . . . . . 3.25.8 First-Order Perturbative Result . . . . . . . . . . . . . . 3.26 Perturbative Approach to Scattering Amplitude . . . . . . . . 3.26.1 Generating Functional . . . . . . . . . . . . . . . . . . . 3.26.2 Application to Scattering Amplitude . . . . . . . . . . . 3.26.3 First Correction to Eikonal Approximation . . . . . . . 3.26.4 Rayleigh-Schrödinger Expansion of Scattering Amplitude 3.27 Functional Determinants from Green Functions . . . . . . . . Appendix 3A Matrix Elements for General Potential . . . . . . . . . Appendix 3B Energy Shifts for gx4 /4-Interaction . . . . . . . . . . . Appendix 3C Recursion Relations for Perturbation Coefficients . . . 3C.1 One-Dimensional Interaction x4 . . . . . . . . . . . . . . 3C.2 General One-Dimensional Interaction . . . . . . . . . . . 3C.3 Cumulative Treatment of Interactions x4 and x3 . . . . . 3C.4 Ground-State Energy with External Current . . . . . . . 3C.5 Recursion Relation for Effective Potential . . . . . . . . 3C.6 Interaction r 4 in D-Dimensional Radial Oscillator . . . . 3C.7 Interaction r 2q in D Dimensions . . . . . . . . . . . . . . 3C.8 Polynomial Interaction in D Dimensions . . . . . . . . . Appendix 3D Feynman Integrals for T 6= 0 . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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318 320 323 325 326 329 330 331 332 333 335 337 339 339 340 340 341 343 349 350 352 352 355 355 357 359 362 363 363 363 366

4 Semiclassical Time Evolution Amplitude 368 4.1 Wentzel-Kramers-Brillouin (WKB) Approximation . . . . . . . . 368 4.2 Saddle Point Approximation . . . . . . . . . . . . . . . . . . . . 373 4.2.1 Ordinary Integrals . . . . . . . . . . . . . . . . . . . . . . 373 4.2.2 Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . 376 4.3 Van Vleck-Pauli-Morette Determinant . . . . . . . . . . . . . . . 382 4.4 Fundamental Composition Law for Semiclassical Time Evolution Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 4.5 Semiclassical Fixed-Energy Amplitude . . . . . . . . . . . . . . 388 4.6 Semiclassical Amplitude in Momentum Space . . . . . . . . . . . 390 4.7 Semiclassical Quantum-Mechanical Partition Function . . . . . . 392 4.8 Multi-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . 397

xxii 4.9

Quantum Corrections to Classical Density of States . . . . . . . 4.9.1 One-Dimensional Case . . . . . . . . . . . . . . . . . . . . 4.9.2 Arbitrary Dimensions . . . . . . . . . . . . . . . . . . . . 4.9.3 Bilocal Density of States . . . . . . . . . . . . . . . . . . . 4.9.4 Gradient Expansion of Tracelog of Hamiltonian Operator . 4.9.5 Local Density of States on Circle . . . . . . . . . . . . . . 4.9.6 Quantum Corrections to Bohr-Sommerfeld Approximation 4.10 Thomas-Fermi Model of Neutral Atoms . . . . . . . . . . . . . . 4.10.1 Semiclassical Limit . . . . . . . . . . . . . . . . . . . . . . 4.10.2 Self-Consistent Field Equation . . . . . . . . . . . . . . . 4.10.3 Energy Functional of Thomas-Fermi Atom . . . . . . . . . 4.10.4 Calculation of Energies . . . . . . . . . . . . . . . . . . . 4.10.5 Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . 4.10.6 Exchange Energy . . . . . . . . . . . . . . . . . . . . . . . 4.10.7 Quantum Correction Near Origin . . . . . . . . . . . . . . 4.10.8 Systematic Quantum Corrections to Thomas-Fermi Energies 4.11 Classical Action of Coulomb System . . . . . . . . . . . . . . . . 4.12 Semiclassical Scattering . . . . . . . . . . . . . . . . . . . . . . . 4.12.1 General Formulation . . . . . . . . . . . . . . . . . . . . . 4.12.2 Semiclassical Cross Section of Mott Scattering . . . . . . . Appendix 4A Semiclassical Quantization for Pure Power Potentials . . Appendix 4B Derivation of Semiclassical Time Evolution Amplitude . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

402 403 405 406 408 412 413 416 416 417 419 421 424 424 426 428 432 441 441 445 446 448 452

5 Variational Perturbation Theory 368 5.1 Variational Approach to Effective Classical Partition Function . 368 5.2 Local Harmonic Trial Partition Function . . . . . . . . . . . . . 369 5.3 Optimal Upper Bound . . . . . . . . . . . . . . . . . . . . . . . 374 5.4 Accuracy of Variational Approximation . . . . . . . . . . . . . . 375 5.5 Weakly Bound Ground State Energy in Finite-Range Potential Well 377 5.6 Possible Direct Generalizations . . . . . . . . . . . . . . . . . . . 379 5.7 Effective Classical Potential for Anharmonic Oscillator . . . . . 380 5.8 Particle Densities . . . . . . . . . . . . . . . . . . . . . . . . . . 386 5.9 Extension to D Dimensions . . . . . . . . . . . . . . . . . . . . 389 5.10 Application to Coulomb and Yukawa Potentials . . . . . . . . . 391 5.11 Hydrogen Atom in Strong Magnetic Field . . . . . . . . . . . . . 394 5.11.1 Weak-Field Behavior . . . . . . . . . . . . . . . . . . . . . 397 5.11.2 Effective Classical Hamiltonian . . . . . . . . . . . . . . . 398 5.12 Variational Approach to Excitation Energies . . . . . . . . . . . 401 5.13 Systematic Improvement of Feynman-Kleinert Approximation . . . 405 5.14 Applications of Variational Perturbation Expansion . . . . . . . 408 5.14.1 Anharmonic Oscillator at T = 0 . . . . . . . . . . . . . . . 408 5.14.2 Anharmonic Oscillator for T > 0 . . . . . . . . . . . . . . 410 5.15 Convergence of Variational Perturbation Expansion . . . . . . . 414 H. Kleinert, PATH INTEGRALS

xxiii 5.16 5.17 5.18

Variational Perturbation Theory for Strong-Coupling Expansion General Strong-Coupling Expansions . . . . . . . . . . . . . . . Variational Interpolation between Weak and Strong-Coupling Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Systematic Improvement of Excited Energies . . . . . . . . . . . 5.20 Variational Treatment of Double-Well Potential . . . . . . . . . 5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21.1 Evaluation of Path Integrals . . . . . . . . . . . . . . . . . 5.21.2 Higher-Order Smearing Formula in D Dimensions . . . . . 5.21.3 Isotropic Second-Order Approximation to Coulomb Problem 5.21.4 Anisotropic Second-Order Approximation to Coulomb Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21.5 Zero-Temperature Limit . . . . . . . . . . . . . . . . . . . 5.22 Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.22.1 Partition Function . . . . . . . . . . . . . . . . . . . . . . 5.22.2 Harmonic Trial System . . . . . . . . . . . . . . . . . . . 5.22.3 Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . 5.22.4 Second-Order Correction . . . . . . . . . . . . . . . . . . . 5.22.5 Polaron in Magnetic Field, Bipolarons, etc. . . . . . . . . 5.22.6 Variational Interpolation for Polaron Energy and Mass . . 5.23 Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23.1 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 5.23.2 Variational Perturbation Theory for Density Matrices . . . 5.23.3 Smearing Formula for Density Matrices . . . . . . . . . . 5.23.4 First-Order Variational Approximation . . . . . . . . . . . 5.23.5 Smearing Formula in Higher Spatial Dimensions . . . . . . 5.23.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 5A Feynman Integrals for T 6= 0 without Zero Frequency . Appendix 5B Proof of Scaling Relation for the Extrema of WN . . . . Appendix 5C Second-Order Shift of Polaron Energy . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Path Integrals with Topological Constraints 6.1 Point Particle on Circle . . . . . . . . . . . . . . . . 6.2 Infinite Wall . . . . . . . . . . . . . . . . . . . . . . 6.3 Point Particle in Box . . . . . . . . . . . . . . . . . 6.4 Strong-Coupling Theory for Particle in Box . . . . . 6.4.1 Partition Function . . . . . . . . . . . . . . . 6.4.2 Perturbation Expansion . . . . . . . . . . . . 6.4.3 Variational Strong-Coupling Approximations 6.4.4 Special Properties of Expansion . . . . . . . . 6.4.5 Exponentially Fast Convergence . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . .

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421 424 427 428 429 432 432 434 435 437 438 442 444 446 451 452 453 453 456 457 458 460 463 467 469 478 480 482 483 489 489 493 497 500 501 501 503 505 506 507

xxiv 7 Many Particle Orbits — Statistics and Second Quantization 7.1 Ensembles of Bose and Fermi Particle Orbits . . . . . . . . . . . 7.2 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . 7.2.1 Free Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Bose Gas in Finite Box . . . . . . . . . . . . . . . . . . . 7.2.3 Effect of Interactions . . . . . . . . . . . . . . . . . . . . . 7.2.4 Bose-Einstein Condensation in Harmonic Trap . . . . . . 7.2.5 Thermodynamic Functions . . . . . . . . . . . . . . . . . 7.2.6 Critical Temperature . . . . . . . . . . . . . . . . . . . . . 7.2.7 More General Anisotropic Trap . . . . . . . . . . . . . . . 7.2.8 Rotating Bose-Einstein Gas . . . . . . . . . . . . . . . . . 7.2.9 Finite-Size Corrections . . . . . . . . . . . . . . . . . . . . 7.2.10 Entropy and Specific Heat . . . . . . . . . . . . . . . . . . 7.2.11 Interactions in Harmonic Trap . . . . . . . . . . . . . . . 7.3 Gas of Free Fermions . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Statistics Interaction . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Fractional Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Second-Quantized Bose Fields . . . . . . . . . . . . . . . . . . . 7.7 Fluctuating Bose Fields . . . . . . . . . . . . . . . . . . . . . . . 7.8 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Second-Quantized Fermi Fields . . . . . . . . . . . . . . . . . . 7.10 Fluctuating Fermi Fields . . . . . . . . . . . . . . . . . . . . . . 7.10.1 Grassmann Variables . . . . . . . . . . . . . . . . . . . . . 7.10.2 Fermionic Functional Determinant . . . . . . . . . . . . . 7.10.3 Coherent States for Fermions . . . . . . . . . . . . . . . . 7.11 Hilbert Space of Quantized Grassmann Variable . . . . . . . . . 7.11.1 Single Real Grassmann Variable . . . . . . . . . . . . . . 7.11.2 Quantizing Harmonic Oscillator with Grassmann Variables 7.11.3 Spin System with Grassmann Variables . . . . . . . . . . 7.12 External Sources in a∗ , a -Path Integral . . . . . . . . . . . . . . 7.13 Generalization to Pair Terms . . . . . . . . . . . . . . . . . . . . 7.14 Spatial Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . 7.14.1 Grand-Canonical Ensemble of Particle Orbits from Free Fluctuating Field . . . . . . . . . . . . . . . . . . . . . . . 7.14.2 First versus Second Quantization . . . . . . . . . . . . . . 7.14.3 Interacting Fields . . . . . . . . . . . . . . . . . . . . . . . 7.14.4 Effective Classical Field Theory . . . . . . . . . . . . . . . 7.15 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.15.1 Collective Field . . . . . . . . . . . . . . . . . . . . . . . . 7.15.2 Bosonized versus Original Theory . . . . . . . . . . . . . . Appendix 7A Treatment of Singularities in Zeta-Function . . . . . . . 7A.1 Finite Box . . . . . . . . . . . . . . . . . . . . . . . . . . 7A.2 Harmonic Trap . . . . . . . . . . . . . . . . . . . . . . . .

509 510 517 517 525 527 533 533 535 538 539 540 541 544 548 553 558 559 562 568 572 572 572 575 579 581 581 584 585 590 592 594 594 596 596 597 599 600 602 604 605 607


xxv Appendix 7B Experimental versus Theoretical Would-be Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

609 610

8 Path Integrals in Polar and Spherical Coordinates 615 8.1 Angular Decomposition in Two Dimensions . . . . . . . . . . . . 615 8.2 Trouble with Feynman’s Path Integral Formula in Radial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 8.3 Cautionary Remarks . . . . . . . . . . . . . . . . . . . . . . . . 622 8.4 Time Slicing Corrections . . . . . . . . . . . . . . . . . . . . . . 625 8.5 Angular Decomposition in Three and More Dimensions . . . . . 629 8.5.1 Three Dimensions . . . . . . . . . . . . . . . . . . . . . . 630 8.5.2 D Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 632 8.6 Radial Path Integral for Harmonic Oscillator and Free Particle . . . 638 8.7 Particle near the Surface of a Sphere in D Dimensions . . . . . . 639 8.8 Angular Barriers near the Surface of a Sphere . . . . . . . . . . 642 8.8.1 Angular Barriers in Three Dimensions . . . . . . . . . . . 642 8.8.2 Angular Barriers in Four Dimensions . . . . . . . . . . . . 647 8.9 Motion on a Sphere in D Dimensions . . . . . . . . . . . . . . . 652 8.10 Path Integrals on Group Spaces . . . . . . . . . . . . . . . . . . 656 8.11 Path Integral of Spinning Top . . . . . . . . . . . . . . . . . . . 659 8.12 Path Integral of Spinning Particle . . . . . . . . . . . . . . . . . 660 8.13 Berry Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 8.14 Spin Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 9 Wave Functions 9.1 Free Particle in D Dimensions . . . . . . . . 9.2 Harmonic Oscillator in D Dimensions . . . . 9.3 Free Particle from ω → 0 -Limit of Oscillator 9.4 Charged Particle in Uniform Magnetic Field 9.5 Dirac δ-Function Potential . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . .

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669 669 672 678 680 687 689

10 Spaces with Curvature and Torsion 690 10.1 Einstein’s Equivalence Principle . . . . . . . . . . . . . . . . . . 691 10.2 Classical Motion of Mass Point in General Metric-Affine Space 692 10.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . 692 10.2.2 Nonholonomic Mapping to Spaces with Torsion . . . . . . 695 10.2.3 New Equivalence Principle . . . . . . . . . . . . . . . . . . 701 10.2.4 Classical Action Principle for Spaces with Curvature and Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 10.3 Path Integral in Metric-Affine Space . . . . . . . . . . . . . . . . 706 10.3.1 Nonholonomic Transformation of Action . . . . . . . . . . 706

xxvi 10.3.2 Measure of Path Integration . . . . . . . . . . . . . . . . . 711 10.4 Completing Solution of Path Integral on Surface of Sphere . . . 717 10.5 External Potentials and Vector Potentials . . . . . . . . . . . . . 719 10.6 Perturbative Calculation of Path Integrals in Curved Space . . . 721 10.6.1 Free and Interacting Parts of Action . . . . . . . . . . . . 721 10.6.2 Zero Temperature . . . . . . . . . . . . . . . . . . . . . . 724 10.7 Model Study of Coordinate Invariance . . . . . . . . . . . . . . 726 10.7.1 Diagrammatic Expansion . . . . . . . . . . . . . . . . . . 728 10.7.2 Diagrammatic Expansion in d Time Dimensions . . . . . . 730 10.8 Calculating Loop Diagrams . . . . . . . . . . . . . . . . . . . . . 731 10.8.1 Reformulation in Configuration Space . . . . . . . . . . . 738 10.8.2 Integrals over Products of Two Distributions . . . . . . . 739 10.8.3 Integrals over Products of Four Distributions . . . . . . . 740 10.9 Distributions as Limits of Bessel Function . . . . . . . . . . . . 742 10.9.1 Correlation Function and Derivatives . . . . . . . . . . . . 742 10.9.2 Integrals over Products of Two Distributions . . . . . . . 744 10.9.3 Integrals over Products of Four Distributions . . . . . . . 745 10.10 Simple Rules for Calculating Singular Integrals . . . . . . . . . . 747 10.11 Perturbative Calculation on Finite Time Intervals . . . . . . . . 752 10.11.1 Diagrammatic Elements . . . . . . . . . . . . . . . . . . . 753 10.11.2 Cumulant Expansion of D-Dimensional Free-Particle Amplitude in Curvilinear Coordinates . . . . . . . . . . . . . 754 10.11.3 Propagator in 1 − ε Time Dimensions . . . . . . . . . . . 756 10.11.4 Coordinate Independence for Dirichlet Boundary Conditions 757 10.11.5 Time Evolution Amplitude in Curved Space . . . . . . . . 763 10.11.6 Covariant Results for Arbitrary Coordinates . . . . . . . . 769 10.12 Effective Classical Potential in Curved Space . . . . . . . . . . . 774 10.12.1 Covariant Fluctuation Expansion . . . . . . . . . . . . . . 775 10.12.2 Arbitrariness of q0µ . . . . . . . . . . . . . . . . . . . . . . 778 10.12.3 Zero-Mode Properties . . . . . . . . . . . . . . . . . . . . 779 10.12.4 Covariant Perturbation Expansion . . . . . . . . . . . . . 782 10.12.5 Covariant Result from Noncovariant Expansion . . . . . . 783 10.12.6 Particle on Unit Sphere . . . . . . . . . . . . . . . . . . . 786 10.13 Covariant Effective Action for Quantum Particle with CoordinateDependent Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 10.13.1 Formulating the Problem . . . . . . . . . . . . . . . . . . 789 10.13.2 Gradient Expansion . . . . . . . . . . . . . . . . . . . . . 792 Appendix 10A Nonholonomic Gauge Transformations in Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792 10A.1 Gradient Representation of Magnetic Field of Current Loops 793 10A.2 Generating Magnetic Fields by Multivalued Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 797 10A.3 Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . 798 H. Kleinert, PATH INTEGRALS

xxvii 10A.4

Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations . . . . . . . . . . . . . . . . . . . 800 10A.5 Gauge Field Representation of Current Loops and Monopoles 801 Appendix 10B Comparison of Multivalued Basis Tetrads with Vierbein Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803 Appendix 10C Cancellation of Powers of δ(0) . . . . . . . . . . . . . . 805 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 11 Schr¨ odinger Equation in General Metric-Affine Spaces 11.1 Integral Equation for Time Evolution Amplitude . . . . . 11.1.1 From Recursion Relation to Schrödinger Equation . 11.1.2 Alternative Evaluation . . . . . . . . . . . . . . . . 11.2 Equivalent Path Integral Representations . . . . . . . . . 11.3 Potentials and Vector Potentials . . . . . . . . . . . . . . 11.4 Unitarity Problem . . . . . . . . . . . . . . . . . . . . . . 11.5 Alternative Attempts . . . . . . . . . . . . . . . . . . . . 11.6 DeWitt-Seeley Expansion of Time Evolution Amplitude . Appendix 11A Cancellations in Effective Potential . . . . . . . . Appendix 11B DeWitt’s Amplitude . . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

12 New Path Integral Formula for Singular Potentials 12.1 Path Collapse in Feynman’s formula for the Coulomb System 12.2 Stable Path Integral with Singular Potentials . . . . . . . . . 12.3 Time-Dependent Regularization . . . . . . . . . . . . . . . . 12.4 Relation to Schrödinger Theory. Wave Functions . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . .

811 811 812 815 818 822 823 826 827 830 833 833

. . . . .

835 835 838 843 845 847

13 Path Integral of Coulomb System 848 13.1 Pseudotime Evolution Amplitude . . . . . . . . . . . . . . . . . 848 13.2 Solution for the Two-Dimensional Coulomb System . . . . . . . 850 13.3 Absence of Time Slicing Corrections for D = 2 . . . . . . . . . . 855 13.4 Solution for the Three-Dimensional Coulomb System . . . . . . 860 13.5 Absence of Time Slicing Corrections for D = 3 . . . . . . . . . . 866 13.6 Geometric Argument for Absence of Time Slicing Corrections . . 868 13.7 Comparison with Schrödinger Theory . . . . . . . . . . . . . . . 869 13.8 Angular Decomposition of Amplitude, and Radial Wave Functions 874 13.9 Remarks on Geometry of Four-Dimensional uµ -Space . . . . . . 878 13.10 Solution in Momentum Space . . . . . . . . . . . . . . . . . . . 880 13.10.1 Gauge-Invariant Canonical Path Integral . . . . . . . . . . 881 13.10.2 Another Form of Action . . . . . . . . . . . . . . . . . . . 884 13.10.3 Absence of Extra R-Term . . . . . . . . . . . . . . . . . . 885 Appendix 13A Dynamical Group of Coulomb States . . . . . . . . . . . 885 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889

xxviii 14 Solution of Further Path Integrals by Duru-Kleinert Method 891 14.1 One-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . 891 14.2 Derivation of the Effective Potential . . . . . . . . . . . . . . . . 895 14.3 Comparison with Schrödinger Quantum Mechanics . . . . . . . . 899 14.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900 14.4.1 Radial Harmonic Oscillator and Morse System . . . . . . 900 14.4.2 Radial Coulomb System and Morse System . . . . . . . . 902 14.4.3 Equivalence of Radial Coulomb System and Radial Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903 14.4.4 Angular Barrier near Sphere, and Rosen-Morse Potential 911 14.4.5 Angular Barrier near Four-Dimensional Sphere, and General Rosen-Morse Potential . . . . . . . . . . . . . . . . . 913 14.4.6 Hulthén Potential and General Rosen-Morse Potential . . 916 14.4.7 Extended Hulthén Potential and General Rosen-Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 14.5 D-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . 919 14.6 Path Integral of the Dionium Atom . . . . . . . . . . . . . . . . 921 14.6.1 Formal Solution . . . . . . . . . . . . . . . . . . . . . . . 922 14.6.2 Absence of Time Slicing Corrections . . . . . . . . . . . . 926 14.7 Time-Dependent Duru-Kleinert Transformation . . . . . . . . . 929 Appendix 14A Affine Connection of Dionium Atom . . . . . . . . . . . 932 Appendix 14B Algebraic Aspects of Dionium States . . . . . . . . . . . 933 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933 15 Path Integrals in Polymer Physics 935 15.1 Polymers and Ideal Random Chains . . . . . . . . . . . . . . . . 935 15.2 Moments of End-to-End Distribution . . . . . . . . . . . . . . . 937 15.3 Exact End-to-End Distribution in Three Dimensions . . . . . . . 940 15.4 Short-Distance Expansion for Long Polymer . . . . . . . . . . . 942 15.5 Saddle Point Approximation to Three-Dimensional End-to-End Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944 15.6 Path Integral for Continuous Gaussian Distribution . . . . . . . 945 15.7 Stiff Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948 15.7.1 Sliced Path Integral . . . . . . . . . . . . . . . . . . . . . 950 15.7.2 Relation to Classical Heisenberg Model . . . . . . . . . . . 951 15.7.3 End-to-End Distribution . . . . . . . . . . . . . . . . . . . 953 15.7.4 Moments of End-to-End Distribution . . . . . . . . . . . . 953 15.8 Continuum Formulation . . . . . . . . . . . . . . . . . . . . . . 954 15.8.1 Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . 954 15.8.2 Correlation Functions and Moments . . . . . . . . . . . . 955 15.9 Schrödinger Equation and Recursive Solution for Moments . . . 959 15.9.1 Setting up the Schrödinger Equation . . . . . . . . . . . . 959 15.9.2 Recursive Solution of Schrödinger Equation. . . . . . . . . 960 15.9.3 From Moments to End-to-End Distribution for D = 3 . . 963 H. Kleinert, PATH INTEGRALS

xxix 15.9.4 Large-Stiffness Approximation to End-to-End Distribution 15.9.5 Higher Loop Corrections . . . . . . . . . . . . . . . . . . 15.10 Excluded-Volume Effects . . . . . . . . . . . . . . . . . . . . . . 15.11 Flory’s Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 15.12 Polymer Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 15.13 Fermi Fields for Self-Avoiding Lines . . . . . . . . . . . . . . . . Appendix 15A Basic Integrals . . . . . . . . . . . . . . . . . . . . . . . Appendix 15B Loop Integrals . . . . . . . . . . . . . . . . . . . . . . . Appendix 15C Integrals Involving Modified Green Function . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

965 970 978 986 986 994 994 995 997 998

16 Polymers and Particle Orbits in Multiply Connected Spaces 1000 16.1 Simple Model for Entangled Polymers . . . . . . . . . . . . . . . 1000 16.2 Entangled Fluctuating Particle Orbit: Aharonov-Bohm Effect . 1004 16.3 Aharonov-Bohm Effect and Fractional Statistics . . . . . . . . . 1012 16.4 Self-Entanglement of Polymer . . . . . . . . . . . . . . . . . . . 1017 16.5 The Gauss Invariant of Two Curves . . . . . . . . . . . . . . . . 1031 16.6 Bound States of Polymers and Ribbons . . . . . . . . . . . . . . 1033 16.7 Chern-Simons Theory of Entanglements . . . . . . . . . . . . . . 1040 16.8 Entangled Pair of Polymers . . . . . . . . . . . . . . . . . . . . 1043 16.8.1 Polymer Field Theory for Probabilities . . . . . . . . . . . 1045 16.8.2 Calculation of Partition Function . . . . . . . . . . . . . . 1046 16.8.3 Calculation of Numerator in Second Moment . . . . . . . 1048 16.8.4 First Diagram in Fig. 16.23 . . . . . . . . . . . . . . . . . 1050 16.8.5 Second and Third Diagrams in Fig. 16.23 . . . . . . . . . 1051 16.8.6 Fourth Diagram in Fig. 16.23 . . . . . . . . . . . . . . . . 1052 16.8.7 Second Topological Moment . . . . . . . . . . . . . . . . . 1053 16.9 Chern-Simons Theory of Statistical Interaction . . . . . . . . . . 1053 16.10 Second-Quantized Anyon Fields . . . . . . . . . . . . . . . . . . 1056 16.11 Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . 1059 16.12 Anyonic Superconductivity . . . . . . . . . . . . . . . . . . . . . 1063 16.13 Non-Abelian Chern-Simons Theory . . . . . . . . . . . . . . . . 1065 Appendix 16A Calculation of Feynman Diagrams in Polymer Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Appendix 16B Kauffman and BLM/Ho polynomials . . . . . . . . . . 1069 Appendix 16C Skein Relation between Wilson Loop Integrals . . . . . 1069 Appendix 16D London Equations . . . . . . . . . . . . . . . . . . . . . 1072 Appendix 16E Hall Effect in Electron Gas . . . . . . . . . . . . . . . . 1074 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 17 Tunneling 17.1 Double-Well Potential . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Classical Solutions — Kinks and Antikinks . . . . . . . . . . . . 17.3 Quadratic Fluctuations . . . . . . . . . . . . . . . . . . . . . . .

1080 1080 1083 1087

xxx 17.3.1 Zero-Eigenvalue Mode . . . . . . . . . . . . . . . . . . . . 17.3.2 Continuum Part of Fluctuation Factor . . . . . . . . . . . 17.4 General Formula for Eigenvalue Ratios . . . . . . . . . . . . . . 17.5 Fluctuation Determinant from Classical Solution . . . . . . . . . 17.6 Wave Functions of Double-Well . . . . . . . . . . . . . . . . . . 17.7 Gas of Kinks and Antikinks and Level Splitting Formula . . . . 17.8 Fluctuation Correction to Level Splitting . . . . . . . . . . . . . 17.9 Tunneling and Decay . . . . . . . . . . . . . . . . . . . . . . . . 17.10 Large-Order Behavior of Perturbation Expansions . . . . . . . . 17.10.1 Growth Properties of Expansion Coefficients . . . . . . . . 17.10.2 Semiclassical Large-Order Behavior . . . . . . . . . . . . . 17.10.3 Fluctuation Correction to the Imaginary Part and LargeOrder Behavior . . . . . . . . . . . . . . . . . . . . . . . . 17.10.4 Variational Approach to Tunneling. Perturbation Coefficients to All Orders . . . . . . . . . . . . . . . . . . . . . 17.10.5 Convergence of Variational Perturbation Expansion . . . . 17.11 Decay of Supercurrent in Thin Closed Wire . . . . . . . . . . . . 17.12 Decay of Metastable Thermodynamic Phases . . . . . . . . . . . 17.13 Decay of Metastable Vacuum State in Quantum Field Theory . 17.14 Crossover from Quantum Tunneling to Thermally Driven Decay Appendix 17A Feynman Integrals for Fluctuation Correction . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Nonequilibrium Quantum Statistics 18.1 Linear Response and Time-Dependent Green Functions for T 18.2 Spectral Representations of Green Functions for T 6= 0 . . 18.3 Other Important Green Functions . . . . . . . . . . . . . . 18.4 Hermitian Adjoint Operators . . . . . . . . . . . . . . . . . 18.5 Harmonic Oscillator Green Functions for T 6= 0 . . . . . . 18.5.1 Creation Annihilation Operators . . . . . . . . . . . 18.5.2 Real Field Operators . . . . . . . . . . . . . . . . . . 18.6 Nonequilibrium Green Functions . . . . . . . . . . . . . . . 18.7 Perturbation Theory for Nonequilibrium Green Functions . 18.8 Path Integral Coupled to Thermal Reservoir . . . . . . . . 18.9 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . 18.9.1 Canonical Path Integral for Probability Distribution 18.9.2 Solving the Operator Ordering Problem . . . . . . . 18.9.3 Strong Damping . . . . . . . . . . . . . . . . . . . . 18.10 Langevin Equations . . . . . . . . . . . . . . . . . . . . . . 18.11 Stochastic Quantization . . . . . . . . . . . . . . . . . . . 18.12 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . 18.12.1 Kubo’s stochastic Liouville equation . . . . . . . . . 18.12.2 From Kubo’s to Fokker-Planck Equations . . . . . . 18.12.3 Itô’s Lemma . . . . . . . . . . . . . . . . . . . . . .

6 0 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1093 1097 1099 1101 1105 1106 1110 1115 1123 1124 1127 1132 1135 1143 1151 1163 1170 1171 1173 1175 1178 1178 1181 1184 1187 1188 1188 1191 1193 1202 1205 1211 1212 1213 1219 1222 1226 1229 1229 1230 1233


xxxi 18.13 18.14 18.15 18.16 18.17

Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Quantum Liouville Equation . . . . . . . . . . . . . . Master Equation for Time Evolution . . . . . . . . . . . . . . . Relation to Quantum Langevin Equation . . . . . . . . . . . . . Electromagnetic Dissipation and Decoherence . . . . . . . . . . 18.17.1 Forward–Backward Path Integral . . . . . . . . . . . . . . 18.17.2 Master Equation for Time Evolution in Photon Bath . . 18.17.3 Line Width . . . . . . . . . . . . . . . . . . . . . . . . . . 18.17.4 Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . . . 18.17.5 Langevin Equations . . . . . . . . . . . . . . . . . . . . . 18.18 Fokker-Planck Equation in Spaces with Curvature and Torsion . 18.19 Stochastic Interpretation of Quantum-Mechanical Amplitudes . 18.20 Stochastic Equation for Schrödinger Wave Function . . . . . . . 18.21 Real Stochastic and Deterministic Equation for Schr¨ odinger Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.21.1 Stochastic Differential Equation . . . . . . . . . . . . . . . 18.21.2 Equation for Noise Average . . . . . . . . . . . . . . . . . 18.21.3 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 18.21.4 General Potential . . . . . . . . . . . . . . . . . . . . . . . 18.21.5 Deterministic Equation . . . . . . . . . . . . . . . . . . . 18.22 Heisenberg Picture for Probability Evolution . . . . . . . . . . . Appendix 18A Inequalities for Diagonal Green Functions . . . . . . . . Appendix 18B General Generating Functional . . . . . . . . . . . . . . Appendix 18C Wick Decomposition of Operator Products . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Relativistic Particle Orbits 19.1 Special Features of Relativistic Path Integrals . . . . . . . . . 19.2 Proper Action for Fluctuating Relativistic Particle Orbits . . . 19.2.1 Gauge-Invariant Formulation . . . . . . . . . . . . . . . 19.2.2 Simplest Gauge Fixing . . . . . . . . . . . . . . . . . . . 19.2.3 Partition Function of Ensemble of Closed Particle Loops 19.2.4 Fixed-Energy Amplitude . . . . . . . . . . . . . . . . . . 19.3 Tunneling in Relativistic Physics . . . . . . . . . . . . . . . . . 19.3.1 Decay Rate of Vacuum in Electric Field . . . . . . . . . 19.3.2 Birth of Universe . . . . . . . . . . . . . . . . . . . . . . 19.3.3 Friedmann Model . . . . . . . . . . . . . . . . . . . . . . 19.3.4 Tunneling of Expanding Universe . . . . . . . . . . . . . 19.4 Relativistic Coulomb System . . . . . . . . . . . . . . . . . . . 19.5 Relativistic Particle in Electromagnetic Field . . . . . . . . . . 19.5.1 Action and Partition Function . . . . . . . . . . . . . . 19.5.2 Perturbation Expansion . . . . . . . . . . . . . . . . . . 19.5.3 Lowest-Order Vacuum Polarization . . . . . . . . . . . . 19.6 Path Integral for Spin-1/2 Particle . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

1236 1240 1242 1244 1245 1245 1249 1250 1252 1256 1257 1259 1261 1263 1263 1264 1265 1265 1266 1267 1270 1274 1278 1279 1284 1286 1289 1289 1291 1293 1294 1295 1295 1304 1309 1314 1314 1318 1318 1319 1321 1325

xxxii 19.6.1 Dirac Theory . . . . . . . . . . . . . . . . . 19.6.2 Path Integral . . . . . . . . . . . . . . . . . 19.6.3 Amplitude with Electromagnetic Interaction 19.6.4 Effective Action in Electromagnetic Field . 19.6.5 Perturbation Expansion . . . . . . . . . . . 19.6.6 Vacuum Polarization . . . . . . . . . . . . . 19.7 Supersymmetry . . . . . . . . . . . . . . . . . . . 19.7.1 Global Invariance . . . . . . . . . . . . . . . 19.7.2 Local Invariance . . . . . . . . . . . . . . . Notes and References . . . . . . . . . . . . . . . . . . . . . 20 Path Integrals and Financial Markets 20.1 Fluctuation Properties of Financial Assets . . . 20.1.1 Harmonic Approximation to Fluctuations 20.1.2 Lévy Distributions . . . . . . . . . . . . . 20.1.3 Truncated Lévy Distributions . . . . . . . 20.1.4 Asymmetric Truncated Lévy Distributions Index

. . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . .

1325 1329 1332 1334 1335 1337 1338 1338 1340 1341

. . . . .

1343 1343 1345 1347 1349 1354 1343


List of Figures 1.1 1.2 1.3 1.4

Probability distribution of particle behind a double slit . . . . . P 2πiµn Relevant function N in Poisson’s summation formula n=−N e Illustration of time-ordering procedure . . . . . . . . . . . . . . . Triangular closed contour for Cauchy integral . . . . . . . . . . .

. . . .

12 30 36 84

2.1 2.2 2.3 2.4

Zigzag paths, along which a point particle fluctuates . . Solution of equation of motion . . . . . . . . . . . . . . . Illustration of eigenvalues of fluctuation matrix . . . . . Finite-lattice effects in internal energy E and specific heat

. . . .

98 121 143 175

3.1 3.2

Pole in Fourier transform of Green functions Gp,a 220 ω (t) . . . . . . . . p Subtracted periodic Green function Gω,e(τ ) − 1/ω and antiperiodic Green function Gaω,e(τ ) for frequencies ω = (0, 5, 10)/¯ hβ . . . . . . 221 p,a Two poles in Fourier transform of Green function Gω2 (t) . . . . . 222 Subtracted periodic Green function Gpω2 ,e (τ ) − 1/¯ hβω 2 and antiperia odic Green function Gω2 ,e (τ ) for frequencies ω = (0, 5, 10)/¯ hβ . . . 243 Poles in complex β-plane of Fourier integral . . . . . . . . . . . . 270 Density of states for weak and strong damping in natural units . . 271 Perturbation expansion of free energy up to order g 3 . . . . . . . . 283 Diagrammatic solution of recursion relation for the generating functional W [j[ of all connected correlation functions . . . . . . . . . 290 Diagrammatic representation of functional differential equation . . 295 Diagrammatic representation of recursion relation . . . . . . . . . 297 Vacuum diagrams up to five loops and their multiplicities . . . . . 298 Diagrammatic differentiations for deriving tree decomposition of connected correlation functions . . . . . . . . . . . . . . . . . . . . 303 Effective potential for ω 2 > 0 and ω 2 < 0 in mean-field approximation 309 Local fluctuation width of harmonic oscillator . . . . . . . . . . . 327 Magnetization curves in double-well potential . . . . . . . . . . . 336 Plot of reduced Feynman integrals aˆ2L 365 V (x) . . . . . . . . . . . . . .

3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 4.1 4.2 4.3 4.4

. . . . . . C

. . . .

. . . .

Left: Determination of energy eigenvalues E (n) in semiclassical expansion; Right: Comparison between exact and semiclassical energies 415 Solution for screening function f (ξ) in Thomas-Fermi model . . . . 419 Orbits in Coulomb potential . . . . . . . . . . . . . . . . . . . . . 435 Circular orbits in momentum space for E > 0 . . . . . . . . . . . . 438 xxxiii

xxxiv 4.5 4.6 4.7

Geometry of scattering in momentum space . . . . . . . . . . . . . Classical trajectories in Coulomb potential . . . . . . . . . . . . . Oscillations in differential Mott scattering cross section . . . . . .

439 445 446

5.1 5.2 5.3 5.4 5.5

Illustration of convexity of exponential function e−x . . . . . . . . Approximate free energy F1 of anharmonic oscillator . . . . . . . Effective classical potential of double well . . . . . . . . . . . . . . Free energy F1 in double-well potential . . . . . . . . . . . . . . . Comparison of approximate effective classical potentials W1 (x0 ) and W3 (x0 ) with exact V eff cl (x0 ) . . . . . . . . . . . . . . . . . . . . . . Effective classical potential W1 (x0 ) for double-well potential and various numbers of time slices . . . . . . . . . . . . . . . . . . . . . . Approximate particle density of anharmonic oscillator . . . . . . . Particle density in double-well potential . . . . . . . . . . . . . . Approximate effective classical potential W1 (r) of Coulomb system at various temperatures . . . . . . . . . . . . . . . . . . . . . . . . Particle distribution in Coulomb potential at different T 6= 0 . . . First-order variational result for binding energy of atom in strong magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective classical potential of atom in strong magnetic field . . . One-particle reducible vacuum diagram . . . . . . . . . . . . . . . Typical Ω-dependence of approximations W1,2,3 at T = 0 . . . . . . Typical Ω-dependence of Nth approximations WN at T = 0 . . . New plateaus in WN developing for higher orders N ≥ 15 . . . . . Trial frequencies ΩN extremizing variational approximation WN at T = 0 for odd N ≤ 91 . . . . . . . . . . . . . . . . . . . . . . . . Extremal and turning point frequencies ΩN in variational approximation WN at T = 0 for even and odd N ≤ 30 . . . . . . . . . . . Difference between approximate ground state energies E = WN and exact energies Eex . . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithmic plot of kth terms in re-expanded perturbation series . Logarithmic plot of N-behavior of strong-coupling expansion coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oscillations of approximate strong-coupling expansion coefficient b0 as a function of N . . . . . . . . . . . . . . . . . . . . . . . . . . Ratio of approximate and exact ground state energy of anharmonic oscillator from lowest-order variational interpolation . . . . . . . . Lowest two energies in double-well potential as function of coupling strength g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isotropic approximation to effective classical potential of Coulomb system in first and second order . . . . . . . . . . . . . . . . . . . Isotropic and anisotropic approximations to effective classical potential of Coulomb system in first and second order . . . . . . . . . .

370 381 383 384

5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26

385 386 387 388 392 393 396 400 407 410 415 416 417 417 418 420 422 422 428 431 437 439


xxxv 5.27 Approach of the variational approximations of first, second, and third order to the correct ground state energy . . . . . . . . . . . 5.28 Variational interpolation of polaron energy . . . . . . . . . . . . . 5.29 Variational interpolation of polaron effective mass . . . . . . . . . 5.30 Temperature dependence of fluctuation widths of any point x(τ ) on the path in a harmonic oscillator . . . . . . . . . . . . . . . . . . . (n) 5.31 Temperature-dependence of first 9 functions Cβ , where β = 1/kB T . ˜ 1Ω,xm (xa ) to the effective classical 5.32 Plots of first-order approximation W potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ 1 (xa ) . . 5.33 First-order approximation to effective classical potential W 5.34 Trial frequency Ω(xa ) and minimum of trial oscillator xm (xa ) at different temperatures and coupling strength g = 0.1 . . . . . . . 5.35 Trial frequency Ω(xa ) and minimum of trial oscillator xm (xa ) at different temperatures and coupling strength g = 10 . . . . . . . . 5.36 First-order approximation to particle density . . . . . . . . . . . . 5.37 First-order approximation to particle densities of the double-well for g = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.38 Second-order approximation to particle density (dashed) compared to exact results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.39 Radial distribution function for an electron-proton pair . . . . . . 5.40 Plot of reduced Feynman integrals aˆ2L V (x) . . . . . . . . . . . . . .

441 455 456 459 464 470 471 472 472 473 474 475 477 480

6.1 6.2 6.3 6.4 6.5 6.6

Path with jumps in cyclic variable redrawn in extended zone scheme 493 Illustration of path counting near reflecting wall . . . . . . . . . . 496 Illustration of path counting in a box . . . . . . . . . . . . . . . . 499 Equivalence of paths in a box and paths on a circle with infinite wall 499 Variational functions fN (c) for particle between walls up to N = 16 504 Exponentially fast convergence of strong-coupling approximations . 505

7.1 7.2 7.3

512 512

Paths summed in partition function (7.9) . . . . . . . . . . . . . . Periodic representation of paths summed in partition function (7.9) Among the w! permutations of the different windings around the cylinder, (w − 1)! are connected . . . . . . . . . . . . . . . . . . . 7.4 Plot of the specific heat of free Bose gas . . . . . . . . . . . . . . . 7.5 Plot of functions ζν (z) appearing in Bose-Einstein thermodynamics 7.6 Specific heat of ideal Bose gas with phase transition at Tc . . . . . 7.7 Reentrant transition in phase diagram of Bose-Einstein condensation for different interaction strengths . . . . . . . . . . . . . . . . . . 7.8 Energies of elementary excitations of superfluid 4 He . . . . . . . . 7.9 Condensate fraction Ncond /N ≡ 1 − Nn /N as function of temperature 7.10 Peak of specific heat in harmonic trap . . . . . . . . . . . . . . . . 7.11 Temperature behavior of specific heat of free Fermi gas . . . . . . 7.12 Finite-size corrections to the critical temperature for N = 300 to infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

514 515 520 525 531 532 537 544 552 607

xxxvi 7.13 10.1 10.2 10.3 10.4 10.5 10.6 13.1 15.1 15.2 15.3 15.4 15.5 15.6 15.7 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 16.13 16.14 16.15

Plots of condensate fraction and its second derivative for simple Bose gas in a finite box. . . . . . . . . . . . . . . . . . . . . . . . . . .

610

Edge dislocation in crystal associated with missing semi-infinite plane of atoms as source of torsion . . . . . . . . . . . . . . . . . . 699 Edge disclination in crystal associated with missing semi-infinite section of atoms as source of curvature . . . . . . . . . . . . . . . . . 700 Images under holonomic and nonholonomic mapping of δ-function variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 Green functions for perturbation expansions in curvilinear coordinates 725 Infinitesimally thin closed current loop L and magnetic field . . . . 794 Coordinate system q µ and the two sets of local nonholonomic coordinates dxα and dxa . . . . . . . . . . . . . . . . . . . . . . . . . . 805 Illustration of associated final points in u-space, to be summed in the harmonic-oscillator amplitude . . . . . . . . . . . . . . . . . . Random chain of N links . . . . . . . . . . . . . . . . . . . . . . . End-to-end distribution PN (R) of random chain with N links . . Neighboring links for the calculation of expectation values . . . . . Paramters k, β, and m for a best fit of end-to-end distribution . . Structure functions for different persistence lengths following from the end-to-end distributions . . . . . . . . . . . . . . . . . . . . . . Normalized end-to-end distribution of stiff polymer . . . . . . . . . Comparison of critical exponent ν in Flory approximation with result of quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . Second virial coefficient B2 as function of flux µ0 . . . . . . . . . Lefthanded trefoil knot in polymer . . . . . . . . . . . . . . . . . . Nonprime knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of multiplication law in knot group . . . . . . . . . . . Inequivalent compound knots possessing isomorphic knot groups . Reidemeister moves in projection image of knot . . . . . . . . . . Simple knots with up to 8 minimal crossings . . . . . . . . . . . . Labeling of underpasses for construction of Alexander polynomial . Exceptional knots found by Kinoshita and Terasaka, Conway, and Seifert, all with same Alexander polynomial as trivial knot . . . . Graphical rule for removing crossing in generating Kauffman polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kauffman decomposition of trefoil knot . . . . . . . . . . . . . . . Skein operations relating higher knots to lower ones . . . . . . . . Skein operations for calculating Jones polynomial of two disjoint unknotted loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . Skein operation for calculating Jones polynomial of trefoil knot . . Skein operation for calculating Jones polynomial of Hopf link . . .

853 936 942 952 964 965 968 993 1016 1017 1018 1018 1019 1020 1021 1022 1024 1025 1026 1027 1028 1028 1028


xxxvii 16.16 Knots with 10 and 13 crossings, not distinguished byJonespolynomials1030 16.17 Fraction fN of unknotted closed polymers in ensemble of fixed length L = Na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 16.18 Idealized view of circular DNA . . . . . . . . . . . . . . . . . . . . 1034 16.19 Supercoiled DNA molecule . . . . . . . . . . . . . . . . . . . . . . 1034 16.20 Simple links of two polymers up to 8 crossings . . . . . . . . . . . 1035 16.21 Illustration of Calagareau-White relation . . . . . . . . . . . . . . 1039 16.22 Closed polymers along the contours C1 , C2 respectively . . . . . . . 1043 16.23 Four diagrams contributing to functional integral . . . . . . . . . 1050 16.24 Values of parameter ν at which plateaus in fractional quantum Hall resistance h/e2 ν are expected theoretically . . . . . . . . . . . . . 1062 16.25 Trivial windings LT + and LT − . Their removal by means of Reidemeister move of type I decreases or increases writhe w . . . . . . . 1069 17.1 Plot of symmetric double-well potential . . . . . . . . . . . . . . . 17.2 Classical kink solution in double-well potential connecting two degenerate maxima in reversed potential . . . . . . . . . . . . . . . . 17.3 Reversed double-well potential governing motion of position x as function of imaginary time τ . . . . . . . . . . . . . . . . . . . . . 17.4 Potential for quadratic fluctuations around kink solution . . . . . . 17.5 Vertices and lines of Feynman diagrams for correction factor C in Eq. (17.225) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Positions of extrema xex in asymmetric double-well potential . . . 17.7 Classical bubble solution in reversed asymmetric quartic potential 17.8 Action of deformed bubble solution as function of deformation parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.9 Sequence of paths as function of parameter ξ . . . . . . . . . . . . 17.10 Lines of constant Re (t2 + t3 ) in complex t-plane and integration contours Ci which maintain convergence of fluctuation integral . . 17.11 Potential of anharmonic oscillator for small negative coupling . . . 17.12 Rosen-Morse Potential for fluctuations around the classical bubble solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.13 Reduced imaginary part of lowest three energy levels of anharmonic oscillator for negative couplings . . . . . . . . . . . . . . . . . . . 17.14 Energies of anharmonic oscillator as function of g ′ ≡ g/ω 3, obtained from the variational imaginary part . . . . . . . . . . . . . . . . . 17.15 Reduced imaginary part of ground state energy of anharmonic oscillator from variational perturbation theory . . . . . . . . . . . . 17.16 Cuts in complex gˆ-plane whose moments with respect to inverse coupling constant determine re-expansion coefficients . . . . . . . . 17.17 Theoretically obtained convergence behavior of Nth approximants for α0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.18 Theoretically obtained oscillatory behavior around exponentially fast asymptotic approach of α0 to its exact value . . . . . . . . .

1081 1084 1085 1088 1113 1115 1117 1119 1120 1121 1129 1130 1138 1141 1142 1145 1149 1149

xxxviii 17.19 Comparison of ratios Rn between successive expansion coefficients of the strong-coupling expansion with ratios Rnas . . . . . . . . . . 17.20 Strong-Coupling Expansion of ground state energy in comparison with exact values and perturbative results of 2nd and 3rd order . . 17.21 Renormalization group trajectories for physically identical superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.22 Potential V (ρ) = −ρ2 + ρ4 /2 − j 2 /ρ2 showing barrier in superconducting wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.23 Condensation energy as function of velocity parameter kn = 2πn/L 17.24 Order parameter of superconducting thin circular wire . . . . . . . 17.25 Extremal excursion of order parameter in superconducting wire . . 17.26 Infinitesimal translation of the critical bubble yields antisymmetric wave function of zero energy . . . . . . . . . . . . . . . . . . . . . 17.27 Logarithmic plot of resistance of thin superconducting wire as function of temperature at current 0.2µA . . . . . . . . . . . . . . . . 17.28 Bubble energy as function of its radius R . . . . . . . . . . . . . . 17.29 Qualitative behavior of critical bubble solution as function of its radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.30 Decay of metastable false vacuum in Minkowski space . . . . . . .

1150 1151 1153 1157 1158 1159 1160 1161 1162 1163 1165 1170

18.1 18.2

Closed-time contour in forward–backward path integrals . . . . . . 1196 Behavior of function 6J(z)/π 2 in finite-temperature Lamb shift . 1255

19.1 19.2

Spacetime picture of pair creation . . . . . . . . . . . . . . . . . . 1296 Potential of closed Friedman universe as a function of the radius a/amax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 Radius of universe as a function of time in Friedman universe . . 1312

19.3 20.1 20.2 20.3 20.4 20.5 20.6

20.7 20.8 20.9

Periods of exponential growth of price index averaged over major industrial stocks in the United States over 60 years . . . . . . . . . 1343 Index S&P 500 for 13-year period Jan. 1, 1984 — Dec. 14, 1996, recorded every minute, and volatility in time intervals 30 minutes. 1344 Comparison of best log-normal and Gaussian fits to volatilities over 300 min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344 Fluctuation spectrum of exchange rate DM/US$ . . . . . . . . . . 1345 Behavior of logarithm of stock price following the stochastic differential equation (20.1) . . . . . . . . . . . . . . . . . . . . . . . . . 1346 Left: Lévy tails of the S&P 500 index (1 minute log-returns) plotted against z/δ. Right: Double-logarithmic plot exhibiting the powerlike falloffs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348 Best fit of cumulative versions (20.36) of truncated Lévy distribution 1352 Change in shape of truncated Lévy distributions of width σ = 1 with increasing kurtoses κ = 0 (Gaussian, solid curve), 1, 2 , 5, 10 1353 Change in shape of truncated Lévy distributions of width σ = 1 and kurtosis κ = 1 with increasing skewness s = 0 (solid curve), 0.4, 0.8 1356 H. Kleinert, PATH INTEGRALS

List of Tables 3.1 3.2 3.3 4.1

Expansion coefficients for the ground-state energy of the oscillator with cubic and quartic anharmonicity . . . . . . . . . . . . . . . . . 358 Expansion coefficients for the ground-state energy of the oscillator with cubic and quartic anharmonicity in presence of an external current 359 Effective potential for the oscillator with cubic and quartic anharmonicity, expanded in the coupling constant g . . . . . . . . . . . . 361 Particle energies in purely anharmonic potential gx4 /4 for n = 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 5.2 5.3

Comparison of variational energy with exact ground state energy . Example for competing leading six terms in large-B expansion . . . Perturbation coefficients up to order B 6 in weak-field expansions of variational parameters, and binding energy . . . . . . . . . . . . . . 5.4 Approach of variational energies to Bohr-Sommerfeld approximation 5.5 Energies of the nth excited states of anharmonic oscillator for various coupling strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Second- and third-order approximations to ground state energy of anharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Free energy of anharmonic oscillator for various coupling strengths and temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Comparison of the variational approximations WN at T = 0 for increasing N with the exact ground state energy . . . . . . . . . . . 5.9 Coefficients bn of strong-coupling expansion of ground state energy of anharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Equations determining coefficients bn in strong-coupling expansion . 5.11 Higher approximations to excited energy with n = 8 of anharmonic oscillator at various coupling constants g . . . . . . . . . . . . . . . 5.12 Numerical results for variational parameters and energy . . . . . . . 6.1

First eight variational functions fN (c) . . . . . . . . . . . . . . . . .

415 377 396 398 403 404 409 414 419 423 426 430 451 504

16.1 Numbers of simple and compound knots . . . . . . . . . . . . . . . 1020 16.2 Tables of underpasses and directions of overpassing lines for trefoil knot and knot 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022 16.3 Alexander, Jones, and HOMFLY polynomials for smallest simple knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 xxxix

xl 16.4 Kauffman polynomials in decomposition of trefoil knot . . . . . . . 1026 16.5 Alexander polynomials A(s, t) and HOMFLY polynomials H(t, α) for simple links of two closed curves up to 8 minimal crossings . . . . . 1037 17.1 Comparison between exact perturbation coefficients, semiclassical ones, and those from our variational approximation . . . . . . . . . 1140 17.2 Coefficients of semiclassical expansion around classical solution . . . 1143


H. Kleinert, PATH INTEGRALS June 16, 2007 ( /home/kleinert/kleinert/books/pathis/pthic1.tex)

Ay, call it holy ground, The soil where first they trod! F. D. Hemans (1793-1835), Landing of the Pilgrim Fathers

1 Fundamentals Path integrals deal with fluctuating line-like structures. These appear in nature in a variety of ways, for instance, as particle orbits in spacetime continua, as polymers in solutions, as vortex lines in superfluids, as defect lines in crystals and liquid crystals. Their fluctuations can be of quantum-mechanical, thermodynamic, or statistical origin. Path integrals are an ideal tool to describe these fluctuating line-like structures, thereby leading to a unified understanding of many quite different physical phenomena. In developing the formalism we shall repeatedly invoke well-known concepts of classical mechanics, quantum mechanics, and statistical mechanics, to be summarized in this chapter. In Section 1.13, we emphasize some important problems of operator quantum mechanics in spaces with curvature and torsion. These problems will be solved in Chapters 10 and 8 by means of path integrals.1

1.1

Classical Mechanics

The orbits of a classical-mechanical system are described by a set of time-dependent generalized coordinates q1 (t), . . . , qN (t). A Lagrangian L(qi , q˙i , t)

(1.1)

depending on q1 , . . . , qN and the associated velocities q˙1 , . . . , q˙N governs the dynamics of the system. The dots denote the time derivative d/dt. The Lagrangian is at most a quadratic function of q˙i . The time integral A[qi ] =

Z

tb

ta

dt L(qi (t), q˙i (t), t)

(1.2)

of the Lagrangian along an arbitrary path qi (t) is called the action of this path. The path being actually chosen by the system as a function of time is called the classical path or the classical orbit qicl (t). It has the property of extremizing the action in comparison with all neighboring paths qi (t) = qicl (t) + δqi (t) 1

Readers familiar with the foundations may start directly with Section 1.13.

1

(1.3)

2

1 Fundamentals

having the same endpoints q(tb ), q(ta ). To express this property formally, one introduces the variation of the action as the linear term in the Taylor expansion of A[qi ] in powers of δqi (t): δA[qi ] ≡ {A[qi + δqi ] − A[qi ]}lin .

(1.4)

The extremal principle for the classical path is then

δA[qi ]

qi (t)=qicl (t)

=0

(1.5)

for all variations of the path around the classical path, δqi (t) ≡ qi (t) − qicl (t), which vanish at the endpoints, i.e., which satisfy δqi (ta ) = δqi (tb ) = 0.

(1.6)

Since the action is a time integral of a Lagrangian, the extremality property can be phrased in terms of differential equations. Let us calculate the variation of A[qi ] explicitly: δA[qi ] = {A[qi + δqi ] − A[qi ]}lin =

Z

=

Z

=

Z

tb ta tb ta tb ta

dt {L (qi (t) + δqi (t), q˙i (t) + δ q˙i (t), t) − L (qi (t), q˙i (t), t)}lin )

dt

(

∂L ∂L δqi (t) + δ q˙i (t) ∂qi ∂ q˙i

dt

(

tb d ∂L ∂L ∂L − δqi (t) + δqi (t) . ∂qi dt ∂ q˙i ∂ q˙i ta

)

(1.7)

The last expression arises from a partial integration of the δ q˙i term. Here, as in the entire text, repeated indices are understood to be summed (Einstein’s summation convention). The endpoint terms (surface or boundary terms) with the time t equal to ta and tb may be dropped, due to (1.6). Thus we find for the classical orbit qicl (t) the Euler-Lagrange equations: d ∂L ∂L = . (1.8) dt ∂ q˙i ∂qi There is an alternative formulation of classical dynamics which is based on a Legendre-transformed function of the Lagrangian called the Hamiltonian H≡

∂L q˙i − L(qi , q˙i , t). ∂ q˙i

(1.9)

Its value at any time is equal to the energy of the system. According to the general theory of Legendre transformations [1], the natural variables which H depends on are no longer qi and q˙i , but qi and the generalized momenta pi , the latter being defined by the N equations ∂ pi ≡ L(qi , q˙i , t). (1.10) ∂ q˙i H. Kleinert, PATH INTEGRALS

3

1.1 Classical Mechanics

In order to express the Hamiltonian H (pi , qi , t) in terms of its proper variables pi , qi , the equations (1.10) have to be solved for q˙i , q˙i = vi (pi , qi , t).

(1.11)

This is possible provided the Hessian metric Hij (qi , q˙i , t) ≡

∂2 L(qi , q˙i , t) ∂ q˙i ∂ q˙j

(1.12)

is nonsingular. The result is inserted into (1.9), leading to the Hamiltonian as a function of pi and qi : H (pi , qi , t) = pi vi (pi , qi , t) − L (qi , vi (pi , qi , t) , t) .

(1.13)

In terms of this Hamiltonian, the action is the following functional of pi (t) and qi (t): A[pi , qi ] =

Z

tb

ta

h

i

dt pi (t)q˙i (t) − H(pi (t), qi (t), t) .

(1.14)

This is the so-called canonical form of the action. The classical orbits are now speccl ified by pcl i (t), qi (t). They extremize the action in comparison with all neighboring orbits in which the coordinates qi (t) are varied at fixed endpoints [see (1.3), (1.6)] whereas the momenta pi (t) are varied without restriction: qi (t) = qicl (t) + δqi (t),

δqi (ta ) = δqi (tb ) = 0,

(1.15)

pi (t) = pcl i (t) + δpi (t). In general, the variation is δA[pi , qi ] =

Z

tb

=

Z

tb

ta

ta

"

∂H ∂H dt δpi (t)q˙i (t) + pi (t)δ q˙i (t) − δpi − δqi ∂pi ∂qi dt

#

("

"

#

∂H ∂H q˙i (t) − δpi − p˙i (t) + δqi ∂pi ∂qi

)

#

(1.16)

tb

+ pi (t)δqi (t) . tb

cl Since this variation has to vanish for the classical orbits, we find that pcl i (t), qi (t) must be solutions of the Hamilton equations of motion

∂H , ∂qi ∂H = . ∂pi

p˙i = − q˙i

(1.17)

These agree with the Euler-Lagrange equations (1.8) via (1.9) and (1.10), as can easily be verified. The 2N-dimensional space of all pi and qi is called the phase space.

4

1 Fundamentals

As a particle moves along a classical trajectory, the action changes as a function of the end positions (1.16) by δA[pi, qi ] = pi (tb )δqi (tb ) − pi (ta )δqi (ta ).

(1.18)

An arbitrary function O(pi (t), qi (t), t) changes along an arbitrary path as follows: d ∂O ∂O ∂O p˙i + q˙i + O (pi (t), qi (t), t) = . dt ∂pi ∂qi ∂t

(1.19)

If the path coincides with a classical orbit, we may insert (1.17) and find ∂H ∂O ∂O ∂H ∂O dO = − + dt ∂pi ∂qi ∂pi ∂qi ∂t ∂O ≡ {H, O} + . ∂t

(1.20)

Here we have introduced the symbol {. . . , . . .} called Poisson brackets: {A, B} ≡

∂A ∂B ∂B ∂A − , ∂pi ∂qi ∂pi ∂qi

(1.21)

again with the Einstein summation convention for the repeated index i. The Poisson brackets have the obvious properties {A, B} = − {B, A} {A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0

antisymmetry,

(1.22)

Jacobi identity.

(1.23)

If two quantities have vanishing Poisson brackets, they are said to commute. The original Hamilton equations are a special case of (1.20): d pi = {H, pi} = dt d qi = {H, qi } = dt

∂H ∂pi ∂pi ∂H ∂H − =− , ∂pj ∂qj ∂pj ∂qj ∂qi ∂H ∂qi ∂qi ∂H ∂H − = . ∂pj ∂qj ∂pj ∂qj ∂pi

(1.24)

By definition, the phase space variables pi , qi satisfy the Poisson brackets {pi , qj } = δij ,

{pi , pj } = 0,

(1.25)

{qi , qj } = 0.

A function O(pi, qi ) which has no explicit dependence on time and which, moreover, commutes with H (i.e., {O, H} = 0), is a constant of motion along the classical path, due to (1.20). In particular, H itself is often time-independent, i.e., of the form H = H(pi , qi ).

(1.26) H. Kleinert, PATH INTEGRALS

5


Then, since H commutes with itself, the energy is a constant of motion. The Lagrangian formalism has the virtue of being independent of the particular choice of the coordinates qi . Let Qi be any other set of coordinates describing the system which is connected with qi by what is called a local 2 or point transformation qi = fi (Qj , t).

(1.27)

Certainly, to be of use, this relation must be invertible, at least in some neighborhood of the classical path, (1.28) Qi = f −1 i (qj , t). Otherwise Qi and qi could not both parametrize the same system. Therefore, fi must have a nonvanishing Jacobi determinant: det

∂fi ∂Qj

!

6= 0.

(1.29)

In terms of Qi , the initial Lagrangian takes the form L′ Qj , Q˙ j , t ≡ L fi (Qj , t) , f˙i (Qj , t) , t and the action reads

A =

Z

=

Z

tb

ta tb ta

(1.30)

dt L′ Qj (t), Q˙ j (t), t

(1.31)

dt L fi (Qj (t), t) , f˙i (Qj (t), t) , t .

By varying the upper expression with respect to δQj (t), δ Q˙ j (t) while keeping δQj (ta ) = δQj (tb ) = 0, we find the equations of motion ∂L′ d ∂L′ − = 0. dt ∂ Q˙ j ∂Qj

(1.32)

The variation of the lower expression, on the other hand, gives δA =

Z

tb

=

Z

tb

ta

ta

dt dt

∂L ∂L ˙ δfi + δ fi ∂qi ∂ q˙i !

!

∂L d ∂L ∂L tb − δfi + δfi . ta ∂qi dt ∂ q˙i ∂ q˙i

(1.33)

If δqi is arbitrary, then so is δfi . Moreover, with δqi (ta ) = δqi (tb ) = 0, also δfi vanishes at the endpoints. Hence the extremum of the action is determined equally well by the Euler-Lagrange equations for Qj (t) [as it was by those for qi (t)]. 2

The word local means here at a specific time. This terminology is of common use in field theory where local means, more generally, at a specific spacetime point .

6

1 Fundamentals

Note that the locality property is quite restrictive for the transformation of the generalized velocities q˙i (t). They will necessarily be linear in Q˙ j : ∂fi ˙ ∂fi q˙i = f˙i (Qj , t) = Qj + . ∂Qj ∂t

(1.34)

In phase space, there exists also the possibility of performing local changes of the canonical coordinates pi , qi to new ones Pj , Qj . Let them be related by pi = pi (Pj , Qj , t),

(1.35)

qi = qi (Pj , Qj , t), with the inverse relations

Pj = Pj (pi , qi , t),

(1.36)

Qj = Qj (pi , qi , t).

However, while the Euler-Lagrange equations maintain their form under any local change of coordinates, the Hamilton equations do not hold, in general, for any transformed coordinates Pj (t), Qj (t). The local transformations pi (t), qi (t) → Pj (t), Qj (t) for which they hold, are referred to as canonical . They are characterized by the form invariance of the action, up to an arbitrary surface term, Z

tb

ta

dt [pi q˙i − H(pi , qi , t)] =

Z

tb

ta

h

dt Pj Q˙ j − H ′ (Pj , Qj , t) tb + F (Pj , Qj , t)

ta

i

(1.37)

,

where H ′ (Pj , Qj , t) is some new Hamiltonian. Its relation with H(pi, qi , t) must be chosen in such a way that the equality of the action holds for any path pi (t), qi (t) connecting the same endpoints (at least any in some neighborhood of the classical orbits). If such an invariance exists then a variation of this action yields for Pj (t) and Qj (t) the Hamilton equations of motion governed by H ′ : ′

∂H , P˙i = − ∂Qi ∂H ′ ˙ Qi = . ∂Pi

(1.38)

The invariance (1.37) can be expressed differently by rewriting the integral on the left-hand side in terms of the new variables Pj (t), Qj (t), Z

tb

ta

dt

(

pi

∂qi ˙ ∂qi ˙ ∂qi Pj + Qj + ∂Pj ∂Qj ∂t

!

)

− H(pi (Pj , Qj , t), qi (Pj , Qj , t), t) ,

(1.39)

and subtracting it from the right-hand side, leading to Z

tb

ta

(

∂qi P j − pi ∂Qj

!

dQj − pi

∂qi dPj ∂P!j )

∂qi − H + pi − H dt ∂t ′

=

tb −F (Pj , Qj , t) .

(1.40)

ta


7


The integral is now a line integral along a curve in the (2N + 1)-dimensional space, consisting of the 2N-dimensional phase space variables pi , qi and of the time t. The right-hand side depends only on the endpoints. Thus we conclude that the integrand on the left-hand side must be a total differential. As such it has to satisfy the standard Schwarz integrability conditions [2], according to which all second derivatives have to be independent of the sequence of differentiation. Explicitly, these conditions are ∂pi ∂qi ∂qi ∂pi − = δkl , ∂Pk ∂Ql ∂Pk ∂Ql ∂pi ∂qi ∂qi ∂pi − = 0, ∂Pk ∂Pl ∂Pk ∂Pl ∂pi ∂qi ∂qi ∂pi − ∂Qk ∂Ql ∂Qk ∂Ql

(1.41)

= 0,

and

∂pi ∂qi ∂qi ∂pi ∂(H ′ − H) − = , ∂t ∂Pl ∂t ∂Pl ∂Pl (1.42) ∂pi ∂qi ∂qi ∂pi ∂(H ′ − H) − = . ∂t ∂Ql ∂t ∂Ql ∂Ql The first three equations define the so-called Lagrange brackets in terms of which they are written as (Pk , Ql ) = δkl , (Pk , Pl ) = 0, (Qk , Ql ) = 0.

(1.43)

Time-dependent coordinate transformations satisfying these equations are called symplectic. After a little algebra involving the matrix of derivatives 

its inverse

J= 

J −1 = 

∂Pi /∂pj

∂Pi /∂qj

∂Qi /∂pj

∂Qi /∂qj

∂pi /∂Pj

∂pi /∂Qj

∂qi /∂Pj

∂qi /∂Qj

and the symplectic unit matrix

E=

0 −δij

δij 0

!

,



(1.44)



(1.45)

, ,

(1.46)

we find that the Lagrange brackets (1.43) are equivalent to the Poisson brackets {Pk , Ql } = δkl , {Pk , Pl } = 0, {Qk , Ql } = 0.

(1.47)

8

1 Fundamentals

This follows from the fact that the 2N × 2N matrix formed from the Lagrange brackets   −(Qi , Pj ) −(Qi , Qj )  L≡ (1.48) (Pi , Pj ) (Pi , Qj ) can be written as (E −1 J −1 E)T J −1 , while an analogous matrix formed from the Poisson brackets   {Pi , Qj } − {Pi , Pj }  P ≡ (1.49) {Qi , Qj } − {Qi , Pj }

is equal to J(E −1 JE)T . Hence L = P −1 , so that (1.43) and (1.47) are equivalent to each other. Note that the Lagrange brackets (1.43) [and thus the Poisson brackets (1.47)] ensure pi q˙i − Pj Q˙ j to be a total differential of some function of Pj and Qj in the 2N-dimensional phase space: d pi q˙i − Pj Q˙ j = G(Pj , Qj , t). dt

(1.50)

The Poisson brackets (1.47) for Pi , Qi have the same form as those in Eqs. (1.25) for the original phase space variables pi , qi . The other two equations (1.42) relate the new Hamiltonian to the old one. They can always be used to construct H ′ (Pj , Qj , t) from H(pi, qi , t). The Lagrange brackets (1.43) or Poisson brackets (1.47) are therefore both necessary and sufficient for the transformation pi , qi → Pj , Qj to be canonical. A canonical transformation preserves the volume in phase space. This follows from the fact that the matrix product J(E −1 JE)T is equal to the 2N × 2N unit matrix (1.49). Hence det (J) = ±1 and YZ

[dpi dqi ] =

i

YZ

[dPj dQj ] .

(1.51)

j

It is obvious that the process of canonical transformations is reflexive. It may be viewed just as well from the opposite side, with the roles of pi , qi and Pj , Qj exchanged [we could just as well have considered the integrand (1.40) as a complete differential in Pj , Qj , t space]. Once a system is described in terms of new canonical coordinates Pj , Qj , we introduce the new Poisson brackets {A, B}′ ≡

∂A ∂B ∂B ∂A − , ∂Pj ∂Qj ∂Pj ∂Qj

(1.52)

and the equation of motion for an arbitrary observable quantity O (Pj (t), Qj (t), t) becomes with (1.38) dO ∂O ′ = {H ′, O} + , (1.53) dt ∂t H. Kleinert, PATH INTEGRALS

9


by complete analogy with (1.20). The new Poisson brackets automatically guarantee the canonical commutation rules {Pi , Qj }′ {Pi , Pj }′

= δij , = 0,

(1.54)

{Qi , Qj }′ = 0.

A standard class of canonical transformations can be constructed by introducing a generating function F satisfying a relation of the type (1.37), but depending explicitly on half an old and half a new set of canonical coordinates, for instance F = F (qi , Qj , t).

(1.55)

One now considers the equation Z

tb

ta

dt [pi q˙i − H(pi , qi , t)] =

replaces Pj Q˙ j by −P˙j Qj +

Z

tb

ta

"

#

d dt Pj Q˙ j − H (Pj , Qj , t) + F (qi , Qj , t) , (1.56) dt

d PQ, dt j j

′

defines

F (qi , Pj , t) ≡ F (qi , Qj , t) + Pj Qj , and works out the derivatives. This yields Z

tb

ta

=

n

o

dt pi q˙i + P˙j Qj − [H(pi , qi , t) − H ′ (Pj , Qj , t)] Z

tb

ta

(

)

∂F ∂F ∂F (qi , Pj , t) . dt (qi , Pj , t)q˙i + (qi , Pj , t)P˙j + ∂qi ∂Pj ∂t

(1.57)

A comparison between the two sides renders for the canonical transformation the equations ∂ pi = F (qi , Pj , t), ∂qi (1.58) ∂ Qj = F (qi , Pj , t). ∂Pj The second equation shows that the above relation between F (qi , Pj , t) and F (qi , Qj , t) amounts to a Legendre transformation. The new Hamiltonian is H ′ (Pj , Qj , t) = H(pi , qi , t) +

∂ F (qi , Pj , t). ∂t

(1.59)

Instead of (1.55) we could, of course, also have chosen functions with other mixtures of arguments such as F (qi , Pj , t), F (pi, Qj , t), F (pi, Pj , t) to generate simple canonical transformations.

10

1 Fundamentals

A particularly important canonical transformation arises by choosing a generating function F (qi , Pj ) in such a way that it leads to time-independent momenta Pj ≡ αj . Coordinates Qj with this property are called cyclic. To find cyclic coordinates we must search for a generating function F (qj , Pj , t) which makes the transformed H ′ in (1.59) vanish identically. Then all derivatives with respect to the coordinates vanish and the new momenta Pj are trivially constant. Thus we seek for a solution of the equation ∂ F (qi , Pj , t) = −H(pi , qi , t), ∂t

(1.60)

where the momentum variables in the Hamiltonian obey the first equation of (1.58). This leads to the following partial differential equation for F (qi , Pj , t): ∂t F (qi , Pj , t) = −H(∂qi F (qi , Pj , t), qi , t),

(1.61)

called the Hamilton-Jacobi equation. A generating function which achieves this goal is supplied by the action functional (1.14). When following the solutions starting from a fixed initial point and running to all possible final points qi at a time t, the associated actions of these solutions form a function A(qi , t). Due to (1.18), this satisfies precisely the first of the equations (1.58): pi =

∂ A(qi , t). ∂qi

(1.62)

Moreover, the function A(qi , t) has the time derivative d A(qi (t), t) = pi (t)q˙i (t) − H(pi (t), qi (t), t). dt

(1.63)

Together with (1.62) this implies ∂t A(qi , t) = −H(pi , qi , t).

(1.64)

If the momenta pi on the right-hand side are replaced according to (1.62), A(qi , t) is indeed seen to be a solution of the Hamilton-Jacobi differential equation: ∂t A(qi , t) = −H(∂qi A(qi , t), qi , t).

1.2

(1.65)

Relativistic Mechanics in Curved Spacetime

The classical action of a relativistic spinless point particle in a curved fourdimensional spacetime is usually written as an integral A = −Mc2

Z

dτ L(q, q) ˙ = −Mc2

Z

q

dτ gµν q˙µ (τ )q˙ν (τ ),

(1.66)


11

1.3 Quantum Mechanics

where τ is an arbitrary parameter of the trajectory. It can be chosen in the final trajectory to make L(q, q) ˙ ≡ 1, in which case it coincides with the proper time of the particle. For arbitrary τ , the Euler-Lagrange equation (1.8) reads "

#

d 1 1 gµν q˙ν = (∂µ gκλ ) q˙κ q˙λ . dt L(q, q) ˙ 2L(q, q) ˙

(1.67)

If τ is the proper time where L(q, q) ˙ ≡ 1, this simplifies to d 1 (gµν q˙ν ) = (∂µ gκλ ) q˙κ q˙λ , dt 2 or

1 gµν q¨ = ∂µ gκλ − ∂λ gµκ q˙κ q˙λ . 2 At this point one introduces the Christoffel symbol ν

¯ λνµ ≡ 1 (∂λ gνµ + ∂ν gλµ − ∂µ gλν ), Γ 2

(1.68) (1.69)

(1.70)

and the Christoffel symbol of the second kind3 ¯ µ ≡ g µσ Γ ¯ κνσ . Γ κν

(1.71)

¯ κλ µ q˙κ q˙λ = 0. q¨µ + Γ

(1.72)

Then (1.69) can be written as

Since the solutions of this equation minimize the length of a curve in spacetime, they are called geodesics.

1.3

Quantum Mechanics

Historically, the extension of classical mechanics to quantum mechanics became necessary in order to understand the stability of atomic orbits and the discrete nature of atomic spectra. It soon became clear that these phenomena reflect the fact that at a sufficiently short length scale, small material particles such as electrons behave like waves, called material waves. The fact that waves cannot be squeezed into an arbitrarily small volume without increasing indefinitely their frequency and thus their energy, prevents the collapse of the electrons into the nucleus, which would take place in classical mechanics. The discreteness of the atomic states of an electron are a manifestation of standing material waves in the atomic potential well, by analogy with the standing waves of electromagnetism in a cavity. 3

In many textbooks, for instance S. Weinberg, Gravitation and Cosmology, Wiley, New York, 1972, the upper index and the third index in (1.70) stand at the first position. Our notation follows J.A. Schouten, Ricci Calculus, Springer, Berlin, 1954. It will allow for a closer analogy with gauge fields in the construction of the Riemann tensor as a covariant curl of the Christoffel symbol in Chapter 10. See H. Kleinert, Gauge Fields in Condensed Matter , Vol. II Stresses and Defects, World Scientific Publishing Co., Singapore 1989, pp. 744-1443 (http://www.physik.fu-berlin.de/~kleinert/b2).

12

1.3.1

1 Fundamentals

Bragg Reflections and Interference

The most direct manifestation of the wave nature of small particles is seen in diffraction experiments on periodic structures, for example of electrons diffracted by a crystal. If an electron beam of fixed momentum p passes through a crystal, it emerges along sharply peaked angles. These are the well-known Bragg reflections. They look very similar to the interference patterns of electromagnetic waves. In fact, it is possible to use the same mathematical framework to explain these patterns as in electromagnetism. A free particle moving with momentum p = (p1 , p2 , . . . , pD ).

(1.73)

through a D-dimensional Euclidean space spanned by the Cartesian coordinate vectors x = (x1 , x2 , . . . , xD ) (1.74) is associated with a plane wave, whose field strength or wave function has the form Ψp (x, t) = eikx−iωt ,

(1.75)

where k is the wave vector pointing into the direction of p and ω is the wave frequency. Each scattering center, say at x′ , becomes a source of a spherical wave with the spatial behavior eikR /R (with R ≡ |x − x′ | and k ≡ |k|) and the wavelength λ = 2π/k. At the detector, all field strengths have to be added to the total field strength Ψ(x, t). The absolute square of the total field strength, |Ψ(x, t)|2, is proportional to the number of electrons arriving at the detector. The standard experiment where these rules can most simply be applied consists of an electron beam impinging vertically upon a flat screen with two parallel slits a distance d apart. Behind these, one observes the number of particles arriving per unit time (see Fig. 1.1)

dN dt

2 1 1 ∝ eik(R+ 2 d sin ϕ) + eik(R− 2 d sin ϕ)

eikx

Figure 1.1 Probability distribution of particle behind double slit, being proportional to the absolute square of the sum of the two complex field strengths. H. Kleinert, PATH INTEGRALS

13


2 1 1 1 dN ∝ |Ψ1 + Ψ2 |2 ≈ eik(R+ 2 d sin ϕ) + eik(R− 2 d sin ϕ) 2 , (1.76) dt R where ϕ is the angle of deflection from the normal. Conventionally, the wave function Ψ(x, t) is normalized to describe a single particle. Its absolute square gives directly the probability density of the particle at the place x in space, i.e., d3 x |Ψ(x, t)|2 is the probability of finding the particle in the volume element d3 x around x.

1.3.2

Matter Waves

From the experimentally observed relation between the momentum and the size of the angular deflection ϕ of the diffracted beam of the particles, one deduces the relation between momentum and wave vector p=h ¯ k,

(1.77)

where h ¯ is the universal Planck constant whose dimension is equal to that of an action, h = 1.0545919(80) × 10−27 erg sec (1.78) h ¯≡ 2π (the number in parentheses indicating the experimental uncertainty of the last two digits before it). A similar relation holds between the energy and the frequency of the wave Ψ(x, t). It may be determined by an absorption process in which a light wave hits an electron (for example, by kicking it out of the surface of a metal, the well-known photoeffect). From the threshold property of the photoeffect one learns that an electromagnetic wave oscillating in time as e−iωt can transfer to the electron the energy E=h ¯ ω, (1.79) where the proportionality constant h ¯ is the same as in (1.77). The reason for this lies in the properties of electromagnetic waves. On the one hand, their frequency ω and the wave vector k satisfy the relation ω/c = |k|, where c is the light velocity defined to be c ≡ 299 792.458km/s. On the other hand, energy and momentum are related by E/c = |p|. Thus, the quanta of electromagnetic waves, the photons, certainly satisfy (1.77) and the constant h ¯ must be the same as in Eq. (1.79). With matter waves and photons sharing the same relations (1.77), it is suggestive to postulate also the relation (1.79) between energy and frequency to be universal for the waves of all particles, massive and massless ones. All free particle of momentum p are described by a plane wave of wavelength λ = 2π/|k| = 2π¯ h/|p|, with the explicit form (1.80) Ψp (x, t) = N ei(px−Ep t)/¯h , where N is some normalization constant. In a finite volume, the wave function is normalized to unity. In an infinite volume, this normalization makes the wave function vanish. To avoid this, the current density of the particle probability j(x, t) ≡ −i

↔ h ¯ ∗ ψ (x, t) ∇ ψ(x, t) 2m

(1.81)

14

1 Fundamentals ↔

is normalized in some convenient way, where ∇ is a short notation for the difference between right- and left-derivatives ↔

→

←

ψ ∗ (x, t) ∇ ψ(x, t) ≡ ψ ∗ (x, t) ∇ ψ(x, t) − ψ ∗ (x, t) ∇ ψ(x, t) ≡ ψ ∗ (x, t)∇ψ(x, t) − [∇ψ ∗ (x, t)] ψ(x, t).

(1.82)

The energy Ep depends on the momentum of the particle in the classical way, i.e., for nonrelativistic material particles of mass M it is Ep = p2 /2M, for relativistic √ 2 ones Ep = c p + M 2 c2 , and for massless particles such as photons Ep = c|p|. The common relation Ep = h ¯ ω for photons and matter waves is necessary to guarantee conservation of energy in quantum mechanics. In general, momentum and energy of a particle are not defined as well as in the plane-wave function (1.80). Usually, a particle wave is some superposition of plane waves (1.80) Z d3 p f (p)ei(px−Ep t)/¯h . (1.83) Ψ(x, t) = (2π¯ h)3 By the Fourier inversion theorem, f (p) can be calculated via the integral f (p) =

Z

d3 x e−ipx/¯h Ψ(x, 0).

(1.84)

With an appropriate choice of f (p) it is possible to prepare Ψ(x, t) in any desired form at some initial time, say at t = 0. For example, Ψ(x, 0) may be a function sharply centered around a space point x ¯. Then f (p) is approximately a pure phase −ip¯ x/¯ h f (p) ∼ e , and the wave contains all momenta with equal probability. Conversely, if the particle amplitude is spread out in space, its momentum distribution is confined to a small region. The limiting f (p) is concentrated at a specific mo¯ . The particle is found at each point in space with equal probability, with mentum p the amplitude oscillating like Ψ(x, t) ∼ ei(¯px−Ep¯ t)/¯h . In general, the width of Ψ(x, 0) in space and of f (p) in momentum space are inversely proportional to each other: ∆x ∆p ∼ h ¯.

(1.85)

This is the content of Heisenberg’s principle of uncertainty. If the wave is localized in a finite region of space while having at the same time a fairly well-defined average ¯ , it is called a wave packet. The maximum in the associated probability momentum p density can be shown from (1.83) to move with a velocity ¯ = ∂Ep¯ /∂ p ¯. v

(1.86)

¯. This coincides with the velocity of a classical particle of momentum p H. Kleinert, PATH INTEGRALS

15


1.3.3

Schr¨ odinger Equation

Suppose now that the particle is nonrelativistic and has a mass M. The classical Hamiltonian, and thus the energy Ep , are given by H(p) = Ep =

p2 . 2M

(1.87)

We may therefore derive the following identity for the wave field Ψp (x, t): Z

d3 p f (p) [H(p) − Ep ] ei(px−Ep t)/¯h = 0. (2π¯ h)3

(1.88)

The arguments inside the brackets can be removed from the integral by observing that p and Ep inside the integral are equivalent to the differential operators ˆ = −i¯ p h∇,

Eˆ = i¯ h∂t

(1.89)

outside. Then, Eq. (1.88) may be written as the differential equation [H(−i¯ h∇) − i¯ h∂t )]Ψ(x, t) = 0.

(1.90)

This is the Schrödinger equation for the wave function of a free particle. The equation suggests that the motion of a particle with an arbitrary Hamiltonian H(p, x, t) follows the straightforward generalization of (1.90)

ˆ − i¯ H h∂t Ψ(x, t) = 0,

ˆ is the differential operator where H

ˆ ≡ H(−i¯ H h∇, x, t).

(1.91)

(1.92)

ˆ from the classical Hamiltonian H(p, x, t) by the substitution The rule of obtaining H ˆ = −i¯ p→p h∇ will be referred to as the correspondence principle.4 We shall see in Sections 1.13–1.15 that this simple correspondence principle holds only in Cartesian coordinates. The Schrödinger operators (1.89) of momentum and energy satisfy with x and t the so-called canonical commutation relations [ˆ pi , xj ] = −i¯ h,

ˆ t] = 0 = i¯ [E, h.

(1.93)

If the Hamiltonian does not depend explicitly on time, the Hilbert space can be spanned by the energy eigenstates states ΨEn (x, t) = e−iEn t/¯h ΨEn (x), where 4

Our formulation of this principle is slightly stronger than the historical one used in the initial phase of quantum mechanics, which gave certain translation rules between classical and quantummechanical relations. The substitution rule for the momentum runs also under the name Jordan rule.

16

1 Fundamentals

ΨEn (x) are time-independent stationary states, which solve the time-independent Schrödinger equation ˆ p, x)ΨEn (x) = En ΨEn (x). H(ˆ (1.94) The validity of the Schrödinger theory (1.91) is confirmed by experiment, most notably for the Coulomb Hamiltonian e2 p2 − , (1.95) 2M r which governs the quantum mechanics of the hydrogen atom in the center-of-mass coordinate system of electron and proton, where M is the reduced mass of the two particles. Since the square of the wave function, |Ψ(x, t)|2 , is interpreted as the probability density of a single particle in a finite volume, the integral over the entire volume must be normalized to unity: H(p, x) =

Z

d3 x |Ψ(x, t)|2 = 1.

(1.96)

For a stable particle, this normalization must remain the same at all times. If Ψ(x, t) is to follow the Schrödinger equation (1.91), this is assured if and only if the Hamiltonian operator is Hermitian,5 i.e., if it satisfies for arbitrary wave functions Ψ1 , Ψ2 the equality Z

ˆ 2 (x, t)]∗ Ψ1 (x, t) = d3 x [HΨ

Z

ˆ 1 (x, t). d3 x Ψ∗2 (x, t)HΨ

(1.97)

ˆ † of the operator H, ˆ which satThe left-hand side defines the Hermitian-adjoint H isfies for all wave functions Ψ1 (x, t), Ψ2 (x, t) the identity Z

d

3

ˆ † Ψ1 (x, t) x Ψ∗2 (x, t)H

≡

Z

ˆ 2 (x, t)]∗ Ψ1 (x, t). d3 x [HΨ

ˆ is Hermitian, if it coincides with its Hermitian-adjoint H ˆ †: An operator H ˆ =H ˆ †. H Let Z

(1.98)

(1.99)

us calculate the time change of the integral over two arbitrary wave functions, d x Ψ∗2 (x, t)Ψ1 (x, t). With the Schrödinger equation (1.91), this time change vanˆ is Hermitian: ishes indeed as long as H 3

d i¯ h dt =

Z

Z

d3 x Ψ∗2 (x, t)Ψ1 (x, t) d

3

ˆ 1 (x, t) x Ψ∗2 (x, t)HΨ

−

Z

(1.100) ˆ 2 (x, t)]∗ Ψ1 (x, t) = 0. d x [HΨ 3

5

Problems arising from unboundedness or discontinuities of the Hamiltonian and other quantum-mechanical operators, such as restrictions of the domains of definition, are ignored here since they are well understood. Correspondingly we do not distinguish between Hermitian and selfadjoint operators (see J. v. Neumann, Mathematische Grundlagen der Quantenmechanik , Springer, Berlin, 1932). Some quantum-mechanical operator subtleties will manifest themselves in this book as problems of path integration to be solved in Chapter 12. The precise relationship between the two calls for further detailed investigations. H. Kleinert, PATH INTEGRALS

17


This also implies the time independence of the normalization integral R 3 d x |Ψ(x, t)|2 = 1. ˆ is not Hermitian, one can always find an eigenstate of H ˆ whose Conversely, if H † norm changes with time: any eigenstate of (H − H )/i has this property. ˆ will automatically ˆ = −i¯ Since p h∇ and x are themselves Hermitian operators, H be a Hermitian operator if it is a sum of a kinetic and a potential energy: H(p, x, t) = T (p, t) + V (x, t).

(1.101)

This is always the case for nonrelativistic particles in Cartesian coordinates x. If p and x appear in one and the same term of H, for instance as p2 x2 , the corresponˆ Then dence principle does not lead to a unique quantum-mechanical operator H. there seem to be, in principle, several Hermitian operators which, in the above examˆ and two x ˆ operators [for instance ple, can be constructed from the product of two p ˆ2p ˆx ˆ 2p ˆ 2 +β x ˆ 2 +γ p ˆ with α+β+γ = 1]. They all correspond to the same classical αˆ p2 x p2 x2 . At first sight it appears as though only a comparison with experiment could select the correct operator ordering. This is referred to as the operator-ordering problem of quantum mechanics which has plagued many researchers in the past. If the ordering problem is caused by the geometry of the space in which the particle moves, there exists a surprisingly simple geometric principle which specifies the ordering in the physically correct way. Before presenting this in Chapter 10 we shall avoid ambiguities by assuming H(p, x, t) to have the standard form (1.101), unless otherwise stated.

1.3.4

Particle Current Conservation

The conservation of the total probability (1.96) is a consequence of a more general local conservation law linking the current density of the particle probability j(x, t) ≡ −i

↔ h ¯ ψ(x, t) ∇ ψ(x, t) 2m

(1.102)

with the probability density ρ(x, t) = ψ ∗ (x, t)ψ(x, t)

(1.103)

∂t ρ(x, t) = −∇ · j(x, t).

(1.104)

via the relation By integrating this current conservation law over a volume V enclosed by a surface S, and using Green’s theorem, one finds Z

V

3

d x ∂t ρ(x, t) = −

Z

3

V

d x ∇ · j(x, t) = −

Z

S

dS · j(x, t),

(1.105)

where dS are the directed infinitesimal surface elements. This equation states that the probability in a volume decreases by the same amount by which probability leaves the surface via the current j(x, t).

18

1 Fundamentals

By extending the integral (1.105) over the entire space and assuming the currents to vanish at spatial infinity, we recover the conservation of the total probability (1.96). More general dynamical systems with N particles in Euclidean space are parametrized in terms of 3N Cartesian coordinates xν (ν = 1, . . . , N). The Hamiltonian has the form N X p2ν H(pν , xν , t) = + V (xν , t), (1.106) ν=1 2Mν where the arguments pν , xν in H and V stand for all pν ’s, xν with ν = 1, 2, 3, . . . , N. The wave function Ψ(xν , t) satisfies the N-particle Schrödinger equation (

1.4

−

N X

ν=1

"

h ¯2 ∂x 2 + V (xν , t) 2Mν ν

#)

Ψ(xν , t) = i¯h∂t Ψ(xν , t).

(1.107)

Dirac’s Bra-Ket Formalism

Mathematically speaking, the wave function Ψ(x, t) may be considered as a vector in an infinite-dimensional complex vector space called Hilbert space. The configuration space variable x plays the role of a continuous “index” of these vectors. An obvious contact with the usual vector notation may be established, in which a D-dimensional vector v is given in terms of its components vi with a subscript i = 1, . . . D, by writing the argument x of Ψ(x, t) as a subscript: Ψ(x, t) ≡ Ψx (t).

(1.108)

The usual norm of a complex vector is defined by |v|2 =

X

vi∗ vi .

(1.109)

Z

(1.110)

i

The continuous version of this is 2

|Ψ| =

Z

d

3

x Ψ∗x (t)Ψx (t)

=

d3 x Ψ∗ (x, t)Ψ(x, t).

The normalization condition (1.96) requires that the wave functions have the norm |Ψ| = 1, i.e., that they are unit vectors in the Hilbert space.

1.4.1

Basis Transformations

In a vector space, there are many possible choices of orthonormal basis vectors bi a labeled by a = 1, . . . , D, in terms of which6 vi =

X

bi a va ,

(1.111)

a

P (b) Mathematicians would expand more precisely vi = a bi a va , but physicists prefer to shorten the notation by distinguishing the different components via different types of subscripts, using for the initial components i, j, k, . . . and for the b-transformed components a, b, c, . . . . 6


19

1.4 Dirac’s Bra-Ket Formalism

with the components va given by the scalar products va ≡

bi a∗ vi .

X

(1.112)

i

The latter equation is a consequence of the orthogonality relation 7 X

′

′

bi a∗ bi a = δ aa ,

(1.113)

i

which in a finite-dimensional vector space implies the completeness relation X

bi a∗ bj a = δ ij .

(1.114)

a

In the space of wave functions (1.108) there exists a special set of basis functions called local basis functions is of particular importance. It may be constructed in the following fashion: Imagine the continuum of space points to be coarse-grained into a cubic lattice of mesh size ǫ, at positions n1,2,3 = 0, ±1, ±2, . . . .

xn = (n1 , n2 , n3 )ǫ,

(1.115)

Let hn (x) be a function that vanishes everywhere in space, except in a cube of size ǫ3 centered around xn , i.e., for each component xi of x, ( √ |xi − xn i | ≤ ǫ/2, i = 1, 2, 3. 1/ ǫ3 n (1.116) h (x) = 0 otherwise. These functions are certainly orthonormal: Z

′

′

d3 x hn (x)∗ hn (x) = δ nn .

(1.117)

Consider now the expansion Ψ(x, t) =

X

hn (x)Ψn (t)

(1.118)

n

with the coefficients Ψn (t) =

Z

d3 x hn (x)∗ Ψ(x, t) ≈

√

ǫ3 Ψ(xn , t).

(1.119)

It provides an excellent approximation to the true wave function Ψ(x, t), as long as the mesh size ǫ is much smaller than the scale over which Ψ(x, t) varies. In fact, if Ψ(x, t) is integrable, the integral over the sum (1.118) will always converge to Ψ(x, t). The same convergence of discrete approximations is found in any scalar product, and thus in any observable probability amplitudes. They can all be calculated with 7

An orthogonality relation implies usually a unit norm and is thus really an orthonormality relation but this name is rarely used.

20

1 Fundamentals

arbitrary accuracy knowing the discrete components of the type (1.119) in the limit ǫ → 0. The functions hn (x) may therefore be used as an approximate basis in the same way as the previous basis functions f a (x), g b(x), with any desired accuracy depending on the choice of ǫ. In general, there are many possible orthonormal basis functions f a (x) in the Hilbert space which satisfy the orthonormality relation Z

′

′

(1.120)

f a (x)Ψa (t),

(1.121)

d3 x f a (x)∗ f a (x) = δ aa ,

in terms of which we can expand Ψ(x, t) =

X a

with the coefficients Ψa (t) =

Z

d3 x f a (x)∗ Ψ(x, t).

(1.122)

Suppose we use other orthonormal basis f˜b (x) with the orthonormality relation Z

′ ′ d3 x f˜b (x)∗ f˜b (x) = δ bb ,

X b

f˜b (x)f˜b (x)∗ = δ (3) (x − x′ ),

to re-expand

(1.123)

˜ b (t), f˜b (x)Ψ

(1.124)

d3 x f˜b (x)∗ Ψ(x, t).

(1.125)

X

Ψ(x, t) =

b

with the components ˜ b (t) = Ψ

Z

Inserting (1.121) shows that the components are related to each other by ˜ b (t) = Ψ

X Z

d x f˜b (x)∗ f a (x) Ψa (t). 3

a

1.4.2

(1.126)

Bracket Notation

It is useful to write the scalar products between two wave functions occurring in the above basis transformations in the so-called bracket notation as h˜b|ai ≡

Z

d3 x f˜b (x)∗ f a (x).

(1.127)

In this notation, the components of the state vector Ψ(x, t) in (1.122), (1.125) are Ψa (t) = ha|Ψ(t)i, ˜ b (t) = h˜b|Ψ(t)i. Ψ

(1.128)

The transformation formula (1.126) takes the form h˜b|Ψ(t)i =

X a

h˜b|aiha|Ψ(t)i.

(1.129)


21


The right-hand side of this equation may be formally viewed as a result of inserting the abstract relation X |aiha| = 1 (1.130) a

between h˜b| and |Ψ(t)i on the left-hand side: X h˜b|Ψ(t)i = h˜b|1|Ψ(t)i = h˜b|aiha|Ψ(t)i.

(1.131)

a

Since this expansion is only possible if the functions f b (x) form a complete basis, the relation (1.130) is alternative, abstract way of stating the completeness of the basis functions. It may be referred to as completeness relation à la Dirac. Since the scalar products are written in the form of brackets ha|a′ i, Dirac called the formal objects ha| and |a′ i, from which the brackets are composed, bra and ket, respectively. In the bracket notation, the orthonormality of the basis f a (x) and g b (x) may be expressed as follows: Z

ha|a′ i =

Z

h˜b|˜b′ i =

′

′

d3 x f a (x)∗ f a (x) = δ aa , ′ ′ d x f˜b (x)∗ f˜b (x) = δ bb .

(1.132)

3

In the same spirit we introduce abstract bra and ket vectors associated with the basis functions hn (x) of Eq. (1.116), denoting them by hxn | and |xn i, respectively, and writing the orthogonality relation (1.117) in bracket notation as hxn |x i ≡ n′

Z

′

d3 x hn (x)∗ hn (x) = δnn′ .

The components Ψn (t) may be considered as the scalar products √ Ψn (t) ≡ hxn |Ψ(t)i ≈ ǫ3 Ψ(xn , t).

(1.133)

(1.134)

Changes of basis vectors, for instance from |xn i to the states |ai, can be performed according to the rules developed above by inserting a completeness relation à la Dirac of the type (1.130). Thus we may expand Ψn (t) = hxn |Ψ(t)i =

X a

hxn |aiha|Ψ(t)i.

Also the inverse relation is true: X ha|Ψ(t)i = ha|xn ihxn |Ψ(t)i. n

(1.135)

(1.136)

This is, of course, just an approximation to the integral Z

d3 x hn (x)∗ hx|Ψ(t)i.

(1.137)

The completeness of the basis hn (x) may therefore be expressed via the abstract relation X |xn ihxn | ≈ 1. (1.138) n

The approximate sign turns into an equality sign in the limit of zero mesh size, ǫ → 0.

22

1.4.3

1 Fundamentals

Continuum Limit

In ordinary calculus, finer and finer sums are eventually replaced by integrals. The same thing is done here. We define new continuous scalar products 1 hx|Ψ(t)i ≈ √ hxn |Ψ(t)i, ǫ3

(1.139)

where xn are the lattice points closest to x. With (1.134), the right-hand side is equal to Ψ(xn , t). In the limit ǫ → 0, x and xn coincide and we have hx|Ψ(t)i ≡ Ψ(x, t).

(1.140)

The completeness relation can be used to write ha|Ψ(t)i ≈

X

ha|xn ihxn |Ψ(t)i

≈

X

ǫ3 ha|xihx|Ψ(t)i

n

n

which in the limit ǫ → 0 becomes ha|Ψ(t)i =

Z

x=xn

d3 x ha|xihx|Ψ(t)i.

(1.141) ,

(1.142)

This may be viewed as the result of inserting the formal completeness relation of the limiting local bra and ket basis vectors hx| and |xi, Z

d3 x |xihx| = 1,

(1.143)

evaluated between the vectors ha| and |Ψ(t)i. With the limiting local basis, the wave functions can be treated as components of the state vectors |Ψ(t)i with respect to the local basis |xi in the same way as any other set of components in an arbitrary basis |ai. In fact, the expansion ha|Ψ(t)i =

Z

d3 x ha|xihx|Ψ(t)i

(1.144)

may be viewed as a re-expansion of a component of |Ψ(t)i in one basis, |ai, into those of another basis, |xi, just as in (1.129). In order to express all these transformation properties in a most compact notation, it has become customary to deal with an arbitrary physical state vector in a basis-independent way and denote it by a ket vector |Ψ(t)i. This vector may be specified in any convenient basis by multiplying it with the corresponding completeness relation X |aiha| = 1, (1.145) a

resulting in the expansion

|Ψ(t)i =

X a

|aiha|Ψ(t)i.

(1.146)


23


This can be multiplied with any bra vector, say hb|, from the left to obtain the expansion formula (1.131): hb|Ψ(t)i =

X a

hb|aiha|Ψ(t)i.

(1.147)

The continuum version of the completeness relation (1.138) reads Z

d3 x |xihx| = 1,

(1.148)

and leads to the expansion |Ψ(t)i =

Z

d3 x |xihx|Ψ(t)i,

(1.149)

in which the wave function Ψ(x, t) = hx|Ψ(t)i plays the role of an xth component of the state vector |Ψ(t)i in the local basis |xi. This, in turn, is the limit of the discrete basis vectors |xn i, 1 (1.150) |xi ≈ √ |xn i , ǫ3 with xn being the lattice points closest to x. A vector can be described equally well in bra or in ket form. To apply the above formalism consistently, we observe that the scalar products ha|˜bi = h˜b|ai = satisfy the identity

Z Z

d3 x f a (x)∗ f˜b (x), d x f˜b (x)∗ f a (x)

(1.151)

3

h˜b|ai ≡ ha|˜bi∗ .

(1.152)

Therefore, when expanding a ket vector as |Ψ(t)i =

X

|aiha|Ψ(t)i,

(1.153)

hΨ(t)| =

X

hΨ(t)|aiha|,

(1.154)

a

or a bra vector as a

a multiplication of the first equation with the bra hx| and of the second with the ket |xi produces equations which are complex-conjugate to each other.

1.4.4

Generalized Functions

Dirac’s bra-ket formalism is elegant and easy to handle. As far as the vectors |xi are concerned there is, however, one inconsistency with some fundamental postulates of quantum mechanics: When introducing state vectors, the norm was required to be unity in order to permit a proper probability interpretation of single-particle states.

24

1 Fundamentals

The limiting states |xi introduced above do not satisfy this requirement. In fact, the scalar product between two different states hx| and |x′ i is hx|x′ i ≈

1 1 ′i = hx |x δnn′ , n n ǫ3 ǫ3

(1.155)

where xn and xn′ are the lattice points closest to x and x′ . For x 6= x′ , the states are orthogonal. For x = x′ , on the other hand, the limit ǫ → 0 is infinite, approached in such a way that X 1 ǫ3 δ ′ = 1. (1.156) 3 nn ǫ ′ n Therefore, the limiting state |xi is not a properly normalizable vector in the Hilbert space. For the sake of elegance, it is useful to weaken the requirement of normalizability (1.96) by admitting the limiting states |xi to the physical Hilbert space. In fact, one admits all states which can be obtained by a limiting sequence from properly normalized state vectors. The scalar product between states hx|x′ i is not a proper function. It is denoted by the symbol δ (3) (x − x′ ) and called Dirac δ-function: hx|x′ i ≡ δ (3) (x − x′ ).

(1.157)

The right-hand side vanishes everywhere, except in the infinitely small box of width ǫ around x ≈ x′ . Thus the δ-function satisfies δ (3) (x − x′ ) = 0

for

x 6= x′ .

(1.158)

At x = x′ , it is so large that its volume integral is unity: Z

d3 x′ δ (3) (x − x′ ) = 1.

(1.159)

Obviously, there exists no proper function that can satisfy both requirements, (1.158) and (1.159). Only the finite-ǫ approximation in (1.155) to the δ-function are proper functions. In this respect, the scalar product hx|x′ i behaves just like the states |xi themselves: Both are ǫ → 0 -limits of properly defined mathematical objects. Note that the integral Eq. (1.159) implies the following property of the δ function: 1 (3) δ (3) (a(x − x′ )) = δ (x − x′ ). (1.160) |a| In one dimension, this leads to the more general relation δ(f (x)) =

X i

1 |f ′ (xi )|

δ(x − xi ),

(1.161)

where xi are the simple zeros of f (x). H. Kleinert, PATH INTEGRALS

25


In mathematics, one calls the δ-function a generalized function or a distribution. It defines a linear functional of arbitrary smooth test functions f (x) which yields its value at any desired place x: δ[f ; x] ≡

Z

d3 x δ (3) (x − x′ )f (x′ ) = f (x).

(1.162)

Test functions are arbitrarily often differentiable functions with a sufficiently fast falloff at spatial infinity. There exist a rich body of mathematical literature on distributions [3]. They form a linear space. This space is restricted in an essential way in comparison with ordinary functions: products of δ-functions or any other distributions remain undefined. In Section 10.8.1 we shall find, however, that physics forces us to go beyond these rules. An important requirement of quantum mechanics is coordinate invariance. If we want to achieve this for the path integral formulation of quantum mechanics, we must set up a definite extension of the existing theory of distributions, which specifies uniquely integrals over products of distributions. In quantum mechanics, the role of the test functions is played by the wave packets Ψ(x, t). By admitting the generalized states |xi to the Hilbert space, we also admit the scalar products hx|x′ i to the space of wave functions, and thus all distributions, although they are not normalizable.

1.4.5

Schr¨ odinger Equation in Dirac Notation

In terms of the bra-ket notation, the Schrödinger equation can be expressed in a basis-independent way as an operator equation ˆ ˆ , t)|Ψ(t)i = i¯ H|Ψ(t)i ≡ H(ˆ p, x h∂t |Ψ(t)i,

(1.163)

to be supplemented by the following specifications of the canonical operators: hx|ˆ p ≡ −i¯ h∇hx|, hx|ˆ x ≡ xhx|.

(1.164) (1.165)

Any matrix element can be obtained from these equations by multiplication from the right with an arbitrary ket vector; for instance with the local basis vector |x′ i: h∇hx|x′ i = −i¯ h∇δ (3) (x − x′ ), hx|ˆ p|x′ i = −i¯

(1.166)

hx|ˆ x|x′ i = xhx|x′ i = xδ (3) (x − x′ ).

(1.167)

The original differential form of the Schrödinger equation (1.91) follows by multiplying the basis-independent Schrödinger equation (1.163) with the bra vector hx| from the left: ˆ , t)|Ψ(t)i = H(−i¯ hx|H(ˆ p, x h∇, x, t)hx|Ψ(t)i = i¯ h∂t hx|Ψ(t)i.

(1.168)

26

1 Fundamentals

ˆ and x ˆ are Hermitian matrices in any basis, Obviously, p

and so is the Hamiltonian

ha|ˆ p|a′ i = ha′ |ˆ p|ai∗ ,

(1.169)

ha|ˆ x|a′ i = ha′ |ˆ x|ai∗ ,

(1.170)

ˆ ′ i = ha′ |H|ai ˆ ∗, ha|H|a

(1.171)

as long as it has the form (1.101). The most general basis-independent operator that can be constructed in the ˆ, x ˆ , t, generalized Hilbert space spanned by the states |xi is some function of p ˆ ≡ O(ˆ ˆ , t). O(t) p, x

(1.172)

In general, such an operator is called Hermitian if all its matrix elements have this property. In the basis-independent Dirac notation, the definition (1.97) of a ˆ † (t) implies the equality of the matrix elements Hermitian-adjoint operator O ∗ ˆ † (t)|a′ i ≡ ha′ |O(t)|ai ˆ ha|O .

(1.173)

Thus we can rephrase Eqs. (1.169)–(1.171) in the basis-independent form ˆ = p ˆ †, p ˆ = x ˆ†, x

(1.174)

ˆ = H ˆ †. H The stationary states in Eq. (1.94) have a Dirac ket representation |En i, and satisfy the time-independent operator equation ˆ n i = En |En i. H|E

1.4.6

(1.175)

Momentum States

ˆ . Its eigenstates are given by the eigenvalue Let us now look at the momentum p equation ˆ |pi = p|pi. p (1.176) By multiplying this with hx| from the left and using (1.164), we find the differential equation (1.177) hx|ˆ p|pi = −i¯ h∂x hx|pi = phx|pi. The solution is hx|pi ∝ eipx/¯h .

(1.178)

Up to a normalization factor, this is just a plane wave introduced before in Eq. (1.75) to describe free particles of momentum p. H. Kleinert, PATH INTEGRALS

27


In order for the states |pi to have a finite norm, the system must be confined to a finite volume, say a cubic box of length L and volume L3 . Assuming periodic boundary conditions, the momenta are discrete with values pm =

2π¯ h (m1 , m2 , m3 ), L

mi = 0, ±1, ±2, . . . .

(1.179)

Then we adjust the factor in front of exp (ipm x/¯ h) to achieve unit normalization 1 hx|pm i = √ exp (ipm x/¯ h) , L3

(1.180)

and the discrete states |pm i satisfy Z

d3 x |hx|pm i|2 = 1.

(1.181)

The states |pm i are complete: X m

|pm ihpm | = 1.

(1.182)

We may use this relation and the matrix elements hx|pm i to expand any wave function within the box as Ψ(x, t) = hx|Ψ(t)i =

X m

hx|pm ihpm |Ψ(t)i.

(1.183)

If the box is very large, the sum over the discrete momenta pm can be approximated by an integral over the momentum space [4]. X m

≈

Z

d3 pL3 . (2π¯ h)3

(1.184)

In this limit, the states |pm i may be used to define a continuum of basis vectors with an improper normalization √ |pi ≈ L3 |pm i, (1.185) √ in the same way as |xn i was used in (1.150) to define |xi ∼ (1/ ǫ3 )|xn i. The momentum states |pi satisfy the orthogonality relation hp|p′ i = (2π¯ h)3 δ (3) (p − p′ ),

(1.186)

with δ (3) (p−p′ ) being again the Dirac δ-function. Their completeness relation reads Z

d3 p |pihp| = 1, (2π¯ h)3

(1.187)

28

1 Fundamentals

such that the expansion (1.183) becomes Ψ(x, t) =

d3 p hx|pihp|Ψ(t)i, (2π¯ h)3

Z

(1.188)

with the momentum eigenfunctions hx|pi = eipx/¯h .

(1.189)

This coincides precisely with the Fourier decomposition introduced above in the description of a general particle wave Ψ(x, t) in (1.83), (1.84), with the identification hp|Ψ(t)i = f (p)e−iEp t/¯h .

(1.190)

The bra-ket formalism accommodates naturally the technique of Fourier transforms. The Fourier inversion formula is found by simply inserting into hp|Ψ(t)i a R 3 completeness relation d x|xihx| = 1 which yields hp|Ψ(t)i = =

Z Z

d3 x hp|xihx|Ψ(t)i 3

−ipx/¯ h

d xe

(1.191)

Ψ(x, t).

The amplitudes hp|Ψ(t)i are referred to as momentum space wave functions. By inserting the completeness relation Z

d3 x|xihx| = 1

(1.192)

between the momentum states on the left-hand side of the orthogonality relation (1.186), we obtain the Fourier representation of the δ-function ′

hp|p i = =

1.4.7

Z Z

d3 x hp|xihx|p′i 3

−i(p−p′ )x/¯ h

d xe

(1.193)

.

Incompleteness and Poisson’s Summation Formula

For many physical applications it is important to find out what happens to the completeness relation (1.148) if one restrict the integral so a subset of positions. Most relevant will be the one-dimensional integral, Z

dx |xihx| = 1,

(1.194)

restricted to a sum over equally spaced points xn = na: N X

n=−N

|xn ihxn |.

(1.195)


29


Taking this sum between momentum eigenstates |pi, we obtain N X

n=−N

hp|xn ihxn |p′ i =

N X

n=−N

hp|xn ihxn |p′ i =

N X

′

ei(p−p )na/¯h

(1.196)

n=−N

For N → ∞ we can perform the sum with the help of Poisson’s summation formula ∞ X

2πiµn

e

=

n=−∞

∞ X

m=−∞

δ(µ − m).

(1.197)

Identifying µ with (p − p′ )a/2π¯ h, we find using Eq. (1.160): ∞ X

(p − p′ )a 2π¯ h 2π¯ hm hp|xn ihxn |p i = δ −m = δ p − p′ − . 2π¯ h a a n=−∞ !

′

!

(1.198)

In order to prove the Poisson formula (1.197), we observe that the sum s(µ) ≡ side is periodic in µ with a unit period and has m δ(µ − m) on the right-hand P 2πiµn the Fourier series s(µ) = ∞ . The Fourier coefficients are given by n=−∞ sn e R 1/2 −2πiµn ≡ 1. These are precisely the Fourier coefficients on the sn = −1/2 dµ s(µ)e left-hand side. For a finite N, the sum over n on the left-hand side of (1.197) yields

P

N X

e2πiµn = 1 + e2πiµ + e2·2πiµ + . . . + eN ·2πiµ + cc

n=−N

1 − e−2πiµ(N +1) = −1 + + cc 1 − e−2πiµ = 1+

!

(1.199)

sin πµ(2N + 1) e−2πiµ − e−2πiµ(N +1) + cc = . 1 − e−2πiµ sin πµ

This function is well known in wave optics (see Fig. 2.4). It determines the diffraction pattern of light behind a grating with 2N + 1 slits. It has large peaks at µ = 0, ±1, ±2, ±3, . . . and N − 1 small maxima between each pair of neighboring peaks, at ν = (1 + 4k)/2(2N + 1) for k = 1, . . . , N − 1. There are zeros at ν = (1 + 2k)/(2N + 1) for k = 1, . . . , N − 1. Inserting µ = (p − p′ )a/2π¯ h into (1.199), we obtain sin (p − p′ )a(2N + 1)/2¯ h hp|xn ihxn |p i = . ′ sin (p − p )a/2¯ h n=−N N X

′

(1.200)

Let us see how the right-hand side of (1.199) turns into the right-hand side of (1.197) in the limit N → ∞. In this limit, the area under each large peak can be calculated by an integral over the central large peak plus a number n of small maxima next to it: Z

n/2N

−n/2N

dµ

sin πµ(2N + 1) = sin πµ

Z

n/2N −n/2N

dµ

sin 2πµN cos πµ+cos 2πµN sin πµ . sin πµ (1.201)

30

1 Fundamentals

2πiµn in Poisson’s summation formula. In the Figure 1.2 Relevant function N n=−N e limit N → ∞, µ is squeezed to the integer values.

P

Keeping keeping a fixed ratio n/N ≪ 1, we we may replace in the integrand sin πµ by πµ and cos πµ by 1. Then the integral becomes, for N → ∞ at fixed n/N, n/2N sin πµ(2N + 1) N →∞ n/2N sin 2πµN + dµ − −−→ dµ dµ cos 2πµN sin πµ πµ −n/2N −n/2N −n/2N Z πn Z πn N →∞ 1 N →∞ sin x 1 − −−→ dx + dx cos x − −−→ 1, (1.202) π −πn x 2πN −πn

Z

n/2N

Z

Z

where we have used the integral formula Z

∞

−∞

dx

sin x = π. x

(1.203)

In the limit N → ∞, we find indeed (1.197) and thus (1.205), as well as the expression (2.458) for the free energy. There exists another useful way of expressing Poisson’s formula. Consider a an arbitrary smooth function f (µ) which possesses a convergent sum ∞ X

f (m).

(1.204)

m=−∞

Then Poisson’s formula (1.197) implies that the sum can be rewritten as an integral and an auxiliary sum: ∞ X

m=−∞

f (m) =

Z

∞

−∞

dµ

∞ X

e2πiµn f (µ).

(1.205)

n=−∞

The auxiliary sum over n squeezes µ to the integer numbers. H. Kleinert, PATH INTEGRALS

31

1.5 Observables

1.5

Observables

Changes of basis vectors are an important tool in analyzing the physically observable content of a wave vector. Let A = A(p, x) be an arbitrary time-independent real function of the phase space variables p and x. Important examples for such an A are p and x themselves, the Hamiltonian H(p, x), and the angular momentum L = x × p. Quantum-mechanically, there will be an observable operator associated with each such quantity. It is obtained by simply replacing the variables p and x in ˆ and x ˆ: A by the corresponding operators p ˆ ). Aˆ ≡ A(ˆ p, x

(1.206)

This replacement rule is the extension of the correspondence principle for the Hamiltonian operator (1.92) to more general functions in phase space, converting them into observable operators. It must be assumed that the replacement leads to a unique Hermitian operator, i.e., that there is no ordering problem of the type discussed in context with the Hamiltonian (1.101).8 If there are ambiguities, the naive correspondence principle is insufficient to determine the observable operator. Then the correct ordering must be decided by comparison with experiment, unless it can be specified by means of simple geometric principles. This will be done for the Hamiltonian operator in Chapter 8. Once an observable operator Aˆ is Hermitian, it has the useful property that the set of all eigenvectors |ai obtained by solving the equation ˆ = a|ai A|ai

(1.207)

can be used as a basis to span the Hilbert space. Among the eigenvectors, there is always a choice of orthonormal vectors |ai fulfilling the completeness relation X a

|aiha| = 1.

(1.208)

The vectors |ai can be used to extract physical information concerning the observable A from arbitrary state vector |Ψ(t)i. For this we expand this vector in the basis |ai: X |Ψ(t)i = |aiha|Ψ(t)i. (1.209) a

The components

ha|Ψ(t)i

(1.210)

yield the probability amplitude for measuring the eigenvalue a for the observable quantity A. The wave function Ψ(x, t) itself is an example of this interpretation. If we write it as Ψ(x, t) = hx|Ψ(t)i, (1.211) 8

Note that this is true for the angular momentum

L

= x × p.

32

1 Fundamentals

it gives the probability amplitude for measuring the eigenvalues x of the position ˆ , i.e., |Ψ(x, t)|2 is the probability density in x-space. operator x The expectation value of the observable operator (1.206) in the state |Ψ(t)i is defined as the matrix element ˆ hΨ(t)|A|Ψ(t)i ≡

1.5.1

Z

d3 xhΨ(t)|xiA(−i¯ h∇, x)hx|Ψ(t)i.

(1.212)

Uncertainty Relation

We have seen before [see the discussion after (1.83), (1.84)] that the amplitudes in real space and those in momentum space have widths inversely proportional to each other, due to the properties of Fourier analysis. If a wave packet is localized in real space with a width ∆x, its momentum space wave function has a width ∆p given by ∆x ∆p ∼ h ¯. (1.213) From the Hilbert space point of view this uncertainty relation can be shown to be ˆ and p ˆ do not commute with each a consequence of the fact that the operators x other, but the components satisfy the canonical commutation rules [ˆ pi , xˆj ] = −i¯ hδij , [ˆ xi , xˆj ] = 0, [ˆ pi , pˆj ] = 0.

(1.214)

In general, if an observable operator Aˆ is measured sharply to have the value a in one state, this state must be an eigenstate of Aˆ with an eigenvalue a: ˆ = a|ai. A|ai

(1.215)

This follows from the expansion |Ψ(t)i =

X a

|aiha|Ψ(t)i,

(1.216)

in which |ha|Ψ(t)i|2 is the probability to measure an arbitrary eigenvalue a. If this probability is sharply focused at a specific value of a, the state necessarily coincides with |ai. ˆ we may ask under what circumstances Given the set of all eigenstates |ai of A, ˆ can be measured sharply in each of these states. The another observable, say B, ˆ requirement implies that the states |ai are also eigenstates of B, ˆ B|ai = ba |ai,

(1.217)

with some a-dependent eigenvalue ba . If this is true for all |ai, ˆ A|ai ˆ = ba a|ai = aba |ai = AˆB|ai, ˆ B


33

1.5 Observables

ˆ necessarily commute: the operators Aˆ and B ˆ B] ˆ = 0. [A,

(1.219)

Conversely, it can be shown that a vanishing commutator is also sufficient for two observable operators to be simultaneously diagonalizable and thus to allow for simultaneous sharp measurements.

1.5.2

Density Matrix and Wigner Function

An important object for calculating observable properties of a quantum-mechanical system is the quantum mechanical density operator associated with a pure state ρˆ(t) ≡ |Ψ(t)ihΨ(t)|,

(1.220)

and the associated density matrix associated with a pure state ρ(x1 , x2 ; t) = hx1 |Ψ(t)ihΨ(t)|x2 i.

(1.221)

ˆ ) can be calculated from the trace The expectation value of any function f (x, p ˆ )|Ψ(t)i = tr [f (x, p ˆ )ˆ hΨ(t)|f (x, p ρ(t)] =

Z

d3 xhΨ(t)|xif (x, −i¯ h∇)hx|Ψ(t)i.

(1.222) If we decompose the states |Ψ(t)i into stationary eigenstates |En i of the Hamiltonian ˆ [recall (1.175)], |Ψ(t)i = Pn |En ihEn |Ψ(t)i, then the density matrix has operator H the expansion ρˆ(t) ≡

X

n,m

|En iρnm (t)hEm | =

X

n,m

|En ihEn |Ψ(t)ihΨ(t)|Em ihEm |.

(1.223)

Wigner showed that the Fourier transform of the density matrix, the Wigner function d3 ∆x ip∆x/¯h e ρ(X + ∆x/2, X − ∆x/2; t) (1.224) (2π¯ h)3 satisfies, for a single particle of mass M in a potential V (x), the Wigner-Liouville equation p ∂t + v · ∇X W (X, p; t) = Wt (X, p; t), v ≡ , (1.225) M where Z Z d3 q 2 Wt (X, p; t) ≡ W (X, p − q; t) d3 ∆x V (X − ∆x/2)eiq∆x/¯h . (1.226) h ¯ (2π¯ h)3 W (X, p; t) ≡

Z

In the limit h ¯ → 0, we may expand W (X, p − q; t) in powers of q, and V (X − ∆x/2) in powers of ∆x, which we rewrite in front of the exponential eiq∆x/¯h as powers of −i¯ h∇q . Then we perform the integral over ∆x to obtain (2π¯ h)3 δ (3) (q), and perform the integral over q to obtain the classical Liouville equation for the probability density of the particle in phase space p ∂t + v · ∇X W (X, p; t) = −F (X)∇p W (X, p; t), v ≡ , (1.227) M where F (X) ≡ −∇X V (X) is the force associated with the potential V (X).

34

1 Fundamentals

1.5.3

Generalization to Many Particles

All this development can be extended to systems of N distinguishable mass points with Cartesian coordinates freedom x1 , . . . , xN . If H(pν , xν , t) is the Hamiltonian, the Schrödinger equation becomes ˆ ν , t)|Ψ(t)i = i¯ H(ˆ pν , x h∂t |Ψ(t)i.

(1.228)

We may introduce a complete local basis |x1 , . . . , xN i with the properties hx1 , . . . , xN |x′1 , . . . , x′N i = δ (3) (x1 − x′1 ) · · · δ (3) (xN − x′N ), Z

and define

d3 x1 · · · d3 xN |x1 , . . . , xN ihx1 , . . . , xN | = 1, hx1 , . . . , xN |ˆ pν = −i¯ h∂xν hx1 , . . . , xN |,

(1.229)

(1.230)

hx1 , . . . , xN |ˆ xν = xν hx1 , . . . , xN |.

The Schrödinger equation for N particles (1.107) follows from (1.228) by multiplying it from the left with the bra vectors hx1 , . . . , xN |. In the same way, all other formulas given above can be generalized to N-body state vectors.

1.6

Time Evolution Operator

If the Hamiltonian operator possesses no explicit time dependence, the basisindependent Schrödinger equation (1.163) can be integrated to find the wave function |Ψ(t)i at any time tb from the state at any other time ta ˆ

The operator

|Ψ(tb )i = e−i(tb −ta )H/¯h |Ψ(ta )i.

(1.231)

ˆ Uˆ (tb , ta ) = e−i(tb −ta )H/¯h

(1.232)

is called the time evolution operator . It satisfies the differential equation ˆ Uˆ (tb , ta ). i¯ h∂tb Uˆ (tb , ta ) = H

(1.233)

Its inverse is obtained by interchanging the order of tb and ta : ˆ ˆ a , tb ). Uˆ −1 (tb , ta ) ≡ ei(tb −ta )H/¯h = U(t

(1.234)

As an exponential of i times a Hermitian operator, Uˆ is a unitary operator satisfying Uˆ † = Uˆ −1 . Indeed,

ˆ† Uˆ † (tb , ta ) = ei(tb −ta )H /¯h ˆ ˆ −1 (tb , ta ). = ei(tb −ta )H/¯h = U

(1.235)

(1.236)


35

1.6 Time Evolution Operator

ˆ , t) depends explicitly on time, the integration of the Schrödinger equation If H(ˆ p, x (1.163) is somewhat more involved. The solution may be found iteratively: For tb > ta , the time interval is sliced into a large number N + 1 of small pieces of thickness ǫ with ǫ ≡ (tb − ta )/(N + 1), slicing once at each time tn = ta + nǫ for n = 0, . . . , N + 1. We then use the Schrödinger equation (1.163) to relate the wave function in each slice approximately to the previous one:

|Ψ(ta + ǫ)i ≈

|Ψ(ta + 2ǫ)i ≈ .. . |Ψ(ta + (N + 1)ǫ)i ≈

E i Z ta +ǫ ˆ 1− dt H(t) Ψ(ta ) , h ¯ ta

i 1− h ¯

Z

ta +2ǫ

i 1− h ¯

Z

ta +(N +1)ǫ

ta +ǫ

ˆ dt H(t) |Ψ(ta + ǫ)i,

ta +N ǫ

(1.237)

!

ˆ dt H(t) |Ψ(ta + Nǫ)i.

From the combination of these equations we extract the evolution operator as a product i Uˆ (tb , ta ) ≈ 1 − h ¯

Z

tb

tN

dt′N +1

ˆ ′ ) ×···× 1− i H(t N +1 h ¯

Z

t1

ta

dt′1

ˆ ′) . H(t 1

(1.238)

By multiplying out the product and going to the limit N → ∞ we find the series i Uˆ (tb , ta ) = 1 − h ¯

Z

tb

ta

dt′1

ˆ ′ ) + −i H(t 1 h ¯

2 Z

tb

ta

dt′2

Z

t2

ta

ˆ ′ )H(t ˆ ′) dt′1 H(t 2 1

(1.239) tb t3 t2 −i ′ ′ ′ ˆ ′ ˆ ′ ˆ ′ dt3 dt2 dt1 H(t3 )H(t2 )H(t1 ) + . . . , + h ¯ ta ta ta known as the Neumann-Liouville expansion or Dyson series. An interesting modification of this is the so-called Magnus expansion to be derived in Eq. (2A.25). Note that each integral has the time arguments in the Hamilton operators ordered causally: Operators with later times stand to left of those with earlier times. It is useful to introduce a time-ordering operator which, when applied to an arbitrary product of operators, ˆ n (tn ) · · · O ˆ 1 (t1 ), O (1.240)

3 Z

Z

Z

reorders the times successively. More explicitly we define

ˆ n (tn ) · · · O ˆ 1 (t1 )) ≡ O ˆ in (tin ) · · · O ˆ i1 (ti1 ), Tˆ(O

(1.241)

where tin , . . . , ti1 are the times tn , . . . , t1 relabeled in the causal order, so that tin > tin−1 > . . . > ti1 .

(1.242)

Any c-number factors in (1.241) can be pulled out in front of the Tˆ operator. With this formal operator, the Neumann-Liouville expansion can be rewritten in a more compact way. Take, for instance, the third term in (1.239) Z

tb

ta

dt2

Z

t2

ta

ˆ 2 )H(t ˆ 1 ). dt1 H(t

(1.243)

36

1 Fundamentals

tb t2

ta

ta

t1

tb

Figure 1.3 Illustration of time-ordering procedure in Eq. (1.243).

The integration covers the triangle above the diagonal in the square t1 , t2 ∈ [ta , tb ] in the (t1 , t2 ) plane (see Fig. 1.2). By comparing this with the missing integral over the lower triangle Z tb Z tb ˆ 2 )H(t ˆ 1) dt2 dt1 H(t (1.244) ta

t2

we see that the two expressions coincide except for the order of the operators. This can be corrected with the use of a time-ordering operator Tˆ. The expression Z

Tˆ

tb

ta

Z

dt2

tb t2

ˆ 2 )H(t ˆ 1) dt1 H(t

(1.245)

is equal to (1.243) since it may be rewritten as Z

tb

ta

dt2

Z

tb

ˆ 1 )H(t ˆ 2) dt1 H(t

t2

(1.246)

or, after interchanging the order of integration, as Z

tb

ta

dt1

Z

t1

ta

ˆ 1 )H(t ˆ 2 ). dt2 H(t

(1.247)

Apart from the dummy integration variables t2 ↔ t1 , this double integral coincides with (1.243). Since the time arguments are properly ordered, (1.243) can trivially be multiplied with the time-ordering operator. The conclusion of this discussion is that (1.243) can alternatively be written as Z tb 1 ˆ Z tb ˆ 2 )H(t ˆ 1 ). T dt2 dt1 H(t 2 ta ta

(1.248)

On the right-hand side, the integrations now run over the full square in the t1 , t2 plane so that the two integrals can be factorized into 1ˆ T 2

Z

tb

ta

ˆ dt H(t)

2

.


37

1.7 Properties of Time Evolution Operator

Similarly, we may rewrite the nth-order term of (1.239) as 1 ˆ T n!

Z

tb

ta

dtn

Z

tb

ta

dtn−1 · · ·

Z

tb

ta

ˆ n )H(t ˆ n−1 ) · · · H(t ˆ 1) dt1 H(t

(1.250)

" #n 1 ˆ Z tb ˆ = T dt H(t) . n! ta

The time evolution operator Uˆ (tb , ta ) has therefore the series expansion i ˆ Z tb ˆ + 1 −i ˆ dt H(t) U(tb , ta ) = 1 − T h ¯ ta 2! h ¯

1 −i +...+ n! h ¯

n

Tˆ

Z

tb ta

2

Tˆ

tb

Z

ˆ dt H(t)

ta

n

ˆ dt H(t)

2

(1.251)

+ ... .

The right-hand side of Tˆ contains simply the power series expansion of the exponential so that we can write i Uˆ (tb , ta ) = Tˆ exp − h ¯

Z

tb

ta

ˆ dt H(t) .

(1.252)

ˆ does not depend on the time, the time-ordering operation is superfluous, the If H integral can be done trivially, and we recover the previous result (1.232). ˆ ˆ Note that a small variation δ H(t) of H(t) changes Uˆ (tb , ta ) by (

tb i tb ˆ b , ta ) = − i ˆ ˆ ′ ) Tˆ exp − i δ U(t dt′ Tˆ exp − dt H(t) δ H(t h ¯ ta h ¯ t′ h ¯ Z tb i ˆ ′ ) U(t ˆ ′ , ta ). =− dt′ Uˆ (tb , t′ ) δ H(t h ¯ ta

Z

Z

Z

t′

ta

ˆ dt H(t)

)

(1.253)

A simple application for this relation is given in Appendix 1A.

1.7

Properties of Time Evolution Operator

ˆ b , ta ) has some important properties: By construction, U(t a) Fundamental composition law ˆ If two time translations are performed successively, the corresponding operators U are related by ˆ b , ta ) = Uˆ (tb , t′ )U(t ˆ ′ , ta ), U(t t′ ∈ (ta , tb ). (1.254) This composition law makes the operators Uˆ a representation of the abelian group ˆ b , ta ) given by of time translations. For time-independent Hamiltonians with U(t

38

1 Fundamentals

(1.232), the proof of (1.254) is trivial. In the general case (1.252), it follows from the simple manipulation valid for tb > ta : i Tˆ exp − h ¯

Z

tb

t′

"

ˆ dt Tˆ exp − i H(t) h ¯

i = Tˆ exp − h ¯

i = Tˆ exp − h ¯

tb

Z

t′

Z

tb

ta

t′

Z

ta

!

ˆ dt H(t)

ˆ dt exp − i H(t) h ¯

Z

t′

ta

!#

ˆ dt H(t)

(1.255)

ˆ dt . H(t)

b) Unitarity The expression (1.252) for the time evolution operator Uˆ (tb , ta ) was derived only for the causal (or retarded ) time arguments, i.e., for tb later than ta . We may, however, define Uˆ (tb , ta ) also for the anticausal (or advanced ) case where tb lies before ta . To be consistent with the above composition law (1.254), we must have ˆ a , tb )−1 . Uˆ (tb , ta ) ≡ U(t

(1.256)

Indeed, when considering two states at successive times |Ψ(ta )i = Uˆ (ta , tb )|Ψ(tb )i,

(1.257)

ˆ −1 (ta , tb ): the order of succession is inverted by multiplying both sides by U |Ψ(tb )i = Uˆ (ta , tb )−1 |Ψ(ta )i,

tb < ta .

(1.258)

The operator on the right-hand side is defined to be the time evolution operator ˆ U (tb , ta ) from the later time ta to the earlier time tb . If the Hamiltonian is independent of time, with the time evolution operator being ˆ Uˆ (ta , tb ) = e−i(ta −tb )H/¯h ,

ta > tb ,

(1.259)

tb < ta .

(1.260)

ˆ b , ta ) is obvious: the unitarity of the operator U(t ˆ b , ta )−1 , Uˆ † (tb , ta ) = U(t

Let us verify this property for a general time-dependent Hamiltonian. There, a direct solution of the Schrödinger equation (1.163) for the state vector shows that the operator Uˆ (tb , ta ) for tb < ta has a representation just like (1.252), except for a reversed time order of its arguments. One writes this in the form [compare (1.252)] Uˆ (tb , ta ) = Tˆ exp

i h ¯

Z

tb

ta

ˆ dt , H(t)

(1.261)

where Tˆ denotes the time-antiordering operator, with an obvious definition analogous to (1.241), (1.242). This operator satisfies the relation h

ˆ 1 (t1 )O ˆ 2(t2 ) Tˆ O

i†

ˆ 2† (t2 )O ˆ 1† (t1 ) , = Tˆ O


39

1.8 Heisenberg Picture of Quantum Mechanics

with an obvious generalization to the product of n operators. We can therefore conclude right away that ˆ † (tb , ta ) = Uˆ (ta , tb ), U

tb > ta .

(1.263)

With Uˆ (ta , tb ) ≡ Uˆ (tb , ta )−1 , this proves the unitarity relation (1.260), in general. c) Schrödinger equation for Uˆ (tb , ta ) Since the operator Uˆ (tb , ta ) rules the relation between arbitrary wave functions at different times, ˆ b , ta )|Ψ(ta )i, |Ψ(tb )i = U(t (1.264) the Schrödinger equation (1.228) implies that the operator Uˆ (tb , ta ) satisfies the corresponding equations ˆ Uˆ (t, ta ), i¯ h∂t Uˆ (t, ta ) = H −1 −1 ˆ i¯ h∂t Uˆ (t, ta ) = −Uˆ (t, ta ) H, with the initial condition

1.8

Uˆ (ta , ta ) = 1.

(1.265) (1.266) (1.267)

Heisenberg Picture of Quantum Mechanics

ˆ ta ) may be used to give a different formuThe unitary time evolution operator U(t, lation of quantum mechanics bearing the closest resemblance to classical mechanics. This formulation, called the Heisenberg picture of quantum mechanics, is in a ways closer related to to classical mechanics than the Schrödinger formulation. Many classical equations remain valid by simply replacing the canonical variables pi (t) and qi (t) in phase space by Heisenberg operators, to be denoted by pHi (t), qHi (t). Originally, Heisenberg postulated that they are matrices, but later it became clear that these matrices had to be functional matrix elements of operators, whose indices can be partly continuous. The classical equations hold for the Heisenberg operators and as long as the canonical commutation rules (1.93) are respected at any given time. In addition, qi (t) must be Cartesian coordinates. In this case we shall always use the letter x for the position variable, as in Section 1.4, rather than q, the corresponding Heisenberg operators being xHi (t). Suppressing the subscripts i, the canonical equal-time commutation rules are [pH (t), xH (t)] = −i¯ h, [pH (t), pH (t)] = 0,

(1.268)

[xH (t), xH (t)] = 0. According to Heisenberg, classical equations involving Poisson brackets remain valid if the Poisson brackets are replaced by i/¯ h times the matrix commutators at equal times. The canonical commutation relations (1.268) are a special case of this

40

1 Fundamentals

rule, recalling the fundamental Poisson brackets (1.25). The Hamilton equations of motion (1.24) turn into the Heisenberg equations i [HH , pH (t)] , h ¯ i x˙ H (t) = [HH , xH (t)] , h ¯

(1.269)

HH ≡ H(pH (t), xH (t), t)

(1.270)

OH (t) ≡ O(pH (t), xH (t), t),

(1.271)

p˙H (t) =

where is the Hamiltonian in the Heisenberg picture. Similarly, the equation of motion for arbitrary observable function O(pi(t), xi (t), t) derived in (1.20) goes over into the matrix commutator equation for the Heisenberg observable

namely,

d i ∂ OH = [HH , OH ] + OH . (1.272) dt h ¯ ∂t These rules are referred to as Heisenberg’s correspondence principle. The relation between Schrödinger’s and Heisenberg’s picture is supplied by the ˆ be an arbitrary observable in the Schrödinger detime evolution operator. Let O scription ˆ ≡ O(ˆ O(t) p, xˆ, t). (1.273) If the states |Ψa (t)i are an arbitrary complete set of solutions of the Schrödinger ˆ equation, where a runs through discrete and continuous indices, the operator O(t) can be specified in terms of its functional matrix elements ˆ Oab (t) ≡ hΨa (t)|O(t)|Ψ b (t)i.

(1.274)

We can now use the unitary operator Uˆ (t, 0) to go to a new time-independent basis |ΨH a i, defined by |Ψa (t)i ≡ Uˆ (t, 0)|ΨH a i. (1.275)

Simultaneously, we transform the Schrödinger operators of the canonical coordinates pˆ and xˆ into the time-dependent canonical Heisenberg operators pˆH (t) and xˆH (t) via ˆ 0), pˆH (t) ≡ Uˆ (t, 0)−1 pˆ U(t, xˆH (t) ≡ Uˆ (t, 0)−1 xˆ Uˆ (t, 0).

(1.276) (1.277)

At the time t = 0, the Heisenberg operators pˆH (t) and xˆH (t) coincide with the timeindependent Schrödinger operators pˆ and xˆ, respectively. An arbitrary observable ˆ is transformed into the associated Heisenberg operator as O(t) ˆ H (t) ≡ Uˆ (t, ta )−1 O(ˆ O p, xˆ, t)Uˆ (t, ta ) ≡ O (ˆ pH (t), xˆH (t), t) .

(1.278)


41

1.8 Heisenberg Picture of Quantum Mechanics

The Heisenberg matrices OH (t)ab are then obtained from the Heisenberg operators ˆ H (t) by sandwiching O ˆ H (t) between the time-independent basis vectors |ΨH a i: O ˆ H (t)|ΨH b i. OH (t)ab ≡ hΨH a |O

(1.279)

Note that the time dependence of these matrix elements is now completely due to the time dependence of the operators, d d ˆ OH (t)ab ≡ hΨH a | O H (t)|ΨH b i. dt dt

(1.280)

This is in contrast to the Schrödinger representation (1.274), where the right-hand side would have contained two more terms from the time dependence of the wave functions. Due to the absence of such terms in (1.280) it is possible to study the equation of motion of the Heisenberg matrices independently of the basis by considering directly the Heisenberg operators. It is straightforward to verify that they do indeed satisfy the rules of Heisenberg’s correspondence principle. Consider the time ˆ H (t), derivative of an arbitrary observable O !

d ˆ −1 ˆ Uˆ (t, ta ) U (t, ta ) O(t) dt ! ! ∂ d −1 −1 ˆ (t, ta ) ˆ ˆ (t, ta )O(t) ˆ + U O(t) Uˆ (t, ta ) , Uˆ (t, ta ) + U ∂t dt

d ˆ OH (t) = dt

which can be rearranged as "

!

#

d ˆ −1 ˆ −1 (t, ta )O(t) ˆ Uˆ (t, ta ) U (t, ta ) Uˆ (t, ta ) U (1.281) dt ! h i d ∂ −1 −1 −1 ˆ ˆ Uˆ (t, ta ) U ˆ (t, ta ) Uˆ (t, ta ) + Uˆ (t, ta ) + Uˆ (t, ta )O(t) O(t) Uˆ (t, ta ). dt ∂t

Using (1.265), we obtain !

i h ˆ −1 ˆ ˆ ˆ i ˆ −1 ∂ ˆ d ˆ OH (t) = U H U, OH + U O(t) Uˆ . dt h ¯ ∂t

(1.282)

After inserting (1.278), we find the equation of motion for the Heisenberg operator: i d ˆ i hˆ ˆ ∂ ˆ OH (t) = HH , OH (t) + O dt h ¯ ∂t

!

(t).

(1.283)

H

By sandwiching this equation between the complete time-independent basis states |Ψa i in the Hilbert space, it holds for the matrices and turns into the Heisenberg equation of motion. For the phase space variables pH (t), xH (t) themselves, these equations reduce, of course, to the Hamilton equations of motion (1.269). Thus we have shown that Heisenberg’s matrix quantum mechanics is completely equivalent to Schrödinger’s quantum mechanics, and that the Heisenberg matrices obey the same Hamilton equations as the classical observables.

42

1 Fundamentals

1.9

Interaction Picture and Perturbation Expansion

For some physical systems, the Hamiltonian operator can be split into two contributions ˆ =H ˆ 0 + Vˆ , H (1.284) ˆ 0 is a so-called free Hamiltonian operator for which the Schrödinger equation where H ˆ H0 |ψ(t)i = i¯ h∂t |ψ(t)i can be solved, and Vˆ is an interaction potential which perturbs these solutions slightly. In this case it is useful to describe the system in Dirac’s interaction picture. We remove the time evolution of the unperturbed Schrödinger solutions and define the states ˆ

|ψI (t)i ≡ eiH0 t/¯h |ψ(t)i.

(1.285)

Their time evolution comes entirely from the interaction potential Vˆ . It is governed by the time evolution operator UÎ (tb , ta ) ≡ eiH0 tb /¯h e−iHtb /¯h eiHta /¯h e−iH0 ta /¯h , and reads

(1.286)

|ψI (tb )i = UÎ (tb , ta )|ψI (ta )i.

(1.287)

Î (tb , ta ) = VI (tb )U Î (tb , ta ), i¯ h∂tb U

(1.288)

If Vˆ = 0, the states |ψI (tb )i are time-independent and coincide with the Heisenberg ˆ 0. states (1.275) of the operator H The operator UÎ (tb , ta ) satisfies the equation of motion

where

VÎ (t) ≡ eiH0 t/¯h Vˆ e−iH0 t/¯h

(1.289)

is the potential in the interaction picture. This equation of motion can be turned into an integral equation i Z tb UÎ (tb , ta ) = 1 − dtVI (t)UÎ (t, ta ). h ¯ ta

(1.290)

Inserting Eq. (1.289), this reads i Z tb ˆ ˆ ˆ Î (t, ta ). UI (tb , ta ) = 1 − dt eiH0 t/¯h V e−iH0 t/¯h U h ¯ ta

(1.291)

This equation can be iterated to find a perturbation expansion for the operator Î (tb , ta ) in powers of the interaction potential: U i Z tb ˆ ˆ ˆ UI (tb , ta ) = 1 − dt eiH0 t/¯h V e−iH0 t/¯h h ¯ ta Z Z t i 2 tb ′ ˆ ˆ ˆ ′ + − dt dt′ eiH0 t/¯h V e−iH0 (t−t )/¯h V e−iH0 t /¯h + . . . . h ¯ ta ta

(1.292)


43

1.10 Time Evolution Amplitude

Inserting on the left-hand side the operator (1.286), this can also be rewritten as −iH(tb −ta )/¯ h

e

+ −

i h ¯

−iH0 (tb −ta )/¯ h

=e

2 Z

tb

ta

dt

Z

t

ta

i − h ¯

Z

tb

ta

ˆ

ˆ

dt e−iH0 (tb −t)/¯h V e−iH0 (t−ta )/¯h

ˆ

ˆ

ˆ

′

′

dt′ e−iH0 (tb −t)/¯h V e−iH0 (t−t )/¯h V e−iH0 (t −ta )/¯h + . . . .

(1.293)

This expansion is seen to be the recursive solution of the integral equation −iH(tb −ta )/¯ h

e

−iH0 (tb −ta )/¯ h

=e

i − h ¯

Z

tb

ta

ˆ

ˆ

dt e−iH0 (tb −t)/¯h V e−iH(t−ta )/¯h .

(1.294)

Note that the lowest-order correction agrees with the previous formula (1.253)

1.10

Time Evolution Amplitude

In the subsequent development, an important role will be played by the matrix elements of the time evolution operator in the localized basis states, ˆ b , ta )|xa i. (xb tb |xa ta ) ≡ hxb |U(t

(1.295)

They are referred to as time evolution amplitudes. The functional matrix (xb tb |xa ta ) is also called the propagator of the system. For a system with a time-independent Hamiltonian operator where Uˆ (tb , ta ) is given by (1.259), the propagator is simply ˆ b − ta )/¯ (xb tb |xa ta ) = hxb | exp[−iH(t h]|xa i.

(1.296)

Due to the operator equations (1.265), the propagator satisfies the Schrödinger equation [H(−i¯ h∂xb , xb , tb ) − i¯ h∂tb ] (xb tb |xa ta ) = 0. (1.297) In the quantum mechanics of nonrelativistic particles, only the propagators from earlier to later times will be relevant. It is therefore customary to introduce the so-called causal time evolution operator or retarded time evolution operator :9 ˆR

U (tb , ta ) ≡

(

Uˆ (tb , ta ), 0,

tb ≥ ta , tb < ta ,

(1.298)

and the associated causal time evolution amplitude or retarded time evolution amplitude ˆ R (tb , ta )|xa i. (xb tb |xa ta )R ≡ hxb |U (1.299)

Since this differs from (1.295) only for tb < ta , and since all formulas in the subsequent text will be used only for tb > ta , we shall often omit the superscript R. To abbreviate the case distinction in (1.298), it is convenient to use the Heaviside function defined by 1 for t > 0, Θ(t) ≡ (1.300) 0 for t ≤ 0, 9

Compare this with the retarded Green functions to be introduced in Section 18.1

44

1 Fundamentals

and write ˆ b , ta ), U R (tb , ta ) ≡ Θ(tb − ta )U(t

(xb tb |xa ta )R ≡ Θ(tb − ta )(xb tb |xa ta ). (1.301)

There exists also another Heaviside function which differs from (1.300) only by the value at tb = ta : 1 for t ≥ 0, R Θ (t) ≡ (1.302) 0 for t < 0. Both Heaviside functions have the property that their derivative yields Dirac’s δ-function ∂t Θ(t) = δ(t). (1.303) If it is not important which Θ-function is used we shall ignore the superscript. The retarded propagator satisfies the Schrödinger equation h

i

H(−i¯ h∂xb , xb , tb )R − i¯ h∂tb (xb tb |xa ta )R = −i¯ hδ(tb − ta )δ (3) (xb − xa ).

(1.304)

The nonzero right-hand side arises from the extra term −i¯ h [∂tb Θ(tb − ta )] hxb tb |xa ta i = −i¯ hδ(tb − ta )hxb tb |xa ta i = −i¯ hδ(tb − ta )hxb ta |xa ta i (1.305) and the initial condition hxb ta |xa ta i = hxb |xa i, due to (1.267). If the Hamiltonian does not depend on time, the propagator depends only on the time difference t = tb − ta . The retarded propagator vanishes for t < 0. Functions f (t) with this property have a characteristic Fourier transform. The integral ˜ f(E) ≡

Z

∞

0

dt f (t)eiEt/¯h

(1.306)

is an analytic function in the upper half of the complex energy plane. This analyticity property is necessary and sufficient to produce a factor Θ(t) when inverting the Fourier transform via the energy integral f (t) ≡

Z

∞

−∞

dE ˜ f (E)e−iEt/¯h . 2π¯ h

(1.307)

For t < 0, the contour of integration may be closed by an infinite semicircle in the upper half-plane at no extra cost. Since the contour encloses no singularities, it can be contracted to a point, yielding f (t) = 0. The Heaviside function Θ(t) itself is the simplest retarded function, with a Fourier representation containing just a single pole just below the origin of the complex energy plane: Z ∞ dE i e−iEt , (1.308) Θ(t) = 2π E + iη −∞ where η is an infinitesimally small positive number. The integral representation is undefined for t = 0 and there are, in fact, infinitely many possible definitions for the Heaviside function depending on the value assigned to the function at the origin. A H. Kleinert, PATH INTEGRALS

45

1.11 Fixed-Energy Amplitude

special role is played by the average of the Heaviside functions (1.302) and (1.300), which is equal to 1/2 at the origin:  1

for t > 0, ¯ Θ(t) ≡  12 for t = 0, 0 for t < 0.

(1.309)

Usually, the difference in the value at the origin does not matter since the Heaviside function appears only in integrals accompanied by some smooth function f (t). This makes the Heaviside function a distribution with respect to smooth test functions ¯ f (t) as defined in Eq. (1.162). All three distributions Θr (t), Θl (t), and Θ(t) define the same linear functional of the test functions by the integral Θ[f ] =

Z

dt Θ(t − t′ )f (t′ ),

(1.310)

and this is an element in the linear space of all distributions. As announced after Eq. (1.162), path integrals will specify, in addition, integrals over products of distribution and thus give rise to an important extension of the ¯ − t′ ) plays theory of distributions in Chapter 10. In this, the Heaviside function Θ(t the main role. While discussing the concept of distributions let us introduce, for later use, the closely related distribution ¯ − t′ ) − Θ(t ¯ ′ − t), ǫ(t − t′ ) ≡ Θ(t − t′ ) − Θ(t′ − t) = Θ(t

(1.311)

which is a step function jumping at the origin from −1 to 1 as follows:   

1.11

1 ′ ǫ(t − t ) = 0   −1

t > t′ , t = t′ , t < t′ .

for for for

(1.312)

Fixed-Energy Amplitude

The Fourier-transform of the retarded time evolution amplitude (1.299) (xb |xa )E =

Z

∞

−∞

iE(tb −ta )/¯ h

dtb e

R

(xb tb |xb ta ) =

Z

∞

ta

dtb eiE(tb −ta )/¯h (xb tb |xb ta ) (1.313)

is called the fixed-energy amplitudes. If the Hamiltonian does not depend on time, we insert here Eq. (1.296) and find that the fixed-energy amplitudes are matrix elements ˆ (xb |xa )E = hxb |R(E)|x ai

(1.314)

of the so-called of the so-called resolvent operator ˆ R(E) =

i¯ h , ˆ + iη E−H

(1.315)

46

1 Fundamentals

which is the Fourier transform of the retarded time evolution operator (1.298): ˆ R(E) =

Z

∞

−∞

iE(tb −ta )/¯ h

dtb e

ˆ R (tb , ta ) = U

Z

∞

ta

dtb eiE(tb −ta )/¯h Uˆ (tb , ta ).

(1.316)

Let us suppose that the time-independent Schrödinger equation is completely solved, i.e., that one knows all solutions |ψn i of the equation ˆ n i = En |ψn i. H|ψ

(1.317)

These satisfy the completeness relation X n

|ψn ihψn | = 1,

(1.318)

which can be inserted on the right-hand side of (1.296) between the Dirac brackets leading to the spectral representation (xb tb |xa ta ) =

X n

ψn (xb )ψn∗ (xa ) exp [−iEn (tb − ta )/¯ h] ,

(1.319)

with ψn (x) = hx|ψn i

(1.320)

being the wave functions associated with the eigenstates |ψn i. Applying the Fourier transform (1.313), we obtain (xb |xa )E =

X n

ψn (xb )ψn∗ (xa )Rn (E) =

X

ψn (xb )ψn∗ (xa )

n

i¯ h . E − En + iη

(1.321)

The fixed-energy amplitude (1.313) contains as much information on the system as the time evolution amplitude, which is recovered from it by the inverse Fourier transformation Z ∞ dE −iE(tb −ta )/¯h (xb ta |xa ta ) = e (xb |xa )E . (1.322) h −∞ 2π¯ The small iη-shift in the energy E in (1.321) may be thought of as being attached to each of the energies En , which are thus placed by an infinitesimal piece below the real energy axis. Then the exponential behavior of the wave functions is slightly damped, going to zero at infinite time: e−i(En −iη)t/¯h → 0.

(1.323)

This so-called ensures the causality of the Fourier representation (1.322). When doing the Fourier integral (1.322), the exponential eiE(tb −ta )/¯h makes it always possible to close the integration contour along the energy axis by an infinite semicircle in the complex energy plane, which lies in the upper half-plane for tb < ta and in the lower half-plane for tb > ta . The iη-prescription guarantees that for tb < ta , there is no pole inside the closed contour making the propagator vanish. For tb > ta , on the other hand, the poles in the lower half-plane give, via Cauchy’s residue theorem, the H. Kleinert, PATH INTEGRALS

47

1.12 Free-Particle Amplitudes

spectral representation (1.319) of the propagator. An iη-prescription will appear in another context in Section 2.3. If the eigenstates are nondegenerate, the residues at the poles of (1.321) render directly the products of eigenfunctions (barring degeneracies which must be discussed separately). For a system with a continuum of energy eigenvalues, there is a cut in the complex energy plane which may be thought of as a closely spaced sequence of poles. In general, the wave functions are recovered from the discontinuity of the amplitudes (xb |xa )E across the cut, using the formula disc

i¯ h E − En

!

≡

i¯ h i¯ h − = 2π¯ hδ(E − En ). E − En + iη E − En − iη

(1.324)

Here we have used the general relation to be used in integrals over E: P 1 = ∓ iπδ(E − En ), E − En ± iη E − En

(1.325)

where P indicates that the principal value of the integral. The energy integral over the discontinuity of the fixed-energy amplitude (1.321) (xb |xa )E reproduces the completeness relation (1.318) taken between the local states hxb | and |xa i, Z

∞

−∞

X dE disc (xb |xa )E = ψn (xb )ψn∗ (xa ) = hxb |xa i = δ (D) (xb − xa ). 2π¯ h n

(1.326)

The completeness relation reflects the following property of the resolvent operator: Z

∞

−∞

dE ˆ disc R(E) = ˆ1. 2π¯ h

(1.327)

In general, the system possesses also a continuous spectrum, in which case the completeness relation contains a spectral integral and (1.318) has the form X n

|ψn ihψn | +

Z

dν |ψν ihψν | = 1.

(1.328)

The continuum causes a branch cut along in the complex energy plane, and (1.326) includes an integral over the discontinuity along the cut. The cut will mostly be omitted, for brevity.

1.12

Free-Particle Amplitudes

ˆ =p ˆ 2 /2M, the spectrum is conFor a free particle with a Hamiltonian operator H tinuous. The eigenfunctions are (1.189) with energies E(p) = p2 /2M. Inserting the completeness relation (1.187) into Eq. (1.296), we obtain for the time evolution amplitude of a free particle the Fourier representation (xb tb |xa ta ) =

Z

dD p i p2 exp p(x − x ) − (tb − ta ) b a (2π¯ h)D h ¯ 2M (

"

#)

.

(1.329)

48

1 Fundamentals

The momentum integrals can easily be done. First we perform a quadratic completion in the exponent and rewrite it as 2

M (xb − xa )2 . 2 tb − ta (1.330) ′ Then we replace the integration variables by the shifted momenta p = p − (xb − xa )/(tb − ta )M , and the amplitude (1.329) becomes p(xb − xa ) −

1 2 1 1 xb − xa p (tb − ta ) = p− 2M 2M M tb − ta

(tb − ta ) −

i M (xb − xa )2 (xb tb |xa ta ) = F (tb − ta ) exp , h ¯ 2 tb − ta "

#

(1.331)

d D p′ i p′ 2 exp − (tb − ta ) . (2π¯ h)D h ¯ 2M

(1.332)

where F (tb − ta ) is the integral over the shifted momenta F (tb − ta ) ≡

Z

(

)

This can be performed using the Fresnel integral formula ( √ Z ∞ dp a 2 1 i, a > 0, √ √ exp i p = q (1.333) a < 0. 2 −∞ 2π |a| 1/ i, √ Here the square root i denotes the phase factor eiπ/4 : This follows from the Gauss formula Z ∞ dp α 2 1 √ exp − p = √ , Re α > 0, (1.334) 2 α −∞ 2π by continuing α analytically from positive values into the right complex half-plane. As long as Re α > 0, this is straightforward. On the boundaries, i.e., on the positive and negative imaginary axes, one has to be careful. At α = ±ia + η with a > 0 and < infinitesimal η > 0, the integral is certainly convergent yielding (1.333). But the integral also converges for η = 0, as can easily be seen by substituting x2 = z. See Appendix 1B. Note that differentiation of Eq. (1.334) with respect to α yields the more general Gaussian integral formula Z

∞

−∞

dp α 1 (2n − 1)!! √ p2n exp − p2 = √ 2 α αn 2π

Re α > 0,

(1.335)

where (2n − 1)!! is defined as the product (2n − 1) · (2n − 3) · · · 1. For odd powers p2n+1 , the integral vanishes. In the Fresnel formula (1.333), an extra integrand p2n produces a factor (i/a)n . Since the Fresnel formula is a special analytically continued case of the Gauss formula, we shall in the sequel always speak of Gaussian integrations and use Fresnel’s name only if the imaginary nature of the quadratic exponent is to be emphasized. Applying this formula to (1.332), we obtain 1 F (tb − ta ) = q D, 2πi¯ h(tb − ta )/M

(1.336)


49

1.12 Free-Particle Amplitudes

so that the full time evolution amplitude of a free massive point particle is i M (xb − xa )2 exp . h ¯ 2 tb − ta "

1

(xb tb |xa ta ) = q D 2πi¯ h(tb − ta )/M

#

(1.337)

In the limit tb → ta , the left-hand side becomes the scalar product hxb |xa i = δ (D) (xb − xa ), implying the following limiting formula for the δ-function δ

(D)

(xb − xa ) =

1

lim q D 2πi¯ h(tb − ta )/M

tb −ta →0

i M (xb − xa )2 exp . h ¯ 2 tb − ta "

#

(1.338)

Inserting Eq. (1.331) into (1.313), we have for the fixed-energy amplitude the integral representation dD p i p2 (xb |xa )E = d(tb − ta ) exp p(x − x ) + (t − t ) E − . b a b a (2π¯ h)D h ¯ 2M 0 (1.339) Performing the time integration yields Z

∞

(

Z

(xb |xa )E =

Z

"

!#)

i¯ h dD p exp [ip(x − x )] , b a (2π¯ h)D E − p2 /2M + iη

(1.340)

where we have inserted a damping factor e−η(tb −ta ) into the integral to ensure convergence at large tb − ta . For a more explicit result it is more convenient to calculate the Fourier transform (1.337): (xb |xa )E =

Z

1

∞ 0

d(tb − ta ) q D 2πi¯ h(tb − ta )/M

M (xb −xa )2 i exp E(tb − ta ) + h ¯ 2 tb − ta (

"

#)

.

(1.341)

For E < 0, we set and using the formula10 Z

0

∞

κ≡

dttν−1 e−iγt+iβ/t

q

−2ME/¯ h2 ,

β =2 γ

!ν/2

(1.342)

q

e−iνπ/2 K−ν (2 βγ),

(1.343)

where Kν (z) = K−ν (z) is the modified Bessel function, we find (xb |xa )E = −i 10

2M κD−2 KD/2−1 (κR) , h ¯ (2π)D/2 (κR)D/2−1

(1.344)

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980, Formulas 3.471.10, 3.471.11, and 8.432.6

50

1 Fundamentals

where R ≡ |xb − xa |. The simplest modified Bessel function is11 K1/2 (z) = K−1/2 (z) =

r

π −z e , 2z

(1.345)

so that we find for D = 1, 2, 3, the amplitudes −i

M 1 −κR e , h ¯ κ

−i

M1 K0 (κR), h ¯ π

−i

M 1 −κR e . h ¯ 2πR

(1.346)

At R = 0, the amplitude (1.344) is finite for all D ≤ 2, where we can use small-argument behavior of the associated Bessel function12 Kν (z) = K−ν (z) ≈

−ν

1 z Γ(ν) 2 2

for Re ν > 0,

(1.347)

to obtain (x|x)E = −i

2M κD−2 Γ(1 − D/2). h ¯ (4π)D/2

(1.348)

This result can be continued analytically to D > 2, which will be needed later (for example in Subsection 4.9.4). For E > 0 we set q h2 (1.349) k ≡ 2ME/¯ and use the formula13 Z

0

∞

ν−1 iγt+iβ/t

dtt

e

β = iπ γ

!ν/2

(1)

q

e−iνπ/2 H−ν (2 βγ),

(1.350)

where Hν(1) (z) is the Hankel function, to find (xb |xa )E =

Mπ k D−2 HD/2−1 (kR) . h ¯ (2π)D/2 (kR)D/2−1

(1.351)

The relation14 Kν (−iz) =

π iνπ/2 (1) ie Hν (z) 2

(1.352)

connects the two formulas with each other when continuing the energy from negative to positive values, which replaces κ by e−iπ/2 k = −ik. 11

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965, Formula 10.2.17. 12 ibid., Formula 9.6.9. 13 ibid., Formulas 3.471.11 and 8.421.7. 14 ibid., Formula 8.407.1. H. Kleinert, PATH INTEGRALS

1.13 Quantum Mechanics of General Lagrangian Systems

51

For large distances, the asymptotic behavior15 Kν (z) ≈

r

π −z e , 2z

Hν(1) (z) ≈

s

2 i(z−νπ/2−π/4) e πz

(1.353)

shows that the fixed-energy amplitude behaves for E < 0 like (xb |xa )E ≈ −i

M D−2 1 1 κ e−κR/¯h , h ¯ (2π)(D−1)/2 (κR)(D−1)/2

(1.354)

and for E > 0 like (xb |xa )E ≈

1 1 M D−2 k eikR/¯h . (D−1)/2 (D−1)/2 h ¯ (2πi) (kR)

(1.355)

For D = 1 and 3, these asymptotic expressions hold for all R.

1.13

Quantum Mechanics of General Lagrangian Systems

An extension of the quantum-mechanical formalism to systems described by a set of completely general Lagrange coordinates q1 , . . . , qN is not straightforward. Only in the special case of qi (i = 1, . . . , N) being merely a curvilinear reparametrization of a D-dimensional Euclidean space are the above correspondence rules sufficient to quantize the system. Then N = D and a variable change from xi to qj in the Schrödinger equation leads to the correct quantum mechanics. It will be useful to label the curvilinear coordinates by Greek superscripts and write q µ instead of qj . This will help writing all ensuing equations in a form which is manifestly covariant under coordinate transformations. In the original definition of generalized coordinates in Eq. (1.1), this was unnecessary since transformation properties were ignored. For the Cartesian coordinates we shall use Latin indices alternatively as sub- or superscripts. The coordinate transformation xi = xi (q µ ) implies the relation between the derivatives ∂µ ≡ ∂/∂q µ and ∂i ≡ ∂/∂xi : ∂µ = ei µ (q)∂i ,

(1.356)

ei µ (q) ≡ ∂µ xi (q)

(1.357)

with the transformation matrix

called basis D-ad (in 3 dimensions triad, in 4 dimensions tetrad, etc.). Let ei µ (q) = ∂q µ /∂xi be the inverse matrix (assuming it exists) called the reciprocal D-ad , satisfying with ei µ the orthogonality and completeness relations ei µ ei ν = δµ ν ,

ei µ ej µ = δ i j .

(1.358)

Then, (1.356) is inverted to ∂i = ei µ (q)∂µ 15

ibid., Formulas 8.451.6 and 8.451.3.

(1.359)

52

1 Fundamentals

and yields the curvilinear transform of the Cartesian quantum-mechanical momentum operators pî = −i¯ h∂i = −i¯ hei µ (q)∂µ . (1.360) The free-particle Hamiltonian operator h ¯2 2 ˆ 0 = Tˆ = 1 p ˆ2 = − H ∇ 2M 2M

(1.361)

goes over into

h ¯2 ˆ H0 = − ∆, 2M where ∆ is the Laplacian expressed in curvilinear coordinates: ∆ = ∂i2 = eiµ ∂µ ei ν ∂ν = eiµ ei ν ∂µ ∂ν + (eiµ ∂µ ei ν )∂ν .

(1.362)

(1.363)

At this point one introduces the metric tensor gµν (q) ≡ eiµ (q)ei ν (q),

(1.364)

g µν (q) = eiµ (q)ei ν (q),

(1.365)

its inverse defined by g µν gνλ = δ µ λ , and the so-called affine connection Γµν λ (q) = −ei ν (q)∂µ ei λ (q) = ei λ (q)∂µ ei ν (q).

(1.366)

Then the Laplacian takes the form ∆ = g µν (q)∂µ ∂ν − Γµ µν (q)∂ν ,

(1.367)

with Γµ λν being defined as the contraction Γµ λν ≡ g λκ Γµκ ν .

(1.368)

The reason why (1.364) is called a metric tensor is obvious: An infinitesimal square distance between two points in the original Cartesian coordinates ds2 ≡ dx2

(1.369)

becomes in curvilinear coordinates ds2 =

∂x ∂x µ ν dq dq = gµν (q)dq µ dq ν . µ ν ∂q ∂q

(1.370)

The infinitesimal volume element dD x is given by dD x =

√

g dD q,


53


where g(q) ≡ det (gµν (q))

(1.372)

1 Γµ ≡ g −1/2 (∂µ g 1/2 ) = g λκ (∂µ gλκ ) 2

(1.373)

is the determinant of the metric tensor. Using this determinant, we form the quantity

and see that it is equal to the once-contracted connection Γµ = Γµλ λ .

(1.374)

With the inverse metric (1.365) we have furthermore Γµ µν = −∂µ g µν − Γµ νµ .

(1.375)

We now take advantage of the fact that the derivatives ∂µ , ∂ν applied to the coordinate transformation xi (q) commute causing Γµν λ to be symmetric in µν, i.e., Γµν λ = Γνµ λ and hence Γµ νµ = Γν . Together with (1.373) we find the rotation 1 √ Γµ µν = − √ (∂µ g µν g), g

(1.376)

which allows the Laplace operator ∆ to be rewritten in the more compact form 1 √ ∆ = √ ∂µ g µν g∂ν . g

(1.377)

This expression is called the Laplace-Beltrami operator .16 Thus we have shown that for a Hamiltonian in a Euclidean space H(ˆ p, x) =

1 2 ˆ + V (x), p 2M

(1.378)

the Schrödinger equation in curvilinear coordinates becomes h ¯2 ˆ ∆ + V (q) ψ(q, t) = i¯ Hψ(q, t) ≡ − h∂t ψ(q, t), 2M "

#

(1.379)

where V (q) is short for V (x(q)). The scalar product of two wave functions dD xψ2∗ (x, t)ψ1 (x, t), which determines the transition amplitudes of the system, transforms into Z √ dD q g ψ2∗ (q, t)ψ1 (q, t). (1.380)

R

It is important to realize that this Schrödinger equation would not be obtained by a straightforward application of the canonical formalism to the coordinatetransformed version of the Cartesian Lagrangian ˙ = L(x, x) 16

M 2 x˙ − V (x). 2

More details will be given later in Eqs. (11.12)–(11.18).

(1.381)

54

1 Fundamentals

With the velocities transforming as x˙ i = ei µ (q)q˙µ ,

(1.382)

the Lagrangian becomes L(q, q) ˙ =

M gµν (q)q˙µ q˙ν − V (q). 2

(1.383)

Up to a factor M, the metric is equal to the Hessian metric of the system, which depends here only on q µ [recall (1.12)]: Hµν (q) = Mgµν (q).

(1.384)

The canonical momenta are pµ ≡

∂L = Mgµν q˙ν . µ ∂ q˙

(1.385)

The associated quantum-mechanical momentum operators pˆµ have to be Hermitian in the scalar product (1.380) and must satisfy the canonical commutation rules (1.268): [ˆ pµ , qˆν ] = −i¯ hδµ ν , [ˆ q µ , qˆν ] = 0, [ˆ pµ , pˆν ] = 0.

(1.386)

An obvious solution is pˆµ = −i¯ hg −1/4 ∂µ g 1/4 ,

qˆµ = q µ .

(1.387)

The commutation rules are true for −i¯ hg −z ∂µ g z with any power z, but only z = 1/4 produces a Hermitian momentum operator: Z

√ d3 q g Ψ∗2 (q, t)[−i¯ hg −1/4 ∂µ g 1/4 Ψ1 (q, t)] = =

Z

Z

d3 q g 1/4 Ψ∗2 (q, t)[−i¯ h∂µ g 1/4 Ψ1 (q, t)]

√ d3 q g [−i¯ hg −1/4 ∂µ g 1/4 Ψ2 (q, t)]∗ Ψ1 (q, t),

(1.388)

as is easily verified by partial integration. In terms of the quantity (1.373), this can also be rewritten as pˆµ = −i¯ h(∂µ + 12 Γµ ).

(1.389)

Consider now the classical Hamiltonian associated with the Lagrangian (1.383), which by (1.385) is simply H = pµ q˙µ − L =

1 gµν (q)pµ pν + V (q). 2M


55


When trying to turn this expression into a Hamiltonian operator, we encounter the operator-ordering problem discussed in connection with Eq. (1.101). The correspondence principle requires replacing the momenta pµ by the momentum operators pˆµ , but it does not specify the position of these operators with respect to the coordinates q µ contained in the inverse metric g µν (q). An important constraint is provided by the required Hermiticity of the Hamiltonian operator, but this is not sufficient for a unique specification. We may, for instance, define the canonical Hamiltonian operator as ˆ can ≡ 1 pˆµ gµν (q)ˆ H pν + V (q), (1.391) 2M in which the momentum operators have been arranged symmetrically around the inverse metric to achieve Hermiticity. This operator, however, is not equal to the correct Schrödinger operator in (1.379). The kinetic term contains what we may call the canonical Laplacian ∆can = (∂µ + 12 Γµ ) g µν (q) (∂ν + 12 Γν ).

(1.392)

It differs from the Laplace-Beltrami operator (1.377) in (1.379) by ∆ − ∆can = − 12 ∂µ (g µν Γν ) − 14 g µν Γν Γµ .

(1.393)

The correct Hamiltonian operator could be obtained by suitably distributing pairs of dummy factors of g 1/4 and g −1/4 symmetrically between the canonical operators [5]: ˆ = 1 g −1/4 pˆµ g 1/4 g µν (q)g 1/4 pˆν g −1/4 + V (q). H 2M

(1.394)

This operator has the same classical limit (1.390) as (1.391). Unfortunately, the correspondence principle does not specify how the classical factors have to be ordered before being replaced by operators. The simplest system exhibiting the breakdown of the canonical quantization rules is a free particle in a plane described by radial coordinates q 1 = r, q 2 = ϕ: x1 = r cos ϕ, x2 = r sin ϕ.

(1.395)

Since the infinitesimal square distance is ds2 = dr 2 + r 2 dϕ2 , the metric reads gµν =

1 0 0 r2

!

.

(1.396)

µν

It has a determinant g = r2 and an inverse g

µν

=

1 0 0 r −2

(1.397) !µν

.

(1.398)

56

1 Fundamentals

The Laplace-Beltrami operator becomes 1 1 ∆ = ∂r r∂r + 2 ∂ϕ 2 . r r

(1.399)

The canonical Laplacian, on the other hand, reads 1 2 ∂ϕ r2 1 1 1 = ∂r 2 + ∂r − 2 + 2 ∂ϕ 2 . r 4r r

∆can = (∂r + 1/2r)2 +

(1.400)

The discrepancy (1.393) is therefore ∆can − ∆ = −

1 . 4r 2

(1.401)

Note that this discrepancy arises even though there is no apparent ordering problem in the naively quantized canonical expression pˆµ gµν (q) pˆν in (1.400). Only the need to introduce dummy g 1/4 - and g −1/4 -factors creates such problems, and a specification of the order is required to obtain the correct result. If the Lagrangian coordinates qi do not merely reparametrize a Euclidean space but specify the points of a general geometry, we cannot proceed as above and derive the Laplace-Beltrami operator by a coordinate transformation of a Cartesian Laplacian. With the canonical quantization rules being unreliable in curvilinear coordinates there are, at first sight, severe difficulties in quantizing such a system. This is why the literature contains many proposals for handling this problem [6]. Fortunately, a large class of non-Cartesian systems allows for a unique quantummechanical description on completely different grounds. These systems have the common property that their Hamiltonian can be expressed in terms of the generators of a group of motion in the general coordinate frame. For symmetry reasons, the correspondence principle must then be imposed not on the Poisson brackets of the canonical variables p and q, but on those of the group generators and the coordinates. The brackets containing two group generators specify the structure of the group, those containing a generator and a coordinate specify the defining representation of the group in configuration space. The replacement of these brackets by commutation rules constitutes the proper generalization of the canonical quantization from Cartesian to non-Cartesian coordinates. It is called group quantization. The replacement rule will be referred to as group correspondence principle. The canonical commutation rules in Euclidean space may be viewed as a special case of the commutation rules between group generators, i.e., of the Lie algebra of the group. In a Cartesian coordinate frame, the group of motion is the Euclidean group containing translations and rotations. The generators of translations and rotations are the momenta and the angular momenta, respectively. According to the group correspondence principle, the Poisson brackets between the generators and the coordinates are to be replaced by commutation rules. Thus, in a Euclidean space, the H. Kleinert, PATH INTEGRALS

57

1.14 Particle on the Surface of a Sphere

commutation rules between group generators and coordinates lead to the canonical quantization rules, and this appears to be the deeper reason why the canonical rules are correct. In systems whose energy depends on generators of the group of motion other than those of translations, for instance on the angular momenta, the commutators between the generators have to be used for quantization rather than the canonical commutators between positions and momenta. The prime examples for such systems are a particle on the surface of a sphere or a spinning top whose quantization will now be discussed.

1.14

Particle on the Surface of a Sphere

For a particle moving on the surface of a sphere of radius r with coordinates x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, x3 = r cos θ,

(1.402)

the Lagrangian reads Mr 2 ˙2 L= (θ + sin2 θ ϕ˙ 2 ). 2

(1.403)

The canonical momenta are ˙ pθ = Mr 2 θ,

pϕ = Mr 2 sin2 θ ϕ, ˙

(1.404)

and the classical Hamiltonian is given by 1 2 1 H= p2θ + p . 2 2Mr sin2 θ ϕ

(1.405)

According to the canonical quantization rules, the momenta should become operators 1 h 1/2 ∂θ sin1/2 θ, pˆϕ = −i¯h∂ϕ . (1.406) pˆθ = −i¯ sin θ But as explained in the previous section, these momentum operators are not expected to give the correct Hamiltonian operator when inserted into the Hamiltonian (1.405). Moreover, there exists no proper coordinate transformation from the surface of the sphere to Cartesian coordinates17 such that a particle on a sphere cannot be treated via the safe Cartesian quantization rules (1.268): [ˆ pi , xˆj ] = −i¯ hδi j , [ˆ xi , xˆj ] = 0, [ˆ pi , pˆj ] = 0. 17

(1.407)

There exist, however, certain infinitesimal nonholonomic coordinate transformations which are multivalued and can be used to transform infinitesimal distances in a curved space into those in a flat one. They are introduced and applied in Sections 10.2 and Appendix 10A, leading once more to the same quantum mechanics as the one described here.

58

1 Fundamentals

The only help comes from the group properties of the motion on the surface of the sphere. The angular momentum L=x×p

(1.408)

can be quantized uniquely in Cartesian coordinates and becomes an operator ˆ =x ˆ×p ˆ L

(1.409)

whose components satisfy the commutation rules of the Lie algebra of the rotation group î, L ˆ j ] = i¯ ˆk [L hL

(i, j, k cyclic).

(1.410)

Note that there is no factor-ordering problem since the xî ’s and the pî ’s appear with different indices in each Lˆk . An important property of the angular momentum operator is its homogeneity in x. It has the consequence that when going from Cartesian to spherical coordinates x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, x3 = r cos θ,

(1.411)

the radial coordinate cancels making the angular momentum a differential operator involving only the angles θ, ϕ: ˆ1 = L i¯ h (sin ϕ ∂θ + cot θ cos ϕ ∂ϕ ) , ˆ 2 = −i¯ h (cos ϕ ∂θ − cot θ sin ϕ ∂ϕ ) , L ˆ 3 = −i¯ L h∂ϕ .

(1.412)

There is then a natural way of quantizing the system which makes use of these ˆ i . We re-express the classical Hamiltonian (1.405) in terms of the classical operators L angular momenta L1 = Mr 2 − sin ϕ θ˙ − sin θ cos θ cos ϕ ϕ˙ ,

L2 = Mr 2 cos ϕ θ˙ − sin θ cos θ sin ϕ ϕ˙ ,

L3 = Mr 2 sin2 θ ϕ˙

(1.413)

as

1 L2 , (1.414) 2Mr 2 and replace the angular momenta by the operators (1.412). The result is the Hamiltonian operator: H=

ˆ = H

1 ˆ2 h ¯2 1 1 L = − ∂θ (sin θ ∂θ ) + ∂2 . 2 2 2Mr 2Mr sin θ sin2 θ ϕ

(1.415)


59

1.15 Spinning Top

ˆ 2 are well known. The eigenfunctions diagonalizing the rotation-invariant operator L ˆ i , for instance They can be chosen to diagonalize simultaneously one component of L ˆ 3 , in which case they are equal to the spherical harmonics the third one, L m

Ylm (θ, ϕ) = (−1)

"

2l + 1 (l − m)! 4π (l + m)!

#1/2

Plm (cos θ)eimϕ ,

(1.416)

with Plm (z) being the associated Legendre polynomials Plm (z) =

l+m 1 2 m/2 d ) (z 2 − 1)l . (1 − z 2l l! dxl+m

(1.417)

The spherical harmonics are orthonormal with respect to the rotation-invariant scalar product Z

0

π

dθ sin θ

Z

2π 0

∗ dϕ Ylm (θ, ϕ)Yl′m′ (θ, ϕ) = δll′ δmm′ .

(1.418)

Two important lessons can be learned from this group quantization. First, the correct Hamiltonian operator (1.415) does not agree with the canonically quantized one which would be obtained by inserting Eqs. (1.406) into (1.405). The correct result would, however, arise by distributing dummy factors g −1/4 = r −1 sin−1/2 θ,

g 1/4 = r sin1/2 θ

(1.419)

between the canonical momentum operators as observed earlier in Eq. (1.394). Second, just as in the case of polar coordinates, the correct Hamiltonian operator is equal to ¯2 ˆ =− h H ∆, (1.420) 2M where ∆ is the Laplace-Beltrami operator associated with the metric gµν = r i.e., ∆=

1.15

2

1 0 0 sin2 θ

!

,

(1.421)

1 1 1 ∂2 . ∂ (sin θ∂ ) + θ θ r 2 sin θ sin2 θ ϕ

(1.422)

Spinning Top

For a spinning top, the optimal starting point is again not the classical Lagrangian but the Hamiltonian expressed in terms of the classical angular momenta. In the symmetric case in which two moments of inertia coincide, it is written as H=

1 1 (Lξ 2 + Lη 2 ) + Lζ 2 , 2Iξ 2Iζ

(1.423)

60

1 Fundamentals

where Lξ , Lη , Lζ are the components of the orbital angular momentum in the directions of the principal body axes with Iξ , Iη ≡ Iξ , Iζ being the corresponding moments of inertia. The classical angular momentum of an aggregate of mass points is given by X L= xν × pν , (1.424) ν

where the sum over ν runs over all mass points. The angular momentum possesses a unique operator X ˆ= ˆν × p ˆν , L x (1.425) ν

ˆ i . Since rotations with the commutation rules (1.410) between the components L do not change the distances between the mass points, they commute with the constraints of the rigid body. If the center of mass of the rigid body is placed at the origin, the only dynamical degrees of freedom are the orientations in space. They can uniquely be specified by the rotation matrix which brings the body from some standard orientation to the actual one. We may choose the standard orientation to have the principal body axes aligned with the x, y, z-directions, respectively. An arbitrary orientation is obtained by applying all finite rotations to each point of the body. They are specified by the 3 × 3 orthonormal matrices Rij . The space of these matrices has three degrees of freedom. It can be decomposed, omitting the matrix indices as R(α, β, γ) = R3 (α)R2 (β)R3 (γ), (1.426) where R3 (α), R3 (γ) are rotations around the z-axis by angles α, γ, respectively, and R2 (β) is a rotation around the y-axis by β. These rotation matrices can be expressed as exponentials (1.427) Ri (δ) ≡ e−iδLi /¯h , where δ is the rotation angle and Li are the 3 × 3 matrix generators of the rotations with the elements (Li )jk = −i¯ hǫijk . (1.428) It is easy to check that these generators satisfy the commutation rules (1.410) of angular momentum operators. The angles α, β, γ are referred to as Euler angles. The 3 × 3 rotation matrices make it possible to express the infinitesimal rotations around the three coordinate axes as differential operators of the three Euler angles. Let ψ(R) be the wave function of the spinning top describing the probability amplitude of the different orientations which arise from a standard orientation by the rotation matrix R = R(α, β, γ). Under a further rotation by R(α′ , β ′ , γ ′ ), the wave function goes over into ψ ′ (R) = ψ(R−1 (α′ , β ′ , γ ′ )R). The transformation may be described by a unitary differential operator ′ˆ ′ˆ ′ˆ Uˆ (α′ , β ′, γ ′ ) ≡ e−iα L3 e−iβ L2 e−iγ L3 ,


61

1.15 Spinning Top

ˆ i is the representation of the generators in terms of differential operators. where L To calculate these we note that the 3 × 3 -matrix R−1 (α, β, γ) has the following derivatives −i¯ h∂α R−1 = R−1 L3 , −i¯ h∂β R−1 = R−1 (cos α L2 − sin α L1 ), −i¯ h∂γ R−1 = R−1 [cos β L3 + sin β(cos α L1 + sin α L2 )] .

(1.430)

The first relation is trivial, the second follows from the rotation of the generator e−iαL3 /¯h L2 eiαL3 /¯h = cos α L2 − sin α L1 ,

(1.431)

which is a consequence of Lie’s expansion formula e−iA BeiA = 1 − i[A, B] +

i2 [A, [A, B]] + . . . , 2!

(1.432)

together with the commutation rules (1.428) of the 3 × 3 matrices Li . The third requires, in addition, the rotation e−iβL2 /¯h L3 eiβL2 /¯h = cos βL3 + sin βL1 .

(1.433)

Inverting the relations (1.430), we find the differential operators generating the rotations [7]: ˆ1 L ˆ2 L

!

cos α ∂γ , = i¯ h cos α cot β ∂α + sin α ∂β − sin β ! sin α = i¯ h sin α cot β ∂α − cos α ∂β − ∂γ , sin β

(1.434)

ˆ 3 = −i¯ L h∂α . After exponentiating these differential operators we derive ˆ ′ , β ′ , γ ′ )R−1 Uˆ −1 (α′ , β ′ , γ ′ )(α, β, γ) = R−1 (α, β, γ)R(α′, β ′ , γ ′ ), U(α ˆ −1 (α′ , β ′, γ ′ ) = R−1 (α′, β ′ , γ ′ )R(α, β, γ), (1.435) Uˆ (α′ , β ′, γ ′ )R(α, β, γ)U so that Uˆ (α′ , β ′, γ ′ )ψ(R) = ψ ′ (R), as desired. ˆ along the body axes. In the Hamiltonian (1.423), we need the components of L They are obtained by rotating the 3 × 3 matrices Li by R(α, β, γ) into Lξ = RL1 R−1 = cos γ cos β(cos α L1 + sin α L2 ) + sin γ(cos α L2 − sin α L1 ) − cos γ sin β L3 ,

Lη = RL2 R−1 = − sin γ cos β(cos α L1 + sin α L2 ) + cos γ(cos α L2 − sin α L1 ) + sin γ sin β L3 ,

Lζ = RL3 R−1 = cos β L3 + sin β(cos α L1 + sin α L2 ),

(1.436)

62

1 Fundamentals

ˆ i in the final expressions. Inserting (1.434), we find the operand replacing Li → L ators !

cos γ = i¯ h − cos γ cot β ∂γ − sin γ ∂β + ∂α , sin β ) sin γ = i¯ h sin γ cot β ∂γ − cos γ ∂β − ∂α , sin β

ˆξ L ˆη L

(1.437)

ˆ ζ = −i¯ L h∂γ . Note that these commutation rules have an opposite sign with respect to those in ˆ i :18 Eqs. (1.410) of the operators L ˆξ , L ˆ η ] = −i¯ ˆζ , [L hL

ξ, η, ζ = cyclic.

(1.438)

The sign is most simply understood by writing ˆ ξ = ai L ˆ L ξ i,

ˆ η = ai L ˆ L η i,

ˆ ζ = ai L ˆ L ζ i,

(1.439)

where aiξ , aiη , aiζ , are the components of the body axes. Under rotations these behave ˆ i , aj ] = i¯ hǫijk akξ , i.e., they are vector operators. It is easy to check that this like [L ξ property produces the sign reversal in (1.438) with respect to (1.410). The correspondence principle is now applied to the Hamiltonian in Eq. (1.423) by placing operator hats on the La ’s. The energy spectrum and the wave functions ˆξ, L ˆη, L ˆ ζ . The can then be obtained by using only the group commutators between L spectrum is " ! # 1 1 1 2 ELΛ = h ¯ L(L + 1) + − Λ2 , (1.440) 2Iξ 2Iζ 2Iξ ˆ 2 , and Λ = −L, . . . , L where L(L + 1) with L = 0, 1, 2, . . . are the eigenvalues of L ˆ ζ . The wave functions are the representation functions of are the eigenvalues of L the rotation group. If the Euler angles α, β, γ are used to specify the body axes, the wave functions are L ψLΛm (α, β, γ) = DmΛ (−α, −β, −γ). (1.441) ˆ 3 , the magnetic quantum numbers, and D L (α, β, γ) Here m′ are the eigenvalues of L mΛ are the representation matrices of angular momentum L. In accordance with (1.429), one may decompose ′

L −i(mα+m γ) L Dmm dmm′ (β), ′ (α, β, γ) = e

(1.442)

with the matrices dLmm′ (β)

(L + m′ )!(L − m′ )! = (L + m)!(L − m)! "

×

β cos 2

!m+m′

#1/2

β − sin 2

!m−m′

(m′ −m,m′ +m)

PL−m′

(cos β).

(1.443)

18

When applied to functions not depending on α, then, after replacing β → θ and γ → ϕ, the ˆ 1. operators agree with those in (1.412), up to the sign of L H. Kleinert, PATH INTEGRALS

63

1.15 Spinning Top

For j = 1/2, these form the spinor representation of the rotations around the y-axis 1/2 dm′ m (β)

=

cos β/2 − sin β/2 sin β/2 cos β/2

!

.

(1.444)

The indices have the order +1/2, −1/2. The full spinor representation function D 1/2 (α, β, γ) in (1.442) is most easily obtained by inserting into the general expresˆ i with the sion (1.429) the representation matrices of spin 1/2 for the generators L commutation rules (1.410) the famous Pauli spin matrices: σ1 =

0 1 1 0

!

, σ2 =

0 −i i 0

!

, σ3 =

1 0 0 −1

!

.

(1.445)

Thus we can write D 1/2 (α, β, γ) = e−iασ3 /2 e−iβσ2 /2 e−iγσ3 /2 .

(1.446)

The first and the third factor yield the pure phase factors in (1.442). The function 2 1/2 dm′ m (β) is obtained by a simple power series expansion of e−iβσ /2 , using the fact that (σ 2 )2n = 1 and (σ 2 )2n+1 = σ 2 : e−iβσ

2 /2

= cos β/2 − i sin β/2 σ 2,

(1.447)

which is equal to (1.444). For j = 1, the representation functions (1.443) form the vector representation 

d1m′ m (β) =   

1 (1 + cos β) 2 √1 sin β 2 1 (1 − cos β) 2

− √12 sin β 12 (1 − cos β)  . cos β − √12 sin β  √1 sin β 1 (1 + cos β) 2 2 

(1.448)

where the indices have the order +1/2, −1/2. The vector representation goes over into the ordinary rotation matrices Rij (β) by mapping the states |1mi onto the spherical unit vectors (0) = zˆ, (±1) = ∓(ˆ x ± iˆ y)/2 using the matrix elements P1 i ′ 1 hi|1mi = ǫ (m). Hence R(β) (m) = m′ =−1 (m )dm′ m (β). The representation functions D 1 (α, β, γ) can also be obtained by inserting into the general exponential (1.429) the representation matrices of spin 1 for the generaˆ i with the commutation rules (1.410). In Cartesian coordinates, these are tors L ˆ i )jk = −iǫijk , where ǫijk is the completely antisymmetric tensor with simply (L ˆ i )mm′ = hm|ii(L ˆ i )ij hj|m′ i = ǫ123 = 1. In the spherical basis, these become (L ˆ i )ij j (m′ ). The exponential (e−iβ Lˆ 2 )mm′ is equal to (1.448). ǫ∗i (m)(L (α,β) The functions Pl (z) are the Jacobi polynomials [8], which can be expressed in terms of hypergeometric functions as (α,β)

Pl

≡

(−1)l Γ(l + β + 1) F (−l, l + 1 + α + β; 1 + β; (1 + z)/2), l! Γ(β + 1)

(1.449)

64

1 Fundamentals

where F (a, b; c; z) ≡ 1 +

a(a + 1) b(b + 1) z 2 ab z+ + ... . c c(c + 1) 2!

(1.450)

The rotation functions dLmm′ (β) satisfy the differential equation d2 d m2 + m′ 2 − 2mm′ cos β L − 2 − cot β dmm′ (β) = L(L + 1)dLmm′ (β). (1.451) + dβ dβ sin2 β !

The scalar products of two wave functions have to be calculated with a measure of integration which is invariant under rotations: hψ2 |ψ1 i ≡

Z

2π

0

Z

0

π

Z

2π 0

dαdβ sin βdγ ψ2∗ (α, β, γ)ψ1 (α, β, γ).

(1.452)

The above eigenstates (1.442) satisfy the orthogonality relation Z

0

2π

Z

0

π

Z

2π 0

L1 ∗ L2 dαdβ sin βdγ Dm ′ m (α, β, γ)Dm′ m (α, β, γ) 1 2 1

= δm′1 m′2 δm1 m2 δL1 L2

2

8π 2 . 2L1 + 1

(1.453)

Let us also contrast in this example the correct quantization via the commutation rules between group generators with the canonical approach which would start out with the classical Lagrangian. In terms of Euler angles, the Lagrangian reads 1 L = [Iξ (ωξ 2 + ωη 2 ) + Iζ ωζ 2 ], 2

(1.454)

where ωξ , ωη , ωζ are the angular velocities measured along the principal axes of the top. To find these we note that the components in the rest system ω1 , ω2, ω3 are obtained from the relation ˙ −1 ωk Lk = iRR (1.455) as ω1 = −β˙ sin α + γ˙ sin β cos α, ω2 = β˙ cos α + γ˙ sin β sin α, ω3 = γ˙ cos β + α. ˙

(1.456)

After the rotation (1.436) into the body-fixed system, these become ωξ = β˙ sin γ − α˙ sin β cos γ, ωη = β˙ cos γ + α˙ sin β sin γ, ωζ = α˙ cos β + γ. ˙

(1.457)

Explicitly, the Lagrangian is 1 L = [Iξ (β˙ 2 + α˙ 2 sin2 β) + Iζ (α˙ cos β + γ) ˙ 2 ]. 2

(1.458)


65

1.15 Spinning Top

Considering α, β, γ as Lagrange coordinates q µ with µ = 1, 2, 3, this can be written in the form (1.383) with the Hessian metric [recall (1.12) and (1.384)]: gµν whose determinant is

Iξ sin2 β + Iζ cos2 β 0 Iζ cos β   0 Iξ 0 = , Iζ cos β 0 Iζ 



(1.459)

g = Iξ2 Iζ sin2 β.

√

(1.460)

Hence the measure d3 q g in the scalar product (1.380) agrees with the rotationinvariant measure (1.452) up to a trivial constant factor. Incidentally, this is also true for the asymmetric top with Iξ 6= Iη 6= Iζ , where g = Iξ2 Iζ sin2 β, although the metric gµν is then much more complicated (see Appendix 1C). The canonical momenta associated with the Lagrangian (1.454) are, according to (1.383), R

pα = ∂L/∂ α˙ = Iξ α˙ sin2 β + Iζ cos β(α˙ cos β + γ), ˙ ˙ ˙ pβ = ∂L/∂ β = Iξ β, pγ = ∂L/∂ γ˙ = Iζ (α˙ cos β + γ). ˙

(1.461)

After inverting the metric to g µν

µν



1 0 − cos β 1   2 = 0 sin β 0   2 Iξ sin β − cos β 0 cos2 β + Iξ sin2 β/Iζ

,

(1.462)

we find the classical Hamiltonian

1 1 2 cos2 β 1 H= pβ + + 2 2 Iξ Iξ sin β Iζ "

!

pγ

2

#

1 2 cos β 2 + pα pγ . 2 pα − Iξ sin β Iξ sin2 β

(1.463)

This Hamiltonian has no apparent ordering problem. One is therefore tempted to replace the momenta simply by the corresponding Hermitian operators which are, according to (1.387), pˆα = −i¯ h∂α ,

pˆβ = −i¯ h(sin β)−1/2 ∂β (sin β)1/2 = −i¯ h(∂β +

1 cot β), 2

pˆγ = −i¯ h∂γ .

(1.464)

Inserting these into (1.463) gives the canonical Hamiltonian operator ˆ can = H ˆ +H ˆ discr , H

(1.465)

with h ¯2 Iξ ˆ H ≡ − ∂β 2 + cot β∂β + + cot2 β ∂γ 2 2Iξ Iζ # 1 2 cos β 2 + ∂α − ∂α ∂γ sin2 β sin2 β "

!

(1.466)

66 and

1 Fundamentals

1 3 ˆ discr ≡ 1 (∂β cot β) + 1 cot2 β = H − . 2 2 4 4 sin β 4

(1.467)

ˆ agrees with the correct quantum-mechanical operator derived The first term H above. Indeed, inserting the differential operators for the body-fixed angular moˆ The term H ˆ discr is the menta (1.437) into the Hamiltonian (1.423), we find H. discrepancy between the canonical and the correct Hamiltonian operator. It exists even though there is no apparent ordering problem, just as in the radial coordinate expression (1.400). The correct Hamiltonian could be obtained by replacing the classical pβ 2 term in H by the operator g −1/4 pˆβ g 1/2 pˆβ g −1/4 , by analogy with the ˆ of Eq. (1.394). treatment of the radial coordinates in H As another similarity with the two-dimensional system in radial coordinates and the particle on the surface of the sphere, we observe that while the canonical quantization fails, the Hamiltonian operator of the symmetric spinning top is correctly given by the Laplace-Beltrami operator (1.377) after inserting the metric (1.459) and the inverse (1.462). It is straightforward although tedious to verify that this is also true for the completely asymmetric top [which has quite a complicated metric given in Appendix 1C, see Eqs. (1C.2), and (1C.4)]. This is an important nontrivial result, since for a spinning top, the Lagrangian cannot be obtained by reparametrizing a particle in a Euclidean space with curvilinear coordinates. The result suggests that a replacement h2 ∆ (1.468) gµν (q)pµ pν → −¯ produces the correct Hamiltonian operator in any non-Euclidean space.19 What is the characteristic non-Euclidean property of the α, β, γ space? As we shall see in detail in Chapter 10, the relevant quantity is the curvature scalar R. The exact definition will be found in Eq. (10.42). For the asymmetric spinning top we find (see Appendix 1C) (Iξ + Iη + Iζ )2 − 2(Iξ2 + Iη2 + Iζ2 ) R= . 2Iξ Iη Iζ

(1.469)

Thus, just like a particle on the surface of a sphere, the spinning top corresponds to a particle moving in a space with constant curvature. In this space, the correct correspondence principle can also be deduced from symmetry arguments. The geometry is most easily understood by observing that the α, β, γ space may be considered as the surface of a sphere in four dimensions, as we shall see in more detail in Chapter 8. An important non-Euclidean space of physical interest is encountered in the context of general relativity. Originally, gravitating matter was assumed to move in a spacetime with an arbitrary local curvature. In newer developments of the theory one also allows for the presence of a nonvanishing torsion. In such a general situation, 19

If the space has curvature and no torsion, this is the correct answer. If torsion is present, the correct answer will be given in Chapters 10 and 8. H. Kleinert, PATH INTEGRALS

67

1.16 Scattering

where the group quantization rule is inapplicable, the correspondence principle has always been a matter of controversy [see the references after (1.401)] to be resolved in this text. In Chapters 10 and 8 we shall present a new quantum equivalence principle which is based on an application of simple geometrical principles to path integrals and which will specify a natural and unique passage from classical to quantum mechanics in any coordinate frame.20 The configuration space may carry curvature and a certain class of torsions (gradient torsion). Several arguments suggest that our principle is correct. For the above systems with a Hamiltonian which can be expressed entirely in terms of generators of a group of motion in the underlying space, the new quantum equivalence principle will give the same results as the group quantization rule.

1.16

Scattering

Most observations of quantum phenomena are obtained from scattering processes of fundamental particles.

1.16.1

Scattering Matrix

Consider a particle impinging with a momentum pa and energy E = Ea = p2a /2M upon a nonzero potential concentrated around the origin. After a long time, it will be found far from the potential with some momentum pb . The energy will be unchanged: E = Eb = p2b /2M. The probability amplitude for such a process is given by the time evolution amplitude in the momentum representation ˆ

(pb tb |pa ta ) ≡ hpb |e−iH(tb −ta )/¯h |pa i,

(1.470)

where the limit tb → ∞ and ta → −∞ has to be taken. Long before and after the collision, this amplitude oscillates with a frequency ω = E/¯ h characteristic for free particles of energy E. In order to have a time-independent limit, we remove these oscillations, from (1.470), and define the scattering matrix (S-matrix) by the limit ˆ ai ≡ hpb |S|p

lim

tb −ta →∞

ˆ

ei(Eb tb −Ea ta )/¯h hpb |e−iH(tb −ta )/¯h |pa i.

(1.471)

Most of the impinging particles will not scatter at all, so that this amplitude must contain a leading term, which is separated as follows: ˆ a i = hpb |pa i + hpb |S|p ˆ a i′ , hpb |S|p where

ˆ

hpb |pa i = hpb |e−iH(tb −ta )/¯h |pa i = (2π¯ h)3 δ (3) (pb − pa )

(1.472) (1.473)

shows the normalization of the states [recall (1.186)]. This leading term is commonly subtracted from (1.471) to find the true scattering amplitude. Moreover, 20

H. Kleinert, Mod. Phys. Lett. A 4 , 2329 (1989) (http://www.physik.fu-berlin.de/ ~kleinert/199); Phys. Lett. B 236 , 315 (1990) (ibid.http/202).

68

1 Fundamentals

since potential scattering conserves energy, the remaining amplitude contains a δfunction ensuring energy conservation, and it is useful to divide this out, defining the so-called T -matrix by the decomposition by ˆ a i ≡ (2π¯ hpb |S|p h)3 δ (3) (pa − pa ) − 2π¯ hiδ(Eb − Ea )hpb |Tˆ |pa i.

(1.474)

ˆ it follows that scattering matrix From the definition (1.471) and the hermiticity of H is a unitary matrix. This expresses the physical fact that the total probability of an incident particle to re-emerge at some time is unity (in quantum field theory the situation is more complicated due to emission and absorption processes). In the basis states |pm i introduced in Eq. (1.180) which satisfy the completeness relation (1.182) and are normalized to unity in a finite volume V , the unitarity is expressed as X m′

′ ′ ˆ m′′ i = hpm |Sˆ† |pm ihpm |S|p

X m′

ˆ m′ ihpm′ |Sˆ† |pm′′ i = 1. hpm |S|p

(1.475)

Remembering the relation (1.185) between the discrete states |pm i and their continuous limits |pi, we see that ′ ˆ a m i ≈ 1 hpb |S|p ˆ a i, hpb m |S|p L3

(1.476)

m where L3 is the spatial volume, and pm b and pa are the discrete momenta closest to pb and pa . In the continuous basis |pi, the unitarity relation reads

Z

1.16.2

Z d3 p d3 p † ˆ ˆ ˆ hpb |S |pihp|S|pa i = hpb |S|pihp| Sˆ† |pa i = 1. 3 3 (2π¯ h) (2π¯ h)

(1.477)

Cross Section

ˆ a i gives the probability Pp ←p for the scattering The absolute square of hpb |S|p b a from the initial momentum state pa to the final momentum state pb . Omitting the unscattered particles, we have 1 2π¯ hδ(0) 2π¯ hδ(Eb − Ea )|hpb |Tˆ |pa i|2 . (1.478) L6 The factor δ(0) at zero energy is made finite by imagining the scattering process to take place with an incident time-independent plane wave over a finite total time T . R Then 2π¯ hδ(0) = dt eiEt/¯h |E=0 = T , and the probability is proportional to the time T: 1 Ppb ←pa = 6 T 2π¯ hδ(Eb − Ea )|hpb |Tˆ |pa i|2. (1.479) L By summing this over all discrete final momenta, or equivalently, by integrating this over the phase space of the final momenta [recall (1.184)], we find the total probability per unit time for the scattering to take place Ppb ←pa =

dP 1 = 6 dt L

Z

d3 pb L3 2π¯ hδ(Eb − Ea )|hpb |Tˆ |pa i|2 . (2π¯ h)3

(1.480)


69

1.16 Scattering

The momentum integral can be split into an integral over the final energy and the final solid angle. For non-relativistic particles, this goes as follows Z

d 3 pb M 1 = 3 3 (2π¯ h) (2π¯ h) (2π¯ h)3

Z

dΩ

Z

0

∞

dEb pb ,

(1.481)

where dΩ = dφb d cos θb is the element of solid angle into which the particle is scattered. The energy integral removes the δ-function in (1.480), and makes pb equal to pa . The differential scattering cross section dσ/dΩ is defined as the probability that a single impinging particle ends up in a solid angle dΩ per unit time and unit current density. From (1.480) we identify dP˙ 1 1 Mp dσ 21 = = 3 , 2π¯ h |T | p p b a dΩ dΩ j L (2π¯ h)3 j where we have set

hpb |Tˆ |pa i ≡ Tpb pa ,

(1.482)

(1.483)

for brevity. In a volume L3 , the current density of a single impinging particle is given by the velocity v = p/M as j=

1 p , L3 M

(1.484)

so that the differential cross section becomes dσ M2 = |Tp p |2 . dΩ (2π¯ h)2 b a

(1.485)

If the scattered particle√moves relativistically, we have to replace the constant mass M in (1.481) by E = p2 + M 2 inside the momentum integral, where p = |p|, so that Z

∞ d3 p 1 = dΩ dp p2 3 3 (2π¯ h) (2π¯ h) 0 Z Z ∞ 1 = dΩ dEE p. (2π¯ h)3 0

Z

Z

(1.486)

In the relativistic case, the initial current density is not proportional to p/M but to the relativistic velocity v = p/E so that 1 p . L3 E

(1.487)

dσ E2 = |Tp p |2 . dΩ (2π¯ h)2 b a

(1.488)

j= Hence the cross section becomes

70

1 Fundamentals

1.16.3

Born Approximation

To lowest order in the interaction strength, the operator Sˆ in (1.471) is Sˆ ≈ 1 − iVˆ /¯ h.

(1.489)

For a time-independent scattering potential, this implies Tpb pa ≈ Vpb pa /¯ h,

(1.490)

where Vpb pa ≡ hpb |Vˆ |pa i =

Z

d3 x ei(pb −pa )x/¯h V (x) = V˜ (pb − pa )

(1.491)

is a function of the momentum transfer q ≡ pb − pa only. Then (1.488) reduces to the so called Born approximation (Born 1926) dσ E2 ≈ |Vp p |2 . dΩ (2π¯ h)2h ¯2 b a

(1.492)

The amplitude whose square is equal to the differential cross section is usually denoted by fpb pa , i.e., one writes dσ = |fpb pa |2 . dΩ

(1.493)

By comparison with (1.492) we identify fpb pa ≡ −

M Rp p , 2π¯ h b a

(1.494)

where we have chosen the sign to agree with the convention in the textbook by Landau and Lifshitz [9].

1.16.4

Partial Wave Expansion and Eikonal Approximation

The scattering amplitude is usually expanded in partial waves with the help of Legendre polynomials Pl (z) ≡ Pl0(z) [see (1.417)] as fpb pa =

∞ h ¯ X (2l + 1)Pl (cos θ) e2i∂l (p) − 1 2ip l=0

(1.495)

where p ≡ |p| = |pb | = |pa | and θ is the scattering defined by cos θ ≡ pb pb /|pb ||pa |. In terms of θ, the momentum transfer q = pb − pa has the size |q| = 2p sin(θ/2). For small θ, we can use the asymptotic form of the Legendre polynomials21 Pl−m(cos θ) ≈ 21

1 Jm (lθ), lm

(1.496)

M. Abramowitz and I. Stegun, op. cit., Formula 9.1.71. H. Kleinert, PATH INTEGRALS

71

1.16 Scattering

to rewrite (1.495) approximately as an integral fpeib pa =

n h i o p Z db b J0 (qb) exp 2iδpb/¯h (p) − 1 , i¯ h

(1.497)

where b ≡ l¯ h/p is the so called impact parameter of the scattering process. This is the eikonal approximation to the scattering amplitude. As an example, consider 2 Coulomb scattering where V (r) = Ze /r and (2.743) yields χei b,P [v]

Ze2 M 1 =− |P| h ¯

Z

∞

−∞

dz √

b2

1 . + z2

(1.498)

The integral diverges logarithmically, but in a physical sample, the potential is screened at some distance R by opposite charges. Performing the integral up to R yields √ Z Ze2 M 1 R 1 Ze2 M 1 R + R2 − b2 ei dr √ 2 =− χb,P [v] = − log |P| h ¯ b |P| h ¯ b r − b2 2 2R Ze M 1 log . (1.499) ≈ −2 |P| h ¯ b This implies exp where

χei b,P

≈

b 2R

!2iγ

,

Ze2 M 1 γ≡ |P| h ¯

(1.500)

(1.501)

is a dimensionless quantity since e2 = h ¯ cα where α is the dimensionless fine-structure 22 constant e2 α= = 1/137.035 997 9 . . . . (1.502) h ¯c The integral over the impact parameter in (1.497) can now be performed and yields fpeib pa ≈

h ¯ 1 Γ(1 + iγ) −2iγ log(2pR/¯h) e . 2+2iγ 2ip sin (θ/2) Γ(−iγ)

(1.503)

Remarkably, this is the exact quantum mechanical amplitude of Coulomb scattering, except for the last phase factor which accounts for a finite screening length. This amplitude contains poles at momentum variables p = pn whenever iγn ≡ 22

Ze2 M¯ h = −n, pn

n = 1, 2, 3, . . . .

(1.504)

Throughout this book we use electromagnetic units where the electric field E = −∇φ has the energy density H = E2 /8π + ρφ, where ρ is the charge density, so that ∇ · E = 4πρ and e2 = h ¯ c α. The fine-structure constant is measured most precisely via the quantum Hall effect , see M.E. Cage et al., IEEE Trans. Instrum. Meas. 38 , 284 (1989). The magnetic field satisfies Ampère’s law ∇ × B = 4πj, where j is the current density.

72

1 Fundamentals

This corresponds to energies E

(n)

MZ 2 e4 1 p2n =− =− , 2M h ¯ 2 2n2

(1.505)

which are the well-known energy values of hydrogen-like atoms with nuclear charge Ze. The prefactor EH ≡ e2 /aH = Me4 /¯ h2 = 4.359 × 10−11 erg = 27.210 eV, is equal to twice the Rydberg energy (see also p. 871).

1.16.5

Scattering Amplitude from Time Evolution Amplitude

There exists a heuristic formula expressing the scattering amplitude as a limit of the time evolution amplitude. For this we express the δ-function in the energy as a large-time limit M M tb δ(Eb − Ea ) = δ(pb − pa ) = lim t →∞ pb pb b 2π¯ hM/i

!1/2

i tb exp − (pb − pa )2 , h ¯ 2M (1.506) where pb = |pb |. Inserting this into Eq. (1.474) and setting sloppily pb = pa for elastic scattering, the δ-function is removed and we obtain the following expression for the scattering amplitude fpb pa

pb = M

q

2π¯ hM/i

3

lim

1

tb →∞ t1/2 b

(2π¯ h)3

eiEb (tb −ta )/¯h [(pb tb |pa ta )−hpb |pa i] .

(1.507)

This treatment of a δ-function is certainly unsatisfactory. A satisfactory treatment will be given in the path integral formulation in Section 2.22. At the present stage, we may proceed with more care with the following operator calculation. We rewrite the limit (1.471) with the help of the time evolution operator (2.5) as follows: ˆ ai ≡ hpb |S|p =

lim

ei(Eb tb −Ea ta )/¯h (pb tb |pa ta )

lim

hpb |UÎ (tb , ta )|pa i,

tb −ta →∞

(1.508)

tb ,−ta →∞

Î (tb , ta ) is the time evolution operator in Dirac’s interaction picture (1.286). where U

1.16.6

Lippmann-Schwinger Equation

From the definition (1.286) it follows that the operator UÎ (tb , ta ) satisfies the same ˆ ta ): composition law (1.254) as the ordinary time evolution operator U(t, UÎ (t, ta ) = UÎ (t, tb )UÎ (tb , ta ).

(1.509)

Now we observe that Î (t, ta ) = e−iHt/¯h U Î (0, ta ) = UÎ (0, ta − t)e−iH0 t/¯h , e−iH0 t/¯h U

(1.510)


73

1.16 Scattering

so that in the limit ta → −∞ Î (t, ta ) = e−iHt/¯h U Î (0, ta ) − Î (0, ta )e−iH0 t/¯h , e−iH0 t/¯h U − −→ U

(1.511)

and therefore Î (0, ta )e−iH0 tb /¯h, lim UÎ (tb , ta ) = lim eiH0 tb /¯h e−iHtb /¯h UÎ (0, ta ) = lim eiH0 tb /¯h U

ta →−∞

ta →−∞

(1.512)

ta →−∞

which allows us to rewrite the scattering matrix (1.508) as ˆ ai ≡ hpb |S|p

lim

tb ,−ta →∞

ei(Eb −Ea )tb /¯h hpb |UÎ (0, ta )|pa i.

(1.513)

Note that in contrast to (1.471), the time evolution of the initial state goes now only over the negative time axis rather than the full one. Taking the matrix elements of Eq. (1.291) between free-particle states hpb | and |pb i, and using Eqs. (1.291) and (1.511), we obtain at tb = 0 Z 0 Î (0, ta )|pb i = hpb |pb i − i dt ei(Eb −Ea −iη)t/¯h hpb |Vˆ UÎ (0, ta )|pb i. (1.514) hpb |U h ¯ −∞

A small damping factor eηt/¯h is inserted to ensure convergence at t = −∞. For a time-independent potential, the integral can be done and yields Î (0, ta )|pb i = hpb |pb i − hpb |U

1 hpb |Vˆ UÎ (0, ta )|pb i. Eb − Ea − iη

(1.515)

This is the famous Lippmann-Schwinger equation. Inserting this into (1.513), we obtain the equation for the scattering matrix ˆ ai = hpb |S|p

lim

tb ,−ta →∞

i(Eb −Ea )tb

e

"

#

1 hpb |Vˆ UÎ (0, ta )|pb i . (1.516) hpb |pa i − Eb − Ea − iη

The first term in brackets is nonzero only if the momenta pa and pb are equal, in which case also the energies are equal, Eb = Ea , so that the prefactor can be set equal to one. In front of the second term, the prefactor oscillates rapidly as the time tb grows large, making any finite function of Eb vanish, as a consequence of the Riemann-Lebesgue lemma. The second term contains, however, a pole at Eb = Ea for which the limit has to be done more carefully. The prefactor has the property ei(Eb −Ea )tb /¯h lim = tb →∞ Eb − Ea − iη

(

0, i/η,

Eb = 6 Ea , Eb = Ea .

(1.517)

It is easy to see that this property defines a δ-function in the energy: ei(Eb −Ea )tb /¯h lim = 2πiδ(Eb − Ea ). tb →∞ Eb − Ea − iη

(1.518)

74

1 Fundamentals

Indeed, let us integrate the left-hand side together with a smooth function f (Eb ), and set Eb ≡ Ea + ξ/tb .

(1.519)

Then the Eb -integral is rewritten as Z

∞

dξ

−∞

eiξ f (Ea + ξ/ta ) . ξ + iη

(1.520)

In the limit of large ta , the function f (Ea ) can be taken out of the integral and the contour of integration can then be closed in the upper half of the complex energy plane, yielding 2πi. Thus we obtain from (1.516) the formula (1.474), with the T -matrix 1 hpb |Tˆ |pa i = hpb |Vˆ UÎ (0, ta )|pb i. (1.521) h ¯ For a small potential Vˆ , we approximate UÎ (0, ta ) ≈ 1, and find the Born approximation (1.490). The the Lippmann-Schwinger equation can be recast as an integral equation for the T -matrix. Multiplying the original equation (1.515) by the matrix hpb |Vˆ |pa i = Vpb pc from the left, we obtain Tpb pa = Vpb pa −

Z

d 3 pc 1 V Tp p . p p b c (2π¯ h)3 Ec − Ea − iη c a

(1.522)

To extract physical information from the T -matrix (1.521) it is useful to analyze the behavior of the interacting state UÎ (0, ta )|pa i in x-space. From Eq. (1.511), ˆ with the initial we see that it is an eigenstate of the full Hamiltonian operator H energy Ea . Multiplying this state by hx| from the left, and inserting a complete set of momentum eigenstates, we calculate Î (0, ta )|pa i = hx|U

Z

d3 p Î (0, ta )|pa i = hx|pihp|U (2π¯ h)3

d3 p Î (0, ta )|pa i. hx|pihp|U (2π¯ h)3

Z

Using Eq. (1.515), this becomes Î (0, ta )|pa i = hx|pa i + hx|U

Z

Z

d3 x′

′

d 3 pb eipb (x−x )/¯h Î (0, ta )|pa i. V (x′ )hx′ |U (2π¯ h)3 Ea −p2b /2M +iη (1.523)

The function (x|x′ )Ea =

Z

d3 pb ipb (x−x′ )/¯h i¯ h e (2π¯ h)3 Ea − p2 /2M + iη

(1.524)

is recognized as the fixed-energy amplitude (1.340) of the free particle. In three dimensions it reads [see (1.355)] ′

(x|x′ )Ea = −

2Mi eipa |x−x |/¯h , h ¯ 4π|x − x′ |

pa =

q

2MEa .

(1.525)


75

1.16 Scattering

In order to find the scattering amplitude, we consider the wave function (1.523) far away from the scattering center, i.e., at large |x|. Under the assumption that V (x′ ) ˆ x′ , where x ˆ is the unit is nonzero only for small x′ , we approximate |x − x′ | ≈ r − x vector in the direction of x, and (1.523) becomes eipa r Z 4 ′ −ipa xˆ x′ 2M ipa x/¯ h ′ ′ ˆ ˆ hx|UI (0, ta )|pa i ≈ e − d xe 2 V (x )hx |UI (0, ta )|pa i. 4πr h ¯

(1.526)

In the limit ta → −∞, the factor multiplying the spherical wave factor eipa r/¯h /r is the scattering amplitude f (ˆ x)pa , whose absolute square gives the cross section. For scattering to a final momentum pb , the outgoing particles are detected far away ˆ = pˆb . Because of energy conservation, from the scattering center in the direction x ˆ = pb and obtain the formula we may set pa x fpb pa

M = lim − ta →−∞ 2π¯ h2

Z

Î (0, ta )|pa i. d4 xb e−ipb xb V (xb )hxb |U

(1.527)

By studying the interacting state UÎ (0, ta )|pa i in x-space, we have avoided the singular δ-function of energy conservation. We are now prepared to derive formula (1.507) for the scattering amplitude. We Î (0, ta )|pa i can be obtained observe that in the limit ta → −∞, the amplitude hxb |U from the time evolution amplitude (xb tb |xa ta ) as follows: Î (0, ta )|pa i = hxb |U(0, ˆ ta )|pa ie−iEa ta /¯h hxb |U =

lim

ta →−∞

−2πi¯ hta M

!3/2

(1.528)

2

(xb tb |xa ta )ei(pa xa −pa ta /2M )/¯h

xa =pa ta /M

.

This follows directly from the Fourier transformation hxb |Uˆ (0, ta )|pa ie−iEa ta /¯h =

Z

2

d3 xa (xb tb |xa ta )ei(pa xa −pa ta /2M )/¯h ,

(1.529)

by substituting the dummy integration variable xa by pta /M. Then the right-hand side becomes

−ta M

3 Z

2

d3 p (xb 0|pta ta )ei(pa p−pa )ta /2M ¯h .

(1.530)

Now, for large −ta , the momentum integration is squeezed to p = pa , and we obtain (1.528). The appropriate limiting formula for the δ-function δ

(D)

(−ta )D/2 i ta (pb − pa ) = lim √ (pb − pa )2 D exp − ta →−∞ h ¯ 2M 2πi¯ hM

(1.531)

is easily obtained from Eq. (1.338) by an obvious substitution of variables. Its complex conjugate for D = 1 was written down before in Eq. (1.506) with ta replaced

76

1 Fundamentals

by −tb . The exponential on the right-hand side can just as well be multiplied by a 2 2 2 factor ei(pb −pa ) /2M ¯h which is unity when both sides are nonzero, so that it becomes 2 e−i(pa p−pa )ta /2M ¯h . In this way we obtain a representation of the δ-function by which 2 the Fourier integral (1.530) goes over into (1.528). The phase factor ei(pa xa −pa ta /2M )/¯h on the right-hand side of Eq. (1.528), which is unity in the limit performed in that equation, is kept in Eq. (4.543) for later convenience. Formula (1.528) is a reliable starting point for extracting the scattering amplitude fpb pa from the time evolution amplitude in x-space (xb 0|xa ta ) at xa = pa ta /M by extracting the coefficient of the outcoming spherical wave eipa r/¯h /r. As a cross check we insert the free-particle amplitude (1.337) into (1.528) and obtain the free undisturbed wave function eipa x , which is the correct first term in Eq. (1.523) associated with unscattered particles.

1.17

Classical and Quantum Statistics

Consider a physical system with a constant number of particles N whose Hamiltonian has no explicit time dependence. If it is brought in contact with a thermal reservoir at a temperature T and has reached equilibrium, its thermodynamic properties can be obtained through the following rules: At the level of classical mechanics, each volume element in phase space dp dq dp dq = h 2π¯ h

(1.532)

is occupied with a probability proportional to the Boltzmann factor e−H(p,q)/kB T ,

(1.533)

where kB is the Boltzmann constant, kB = 1.3806221(59) × 10−16 erg/Kelvin.

(1.534)

The number in parentheses indicates the experimental uncertainty of the two digits in front of it. The quantity 1/kB T has the dimension of an inverse energy and is commonly denoted by β. It will be called the inverse temperature, forgetting about the factor kB . In fact, we shall sometimes take T to be measured in energy units kB times Kelvin rather than in Kelvin. Then we may drop kB in all formulas. The integral over the Boltzmann factors of all phase space elements,23 Zcl (T ) ≡

Z

dp dq −H(p,q)/kB T e , 2π¯ h

(1.535)

is called the classical partition function. It contains all classical thermodynamic information of the system. Of course, for a general Hamiltonian system with many 23

In the sequel we shall always work at a fixed volume V and therefore suppress the argument V everywhere. H. Kleinert, PATH INTEGRALS

77

1.17 Classical and Quantum Statistics

degrees of freedom, the phase space integral is

YZ

dpn dqn /2π¯ h. The reader may

n

wonder why an expression containing Planck’s quantum h ¯ is called classical . The reason is that h ¯ can really be omitted in calculating any thermodynamic average. In classical statistics it merely supplies us with an irrelevant normalization factor which makes Z dimensionless.

1.17.1

Canonical Ensemble

ˆ and the integral In quantum statistics, the Hamiltonian is replaced by the operator H over phase space by the trace in the Hilbert space. This leads to the quantumstatistical partition function

ˆ

Z(T ) ≡ Tr e−H/kB T ≡ Tr e−H(ˆp,ˆx)/kB T ,

(1.536)

ˆ denotes the trace of the operator O. ˆ If H ˆ is an N-particle Schrödinger where Tr O Hamiltonian, the quantum-statistical system is referred to as a canonical ensemble. The right-hand side of (1.536) contains the position operator xˆ in Cartesian coordinates rather than qˆ to ensure that the system can be quantized canonically. In cases such as the spinning top, the trace formula is also valid but the Hilbert space is spanned by the representation states of the angular momentum operators. In more general Lagrangian systems, the quantization has to be performed differently in the way to be described in Chapters 10 and 8. At this point we make an important observation: The quantum partition function is related in a very simple way to the quantum-mechanical time evolution operator. To emphasize this relation we shall define the trace of this operator for time-independent Hamiltonians as the quantum-mechanical partition function: ˆ ZQM (tb − ta ) ≡ Tr Uˆ (tb , ta ) = Tr e−i(tb −ta )H/¯h .

(1.537)

Obviously the quantum-statistical partition function Z(T ) may be obtained from the quantum-mechanical one by continuing the time interval tb − ta to the negative imaginary value i¯ h tb − ta = − ≡ −i¯ hβ. (1.538) kB T This simple formal relation shows that the trace of the time evolution operator contains all information on the thermodynamic equilibrium properties of a quantum system.

1.17.2

Grand-Canonical Ensemble

For systems containing many bodies it is often convenient to study their equilibrium properties in contact with a particle reservoir characterized by a chemical potential µ. For this one defines what is called the grand-canonical quantum-statistical partition function ˆ ˆ ZG (T, µ) = Tr e−(H−µN )/kB T . (1.539)

78

1 Fundamentals

ˆ is the operator counting the number of particles in each state of the ensemble. Here N The combination of operators in the exponent, ˆG = H ˆ − µN ˆ, H

(1.540)

is called the grand-canonical Hamiltonian. Given a partition function Z(T ) at a fixed particle number N, the free energy is defined by F (T ) = −kB T log Z(T ). (1.541) Its grand-canonical version at a fixed chemical potential is FG (T, µ) = −kB T log ZG (T, µ).

(1.542)

The average energy or internal energy is defined by ˆ BT ˆ −H/k E = Tr He

.

ˆ

(1.543)

F (T ).

(1.545)

Tr e−H/kB T .

It may be obtained from the partition function Z(T ) by forming the temperature derivative ∂ ∂ Z(T ) = kB T 2 log Z(T ). (1.544) E = Z −1 kB T 2 ∂T ∂T In terms of the free energy (1.541), this becomes ∂ ∂ (−F (T )/T ) = 1 − T E=T ∂T ∂T 2

!

For a grand-canonical ensemble we may introduce an average particle number defined by . ˆ ˆ )/kB T ˆ ˆ N ˆ −(H−µ N = Tr Ne Tr e−(H−µN )/kB T . (1.546) This can be derived from the grand-canonical partition function as N = ZG −1 (T, µ)kB T

∂ ∂ ZG (T, µ) = kB T log ZG (T, µ), ∂µ ∂µ

(1.547)

or, using the grand-canonical free energy, as N =−

∂ FG (T, µ). ∂µ

(1.548)

The average energy in a grand-canonical system, ˆ ˆ )/kB T N ˆ −(H−µ E = Tr He

.

ˆ

ˆ

Tr e−(H−µN )/kB T ,

(1.549)

can be obtained by forming, by analogy with (1.544) and (1.545), the derivative E − µN = ZG −1 (T, µ)kB T 2 =

∂ 1−T ∂T

!

∂ ZG (T, µ) ∂T

(1.550)

FG (T, µ). H. Kleinert, PATH INTEGRALS

79

1.17 Classical and Quantum Statistics

For a large number of particles, the density is a rapidly growing function of energy. For a system of N free particles, for example, the number of states up to energy E is given by N(E) =

X pi

Θ(E −

N X

p2i /2M),

(1.551)

i=1

where each of the particle momenta pi is summed over all discrete momenta pm in (1.179) available to a single particle in a finite box of volume V = L3 . For a large V , the sum can be converted into an integral24 N(E) = V

N

" N Z Y

N X d 3 pi Θ(E − p2i /2M), 3 (2π¯ h) i=1

#

i=1 N

which is h)3 ] √ simply [V /(2π¯ radius 2ME:

(1.552)

times the volume Ω3N of a 3N-dimensional sphere of

N(E) =

"

V (2π¯ h)3

#N

≡

"

V (2π¯ h)3

#N

Ω3N (2πME)3N/2 Γ

3 N 2

+1

(1.553) .

Recall the well-known formula for the volume of a unit sphere in D dimensions: ΩD = π D/2 /Γ(D/2 + 1).

(1.554)

The surface is [see Eqs. (8.116) and (8.117) for a derivation] SD = 2π D/2 /Γ(D/2).

(1.555)

Therefore, the density per energy ρ = ∂N /∂E is given by "

V ρ(E) = (2π¯ h)3

#N

2πM

(2πME)3N/2−1 . Γ( 32 N)

(1.556)

It grows with the very large power E 3N/2 in the energy. Nevertheless, the integral for the partition function (1.577) is convergent, due to the overwhelming exponential falloff of the Boltzmann factor, e−E/kB T . As the two functions ρ(e) and e−e/kB T are multiplied with each other, the result is a function which peaks very sharply at the average energy E of the system. The position of the peak depends on the temperature T . For the free N particle system, for example, ρ(E)e−E/kB T ∼ e(3N /2−1) log E−E/kB T .

(1.557)

This function has a sharp peak at E(T ) = kB T 24

3N 3N − 1 ≈ kB T . 2 2

(1.558)

Remember, however, the exception noted in the footnote to Eq. (1.184) for systems possessing a condensate.

80

1 Fundamentals

The width of the peak is found by expanding (1.557) in δE = E − E(T ): (

)

3N E(T ) 1 3N exp log E(T ) − − (δE)2 + . . . . 2 2 kB T 2E (T ) 2

(1.559)

√ Thus, as soon as δE gets to be of the order of E(T )/ N , the exponential is reduced by a factor √ two with respect to E(T ) ≈ kB T 3N/2. The deviation is of a relative order 1/ N , i.e., the√peak is very sharp. With N being very large, the peak at E(T ) of width E(T )/ N can be idealized by a δ-function, and we may write ρ(E)e−E/kB T ≈ δ(E − E(T ))N(T )e−E(T )/kB T .

(1.560)

The quantity N(T ) measures the total number of states over which the system is distributed at the temperature T . The entropy S(T ) is now defined in terms of N(T ) by N(T ) = eS(T )/kB .

(1.561)

Inserting this with (1.560) into (1.577), we see that in the limit of a large number of particles N: Z(T ) = e−[E(T )−T S(T )]/kB T . (1.562) Using (1.541), the free energy can thus be expressed in the form F (T ) = E(T ) − T S(T ).

(1.563)

By comparison with (1.545) we see that the entropy may be obtained from the free energy directly as ∂ (1.564) S(T ) = − F (T ). ∂T For grand-canonical ensembles we may similarly consider ZG (T, µ) =

Z

dE dn ρ(E, n)e−(E−µn)/kB T ,

(1.565)

where ρ(E, n)e−(E−µn)/kB T

(1.566)

is now strongly peaked at E = E(T, µ), n = N(T, µ) and can be written approximately as ρ(E, n)e−(E−µn)/kB T ≈ δ (E − E(T, µ)) δ (n − N(T, µ)) × eS(T,µ)/kB e−[E(T,µ)−µN (T,µ)]/kB T .

(1.567)

Inserting this back into (1.565) we find for large N ZG (T, µ) = e−[E(T,µ)−µN (T,µ)−T S(T,µ)]/kB T .


81

1.18 Density of States and Tracelog

For the grand-canonical free energy (1.542), this implies the relation FG (T, µ) = E(T, µ) − µN(T, µ) − T S(T, µ).

(1.569)

By comparison with (1.550) we see that the entropy can be calculated directly from the derivative of the grand-canonical free energy S(T, µ) = −

∂ FG (T, µ). ∂T

(1.570)

The particle number is, of course, found from the derivative (1.548) with respect to the chemical potential, as follows directly from the definition (1.565). The canonical free energy and the entropy appearing in the above equations depend on the particle number N and the volume V of the system, i.e., they are more explicitly written as F (T, N, V ) and S(T, N, V ), respectively. In the arguments of the grand-canonical quantities, the particle number N is replaced by the chemical potential µ. Among the arguments of the grand-canonical energy FG (T, µ, V ), the volume V is the only one which grows with the system. Thus FG (T, µ, V ) must be directly proportional to V . The proportionality constant defines the pressure p of the system: FG (T, µ, V ) ≡ −p(T, µ, V )V.

(1.571)

Under infinitesimal changes of the three variables, FG (T, µ, V ) changes as follows: dFG (T, µ, V ) = −SdT + µdN − pdV.

(1.572)

The first two terms on the right-hand side follow from varying Eq. (1.569) at a fixed volume. When varying the volume, the definition (1.571) renders the last term. Inserting (1.571) into (1.569), we find Euler’s relation: E = T S + µN − pV.

(1.573)

The energy has S, N, V as natural variables. Equivalently, we may write F = −µN − pV,

(1.574)

where T, N, V are the natural variables.

1.18

Density of States and Tracelog

In many thermodynamic calculations, a quantity of fundamental interest is the density of states To define it, we express the canonical partition function

ˆ

(1.575)

e−En /kB T .

(1.576)

Z(T ) = Tr e−H/kB T

as a sum over the Boltzmann factors of all eigenstates |ni of the Hamiltonian:, i.e. Z(T ) =

X n

82

1 Fundamentals

This can be rewritten as an integral: Z(T ) =

Z

dE ρ(E)e−E/kB T .

(1.577)

The quantity ρ(E) =

X n

δ(E − En )

(1.578)

specifies the density of states of the system in the energy interval (E, E + dE). It may also be written formally as a trace of the density of states operator ρˆ(E): ˆ ρ(E) = Tr ρˆ(E) ≡ Tr δ(E − H).

(1.579)

The density of states is obviously the Fourier transform of the canonical partition function (1.575): ρ(E) =

Z

∞

−i∞

Z ∞ dβ βE dβ βE ˆ e Tr e−β H = e Z(1/kB β). 2πi −i∞ 2πi

The integral N(E) =

E

Z

dE ′ ρ(E ′ )

(1.580)

(1.581)

is the number of states up to energy E. The integration may start anywhere below the ground state energy. The function N(E) is a sum of Heaviside step functions (1.309): X Θ(E − En ). (1.582) N(E) = n

This equation is correct only with the Heaviside function which is equal to 1/2 at the origin, not with the one-sided version (1.302), as we shall see later. Indeed, if integrated to the energy of a certain level En , the result is N(En ) = (n + 1/2).

(1.583)

This formula will serve to determine the energies of bound states from approximations to ω(E) in Section 4.7, for instance from the Bohr-Sommerfeld condition (4.184) via the relation (4.204). In order to apply this relation one must be sure that all levels have different energies. Otherwise N(E) jumps at En by half the degeneracy of this level. In Eq. (4A.9) we shall exhibit an example for this situation. An important quantity related to ρ(E) which will appear frequently in this text ˆ − E. is the trace of the logarithm, short tracelog, of the operator H ˆ − E) = Tr log(H

X n

log(En − E).

(1.584)

It may be expressed in terms of the density of states (1.579) as ˆ − E) = Tr Tr log(H

Z

∞

−∞

ˆ log(E ′ − E) = dE ′ δ(E ′ − H)

Z

∞

−∞

dE ′ ρ(E ′ ) log(E ′ − E).

(1.585)


Appendix 1A

83

Simple Time Evolution Operator

The tracelog of the Hamiltonian operator itself can be viewed as a limit of an operator ˆ zeta function associated with H: ˆ −ν , ζˆHˆ (ν) = Tr H

(1.586)

whose trace is the generalized zeta-function ˆ −ν ) = ζHˆ (ν) ≡ Tr ζˆHˆ (ν) = Tr (H h

i

X

En−ν .

(1.587)

n

For a linearly spaced spectrum En = n with n = 1, 2, 3 . . . , this reduces to Riemann’s zeta function (2.513). From the generalized zeta function we can obtain the tracelog by forming the derivative ˆ = −∂ν ζ ˆ (ν)| . Tr log H H ν=0

(1.588)

By differentiating the tracelog (1.584) with respect to E, we find the trace of the resolvent (1.315): ˆ − E) = Tr ∂E Tr log(H

X 1 X 1 1 1 ˆ = Rn (E) = Tr R(E). = ˆ E − E i¯ h i¯ h E −H n n n

(1.589)

Recalling Eq. (1.325) we see that the imaginary part of this quantity slightly above the real E-axis yields the density of states X 1 ˆ − E − iη) = − Im ∂E Tr log(H δ(E − En ) = ρ(E). π n

(1.590)

By integrating this over the energy we obtain the number of state function N(E) of Eq. (1.581): X 1 ˆ = − Im Tr log(E − H) Θ(E − En ) = N(E). (1.591) π n

Appendix 1A

Simple Time Evolution Operator

Consider the simplest nontrivial time evolution operator of a spin-1/2 particle in a magnetic field ˆ 0 = −B · /2, so that the time evolution operator reads, B. The reduced Hamiltonian operator is H in natural units with h ¯ = 1, ˆ e−iH0 (tb −ta ) = ei(tb −ta )B· /2 . (1A.1) Expanding this as in (1.293) and using the fact that (B· )2n = B 2n and (B· )2n+1 = B 2n (B· ), we obtain ˆ ˆ · sin B(tb − ta )/2 , e−iH0 (tb −ta ) = cos B(tb − ta )/2 + iB (1A.2) ˆ ≡ B/|B|. Suppose now the magnetic field is not constant but has a small time-dependent where B variation δB(t). Then we obtain from (1.253) [or the lowest expansion term in (1.293)] δe

ˆ 0 (tb −ta ) −iH

=

Z

tb

ta

ˆ

ˆ

dt e−iH0 (tb −t) δB(t) · e−iH0 (t−ta ) .

(1A.3)

84

1 Fundamentals

Using (1A.2), the integrand on the right-hand side becomes h h i ˆ· ˆ · sin B(tb −t)/2 δB(t) · cos B(t−ta )/2+iB cos B(tb −t)/2+iB We simplify this with the help of the formula [recall (1.445)]

i sin B(t−ta )/2 . (1A.4)

σ i σ j = δij + iǫijk σ k

(1A.5)

so that ˆ· B

δB(t) ·

ˆ · δB(t) + i[B ˆ × δB(t)] · , δB(t) · =B

ˆ· B

ˆ · δB(t) − i[B ˆ × δB(t)] · , (1A.6) =B

and ˆ· B

δB(t) ·

ˆ· B

h i ˆ · δB(t) B ˆ· = B

ˆ × δB(t)] · B ˆ· + i[B n o ˆ × δB(t)] · B ˆ + [B ˆ · δB(t)]B ˆ − [B ˆ × δB(t)] × B ˆ · . (1A.7) = i[B

ˆ 2 = 1. Thus The first term on the right-hand side vanishes, the second term is equal to δB, since B we find for the integrand in (1A.4): cos B(tb − t)/2 cos B(t − ta )/2 δB(t) · ˆ · δB(t) + i[B ˆ × δB(t)] · } +i sin B(tb − t)/2 cos B(t − ta )/2{B ˆ · δB(t) − i[B ˆ × δB(t)] · } +i cos B(tb − t)/2 sin B(t − ta )/2{B + sin B(tb − t)/2 sin B(t − ta )/2 δB ·

(1A.8)

which can be combined to n o ˆ × δB(t)] · +i sin B(tb −ta )/2 B·δB(t).(1A.9) ˆ cos B[(tb +ta )/2−t] δB(t)−sin B[(tb +ta )/2−t] [B Integrating this from ta to tb we obtain the to variation (1A.3).

Appendix 1B

Convergence of Fresnel Integral

Here we prove the convergence of the Fresnel integrals (1.333) by relating it to the Gauss integral. According to Cauchy’s integral theorem, the sum of the integrals along the three pieces of the

Figure 1.4 Triangular closed contour for Cauchy integral 2

closed contour shown in Fig. 1.4 vanishes since the integrand e−z is analytic in the triangular domain: I Z A Z B Z O 2 −z 2 −z 2 −z 2 dze = dze + dze + dze−z = 0. (1B.1) 0

A

B

Let R be the radius of the arc. Then we substitute in the three integrals the variable z as follows: H. Kleinert, PATH INTEGRALS

Appendix 1C

85

The Asymmetric Top 0 A: B 0: AB:

dz = dp, z 2 = p2 iπ/4 dz = dp e , z 2 = ip2 dz = i Rdp, z 2 = p2 ,

z = p, z = peiπ/4 , z = R eiϕ ,

and obtain the equation Z

R

2

dp e−p + eiπ/4

0

Z

0

2

dp e−ip +

R

Z

π/4

2

dϕ iR e−R

(cos 2ϕ+i sin 2ϕ)+iϕ

= 0.

(1B.2)

0

√ The first integral converges rapidly to π/2 for R → ∞. The last term goes to zero in this limit. To see this we estimate its absolute value as follows: Z Z π/4 π/4 2 −R2 (cos 2ϕ+i sin 2ϕ)+iϕ dϕ iR e dϕ e−R cos 2ϕ . (1B.3) 0, one certainly has sin 2ϕ > sin 2α, so that R

Z

π/4

dϕ e

−R2 cos 2ϕ

α

ta .

(2.1)

For simplicity, we shall at first assume the space to be one-dimensional. The extension to D Cartesian dimensions will be given later. The introduction of curvilinear coordinates will require a little more work. A further generalization to spaces with a nontrivial geometry, in which curvature and torsion are present, will be described in Chapters 10–11.

2.1.1

Sliced Time Evolution Amplitude

We shall be interested mainly in the causal or retarded time evolution amplitudes [see Eq. (1.299)]. These contain all relevant quantum-mechanical information and 1

For the historical development, see Notes and References at the end of this chapter.

89

90

2 Path Integrals — Elementary Properties and Simple Solutions

possess, in addition, pleasant analytic properties in the complex energy plane [see the remarks after Eq. (1.306)]. This is why we shall always assume, from now on, the causal sequence of time arguments tb > ta . Feynman realized that due to the fundamental composition law of the time evolution operator (see Section 1.7), the amplitude (2.1) could be sliced into a large number, say N + 1, of time evolution operators, each acting across an infinitesimal time slice of width ǫ ≡ tn − tn−1 = (tb − ta )/(N + 1)> 0: ˆ b , tN )U(t ˆ N , tN −1 ) · · · U(t ˆ n , tn−1 ) · · · U(t ˆ 2 , t1 )Uˆ (t1 , ta )|xa i. (2.2) (xb tb |xa ta ) = hxb |U(t When inserting a complete set of states between each pair of Uˆ ’s, Z

∞

−∞

dxn |xn ihxn | = 1,

n = 1, . . . , N,

(2.3)

the amplitude becomes a product of N-integrals (xb tb |xa ta ) =

N Z Y

∞

−∞

n=1

dxn

NY +1 n=1

(xn tn |xn−1 tn−1 ),

(2.4)

where we have set xb ≡ xN +1 , xa ≡ x0 , tb ≡ tN +1 , ta ≡ t0 . The symbol Π[· · ·] denotes the product of the integrals within the brackets. The integrand is the product of the amplitudes for the infinitesimal time intervals ˆ

(xn tn |xn−1 tn−1 ) = hxn |e−iǫH(tn )/¯h |xn−1 i,

(2.5)

with the Hamiltonian operator ˆ H(t) ≡ H(ˆ p, xˆ, t).

(2.6)

The further development becomes simplest under the assumption that the Hamiltonian has the standard form, being the sum of a kinetic and a potential energy: H(p, x, t) = T (p, t) + V (x, t).

(2.7)

For a sufficiently small slice thickness, the time evolution operator ˆ

ˆ

ˆ

e−iǫH/¯h = e−iǫ(T +V )/¯h

(2.8)

is factorizable as a consequence of the Baker-Campbell-Hausdorff formula (to be proved in Appendix 2A) ˆ

ˆ

ˆ

ˆ

e−iǫ(T +V )/¯h = e−iǫV /¯h e−iǫT /¯h e−iǫ

2 X/¯ ˆ h2

,

(2.9)

ˆ has the expansion where the operator X ˆ ≡ i [Vˆ , Tˆ ] − ǫ 1 [Vˆ , [Vˆ , Tˆ ]] − 1 [[Vˆ , Tˆ ], Tˆ] + O(ǫ2 ) . X 2 h ¯ 6 3

(2.10)


91

2.1 Path Integral Representation of Time Evolution Amplitudes

The omitted terms of order ǫ4 , ǫ5 , . . . contain higher commutators of Vˆ and Tˆ . If we ˆ neglect, for the moment, the X-term which is suppressed by a factor ǫ2 , we calculate ˆ for the local matrix elements of e−iǫH/¯h the following simple expression: Z

hxn |e−iǫH(ˆp,ˆx,tn )/¯h |xn−1 i ≈ =

Z

∞ −∞

∞

−∞

−iǫV (ˆ x,tn )/¯ h

dxhxn |e

dxhxn |e−iǫV (ˆx,tn )/¯h |xihx|e−iǫT (ˆp,tn )/¯h |xn−1 i

|xi

Z

dpn ipn (x−xn−1 )/¯h −iǫT (pn ,tn )/¯h e e . 2π¯ h

∞

−∞

(2.11)

Evaluating the local matrix elements, hxn |e−iǫV (ˆx,tn )/¯h |xi = δ(xn − x)e−iǫV (xn ,tn )/¯h ,

(2.12)

this becomes hxn |e−iǫH(ˆp,ˆx,tn )/¯h |xn−1 i ≈ Z

∞

−∞

dpn exp {ipn (xn − xn−1 )/¯ h − iǫ[T (pn , tn ) + V (xn , tn )]/¯ h} . 2π¯ h

(2.13)

Inserting this back into (2.4), we obtain Feynman’s path integral formula, consisting of the multiple integral N Z Y

(xb tb |xa ta ) ≈

dxn

" NY +1 Z n=1

∞ −∞

#

dpn i N exp A , 2π¯ h h ¯

N +1 X

AN =

2.1.2

−∞

n=1

where AN is the sum

∞

n=1

[pn (xn − xn−1 ) − ǫH(pn , xn , tn )].

(2.14)

(2.15)

Zero-Hamiltonian Path Integral

Note that the path integral (2.14) with zero Hamiltonian produces the Hilbert space structure of the theory via a chain of scalar products: (xb tb |xa ta ) ≈

N Z Y

n=1

∞ −∞

dxn

" NY +1 Z n=1

∞

−∞

#

dpn i PN+1 pn (xn −xn−1 )/¯h e n=1 , 2π¯ h

(2.16)

which is equal to (xb tb |xa ta ) ≈

N Z Y

n=1

∞

−∞

dxn

NY +1 n=1

= δ(xb − xa ),

hxn |xn−1 i =

N Z Y

n=1

∞

−∞

dxn

NY +1 n=1

δ(xn − xn−1 ) (2.17)

whose continuum limit is (xb tb |xa ta ) =

Z

Dx

Z

Dp i R dtp(t)x(t)/¯ ˙ h e = hxb |xa i = δ(xb − xa ). 2π¯ h

(2.18)

92


At this point we make the important observation that a momentum variable pn inside the time-sliced version (2.17) at the time tn can be generated by a derivative pˆn ≡ −i¯ h∂xn . In the continuum limit (2.18), this becomes an operator p(t) = −i¯ h∂x(t) . Its commutator with x(t) is [ˆ p(t), x(t)] = −i¯ h, (2.19) which is the famous equal-time canonical commutation rule of Heisenberg. This observation forms the basis for deriving, from the path integral (2.14), the Schrödinger equation for the time evolution amplitude.

2.1.3

Schr¨ odinger Equation for Time Evolution Amplitude

Let us split from the product of integrals in (2.14) the final time slice as a factor, so that we obtain the recursion relation (xb tb |xa ta ) ≈

∞

Z

−∞

where

dxN (xb tb |xN tN ) (xN tN |xa ta ),

(2.20)

dpn (i/¯h)[pb (xb −xN )−ǫH(pb ,xb ,tb )] . (2.21) e h −∞ 2π¯ The momentum pb inside the integral can be generated by a differential operator pˆb ≡ −i¯ h∂xb . The same is true for any function of pb , so that the Hamiltonian can be moved before the momentum integral as follows: (xb tb |xN tN ) ≈

Z

∞

dpn ipb (xb −xN )/¯h −iǫH(−i¯h∂x ,xb,tb )/¯h b e =e δ(xb −xN ). h −∞ 2π¯ (2.22) Inserting this back into (2.20) we obtain Z

−iǫH(−i¯ h∂xb ,xb ,tb )/¯ h

(xb tb |xN tN ) ≈ e

∞

(xb tb |xa ta ) ≈ e−iǫH(−i¯h∂xb ,xb ,tb )/¯h (xb tb −ǫ|xa ta ),

(2.23)

i 1 1 h −iǫH(−i∂x ,xb ,tb )/¯h b e − 1 (xb tb |xa ta ). [(xb tb +ǫ|xa ta ) − (xb tb |xa ta )] ≈ ǫ ǫ

(2.24)

or

In the limit ǫ → 0, this goes over into the differential equation for the time evolution amplitude i¯ h∂tb (xb tb |xa ta ) = H(−i¯ h∂xb , xb , tb )(xb tb |xa ta ),

(2.25)

which is precisely the Schrödinger equation (1.297) of operator quantum mechanics.

2.1.4

Convergence of Sliced Time Evolution Amplitude

Some remarks are necessary concerning the convergence of the time-sliced expression (2.14) to the quantum-mechanical amplitude in the continuum limit, where the thickness of the time slices ǫ = (tb − ta )/(N + 1) → 0 goes to zero and the number H. Kleinert, PATH INTEGRALS

93


N of slices tends to ∞. This convergence can be proved for the standard kinetic energy T = p2 /2M only if the potential V (x, t) is sufficiently smooth. For timeindependent potentials this is a consequence of the Trotter product formula which reads ˆ

e−i(tb −ta )H/¯h = lim

N →∞

ˆ

ˆ

e−iǫV /¯h e−iǫT /¯h

N +1

.

(2.26)

For c-numbers T and V , this is trivially true. For operators Tˆ, Vˆ , we use Eq. (2.9) to rewrite the left-hand side of (2.26) as ˆ

ˆ

ˆ

e−i(tb −ta )H/¯h ≡ e−iǫ(T +V )/¯h

N +1

ˆ

ˆ

≡ e−iǫV /¯h e−iǫT /¯h e−iǫ

2 X/¯ ˆ h2

N +1

.

ˆ proportional to ǫ2 does The Trotter formula implies that the commutator term X not contribute in the limit N → ∞. The mathematical conditions ensuring this require functional analysis too technical to be presented here (for details, see the literature quoted at the end of the chapter). For us it is sufficient to know that the Trotter formula holds for operators which are bounded from below and that for most physically interesting potentials, it cannot be used to derive Feynman’s time-sliced path integral representation (2.14), even in systems where the formula is known to be valid. In particular, the short-time amplitude may be different from (2.13). Take, for example, an attractive Coulomb potential V (x) ∝ −1/|x| for which the Trotter formula has been proved to be valid. Feynman’s time-sliced formula, however, diverges even for two time slices. This will be discussed in detail in Chapter 12. Similar problems will be found for other physically relevant potentials ¯ 2 /sin2 θ such as V (x) ∝ l(l + D − 2)¯ h2 /|x|2 (centrifugal barrier) and V (θ) ∝ m2 h (angular barrier near the poles of a sphere). In all these cases, the commutators ˆ become more and more singular. In fact, as we shall in the expansion (2.10) of X see, the expansion does not even converge, even for an infinitesimally small ǫ. All atomic systems contain such potentials and the Feynman formula (2.14) cannot be used to calculate an approximation for the transition amplitude. A new path integral formula has to be found. This will be done in Chapter 12. Fortunately, it is possible to eventually reduce the more general formula via some transformations back to a Feynman type formula with a bounded potential in an auxiliary space. Thus the above derivation of Feynman’s formula for such potentials will be sufficient for the further development in this book. After this it serves as an independent starting point for all further quantum-mechanical calculations. In the sequel, the symbol ≈ in all time-sliced formulas such as (2.14) will imply that an equality emerges in the continuum limit N → ∞, ǫ → 0 unless the potential has singularities of the above type. In the action, the continuum limit is without subtleties. The sum AN in (2.15) tends towards the integral A[p, x] =

Z

tb

ta

dt [p(t)x(t) ˙ − H(p(t), x(t), t)]

(2.27)

under quite general circumstances. This expression is recognized as the classical canonical action for the path x(t), p(t) in phase space. Since the position variables

94


xN +1 and x0 are fixed at their initial and final values xb and xa , the paths satisfy the boundary condition x(tb ) = xb , x(ta ) = xa . In the same limit, the product of infinitely many integrals in (2.14) will be called a path integral . The limiting measure of integration is written as lim

N →∞

N Z Y

n=1

∞

−∞

dxn

" NY +1 Z

∞

−∞

n=1

#

dpn ≡ 2π¯ h

Z

x(tb )=xb

x(ta )=xa

′

Dx

Z

Dp . 2π¯ h

(2.28)

By definition, there is always one more pn -integral than xn -integrals in this product. While x0 and xN +1 are held fixed and the xn -integrals are done for n = 1, . . . , N, each pair (xn , xn−1 ) is accompanied by one pn -integral for n = 1, . . . , N +1. The situation is recorded by the prime on the functional integral D ′ x. With this definition, the amplitude can be written in the short form (xb tb |xa ta ) =

Z

x(tb )=xb

x(ta )=xa

D′x

Z

Dp iA[p,x]/¯h e . 2π¯ h

(2.29)

The path integral has a simple intuitive interpretation: Integrating over all paths corresponds to summing over all histories along which a physical system can possibly evolve. The exponential eiA[p,x]/¯h is the quantum analog of the Boltzmann factor e−E/kB T in statistical mechanics. Instead of an exponential probability, a pure phase factor is assigned to each possible history: The total amplitude for going from xa , ta to xb , tb is obtained by adding up the phase factors for all these histories, (xb tb |xa ta ) =

X

eiA[p,x]/¯h ,

(2.30)

all histories (xa ,ta ) ; (xb ,tb )

where the sum comprises all paths in phase space with fixed endpoints xb , xa in x-space.

2.1.5

Time Evolution Amplitude in Momentum Space

The above observed asymmetry in the functional integrals over x and p is a consequence of keeping the endpoints fixed in position space. There exists the possibility of proceeding in a conjugate way keeping the initial and final momenta pb and pa fixed. The associated time evolution amplitude can be derived going through the same steps as before but working in the momentum space representation of the Hilbert space, starting from the matrix elements of the time evolution operator ˆ b , ta )|pa i. (pb tb |pa ta ) ≡ hpb |U(t

(2.31)

The time slicing proceeds as in (2.2)–(2.4), with all x’s replaced by p’s, except in the completeness relation (2.3) which we shall take as Z

∞

−∞

dp |pihp| = 1, 2π¯ h


95


corresponding to the choice of the normalization of states [compare (1.186)] hδ(pb − pa ). hpb |pa i = 2π¯

(2.33)

In the resulting product of integrals, the integration measure has an opposite asymmetry: there is now one more xn -integral than pn -integrals. The sliced path integral reads (pb tb |pa ta ) ≈

"Z N Y

N Z ∞ dpn Y dxn 2π¯ h n=0 −∞ #

∞

−∞

n=1

N i X × exp [−xn (pn+1 − pn ) − ǫH(pn , xn , tn )] . h ¯ n=0

(

)

(2.34)

The relation between this and the x-space amplitude (2.14) is simple: By taking in (2.14) the first and last integrals over p1 and pN +1 out of the product, renaming P +1 them as pa and pb , and rearranging the sum N n=1 pn (xn − xn−1 ) as follows N +1 X n=1

pn (xn − xn−1 ) = pN +1 (xN +1 − xN ) + pN (xN − xN −1 ) + . . . . . . + p2 (x2 − x1 ) + p1 (x1 − x0 ) = pN +1 xN +1 − p1 x0 −(pN +1 − pN )xN − (pN − pN −1 )xN −1 − . . . − (p2 − p1 )x1 = pN +1 xN +1 − p1 x0 −

N X

(pn+1 − pn )xn ,

(2.35)

n=1

the remaining product of integrals looks as in Eq. (2.34), except that the lowest index n is one unit larger than there. In the limit N → ∞ this does not matter, and we obtain the Fourier transform (xb tb |xa ta ) =

Z

dpb ipb xb /¯h e 2π¯ h

Z

dxb e−ipb xb /¯h

Z

dpa −ipa xa /¯h e (pb tb |pa ta ). 2π¯ h

(2.36)

dxa eipa xa /¯h (xb tb |xa ta ).

(2.37)

The inverse relation is (pb tb |pa ta ) =

Z

In the continuum limit, the amplitude (2.34) can be written as a path integral (pb tb |pa ta ) =

Z

p(tb )=pb

p(ta )=pa

D′p 2π¯ h

Z

¯

DxeiA[p,x]/¯h,

(2.38)

where ¯ x] = A[p,

Z

tb

ta

dt [−p(t)x(t) ˙ − H(p(t), x(t), t)] = A[p, x] − pb xb + pa xa .

(2.39)

96


2.1.6

Quantum-Mechanical Partition Function

A path integral symmetric in p and x arises when considering the quantummechanical partition function defined by the trace (recall Section 1.17) ˆ

ZQM (tb , ta ) = Tr e−i(tb −ta )H/¯h .

(2.40)

In the local basis, the trace becomes an integral over the amplitude (xb tb |xa ta ) with xb = xa : ZQM (tb , ta ) =

∞

Z

−∞

dxa (xa tb |xa ta ).

(2.41)

The additional trace integral over xN +1 ≡ x0 makes the path integral for ZQM symmetric in pn and xn : Z

∞

−∞

dxN +1

N Z Y

∞

−∞

n=1

dxn

" NY +1 Z

∞

−∞

n=1

NY +1 dpn = 2π¯ h n=1

#

"ZZ

∞

−∞

#

dxn dpn . 2π¯ h

(2.42)

In the continuum limit, the right-hand side is written as lim

N →∞

NY +1 n=1

"ZZ

#

I Z dxn dpn Dp ≡ Dx , 2π¯ h 2π¯ h

∞

−∞

(2.43)

and the measures are related by Z

∞

−∞

dxa

x(tb )=xb

Z

x(ta )=xa

D′x

Dp ≡ 2π¯ h

Z

I

Dx

Dp . 2π¯ h

Z

(2.44)

H

The symbol indicates the periodic boundary condition x(ta ) = x(tb ). In the momentum representation we would have similarly Z

∞ −∞

dpa 2π¯ h

Z

p(tb )=pb

p(ta )=pa

D′p 2π¯ h

Z

Dx ≡

I

Dp 2π¯ h

Z

Dx ,

(2.45)

with the periodic boundary condition p(ta ) = p(tb ), and the same right-hand side. Hence, the quantum-mechanical partition function is given by the path integral ZQM (tb , ta ) =

I

Dx

Z

Dp iA[p,x]/¯h I Dp Z ¯ e = DxeiA[p,x]/¯h. 2π¯ h 2π¯ h

(2.46)

¯ x] can be replaced by A[p, x], since In the right-hand exponential, the action A[p, the extra terms in (2.39) are removed by the periodic boundary conditions. In the time-sliced expression, the equality is easily derived from the rearrangement of the sum (2.35), which shows that N +1 X n=1

pn (xn

− xn−1 )

xN+1 =x0

= −

N X

(pn+1

n=0

− pn )xn

.

(2.47)

pN+1 =p0


97


In the path integral expression (2.46) for the partition function, the rules of quantum mechanics appear as a natural generalization of the rules of classical statistical mechanics, as formulated by Planck. According to these rules, each volume element in phase space dxdp/h is occupied with the exponential probability e−E/kB T . In the path integral formulation of quantum mechanics, each volume element in the path Q phase space n dx(tn )dp(tn )/h is associated with a pure phase factor eiA[p,x]/¯h. We see here a manifestation of the correspondence principle which specifies the transition from classical to quantum mechanics. In path integrals, it looks somewhat more natural than in the historic formulation, where it requires the replacement of all classical phase space variables p, x by operators, a rule which was initially hard to comprehend.

2.1.7

Feynman’s Configuration Space Path Integral

Actually, in his original paper, Feynman did not give the path integral formula in the above phase space formulation. Since the kinetic energy in (2.7) has usually the form T (p, t) = p2 /2M, he focused his attention upon the Hamiltonian H=

p2 + V (x, t), 2M

(2.48)

for which the time-sliced action (2.15) becomes N

A =

N +1 X n=1

"

p2 pn (xn − xn−1 ) − ǫ n − ǫV (xn , tn ) . 2M #

(2.49)

It can be quadratically completed to N

A =

N +1 X n=1

"

2

ǫ xn − xn−1 − pn − M 2M ǫ

M xn − xn−1 + ǫ 2 ǫ

#

2

− ǫV (xn , tn ) . (2.50)

The momentum integrals in (2.14) may then be performed using the Fresnel integral formula (1.333), yielding Z

∞

−∞

"

dpn i ǫ xn − xn−1 exp − pn − M 2π¯ h h ¯ 2M ǫ

2 #

and we arrive at the alternative representation 1

(xb tb |xa ta ) ≈ q 2π¯ hiǫ/M

N Y

n=1

where AN is now the sum N

A =ǫ

N +1 X n=1

"

M 2

 Z 

dxn

∞

−∞

1 =q , 2π¯ hiǫ/M

q

xn − xn−1 ǫ

2π¯ hiǫ/M

2



 exp

#

− V (xn , tn ) ,

i N A , h ¯

(2.51)

(2.52)

(2.53)

98


Figure 2.1 Zigzag paths, along which a point particle explores all possible ways of reaching the point xb at a time tb , starting from xa at ta . The time axis is drawn from right to left to have the same direction as the operator order in Eq. (2.2).

with xN +1 = xb and x0 = xa . Here the integrals run over all paths in configuration space rather than phase space. They account for the fact that a quantum-mechanical particle starting from a given initial point xa will explore all possible ways of reaching a given final point xb . The amplitude of each path is exp(iAN /¯ h). See Fig. 2.1 for a geometric illustration of the path integration. In the continuum limit, the sum (2.53) converges towards the action in the Lagrangian form: A[x] =

tb

Z

ta

dtL(x, x) ˙ =

Z

tb

ta

M 2 dt x˙ − V (x, t) . 2

(2.54)

Note that this action is a local functional of x(t) in the temporal sense as defined in Eq. (1.27).2 For the time-sliced Feynman path integral, one verifies the Schrödinger equation as follows: As in (2.20), one splits off the last slice as follows: (xb tb |xa ta ) ≈ =

Z

∞

−∞ Z ∞ −∞

dxN (xb tb |xN tN ) (xN tN |xa ta ) d∆x (xb tb |xb −∆x tb −ǫ) (xb −∆x tb −ǫ|xa ta ),

(2.55)

where 1

2

(

"

i M (xb tb |xb −∆x tb −ǫ) ≈ q exp ǫ h ¯ 2 2π¯ hiǫ/M

∆x ǫ

2

A functional F [x] is called R local if it can be written as an integral ultra-local if it has the form dtf (x(t)).

− V (xb , tb ) R

#)

.

(2.56)

dtf (x(t), x(t)); ˙ it is called


99


We now expand the amplitude in the integral of (2.55) in a Taylor series 1 (xb −∆x tb −ǫ|xa ta ) = 1 − ∆x ∂xb + (∆x)2 ∂x2b + . . . (xb , tb −ǫ|xa ta ). 2

(2.57)

Inserting this into (2.55), the odd powers of ∆x do not contribute. For the even powers, we perform the integrals using the Fresnel version of formula (1.335), and obtain zero for odd powers of ∆x, and Z

d∆x

∞

−∞

q

2π¯ hiǫ/M

(∆x)

2n

(

iM exp ǫ h ¯ 2

∆x ǫ

2 )

h ¯ǫ = i M

!n

(2.58)

for even powers, so that the integral in (2.55) becomes "

i¯ h 2 ∂ + O(ǫ2 ) (xb tb |xa ta ) = 1 + ǫ 2M xb

#

i 1 − ǫ V (xb , tb ) + O(ǫ2 ) (xb , tb −ǫ|xa ta ). (2.59) h ¯

In the limit ǫ → 0, this yields again the Schrödinger equation. (2.23). In the continuum limit, we write the amplitude (2.52) as a path integral (xb tb |xa ta ) ≡

Z

x(tb )=xb

x(ta )=xa

Dx eiA[x]/¯h .

(2.60)

This is Feynman’s original formula for the quantum-mechanical amplitude (2.1). It consists of a sum over all paths in configuration space with a phase factor containing the form of the action A[x]. We have used the same measure symbol Dx for the paths in configuration space as for the completely different paths in phase space in the expressions (2.29), (2.38), (2.44), (2.45). There should be no danger of confusion. Note that the q extra dpn integration in the phase space formula (2.14) results now in one extra 1/ 2π¯ hiǫ/M factor in (2.52) which is not accompanied by a dxn -integration. The Feynman amplitude can be used to calculate the quantum-mechanical partition function (2.41) as a configuration space path integral ZQM ≈

I

Dx eiA[x]/¯h .

(2.61)

As in (2.43), (2.44), the symbol Dx indicates that the paths have equal endpoints x(ta ) = x(tb ), the path integral being the continuum limit of the product of integrals H

I q

Dx ≈

NY +1 Z ∞ n=1

−∞

dxn q

2πi¯ hǫ/M

.

(2.62)

There is no extra 1/ 2πi¯ hǫ/M factor as in (2.52) and (2.60), due to the integration over the initial (= final) position Hxb = xa representing the quantum-mechanical trace. The use of the same symbol Dx as in (2.46) should not cause any confusion R since (2.46) is always accompanied by an integral Dp.

100


For the sake of generality we might point out that it is not necessary to slice the time axis in an equidistant way. In the continuum limit N → ∞, the canonical path integral (2.14) is indifferent to the choice of the infinitesimal spacings ǫn = tn − tn−1 .

(2.63)

The configuration space formula contains the different spacings ǫn in the following way: When performing the pn integrations, we obtain a formula of the type (2.52), with each ǫ replaced by ǫn , i.e., 1

(xb tb |xa ta ) ≈ q 2π¯ hiǫb /M (

i × exp h ¯

N Y

n=1

" N +1 X M n=1

 Z 

dxn

∞

−∞

q

2πi¯ hǫn /M

 

(xn − xn−1 )2 − ǫn V (xn , tn ) 2 ǫn

#)

.

(2.64)

To end this section, an important remark is necessary: It would certainly be possible to define the path integral for the time evolution amplitude (2.29) without going through Feynman’s time-slicing procedure as the solution of the Schrödinger differential equation [see Eq. (1.304))]: ˆ h∂t ](x t|xa ta ) = −i¯ hδ(t − ta )δ(x − xa ). [H(−i¯ h∂x , x) − i¯

(2.65)

If one possesses an orthonormal and complete set of wave functions ψn (x) solving ˆ n (x)=En ψn (x), this solution is given the time-independent Schrödinger equation Hψ by the spectral representation (1.319) (xb tb |xa ta ) = Θ(tb − ta )

X

ψn (xb )ψn∗ (xa )e−iEn (tb −ta )/¯h ,

(2.66)

n

where Θ(t) is the Heaviside function (1.300). This definition would, however, run contrary to the very purpose of Feynman’s path integral approach, which is to understand a quantum system from the global all-time fluctuation point of view. The goal is to find all properties from the globally defined time evolution amplitude, in particular the Schrödinger wave functions.3 The global approach is usually more complicated than Schrödinger’s and, as we shall see in Chapters 8 and 12–14, contains novel subtleties caused by the finite time slicing. Nevertheless, it has at least four important advantages. First, it is conceptually very attractive by formulating a quantum theory without operators which describe quantum fluctuations by close analogy with thermal fluctuations (as will be seen later in this chapter). Second, it links quantum mechanics smoothly with classical mechanics (as will be shown in Chapter 4). Third, it offers new variational procedures for the approximate study of complicated quantum-mechanical and -statistical systems (see Chapter 5). Fourth, 3

Many publications claiming to have solved the path integral of a system have violated this rule by implicitly using the Schrödinger equation, although camouflaged by complicated-looking path integral notation. H. Kleinert, PATH INTEGRALS

101

2.2 Exact Solution for Free Particle

it gives a natural geometric access to the dynamics of particles in spaces with curvature and torsion (see Chapters 10–11). This has recently led to results where the operator approach has failed due to operator-ordering problems, giving rise to a unique and correct description of the quantum dynamics of a particle in spaces with curvature and torsion. From this it is possible to derive a unique extension of Schrödinger’s theory to such general spaces whose predictions can be tested in future experiments.4

2.2

Exact Solution for Free Particle

In order to develop some experience with Feynman’s path integral formula we consider in detail the simplest case of a free particle, which in the canonical form reads (xb tb |xa ta ) =

Z

x(tb )=xb

′

Dx

x(ta )=xa

Z

"

i Dp exp 2π¯ h h ¯

tb

Z

ta

p2 dt px˙ − 2M

!#

,

(2.67)

and in the pure configuration form: (xb tb |xa ta ) =

Z

x(tb )=xb

x(ta )=xa

i Dx exp h ¯

Z

tb

ta

M dt x˙ 2 . 2

(2.68)

Since the integration limits are obvious by looking at the left-hand sides of the equations, they will be omitted from now on, unless clarity requires their specification.

2.2.1

Direct Solution

The problem is solved most easily in the configuration form. The time-sliced expression to be integrated is given by Eqs. (2.52), (2.53) where we have to set V (x) = 0. The resulting product of Gaussian integrals can easily be done successively using formula (1.333), from which we derive the simple rule Z

i M (x′′ − x′ )2 1 i M (x′ − x)2 q dx q exp exp h ¯ 2 Aǫ h ¯ 2 Bǫ 2πi¯ hAǫ/M 2πi¯ hBǫ/M ′

"

1

#

"

i M (x′′ − x)2 exp =q , h ¯ 2 (A + B)ǫ 2πi¯ h(A + B)ǫ/M "

1

#

#

(2.69)

which leads directly to the free-particle amplitude

i M (xb − xa )2 q (xb tb |xa ta ) = exp . h ¯ 2 (N + 1)ǫ 2πi¯ h(N + 1)ǫ/M 1

After inserting (N + 1)ǫ = tb − ta , this reads

#

i M (xb − xa )2 q . (xb tb |xa ta ) = exp h ¯ 2 tb − ta 2πi¯ h(tb − ta )/M 1

4

"

"

#

(2.70)

(2.71)

H. Kleinert, Mod. Phys. Lett. A 4 , 2329 (1989) (http://www.physik.fu-berlin.de/~kleinert/199); Phys. Lett. B 236 , 315 (1990) (ibid.http/202).

102


Note that the free-particle amplitude happens to be independent of the number N + 1 of time slices. The amplitude (2.71) agrees, of course, with the Schrödinger result (1.337) for D = 1.

2.2.2

Fluctuations around Classical Path

There exists another method of calculating this amplitude which is somewhat more involved than the simple case at hand, but which turns out to be useful for the treatment of a certain class of nontrivial path integrals, after a suitable generalization. This method is based on all paths with respect to the classical path, i.e., all paths are split into the classical path xcl (t) = xa +

xb − xa (t − ta ), tb − ta

(2.72)

along which the free particle would run following the equation of motion x¨cl (t) = 0,

(2.73)

x(t) = xcl (t) + δx(t).

(2.74)

plus deviations δx(t):

Since initial and final points are fixed at xa , xb , respectively, the deviations vanish at the endpoints: δx(ta ) = δx(tb ) = 0.

(2.75)

The deviations δx(t) are referred to as the quantum fluctuations of the particle orbit. In mathematics, the boundary conditions (2.75) are referred to as Dirichlet boundary conditions. When inserting the decomposition (2.74) into the action we observe that due to the equation of motion (2.73) for the classical path, the action separates into the sum of a classical and a purely quadratic fluctuation term o M Z tb n 2 2 dt x˙ cl (t) + 2x˙ cl (t)δ x(t) ˙ + [δ x(t)] ˙ 2 ta Z Z tb Z tb M tb M tb 2 = dtx˙ cl + M xδx ˙ −M dt¨ xcl δx + dt(δ x) ˙ 2 t a 2 ta 2 ta ta Z t Z tb b M = dtx˙ 2cl + dt(δ x) ˙ 2 . 2 ta ta

The absence of a mixed term is a general consequence of the extremality property of the classical path, δA = 0. (2.76) x(t)=xcl (t)

It implies that a quadratic fluctuation expansion around the classical action Acl ≡ A[xcl ]


103


can have no linear term in δx(t), i.e., it must start as δ2A 1 Z tb Z tb ′ ′ A = Acl + dt dt δx(t)δx(t ) + ... . 2 ta δx(t)δx(t′ ) ta x(t)=xcl (t)

(2.78)

With the action being a sum of two terms, the amplitude factorizes into the product of a classical amplitude eiAcl /¯h and a fluctuation factor F0 (tb − ta ), (xb tb |xa ta ) =

Z

Dx eiA[x]/¯h = eiAcl /¯h F0 (tb , ta ).

(2.79)

For the free particle with the classical action Acl =

Z

tb

ta

dt

M 2 x˙ , 2 cl

(2.80)

the function factor F0 (tb − ta ) is given by the path integral Z

F0 (tb − ta ) =

Dδx(t) exp

i h ¯

Z

tb ta

dt

M (δ x) ˙ 2 . 2

(2.81)

Due to the vanishing of δx(t) at the endpoints, this does not depend on xb , xa but only on the initial and final times tb , ta . The time translational invariance reduces this dependence further to the time difference tb −ta . The subscript zero of F0 (tb −ta ) indicates the free-particle nature of the fluctuation factor. After inserting (2.72) into (2.80), we find immediately M (xb − xa )2 . Acl = 2 tb − ta

(2.82)

The fluctuation factor, on the other hand, requires the evaluation of the multiple integral F0N (tb

− ta ) =

1 q

2π¯ hiǫ/M

N Y

n=1

 Z 

dδxn

∞

−∞

q

2π¯ hiǫ/M



 exp

i N A , h ¯ fl

(2.83)

where AN fl is the time-sliced fluctuation action +1 M NX δxn − δxn−1 AN = ǫ fl 2 n=1 ǫ

!2

.

At the end, we have to take the continuum limit N → ∞,

ǫ = (tb − ta )/(N + 1) → 0.

(2.84)

104

2.2.3


Fluctuation Factor

The remainder of this section will be devoted to calculating the fluctuation factor (2.83). Before doing this, we shall develop a general technique for dealing with such time-sliced expressions. Due to the frequent appearance of the fluctuating δx-variables, we shorten the notation by omitting all δ’s and working only with x-variables. A useful device for manipulating sums on a sliced time axis such as (2.84) is the difference operator ∇ and its conjugate ∇, defined by 1 ∇x(t) ≡ [x(t) − x(t − ǫ)]. ǫ

1 ∇x(t) ≡ [x(t + ǫ) − x(t)], ǫ

(2.85)

They are two different discrete versions of the time derivative ∂t , to which both reduce in the continuum limit ǫ → 0: ǫ→0

∇, ∇ − −−→ ∂t ,

(2.86)

if they act upon differentiable functions. Since the discretized time axis with N + 1 steps constitutes a one-dimensional lattice, the difference operators ∇, ∇ are also called lattice derivatives. For the coordinates xn = x(tn ) at the discrete times tn we write 1 (xn+1 − xn ), ǫ 1 (xn − xn−1 ), = ǫ

∇xn =

N ≥ n ≥ 0,

∇xn

N + 1 ≥ n ≥ 1.

(2.87)

The time-sliced action (2.84) can then be expressed in terms of ∇xn or ∇xn as (writing xn instead of δxn ) AN fl =

N +1 M X M NX ǫ (∇xn )2 = ǫ (∇xn )2 . 2 n=0 2 n=1

(2.88)

In this notation, the limit ǫ → 0 is most obvious: The sum ǫ n goes into the R tb integral ta dt, whereas both (∇xn )2 and (∇xn )2 tend to x˙ 2 , so that P

AN fl →

Z

tb

ta

dt

M 2 x˙ . 2

(2.89)

Thus, the time-sliced action becomes the Lagrangian action. Lattice derivatives have properties quite similar to ordinary derivatives. One only has to be careful in distinguishing ∇ and ∇. For example, they allow for a useful operation summation by parts which is analogous to the integration by parts. Recall the rule for the integration by parts Z

tb

ta

tb dtg(t)f˙(t) = g(t)f (t) −

ta

Z

tb

ta

dtg(t)f ˙ (t).


105


On the lattice, this relation yields for functions f (t) → xn and g(t) → pn : ǫ

N +1 X n=1

+1 pn ∇xn = pn xn |N −ǫ 0

N X

(∇pn )xn .

(2.91)

n=0

This follows directly by rewriting (2.35). For functions vanishing at the endpoints, i.e., for xN +1 = x0 = 0, we can omit the surface terms and shift the range of the sum on the right-hand side to obtain the simple formula [see also Eq. (2.47)] N +1 X n=1

pn ∇xn = −

N X

(∇pn )xn = −

n=0

N +1 X

(∇pn )xn .

(2.92)

n=1

The same thing holds if both p(t) and x(t) are periodic in the interval tb − ta , so that p0 = pN +1 , x0 = xN +1 . In this case, it is possible to shift the sum on the right-hand side by one unit arriving at the more symmetric-looking formula N +1 X n=1

pn ∇xn = −

N +1 X

(∇pn )xn .

(2.93)

n=1

In the time-sliced action (2.84) the quantum fluctuations xn (=δx ˆ n ) vanish at the ends, so that (2.92) can be used to rewrite N +1 X n=1

(∇xn )2 = −

N X

n=1

xn ∇∇xn .

(2.94)

In the ∇xn -form of the action (2.88), the same expression is obtained by applying formula (2.92) from the right- to the left-hand side and using the vanishing of x0 and xN +1 : N X

(∇xn )2 = −

n=0

N +1 X n=1

xn ∇∇xn = −

N X

n=1

xn ∇∇xn .

(2.95)

The right-hand sides in (2.94) and (2.95) can be written in matrix form as − −

N X

n=1 N X

n=1

xn ∇∇xn ≡ − xn ∇∇xn ≡ −

with the same N × N -matrix

∇∇ ≡ ∇∇ ≡

1 ǫ2

        

N X

xn (∇∇)nn′ xn′ ,

n,n′ =1 N X

xn (∇∇)nn′ xn′ ,

−2 1 0 ... 0 1 −2 1 . . . 0 .. . 0 0

(2.96)

n,n′ =1

0 0

0 0 .. .

0 0 . . . 1 −2 1 0 0 ... 0 1 −2



    .   

(2.97)

106


This is obviously the lattice version of the double time derivative ∂t2 , to which it reduces in the continuum limit ǫ → 0. It will therefore be called the lattice Laplacian. A further common property of lattice and ordinary derivatives is that they can both be diagonalized by going to Fourier components. When decomposing x(t) =

Z

∞

−∞

dωe−iωt x(ω),

(2.98)

and applying the lattice derivative ∇, we find 1 −iω(tn +ǫ) e − e−iωtn x(ω) ǫ −∞ Z ∞ 1 dωe−iωtn (e−iωǫ − 1) x(ω). = ǫ −∞

∇x(tn ) =

Z

∞

dω

(2.99)

Hence, on the Fourier components, ∇ has the eigenvalues 1 −iωǫ (e − 1). ǫ

(2.100)

In the continuum limit ǫ → 0, this becomes the eigenvalue of the ordinary time derivative ∂t , i.e., −i times the frequency of the Fourier component ω. As a reminder of this we shall denote the eigenvalue of i∇ by Ω and have i (i∇x)(ω) = Ω x(ω) ≡ (e−iωǫ − 1) x(ω). ǫ

(2.101)

For the conjugate lattice derivative we find similarly i (i∇x)(ω) = Ω x(ω) ≡ − (eiωǫ − 1) x(ω), ǫ

(2.102)

where Ω is the complex-conjugate number of Ω, i.e., Ω ≡ Ω∗ . As a consequence, the eigenvalues of the negative lattice Laplacian −∇∇≡−∇∇ are real and nonnegative: i −iωǫ i 1 − 1) (1 − eiωǫ ) = 2 [2 − 2 cos(ωǫ)] ≥ 0. (e ǫ ǫ ǫ

(2.103)

¯ have the same continuum limit ω. Of course, Ω and Ω When decomposing the quantum fluctuations x(t) [=δx(t)] ˆ into their Fourier components, not all eigenfunctions occur. Since x(t) vanishes at the initial time t = ta , the decomposition can be restricted to the sine functions and we may expand x(t) =

Z

0

∞

dω sin ω(t − ta ) x(ω).

(2.104)

The vanishing at the final time t = tb is enforced by a restriction of the frequencies ω to the discrete values πm πm νm = = . (2.105) tb − ta (N + 1)ǫ H. Kleinert, PATH INTEGRALS

107


Thus we are dealing with the Fourier series x(t) =

∞ X

m=1

s

2 sin νm (t − ta ) x(νm ) (tb − ta )

(2.106)

with real Fourier components x(νm ). A further restriction arises from the fact that for finite ǫ, the series has to represent x(t) only at the discrete points x(tn ), n = 0, . . . , N + 1. It is therefore sufficient to carry the sum only up to m = N and to expand x(tn ) as x(tn ) =

N X

m=1

s

2 sin νm (tn − ta ) x(νm ), N +1

(2.107)

√ where a factor ǫ has been removed from the Fourier components, for convenience. The expansion functions are orthogonal, N 2 X sin νm (tn − ta ) sin νm′ (tn − ta ) = δmm′ , N + 1 n=1

(2.108)

N 2 X sin νm (tn − ta ) sin νm (tn′ − ta ) = δnn′ N + 1 m=1

(2.109)

and complete:

(where 0 < m, m′ < N + 1). The orthogonality relation follows by rewriting the left-hand side of (2.108) in the form N +1 X 2 1 iπ(m − m′ ) iπ(m + m′ ) Re exp n − exp n N +12 N +1 N +1 n=0

(

"

#

"

#)

,

(2.110)

with the sum extended without harm by a trivial term at each end. Being of the geometric type, this can be calculated right away. For m = m′ the sum adds up to 1, while for m 6= m′ it becomes ′

′

1 − eiπ(m−m ) eiπ(m−m )/(N +1) 2 1 Re − (m′ → −m′ ) . N +12 1 − eiπ(m−m′ )/(N +1) "

#

(2.111)

The first expression in the curly brackets is equal to 1 for even m − m′ 6= 0; while being imaginary for odd m − m′ [since (1 + eiα )/(1 − eiα ) is equal to (1 + eiα )(1 − e−iα )/|1 − eiα |2 with the imaginary numerator eiα − e−iα ]. For the second term the same thing holds true for even and odd m + m′ 6= 0, respectively. Since m − m′ and m + m′ are either both even or both odd, the right-hand side of (2.108) vanishes for m 6= m′ [remembering that m, m′ ∈ [0, N + 1] in the expansion (2.107), and thus in (2.111)]. The proof of the completeness relation (2.109) can be carried out similarly. Inserting now the expansion (2.107) into the time-sliced fluctuation action (2.84), the orthogonality relation (2.108) yields AN fl =

N +1 M X M NX ǫ (∇xn )2 = ǫ x(νm )Ωm Ωm x(νm ). 2 n=0 2 m=1

(2.112)

108


Thus the action decomposes into a sum of independent quadratic terms involving the discrete set of eigenvalues Ωm Ωm =

1 1 πm [2 − 2 cos(ν ǫ)] = 2 − 2 cos m ǫ2 ǫ2 N +1

,

(2.113)

and the fluctuation factor (2.83) becomes 1

F0N (tb

− ta ) = q 2π¯ hiǫ/M N Y

N Y

n=1

 

Z

dxn

∞

−∞

q

2π¯ hiǫ/M

 

iM × exp ǫΩm Ωm [x(νm )]2 . h ¯ 2 m=1

(2.114)

Before performing the integrals, we must transform the measure of integration from the local variables xn to the Fourier components x(νm ). Due to the orthogonality relation (2.108), the transformation has a unit determinant implying that N Y

dxn =

n=1

N Y

dx(νm ).

(2.115)

m=1

With this, Eq. (2.114) can be integrated with the help of Fresnel’s formula (1.333). The result is N Y 1 1 q . F0N (tb − ta ) = q 2π¯ hiǫ/M m=1 ǫ2 Ωm Ωm

(2.116)

To calculate the product we use the formula5 N Y

m=1

mπ x2(N +1) − 1 1 + x − 2x cos = . N +1 x2 − 1

2

(2.117)

Taking the limit x → 1 gives N Y

m=1

ǫ2 Ωm Ωm =

N Y

m=1

2 1 − cos

mπ = N + 1. N +1

(2.118)

The time-sliced fluctuation factor of a free particle is therefore simply 1 F0N (tb − ta ) = q , 2πi¯ h(N + 1)ǫ/M

(2.119)

or, expressed in terms of tb − ta ,

5

1 F0 (tb − ta ) = q . 2πi¯ h(tb − ta )/M

(2.120)

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.396.2.


109


As in the amplitude (2.71) we have dropped the superscript N since this final result is independent of the number of time slices. Note that the dimension of the fluctuation factor is 1/length. In fact, one may introduce a length scale associated with the time interval tb − ta , l(tb − ta ) ≡

q

2π¯ h(tb − ta )/M,

and write F0 (tb − ta ) = √

1 il(tb − ta )

(2.121)

.

(2.122)

With (2.120) and (2.82), the full time evolution amplitude of a free particle (2.79) is again given by (2.71) i M (xb − xa )2 (xb tb |xa ta ) = q exp . h ¯ 2 tb − ta 2πi¯ h(tb − ta )/M "

1

#

(2.123)

It is straightforward to generalize this result to a point particle moving through any number D of Cartesian space dimensions. If x = (x1 , . . . , xD ) denotes the spatial coordinates, the action is M A[x] = 2

Z

tb

ta

dt x˙ 2 .

(2.124)

Being quadratic in x, the action is the sum of the actions for each component. Hence, the amplitude factorizes and we find 1

(xb tb |xa ta ) = q D 2πi¯ h(tb − ta )/M

i M (xb − xa )2 exp , h ¯ 2 tb − ta "

#

(2.125)

in agreement with the quantum-mechanical result in D dimensions (1.337). It is instructive to present an alternative calculation of the product of eigenvalues in (2.116) which does not make use of the Fourier decomposition and works entirely in configuration space. We observe that the product N Y

ǫ2 Ωm Ωm

(2.126)

m=1

is the determinant of the diagonalized N × N -matrix −ǫ2 ∇∇. This follows from the fact that for any matrix, the determinant is the product of its eigenvalues. The product (2.126) is therefore also called the fluctuation determinant of the free particle and written N Y

m=1

ǫ2 Ωm Ωm ≡ detN (−ǫ2 ∇∇).

(2.127)

110


With this notation, the fluctuation factor (2.116) reads h i−1/2 1 detN (−ǫ2 ∇∇) . F0N (tb − tb ) = q 2π¯ hiǫ/M

(2.128)

Now one realizes that the determinant of ǫ2 ∇∇ can be found very simply from the explicit N × N matrix (2.97) by induction: For N = 1 we see directly that detN =1 (−ǫ2 ∇∇) = |2| = 2.

(2.129)

For N = 2, the determinant is

2 −1 = 3. detN =2 (−ǫ ∇∇) = −1 2 2

(2.130)

A recursion relation is obtained by developing the determinant twice with respect to the first row: detN (−ǫ2 ∇∇) = 2 detN −1 (−ǫ2 ∇∇) − detN −2 (−ǫ2 ∇∇).

(2.131)

With the initial condition (2.129), the solution is detN (−ǫ2 ∇∇) = N + 1,

(2.132)

in agreement with the previous result (2.118). Let us also find the time evolution amplitude in momentum space. A simple Fourier transform of initial and final positions according to the rule (2.37) yields (pb tb |pa ta ) =

Z

dD xb e−ipb xb /¯h

Z

dD xa eipa xa /¯h (xb tb |xa ta ) 2

= (2π)D δ (D) (pb − pa )e−ipb (tb −ta )/¯h .

2.2.4

(2.133)

Finite Slicing Properties of Free-Particle Amplitude

The time-sliced free-particle time evolution amplitudes (2.70) happens to be independent of the number N of time slices used for their calculation. We have pointed this out earlier for the fluctuation factor (2.119). Let us study the origin of this independence for the classical action in the exponent. The difference equation of motion −∇∇x(t) = 0

(2.134)

is solved by the same linear function x(t) = At + B,

(2.135)

as in the continuum. Imposing the initial conditions gives xcl (tn ) = xa + (xb − xa )

n . N +1


111

2.3 Exact Solution for Harmonic Oscillator

The time-sliced action of the fluctuations is calculated, via a summation by parts on the lattice [see (2.91)]. Using the difference equation ∇∇xcl = 0, we find N +1 X

Acl = ǫ

M (∇xcl )2 2

n=1

M = 2

N +1 xcl ∇xcl n=0

(2.137) −ǫ

N X

n=0

xcl ∇∇xcl

!

N +1 M M (xb − xa )2 = xcl ∇xcl = . n=0 2 2 tb − ta

This coincides with the continuum action for any number of time slices. In the operator formulation of quantum mechanics, the ǫ-independence of the amplitude of the free particle follows from the fact that in the absence of a potential V (x), the two sides of the Trotter formula (2.26) coincide for any N.

2.3

Exact Solution for Harmonic Oscillator

A further problem to be solved along similar lines is the time evolution amplitude of the linear oscillator Dp i (xb tb |xa ta ) = Dx exp A[p, x] 2π¯ h h ¯ Z i = Dx exp A[x] , h ¯ Z

′

Z

(2.138)

with the canonical action A[p, x] =

Z

tb

ta

1 2 Mω 2 2 dt px˙ − p − x , 2M 2

(2.139)

M 2 (x˙ − ω 2 x2 ). 2

(2.140)

!

and the Lagrangian action A[x] =

2.3.1

Z

tb

ta

dt

Fluctuations around Classical Path

As before, we proceed with the Lagrangian path integral, starting from the timesliced form of the action AN = ǫ

+1 h i M NX (∇xn )2 − ω 2 x2n . 2 n=1

(2.141)

The path integral is again a product of Gaussian integrals which can be evaluated successively. In contrast to the free-particle case, however, the direct evaluation is now quite complicated; it will be presented in Appendix 2B. It is far easier to

112


employ the fluctuation expansion, splitting the paths into a classical path xcl (t) plus fluctuations δx(t). The fluctuation expansion makes use of the fact that the action is quadratic in x = xcl + δx and decomposes into the sum of a classical part Z tb M Acl = dt (x˙ 2cl − ω 2 x2cl ), (2.142) 2 ta and a fluctuation part Z tb M dt [(δ x) (2.143) Afl = ˙ 2 − ω 2 (δx)2 ], 2 ta with the boundary condition δx(ta ) = δx(tb ) = 0.

(2.144)

There is no mixed term, due to the extremality of the classical action. The equation of motion is x¨cl = −ω 2 xcl .

(2.145)

Thus, as for a free-particle, the total time evolution amplitude splits into a classical and a fluctuation factor: Z (xb tb |xa ta ) =

Dx eiA[x]/¯h = eiAcl /¯h Fω (tb − ta ).

(2.146)

The subscript of Fω records the frequency of the oscillator. The classical orbit connecting initial and final points is obviously xb sin ω(t − ta ) + xa sin ω(tb − t) xcl (t) = . (2.147) sin ω(tb − ta ) Note that this equation only makes sense if tb − ta is not equal to an integer multiple of π/ω which we shall always assume from now on.6 After an integration by parts we can rewrite the classical action Acl as Z tb tb i Mh M Acl = dt xcl (−¨ xcl − ω 2 xcl ) + xcl x˙ cl . (2.148) ta 2 2 ta The first term vanishes due to the equation of motion (2.145), and we obtain the simple expression M Acl = [xcl (tb )x˙ cl (tb ) − xcl (ta )x˙ cl (ta )]. (2.149) 2 Since ω x˙ cl (ta ) = [xb − xa cos ω(tb − ta )], (2.150) sin ω(tb − ta ) ω x˙ cl (tb ) = [xb cos ω(tb − ta ) − xa ], (2.151) sin ω(tb − ta ) we can rewrite the classical action as h i Mω Acl = (x2b + x2a ) cos ω(tb − ta ) − 2xb xa . (2.152) 2 sin ω(tb − ta ) 6

For subtleties in the immediate neighborhood of the singularities which are known as caustic phenomena, see Notes and References at the end of the chapter, as well as Section 4.8. H. Kleinert, PATH INTEGRALS

113


2.3.2

Fluctuation Factor

We now turn to the fluctuation factor. With the matrix notation for the lattice operator −∇∇ − ω 2 , we have to solve the multiple integral 1

FωN (tb , ta ) = q 2π¯ hiǫ/M

N Y

n,n′ =1

 Z 

dδxn

∞

−∞

q

2π¯ hiǫ/M

 

N iM X × exp δxn [−∇∇ − ω 2 ]nn′ δxn′ . ǫ h ¯ 2 n=1

(

)

(2.153)

When going to the Fourier components of the paths, the integral factorizes in the same way as for the free-particle expression (2.114). The only difference lies in the eigenvalues of the fluctuation operator which are now Ωm Ωm − ω 2 =

1 [2 − 2 cos(νm ǫ)] − ω 2 ǫ2

(2.154)

instead of Ωm Ωm . For times tb , ta where all eigenvalues are positive (which will be specified below) we obtain from the upper part of the Fresnel formula (1.333) directly N Y 1 1 q FωN (tb , ta ) = q . 2π¯ hiǫ/M m=1 ǫ2 Ωm Ωm − ǫ2 ω 2

(2.155)

The product of these eigenvalues is found by introducing an auxiliary frequency ω ˜ satisfying sin

ǫ˜ ω ǫω ≡ . 2 2

(2.156)

Then we decompose the product as N Y

2

2

2

[ǫ Ωm Ωm − ǫ ω ] =

m=1

=

N h Y

ǫ2 Ωm Ωm

m=1

i



N h Y

2

ǫ Ωm Ωm

m=1



N i Y

m=1

"

ǫ2 Ωm Ωm − ǫ2 ω 2 ǫ2 Ωm Ωm



sin2 ǫ˜2ω  1 −  . 2 mπ sin m=1 2(N +1) N Y

#

(2.157)

The first factor is equal to (N + 1) by (2.118). The second factor, the product of the ratios of the eigenvalues, is found from the standard formula7 N Y

7





sin2 x  1 sin[2(N + 1)x] 1 − = . 2 mπ sin 2x (N + 1) sin 2(N +1) m=1 I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.391.1.

(2.158)

114


With x = ω ˜ ǫ/2, we arrive at the fluctuation determinant detN (−ǫ2 ∇∇ − ǫ2 ω 2) =

N Y

[ǫ2 Ωm Ωm − ǫ2 ω 2] =

m=1

sin ω ˜ (tb − ta ) , sin ǫ˜ ω

(2.159)

and the fluctuation factor is given by FωN (tb , ta )

1

=q 2πi¯ h/M

s

sin ω ˜ǫ , ǫ sin ω ˜ (tb − ta )

where, as we have agreed earlier in Eq. (1.333), larger than zero.

2.3.3

√

tb − ta < π/˜ ω,

(2.160)

i means eiπ/4 , and tb − ta is always

The iη -Prescription and Maslov-Morse Index

The result (2.160) is initially valid only for tb − ta < π/˜ ω.

(2.161)

In this time interval, all eigenvalues in the fluctuation determinant (2.159) are positive, and the upper version of the Fresnel formula (1.333) applies to each of the integrals in (2.153) [this was assumed in deriving (2.155)]. If tb − ta grows larger than π/˜ ω , the smallest eigenvalue Ω1 Ω1 − ω 2 becomes negative and the integration over the associated Fourier component has to be done according to the lower case of the Fresnel formula (1.333). The resulting amplitude carries an extra phase factor e−iπ/2 and remains valid until tb − ta becomes larger than 2π/˜ ω , where the second eigenvalue becomes negative introducing a further phase factor e−iπ/2 . All phase factors emerge naturally if we associate with the oscillator frequency ω an infinitesimal negative imaginary part, replacing everywhere ω by ω − iη with an infinitesimal η > 0. This is referred to as the iη-prescription. Physically, it amounts to attaching an infinitesimal damping term to the oscillator, so that the amplitude behaves like e−iωt−ηt and dies down to zero after a very long time (as opposed to an unphysical antidamping term which would make it diverge after a long time). Now, each time that tb − ta passes an integer multiple of π/˜ ω , the square root of sin ω ˜ (tb −ta ) in (2.160) passes a singularity in a specific way which ensures the proper phase.8 With such an iη-prescription it will be superfluous to restrict tb − ta to the range (2.161). Nevertheless it will sometimes be useful to exhibit the phase factor arising in this way in the fluctuation factor (2.160) for tb − ta > π/˜ ω by writing FωN (tb , ta ) 8

1

=q 2πi¯ h/M

s

sin ω ˜ǫ e−iνπ/2 , ǫ| sin ω ˜ (tb − ta )|

(2.162)

In the square root, we may equivalently assume tb − ta to carry a small negative imaginary part. For a detailed discussion of the phases of the fluctuation factor in the literature, see Notes and References at the end of the chapter. H. Kleinert, PATH INTEGRALS

115


where ν is the number of zeros encountered in the denominator along the trajectory. This number is called the Maslov-Morse index of the trajectory9 .

2.3.4

Continuum Limit

Let us now go to the continuum limit, ǫ → 0. Then the auxiliary frequency ω ˜ tends to ω and the fluctuation determinant becomes ǫ→0

detN (−ǫ2 ∇∇ − ǫ2 ω 2 ) − −−→

sin ω(tb − ta ) . ωǫ

(2.163)

The fluctuation factor FωN (tb − ta ) goes over into Fω (tb − ta ) =

1 q

2πi¯ h/M

s

ω , sin ω(tb − ta )

(2.164)

with the phase for tb − ta > π/ω determined as above. In the limit ω → 0, both fluctuation factors agree, of course, with the free-particle result (2.120). In the continuum limit, the ratios of eigenvalues in (2.157) can also be calculated in the following simple way. We perform the limit ǫ → 0 directly in each factor. This gives ǫ2 Ωm Ωm − ǫ2 ω 2 ǫ2 Ωm Ωm

ǫ2 ω 2 2 − 2 cos(νm ǫ) ǫ→0 ω 2 (tb − ta )2 − −−→ 1 − . π 2 m2 =

1−

(2.165)

As the number N goes to infinity we wind up with an infinite product of these factors. Using the well-known infinite-product formula for the sine function10 sin x = x

∞ Y

m=1

x2 1− 2 2 , mπ !

(2.166)

we find, with x = ω(tb − ta ), Y m

∞ 2 Y ǫ→0 Ωm Ωm νm ω(tb − ta ) − − −→ = , 2 2 sin ω(tb − ta ) Ωm Ωm − ω 2 m=1 νm − ω

(2.167)

and obtain once more the fluctuation factor in the continuum (2.164). 9

V.P. Maslov and M.V. Fedoriuk, Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston, 1981. 10 I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.431.1.

116


Multiplying the fluctuation factor with the classical amplitude, the time evolution amplitude of the linear oscillator in the continuum reads i Z tb M 2 dt (x˙ − ω 2 x2 ) h ¯ ta 2 s 1 ω = q 2πi¯ h/M sin ω(tb − ta )

(xb tb |xa ta ) =

Z

Dx(t) exp

(2.168)

(

)

Mω i × exp [(x2b + x2a ) cos ω(tb − ta ) − 2xb xa ] . 2¯ h sin ω(tb − ta )

The result can easily be extended to any number D of dimensions, where the action is A=

Z

tb ta

dt

M 2 x˙ − ω 2 x2 . 2

(2.169)

Being quadratic in x, the action is the sum of the actions of each component leading to the factorized amplitude: (xb tb |xa ta ) =

D Y

i=1

xib tb |xia ta

(

1

=q D 2πi¯ h/M

s

ω sin ω(tb − ta )

D

)

i Mω × exp [(x2 + x2a ) cos ω(tb − ta ) − 2xb xa ] , 2¯ h sin ω(tb − ta ) b

(2.170)

where the phase of the second square root for tb − ta > π/ω is determined as in the one-dimensional case [see Eq. (1.543)].

2.3.5

Useful Fluctuation Formulas

It is worth realizing that when performing the continuum limit in the ratio of eigenvalues (2.167), we have actually calculated the ratio of the functional determinants of the differential operators det(−∂t2 − ω 2 ) . det(−∂t2 )

(2.171)

Indeed, the eigenvalues of −∂t2 in the space of real fluctuations vanishing at the endpoints are simply 2 πm 2 νm = , (2.172) tb − ta so that the ratio (2.171) is equal to the product ∞ 2 Y det(−∂t2 − ω 2 ) νm − ω2 = , 2 2 det(−∂t ) νm m=1

(2.173)

which is the same as (2.167). This observation should, however, not lead us to believe that the entire fluctuation factor Z tb Z i M 2 2 2 Fω (tb − ta ) = Dδx exp dt [(δ x) ˙ − ω (δx) ] (2.174) ¯h ta 2 H. Kleinert, PATH INTEGRALS

117

2.3 Exact Solution for Harmonic Oscillator could be calculated via the continuum determinant ǫ→0 1 1 p Fω (tb , ta ) − −−→ p 2π¯hiǫ/M det(−∂t2 − ω 2 )

(false).

(2.175)

The product of eigenvalues in det(−∂t2 − ω 2 ) would be a strongly divergent expression det(−∂t2 − ω 2 ) = =

∞ Y

m=1

2 νm

∞ Y

2 (νm − ω2)

(2.176)

m=1 ∞ 2 Y νm

m=1

∞ Y − ω2 π 2 m2 sin ω(tb − ta ) = × . 2 2 νm (t − t ) ω(tb − ta ) b a m=1

Only ratios of determinants −∇∇ − ω 2 with different ω’s can be replaced by their differential limits. Then the common divergent factor in (2.176) cancels. Let us look at the origin of this strong divergence. The eigenvalues on the lattice and their continuum approximation start both out for small m as 2 ≈ Ωm Ωm ≈ νm

π 2 m2 . (tb − ta )2

(2.177)

2 ’s keep For large m ≤ N , the eigenvalues on the lattice saturate at Ωm Ωm → 2/ǫ2 , while the νm growing quadratically in m. This causes the divergence. The correct time-sliced formulas for the fluctuation factor of a harmonic oscillator is summarized by the following sequence of equations: "Z # N Y 1 dδxn iM T N p Fω (tb − ta ) = p exp δx (−ǫ2 ∇∇ − ǫ2 ω 2 )δx ¯h 2ǫ 2π¯hiǫ/M n=1 2π¯hiǫ/M

=

1 1 p q , 2π¯hiǫ/M det (−ǫ2 ∇∇ − ǫ2 ω 2 ) N

(2.178)

where in the first expression, the exponent is written in matrix notation with xT denoting the transposed vector x whose components are xn . Taking out a free-particle determinant detN (−ǫ2 ∇∇), formula (2.132), leads to the ratio formula

which yields

1 FωN (tb − ta ) = p 2π¯hi(tb − ta )/M 1

FωN (tb − ta ) = p 2πi¯h/M

detN (−ǫ2 ∇∇ − ǫ2 ω 2 ) detN (−ǫ2 ∇∇) s

−1/2

sin ω ˜ǫ , ǫ sin ω ˜ (tb − ta )

,

(2.179)

(2.180)

If with ω ˜ of Eq. (2.156). If we are only interested in the continuum limit, we may let ǫ go to zero on the right-hand side of (2.179) and evaluate Fω (tb − ta )

= =

=

−1/2 det(−∂t2 − ω 2 ) det(−∂t2 ) −1/2 ∞ 2 Y 1 νm − ω 2 p 2 νm 2π¯hi(tb − ta )/M m=1 s 1 ω(tb − ta ) p . 2π¯hi(tb − ta )/M sin ω(tb − ta ) 1 p 2π¯hi(tb − ta )/M

(2.181)

118


Let us calculate also here the time evolution amplitude in momentum space. The Fourier transform of initial and final positions of (2.170) [as in (2.133)] yields Z Z (pb tb |pa ta ) = dD xb e−ipb xb /¯h dD xa eipa xa /¯h (xb tb |xa ta ) (2π¯h)D 1 = √ D p D 2πi¯h M ω sin ω(tb − ta ) 2 i 1 (pb + p2a ) cos ω(tb − ta ) − 2pb pa . × exp ¯h 2M ω sin ω(tb − ta )

(2.182)

The limit ω → 0 reduces to the free-particle expression (2.133), not quite as directly as in the x-space amplitude (2.170). Expanding the exponent 2 1 (pb + pa ) cos ω(tb − ta ) − 2pb p2a 2M ω sin ω(tb − ta ) 1 1 2 2 2 2 (p − p ) − (p + p )[ω(t − t )] + . . . , = b a b a a 2M ω 2 (tb − ta ) 2 b

(2.183)

and going to the limit ω → 0, the leading term in (2.182) (2π)D

exp p D 2πiω 2 (tb − ta )¯ hM

i 1 (pb − pa )2 2 ¯h 2M ω (tb − ta )

(2.184)

tends to (2π¯h)D δ (D) (pb − pa ) [recall (1.531)], while the second term in (2.183) yields a factor 2 e−ip (tb −ta )/2M , so that we recover indeed (2.133).

2.3.6

Oscillator Amplitude on Finite Time Lattice

Let us calculate the exact time evolution amplitude for a finite number of time slices. In contrast to the free-particle case in Section 2.2.4, the oscillator amplitude is no longer equal to its continuum limit but ǫ-dependent. This will allow us to study some typical convergence properties of path integrals in the continuum limit. Since the fluctuation factor was initially calculated at a finite ǫ in (2.162), we only need to find the classical action for finite ǫ. To maintain time reversal invariance at any finite ǫ, we work with a slightly different sliced potential term in the action than before in (2.141), using

AN = ǫ

N +1 M X (∇xn )2 − ω 2 (x2n + x2n−1 )/2 , 2 n=1

(2.185)

or, written in another way, AN = ǫ

N M X (∇xn )2 − ω 2 (x2n+1 + x2n )/2 . 2 n=0

(2.186)

This differs from the original time-sliced action (2.141) by having the potential ω 2 x2n replaced by the more symmetric one ω 2 (x2n + x2n−1 )/2. The gradient term is the same in both cases and can be rewritten, after a summation by parts, as N +1 X

(∇xn )2 =

n=1

N X

N X (∇xn )2 = xb ∇xb − xa ∇xa − xn ∇∇xn .

n=0

(2.187)

n=1


119

2.4 Gelfand-Yaglom Formula This leads to a time-sliced action N

A

N M 2 2 M M X 2 xb ∇xb − xa ∇xa − ǫ ω (xb + xa ) − ǫ xn (∇∇ + ω 2 )xn . = 2 4 2 n=1

(2.188)

Since the variation of AN is performed at fixed endpoints xa and xb , the fluctuation factor is the same as in (2.153). The equation of motion on the sliced time axis is (∇∇ + ω 2 )xcl (t) = 0.

(2.189)

Here it is understood that the time variable takes only the discrete lattice values tn . The solution of this difference equation with the initial and final values xa and xb , respectively, is given by 1 xcl (t) = [xb sin ω ˜ (t − ta ) + xa sin ω ˜ (tb − t)] , (2.190) sin ω ˜ (tb − ta )

where ω ˜ is the auxiliary frequency introduced in (2.156). To calculate the classical action on the lattice, we insert (2.190) into (2.188). After some trigonometry, and replacing ǫ2 ω 2 by 4 sin2 (˜ ω ǫ/2), the action resembles closely the continuum expression (2.152): 2 M sin ω ˜ǫ AN (xb + x2a ) cos ω ˜ (tb − ta ) − 2xb xa . (2.191) cl = 2ǫ sin ω ˜ (tb − ta ) The total time evolution amplitude on the sliced time axis is N

(xb tb |xa ta ) = eiAcl /¯h FωN (tb − ta ),

(2.192)

with sliced action (2.191) and the sliced fluctuation factor (2.162).

2.4

Gelfand-Yaglom Formula

In many applications one encounters a slight generalization of the oscillator fluctuation problem: The action is harmonic but contains a time-dependent frequency Ω2 (t) instead of the constant oscillator frequency ω 2 . The associated fluctuation factor is Z i F (tb , ta ) = Dδx(t) exp (2.193) A , h ¯ with the action Z tb M A= dt [(δ x) ˙ 2 − Ω2 (t)(δx)2 ]. (2.194) 2 ta Since Ω(t) may not be translationally invariant in time, the fluctuation factor depends now in general on both the initial and final times. The ratio formula (2.179) holds also in this more general case, i.e., detN (−ǫ2 ∇∇ − ǫ2 Ω2 ) q F (tb , ta ) = detN (−ǫ2 ∇∇) 2π¯ hi(tb − ta )/M N

"

1

#−1/2

.

(2.195)

Here Ω2 (t) denotes the diagonal matrix 

Ω2 (t) =   

with the matrix elements Ω2n = Ω2 (tn ).

Ω2N

..



. Ω21

 , 

(2.196)

120

2.4.1


Recursive Calculation of Fluctuation Determinant

In general, the full set of eigenvalues of the matrix −∇∇ − Ω2 (t) is quite difficult to find, even in the continuum limit. It is, however, possible to derive a powerful difference equation for the fluctuation determinant which can often be used to find its value without knowing all eigenvalues. The method is due to Gelfand and Yaglom.11 Let us denote the determinant of the N × N fluctuation matrix by DN , i.e.,

DN ≡ detN −ǫ2 ∇∇ − ǫ2 Ω2

(2.197)

2 − ǫ2 Ω2N −1 0 ... 0 0 0 2 2 −1 2 − ǫ ΩN −1 −1 . . . 0 0 0 . .. .. ≡ . 0 0 0 . . . −1 2 − ǫ2 Ω22 −1 0 0 0 ... 0 −1 2 − ǫ2 Ω21

.

By expanding this along the first column, we obtain the recursion relation DN = (2 − ǫ2 Ω2N )DN −1 − DN −2 ,

(2.198)

which may be rewritten as 2

ǫ

1 DN − DN −1 DN −1 − DN −2 + Ω2N DN −1 = 0. − ǫ ǫ ǫ

(2.199)

Since the equation is valid for all N, it implies that the determinant DN satisfies the difference equation (∇∇ + Ω2N +1 )DN = 0. (2.200) In this notation, the operator −∇∇ is understood to act on the dimensional label N of the determinant. The determinant DN may be viewed as the discrete values of a function of D(t) evaluated on the sliced time axis. Equation (2.200) is called the Gelfand-Yaglom formula. Thus the determinant as a function of N is the solution of the classical difference equation of motion and the desired result for a given N is obtained from the final value DN = D(tN +1 ). The initial conditions are D1 = (2 − ǫ2 Ω21 ), D2 = (2 − ǫ2 Ω21 )(2 − ǫ2 Ω22 ) − 1.

2.4.2

(2.201)

Examples

As an illustration of the power of the Gelfand-Yaglom formula, consider the known case of a constant Ω2 (t) ≡ ω 2 where the Gelfand-Yaglom formula reads (∇∇ + ω 2 )DN = 0.

(2.202)

This is solved by a linear combination of sin(N ω ˜ ǫ) and cos(N ω ˜ ǫ), where ω ˜ is given by (2.156). The solution satisfying the correct boundary condition is obviously DN = 11

sin(N + 1)ǫ˜ ω . sin ǫ˜ ω

(2.203)

I.M. Gelfand and A.M. Yaglom, J. Math. Phys. 1 , 48 (1960). H. Kleinert, PATH INTEGRALS

121

2.4 Gelfand-Yaglom Formula

Figure 2.2 Solution of equation of motion with zero initial value and unit initial slope. Its value at the final time is equal to 1/ǫ times the fluctuation determinant. Indeed, the two lowest elements are D1

=

2 cos ǫ˜ ω,

D2

=

4 cos2 ǫ˜ ω − 1,

(2.204)

which are the same as (2.201), since ǫ2 Ω2 ≡ ǫ2 ω 2 =2(1 − cos ω ˜ ǫ). The Gelfand-Yaglom formula becomes especially easy to handle in the continuum limit ǫ → 0. Then, by considering the renormalized function Dren (tN ) = ǫDN ,

(2.205)

the initial conditions D1 = 2 and D2 = 3 can be re-expressed as (ǫD)1 = Dren (ta ) = 0, ǫ→0 ǫD2 − ǫD1 = (∇ǫD)1 − −−→ D˙ ren (ta ) = 1. ǫ

(2.206) (2.207)

The difference equation for DN turns into the differential equation for Dren (t): [∂t2 + Ω2 (t)]Dren (t) = 0.

(2.208)

The situation is pictured in Fig. 2.2. The determinant DN is 1/ǫ times the value of the function Dren (t) at tb . This value is found by solving the differential equation starting from ta with zero value and unit slope. As an example, consider once more the harmonic oscillator with a fixed frequency ω. The equation of motion in the continuum limit is solved by Dren (t) =

1 sin ω(t − ta ), ω

(2.209)

which satisfies the initial conditions (2.207). Thus we find the fluctuation determinant to become, for small ǫ, ǫ→0

det(−ǫ2 ∇∇ − ǫ2 ω 2 ) − −−→

1 sin ω(tb − ta ) , ǫ ω

(2.210)

in agreement with the earlier result (2.203). For the free particle, the solution is Dren (t) = t − ta and we obtain directly the determinant detN (−ǫ2 ∇∇) = (tb − ta )/ǫ. For time-dependent frequencies Ω(t), an analytic solution of the Gelfand-Yaglom initial-value problem (2.206), (2.207), and (2.208) can be found only for special classes of functions Ω(t). In fact, (2.208) has the form of a Schrödinger equation of a point particle in a potential Ω2 (t), and the classes of potentials for which the Schrödinger equation can be solved are well-known.

122

2.4.3


Calculation on Unsliced Time Axis

In general, the most explicit way of expressing the solution is by linearly combining Dren =ǫDN from any two independent solutions ξ(t) and η(t) of the homogeneous differential equation [∂t2 + Ω2 (t)]x(t) = 0. (2.211) The solution of (2.208) is found from a linear combination Dren (t) = αξ(t) + βη(t).

(2.212)

The coefficients are determined from the initial condition (2.207), which imply αξ(ta) + βη(ta ) = 0, ˙ a ) + β η(t αξ(t ˙ a ) = 1,

(2.213)

and thus

ξ(t)η(ta ) − ξ(ta )η(t) . (2.214) ˙ a )η(ta ) − ξ(ta )η(t ξ(t ˙ a) The denominator is recognized as the time-independent Wronski determinant of the two solutions Dren (t) =

↔

˙ W ≡ ξ(t) ∂ t η(t) ≡ ξ(t)η(t) ˙ − ξ(t)η(t)

(2.215)

at the initial point ta . The right-hand side is independent of t. The Wronskian is an important quantity in the theory of second-order differential equations. It is defined for all equations of the Sturm-Liouville type "

#

dy(t) d a(t) + b(t)y(t) = 0, dt dt

(2.216)

for which it is proportional to 1/a(t). The Wronskian serves to construct the Green function for all such equations.12 In terms of the Wronskian, Eq. (2.214) has the general form Dren (t) = −

1 [ξ(t)η(ta ) − ξ(ta )η(t)] . W

(2.217)

Inserting t = tb gives the desired determinant Dren = −

1 [ξ(tb )η(ta ) − ξ(ta )η(tb )] . W

(2.218)

Note that the same functional determinant can be found from by evaluating the function ˜ ren (t) = − 1 [ξ(tb )η(t) − ξ(t)η(tb)] D W

(2.219)

12

For its typical use in classical electrodynamics, see J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1975, Section 3.11. H. Kleinert, PATH INTEGRALS

123

2.4 Gelfand-Yaglom Formula

at ta . This also satisfies the homogenous differential equation (2.208), but with the initial conditions ˜˙ ren (tb ) = −1. D

˜ ren (tb ) = 0, D

(2.220)

It will be useful to emphasize at which ends the Gelfand-Yaglom boundary condi˜ ren (t) by Da (t) and Db (t), respectively, tions are satisfied by denoting Dren (t) and D summarizing their symmetric properties as [∂t2 + Ω2 (t)]Da (t) = 0 ; [∂t2 + Ω2 (t)]Db (t) = 0 ;

Da (ta ) = 0, Db (tb ) = 0,

D˙ a (ta ) = 1, D˙ b (tb ) = −1,

(2.221) (2.222)

with the determinant being obtained from either function as Dren = Da (tb ) = Db (ta ).

(2.223)

In contrast to this we see from the explicit equations (2.217) and (2.219) that the time derivatives of two functions at opposite endpoints are in general not related. Only for frequencies Ω(t) with time reversal invariance, one has D˙ a (tb ) = −D˙ b (ta ),

for Ω(t) = Ω(−t).

(2.224)

For arbitrary Ω(t), one can derive a relation D˙ a (tb ) + D˙ b (ta ) = −2

Z

tb

ta

˙ dt Ω(t)Ω(t)D a (t)Db (t).

(2.225)

As an application of these formulas, consider once more the linear oscillator, for which two independent solutions are ξ(t) = cos ωt,

η(t) = sin ωt.

(2.226)

Hence W = ω,

(2.227)

and the fluctuation determinant becomes 1 1 Dren = − (cos ωtb sin ωta − cos ωta sin ωtb ) = sin ω(tb − ta ). ω ω

2.4.4

(2.228)

D’Alembert’s Construction

It is important to realize that the construction of the solutions of Eqs. (2.221) and (2.222) requires only the knowledge of one solution of the homogenous differential equation (2.211), say ξ(t). A second linearly independent solution η(t) can always be found with the help of a formula due to d’Alembert, η(t) = w ξ(t)

Z

t

dt′ , ξ 2 (t′ )

(2.229)

124


where w is some constant. Differentiation yields η˙ =

˙ w ξη + , ξ ξ

η¨ =

¨ ξη . ξ

(2.230)

The second equation shows that with ξ(t), also η(t) is a solution of the homogenous differential equation (2.211). From first equation we find that the Wronski determinant of the two functions is equal to w: ˙ W = ξ(t)η(t) ˙ − ξ(t)η(t) = w.

(2.231)

Inserting the solution (2.229) into the formulas (2.217) and (2.219), we obtain explicit expressions for the Gelfand-Yaglom functions in terms of one arbitrary solution of the homogenous differential equation (2.211): Dren (t) = Da (t) = ξ(t)ξ(ta )

Z

t

ta

dt′ , ξ 2 (t′ )

˜ ren (t) = Db (t) = ξ(tb )ξ(t) D

Z

t

tb

dt′ . (2.232) ξ 2 (t′ )

The desired functional determinant is Dren = ξ(tb )ξ(ta )

2.4.5

Z

tb

ta

dt′ . ξ 2 (t′ )

(2.233)

Another Simple Formula

There exists yet another useful formula for the functional determinant. For this we solve the homogenous differential equation (2.211) for an arbitrary initial position xa and initial velocity x˙ a at the time ta . The result may be expressed as the following linear combination of Da (t) and Db (t): x(xa , x˙ a ; t) =

i 1 h Db (t) − Da (t)D˙ b (ta ) xa + Da (t)x˙ a . Db (ta )

(2.234)

We then see that the Gelfand-Yaglom function Dren (t) = Da (t) can be obtained from the partial derivative ∂x(xa , x˙ a ; t) Dren (t) = . (2.235) ∂ x˙ a This function obviously satisfies the Gelfand-Yaglom initial conditions Dren (ta ) = 0 and D˙ ren (ta ) = 1 of (2.206) and (2.207), which are a direct consequence of the fact that xa and x˙ a are independent variables in the function x(xa , x˙ a ; t), for which ∂xa /∂ x˙ a = 0 and ∂ x˙ a /∂ x˙ a = 1. The fluctuation determinant Dren = Da (tb ) is then given by Dren =

∂xb , ∂ x˙ a

(2.236)

where xb abbreviates the function x(xa , x˙ a ; tb ). It is now obvious that the analogous equations (2.222) are satisfied by the partial derivative Db (t) = −∂x(t)/∂ x˙ b , where x(t) is expressed in terms of the final position xb and velocity x˙ b as x(t) = x(xb , x˙ b ; t) x(xb , x˙ b ; t) =

i 1 h Da (t) + Db (t)D˙ a (tb ) xb − Db (t)x˙ b , Da (tb )

(2.237)


125

2.4 Gelfand-Yaglom Formula so that we obtain the alternative formula Dren = −

∂xa . ∂ x˙ b

(2.238)

These results can immediately be generalized to functional determinants of differential operators of the form −∂t2 δij −Ω2ij (t) where the time-dependent frequency is a D ×D-dimensional matrix Ω2ij (t), (i, j = 1, . . . , D). Then the associated Gelfand-Yaglom function Da (t) becomes a matrix Dij (t) satisfying the initial conditions Dij (ta ) = 0, D˙ ij (tb ) = δij , and the desired functional determinant Dren is equal to the ordinary determinant of Dij (tb ): Dren = Det −∂t2 δij − Ω2ij (t) = det Dij (tb ). (2.239) The homogeneous differential equation and the initial conditions are obviously satisfied by the partial derivative matrix Dij (t) = ∂xi (t)/∂ x˙ ja , so that the explicit representations of Dij (t) in terms of the general solution of the classical equations of motion −∂t2 δij − Ω2ij (t) xj (t) = 0 become ∂xi ∂xi Dren = det jb = −det aj . (2.240) ∂ x˙ a ∂ x˙ b

A further couple of formulas for functional determinants can be found by constructing a solution of the homogeneous differential equation (2.211) which passes through specific initial and final points xa and xb at ta and tb , respectively: x(xb , xa ; t) =

Db (t) Da (t) xa + xb . Db (ta ) Da (tb )

(2.241)

The Gelfand-Yaglom functions Da (t) and Db (t) can therefore be obtained from the partial derivatives Da (t) ∂x(xb , xa ; t) Db (t) ∂x(xb , xa ; t) = , = . (2.242) Da (tb ) ∂xb Db (ta ) ∂xa At the endpoints, Eqs. (2.241) yield x˙ a

=

x˙ b

=

D˙ b (ta ) 1 xa + xb , Db (ta ) Da (tb ) 1 D˙ a (tb ) − xa + xb , Db (ta ) Da (tb )

(2.243) (2.244)

so that the fluctuation determinant Dren = Da (tb ) = Db (ta ) is given by the formulas Dren =

∂ x˙ a ∂xb

−1

=−

∂ x˙ b ∂xa

−1

,

(2.245)

where x˙ a and x˙ b are functions of the independent variables xa and xb . The equality of these expressions with the previous ones in (2.236) and (2.238) is a direct consequence of the mathematical identity for partial derivatives !−1 ∂xb ∂ x˙ a = . (2.246) ∂ x˙ a ∂xb xa

xa

Let us emphasize that all functional determinants calculated in this Chapter apply to the fluctuation factor of paths with fixed endpoints. In mathematics, this property is referred to as Dirichlet boundary conditions. In the context of quantum statistics, we shall also need such determinants for fluctuations with periodic boundary conditions, for which the Gelfand-Yaglom method must be modified. We shall see in Section 2.11 that this causes considerable complications

126


in the lattice derivation, which will make it desirable to find a simpler derivation of both functional determinants. This will be found in Section 3.27 in a continuum formulation. In general, the homogenous differential equation (2.211) with time-dependent frequency Ω(t) cannot be solved analytically. The equation has the same form as a Schrödinger equation for a point particle in one dimension moving in a one dimensional potential Ω2 (t), and there are only a few classes of potentials for which the solutions are known in closed form. Fortunately, however, the functional determinant will usually arise in the context of quadratic fluctuations around classical solutions in time-independent potentials (see in Section 4.3). If such a classical solution is known analytically, it will provide us automatically with a solution of the homogeneous differential equation (2.211). Some important examples will be discussed in Sections 17.4 and 17.11.

2.4.6

Generalization to D Dimensions

The above formulas have an obvious generalization to a D-dimensional version of the fluctuation action (2.194) A=

Z

tb

ta

dt

M ˙ 2 − δxT [(δ x) 2

2

(t)δx],

(2.247)

where 2 (t) is a D × D matrix with elements Ω2ij (t). The fluctuation factor (2.195) generalizes to 1

N

F (tb , ta ) = q D 2π¯ hi(tb − ta )/M

"

detN (−ǫ2 ∇∇ − ǫ2 detN (−ǫ2 ∇∇)

2

)

#−1/2

.

(2.248)

The fluctuation determinant is found by Gelfand-Yaglom’s construction from a formula Dren = det Da (tb ) = det Db (ta ), (2.249) with the matrices Da (t) and Db (t) satisfying the classical equations of motion and initial conditions corresponding to (2.221) and (2.222): [∂t2 + [∂t2 +

2

(t)]Da (t) = 0 ; 2 (t)]Db (t) = 0 ;

Da (ta ) = 0, Db (tb ) = 0,

˙ a (ta ) = 1, D ˙ b (tb ) = −1, D

(2.250) (2.251)

where 1 is the unit matrix in D dimensions. We can then repeat all steps in the last section and find the D-dimensional generalization of formulas (2.245): Dren

2.5

∂ x˙ i = det aj ∂xb

!−1

∂ x˙ i = − det jb ∂xa

!−1

.

(2.252)

Harmonic Oscillator with Time-Dependent Frequency

The results of the last section put us in a position to solve exactly the path integral of a harmonic oscillator with arbitrary time-dependent frequency Ω(t). We shall first do this in coordinate space, later in momentum space. H. Kleinert, PATH INTEGRALS

127

2.5 Harmonic Oscillator with Time-Dependent Frequency

2.5.1

Coordinate Space

Consider the path integral (xb tb |xa ta ) =

Z

Dx exp

i A[x] , h ¯

(2.253)

with the Lagrangian action A[x] =

Z

tb

ta

i Mh 2 x˙ (t) − Ω2 (t)x2 (t) , 2

(2.254)

which is harmonic with a time-dependent frequency. As in Eq. (2.14), the result can be written as a product of a fluctuation factor and an exponential containing the classical action: (xb tb |xa ta ) =

Z

Dx eiA[x]/¯h = FΩ (tb , ta )eiAcl /¯h .

(2.255)

From the discussion in the last section we know that the fluctuation factor is, by analogy with (2.164), and recalling (2.236), 1 1 q FΩ (tb , ta ) = q . 2πi¯ h/M Da (tb )

(2.256)

The determinant Da (tb ) = Dren may be expressed in terms of partial derivatives according to formulas (2.236) and (2.245): 1

∂xb ∂ x˙ a

FΩ (tb − ta ) = q 2πi¯ h/M

!−1/2

1

=q 2πi¯ h/M

∂ x˙ a ∂xb

!1/2

,

(2.257)

where the first partial derivative is calculated from the function x(xa , x˙ a ; t), the second from x(x ˙ b , xa ; t). Equivalently we may use (2.238) and the right-hand part of Eq. (2.245) to write 1

FΩ (tb − ta ) = q 2πi¯ h/M

∂xa − ∂ x˙ b

!−1/2

1

=q 2πi¯ h/M

∂ x˙ b − ∂xa

!1/2

.

(2.258)

It remains to calculate the classical action Acl . This can be done in the same way as in Eqs. (2.148) to (2.152). After a partial integration, we have as before Acl =

M (xb x˙ b − xa x˙ a ). 2

(2.259)

Exploiting the linear dependence of x˙ b and x˙ a on the endpoints xb and xa , we may rewrite this as M Acl = 2

!

∂ x˙ b ∂ x˙ a ∂ x˙ b ∂ x˙ a xb xb − xa xa + xb xa − xa xb . ∂xb ∂xa ∂xa ∂xb

(2.260)

128


Inserting the partial derivatives from (2.243) and (2.244) and using the equality of Da (tb ) and Db (ta ), we obtain the classical action Acl =

i M h 2˙ xb Da (tb ) − x2a D˙ b (ta ) − 2xb xa . 2Da (tb )

(2.261)

Note that there exists another simple formula for the fluctuation determinant Dren : Dren = Da (tb ) = Db (ta ) = −M

∂2 Acl ∂xb ∂xa

!−1

.

(2.262)

For the harmonic oscillator with time-independent frequency ω, the GelfandYaglom function Da (t) of Eq. (2.228) has the property (2.224) due to time reversal invariance, and (2.261) reproduces the known result (2.152). The expressions containing partial derivatives are easily extended to D dimensions: We simply have to replace the partial derivatives ∂xb /∂ x˙ a , ∂ x˙ b /∂ x˙ a , . . . by the corresponding D × D matrices, and write the action as the associated quadratic form. The D-dimensional versions of the fluctuation factors (2.257) are 1

FΩ (tb − ta ) = q D 2πi¯ h/M

∂xi det jb ∂ x˙ a

"

#−1/2

1

=q D 2πi¯ h/M

∂ x˙ i det aj ∂xb

"

#1/2

.

(2.263)

All formulas for fluctuation factors hold initially only for sufficiently short times tb − ta . For larger times, they carry phase factors determined as before in (2.162). The fully defined expression may be written as 1

FΩ (tb − ta ) = q D 2πi¯ h/M

det

−1/2

∂xib ∂ x˙ ja

−iνπ/2

e

1

=q D 2πi¯ h/M

det

1/2

∂ x˙ ia ∂xjb

e−iνπ/2 ,

(2.264) where ν is the Maslov-Morse index. In the one-dimensional case it counts the turning points of the trajectory, in the multidimensional case the number of zeros in determinant det ∂xib /∂ x˙ ja along the trajectory, if the zero is caused by a reduction of the rank of the matrix ∂xib /∂ x˙ ja by one unit. If it is reduced by more than one unit, ν increases accordingly. In this context, the number ν is also called the Morse index of the trajectory. The zeros of the functional determinant are also called conjugate points. They are generalizations of the turning points in one-dimensional systems. The surfaces in x-space, on which the determinant vanishes, are called caustics. The conjugate points are the places where the orbits touch the caustics.13 Note that for infinitesimally short times, all fluctuation factors and classical actions coincide with those of a free particle. This is obvious for the time-independent harmonic oscillator, where the amplitude (2.170) reduces to that of a free particle 13

See M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, Berlin, 1990. H. Kleinert, PATH INTEGRALS

2.5 Harmonic Oscillator with Time-Dependent Frequency

129

in Eq. (2.125) in the limit tb → ta . Since a time-dependent frequency is constant over an infinitesimal time, this same result holds also here. Expanding the solution of the equations of motion for infinitesimally short times as xb ≈ (tb − ta )x˙ a + xa ,

xa ≈ −(tb − ta )x˙ b + xb ,

(2.265)

we have immediately ∂xib = δij (tb − ta ), ∂ x˙ ja

∂xa = −δij (tb − ta ). ∂ x˙ jb

Similarly, the expansions x˙ b ≈ x˙ a ≈ lead to ∂ x˙ ib 1 , j = −δij tb − ta ∂xa

xb − xa tb − ta

∂ x˙ ia 1 . j = δij tb − ta ∂xb

(2.266)

(2.267)

(2.268)

Inserting the expansions (2.266) or (2.267) into (2.259) (in D dimensions), the action reduces approximately to the free-particle action M (xb − xa )2 . 2 tb − ta

Acl ≈

2.5.2

(2.269)

Momentum Space

Let us also find the time evolution amplitude in momentum space. For this we write the classical action (2.260) as a quadratic form M xb (xb , xa ) A Acl = (2.270) xa 2 with a matrix

∂ x˙ b ∂xb



The inverse of this matrix is

 A=  A−1

−

∂ x˙ a ∂xb

∂xb  ∂ x˙ b =  ∂x 

a

∂ x˙ b

−

 ∂ x˙ b ∂xa  . ∂ x˙ 

(2.271)

a

∂xa

− −

 ∂xb ∂ x˙ a  . ∂x 

(2.272)

a

∂ x˙ a

The partial derivatives of xb and xa are calculated from the solution of the homogeneous differential equation (2.211) specified in terms of the final and initial velocities x˙ b and x˙ a : 1 ˙ ˙ Da (tb )Db (ta ) + 1 nh i h i o × Da (t) + Db (t)D˙ a (tb ) x˙ a + −Db (t) + Da (t)D˙ b (ta ) x˙ b ,

x(x˙ b , x˙ a ; t) =

(2.273)

130


which yields xa

=

xb

=

so that A−1

h i 1 Db (ta )D˙ a (ta )x˙ b − Db (ta )x˙ b , D˙ a (tb )D˙ b (ta ) + 1 i h 1 Da (tb )x˙ a + Da (tb )D˙ b (ta )x˙ b , D˙ a (tb )D˙ b (ta ) + 1  D˙ b (ta ) Da (tb )  = D˙ a (tb )D˙ b (ta ) + 1 −1

−1 −D˙ a (tb )

The determinant of A is the Jacobian det A = −

(2.274) (2.275)



.

(2.276)

∂(x˙ b , x˙ a ) D˙ a (tb )D˙ b (ta ) + 1 =− . ∂(xb , xa ) Da (tb )Db (ta )

(2.277)

We can now perform the Fourier transform of the time evolution amplitude and find, via a quadratic completion, Z Z (pb tb |pa ta ) = dxb e−ipb xb /¯h dxa eipa xa /¯h (xb tb |xa ta ) (2.278) s r 2π¯h Da (tb ) = ˙ iM Da (tb )D˙ b (ta ) + 1 h i Da (tb ) i 1 2 2 ˙ ˙ × exp −Db (ta )pb + Da (tb )pa − 2pb pa . ¯h 2M D˙ a (tb )D˙ b (ta ) + 1 Inserting here Da (tb ) = sin ω(tb − ta )/ω and D˙ a (tb ) = cos ω(tb − ta ), we recover the oscillator result (2.182). In D dimensions, the classical action has the same quadratic form as in (2.270) M T T xb Acl = xb , xa A (2.279) xa 2

with a matrix A generalizing (2.271) by having the partial derivatives replaced by the corresponding D × D-matrices. The inverse is the 2D × 2D-version of (2.272), i.e.     ∂ x˙ b ∂ x˙ b ∂xb ∂xb −  ∂xb  ∂ x˙ b ∂xa  ∂ x˙ a  , . A= A−1 =  (2.280)    ∂ x˙ ∂ x˙ a ∂xa ∂xa  a − − − ∂xb ∂xa ∂ x˙ b ∂ x˙ a The determinant of such a block matrix

A=

a c

b d

is calculated after a triangular decomposition a b a 0 1 a−1 b 1 A= = = c d c 1 0 d − ca−1 b 0

(2.281)

b d

a − bd−1 c d−1 c

0 1

in two possible ways as a b det = det a · det (d − ca−1 b) = det (a − bd−1 c) · det d, c d

(2.282)

(2.283)


131

2.6 Free-Particle and Oscillator Wave Functions depending whether det a or det b is nonzero. The inverse is in the first case −1 −1 −1 1 −a−1 bx a 0 a +a bxca−1 −a−1 bx A= = , x ≡(d−ca−1b)−1. 0 x −ca−1 1 −xca−1 x The resulting amplitude in momentum space is Z Z −ipb xb /¯ h (pb tb |pa ta ) = dxb e dxa eipa xa /¯h (xb tb |xa ta ) −1 pb 2π 1 i 1 T T √ exp pb , pa A . = √ pa ¯h 2M 2πi¯hM Dren det A

(2.284)

(2.285)

Also in momentum space, the amplitude (2.285) reduces to the free-particle one in Eq. (2.133) in the limit of infinitesimally short time tb − ta : For the time-independent harmonic oscillator, this was shown in Eq. (2.184), and the time-dependence of Ω(t) becomes irrelevant in the limit of small tb − ta → 0.

2.6

Free-Particle and Oscillator Wave Functions

In Eq. (1.331) we have expressed the time evolution amplitude of the free particle (2.71) as a Fourier integral (xb tb |xa ta ) =

Z

dp ip(x−x′ )/¯h −ip2 (tb −ta )/2M ¯h e e . (2π¯ h)

(2.286)

This expression contains the information on all stationary states of the system. To find these states we have to perform a spectral analysis of the amplitude. Recall that according to Section 1.7, the amplitude of an arbitrary time-independent system possesses a spectral representation of the form (xb tb |xa ta ) =

∞ X


(2.287)

n=0

where En are the eigenvalues and ψn (x) the wave functions of the stationary states. In the free-particle case the spectrum is continuous and the spectral sum is an integral. Comparing (2.287) with (2.286) we see that the Fourier decomposition itself happens to be the spectral representation. If the sum over n is written as an integral over the momenta, we can identify the wave functions as ψp (x) = √

1 eipx . 2π¯ h

(2.288)

For the time evolution amplitude of the harmonic oscillator 1 (xb tb |xa ta ) = q 2πi¯ h sin [ω(tb − ta )] /Mω (

(2.289) )

h i iMω × exp (x2b + x2a ) cos ω(tb − ta ) − 2xb xa , 2¯ h sin [ω(tb − ta )]

132


the procedure is not as straight-forward. Here we must make use of a summation formula for Hermite polynomials (see Appendix 2C) Hn (x) due to Mehler:14 (

)

1 1 √ exp − [(x2 + x′2 )(1 + a2 ) − 4xx′ a] 2) 2 2(1 − a 1−a ∞ X an 2 ′2 Hn (x)Hn (x′ ), = exp(−x /2 − x /2) n n=0 2 n!

(2.290)

with H0 (x) = 1, H1 (x) = 2x, H2 (x) = 4x2 − 2, . . . , Hn (x) = (−1)n ex

2

dn −x2 e . dxn

(2.291)

Identifying x≡ so that

q

x′ ≡

Mω/¯ h xb ,

q

Mω/¯ h xa ,

a ≡ e−iω(tb −ta ) ,

(2.292)

1 + a2 1 + e−2iω(tb −ta ) cos [ω(tb − ta )] = = 2 −2iω(t −t ) a b 1−a 1−e i sin [ω(tb − ta )]

1 a = , 2 1−a 2i sin [ω(tb − ta )]

we arrive at the spectral representation (xb tb |xa ta ) =

∞ X

ψn (xb )ψn (xa )e−i(n+1/2)ω(tb −ta ) .

(2.293)

n=0

From this we deduce that the harmonic oscillator has the energy eigenvalues En = h ¯ ω(n + 1/2)

(2.294)

and the wave functions ψn (x) = Nn λ−1/2 e−x ω

2 /2λ2 ω

Hn (x/λω ).

(2.295)

Here, λω is the natural length scale of the oscillator λω ≡

s

h ¯ , Mω

(2.296)

and Nn the normalization constant

√ Nn = (1/2n n! π)1/2 .

(2.297)

It is easy to check that the wave functions satisfy the orthonormality relation Z

∞

−∞

dx ψn (x)ψn′ (x)∗ = δnn′ ,

(2.298)

using the well-known orthogonality relation of Hermite polynomials15 Z ∞ 1 2 √ dx e−x Hn (x)Hn′ (x) = δn,n′ . n 2 n! π −∞

(2.299)

14

See P.M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, Vol. I, p. 781 (1953). 15 I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 7.374.1. H. Kleinert, PATH INTEGRALS

2.7 General Time-Dependent Harmonic Action

2.7

133

General Time-Dependent Harmonic Action

A simple generalization of the harmonic oscillator with time-dependent frequency allows also for a time-dependent mass, so that the action (2.300) becomes Z tb M g(t)x˙ 2 (t) − Ω2 (t)x2 (t) , (2.300) A[x] = dt 2 ta

with some dimensionless time-dependent factor g(t). This factor changes the measure of path integration so that the time evolution amplitude is no longer given by (2.304). To find the correct measure we must return to the canonical path integral (2.29) which now reads Z x(tb )=xb Z Dp iA[p,x]/¯h (xb tb |xa ta ) = D′ x e , (2.301) 2π¯h x(ta )=xa with the canonical action A[p, x] =

Z

tb

ta

p2 2 2 − M Ω (t)x (t) . dt px˙ − 2M g(t)

(2.302)

Integrating the momentum variables out in the sliced form of this path integral as in Eqs. (2.49)– (2.51) yields "Z # N ∞ Y 1 dxn i N p (xb tb |xa ta ) ≈ p exp A . (2.303) ¯h 2π¯hiǫ/M g(tN +1) n=1 −∞ 2π¯hiǫ/M g(tn) The continuum limit of this path integral is written as Z i √ (xb tb |xa ta ) = Dx g exp A[x] , ¯h

with the action (2.300). The classical orbits solve the equation of motion −∂t g(t)∂t − Ω2 (t) x(t) = 0,

(2.304)

(2.305)

which, by the transformation x ˜(t) =

p g(t)x(t),

2 ˜ 2 (t) = 1 Ω2 (t) + 1 g˙ (t) − 1 g¨(t) , Ω g(t) 4 g(t) 2 g(t)

can be reduced to the previous form h i p ˜ 2 (t) x g(t) −∂t2 − Ω ˜(t) = 0.

The result of the path integration is therefore Z √ (xb tb |xa ta ) = Dx g eiA[x]/¯h = F (xb , tb ; xa , ta )eiAcl /¯h ,

(2.306)

(2.307)

(2.308)

with a fluctuation factor [compare (2.256)] 1 1 p F (xb , tb ; xa , ta ) = p , 2πi¯h/M Da (tb )

(2.309)

where Da (tb ) is found from a generalization of the formulas (2.257)–(2.262). The classical action is Acl =

M (gb xb x˙ b − ga xa x˙ a ), 2

(2.310)

134


where gb ≡ g(tb ), ga ≡ g(ta ). The solutions of the equation of motion can be expressed in terms of modified Gelfand-Yaglom functions (2.221) and (2.222) with the properties [∂t g(t)∂t + Ω2 (t)]Da (t) = 0 ; 2

[∂t g(t)∂t + Ω (t)]Db (t) = 0 ;

Da (ta ) = 0, Db (tb ) = 0,

D˙ a (ta ) = 1/ga , D˙ b (tb ) = −1/gb,

(2.311) (2.312)

as in (2.241): x(xb , xa ; t) =

Db (t) Da (t) xa + xb . Db (ta ) Da (tb )

(2.313)

This allows us to write the classical action (2.310) in the form h i M Acl = gb x2b D˙ a (tb ) − ga x2a D˙ b (ta ) − 2xb xa . 2Da (tb )

(2.314)

From this we find, as in (2.262),

Dren = Da (tb ) = Db (ta ) = −M

∂ 2 Acl ∂xb ∂xa

−1

,

(2.315)

so that the fluctuation factor becomes 1

F (xb , tb ; xa , ta ) = √ 2πi¯h

s

−

∂ 2 Acl . ∂xb ∂xa

(2.316)

As an example take a free particle with a time-dependent mass term, where Z t Z tb Z tb Da (t) = dt′ g −1 (t′ ), Db (t) = dt′ g −1 (t′ ), Dren = Da (tb ) = Db (ta ) = dt′ g −1 (t′ ), (2.317) ta

t

ta

and the classical action reads

M (xb − xa )2 . 2 Da (tb ) The result can easily be generalized to an arbitrary harmonic action Z tb M A= dt g(t)x˙ 2 + 2b(t)xx˙ − Ω2 (t)x2 , 2 ta Acl =

which is extremized by the Euler-Lagrange equation [recall (1.8)] h i ˙ + Ω2 (t) x = 0. ∂t g(t)∂t + b(t)

(2.318)

(2.319)

(2.320)

The solution of the path integral (2.308) is again given by (2.308), with the fluctuation factor (2.316), where Acl is the action (2.319) along the classical path connecting the endpoints. A further generalization to D dimensions is obvious by adapting the procedure in Subsection 2.4.6, which makes Eqs. (2.311)–(2.313). matrix equations.

2.8

Path Integrals and Quantum Statistics

The path integral approach is useful to also understand the thermal equilibrium properties of a system. We assume the system to have a time-independent Hamiltonian and to be in contact with a reservoir of temperature T . As explained in Section 1.7, the bulk thermodynamic quantities can be determined from the quantumstatistical partition function

ˆ

Z = Tr e−H/kB T =

X

e−En /kB T .

(2.321)

n


135

2.8 Path Integrals and Quantum Statistics

This, in turn, may be viewed as an analytic continuation of the quantum-mechanical partition function ˆ (2.322) ZQM = Tr e−i(tb −ta )H/¯h to the imaginary time

tb − ta = −

i¯ h ≡ −i¯ hβ. kB T

(2.323)

In the local particle basis |xi, the quantum-mechanical trace corresponds to an integral over all positions so that the quantum-statistical partition function can be obtained by integrating the time evolution amplitude over xb = xa and evaluating it at the analytically continued time: Z≡

Z

∞

−∞

dx z(x) =

Z

∞

ˆ −β H

−∞

dx hx|e

|xi =

Z

∞ −∞

dx (x tb |x ta )|tb −ta =−i¯hβ .

(2.324)

The diagonal elements ˆ

z(x) ≡ hx|e−β H |xi = (x tb |x ta )|tb −ta =−i¯hβ

(2.325)

play the role of a partition function density. For a harmonic oscillator, this quantity has the explicit form [recall (2.168)] 1

zω (x) = q 2π¯ h/M

s

(

)

Mω ω h ¯ βω 2 exp − tanh x . sinh h ¯ βω h ¯ 2

(2.326)

ˆ

ˆ

By splitting the Boltzmann factor e−β H into a product of N + 1 factors e−ǫH/¯h with ǫ = h ¯ /kB T (N + 1), we can derive for Z a similar path integral representation just as for the corresponding quantum-mechanical partition function in (2.40), (2.46): Z≡

NY +1 Z ∞ n=1

−∞

dxn

(2.327)

ˆ

ˆ

ˆ

ˆ

× hxN +1 |e−ǫH/¯h |xN ihxN |e−ǫH/¯h |xN −1 i × . . . × hx2 |e−ǫH/¯h |x1 ihx1 |e−ǫH/¯h |xN +1 i. ˆ

As in the quantum-mechanical case, the matrix elements hxn |e−ǫH/¯h |xn−1 i are reexpressed in the form ˆ h −ǫH/¯

hxn |e

|xn−1 i ≈

Z

∞

−∞

dpn ipn (xn −xn−1 )/¯h−ǫH(pn ,xn )/¯h e , 2π¯ h

(2.328)

with the only difference that there is now no imaginary factor i in front of the Hamiltonian. The product (2.327) can thus be written as Z≈

NY +1 n=1

"Z

∞

−∞

dxn

Z

∞

−∞

#

dpn 1 exp − AN , 2π¯ h h ¯ e

(2.329)

136


where AN e denotes the sum AN e

=

N +1 X n=1

[−ipn (xn − xn−1 ) + ǫH(pn , xn )] .

(2.330)

In the continuum limit ǫ → 0, the sum goes over into the integral Ae [p, x] =

Z

hβ ¯

0

dτ [−ip(τ )x(τ ˙ ) + H(p(τ ), x(τ ))],

(2.331)

and the partition function is given by the path integral Z Z Dp −Ae [p,x]/¯h Z = Dx e . (2.332) 2π¯ h In this expression, p(τ ), x(τ ) may be considered as paths running along an “imaginary time axis” τ = it. The expression Ae [p, x] is very similar to the mechanical canonical action (2.27). Since it governs the quantum-statistical path integrals it is called quantum-statistical action or Euclidean action, indicated by the subscript e. The name alludes to the fact that a D-dimensional Euclidean space extended by an imaginary-time axis τ = it has the same geometric properties as a D + 1dimensional Euclidean space. For instance, a four-vector in a Minkowski spacetime has a square length dx2 = −(cdt)2 + (dx)2 . Continued to an imaginary time, this becomes dx2 = (cdτ )2 + (dx)2 which is the square distance in a Euclidean fourdimensional space with four-vectors (cτ, x). Just as in the path integralH for the quantum-mechanical partition function (2.46), R the measure of integration Dx Dp/2π¯ h in the quantum-statistical expression (2.332) is automatically symmetric in all p’s and x’s: I

Dx

Z

Dp = 2π¯ h

I

Dp 2π¯ h

Z

Dx =

NY +1 ZZ ∞ n=1

−∞

dxn dpn . 2π¯ h

(2.333)

The symmetry is of course due to the trace integration over all initial ≡ final positions. Most remarks made in connection with Eq. (2.46) carry over to the present case. The above path integral (2.332) is a natural extension of the rules of classical statistical mechanics. According to these, each cell in phase space dxdp/h is occupied with equal statistical weight, with the probability factor e−E/kB T . In quantum statistics, the paths of all particles fluctuate evenly over the cells in path phase Q space n dx(τn )dp(τn )/h (τn ≡ nǫ), each path carrying a probability factor e−Ae /¯h involving the Euclidean action of the system.

2.9

Density Matrix

The partition function does not determine any local thermodynamic quantities. Important local information resides in the thermal analog of the time evolution ampliˆ tude hxb |e−H/kB T |xa i. Consider, for instance, the diagonal elements of this amplitude renormalized by a factor Z −1 : ˆ

ρ(xa ) ≡ Z −1 hxa |e−H/kB T |xa i.


137

2.9 Density Matrix

It determines the thermal average of the particle density of a quantum-statistical system. Due to (2.327), the factor Z −1 makes the spatial integral over ρ equal to unity: Z ∞ dx ρ(x) = 1. (2.335) −∞

By inserting into (2.334) a complete set of eigenfunctions ψn (x) of the Hamiltonian ˆ we find the spectral decomposition operator H, ρ(xa ) =

X n

|ψn (xa )|2 e−βEn

.X

e−βEn .

(2.336)

n

Since |ψn (xa )|2 is the probability distribution of the system in the eigenstate |ni, P while the ratio e−βEn / n e−βEn is the normalized probability to encounter the system in the state |ni, the quantity ρ(xa ) represents the normalized average particle density in space as a function of temperature. Note the limiting properties of ρ(xa ). In the limit T → 0, only the lowest energy state survives and ρ(xa ) tends towards the particle distribution in the ground state T →0

ρ(xa ) − −−→ |ψ0 (xa )|2 .

(2.337)

In the opposite limit of high temperatures, quantum effects are expected to become irrelevant and the partition function should converge to the classical expression (1.535) which is the integral over the phase space of the Boltzmann distribution T →∞

−−→ Zcl = Z−

Z

∞

−∞

dx

Z

∞

−∞

dp −H(p,x)/kB T . e 2π¯ h

(2.338)

We therefore expect the large-T limit of ρ(x) to be equal to the classical particle distribution T →∞

Zcl−1

ρ(x) − −−→ ρcl (x) =

Z

∞

−∞

dp −H(p,x)/kB T e . 2π¯ h

(2.339)

Within the path integral approach, this limit will be discussed in more detail in Section 2.13. At this place we roughly argue as follows: When going in the original time-sliced path integral (2.327) to large T , i.e., small τb − τa = h ¯ /kB T , we may keep only a single time slice and write Z≈

Z

∞

−∞

ˆ

dx hx|e−ǫH/¯h |xi,

(2.340)

with ˆ

hx|e−ǫH |xi ≈

Z

∞

−∞

dpn −ǫH(pn ,x)/¯h . e 2π¯ h

(2.341)

After substituting ǫ = τb − τa this gives directly (2.339). Physically speaking, the path has at high temperatures “no (imaginary) time” to fluctuate, and only one term in the product of integrals needs to be considered.

138


If H(p, x) has the standard form H(p, x) =

p2 + V (x), 2M

(2.342)

the momentum integral is Gaussian in p and can be done using the formula Z

∞

−∞

1 dp −ap2 /2¯h =√ e . 2π¯ h 2π¯ ha

(2.343)

This leads to the pure x-integral for the classical partition function Zcl =

Z

dx

∞

−∞

q

2π¯ h2 /MkB T

e−V (x)/kB T =

Z

∞

−∞

dx −βV (x) e . le (¯ hβ)

(2.344)

In the second expression we have introduced the length q

2π¯ h2 β/M.

hβ) ≡ le (¯

(2.345)

It is the thermal (or Euclidean) analog of the characteristic length l(tb − ta ) introduced before in (2.121). It is called the de Broglie wavelength associated with the temperature T = 1/kB β, or short thermal de Broglie wavelength. Omitting the x-integration in (2.344) renders the large-T limit ρ(x), the classical particle distribution T →∞

−−→ ρcl (x) = Zcl−1 ρ(x) −

1 ¯ e−V (x) . le (¯ hβ)

(2.346)

For a free particle, the integral over x in (2.344) diverges. If we imagine the length of the x-axis to be very large but finite, say equal to L, the partition function is equal to L Zcl = . (2.347) le (¯ hβ) In D dimensions, this becomes Zcl =

VD , D le (¯ hβ)

(2.348)

where VD is the volume of the D-dimensional system. For a harmonic oscillator with potential Mω 2 x2 /2, the integral over x in (2.344) is finite and yields, in the D-dimensional generalization lD Zcl = D ω , (2.349) l (¯ hβ) where lω ≡

s

2π βMω 2


139

2.9 Density Matrix

is the classical length scale defined by the frequency of the harmonic oscillator. It is related to the quantum-mechanical one λω of Eq. (2.296) by lω le (¯ hβ) = 2π λ2ω .

(2.351)

Thus we obtain the mnemonic rule for going over from the partition function of a harmonic oscillator to that of a free particle: we must simply replace lω − −−→ L,

(2.352)

ω→0

or

s

1 βM − −−→ L. ω ω→0 2π The real-time version of this is, of course, 1 − −−→ ω ω→0

s

(2.353)

(tb − ta )M L. 2π¯ h

(2.354)

Let us write down a path integral representation for ρ(x). Omitting in (2.332) the final trace integration over xb ≡ xa and normalizing the expression by a factor Z −1 , we obtain −1

Z

= Z −1

Z

ρ(xa ) = Z

x(¯ hβ)=xb

x(0)=xa x(¯ hβ)=xb x(0)=xa

′

Dx

Z

Dp −Ae [p,x]/¯h e 2π¯ h

Dxe−Ae [x]/¯h .

(2.355)

ˆ is The thermal equilibrium expectation of an arbitrary Hermitian operator O given by X ˆ T ≡ Z −1 ˆ hOi e−βEn hn|O|ni. (2.356) n

In the local basis |xi, this becomes ˆ T = Z −1 hOi

ZZ

∞

−∞

ˆ ˆ b i. dxb dxa hxb |e−β H |xa ihxa |O|x

(2.357)

An arbitrary function of the position operator xˆ has the expectation hf (ˆ x)iT = Z

−1

ZZ

∞

−∞

ˆ −β H

dxb dxa hxb |e

|xa iδ(xb − xa )f (xa ) =

Z

dxρ(x)f (x). (2.358)

The particle density ρ(xa ) determines the thermal averages of local observables. If f depends also on the momentum operator pˆ, then the off-diagonal matrix ˆ elements hxb |e−β H |xa i are also needed. They are contained in the density matrix introduced for pure quantum systems in Eq. (1.221), and reads now in a thermal ensemble of temperature T : ˆ

ρ(xb , xa ) ≡ Z −1 hxb |e−β H |xa i,

(2.359)

140


whose diagonal values coincide with the above particle density ρ(xa ). It is useful to keep the analogy between quantum mechanics and quantum statistics as close as possible and to introduce the time translation operator along the imaginary time axis ˆ Uê (τb , τa ) ≡ e−(τb −τa )H/¯h ,

τb > τa ,

(2.360)

defining its local matrix elements as imaginary or Euclidean time evolution amplitudes ê (τb , τa )|xa i, (xb τb |xa τa ) ≡ hxb |U

τb > τa .

(2.361)

As in the real-time case, we shall only consider the causal time-ordering τb > τa . Otherwise the partition function and the density matrix do not exist in systems with energies up to infinity. Given the imaginary-time amplitudes, the partition function is found by integrating over the diagonal elements Z=

Z

∞

−∞

dx(x h ¯ β|x 0),

(2.362)

and the density matrix ρ(xb , xa ) = Z −1 (xb h ¯ β|xa 0).

(2.363)

For the sake of generality we may sometimes also consider the imaginary-time evolution operators for time-dependent Hamiltonians and the associated amplitudes. They are obtained by time-slicing the local matrix elements of the operator ˆ b , τa ) = Tτ exp − 1 U(τ h ¯

τb

Z

τa

ˆ dτ H(−iτ ) .

(2.364)

Here Tτ is an ordering operator along the imaginary-time axis. It must be emphasized that the usefulness of the operator (2.364) in describing ˆ thermodynamic phenomena is restricted to the Hamiltonian operator H(t) depending very weakly on the physical time t. The system has to remain close to equilibrium at all times. This is the range of validity of the so-called linear response theory (see Chapter 18 for more details). The imaginary-time evolution amplitude (2.361) has a path integral representation which is obtained by dropping the final integration in (2.329) and relaxing the condition xb = xa : (xb τb |xa τa ) ≈

N Z Y

n=1

∞

−∞

dxn

" NY +1 Z n=1

∞

−∞

#

dpn exp −AN /¯ h . e 2π¯ h

(2.365)

The time-sliced Euclidean action is AN e =

N +1 X n=1

[−ipn (xn − xn−1 ) + ǫH(pn , xn , τn )]

(2.366)


141

2.9 Density Matrix

(we have omitted the factor −i in the τ -argument of H). In the continuum limit this is written as a path integral Z

(xb τb |xa τa ) =

′

Dx

1 Dp exp − Ae [p, x] 2π¯ h h ¯

Z

(2.367)

[by analogy with (2.332)]. For a Hamiltonian of the standard form (2.7), p2 + V (x, τ ), 2M

H(p, x, τ ) =

with a smooth potential V (x, τ ), the momenta can be integrated out, just as in (2.51), and the Euclidean version of the pure x-space path integral (2.52) leads to (2.53): (xb τb |xa τa ) =

Z

(

1 Dx exp − h ¯ N Y

1

≈ q 2π¯ hǫ/M

n=1

hβ ¯

Z

 

0

Z

M dτ (∂τ x)2 + V (x, τ ) 2 dxn

∞

−∞

+1 1 NX M × exp − ǫ h ¯ n=1 2

(

"

q

2πβ/M

)

 

xn − xn−1 ǫ

(2.368) 2

+ V (xn , τn )

#)

.

From this we calculate the quantum-statistical partition function Z =

Z

=

Z

∞ −∞

dx (x h ¯ β|x 0)

dx

Z

x(¯ hβ)=x

x(0)=x

−Ae [x]/¯ h

Dx e

=

I

Dx e−Ae [x]/¯h ,

(2.369)

where Ae [x] is the Euclidean version of the Lagrangian action Ae [x] =

Z

τb

τa

M ′2 dτ x + V (x, τ ) . 2

(2.370)

The prime denotes differentiation with respect to the imaginary Htime. As in the quantum-mechanical partition function in (2.61), the path integral Dx now stands for I NY +1 Z ∞ dxn q Dx ≈ . (2.371) 2π¯ hǫ/M n=1 −∞ q

It contains no extra 1/ 2π¯ hǫ/M factor, as in (2.368), due to the trace integration over the exterior x. The condition x(¯ hβ) = x(0) is most easily enforced by expanding x(τ ) into a Fourier series x(τ ) =

∞ X

m=−∞

√

1 e−iωm τ xm , N +1

(2.372)

142


with the Matsubara frequencies h= ωm ≡ 2πmkB T /¯

2πm , h ¯β

m = 0, ±1, ±2, . . . .

(2.373)

When considered as functions on the entire τ -axis, the paths are periodic in h ¯ β at any τ , i.e., x(τ ) = x(τ + h ¯ β). (2.374) Thus the path integral for the quantum-statistical partition function comprises all periodic paths with a period h ¯ β. In the time-sliced path integral (2.368), the coordinates x(τ ) are needed only at the discrete times τn = nǫ. Correspondingly, the sum over m in (2.372) can be restricted to run from m = −N/2 to N/2 for even N and from −(N − 1)/2 to (N + 1)/2 for odd N (see Fig. 2.3). In order to have a real x(τn ), we must require that xm = x∗−m

(modulo N + 1).

(2.375)

Note that the Matsubara frequencies in the expansion of the paths x(τ ) are now twice as big as the frequencies νm in the quantum fluctuations (2.105) (after analytic continuation of tb − ta to −i¯ h/kB T ). Still, they have about the same total number, since they run over positive and negative integers. An exception is the zero frequency ωm = 0, which is included here, in contrast to the frequencies νm in (2.105) which run only over positive m = 1, 2, 3, . . . . This is necessary to describe paths with arbitrary nonzero endpoints xb = xa = x (included in the trace).

2.10

Quantum Statistics of Harmonic Oscillator

The harmonic oscillator is a good example for solving the quantum-statistical path integral. The τ -axis is sliced at τn = nǫ, with ǫ ≡ ¯hβ/(N + 1) (n = 0, . . . , N + 1), and the partition function is given by the N → ∞ -limit of the product of integrals "Z # N ∞ Y dx n p ZωN = exp −AN h , (2.376) e /¯ 2π¯hǫ/M −∞ n=0

where AN e is the time-sliced Euclidean oscillator action AN e

N +1 M X = xn (−ǫ2 ∇∇ + ǫ2 ω 2 )xn . 2ǫ n=1

(2.377)

Integrating out the xn ’s, we find immediately 1 ZωN = q . detN +1 (−ǫ2 ∇∇ + ǫ2 ω 2 )

(2.378)

Let us evaluate the fluctuation determinant via the product of eigenvalues which diagonalize the matrix −ǫ2 ∇∇ + ǫ2 ω 2 in the sliced action (2.377). They are ǫ2 Ωm Ωm + ǫ2 ω 2 = 2 − 2 cos ωm ǫ + ǫ2 ω 2 ,


143

2.10 Quantum Statistics of Harmonic Oscillator

Figure 2.3 Illustration of the eigenvalues (2.379) of the fluctuation matrix in the action (2.377) for even and odd N . with the Matsubara frequencies ωm . For ω = 0, the eigenvalues are pictured in Fig. 2.3. The action (2.377) becomes diagonal after going to the Fourier components xm . To do this we arrange the real and imaginary parts Re xm and Im xm in a row vector (Re x1 , Im x1 ; Re x2 , Im x2 ; . . . ; Re xn , Im xn ; . . .), and see that it is related to the time-sliced positions xn = x(τn ) by a transformation matrix with the rows Tmn xn

=

(Tm )n xn r 2 1 m m √ , cos = 2π · 1, sin 2π · 1, N +1 N +1 N +1 2 m m cos 2π · 2, sin 2π · 2, . . . N +1 N +1 m m . . . , cos 2π · n, sin 2π · n, . . . xn . (2.380) N +1 N +1 n For each row index m = 0, . . . , N, the column index n runs from zero to N/2 for even N , and to (N + 1)/2 for odd N . In the odd case, the last column sin Nm +1 2π · n with n = (N + 1)/2 vanishes identically and must be dropped, so that the number of columns in Tmn is in both cases N + 1, as it should be. For odd N , the second-last column of Tmn is an alternating sequence √ ±1. Thus, for a proper normalization, it has to be multiplied by an extra normalization factor 1/ 2, just as the elements in the first column. An argument similar to (2.110), (2.111) shows that the resulting matrix is orthogonal. Thus, we can diagonalize the sliced action in (2.377) as follows i  h PN/2 2 2 2 2  ω x + 2 (Ω Ω + ω )|x | for N = even, m m m  0 m=1 M  2 2 N ǫ (2.381) Ae = ω x0 + (Ω(N +1)/2 Ω(N +1)/2 + ω 2 )xN2+1 i 2  P(N −1)/2   + 2 m=1 (Ωm Ωm + ω 2 )|xm |2 for N = odd. Q R∞ Thanks to the orthogonality of Tmn , the measure n −∞ dx(τn ) transforms simply into Z

Z

∞

−∞

dx0

Z

N/2

∞

dx0 −∞

∞

−∞

dx(N +1)/2

YZ

m=1

∞

d Re xm −∞

∞

Z

∞

d Im xm

for

N = even,

−∞

(2.382)

(N −1)/2 Z ∞ Y m=1

Z

−∞

d Re xm

−∞

d Im xm

for

N = odd.

144


By performing the Gaussian integrals we obtain the partition function " N #−1/2 Y 2 2 −1/2 2 2 2 N 2 Zω = detN +1 (−ǫ ∇∇ + ǫ ω ) = (ǫ Ωm Ωm + ǫ ω ) m=0

=

(

N Y 2(1 − cos ωm ǫ) + ǫ2 ω 2

m=0

)−1/2 " =

N Y

m=0

# −1/2 ωm ǫ 2 2 4 sin +ǫ ω . 2 2

(2.383)

Thanks to the periodicity of the eigenvalues under the replacement n → n + N + 1, the result has become a unique product expression for both even and odd N . It is important to realize that contrary to the fluctuation factor (2.155) in the real-time amplitude, the partition function (2.383) contains the square root of only positive eigenmodes as a unique result of Gaussian integrations. There are no phase subtleties as in the Fresnel integral (1.333). To calculate the product, we observe that upon decomposing ωm ǫ ωm ǫ ωm ǫ sin2 = 1 + cos 1 − cos , (2.384) 2 2 2 the sequence of first factors

1 + cos

πm ωm ǫ ≡ 1 + cos 2 N +1

(2.385)

runs for m = 1, . . . N through the same values as the sequence of second factors 1 − cos

ωm ǫ πm N +1−m = 1 − cos ≡ 1 + cos π , 2 N +1 N +1

(2.386)

except in an opposite order. Thus, separating out the m = 0 -term, we rewrite (2.383) in the form ZωN

1 = ǫω

"

N Y

ωm ǫ 2 1 − cos 2 m=1

#−1 "

N Y

m=1

ǫ2 ω 2 1+ ǫ 4 sin2 ωm 2

!#−1/2 .

(2.387)

The first factor on the right-hand side is the quantum-mechanical fluctuation determinant of the free-particle determinant detN (−ǫ2 ∇∇) = N + 1 [see (2.118)], so that we obtain for both even and odd N " N !#−1/2 ǫ2 ω 2 kB T Y N Zω = 1+ . (2.388) ǫ ¯hω m=1 4 sin2 ωm 2 To evaluate the remaining product, we must distinguish again between even and odd cases of N . For even N , where every eigenvalue occurs twice (see Fig. 2.3), we obtain  !−1 N/2 2 2 Y k T ǫ ω B   . ZωN = 1+ (2.389) ¯hω m=1 4 sin2 Nmπ +1

For odd N , the term with m = (N + 1)/2 occurs only once and must be treated separately so that  !−1 −1)/2 2 2 1/2 (NY 2 2 kB T  ǫ ω ǫ ω  . ZωN = 1+ 1+ (2.390) 2 πm ¯hω 4 4 sin N +1 m=1 We now introduce the parameter ω ˜ e , the Euclidean analog of (2.156), via the equations sin i

ω ˜eǫ ωǫ ≡i , 2 2

sinh

ω ˜eǫ ωǫ ≡ . 2 2

(2.391)


145

2.10 Quantum Statistics of Harmonic Oscillator In the odd case, the product formula16 # " (N −1)/2 Y sin2 x 2 sin[(N + 1)x] 1− = 2 mπ sin 2x (N + 1) sin (N +1) m=1 [similar to (2.158)] yields, with x = ω ˜ e ǫ/2, −1 1 sinh[(N + 1)˜ ωe ǫ/2] kB T . ZωN = ¯hω sinh(˜ ωe ǫ/2) N +1 In the even case, the formula17 " N/2 Y 1− m=1

sin2 x sin2 (Nmπ +1)

#

=

1 sin[(N + 1)x] , sin x (N + 1)

(2.392)

(2.393)

(2.394)

produces once more the same result as in Eq. (2.393). Inserting Eq. (2.391) leads to the partition function on the sliced imaginary time axis: ZωN =

1 . 2 sinh(¯ hω ˜ e β/2)

(2.395)

The partition function can be expanded into the following series ZωN = e−¯hω˜ e /2kB T + e−3¯hω˜ e /2kB T + e−5¯hω˜ e /2kB T + . . . .

(2.396)

By comparison with the general spectral expansion (2.321), we display the energy eigenvalues of the system: 1 En = n + ¯hω ˜e. (2.397) 2 They show the typical linearly rising oscillator sequence with ω ˜e =

ωǫ 2 arsinh ǫ 2

(2.398)

playing the role of the frequency on the sliced time axis, and ¯hω ˜ e /2 being the zero-point energy. In the continuum limit ǫ → 0, the time-sliced partition function ZωN goes over into the usual oscillator partition function 1 Zω = . (2.399) 2 sinh(β¯hω/2) In D dimensions this becomes, of course, [2 sinh(β¯hω/2)]−D , due to the additivity of the action in each component of x. Note that the continuum limit of the product in (2.388) can also be taken factor by factor. Then Zω becomes " ∞ #−1 kB T Y ω2 Zω = 1+ 2 . (2.400) ¯hω m=1 ωm According to formula (2.166), the product and we find with x = h ¯ ωβ/2 Zω = 16 17

Q∞

m=1

1+

x2 m2 π 2

converges rapidly against sinh x/x

kB T ¯hω/2kB T 1 = . ¯hω sinh(¯ hω/2kB T ) 2 sinh(β¯hω/2)

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.391.1. ibid., formula 1.391.3.

(2.401)

146


As discussed after Eq. (2.176), the continuum limit can be taken in each factor since the product in (2.388) contains only ratios of frequencies. Just as in the quantum-mechanical case, this procedure of obtaining the continuum limit can be summarized in the sequence of equations arriving at a ratio of differential operators −1/2 ZωN = detN +1 (−ǫ2 ∇∇ + ǫ2 ω 2 ) " #−1/2 ′ −1/2 detN +1 (−ǫ2 ∇∇ + ǫ2 ω 2 ) 2 = detN +1 (−ǫ ∇∇) det′N +1 (−ǫ2 ∇∇) −1/2 −1 ∞ 2 ǫ→0 kB T det(−∂τ2 + ω 2 ) kB T Y ωm + ω2 = . (2.402) − −−→ 2 ¯h ¯hω m=1 ωm det′ (−∂τ2 )

In the ω = 0 -determinants, the zero Matsubara frequency is excluded to obtain a finite expression. This is indicated by a prime. The differential operator −∂τ2 acts on real functions which are periodic 2 under the replacement τ → τ +¯ hβ. Remember that each eigenvalue ωm of −∂τ2 occurs twice, except for the zero frequency ω0 = 0, which appears only once. Let us finally mention that the results of this section could also have been obtained directly from the quantum-mechanical amplitude (2.168) [or with the discrete times from (2.192)] by an analytic continuation of the time difference tb − ta to imaginary values −i(τb − τa ): r 1 ω (xb τb |xa τa ) = p 2π¯h/M sinh ω(τb − τa ) 1 Mω 2 2 × exp − [(x + xa ) cosh ω(τb − τa ) − 2xb xa ] . (2.403) 2¯h sinh ω(τb − τa ) b

By setting x = xb = xa and integrating over x, we obtain [compare (2.326)] s Z ∞ 1 ω(τb − τa ) Zω = dx (x τb |x τa ) = p sinh[ω(τ 2π¯h(τb − τa )/M b − τa )] −∞ p 2π¯h sinh[ω(τb − τa )]/ωM 1 × = . 2 sinh[ω(τb − τa )/2] 2 sinh[ω(τb − τa )/2]

(2.404)

Upon equating τb − τa = h ¯ β, we retrieve the partition function (2.399). A similar treatment of the discrete-time version (2.192) would have led to (2.395). The main reason for presenting an independent direct evaluation in the space of real periodic functions was to display the frequency structure of periodic paths and to see the difference with respect to the quantum-mechanical paths with fixed ends. We also wanted to show how to handle the ensuing product expressions. For applications in polymer physics (see Chapter 15) one also needs the partition function of all path fluctuations with open ends s Z ∞ Z ∞ 1 ω(τb − τa ) 2π¯h open = dxb dxa (xb τb |xa τa ) = p Zω 2π¯h(τb − τa )/M sinh[ω(τb − τa )] M ω −∞ −∞ r 2π¯h 1 p = . (2.405) M ω sinh[ω(τb − τa )] √ The prefactor is 2π times the length scale λω of Eq. (2.296).

2.11

Time-Dependent Harmonic Potential

It is often necessary to calculate thermal fluctuation determinants for the case of a time-dependent frequency Ω(τ ) which is periodic under τ → τ + h ¯ β. As in Section 2.3.6, we consider the amplitude Z Z R 2 Dp − ττb dτ [−ipx+p ˙ /2M+MΩ2 (τ )x2 /2]/¯ h a (xb τb |xa τa ) = D′ x e 2π¯h H. Kleinert, PATH INTEGRALS

147

2.11 Time-Dependent Harmonic Potential

=

Z

Dxe

−

R τb τa

dτ [M x˙ 2 +Ω2 (τ )x2 ]/2¯ h

.

(2.406)

The time-sliced fluctuation factor is [compare (2.195)] F N (τa − τb ) = detN +1 [−ǫ2 ∇∇ + ǫΩ2 (τ )]−1/2 ,

(2.407)

with the continuum limit F (τa − τb ) =

kB T ¯h

det(−∂τ2 + Ω2 (τ )) det′ (−∂τ2 )

−1/2

.

(2.408)

Actually, in the thermal case it is preferable to use the oscillator result for normalizing the fluctuation factor, rather than the free-particle result, and to work with the formula −1/2 det(−∂τ2 + Ω2 (τ )) 1 . (2.409) F (τb , τa ) = 2 sinh(β¯hω/2) det(−∂τ2 + ω 2 ) This has the advantage that the determinant in the denominator contains no zero eigenvalue which would require a special treatment as in (2.402); the operator −∂τ2 + ω 2 is positive. As in the quantum-mechanical case, the spectrum of eigenvalues is not known for general Ω(τ ). It is, however, possible to find a differential equation for the entire determinant, analogous to the Gelfand-Yaglom formula (2.202), with the initial condition (2.207), although the derivation is now much more tedious. The origin of the additional difficulties lies in the periodic boundary condition which introduces additional nonvanishing elements −1 in the upper right and lower left corners of the matrix −ǫ2 ∇∇ [compare (2.97)]:   2 −1 0 ... 0 0 −1  −1 2 −1 ... 0 0 0     .. ..  . 2 (2.410) −ǫ ∇∇ =  . .     0 0 0 . . . −1 2 −1  −1 0 0 . . . 0 −1 2 To better understand the relation with the previous result we shall replace the corner elements −1 by −α which can be set equal to zero at the end, for a comparison. Adding to −ǫ2 ∇∇ a time-dependent frequency matrix we then consider the fluctuation matrix   2 + ǫ2 Ω2N +1 −1 0 ... 0 −α   −1 2 + ǫ2 Ω2N −1 . . . 0 0   −ǫ2 ∇∇ + ǫ2 Ω2 =  . .. ..   . . −α

0

. . . −1

0

2 + ǫ2 Ω21

(2.411)

˜ N +1 . Expanding it along the Let us denote the determinant of this (N + 1) × (N + 1) matrix by D first column, it is found to satisfy the equation ˜ N +1 = (2 + ǫ2 Ω2 ) D N +1  2 + ǫ2 Ω2N  .. ×detN  . 0



   +detN   

−1 0 −1 2 + ǫ2 Ω2N −1 0 −1 .. . 0

0

(2.412) −1 0 0

0

...

0 .. .

. . . −1

0 −1 2 + ǫ2 Ω2N −2 0

0

2 + ǫ2 Ω21

0 ... 0 ... −1 . . . 0

0 0 0

   −α 0 0 .. .

. . . −1 2 + ǫ2 Ω21

      

148


N +1

+(−1)



   αdetN   

−1 2 + ǫ2 Ω2N −1 .. .

0 −1 2 + ǫ2 Ω2N −1

0

0

0 ... 0 ... −1 . . . 0

−α 0 0 .. .

0 0 0

. . . 2 + ǫ2 Ω22

−1



   .  

The first determinant was encountered before in Eq. (2.197) (except that there it appeared with −ǫ2 Ω2 instead of ǫ2 Ω2 ). There it was denoted by DN , satisfying the difference equation −ǫ2 ∇∇ + ǫ2 Ω2N +1 DN = 0, (2.413) with the initial conditions

D1

= 2 + ǫ2 Ω21 ,

D2

= (2 + ǫ2 Ω21 )(2 + ǫ2 Ω22 ) − 1.

(2.414)

The second determinant in (2.412) can be expanded with respect to its first column yielding −DN −1 − α.

(2.415)

The third determinant is more involved. When expanded along the first column it gives (−1)N 1 + (2 + ǫ2 Ω2N )HN −1 − HN −2 ,

(2.416)

with the (N − 1) × (N − 1) determinant HN −1 ≡ (−1)N −1 

(2.417)

0  2 + ǫ2 Ω2N −1   −1 ×detN −1   ..  . 0

0 −1 2 + ǫ2 Ω2N −2 0

0 ... 0 ... −1 . . . 0

0 0 0

−α 0 0 .. .

0 0 0

. . . −1 2 + ǫ2 Ω22

−1



   .  

By expanding this along the first column, we find that HN satisfies the same difference equation as DN : (−ǫ2 ∇∇ + ǫ2 Ω2N +1 )HN = 0.

(2.418)

However, the initial conditions for HN are different: 0 −α H2 = = α(2 + ǫ2 Ω22 ), 2 + ǫ2 Ω22 −1 0 0 −α H3 = − 2 2 2 + ǫ Ω −1 0 3 −1 2 + ǫ2 Ω22 −1 = α (2 + ǫ2 Ω22 )(2 + ǫ2 Ω23 ) − 1 .

(2.419)

(2.420)

They show that HN is in fact equal to αDN −1 , provided we shift Ω2N by one lattice unit upwards to Ω2N +1 . Let us indicate this by a superscript +, i.e., we write + HN = αDN −1 .

(2.421)

Thus we arrive at the equation ˜ N +1 D

= (2 + ǫ2 Ω2N )DN − DN −1 − α

+ + −α[1 + (2 + ǫ2 Ω2N )αDN −2 − αDN −3 ].

(2.422)


149

2.11 Time-Dependent Harmonic Potential + Using the difference equations for DN and DN , this can be brought to the convenient form

˜ N +1 = DN +1 − α2 D+ − 2α. D N −1

(2.423)

For quantum-mechanical fluctuations with α = 0, this reduces to the earlier result in Section 2.3.6. For periodic fluctuations with α = 1, the result is ˜ N +1 = DN +1 − D+ − 2. D N −1

(2.424)

+ ˙ In the continuum limit, DN +1 − DN −1 tends towards 2Dren , where Dren (τ ) = Da (t) is the imaginary-time version of the Gelfand-Yaglom function in Section 2.4 solving the homogenous differential equation (2.208), with the initial conditions (2.206) and (2.207), or Eqs. (2.221). The corresponding properties are now: 2 −∂τ + Ω2 (τ ) Dren (τ ) = 0, Dren (0) = 0, D˙ ren (0) = 1. (2.425)

In terms of Dren (τ ), the determinant is given by the Gelfand-Yaglom-like formula ǫ→0

−−→ 2[D˙ ren(¯ hβ) − 1], det(−ǫ2 ∇∇ + ǫΩ2 )T −

(2.426)

and the partition function reads 1 ZΩ = r h i. ˙ 2 Dren (¯ hβ) − 1

(2.427)

The result may be checked by going back to the amplitude (xb tb |xa ta ) of Eq. (2.255), continuing it to imaginary times t = iτ , setting xb = xa = x, and integrating over all x. The result is 1 , ZΩ = q 2 D˙ a (tb ) − 1

tb = i¯hβ,

(2.428)

in agreement with (2.427). As an example, take the harmonic oscillator for which the solution of (2.425) is Dren (τ ) =

1 sinh ωτ ω

(2.429)

[the analytically continued (2.209)]. Then 2[D˙ ren (τ ) − 1] = 2(cosh β¯hω − 1) = 4 sinh2 (β¯hω/2),

(2.430)

and we find the correct partition function: Zω

= =

n o−1/2 2[D˙ ren(τ ) − 1] 1 . 2 sinh(β¯hω/2)

τ =¯ hβ

(2.431)

On a sliced imaginary-time axis, the case of a constant frequency Ω2 ≡ ω 2 is solved as follows. From Eq. (2.203) we take the ordinary Gelfand-Yaglom function DN , and continue it to Euclidean ω ˜ e , yielding the imaginary-time version DN =

sinh(N + 1)ω˜e ǫ . sinh ω˜e ǫ

(2.432)

150


+ Then we use formula (2.424), which simplifies for a constant Ω2 ≡ ω 2 for which DN −1 = DN −1 , and calculate

˜ N +1 D

= =

1 [sinh(N + 2)ω˜e ǫ − sinh N ω˜e ǫ] − 2 sinh ω˜e ǫ 2 [cosh(N + 1)ω˜e ǫ − 1] = 4 sinh2 [(N + 1)ω˜e ǫ/2].

(2.433)

Inserting this into Eq. (2.378) yields the partition function 1 1 Zω = q = , 2 sinh(¯ h ω˜e β/2) ˜ N +1 D

(2.434)

in agreement with (2.395).

2.12

Functional Measure in Fourier Space

There exists an alternative definition for the quantum-statistical path integral which is useful for some applications (for example in Section 2.13 and in Chapter 5). The limiting product formula (2.402) suggests that instead of summing over all zigzag configurations of paths on a sliced time axis, a path integral may be defined with the help of the Fourier components of the paths on a continuous time axis. As in (2.372), but with a slightly different normalization of the coefficients, we expand these paths here as x(τ ) = x0 + η(τ ) ≡ x0 +

∞ X

xm eiωm τ + c.c. ,

m=1

x0 = real,

x−m ≡ x∗m . (2.435)

Note that the temporal integral over the time-dependent fluctuations η(τ ) is zero, dτ η(τ ) = 0, so that the zero-frequency component x0 is the temporal average 0 of the fluctuating paths:

R ¯h/kB T

x0 = x¯ ≡

kB T h ¯

Z

0

h/kB T ¯

dτ x(τ ).

(2.436)

In contrast to (2.372) which was valid on a sliced time axis and was therefore subject to a restriction on the range of the m-sum, the present sum is unrestricted and runs over all Matsubara frequencies ωm = 2πmkB T /¯ h = 2πm/¯ hβ. In terms of xm , the Euclidean action of the linear oscillator is Ae

M ¯h/kB T = dτ (x˙ 2 + ω 2 x2 ) 2 0" # ∞ M¯ h ω2 2 X 2 2 2 x + = (ω + ω )|xm | . kB T 2 0 m=1 m Z

(2.437)

The integration variables of the time-sliced path integral were transformed to the Q R∞ Fourier components xm in Eq. (2.380). The product of integrals n −∞ dx(τn ) H. Kleinert, PATH INTEGRALS

151

2.12 Functional Measure in Fourier Space

turned into the product (2.382) of integrals over real and imaginary parts of xm . In the continuum limit, the result is ∞

Z

−∞

dx0

∞ Z Y

∞

m=1 −∞

d Re xm

Z

∞

−∞

d Im xm .

(2.438)

Placing the exponential e−Ae /¯h with the frequency sum (2.437) into the integrand, 2 the product of Gaussian integrals renders a product of inverse eigenvalues (ωm +ω 2 )−1 for m = 1, . . . , ∞, with some infinite factor. This may be determined by comparison with the known continuous result (2.402) for the harmonic partition function. The infinity is of the type encountered in Eq. (2.176), and must be divided out of the measure (2.438). The correct result (2.400) is obtained from the following measure of integration in Fourier space I

Dx ≡

Z

# "Z ∞ ∞ Z ∞ d Re x d Im x dx0 Y m m . 2 le (¯ hβ) m=1 −∞ −∞ πkB T /Mωm

∞

−∞

(2.439)

2 + ω 2 )−1 discussed after The divergences in the product over the factors (ωm 2 Eq. (2.176) are canceled by the factors ωm in the measure. It will be convenient to introduce a short-hand notation for the measure on the right-hand side, writing it as

I

Dx ≡

Z

∞

−∞

dx0 I ′ D x. le (¯ hβ)

(2.440)

hβ) associated with β The denominator of the x0 -integral is the length scale le (¯ defined in Eq. (2.345). Then we calculate Zωx0 ≡

I

′

−Ae /¯ h

D xe

=

" ∞ Z Y

m=1

∞

−∞

#

d Re xm d Im xm −M ¯h[ω2 x20 /2+P∞ (ωm 2 +ω 2 )|x |2 /k T m ] B m=1 e 2 −∞ πkB T /Mωm

Z

∞

−M ω 2 x20 /2kB T

=e

∞ Y

m=1

"

2 ωm + ω2 2 ωm

#−1

.

(2.441)

The final integral over the zero-frequency component x0 yields the partition function Zω =

I

−Ae /¯ h

Dx e

=

Z

∞

−∞

∞ 2 dx0 x0 kB T Y ωm + ω2 Zω = 2 le (¯ hβ) h ¯ ω m=1 ωm

"

#−1

,

(2.442)

as in (2.402). The same measure can be used for the more general amplitude (2.406), as is obvious from (2.408). With the predominance of the kinetic term in the measure of path integrals [the divergencies discussed after (2.176) stem only from it], it can easily be shown that the same measure is applicable to any system with the standard kinetic term.

152


It is also possible to find a Fourier decomposition of the paths and an associated integration measure for the open-end partition function in Eq. (2.405). We begin by considering the slightly reduced set of all paths satisfying the Neumann boundary conditions x(τ ˙ b ) = vb = 0. (2.443) x(τ ˙ a ) = va = 0, They have the Fourier expansion x(τ ) = x0 + η(τ ) = x0 +

∞ X

n=1

xn cos νn (τ − τa ),

νn = nπ/β.

(2.444)

The frequencies νn are the Euclidean version of the frequencies (3.64) for Dirichlet boundary conditions. Let us calculate the partition function for such paths by analogy with the above periodic case by a Fourier decomposition of the action M Ae = 2

Z

h/kB T ¯

0

∞ M¯ h ω2 2 1 X dτ (x˙ + ω x ) = x0 + (ν 2 + ω 2 )x2n , kB T 2 2 n=1 n 2

"

2 2

#

(2.445)

and of the measure I

Dx ≡

Z

∞

≡

Z

∞

−∞

−∞

∞ ∞ dx0 Y le (¯ hβ) n=1 −∞ I dx0 D ′ x. le (¯ hβ)

"Z

Z

d xn πkB T /2Mνn2

∞

−∞

#

(2.446)

We now perform the path integral over all fluctuations at fixed x0 as in (2.441): ZωN,x0 ≡

I

′

−Ae /¯ h

D xe

=

" ∞ Z Y

n=1

∞

−∞

Z

∞

−∞

−M ω 2 x20 /2kB T

=e

#

P∞ 2 2 d xn −M ¯ h[ω 2 x20 /2+ n=1 (νn +ω )|xn |2 ]/kB T e πkB T /2Mνn2 ∞ Y

n=1

"

νn2 + ω 2 νn2

#−1

.

(2.447)

Using the product formula (2.176), this becomes ZωN,x0

=

s

ω¯ hβ M exp −β ω 2x20 . sinh ω¯hβ 2

(2.448)

The final integral over the zero-frequency component x0 yields the partition function 1 ZωN = le (¯ hβ)

s

2π¯ h 1 √ . Mω sinh ω¯ hβ

(2.449)

We have replaced the denominator in the prefactor 1/le (¯ hβ) by the length scale 1/le (¯ hβ) of Eq. (2.345). Apart from this prefactor, the Neumann partition function coincides precisely with the open-end partition function Zωopen in Eq. (2.405). What is the reason for this coincidence up to a trivial factor, even though the paths satisfying Neumann boundary conditions do not comprise all paths with open H. Kleinert, PATH INTEGRALS

153

2.13 Classical Limit

ends. Moreover, the integrals over the endpoints in the defining equation (2.405) does not force the endpoint velocities, but rather endpoint momenta to vanish. Indeed, recalling Eq. (2.182) for the time evolution amplitude in momentum space we can see immediately that the partition function with open ends Zωopen in Eq. (2.405) is identical to the imaginary-time amplitude with vanishing endpoint momenta: Zωopen = (pb h ¯ β|pa 0)|pb =pa =0 .

(2.450)

Thus, the sum over all paths with arbitrary open ends is equal to the sum of all paths satisfying Dirichlet boundary conditions in momentum space. Only classically, the vanishing of the endpoint momenta implies the vanishing of the endpoint velocities. From the general discussion of the time-sliced path integral in phase space in Section 2.1 we know that fluctuating paths have M x˙ 6= p. The fluctuations of the difference are controlled by a Gaussian exponential of the type (2.51). This leads to the explanation of the trivial factor between Zωopen and ZωN . The difference between M x˙ and p appears only in the last short-time intervals at the ends. But at short time, the potential does not influence the fluctuations in (2.51). This is the reason why the fluctuations at the endpoints contribute only a trivial overall factor le (¯ hβ) N to the partition function Zω .

2.13

Classical Limit

The alternative measure of the last section serves to show, somewhat more convincingly than before, that in the high-temperature limit the path integral representation of any quantum-statistical partition function reduces to the classical partition function as stated in Eq. (2.338). We start out with the Lagrangian formulation (2.368). Inserting the Fourier decomposition (2.435), the kinetic term becomes Z

hβ ¯ 0

dτ

∞ M 2 M¯ h X x˙ = ω 2 |xm |2 , 2 kB T m=1 m

(2.451)

and the partition function reads Z=

I

Z ∞ ∞ X M X 1 ¯h/kB T 2 2 ′ Dx exp − ωm |xm | − dτ V (x0 + xm e−iωm τ ) . (2.452) kB T m=1 h ¯ 0 m=−∞ "

#

The summation symbol with a prime implies the absence of the m = 0 -term. The measure is the product (2.439) of integrals of all Fourier components. We now observe that for large temperatures, the Matsubara frequencies for m 6= 0 diverge like 2πmkB T /¯ h . This has the consequence that the Boltzmann factor for the x fluctuations becomes sharply peaked around xm = 0. The average size ofxm is qm6=0 √ P ′ −iωm τ kB T /M/ωm = h ¯ /2πm MkB T . If the potential V x0 + ′ ∞ is a m=−∞ xm e smooth function of its arguments, we can approximate it by V (x0 ), terms containing higher powers of xm . For large temperatures, these are small on the average and

154


can be ignored. The leading term V (x0 ) is time-independent. Hence we obtain in the high-temperature limit T →∞

Z− −−→

I

∞ M X 1 2 V (x0 ) . Dx exp − ωm |xm |2 − kB T m=1 kB T

"

#

(2.453)

The right-hand side is quadratic in the Fourier components xm . With the measure of integration (2.439), we perform the integrals over xm and obtain T →∞

Z− −−→ Zcl =

Z

∞

−∞

dx0 −V (x0 )/kB T e . le (¯ hβ)

(2.454)

This agrees with the classical statistical partition function (2.344). The derivation reveals an important prerequisite for the validity of the classical limit: It holds only for sufficiently smooth potentials. We shall see in Chapter 8 that for singular potentials such as −1/|x| (Coulomb), 1/|x|2 (centrifugal barrier), 1/ sin2 θ (angular barrier), this condition is not fulfilled and the classical limit is no longer given by (2.454). The particle distribution ρ(x) at a fixed x does not have this problem. It always tends towards the naively expected classical limit (2.346): T →∞

ρ(x) − −−→ Zcl−1 e−V (x)/kB T .

(2.455)

The convergence is nonuniform in x, which is the reason why the limit does not always carry over to the integral (2.454). This will be an important point in deriving in Chapter 12 a new path integral formula valid for singular potentials. At first, we shall ignore such subtleties and continue with the conventional discussion valid for smooth potentials.

2.14

Calculation Techniques on Sliced Time Axis via Poisson Formula

In the previous sections we have used tabulated product formulas such as (2.117), (2.158), (2.166), (2.392), (2.394) to find fluctuation determinants on a finite sliced time axis. With the recent interest in lattice models of quantum field theories, it is useful to possess an efficient calculational technique to derive such product formulas (and related sums). Consider, as a typical example, the quantum-statistical partition function for a harmonic oscillator of frequency ω on a time axis with N + 1 slices of width ǫ, Z=

N Y

[2(1 − cos ωm ǫ) + ǫ2 ω 2 ]−1/2 ,

(2.456)

m=0

with the product running over all Matsubara frequencies ωm = 2πmkB T /¯h. Instead of dealing with this product it is advantageous to consider the free energy F = −kB T log Z =

N X 1 kB T log[2(1 − cos ωm ǫ) + ǫ2 ω 2 ]. 2 m=0

(2.457)


155

2.14 Calculation Techniques on Sliced Time Axis via Poisson Formula

We now observe that by virtue of Poisson’s summation formula (1.205), the sum can be rewritten as the following combination of a sum and an integral: ∞ Z 2π X 1 dλ iλn(N +1) e log[2(1 − cos λ) + ǫ2 ω 2 ]. (2.458) F = kB T (N + 1) 2 2π n=−∞ 0 The sum over n squeezes λ to integer multiples of 2π/(N + 1) = ωm ǫ which is precisely what we want. We now calculate integrals in (2.458): Z 2π dλ iλn(N +1) e log[2(1 − cos λ) + ǫ2 ω 2 ]. (2.459) 2π 0 For this we rewrite the logarithm of an arbitrary positive argument as the limit Z ∞ dτ −τ a/2 log a = lim − e + log(2δ) + γ, δ→0 τ δ where ′

γ ≡ −Γ (1)/Γ(1) = lim

N →∞

N X 1 − log N n n=1

!

≈ 0.5773156649 . . .

is the Euler-Mascheroni constant. Indeed, the function Z ∞ dt −t E1 (x) = e t x

(2.460)

(2.461)

(2.462)

is known as the exponential integral with the small-x expansion18 E1 (x) = −γ − log x −

∞ X (−x)k

k=1

kk!

.

(2.463)

With the representation (2.460) for the logarithm, the free energy can be rewritten as Z ∞ Z ∞ 1 X dτ 2π dλ iλn(N +1)−τ [2(1−cos λ)+ǫ2 ω2 ]/2 F = lim − e − δn0 [log(2δ) + γ] . 2ǫ n=−∞ δ→0 τ 0 2π δ

(2.464)

The integral over λ is now performed19 giving rise to a modified Bessel function In(N +1) (τ ): F =

Z ∞ ∞ 2 2 1 X dτ lim − In(N +1) (τ )e−τ (2+ǫ ω )/2 − δn0 [log(2δ) + γ] . 2ǫ n=−∞ δ→0 τ δ

If we differentiate this with respect to ǫ2 ω 2 ≡ m2 , we obtain ∞ Z ∞ 2 ∂F 1 X = dτ In(N +1) (τ )e−τ (2+m )/2 ∂m2 4ǫ n=−∞ 0 and perform the τ -integral, using the formula valid for Re ν > −1, Re α > Re µ p p Z ∞ α2 − µ2 )−ν , α2 − µ2 )ν −τ α ν (α − −ν (α − p p dτ Iν (µτ )e =µ =µ , α2 − µ2 α2 − µ2 0 18 19

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.214.2. I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 8.411.1 and 8.406.1.

(2.465)

(2.466)

(2.467)

156


to find " #|n|(N +1) p ∞ m2 + 2 − (m2 + 2)2 − 4 ∂F 1 X 1 p = . ∂m2 2ǫ n=−∞ (m2 + 2)2 − 4 2

(2.468)

From this we obtain F by integration over m2 + 1. The n = 0 -term under the sum gives p log[(m2 + 2 + (m2 + 2)2 − 4 )/2] + const (2.469)

and the n 6= 0 -terms: p 1 − [(m2 + 2 + (m2 + 2)2 − 4 )/2]−|n|(N +1) + const , |n|(N + 1)

(2.470)

where the constants of integration can depend on n(N + 1). They are adjusted by going to the limit m2 → ∞ in (2.465). There the integral is dominated by the small-τ regime of the Bessel functions 1 z α Iα (z) ∼ [1 + O(z 2 )], (2.471) |α|! 2 and the first term in (2.465) becomes Z ∞ 1 dτ τ |n|(N +1) −τ m2 /2 e − (|n|(N + 1))! δ τ 2 log m2 + γ + log(2δ) n=0 ≈ . −(m2 )−|n|(N +1) /|n|(N + 1) n = 6 0

(2.472)

The limit m2 → ∞ in (2.469), (2.470) gives, on the other hand, log m2 + const and −(m2 )−|n|(N +1) /|n|(N + 1) + const , respectively. Hence the constants of integration must be zero. We can therefore write down the free energy for N + 1 time steps as F

= =

N 1 X log[2(1 − cos(ωm ǫ)) + ǫ2 ω 2 ] 2β m=0 ( h i p 1 log ǫ2 ω 2 + 2 + (ǫ2 ω 2 + 2)2 − 4 2 2ǫ

(2.473)

∞ i−|n|(N +1) p 2 X 1 h 2 2 − ǫ ω + 2 + (ǫ2 ω 2 + 2)2 − 4 2 N + 1 n=1 n

)

Here it is convenient to introduce the parameter nh i o p ǫ˜ ωe ≡ log ǫ2 ω 2 + 2 + (ǫ2 ω 2 + 2)2 − 4 2 ,

.

(2.474)

which satisfies

cosh(ǫ˜ ωe ) = (ǫ2 ω 2 + 2)/2, or

sinh(ǫ˜ ωe ) =

p (ǫ2 ω 2 + 2)2 − 4/2,

(2.475)

sinh(ǫ˜ ωe /2) = ǫω/2. Thus it coincides with the parameter introduced in (2.391), which brings the free energy (2.473) to the simple form " # ∞ X 2 1 −ǫ˜ωe n(N +1) ¯h F = ω ˜e − e 2 ǫ(N + 1) n=1 n 1 = ¯hω ˜ e + 2kB T log(1 − e−β¯hω˜ e ) 2 1 = log [2 sinh(β¯hω ˜ e /2)] , (2.476) β H. Kleinert, PATH INTEGRALS

2.15 Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization 157 whose continuum limit is ǫ→0

F =

2.15

1 ¯hω 1 log [2 sinh (β¯hω/2)] = + log(1 − e−β¯hω ). β 2 β

(2.477)

Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization

A slight modification of the calculational techniques developed in the last section for the quantum partition function of a harmonic oscillator can be used to define the harmonic path integral in a way which neither requires time slicing, as in the original Feynman expression (2.64), nor a precise specification of the integration measure in terms of Fourier components, as in Section 2.12. The path integral for the partition function Zω =

I

Dxe−

R h¯ β 0

M [x˙ 2 (τ )+ω 2 x2 (τ )]/2

=

I

Dxe−

R h¯ β 0

M x(τ )[−∂τ2 +ω 2 ]x(τ )/2

(2.478)

is formally evaluated as 1 1 2 2 = e− 2 Tr log(−∂τ +ω ) . Zω = q Det(−∂τ2 + ω 2 )

(2.479)

Since the determinant of an operator is the product of all its eigenvalues, we may write, again formally, Y 1 √ 2 Zω = . (2.480) ′ ω + ω2 ω′ The product runs over an infinite set of quantities which grow with ω ′2 , thus being certainly divergent. It may be turned into a divergent sum by rewriting Zω as 1

Zω ≡ e−Fω /kB T = e− 2

P

ω′

log(ω ′ 2 +ω 2 )

.

(2.481)

This expression has two unsatisfactory features. First, it requires a proper definition of the formal sum over a continuous set of frequencies. Second, the logarithm of 2 the dimensionful arguments ωm + ω 2 must be turned into a meaningful expression. The latter problem would be removed if we were able to exchange the logarithm P by log[(ω ′ 2 + ω 2 )/ω 2 ]. This would require the formal sum ω′ log ω 2 to vanish. We shall see below in Eq. (2.506) that this is indeed one of the pleasant properties of analytic regularization. At finite temperatures, the periodic boundary conditions along the imaginarytime axis make the frequencies ω ′ in the spectrum of the differential operator −∂τ2 +ω 2 discrete, and the sum in the exponent of (2.481) becomes a sum over all Matsubara frequencies ωm = 2πkB T /¯ h (m = 0, ±1, ±2, . . .): ∞ 1 X 2 Zω = exp − log(ωm + ω 2) . 2 m=−∞

"

#

(2.482)

158


For the free energy Fω ≡ (1/β) log Zω , this implies ∞ 1 1 X 2 2 2 Fω = Tr log(−∂τ + ω ) = log(ωm + ω 2). per 2β 2β m=−∞

(2.483)

where the subscript per emphasizes the periodic boundary conditions in the τ interval (0, h ¯ β).

2.15.1

Zero-Temperature Evaluation of Frequency Sum

In the limit T → 0, the sum in (2.483) goes over into an integral, and the free energy becomes 1 h ¯ = Fω ≡ Tr log(−∂τ2 + ω 2) ±∞ 2β 2

Z

∞ −∞

dω ′ log(ω ′2 + ω 2), 2π

(2.484)

where the subscript ±∞ indicates the vanishing boundary conditions of the eigenfunctions at τ = ±∞. Thus, at low temperature, we can replace the frequency sum in the exponent of (2.481) by X ω′

− −−→ h ¯β T →0

Z

∞

−∞

dω ′ . 2π

(2.485)

This could have been expected on the basis of Planck’s rules for the phase space invoked earlier on p. 97 to explain the measure of path integration. According to theseRrules, the volume element in the phase space of energy and time has the meaR sure dt dE/h = dt dω/2π. If the integrand is independent of time, the temporal integral produces an overall factor , which for the imaginary-time interval (0, h ¯ β) of statistical mechanics is equal to h ¯β = h ¯ /kB T , thus explaining the integral version of the sum (2.485). The integral on the right-hand side of (2.484) diverges at large ω ′. This is called an ultraviolet divergence (UV-divergence), alluding to the fact that the ultraviolet regime of light waves contains the high frequencies of the spectrum. The important observation is now that the divergent integral (2.484) can be made finite by a mathematical technique called analytic regularization.20 This is based on rewriting the logarithm log(ω ′2 + ω 2 ) in the derivative form:

d log(ω + ω ) = − (ω ′2 + ω 2 )−ǫ . dǫ ǫ=0 ′2

2

(2.486)

Equivalently, we may obtain the logarithm from an ǫ → 0 -limit of the function 1 1 lMS (ǫ) = − (ω ′2 + ω 2 )−ǫ + . ǫ ǫ

(2.487)

20

G. ’t Hooft and M. Veltman, Nucl. Phys. B 44 , 189 (1972). Analytic regularization is at present the only method that allows to renormalize nonabelian gauge theories without destroying gauge invariance. See also the review by G. Leibbrandt, Rev. Mod. Phys. 74 , 843 (1975). H. Kleinert, PATH INTEGRALS

2.15 Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization 159

The subtraction of the pole term 1/ǫ is commonly referred to a minimal subtraction. Indicating this process by a subscript MS, we may write 1 lMS (ǫ) = − (ω ′2 + ω 2 )−ǫ ǫ MS,

.

(2.488)

ǫ→0

Using the derivative formula (2.486), the trace of the logarithm in the free energy (2.484) takes the form 1 d Tr log(−∂τ2 + ω 2 ) = − h ¯β dǫ

Z

∞ −∞

dω ′ ′2 (ω + ω 2)−ǫ . 2π ǫ=0

(2.489)

We now set up a useful integral representation, due to Schwinger, for a power a−ǫ generalizing (2.460). Using the defining integral representation for the Gamma function Z ∞ dτ µ −τ ω2 τ e = ω −µ/2 Γ(µ), (2.490) τ 0 the desired generalization is −ǫ

a

1 = Γ(ǫ)

Z

0

∞

dτ ǫ −τ a τ e . τ

(2.491)

This allows us to re-express (2.489) as Z ∞ ′ Z ∞ 2 1 d 1 dω dτ ′ 2 Tr log(−∂τ2 + ω 2) = − τ ǫ e−τ (ω +ω ) . h ¯β dǫ Γ(ǫ) −∞ 2π 0 τ ǫ=0

(2.492)

As long as ǫ is larger than zero, the τ -integral converges absolutely, so that we can interchange τ - and ω ′ -integrations, and obtain 1 d 1 Z ∞ dτ ǫ Z ∞ dω ′ −τ (ω′ 2 +ω2 ) 2 2 Tr log(−∂τ + ω ) = − τ e . h ¯β dǫ Γ(ǫ) 0 τ −∞ 2π ǫ=0

(2.493)

At this point we can perform the Gaussian integral over ω ′ using formula (1.334), and find

1 d 1 Z ∞ dτ ǫ 1 2 2 2 Tr log(−∂τ + ω ) = − τ √ e−τ ω . h ¯β dǫ Γ(ǫ) 0 τ 2 τ π ǫ=0

(2.494)

For small ǫ, the τ -integral is divergent at the origin. It can, however, be defined by an analytic continuation of the integral starting from the regime ǫ > 1/2, where it converges absolutely, to ǫ = 0. The continuation must avoid the pole at ǫ = 1/2. Fortunately, this continuation is trivial since the integral can be expressed in terms of the Gamma function, whose analytic properties are well-known. Using the integral formula (2.490), we obtain

1 1 d 1 Tr log(−∂τ2 + ω 2) = − √ ω 1−2ǫ Γ(ǫ − 1/2) . h ¯β 2 π dǫ Γ(ǫ) ǫ=0

(2.495)

160


The right-hand side has to be continued analytically from ǫ > 1/2 to ǫ = 0. This is easily done using the defining property of the Gamma function Γ(x) = Γ(1 + x)/x, √ from which we find Γ(−1/2) = −2Γ(1/2) = −2 π, and 1/Γ(ǫ) ≈ ǫ/Γ(1 + ǫ) ≈ ǫ. The derivative with respect to ǫ leads to the free energy of the harmonic oscillator at low temperature via analytic regularization: 1 Tr log(−∂τ2 + ω 2 ) = h ¯β

Z

dω ′ log(ω ′2 + ω 2 ) = ω, 2π

∞

−∞

(2.496)

so that the free energy of the oscillator at zero-temperature becomes h ¯ω . 2

Fω =

(2.497)

This agrees precisely with the result obtained from the lattice definition of the path integral in Eq. (2.399), or from the path integral (3.805) with the Fourier measure (2.439). With the above procedure in mind, we shall often use the sloppy formula expressing the derivative of Eq. (2.491) at ǫ = 0: Z

log a = −

∞

0

dτ −τ a e . τ

(2.498)

This formula differs from the correct one by a minimal subtraction and can be used in all calculations with analytic regularization. Its applicability is based on the possibility of dropping the frequency integral over 1/ǫ in the alternative correct expression 1 Z ∞ dω ′ 1 2 2 Tr log(−∂τ + ω ) = − h ¯β ǫ −∞ 2π

1 ′2 1 (ω + ω 2)−ǫ − ǫ ǫ

.

(2.499)

ǫ→0

In fact, within analytic regularization one may set all integrals over arbitrary pure powers of the frequency equal to zero: Z

0

∞

dω ′ (ω ′ )α = 0 for all α.

(2.500)

This is known as Veltman’s rule.21 It is a special limit of a frequency integral which is a generalization of the integral in (2.489): Z

∞

−∞

dω ′ (ω ′2 )γ Γ(γ + 1/2) 2 γ+1/2−ǫ = (ω ) . ′2 2 ǫ 2π (ω + ω ) 2πΓ(ǫ)

(2.501)

This equation may be derived by rewriting the left-hand side as 1 Γ(ǫ)

Z

∞

−∞

dω ′ ′2 γ (ω ) 2π

Z

0

∞

dτ ǫ −τ (ω′ 2 +ω2 ) τ e . τ

(2.502)

21

See the textbook H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories, World Scientific, Singapore, 2001 (http://www.physik.fu-berlin.de/~kleinert/b8). H. Kleinert, PATH INTEGRALS


The integral over ω ′ is performed as follows: Z

∞

−∞

dω ′ ′2 γ −τ (ω′ 2 +ω2 ) 1 (ω ) e = 2π 2π

Z

∞

0

dω ′ 2 ′2 γ+1/2 −τ (ω′ 2 +ω2 ) τ −γ−1/2 (ω ) e = Γ(γ + 1/2), ω′ 2 2π (2.503)

leading to a τ -integral in (2.502) Z

∞

0

dτ ǫ−γ−1/2 −τ ω2 e = (ω 2 )γ+1/2+ǫ , τ τ

(2.504)

and thus to the formula (2.501). The Veltman rule (2.500) follows from this directly in the limit ǫ → 0, since 1/Γ(ǫ) → 0 on the right-hand side. This implies that the subtracted 1/ǫ term in (2.499) gives no contribution. The vanishing of all integrals over pure powers by Veltman’s rule (2.500) was initially postulated in the process of developing a finite quantum field theory of weak and electromagnetic interactions. It has turned out to be extremely useful for the calculation of critical exponents of second-order phase transitions from field theories.21 An important consequence of Veltman’s rule is to make the logarithms of dimensionful arguments in the partition functions (2.481) and the free energy (2.483) R meaningful quantities. First, since d(ω ′/2π) log ω 2 = 0, we can divide the argument of the logarithm in (2.484) by ω 2 without harm, and make them dimensionless. At finite temperatures, we use the equality of sum and integral over an ωm -independent quantity c kB T

∞ X

c=

m=−∞

Z

∞

−∞

dωm c 2π

(2.505)

to show that also kB T

∞ X

log ω 2 = 0,

(2.506)

m=−∞

so that we have, as a consequence of Veltman’s rule, that the Matsubara frequency sum over the constant log ω 2 vanishes for all temperatures. For this reason, also the argument of the logarithm in the free energy (2.483) can be divided by ω 2 without change, thus becoming dimensionless.

2.15.2

Finite-Temperature Evaluation of Frequency Sum

At finite temperature, the free energy contains an additional term consisting of the difference between the Matsubara sum and the frequency integral kB T ∆Fω = 2

∞ X

2 ωm h ¯ log +1 − 2 ω 2 m=−∞

!

Z

∞

−∞

2 dωm ωm log +1 , 2π ω2

!

(2.507)

162


where we have used dimensionless logarithms as discussed at the end of the last subsection. The sum is conveniently split into a subtracted, manifestly convergent expression ∆1 Fω = kB T

∞ X

m=1

∞ 2 2 X ωm ω2 ωm log + 1 − log = k T log 1 + , (2.508) B 2 ω2 ω2 ωm m=1

"

#

!

!

and a divergent sum ∞ X

∆2 Fω = kB T

log

m=1

2 ωm . ω2

(2.509)

The convergent part is most easily evaluated. Taking the logarithm of the product in Eq. (2.400) and recalling (2.401), we find ∞ Y

m=1

ω2 1+ 2 ωm

and therefore ∆F1 =

!

=

sinh(β¯ hω/2) , β¯ hω/2

(2.510)

1 sinh(β¯ hω/2) log . β β¯ hω/2

(2.511)

The divergent sum (2.509) is calculated by analytic regularization as follows: We rewrite ∞ X

log

m=1

2 ωm ω2

"

=− 2

# ∞ X ωm −ǫ

d dǫ m=1

ω

ǫ→0



= − 2

d dǫ

2π β¯ hω

!−ǫ

∞ X

m=1



m−ǫ 

, (2.512)

ǫ→0

and express the sum over m−ǫ in terms of Riemann’s zeta function ∞ X

ζ(z) =

m−z .

(2.513)

m=1

This sum is well defined for z > 1, and can be continued analytically into the entire complex z-plane. The only singularity of this function lies at z = 1, where in the neighborhood ζ(z) ≈ 1/z. At the origin, ζ(z) is regular, and satisfies22 1 ζ ′(0) = − log 2π, 2

ζ(0) = −1/2,

(2.514)

such that we may approximate 1 ζ(z) ≈ − (2π)z , 2

z ≈ 0.

(2.515)

Hence we find ∞ X

22



ω2 d 2π log m2 = − 2 ω dǫ β¯ hω m=1

!−ǫ



ζ(ǫ)

ǫ→0

d = (β¯ hω)ǫ = log h ¯ ωβ. dǫ ǫ→0

(2.516)




thus determining ∆2 Fω in Eq. (2.509). By combining this with (2.511) and the contribution −¯ hω/2 from the integral (2.507), the finite-temperature part (2.483) of the free energy becomes ∆Fω =

1 log(1 − e−¯hβω ). β

(2.517)

Together with the zero-temperature free energy (2.497), this yields the dimensionally regularized sum formula Fω

∞ 1 1 X h ¯ω 1 2 2 2 = log(ωm + ω 2) = Tr log(−∂τ + ω ) = + log(1 − e−¯hω/kB T ) 2β 2β m=−∞ 2 β

!

1 h ¯ ωβ = log 2 sinh , β 2

(2.518)

in agreement with the properly normalized free energy (2.477) at all temperatures. Note that the property of the zeta function ζ(0) = −1/2 in Eq. (2.514) leads once more to our earlier result (2.506) that the sum Matsubara sum of a constant c vanishes: ∞ X

c=

m=−∞

since

∞ X

−1 X

c+c+

c = 0,

(2.519)

1 = ζ(0) = −1/2.

(2.520)

m=−∞

1=

m=1

−∞ X

∞ X

m=1

m=−1

As mentioned before, this allows us to divide ω 2 out of the out the logarithms in the sum in Eq. (2.518) and rewrite this sum as ∞ ∞ 2 2 1 X ωm 1 X ωm log +1 = log +1 2β m=−∞ ω2 β m=1 ω2

!

2.15.3

!

(2.521)

Quantum-Mechanical Harmonic Oscillator

This observation leads us directly to the analogous quantum-mechanical discussion. Starting point fluctuation factor (2.81) of the free particle which can formally be written as F0 (∆t) =

Z

Dδx(t) exp

i h ¯

Z

tb

ta

dt

M 1 δx(−∂t2 )δx = q , 2 2π¯ hi∆t/M

(2.522)

where ∆t ≡ tb − ta [recall (2.120)]. The path integral of the harmonic oscillator has the fluctuation factor [compare (2.181)] Det(−∂t2 − ω 2) Fω (∆t) = F0 (∆t) Det(−∂t2 ) "

#−1/2

.

(2.523)

164


The ratio of the determinants has the Fourier decomposition ∞ h i X Det(−∂t2 − ω 2 ) 2 2 2 log(ν − ω ) − log ν , = exp n n Det(−∂t2 ) n=1

(

)

(2.524)

where νn = nπ/∆t [recall (2.105)], and was calculated in Eq. (2.181) to be Det(−∂t2 − ω 2 ) sin ω∆t = . 2 Det(−∂t ) ω∆t

(2.525)

This result can be reproduced with the help of formulas (2.521) and (2.518). We raplace β by 2∆t, and use again Σn 1 = ζ(0) = −1/2 to obtain Det(−∂t2

2

−ω ) = =

∞ X

log(νn2

+ω

2

n=1 " ∞ X

)

=

ω→iω

∞ X

n=1

"

νn2 log + 1 + log ω 2 2 ω

ν2 1 log n2 + 1 − log ω 2 ω 2 n=1 !

!

#

ω→iω

#

ω→iω

sin ω∆t = log 2 . (2.526) ω

For ω = 0 this reproduces Formula (2.516). Inserting this and (2.526) into (2.524), we recover the result (2.525). Thus we find the amplitude 1

(xb tb |xa ta ) = q Det πi/M

−1/2

(−∂t2

2

iAcl /¯ h

− ω )e

1

=q πi/M

r

ω eiAcl /¯h , (2.527) 2 sin ω∆t

in agreement with (2.168).

2.15.4

Tracelog of First-Order Differential Operator

The trace of the logarithm in the free energy (2.484) can obviously be split into two terms Tr log(−∂τ2 + ω 2) = Tr log(∂τ + ω) + Tr log(−∂τ + ω). (2.528) Since the left-hand side is equal to β¯ hω by (2.496), and the two integrals must be the same, we obtain the low-temperature result dω log(−iω ′ + ω) = h ¯β 2π −∞ h ¯ βω = . 2

Tr log(∂τ + ω) = Tr log(−∂τ + ω) = h ¯β

Z

∞

dω log(iω ′ + ω) 2π −∞

Z

∞

(2.529)

The same result could be obtained from analytic continuation of the integrals over ∂ǫ (±iω ′ + ω)ǫ to ǫ = 0. For a finite temperature, we may use Eq. (2.518) to find !

1 β¯ hω Tr log(∂τ + ω) = Tr log(−∂τ + ω) = Tr log(−∂τ2 + ω 2 ) = log 2 sinh , (2.530) 2 2 H. Kleinert, PATH INTEGRALS


which reduces to (2.529) for T → 0. The result is also the same if there is an extra factor i in the argument of the tracelog. To see this we consider the case of time-independent frequency where Veltman’s rule (2.500) tells us that it does not matter whether one evaluates integrals over log(iω ′ ∓ ω) or over log(ω ′ ± iω). Let us also replace ω ′ by iω ′ in the zero-temperature tracelog (2.496) of the second-order differential operator (−∂τ2 + ω 2 ). Then we rotate the contour of integration clockwise in the complex plane to find ∞

Z

−∞

dω ′ log(−ω ′2 + ω 2 − iη) = ω, 2π

ω ≥ 0,

(2.531)

where an infinitesimal positive η prescribes how to bypass the singularities at ω ′ = ±ω ∓ iη along the rotated contour of integration. Recall the discussion of the iηprescription in Section 3.3. The integral (2.531) can be split into the integrals Z

∞

−∞

ω dω ′ log[ω ′ ± (ω − iη)] = i , 2π 2

ω ≥ 0.

(2.532)

Hence formula (2.529) can be generalized to arbitrary complex frequencies ω = ωR + iωI as follows: Z ∞ dω ′ ω log(ω ′ ± iω) = ∓ǫ(ωR ) , (2.533) 2 −∞ 2π and Z

∞

−∞

dω ′ ω log(ω ′ ± ω) = −iǫ(ωI ) , 2π 2

(2.534)

where ǫ(x) = Θ(x)−Θ(−x) = x/|x| is the antisymmetric Heaviside function (1.311), which yields the sign of its argument. The formulas (2.533) and (2.534) are the large-time limit of the more complicated sums kB T h ¯

∞ X

"

#

kB T h ¯ω log(ωm ± iω) = log 2ǫ(ωR ) sinh , h ¯ 2kB T m=−∞

(2.535)

and kB T h ¯

∞ X

"

#

kB T h ¯ω log(ωm ± ω) = log −2iǫ(ωI ) sin . h ¯ 2kB T m=−∞

(2.536)

The first expression is periodic in the imaginary part of ω, with period 2πkB T , the second in the real part. The determinants possess a meaningful large-time limit only if the periodic parts of ω vanish. In many applications, however, the fluctuations will involve sums of logarithms (2.536) and (2.535) with different complex frequencies ω, and only the sum of the imaginary or real parts will have to vanish to obtain a meaningful large-time limit. On these occasions we may use the simplified formulas (2.533) and (2.534). Important examples will be encountered in Section 18.9.2.

166


In Subsection 3.3.2, Formula (2.530) will be generalized to arbitrary positive time-dependent frequencies Ω(τ ), where it reads [see (3.133)] (

"

#)

1 ¯hβ ′′ Tr log [±∂τ + Ω(τ )] = log 2 sinh dτ Ω(τ ′′ ) 2 0 R h¯ β ′′ 1 Z ¯hβ ′′ ′′ = dτ Ω(τ ′′ ) + log 1 − e− 0 dτ Ω(τ ) . (2.537) 2 0

2.15.5

Z

Gradient Expansion of One-Dimensional Tracelog

Formula (2.537) may be used to calculate the trace of the logarithm of a secondorder differential equation with arbitrary frequency as a semiclassical expansion. We introduce the Planck constant h ¯ and the potential w(τ ) ≡ h ¯ Ω(τ ), and factorize as in (2.528): h

i

Det −¯ h2 ∂τ2 + w 2(τ ) = Det [−¯ h∂τ − w(τ ¯ )] × Det [¯ h∂τ − w(τ ¯ )] ,

(2.538)

where the function w(τ ¯ ) satisfies Riccati differential equation:23 h ¯ ∂τ w(τ ¯ ) + w¯ 2 (τ ) = w 2 (τ ).

(2.539)

By solving this we obtain the trace of the logarithm from (2.537): Tr log

h

−¯ h2 ∂τ2

2

i

(

2

+ w (τ ) = log 4 sinh

"

1 2¯ h

Z

0

hβ ¯

′

′

dτ w(τ ¯ )

#)

.

(2.540)

The exponential of this yields the functional determinant. For constant w(τ ¯ ) = ω this agrees with the result (2.425) of the Gelfand-Yaglom formula for periodic boundary conditions. This agreement is no coincidence. We can find the solution of any Riccati differential equation if we know how to solve the second-order differential equation (2.425). Imposing the Gelfand-Yaglom boundary conditions in (2.425), we find Dren (τ ) and from this the functional determinant 2[D˙ ren (¯ hβ) − 1]. Comparison with (2.540) shows that solution of the Riccati differential equation (2.539) is given by w(τ ¯ ) = 2¯ h∂τ arsinh

q

(2.541)

w¯n (τ )¯ hn ,

(2.542)

[D˙ ren (τ ) − 1]/2.

For the harmonic oscillator where D˙ ren (τ ) is equal to (2.429), this leads to the constant w(τ ¯ )=h ¯ ω, as it should. If we cannot solve the second-order differential equation (2.425), a solution to the Riccati equation (2.539) can still be found as a power series in h ¯: w(τ ¯ )=

∞ X

n=0 23

Recall the general form of the Riccati differential equation y ′ = f (τ )y + g(τ )y 2 + h(y), which is an inhomogeneous version of the Bernoulli differential equation y ′ = f (τ )y + g(τ )y n for n = 2. H. Kleinert, PATH INTEGRALS


which provides us with a so-called gradient expansion of the trace of the logarithm. The lowest-order coefficient function w¯0 (τ ) is obviously equal to w(τ ). The higher ones obey the recursion relation n−1 X 1 w¯n (τ ) = − w¯˙ n−1 (τ ) + w¯n−k (τ )w¯k (τ ) , 2w(τ ) k=1

!

n ≥ 1.

(2.543)

These are solved for n = 0, 1, 2, 3 by (

p v ′ (τ ) , v(τ ), − 4 v(τ ) −

1105 v ′(τ )

5 v ′ (τ )

2

5/2

32 v(τ )

4

11/2

2048 v(τ )

where v(τ ) ≡ w2 (τ ).

2.15.6

−

+

v ′′ (τ ) 8 v(τ )

−

, 3/2

2

+

221 v ′ (τ ) v ′′ (τ ) 9/2

256 v(τ )

15 v ′ (τ )

+

4

64 v(τ )

19 v ′′ (τ )

−

3

2

7/2

128 v(τ )

−

9 v ′ (τ ) v ′′ (τ ) 32 v(τ )

3

7 v ′ (τ ) v (3) (τ ) 7/2

32 v(τ )

− +

v (3) (τ ) 16 v(τ )

2,

v (4) (τ ) 5/2

32 v(τ )

)

,(2.544)

The series can, of course, be trivially extended any desired orders.

Duality Transformation and Low-Temperature Expansion

There exists another method of calculating the finite-temperature part of the free energy (2.483) which is worth presenting at this place, due to its broad applicability in statistical mechanics. For this we rewrite (2.507) in the form ∞ X

kB T ∆Fω = 2

h ¯ − kB T m=−∞

∞

Z

−∞

!

dωm 2 log(ωm + ω 2 ). 2π

(2.545)

Changing the integration variable to m, this becomes ∞ X

kB T ∆Fω = 2



!

2πkB T − dm log  h ¯ −∞ m=−∞ Z

∞

!2



m2 + ω 2  .

(2.546)

Within analytic regularization, this expression is rewritten with the help of formula (2.498) as kB T ∆Fω = − 2

Z

∞

0

dτ τ

∞ X

m=−∞

−

∞

Z

−∞

!

dm e−τ [(2πkB T /¯h)

2 m2 +ω 2

].

(2.547)

The duality transformation proceeds by performing the sum over the Matsubara R frequencies with the help of Poisson’s formula (1.205) as an integral dµ using an extra sum over integer numbers n. This brings (2.547) to the form (expressing the temperature in terms of β), 1 Z ∞ dτ Z ∞ ∆Fω = − dµ 2β 0 τ −∞ The parentheses contain the sum 2 exponent 2πµni − τ

2π h ¯β

!2

2

µ = −τ

∞ X

2πµni

e

n=−∞

P∞

n=1

2π h ¯β

!

− 1 e−τ [(2π/¯hβ)

2 µ2 +ω 2

].

(2.548)

e2πµni . After a quadratic completion of the

!2 "

n¯ h2 β 2 µ−i 4πτ

#2

−

1 (¯ hβn)2 , 4τ

(2.549)

168


the integral over µ can be performed, with the result h ¯ ∆Fω = − √ 2 π

∞

Z

0

∞ 2 dτ −1/2 X 2 τ e−(n¯hβ) /4τ −τ ω . τ n=1

(2.550)

Now we may use the integral formula [compare (1.343)]24 Z

0

∞

ν

dτ ν −a2 /τ −b2 τ a =2 τ e τ b

K−ν (2ab),

(2.551)

to obtain the sum over modified Bessel functions ∞ √ h ¯ω X ∆Fω = − √ 2 (nβ¯ hω)−1/2 2K1/2 (nβ¯hω). 2 π n=1

(2.552)

The modified Bessel functions with index 1/2 are particularly simple: K1/2 (z) =

r

π −z e . 2z

(2.553)

Inserting this into (2.552), the sum is a simple geometric one, and may be performed as follows: 1 X 1 −β¯hωn 1 ∆Fω = − e = log 1 − e−β¯hω , (2.554) β n=1 n β

in agreement with the previous result (2.517). The effect of the duality tranformation may be rephrased in another way. It converts the original sum over m in the expression (2.508): S(β¯ hω) =

∞ X

2 2 ωm ωm log + 1 − log ω2 ω2

#

(2.555)

∞ X β¯ hω 1 −nβ¯hω − log β¯ hω − e . 2 n=1 n

(2.556)

m=1

"

!

into a sum over n: S(β¯ hω) =

The first sum (2.555) converges fast at high temperatures, where it can be expanded in powers of ω 2 : ∞ X

(−1)k S(β¯ hω) = − k k=1

∞ X

!

1  β¯ hω 2k 2π m=1 m

!2 k  .

(2.557)

The expansion coefficients are equal to Riemann’s zeta function ζ(z) of Eq. (2.513) at even arguments z = 2k, so that we may write ∞ X

(−1)k β¯ hω S(β¯ hω) = − ζ(2k) k 2π m=1 24

!2k

.

(2.558)

I.S. Gradshteyn and I.M. Ryzhik, ibid., Formula 3.471.9. H. Kleinert, PATH INTEGRALS


At even positive arguments, the values of the zeta function are related to the Bernoulli numbers by25 ζ(2n) =

(2π)2n |B2n |. 2(2n)!

(2.559)

The Bernoulli numbers are defined by the expansion ∞ X t tn B = . n et − 1 n=0 n!

(2.560)

The lowest nonzero Bernoulli numbers are B0 = 1, B1 = −1/2, B2 = 1/2, B4 = −1/30, . . . . The Bernoulli numbers determine also the zeta functions at negative odd arguments: ζ(1 − 2n) = −

B2n , 2n

(2.561)

this being a consequence of the general identity26 ζ(z) = 2z π z−1 sin(πz/2)Γ(1 − z)ζ(1 − z) = 2z−1 π z ζ(1 − z)/Γ(z) cos

zπ . (2.562) 2

Typical values of ζ(z) which will be needed here are27 ζ(2) =

π2 π4 π6 , ζ(4) = , ζ(5) = , . . . , ζ(∞) = 1. 6 90 945

(2.563)

In contrast to the original sum (2.555) and its expansion (2.558), the dually transformed sum (2.556) converges rapidly for low temperatures. It converges everywhere except at very large temperatures, where it diverges logarithmically. The precise behavior can be calculated as follows: For large T there exists a large number N which is still much smaller than 1/β¯ hω, such that e−β¯hωN is close to unity. Then we split the sum as ∞ X

−1 ∞ X 1 −nβ¯hω NX 1 1 −nβ¯hω e + e ≈ . n=1 n n=1 n n=N n

(2.564)

Since N is large, the second sum can be approximated by an integral Z

∞

N

dn −nβ¯hω e = n

Z

∞

N β¯ hω

dx −x e , x

which is an exponential integral E1 (Nβ¯ hω) of Eq. (2.462) with the large-argument expansion −γ − log(Nβ¯ hω) of Eq. (2.463). 25

ibid., Formulas 9.542 and 9.535. I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 9.535.2. 27 Other often-needed values are ζ(0) = −1/2, ζ ′ (0) = − log(2π)/2, ζ(−2n) = 0, ζ(3) ≈ 1.202057, ζ(5) ≈ 1.036928, . . . . 26

170


The first sum in (2.564) is calculated with the help of the Digamma function ψ(z) ≡

Γ′ (z) . Γ(z)

(2.565)

This has an expansion28 ψ(z) = −γ −

∞ X

n=0

1 1 , − n+z n+1

(2.566)

which reduces for integer arguments to ψ(N) = −γ +

N −1 X n=1

1 , n

(2.567)

and has the large-z expansion ψ(z) ≈ log z −

∞ X 1 B2n − . 2z n=1 2nz 2n

(2.568)

Combining this with (2.463), the logarithm of N cancels, and we find for the sum in (2.564) the large-T behavior ∞ X

1 −nβ¯hω e ≈ − log β¯ hω + O(β). T →∞ n=1 n

(2.569)

This cancels the logarithm in (2.556). The low-temperature series (2.556) can be used to illustrate the power of analytic regularization. Suppose we want to extract from it the large-T behavior, where the sum ∞ X 1 −nβ¯hω g(β¯ hω) ≡ e (2.570) n=1 n converges slowly. We would like to expand the exponentials in the sum into powers of ω, but this gives rise to sums over positive powers of n. It is possible to make sense of these sums by analytic continuation. For this we introduce a generalization of (2.570): ζν (eβ¯hω ) ≡

∞ X

1 −nβ¯hω e , ν n=1 n

(2.571)

which reduces to g(β¯ hω) for ν = 1. This sum is evaluated by splitting it into an integral over n and a difference between sum and integral: β¯ hω

ζν (e 28

)=

Z

0

∞

Z ∞ ∞ X 1 1 −nβ¯hω dn ν e−nβ¯hω + − e . n nν 0 n=1 !

(2.572)

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.362.1. H. Kleinert, PATH INTEGRALS


The integral is convergent for ν < 1 and yields Γ(1 − ν) (β¯ hω)ν via the integral formula (2.490). For other ν’s it is defined by analytic continuation. The remainder may be expanded sloppily in powers of ω and yields Z ∞ ∞ ∞ X X 1 −nβ¯hω 1 dn ν e + − + n nν k=1 0 n=1

∞ X

(−1)k (β¯ hω)k . k! 0 0 n=1 (2.573) The second term is simply the Riemann zeta function ζ(ν) [recall (2.513)]. Since the additional integral vanishes due to Veltman’ rule (2.500), the zeta function may also be defined by Z

ζν (eβ¯hω ) =

∞

∞ X

Z

−

n=1

∞ 0

!

"

!

1 = ζ(ν), νk

Z

−

∞

!

nk−ν

#

(2.574)

If this formula is applied to the last term in (2.573), we obtain the so-called Robinson expansion 29 β¯ hω

ζν (e

) = Γ(1 − ν)(β¯ hω)

ν−1

+ ζ(ν) +

∞ X

1 (−β¯ hω)k ζ(ν − k). k=1 k!

(2.575)

This expansion will later play an important role in the discussion of Bose-Einstein condensation [see Eq. (7.38)]. For various applications it is useful to record also the auxiliary formula ∞ X

Z

−

n=1

∞

0

!

∞ X enβ¯hω 1 = (−β¯ hω)k ζ(ν − k) ≡ ζ¯ν (eβ¯hω ), nν k! k=1

(2.576)

since in the sum minus the integral, the first Robinson terms are absent and the result can be obtained from a naive Taylor expansion of the exponents enβ¯hω and the summation formula (2.574). From (2.575) we can extract the desired sum (2.570) by going to the limit ν → 1. Close to the limit, the Gamma function has a pole Γ(1−ν) = 1/(1−ν)−γ +O(ν −1). From the identity πz = π 1−z ζ(z) 2z Γ(1 − z)ζ(1 − z) sin (2.577) 2 and (2.514) we see that ζ(ν) behaves near ν = 1 like ζ(ν) =

1 + γ + O(ν − 1) = −Γ(1 − ν) + O(ν − 1). ν−1

(2.578)

Hence the first two terms in (2.575) can be combined to yield for ν → 1 the finite result limν→1 Γ(1 − ν) [(β¯ hω)ν−1 − 1]=− log β¯ hω. The remaining terms contain in the limit the values ζ(0) = −1/2, ζ(−1), ζ(−2), etc. Here we use the property of the zeta function that it vanishes at even negative arguments, and that the function at arbitrary negative argument is related to one at positive argument by the identity 29

J.E. Robinson, Phys. Rev. 83 , 678 (1951).

172


(2.577). This implies for the expansion coefficients in (2.575) with k = 1, 2, 3, . . . in the limit ν → 1: 1 (2p)! ζ(1 − 2p) = (−1)p ζ(2p), p (2π)2p

ζ(−2p) = 0,

p = 1, 2, 3, . . . .

(2.579)

Hence we obtain for the expansion (2.575) in the limit ν → 1: ∞ β¯ hω X (−1)k ) = − log β¯ hω + + ζ(2k) (β¯ hω)2k . 2 k! k=1

β¯ hω

g(β¯ hω) = ζ1 (e

(2.580)

This can now be inserted into Eq. (2.556) and we recover the previous expansion (2.558) for S(β¯ hω) which was derived there by a proper duality transformation. It is interesting to observe what goes wrong if we forget the separation (2.572) of the sum into integral plus sum-minus-integral and its regularizationand re-expands (2.570) directly, and illegally, in powers of ω. Then we obtain for ν = 1 the formal expansion β¯ hω

ζ1 (e

)=

∞ X

p=0

∞ X

p−1

n

n=1

!

∞ X (−1)p (−1)p (β¯ hω)p = −ζ(1) + ζ(1 − p) (β¯ hω)p , p! p! p=1

(2.581)

which contains the infinite quantity ζ(1). The correct result (2.580) is obtained from this by replacing the infinite quantity ζ(1) by − log β¯ hω, which may be viewed as a regularized ζreg (1): ζ(1) → ζreg (1) = − log β¯ hω. (2.582) The above derivation of the Robinson expansion can be supplemented by a dual version as follows. With the help of Poisson’s formula (1.197) we rewrite the sum (2.571) as an integral over n and an auxiliary sum over integer numbers m, after which the integral over n can be performed yielding ζν (eβ¯hω ) ≡

Z ∞ X

m=−∞ 0

∞

dn e(2πim+β¯hω)n

+ Γ(1 − ν) 2 Re

∞ X

m=1

1 = Γ(1 − ν)(−β¯hω)ν−1 nν

(−β¯ hω − 2πim)ν−1 .

(2.583)

The sum can again be expanded in powers of ω 2 Re

∞ X

(−2πim)

ν−1

m=1

=2

∞ X ν

k=0

β¯ hω 1+ 2πim

!ν−1

−1 cos[(1 − ν − k)π/2] (2π)ν−1−k ζ(1 − ν + k) (β¯ hω)k .(2.584) k

Using the relation (2.577) for zeta-functions, the expansion (2.583) is seen to coincide with (2.571). H. Kleinert, PATH INTEGRALS


Note that the representation (2.583) of ζν (eβ¯hω ) is a sum over Matsubara frequencies ωm = 2πm/β [recall Eq. (2.373)]: β¯ hω

ζν (e

) ≡

Z ∞ X

m=−∞ 0

∞

dn e(iωm +¯hω)βn

= Γ(1 − ν)(−β¯ hω)

ν−1

1 nν

"

1 + 2 Re

∞ X

ν−1

(1 + iωm /¯ hω)

m=1

#

. (2.585)

The first term coming from the integral over n in (2.572) is associated with the zero Matsubara frequency. This term represents the high-temperature or classical limit of the expansion. The remainder contains the sum over all nonzero Matsubara frequencies, and thus the effect of quantum fluctuations. It should be mentioned that the first two terms in the low-temperature expansion (2.556) can also be found from the sum (2.555) with the help of the Euler-Maclaurin formula30 for a sum over discrete points t = a + (k + κ)∆ of a function F (t) from k = 0 to K ≡ (b − a)/∆: K X

F (a + k∆) =

k=0

+

1 ∆

Z

b

a

dt F (t) +

1 [F (a) + F (b)] 2

∞ X

i h ∆2p−1 B2p F (2p−1) (b) − F (2p−1) (a) , p=1 (2p)!

(2.586)

or, more generally for t = a + (k + κ)∆, K−1 X

∞ h i X 1 Zb ∆p−1 Bp (κ) F (p−1) (b) − F (p−1) (a) , F (a + (k + κ)∆) = dt F (t) + ∆ a p=1 p! k=0 (2.587)

where Bn (κ) are the Bernoulli functions defined by a generalization of the expansion (2.560): ∞ X teκt tn = B (κ) . (2.588) n et − 1 n=0 n! At κ = 0, the Bernoulli functions start out with the Bernoulli numbers: Bn (0) = Bn . The function B0 (κ) is equal to 1 everywhere. Another way of writing formula (2.589) is K−1 X





∞ X 1 Zb  ∆p F (a + (k + κ)∆) = dt 1 + Bp (κ)∂tp  F (t). ∆ p! a p=0 k=0

(2.589)

This implies that a sum over discrete values of a function can be replaced by an integral over a gradient expansion. of the function. 30

M. Abramowitz and I. Stegun, op. cit., Formulas 23.1.30 and 23.1.32.

174


2 Using the first Euler-Maclaurin formula (2.586) with a = ω12, b = ωM , and ∆ = ω1 , we find M h X

2 log(ωm

2

+ ω )−

2 log(ωm )

m=0

i

(

) m=M

i m=M ω ωm h 2 = π + log(ωm + ω 2)−2 − ω = 0 ω1 ω1 m=1 m=1 n o n o 1 2 + log(ω12 + ω 2 ) + log(ωM + ω 2 ) − ω = 0 . (2.590) 2

For small T , the leading two terms on the right-hand side are π

ω ω2 1 − log 2 , ω1 2 ω1

(2.591)

in agreement with the first two terms in the low-temperature series (2.556). Note that the Euler-Maclaurin formula is unable to recover the exponentially small terms in (2.556), since they are not expandable in powers of T . The transformation of high- into low-temperature expansions is an important tool for analyzing phase transitions in models of statistical mechanics.31

2.16

Finite-N Behavior of Thermodynamic Quantities

Thermodynamic fluctuations in Euclidean path integrals are often imitated in computer simulations. These are forced to employ a sliced time axis. It is then important to know in which way the time-sliced thermodynamic quantities converge to their continuum limit. Let us calculate the internal energy E and the specific heat at constant volume C for finite N from (2.476). Using (2.476) we have ∂(β ω ˜e) ∂β ∂(ǫ˜ ωe ) ∂β

= =

ω , cosh(ǫ˜ ωe /2) 2 tanh(ǫ˜ ωe /2), β

(2.592)

and find the internal energy E

= =

∂ ¯h ∂(β ω ˜e) (βF ) = coth(β¯hω ˜ e /2) ∂β 2 ∂β ¯hω coth(β¯hω ˜ e /2) . 2 cosh(ǫ˜ ωe /2)

(2.593)

The specific heat at constant volume is given by 1 ∂2 ∂ C = −β 2 2 (βF ) = −β 2 E kB ∂β ∂β ǫ 1 1 1 = β 2 ¯h2 ω 2 + coth(β¯ h ω ˜ /2) tanh(ǫ˜ ω /2) . e e 4 ¯hβ cosh2 (ǫ˜ sinh2 (β¯hω ˜ e /2) ωe /2)

(2.594)

Plots are shown in Fig. 2.5 for various N using natural units with h ¯ = 1, kB = 1. At high 31

See H. Kleinert, Gauge Fields in Condensed Matter , Vol. I Superflow and Vortex Lines, World Scientific, Singapore, 1989, pp. 1–742 (http://www.physik.fu-berlin.de/~kleinert/b1). H. Kleinert, PATH INTEGRALS

2.16 Finite-N Behavior of Thermodynamic Quantities

175

Figure 2.4 Finite-lattice effects in internal energy E and specific heat C at constant volume, as a function of the temperature for various numbers N + 1 of time slices. Note the nonuniform way in which the exponential small-T behavior of C ∝ e−ω/T is approached in the limit N → ∞. temperatures, F, E, and C are independent of N : F E C

1 log β, β 1 → = T, β → 1. →

(2.595) (2.596) (2.597)

These limits are a manifestation of the Dulong-Petit law : An oscillator has one kinetic and one potential degree of freedom, each carrying an internal energy T /2 and a specific heat 1/2. At low temperatures, on the other hand, E and C are strongly N -dependent (note that since F and E are different at T = 0, the entropy of the lattice approximation does not vanish as it must in the continuum limit). Thus, the convergence N → ∞ is highly nonuniform. After reaching the limit, the specific heat goes to zero for T → 0 exponentially fast, like e−ω/T . The quantity ω is called activation energy.32 It is the energy difference between ground state and first excited state of the harmonic oscillator. For large but finite N , on the other hand, the specific heat has the large value N + 1 at T = 0. This is due to ω ˜ e and cosh2 (ǫ˜ ωe /2) behaving, for a finite N and T → 0 (where ǫ becomes large) like 1 log(ǫ2 ω 2 ), ǫ cosh(ǫ˜ ωe /2) → ǫω/2. ω ˜e

→

(2.598)

Hence E 32

T →0

− −−→

T →0 1 coth[(N + 1) log(ǫω)] − −−→ 0, β

(2.599)

Note that in a D-dimensional solid the lattice vibrations can be considered as an ensemble of harmonic oscillators with energies ω ranging from zero to the Debye frequency. Integrating over R the corresponding specific heats with the appropriate density of states, dωω D−1 e−ω/kB T , gives the well-known power law at low temperatures C ∝ T D .

176


C

T →0

− −−→ N + 1.

(2.600)

The reason for the nonuniform approach of the N → ∞ limit is obvious: If we expand (2.476) in powers of ǫ, we find 1 2 2 ω ˜e = ω 1 − ǫ ω + . . . . (2.601) 24 When going to low T at finite N the corrections are quite large, as can be seen by writing (2.601), with ǫ = h ¯ β/(N + 1), as 1 ¯h2 ω 2 + ... . (2.602) ω ˜e = ω 1 − 24 kb2 T 2 (N + 1)2 Note that (2.601) contains no corrections of the order ǫ. This implies that the convergence of all thermodynamic quantities in the limit N → ∞, ǫ → 0 at fixed T is quite fast — one order in 1/N faster than we might at first expect [the Trotter formula (2.26) also shows the 1/N 2 -behavior].

2.17

Time Evolution Amplitude of Freely Falling Particle

The gravitational potential of a particle on the surface of the earth is V (x) = V0 + M g · x,

(2.603)

where −g is the earth’s acceleration vector pointing towards the ground, and V0 some constant. The equation of motion reads

which is solved by

¨ = −g, x

(2.604)

g x = xa + va (t − ta ) + (t − ta )2 , 2

(2.605)

with the initial velocity va = Inserting this into the action A=

Z

tb

ta

xb − xa g − (tb − ta ). tb − ta 2

(2.606)

M 2 dt x˙ − V0 − g · x , 2

(2.607)

we obtain the classical action Acl = −V0 (tb − ta ) +

M (xb − xa )2 1 1 − (tb − ta )g · (xb + xa ) − (tb − ta )3 g2 . (2.608) 2 tb − ta 2 24

Since the quadratic part of (2.607) is the same as for a free particle, also the fluctuation factor is the same [see (2.120)], and we find the time evolution amplitude i 1 −h V (t −ta ) ¯ 0 b (xb tb |xa ta ) = q 3e 2πi¯ hω(tb − ta )/M

iM (xb − xa )2 1 × exp − (tb − ta )g · (xb + xa ) − (tb − ta )3 g2 2¯ h tb − ta 12 (

"

#)

. (2.609)


2.17 Time Evolution Amplitude of Freely Falling Particle

177

The potential (2.603) can be considered as a limit of a harmonic potential V (x) = V0 +

M 2 ω (x − x0 )2 2

(2.610)

for ω → 0,

ˆ, x0 = −g/ω 2 → −∞ ≈ g

V0 = −Mx20 /2 = −Mg 2 /2ω 4 → −∞, (2.611)

keeping g = −Mω 2 x0 ,

and

(2.612)

M 2 2 (2.613) ω x0 2 fixed. If we perform this limit in the amplitude (2.170), we find of course (2.609). The wave functions can be obtained most easily by performing this limiting procedure on the wave functions of the harmonic oscillator. In one dimension, we set n = E/ω and find that the spectral representation (2.287) goes over into v0 = V0 +

(xb tb |xa ta ) =

Z

dEAE (xb )A∗E (xa )e−iE(tb −ta )/¯h ,

(2.614)

with the wave functions 1 x E − AE (x) = √ Ai . l ε lε

(2.615)

h2 /2M 2 g)1/3 = ε/Mg are the natural units of Here ε ≡ (¯ h2 g 2 M/2)1/3 and l ≡ (¯ energy and length, respectively, and Ai(z) is the Airy function solving the differential equation Ai′′ (z) = zAi(z), (2.616) For positive z, the Airy function can be expressed in terms of modified Bessel functions Iν (ξ) and Kν (ξ):33 √ r 1 z z Ai(z) = (2.617) [I−1/3 (2z 3/2 /3) − I1/3 (2z 3/2 /3)] = K1/3 2z 3/2 /3 . 2 π 3 For large z, this falls off exponentially: 1 3/2 Ai(z) → √ 1/4 e−2z /3 , 2 πz

z → ∞.

(2.618)

− π < argξ ≤ π/2, π/2 < argξ ≤ π,

(2.619)

For negative z, an analytic continuation34 Iν (ξ) = e−πνi/2 J(eπi/2 ξ), Iν (ξ) = e−πνi/2 J(eπi/2 ξ), 33

A compact description of the properties of Bessel functions is found in M. Abramowitz and I. Stegun, op. cit., Chapter 10. The Airy function is expressed in Formulas 10.4.14. 34 I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 8.406.

178


leads to

i 1√ h z J−1/3 (2(−z)3/2 /3) + J1/3 (2(−z)3/2 /3) , (2.620) 3 where J1/3 (ξ) are ordinary Bessel functions. For large arguments, these oscillate like

Ai(z) =

Jν (ξ) →

s

2 cos(ξ − πν/2 − π/4) + O(ξ −1), πξ

(2.621)

from which we obtain the oscillating part of the Airy function Ai(z) → √

h i 1 3/2 sin 2(−z) /3 + π/4 , πz 1/4

z → −∞.

(2.622)

The Airy function has the simple Fourier representation Ai(x) =

Z

∞

−∞

dk i(xk+k3 /3) e . 2π

(2.623)

In fact, the momentum space wave functions of energy E are hp|Ei =

s

l −i(pE−p3 /6M )l/ε¯h e ε

(2.624)

fulfilling the orthogonality and completeness relations Z

dp hE ′ |pihp|Ei = δ(E ′ − E), 2π¯ h

Z

dE hp′|EihE|pi = 2π¯ hδ(p′ − p).

(2.625)

The Fourier transform of (2.624) is equal to (2.615), due to (2.623).

2.18

Charged Particle in Magnetic Field

Having learned how to solve the path integral of the harmonic oscillator we are ready to study also a more involved harmonic system of physical importance: a charged particle in a magnetic field. This problem was first solved by L.D. Landau in 1930 in Schrödinger theory.35

2.18.1

Action

The magnetic interaction of a particle of charge e is given by Amag

e Z tb ˙ · A(x(t)), = dt x(t) c ta

(2.626)

where A(x) is the vector potential of the magnetic field. The total action is A[x] = 35

Z

tb

ta

M 2 e ˙ · A(x(t)) . dt x˙ (t) + x(t) 2 c

(2.627)

L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965. H. Kleinert, PATH INTEGRALS

179

2.18 Charged Particle in Magnetic Field

Suppose now that the particle moves in a homogeneous magnetic field B pointing along the z-direction. Such a field can be described by a vector potential A(x) = (0, Bx, 0).

(2.628)

But there are other possibilities. The magnetic field B(x) = ∇ × A(x)

(2.629)

as well as the magnetic interaction (2.626) are invariant under gauge transformations A(x) → A(x) + ∇Λ(x),

(2.630)

where Λ(x) are arbitrary single-valued functions of x. As such they satisfy the Schwarz integrability condition [compare (1.41)–(1.42)] (∂i ∂j − ∂j ∂i )Λ(x) = 0.

(2.631)

For instance, the axially symmetric vector potential 1 ˜ A(x) = B×x 2

(2.632)

gives the same magnetic field; it differs from (2.628) by a gauge transformation ˜ A(x) = A(x) + ∇Λ(x),

(2.633)

with the gauge function 1 Λ(x) = − B xy. 2 In the canonical form, the action reads A[p, x] =

Z

tb

ta

(

1 e dt p · x˙ − p − A(x) 2M c

(2.634)

2 )

.

(2.635)

The magnetic interaction of a point particle is thus included in the path integral by the so-called minimal substitution of the momentum variable: e p− − −→ P ≡ p − A(x). (2.636) c For the vector potential (2.628), the action (2.635) becomes A[p, x] =

Z

tb

ta

dt [p · x˙ − H(p, x)] ,

(2.637)

with the Hamiltonian H(p, x) =

p2 1 1 + MωL2 x2 − ωLlz (p, x), 2M 8 2

(2.638)

180


where x = (x, y) and p = (px , py ) and lz (p, x) = (x × p)z = xpy − ypx

(2.639)

is the z-component of the orbital angular momentum. In a Schrödinger equation, the last term in H(p, x) is diagonal on states with a given angular momentum around the z-axis. We have introduced the field-dependent frequency ωL =

e B, Mc

(2.640)

called Landau frequency or cyclotron frequency. This can also be written in terms of the Bohr magneton h ¯e µB ≡ , (2.641) Mc as ωL = µB B/¯ h. (2.642) The first two terms in (2.638) describe a harmonic oscillator in the xy-plane with a field-dependent magnetic frequency ωB ≡

ωL . 2

(2.643)

Note that in the gauge gauge (2.628), the Hamiltonian would have the rotationally noninvariant form H(p, x) =

p2 1 + MωL2 x2 − ωL xpy 2M 2

(2.644)

rather than (2.638), implying oscillations of frequency ωL in the x-direction and a free motion in the y-direction. The time-sliced form of the canonical action (2.635) reads AN e

=

N +1 X n=1

i 1 h 2 2 2 pn (xn − xn−1 ) − p + (py n − Bxn ) + pzn , 2M x n

(2.645)

and the associated tome-evolution amplitude for the particle to run from xa to xb is given by (xb tb |xa ta ) =

N Z Y

n=1

3

d xn

" NY +1 Z n=1

d 3 pn i N A , exp 3 (2π¯ h) h ¯ e #

(2.646)

with the time-sliced action N

A =

N +1 X n=1

i 1 h 2 2 2 pn (xn − xn−1 ) − p + (py n − Bxn ) + pzn . 2M x n

(2.647)


181


2.18.2

Gauge Properties

Note that the time evolution amplitude is not gauge-invariant. If we use the vector potential in some other gauge A′ (x) = A(x) + ∇Λ(x),

(2.648)

the action changes by a surface term ∆A =

e c

Z

tb

ta

dt x˙ · ∇Λ(x) =

e [Λ(xb ) − Λ(xa )]. c

(2.649)

The amplitude is therefore multiplied by a phase factor on both ends (xb tb |xa ta )A → (xb tb |xa ta )A′ = eieΛ(xb )/c¯h (xb tb |xa ta )A e−ieΛ(xa )/c¯h .

(2.650)

For the observable particle distribution (x tb |x ta ), the phase factors are obviously irrelevant. But all other observables of the system must also be independent of the phases Λ(x) which is assured if they correspond to gauge-invariant operators.

2.18.3

Time-Sliced Path Integration

Since the action AN contains the variables yn and zn only in the first term PN +1 n=1 ipn xn , we can perform the yn , zn integrations and find a product of N ∆functions in the y- and z-components of the momenta pn . If the projections of p to the yz-plane are denoted by p′ , the product is

(2π¯ h)2 δ (2) p′N +1 − p′N · · · (2π¯ h)2 δ (2) (p′2 − p′1 ) .

(2.651)

These allow performing all py n , pz n -integrals, except for one overall py , pz . The path integral reduces therefore to (xb tb |xa ta ) = ×

∞

Z

−∞

# " NY N Z ∞ +1 Z dpy dpz Y dpx n dxn (2π¯ h)2 n=1 −∞ 2π¯ h n=1

(2.652)

i p2z exp py (yb − ya ) + pz (zb − za ) − (tb − ta ) h ¯ 2M (

"

#)

i N exp A , h ¯ x

where AN x is the time-sliced action involving only a one-dimensional path integral over the x-component of the path, x(t), with the sliced action AN x

=

N +1 h X n=1

p2x n 1 e px n (xn − xn−1 ) − − py − Bxn 2M 2M c

2 i

.

(2.653)

This is the action of a one-dimensional harmonic oscillator with field-dependent frequency ωB whose center of oscillation depends on py and lies at x0 = py /MωL .

(2.654)

182


The path integral over x(t) is harmonic and known from (2.168): s

MωL 2πi¯ h sin ωL (tb − ta ) nh i i MωL × exp (xb − x0 )2 + (xa − x0 )2 cos ωL (tb − ta ) h ¯ 2 sin ωL (tb − ta )

(xb tb |xa ta )x0 =

− 2(xb − x0 )(xa − x0 )} .

(2.655)

Doing the pz -integral in (2.652), we arrive at the formula (z −z )2 1 iM b a ⊥ (xb tb |xa ta ) = q e 2¯h tb −ta (x⊥ b tb |xa ta ), 2πi¯ h(tb − ta )/M

(2.656)

with the amplitude orthogonal to the magnetic field MωL Z ∞ ⊥ ⊥ (xb tb |xa ta ) ≡ dx0 eiM ωL x0 (yb −ya )/¯h (xb tb |xa ta )x0 . 2π¯ h −∞ After a quadratic completion in x0 , the total exponent in (2.657) reads "

(2.657)

1 iMωL −(x2b + x2a ) tan[ωL (tb − ta )/2] + (xb − xa )2 2¯ h sin ωL (tb − ta )

#

!2

MωL xb + xa yb − ya tan[ωL (tb − ta )/2] x0 − − −i h ¯ 2 2 tan[ωL (tb − ta )/2] " # MωL (xb + xa )2 (yb − ya )2 +i tan[ωL (tb − ta )/2] + 2¯ h 2 2 tan[ωL (tb − ta )/2] MωL (xb + xa )(yb − ya ). +i (2.658) 2¯ h R∞ The integration MωL −∞ dx0 /2π¯ h removes the second term and results in a factor MωL 2π¯ h

s

π¯ h . iMωL tan[ωL (tb − ta )/2]

(2.659)

By rearranging the remaining terms, we arrive at the amplitude (xb tb |xa ta ) =

s

3

M ωL (tb − ta )/2 i exp (Acl + Asf ) , 2πi¯ h(tb − ta ) sin[ωL (tb − ta )/2] h ¯

(2.660)

with an action h i M (zb − za )2 ωL Acl = + cot[ωL (tb − ta )/2] (xb − xa )2 + (yb − ya )2 2 tb − ta 2 +

ωL (xa yb − xb ya )

(2.661)

and the surface term Asf =

b MωL e (xb yb − xa ya ) = B xy . a 2 2c

(2.662)


183


2.18.4

Classical Action

Since the action is harmonic, the amplitude is again a product of a phase eiAcl and a fluctuation factor. A comparison with (2.633) and (2.650) shows that the surface term would be absent if the amplitude (xb tb |xa ta )A˜ were calculated with the vector potential in the axially symmetric gauge (2.632). Thus Acl must be equal to the classical action in this gauge. Indeed, the orthogonal part can be rewritten as A⊥ cl

=

Z

tb

ta

)

(

M M d (xx˙ + y y) ˙ + [x(−¨ x + ωL y) ˙ + y(−¨ y − ωL x)] ˙ . dt 2 dt 2

(2.663)

The equations of motion are ˙ x¨ = ωL y,

y¨ = −ωL x, ˙

(2.664)

reducing the action of a classical orbit to tb M M (xx˙ + y y) ˙ = ([xb x˙ b − xa x˙ a ] + [yb y˙b − ya y˙ a ]) . (2.665) ta 2 2 The orbits are easily determined. By inserting the two equations in (2.664) into each other we see that x˙ and y˙ perform independent oscillations:

A⊥ cl =

x¨˙ + ωL2 x˙ = 0,

y¨˙ + ωL2 y˙ = 0.

(2.666)

The general solution of these equations is 1 x= [(xb − x0 ) sin ωL (t − ta ) − (xa − x0 ) sin ωL (t − tb ) ] + x0 , (2.667) sin ωL (tb − ta ) 1 y= [(yb − y0 ) sin ωL (t − ta ) − (ya − y0 ) sin ωL (t − tb ) ] + y0 , (2.668) sin ωL (tb − ta )

where we have incorporated the boundary condition x(ta,b ) = xa,b , y(ta,b) = ya,b . The constants x0 , y0 are fixed by satisfying (2.664). This gives 1 ωL (xb + xa ) + (yb − ya ) cot (tb − ta ) , 2 2 1 ωL = (yb + ya ) − (xb − xa )cot (tb − ta ) . 2 2

x0 = y0

(2.669) (2.670)

Now we calculate ωL xb [(x0 − xa ) + (xb − x0 ) cos ωL (tb − ta )] , sin ωL (tb − ta ) ωL = xa [(x0 − xa ) cos ωL(tb − ta ) + (xb − x0 )] , sin ωL (tb − ta )

xb x˙ b =

(2.671)

xa x˙ a

(2.672)

and hence

ωL xb x˙ b − xa x˙ a = ωL x0 (xb + xa ) tan (tb − ta ) 2 h i ωL + (x2b + x2a ) cos ωL (tb − ta ) − 2xb xa sin ωL (tb − ta ) ωL ωL 2 = (xb − xa ) cot (tb − ta ) + (xb + xa )(yb − ya ) . 2 2

(2.673)

184


Similarly we find ωL yb y˙ b − ya y˙ a = ωL y0 (yb + ya ) tan (tb − ta ) 2 ωL + [(y 2 + ya2 ) cos ωL (tb − ta ) − 2yb ya ] sin ωL (tb − ta ) b ωL ωL 2 = (yb − ya ) cot (tb − ta ) − (xb − xa )(yb + ya ) . 2 2

(2.674)

Inserted into (2.665), this yields the classical action for the orthogonal motion A⊥ cl

h i M ωL = cot[ωL (tb − ta )/2] (xb − xa )2 + (yb − ya )2 + ωL (xa yb − xb ya ) , 2 2 (2.675)

which is indeed the orthogonal part of the action (2.661).

2.18.5

Translational Invariance

It is interesting to see how the amplitude ensures the translational invariance of all physical observables. The first term in the classical action is trivially invariant. The last term reads ∆A =

MωL (xa yb − xb ya ). 2

(2.676)

Under a translation by a distance d, x → x + d,

(2.677)

MωL MωL [dx (yb − ya ) + dy (xa − xb )] = [(d × x)b − (d × x)a ]z 2 2

(2.678)

this term changes by

causing the amplitude to change by a pure gauge transformation as (xb tb |xa ta ) → eieΛ(xb )/c¯h (xb tb |xa ta )e−ieΛ(xa )/c¯h ,

(2.679)

with the phase Λ(x) = −

MωL h ¯c [d × x]z . 2e

(2.680)

Since observables involve only gauge-invariant quantities, such transformations are irrelevant. This will be done in 2.23.3. H. Kleinert, PATH INTEGRALS

185

2.19 Charged Particle in Magnetic Field plus Harmonic Potential

2.19

Charged Particle in Magnetic Field plus Harmonic Potential

For application in Chapter 5 we generalize the above magnetic system by adding a harmonic oscillator potential, thus leaving the path integral solvable. For simplicity, we consider only the orthogonal part of the resulting system with respect to the magnetic field. Omitting orthogonality symbols, the Hamiltonian is the same as in (2.638). Without much more work we may solve the path integral of a more general system in which the harmonic potential in (2.638) has a different frequency ω 6= ωL , and thus a Hamiltonian H(p, x) =

1 p2 + Mω 2 x2 − ωB lz (p, x). 2M 2

(2.681)

The associated Euclidean action Ae [p, x] =

Z

τb

τa

dτ [−ip · x˙ + H(p, x)]

(2.682)

has the Lagrangian form Ae [x] =

Z

0

hβ ¯

dτ

M 2 1 ˙ )]z . x˙ (τ ) + M(ω 2 − ωB2 )x2 (τ ) − iMωB [x(τ ) × x(τ 2 2 (2.683)

At this point we observe that the system is stable only for ω ≥ ωB . The action (2.683) can be written in matrix notation as Acl =

Z

0

hβ ¯

dτ

"

#

M d M ˙ + xT Dω2 ,B x , (xx) 2 dτ 2

(2.684)

where Dω2 ,B is the 2 × 2 -matrix ′

Dω2 ,B (τ, τ ) ≡

−∂τ2 + ω 2 − ωB2 −2iωB ∂τ 2iωB ∂τ −∂τ2 + ω 2 − ωB2

!

δ(τ − τ ′ ).

(2.685)

Since the path integral is Gaussian, we can immediately calculate the partition function 1 −1/2 Z= det Dω2 ,B . (2.686) (2π¯ h/M)2 By expanding Dω2 ,B (τ, τ ′ ) in a Fourier series Dω2 ,B (τ, τ ′ ) =

∞ 1 X ˜ ω2 ,B (ωm )e−iωm (τ −τ ′ ) , D h ¯ β m=−∞

(2.687)

we find the Fourier components ˜ ω2 ,B (ωm ) = D

2 ωm + ω 2 − ωB2 −2ωB ωm 2 2ωB ωm ωm + ω 2 − ωB2

!

,

(2.688)

186


with the determinants ˜ ω2 ,B (ωm ) = (ω 2 + ω 2 − ω 2 )2 + 4ω 2 ω 2 . det D m B B m

(2.689)

These can be factorized as ˜ ω2 ,B (ωm ) = (ω 2 + ω 2 )(ω 2 + ω 2 ), det D m + m −

(2.690)

with ˜ ω2 ,B (ωm ) are The eigenvectors of D 1 e+ = √ 2

1 i

ω± ≡ ω ± ωB . !

,

with eigenvalues

1 e− = − √ 2

(2.691) 1 −i

!

,

(2.692)

2 d± = ωm + ω 2 ± 2iωm ωB = (ωm + iω± )(ωm − iω∓ ). (2.693) √ Thus the right- and left-circular combinations x± = ±(x ± iy)/ 2 diagonalize the Lagrangian (2.684) to

Acl =

Z

0

hβ ¯

dτ

(

M d ∗ x+ x˙ + + x∗− x˙ − 2 dτ

i Mh ∗ + x+ (−∂τ − ω+ )(−∂τ + ω− )x+ + x∗− (−∂τ − ω− )(−∂τ + ω+ )x− . (2.694) 2 Continued back to real times, the components x± (t) are seen to oscillate independently with the frequencies ω± . The factorization (2.690) makes (2.694) an action of two independent harmonic oscillators of frequencies ω± . The associated partition function has therefore the product form 1 1 Z= . (2.695) 2 sinh (¯ hβω+ /2) 2 sinh (¯ hβω− /2) For the original system of a charged particle in a magnetic field discussed in Section 2.18, the partition function is obtained by going to the limit ω → ωB in the Hamiltonian (2.681). Then ω− → 0 and the partition function (2.695) diverges, since the system becomes translationally invariant in space. From the mnemonic replacement rule (2.353) we see that in this limit we must replace the relevant vanishing inverse frequency by an expression proportional to the volume of the system. The role of ω 2 in (2.353) is played here by the frequency in front of the x2 -term of the Lagrangian (2.683). Since there are two dimensions, we must replace

and thus

1 1 β V2 , − − −→ − − −→ 2 ω 2 − ωB ω→ωB 2ωω− ω− →0 2π/M

(2.696)

1 V2 , (2.697) ω− →0 2 sinh (¯ hβω) λ2ω where λω is the quantum-mechanical length scale in Eqs. (2.296) and (2.351) of the harmonic oscillator. Z − −−→


2.20 Gauge Invariance and Alternative Path Integral Representation

2.20

187

Gauge Invariance and Alternative Path Integral Representation

The action (2.627) of a particle in an external ordinary potential V (x, t) and a vector potential A(x, t) can be rewritten with the help of an arbitrary space- and time-dependent gauge function Λ(x, t) in the following form: A[x] =

Z

tb

dt ta

M 2 e e ˙ x˙ + x(t)[A(x, t) + ∇Λ(x, t)]−V (x, t)+ ∂t Λ(x, t) 2 c c e − [Λ(xb , tb ) − Λ(xa , ta )] . c

(2.698)

The Λ(x, t)-terms inside the integral are canceled by the last two surface terms making the action independent of Λ(x, t). We may now choose a particular function Λ(x, t) equal to c/e-times the classical action A(x, t) which solves the Hamilton-Jacobi equation (1.65), i.e., i2 e 1 h ∇A(x, t) − A(x, t) + ∂t A(x, t) + V (x, t) = 0. 2M c

(2.699)

Then we obtain the following alternative expression for the action: A[x]

=

Z

tb

dt

ta

i2 1 h e M x˙ − ∇A(x, t) + A(x, t) 2M c + A(xb , tb ) − A(xa , ta ).

(2.700)

For two infinitesimally different solutions of the Hamilton-Jacobi equation, the difference between the associated action functions δA satisfies the differential equation v · ∇δA + ∂t δA = 0,

(2.701)

where v is the classical velocity field h i e v(x, t) ≡ (1/M ) ∇A(x, t) − A(x, t) . c

(2.702)

The differential equation (2.701) expresses the fact that two solutions A(x, t) for which the particle energy and momenta at x and t differ by δE and δp, respectively, satisfy the kinematic relation δE = p · δp/M = x˙ cl · ∇δA. This follows directly from E = p2 /2M . The so-constrained variations δE and δp leave the action (2.700) invariant. A sequence of changes δA of this type can be used to make the function A(x, t) coincide with the action A(x, t; xa , ta ) of paths which start out from xa , ta and arrive at x, t. In terms of this action function, the path integral representation of the time evolution amplitude takes the form (xb tb |xa ta ) =

eiA(xb ,tb ;xa ,ta )/¯h

Z

x(tb )=xb

x(ta )=xa

Dx

(2.703)

Z tb i2 i 1 h e ×exp dt M x˙ − ∇A(x, t; xa , ta ) + A(x, t) ¯h ta 2M c or, using v(x(t), t), (xb tb |xa ta ) = e

iA(xb ,tb ;xa ,ta )/¯ h

Z

x(tb )=xb

x(ta )=xa

Z tb i M 2 Dx exp dt (x˙ − v) . ¯h ta 2

(2.704)

188


˙ The fluctuations are now controlled by the deviations of the instantaneous velocity x(t) from local value of the classical velocity field v(x, t). Since the path integral attempts to keep the deviations as small as possible, we call v(x, t) the desired velocity of the particle at x and t. Introducing momentum variables p(t), the amplitude may be written as a phase space path integral (xb tb |xa ta ) = ×

e

iA(xb ,tb ;xa ,ta )/¯ h

Z

x(tb )=xb

x(ta )=xa

′

Dx

Z

Dp 2π¯h

Z tb i 1 2 ˙ exp dt p(t) [x(t) − v(x(t), t)] − p (t) , ¯h ta 2M

(2.705)

which will be used in Section 18.22 to give a stochastic interpretation of quantum processes.

2.21

Velocity Path Integral

There exists yet another form of writing the path integral in which the fluctuating velocities play a fundamental role and which will later be seen to be closely related to path integrals in the so-called stochastic calculus to be introduced in Sections 18.12 and 18.565. We observe that by rewriting the path integral as Z tb Z Z tb i M 2 3 ˙ dt x(t) exp dt x˙ −V (x) , (2.706) (xb tb |xa ta ) = D x δ xb −xa − ¯h ta 2 ta the δ-function allows us to include the last variable xn in the integration measure of the time-sliced version of the path integral. Thus all time-sliced time derivatives (xn+1 − xn )/ǫ for n = 0 to N are integrated over implying that they can be considered as independent fluctuating variables vn . In the potential, the dependence on the velocities can be made explicit by inserting Z tb x(t) = xb − dt v(t), (2.707) t Z t x(t) = xa + dt v(t), (2.708) ta

x(t)

=

1 X+ 2

Z

tb

tb

dt′ v(t′ )ǫ(t′ − t),

(2.709)

where

xb + xa (2.710) 2 is the average position of the endpoints and ǫ(t − t′ ) is the antisymmetric combination of Heaviside functions introduced in Eq. (1.311). In the first replacement, we obtain the velocity path integral Z tb Z Z tb Z tb i M 2 (xb tb |xa ta ) = D3 v δ xb −xa− dt v(t) exp dt v −V xb − dt v(t) . (2.711) ¯h ta 2 ta t X≡

The measure of integration is normalized to make Z tb Z i M 2 3 D v exp dt v = 1. ¯h ta 2

(2.712)

The correctness of this normalization can be verified by evaluating (2.711) for a free particle. Inserting the Fourier representation for the δ-function Z Z tb Z tb d3 p i dt v(t) = δ xb − xa − exp p xb − xa − dt v(t) , (2.713) (2πi)3 ¯h ta ta H. Kleinert, PATH INTEGRALS

189

2.22 Path Integral Representation of Scattering Matrix

we can complete the square in the exponent and integrate out the v-fluctuations using (2.712) to obtain Z d3 p i p2 exp p (x − x ) − (t − t ) . (2.714) (xb tb |xa ta ) = b a b a (2πi)3 ¯h 2M This is precisely the spectral representation (1.329) of the free-particle time evolution amplitude (1.331) [see also Eq. (2.51)]. A more symmetric velocity path integral is obtained by choosing the third replacement (2.709). This leads to the expression Z tb Z Z tb i M 2 3 dt v(t) exp (xb tb |xa ta ) = D v δ ∆x − dt v ¯h ta 2 ta Z tb Z tb i 1 × exp − dt V X + dt′ v(t′ )ǫ(t′ − t) . (2.715) ¯h ta 2 ta The velocity representations are particularly useful if we want to know integrated amplitudes such as Z tb Z Z i M 2 3 3 d xa (xb tb |xa ta ) = D v exp dt v . ¯h ta 2 Z tb Z i 1 tb ′ × exp − dtV xb − dt v(t′ ) , (2.716) ¯h ta 2 t which will be of use in the next section.

2.22

Path Integral Representation of Scattering Matrix

In Section 1.16 we have seen that the description of scattering processes requires several nontrivial limiting procedures on the time evolution amplitude. Let us see what these procedures yield when applied to the path integral representation of this amplitude.

2.22.1

General Development

Formula (1.471) for the scattering matrix expressed in terms of the time evolution operator in momentum space has the following path integral representation: Z Z ˆ ai ≡ hpb |S|p lim ei(Eb tb −Ea ta )/¯h d3 xb d3 xa e−i(pb xb −pa xa )/¯h (xb tb |xa ta ). (2.717) tb −ta →∞

Introducing the momentum transfer q ≡ (pb − pa ), we rewrite e−i(pb xb −pa xa )/¯h as e−iqxb /¯h e−ipa (xb −xa )/¯h , and observe that the amplitude including the exponential prefactor e−ipa (xb −xa )/¯h has the path integral representation: Z tb Z i M 2 −ipa (xb −xa )/¯ h 3 e (xb tb |xa ta ) = D x exp dt x˙ − pa x˙ − V (x) . (2.718) ¯h ta 2 The linear term in x˙ is eliminated by shifting the path from x(t) to y(t) = x(t) −

pa t M

(2.719)

leading to e

−ipa (xb −xb )/¯ h

(xb tb |xa ta ) = e

Z

−ip2a (tb −ta )/2M¯ h

Z tb i M 2 P D y exp dt y˙ −V y+ t . ¯h ta 2 M (2.720) 3

190


Inserting everything into (2.717) we obtain Z Z 2 eiq tb /2M¯h d3 yb e−iqyb /¯h d3 ya tb −ta →∞ Z tb Z M 2 pa i dt y˙ −V y + t . D3 y exp ¯h ta 2 M

ˆ ai ≡ hpb |S|p

lim

×

In the absence of an interaction, the path integral over y(t) gives simply " # Z 2 1 i M (yb − ya ) 3 d ya p exp = 1, ¯h 2 tb − ta ) 2π¯hi(tb − ta )/M

(2.721)

(2.722)

and the integral over ya yields ˆ a i|V ≡0 = hpb |S|p

lim

tb −ta →∞

eiq

2

(tb −ta )/8M¯ h

(2π¯h)3 δ (3) (q) = (2π¯h)3 δ (3) (pb − pa ),

(2.723)

which is the contribution from the unscattered beam to the scattering matrix in Eq. (1.474). The first-order contribution from the interaction reads, after a Fourier decomposition of the potential, Z Z Z d3 Q i iq2 tb /2M¯ h 3 −iqyb /¯ h ˆ lim e d yb e V (Q) d3 ya hpb |S1 |pa i = − ¯h tb −ta →∞ (2π¯h)3 Z Z tb Z tb i M 2 i pa Q ′ 3 ′ ′ × dt exp t D y exp dt y˙ + δ(t −t)Q y . (2.724) ¯h M ¯h ta 2 ta The harmonic path integral was solved in one dimension for an arbitrary source j(t) in Eq. (3.168). For ω = 0 and the particular source j(t) = δ(t′ − t)Q the result reads, in three dimensions, 1 i M (yb − ya )2 exp p 3 ¯h 2 tb − ta 2πi¯h(tb − ta )/M i 1 1 × exp [yb (t′ − ta ) + ya (tb − t′ )] Q − (tb − t′ )(t′ − ta )Q2 . (2.725) ¯h tb − ta 2M Performing here the integral over ya yields i i 1 ′ 2 exp Q yb exp − (tb − t )Q . ¯h ¯h 2M

(2.726)

The integral over yb in (2.724) leads now to a δ-function (2π¯h)3 δ (3) (Q − q), such that the exponential prefactor in (2.724) is canceled by part of the second factor in (2.726). In the limit tb − ta → ∞, the integral over t′ produces a δ-function 2π¯hδ(pb Q/M + Q2 /2M ) = 2π¯hδ(Eb − Ea ) which enforces the conservation of energy. Thus we find the well-known Born approximation hpb |Sˆ1 |pa i = −2πiδ(Eb − Ea )V (q).

(2.727)

In general, we subtract the unscattered particle term (2.723) from (2.721), to obtain a path integral representation for the T -matrix [for the definition recall (1.474)]: Z Z iq2 tb /2M¯ h 3 −iqyb /¯ h ˆ 2π¯hiδ(Eb − Ea )hpb |T |pa i ≡ − lim e d yb e d3 ya tb −ta →∞ Z tb Z Z i M 2 i tb pa 3 × D y exp dt y˙ exp − dt V y + t −1 . (2.728) ¯h ta 2 ¯h ta M H. Kleinert, PATH INTEGRALS

2.22 Path Integral Representation of Scattering Matrix

191

It is preferable to find a formula which does not contain the δ-function of energy conservation as a factor on the left-hand side. In order to remove this we observe that its origin lies in the timetranslational invariance of the path integral in the limit tb − ta → ∞. If we go over to a shifted time variable t → t + t0 , and change simultaneously y → y − pa t0 /M , then the path integral remains the same Rexcept for shifted initial and finalRtimes tb + t0 and ta + t0 . In the limit tb − ta → ∞, the t +t t integrals tab+t00 dt can be replaced again by tab dt. The only place where a t0 -dependence remains is in the prefactor e−iqyb /¯h which changes to e−iqyb /¯h eiqpa t0 /M¯h . Among all path fluctuations, there exists one degree of freedom which is equivalent to a temporal shift of the path. This is equivalent to an integral over t0 which yields a δ-function 2π¯hδ (qpa /M ) = 2π¯hδ (Eb − Ea ). We only must make sure to find the relation between this temporal shift and the corresponding measure in the path ˆ a ≡ pa /|pa |. The formal integral. This is obviously a shift of the path as a whole in the direction p way of isolating this degree of freedom proceeds according to a method developed by Faddeev and Popov36 by inserting into the path integral (2.721) the following integral representation of unity: Z |pa | ∞ dt0 δ (ˆ pa (yb + pa t0 /M )) . (2.729) 1= M −∞ In the following, we shall drop the subscript a of the incoming beam, writing p ≡ pa ,

p ≡ |pa | = |pb |.

(2.730)

After the above shift in the path integral, the δ-function in (2.729) becomes δ (ˆ pa yb ) inside the path integral, with no t0 -dependence. The integral over t0 can now be performed yielding the δ-function in the energy. Removing this from the equation we obtain the path integral representation of the T -matrix Z Z 2 p hpb |Tˆ|pa i ≡ i lim eiq (tb −ta )/8M¯h d3 yb δ (ˆ pa yb ) e−iqyb /¯h d3 ya M tb −ta →∞ Z tb Z Z i M i tb P × D3 y exp dt y˙ 2 exp − dt V y+ t −1 . (2.731) ¯h ta 2 ¯h ta M At this point it is convenient to go over to the velocity representation of the path integral (2.715). This enables us to perform trivially the integral over yb , and we obtain the y version of (2.716). The δ-function enforces a vanishing longitudinal component of yb . The transverse component of yb will be denoted by b: b ≡ yb − (ˆ pa yb )ˆ p/a. (2.732) Hence we find the path integral representation Z 2 p hpb |Tˆ|pa i ≡ i lim eiq tb /2M¯h d2 b e−iqb/¯h M tb −ta →∞ Z tb h Z i i M × D3 v exp dt v2 eiχb,p [v] − 1 , ¯h ta 2 where the effect of the interaction is contained in the scattering phase Z Z tb 1 tb p ′ ′ χb,p [v] ≡ − dt V b + t− dt v(t ) . ¯h ta M t

(2.733)

(2.734)

We can go back to a more conventional path integral by replacing the velocity paths v(t) by Rt ˙ y(t) = − t b v(t). This vanishes at t = tb . Equivalently, we can use paths z(t) with periodic boundary conditions and subtract from these z(tb ) = zb . From hpb |Tˆ|pa i we obtain the scattering amplitude fpb pa , whose square gives the differential cross section, by multiplying it with a factor −M/2π¯h [see Eq. (1.494)]. 36

L.D. Faddeev and V.N. Popov, Phys. Lett. B 25 , 29 (1967).

192


Note that in the velocity representation, the evaluation of the harmonic path integral integrated over ya in (2.724) is much simpler than before where we needed the steps (2.725), (2.726). After the Fourier decomposition of V (x) in (2.734), the relevant integral is Z tb R tb Z i ′ 2 i i M 2 − 2M dtΘ2 (tb−t′)Q2 3 ′ h ¯ ta (2.735) D v exp dt v −Θ(tb− t )Q v = e = e− 2Mh¯ (tb −t )Q . ¯h ta 2 The first factor in (2.726) comes directly from the argument Y in the Fourier representation of the potential Z tb p V yb + t− dt′ v(t′ ) M t in the velocity representation of the S-matrix (2.721).

2.22.2

Improved Formulation 2

The prefactor eiq tb /2M¯h in Formula (2.733) is an obstacle to taking a more explicit limit tb −ta → ∞ on the right-hand side. To overcome this, we represent this factor by an auxiliary path integral37 over some vector field w(t): R tb Z Z i tb M 2 i dt Θ(t)w(t)q/¯ h iq2 tb /2M¯ h 3 e = D w exp − dt w (t) e ta . (2.736) ¯h ta 2 h R i tb −iq b+

dt Θ(t)w(t) /¯ h

ta The last factor changes the exponential e−iqb/¯h in (2.733) into e . Since b Rt is a dummy variable of integration, we can equivalently replace b → bw ≡ b − tab dt Θ(t)w(t) in the scattering phase χb,p [v] and remain with Z Z p 2 −iqb/¯ h fpb pa = lim d be D3 w tb −ta →∞ 2πi¯ h Z ∞ Z i M 2 × D3 v exp dt v − w2 exp iχbw ,p − 1 . (2.737) ¯h −∞ 2

The scattering phase in this expression can be calculated from formula (2.734) with the integral taken over the entire t-axis: Z Z tb 1 ∞ p ′ ′ ′ ′ ′ χbw ,p [v, w] = − dt V b + t− dt [Θ(t −t)v(t ) − Θ(t )w(t )] . (2.738) ¯h −∞ M ta The fluctuations of w(t) are necessary to correct for the fact that the outgoing particle does not run, on the average, with the velocity p/M = pa /M but with velocity pb /M = (p + q)/M . Rt We may also go back to a more conventional path integral by inserting y(t) = − t b v(t) and Rt setting similarly z(t) = − t b w(t). Then we obtain the alternative representation Z Z Z p d2 b e−iqb/¯h d3 ya d3 za fpb pa = lim tb −ta →∞ 2πi¯ h Z tb Z Z i h iχ i M 2 3 3 2 × D y D z exp dt y˙ − z˙ e bz ,p [y] − 1 , (2.739) ¯h ta 2

with

Z 1 tb p dt V b + t + y(t) − z(0) , (2.740) ¯h ta M where the path integrals run over all paths with yb = 0 and zb = 0. In Section 3.26 this path integral will be evaluated perturbatively. χbz ,p [y] ≡ −

37

See R. Rosenfelder, notes of a lecture held at the ETH Z¨ urich in 1979: Pfadintegrale in der Quantenphysik , 126 p., PSI Report 97-12, ISSN 1019-0643, and Lecture held at the 7th Int. Conf. on Path Integrals in Antwerpen, Path Integrals from Quarks to Galaxies, 2002. H. Kleinert, PATH INTEGRALS

193

2.23 Heisenberg Operator Approach to Time Evolution Amplitude

2.22.3

Eikonal Approximation to Scattering Amplitude

To lowest approximation, we neglect the fluctuating variables y(t) and z(t) in (2.740). Since the integral Z tb Z Z Z Z i M 2 dt y˙ − z˙ 2 (2.741) d3 ya d3 za D3 y D3 z exp ¯h ta 2 in (2.739) has unit normalization [recall the calculation of (2.726)], we obtain directly the eikonal approximation to the scattering amplitude Z h i p fpeib pa ≡ d2 b e−iqb/¯h exp iχei (2.742) b ,p − 1 , 2πi¯h with χei b,p ≡ −

1 ¯h

Z

∞

p dt V b + t . M −∞

(2.743)

The time integration can be converted into a line integration along the direction of the incoming particles by introducing a variable z ≡ pt/M . Then we can write Z M1 ∞ ei ˆ z) . χb , p ≡ − dz V (b + p (2.744) p ¯h −∞ If V (x) is rotationally symmetric, it depends only on r ≡ |x|. Then we shall write the potential as V (r) and calculate (2.744) as the integral Z p M1 ∞ 2 + z2 . χei ≡ − b (2.745) dz V b,p p ¯h −∞ Inserting this into (2.742), we can perform the integral over all angles between q and b using the formula Z π 1 i dθ exp qb cos θ = J0 (qb), (2.746) 2π −π ¯h where J0 (ξ) is the Bessel function, and find Z h i p ei fpb pa = db b J0 (qb) exp iχei b ,p − 1 . i¯h

(2.747)

The variable of integration b coincides with the impact parameterb introduced in Eq. (1.497). The result (2.747) is precisely the eikonal approximation (1.497) with χei b,p /2 playing the role of the scattering phases δl (p) of angular momentum l = pb/¯h: χei h (p). b,p = 2iδpb/¯

2.23

(2.748)

Heisenberg Operator Approach to Time Evolution Amplitude

An interesting alternative to the path integral derivation of the time evolution amplitudes of harmonic systems is based on quantum mechanics in the Heisenberg picture. It bears a close similarity with the path integral derivation in that it requires solving the classical equations of motion with given initial and final positions to obtain the exponential of the classical action eiA/¯h . The fluctuation factor, however, which accompanies this exponential is obtained quite differently from commutation rules of the operatorial orbits at different times as we shall now demonstrate.

194

2.23.1


Free Particle

We want to calculate the matrix element of the time evolution operator ˆ

(x t|x′ 0) = hx|e−iHt/¯h |x′ i,

(2.749)

ˆ is the Hamiltonian operator where H ˆ2 p ˆ = H(ˆ . H p) = 2M

(2.750)

We shall calculate the time evolution amplitude (2.749) by solving the differential equation h i ˆ h ′ ˆ ˆ ˆ h ˆ e−iHt/¯ ˆ e−iHt/¯ i¯h∂t hx t|x′ 0i ≡ hx|H |x i = hx|e−iHt/¯h eiHt/¯h H |x′ i =

hx t|H(ˆ p(t))|x′ 0i.

(2.751)

ˆ . The evaluThe argument contains now the time-dependent Heisenberg picture of the operator p ation of the right-hand side will be based on re-expressing the operator H(ˆ p(t)) as a function of initial and final position operators in such a way that all final position operators stand to the left of all initial ones: ˆ = H(ˆ ˆ (0); t). H x(t), x

(2.752)

Then the matrix elements on the right-hand side can immediately be evaluated using the eigenvalue equations ˆ (0)|x′ 0i = x′ |x′ 0i, hx t|ˆ x(t) = xhx t|, x (2.753) as being ˆ (0); t)|ˆ hx t|H(ˆ x(t), x x 0i = H(x, x′ ; t)hx t|x′ 0i,

(2.754)

and the differential equation (2.751) becomes i¯h∂t hx t|x′ 0i ≡ H(x, x′ ; t)hx t|x′ 0i, or hx t|x′ 0i = C(x, x′ )E(x, x′ ; t) ≡ C(x, x′ )e

−i

R

t

(2.755)

dt′ H(x,x′ ;t′ )/¯ h

.

(2.756)

The prefactor C(x, x′ ) contains a possible constant of integration resulting from the time integral in the exponent. The Hamiltonian operator is brought to the time-ordered form (2.752) by solving the Heisenberg equations of motion dˆ x(t) dt dˆ p(t) dt

= =

i p ˆ (t) i hˆ ˆ (t) = H, x , ¯h M h i i ˆ ˆ (t) = 0. H, p ¯h

(2.757) (2.758)

The second equation shows that the momentum is time-independent: ˆ (t) = p ˆ (0), p

(2.759)

so that the first equation is solved by ˆ (t) − x ˆ (0) = t x

ˆ (t) p , M

(2.760)


195

2.23 Heisenberg Operator Approach to Time Evolution Amplitude which brings (2.750) to

ˆ = M [ˆ ˆ (0)]2 . H x(t) − x (2.761) 2t2 This is not yet the desired form (2.752) since there is one factor which is not time-ordered. The ˆ as proper order is achieved by rewriting H 2 ˆ = M x ˆ (t) − 2ˆ ˆ 2 (0) + [ˆ ˆ (0)] , x(t)ˆ x(0) + x x(t), x (2.762) H 2t2 and calculating the commutator from Eq. (2.760) and the canonical commutation rule [ˆ pi , x ˆj ] = −i¯hδij as i¯h ˆ (0)] = − Dt, (2.763) [ˆ x(t), x M so that we find the desired expression M 2 D ˆ = H(ˆ ˆ (0); t) = 2 x ˆ (t) − 2ˆ ˆ 2 (0) − i¯h . H x(t), x x(t)ˆ x(0) + x (2.764) 2t 2t Its matrix elements (2.754) can now immediately be written down: M D 2 (x − x′ ) − i¯h . 2t2 2t From this we find directly the exponential factor in (2.756) R ′ D i M 2 (x − x′ ) − log t . E(x, x′ ; t) = e−i dt H(x,x ;t)/¯h = exp ¯h 2t 2 H(x, x′ ; t) =

(2.765)

(2.766)

Inserting (2.766) into Eq. (2.756), we obtain ′

1

′

hx t|x 0i = C(x, x )

tD/2

iM ′ 2 exp (x − x ) . ¯h 2t

(2.767)

A possible constant of integration in (2.766) depending on x, x′ is absorbed in the prefactor C(x, x′ ). This is fixed by differential equations involving x: h i ˆ ˆ ˆ ˆ ˆ e−iHt/¯h |x′ i = hx t|ˆ −i¯h∇hx t|x′ 0i = hx|ˆ pe−iHt/¯h |x′ i = hx|e−iHt eiHt/¯h p p(t)|x′ 0i. ˆ

ˆ |x′ ihx t|ˆ i¯h∇′ hx t|x′ 0i = hx|e−iHt/¯h p p(0)|x′ 0i.

(2.768)

Inserting (2.760) and using the momentum conservation (2.759), these become M (x − x′ ) hx t|x′ 0i, t M i¯h∇′ hx t|x′ 0i = (x − x′ ) hx t|x′ 0i. t Inserting here the previous result (2.767), we obtain the conditions −i¯h∇hx t|x′ 0i =

−i∇C(x, x′ ) = 0,

i∇′ C(x, x′ ) = 0,

(2.769)

(2.770)

which is solved only by a constant C. The constant, in turn, is fixed by the initial condition lim hx t|x′ 0i = δ (D) (x − x′ ),

t→0

to be r

D

M , 2πi¯h so that we find the correct free-particle amplitude (2.125) r D M iM ′ hx t|x 0i ≡ exp (x − x′ )2 . 2πi¯ht ¯h 2t C=

(2.771)

(2.772)

(2.773)

Note that the fluctuation factor 1/tD/2 emerges in this approach as a consequence of the commutation relation (2.763).

196

2.23.2


Harmonic Oscillator

Here we are dealing with the Hamiltonian operator ˆ2 p M ω2 2 ˆ = H(ˆ ˆ) = H p, x + x , 2M 2

(2.774)

which has to be brought again to the time-ordered form (2.752). We must now solve the Heisenberg equations of motion dˆ x(t) dt dˆ p(t) dt

= =

i p ˆ (t) i hˆ ˆ (t) = H, x , ¯h M h i i ˆ ˆ (t) = −M ω 2 x ˆ (t). H, p ¯h

(2.775) (2.776)

By solving these equations we obtain [compare (2.151)] ˆ (t) = M p

ω ˆ (0)] . [ˆ x(t) cos ωt − x sin ωt

(2.777)

Inserting this into (2.774), we obtain ˆ = H which is equal to ˆ = H

o M ω2 n 2 2 2 ˆ ˆ [ˆ x (t) cos ωt − x (0)] + sin ωt x (t) , 2 sin2 ωt

M ω2 2 ˆ (t) + x ˆ 2 (0) − 2 cos ωt x ˆ (t)ˆ ˆ (0)] . x x(0) + cos ωt [ˆ x(t), x 2 2 sin ωt

(2.778)

(2.779)

ˆ (t), we find the commutator [compare (2.763)] By commuting Eq. (2.777) with x ˆ (0)] = − [ˆ x(t), x

i¯h sin ωt D , M ω

(2.780)

so that we find the matrix elements of the Hamiltonian operator in the form (2.754) [compare (2.765)] H(x, x′ ; t) =

M ω2 2 D ′2 ′ x + x − 2 cos ωt xx − i¯h ω cot ωt. 2 2 2 sin ωt

This has the integral [compare (2.766)] Z i M ω h 2 D sin ωt 2 dt H(x, x′ ; t) = − x + x′ cos ωt − 2 x x′ − i¯h log . 2 sin ωt 2 ω

(2.781)

(2.782)

Inserting this into Eq. (2.756), we find precisely the harmonic oscillator amplitude (2.170), apart from the factor C(x, x′ ). This is again determined by the differential equations (2.768), leaving only a simple normalization factor fixed by the initial condition (2.771) with the result (2.772). Again, the fluctuation factor has its origin in the commutator (2.780).

2.23.3

Charged Particle in Magnetic Field

We now turn to a charged particle in three dimensions in a magnetic field treated in Section 2.18, where the Hamiltonian operator is most conveniently expressed in terms of the operator of the covariant momentum (2.636), e ˆ ≡p ˆ − A(ˆ P x), (2.783) c H. Kleinert, PATH INTEGRALS

197

2.23 Heisenberg Operator Approach to Time Evolution Amplitude as [compare (2.635)]

ˆ2 P ˆ = H(ˆ ˆ) = H p, x . (2.784) 2M In the presence of a magnetic field, its components do not commute but satisfy the commutation rules: e e e¯h e¯h pi , Aˆj ] − [Aî , pˆj ] = i (∇i Aj − ∇j Ai ) = i Bij , [Pî , Pˆj ] = − [ˆ c c c c

(2.785)

where Bij = ǫijk BK is the usual antisymmetric tensor representation of the magnetic field. We now have to solve the Heisenberg equations of motion dˆ x(t) dt ˆ dP(t) dt

i i hˆ ˆ (t) = H, x ¯h i hˆ ˆ i H, P(t) = ¯h

= =

ˆ P(t) M e e¯h ˆ B(ˆ x(t))P(t) +i ∇j Bji (ˆ x(t)), Mc Mc

(2.786) (2.787)

ˆ ˆ In a where B(ˆ x(t))P(t) is understood as the product of the matrix Bij (ˆ x(t)) with the vector P. constant field, where Bij (ˆ x(t)) is a constant matrix Bij , the last term in the second equation is absent and we find directly the solution ˆ ˆ P(t) = eΩL t P(0),

(2.788)

where ΩL is a matrix version of the Landau frequency (2.640) e Bij , Mc

ΩL ij ≡

(2.789)

which can also be rewritten with the help of the Landau frequency vector L

≡

e B Mc

(2.790)

and the 3 × 3-generators of the rotation group (Lk )ij ≡ −iǫkij

(2.791)

as ΩL = i L ·

L.

(2.792)

Inserting this into Eq. (2.786), we find ˆ (t) = x ˆ (0) + x

ˆ eΩL t − 1 P(0) , ΩL M

(2.793)

where the matrix on the right-hand side is again defined by a power series expansion eΩL t − 1 t2 t3 = t + ΩL + Ω2L + . . . . ΩL 2 3!

(2.794)

ˆ P(0) ΩL /2 ˆ (0)] . = e−ΩL t/2 [ˆ x(t) − x M sinh ΩL t/2

(2.795)

ˆ ˆ (0)] , P(t) = M N (ΩL t) [ˆ x(t) − x

(2.796)

We can invert (2.793) to find

Using (2.788), this implies

198


with the matrix N (ΩL t) ≡

ΩL /2 eΩL t/2 . sinh ΩL t/2

(2.797)

By squaring (2.796) we obtain ˆ 2 (t) P M ˆ (0)]T K(ΩL t) [ˆ ˆ (0)] , = [ˆ x(t) − x x(t) − x 2M 2

(2.798)

K(ΩL t) = N T (ΩL t)N (ΩL t).

(2.799)

where Using the antisymmetry of the matrix ΩL , we can rewrite this as K(ΩL t) = N (−ΩL t)N (ΩL t) =

Ω2L /4 . sinh2 ΩL t/2

(2.800)

ˆ (t) at different times is, due to Eq. (2.793), The commutator between two operators x ΩL t i e −1 ˆ j (0)] = − [ˆ xi (t), x , M ΩL ij

(2.801)

and T

i eΩL t − 1 eΩL t − 1 ˆ i (t), x ˆ j (0) + [ˆ ˆ i (0)] = − x xj (t), x + M ΩL ΩTL ΩL t i e − e−ΩL t i sinh ΩL t = − = −2 . M ΩL M ΩL ij ij

!

ij

(2.802)

ˆ (t) and x ˆ (0), thereby time-ordering Respecting this, we can expand (2.798) in powers of operators x the later operators to the left of the earlier ones as follows: M T ˆ (t)K(ΩL t)ˆ ˆ T K(ΩL t)ˆ x x(t) − 2ˆ xT K(ΩL t)ˆ x(0) + x x(0) 2 i¯h ΩL ΩL t − tr coth . 2 2 2

ˆ (0)) = H(ˆ x(t), x

This has to be integrated in t, for which we use the formulas Z Z Ω2L /2 ΩL ΩL t dt K(ΩL t) = dt =− coth , 2 2 2 sinh ΩL t/2

(2.803)

(2.804)

and Z

ΩL ΩL t sinh ΩL t/2 sinh ΩL t/2 1 coth = tr log = tr log + 3 log t, dt tr 2 2 2 ΩL /2 ΩL t/2

(2.805)

these results following again from a Taylor expansion of both sides. The factor 3 in the last term is due to the three-dimensional trace. We can then immediately write down the exponential factor E(x, x′ ; t) in (2.756): iM 1 ΩL ΩL t 1 sinh ΩL t/2 ′ ′ T ′ E(x, x ; t) = 3/2 exp (x−x ) coth (x−x ) − tr log . (2.806) ¯h 2 2 2 2 ΩL t/2 t The last term gives rise to a prefactor −1/2 sinh ΩL t/2 det . ΩL t/2

(2.807)


199

2.23 Heisenberg Operator Approach to Time Evolution Amplitude

As before, the time-independent integration factor C(x, x′ ) in (2.756) is fixed by differential equations in x and x′ , which involve here the covariant derivatives: i h i h e ˆ h ′ ˆ ˆ ˆ h ˆ −iHt/¯ ˆ −iHt/¯ |x i = hx|e−iHt/¯h eiHt/¯h Pe |x′ i −i¯ h∇− A(x) hx t|x′ 0i = hx|Pe c ′ ˆ = hx t|P(t)|x 0i = L(ΩL t)(x − x′ )hx t|x′ 0i, (2.808) h i e ˆ ˆ ′i i¯h∇′ − A(x) hx t|x′ 0i = hx|e−iHt/¯h P|x c ′ ˆ = hx t|P(0)|x 0i = L(ΩL t)(x − x′ )hx t|x′ 0i. (2.809) Calculating the partial derivative we find −i¯h∇hx t|x′ 0i = [−i¯h∇C(x, x′ )]E(x, x′ ; t)+C(x, x′ )[−i¯h∇E(x, x′ ; t)] ΩL ΩL t ′ ′ ′ = [−i¯h∇C(x, x )]E(x, x ; t)+C(x, x )M coth (x−x′ )E(x, x′ ; t). 2 2 Subtracting the right-hand side of (2.808) leads to ΩL ΩL t M M coth (x − x′ ) − M L(ΩL t)(x − x′ ) = − ΩL (x − x′ ), 2 2 2 so that C(x, x′ ) satisfies the time-independent differential equation M e ΩL (x − x′ ) C(x, x′ ) = 0. −i¯h∇ − A(x) − c 2 A similar equation is found from the second equation (2.809): e M ′ ′ i¯h∇ − A(x) − ΩL (x − x ) C(x, x′ ) = 0. c 2

(2.810)

(2.811)

(2.812)

These equations are solved by Z x e M i ′ C(x, x ) = C exp d A( ) + ΩL ( − x ) . ¯h x′ c 2 ′

The contour of integration is arbitrary since the vector field in brackets, e ′ e ΩL e 1 A ( ) ≡ A( ) + ( − x′ ) = A( ) − B × ( − x′ ) c c 2 c 2

(2.813)

(2.814)

has a vanishing curl, ∇ × A′ (x) = 0. We can therefore choose the contour to be a straight line connecting x′ and x, in which case d points in the same direction of x − x′ as − x′ so that the cross product vanishes. Hence we may write for a straight-line connection the ΩL -term Z e x d A( ) . C(x, x′ ) = C exp i (2.815) c x′ Finally, the normalization constant C is fixed by the initial condition (2.771) to have the value (2.772). Collecting all terms, the amplitude is hx t|x′ 0i =

1

det

sinh ΩL t/2 ΩL t/2

−1/2

Z e x exp i d A( ) c x′

q 3 2πi¯h2 t/M iM ΩL ΩL t ′ T ′ × exp (x − x ) coth (x − x ) . ¯h 2 2 2

(2.816)

200


All expressions simplify if we assume the magnetic case the frequency matrix becomes  0 ωL ΩL =  −ωL 0 0 0

so that

field to point in the z-direction, in which  0 0 , 0

(2.817)



cos ωL t/2 0 ΩL t  0 cos ωL t/2 cos = 2 0 0

and

whose determinant is

 0 0 , 1

 0 sin ωL t/2 sinh ΩL t/2  − sin ωL t/2 0 = ΩL t/2 0 0 det

sinh ΩL t/2 = ΩL t/2

sinh ωL t/2 ωL t/2

2

(2.818)

 0 0 , 1 .

(2.819)

(2.820)

Let us calculate the exponential involving the vector potential in (2.816) explicitly. We choose the gauge in which the vector potential points in the y-direction [recall (2.628)], and parametrize the straight line between x′ and x as = x′ + s(x − x′ ),

s ∈ [0, 1].

(2.821)

Then we find Z

x

d A( ) =

x′

=

Z

B(y − y ′ )

1

0 ′ ′

ds [x′ + s(x − x′ )] = B(y − y ′ )(x + x′ )

B(xy − x y ) + B(x′ y − xy ′ ).

(2.822)

Inserting this and (2.820) into (2.756), we recover the earlier result (2.660).

Appendix 2A

Baker-Campbell-Hausdorff Formula and Magnus Expansion

The standard Baker-Campbell-Hausdorff formula, from which our formula (2.9) can be derived, reads ˆ ˆ ˆ eA eB = eC , (2A.1) where ˆ+ Cˆ = B

Z

1

ˆ dtg(ead A t ead B )[A],

(2A.2)

0

and g(z) is the function g(z) ≡

∞ X log z (1 − z)n = , z − 1 n=0 n + 1

(2A.3)

ˆ in the so-called adjoint representation, which is defined and adB is the operator associated with B by ˆ ≡ [B, ˆ A]. ˆ adB[A] (2A.4) H. Kleinert, PATH INTEGRALS

Baker-Campbell-Hausdorff Formula and Magnus Expansion 201

Appendix 2A

ˆ = 1[A] ˆ ≡ A. ˆ By expanding the exponentials One also defines the trivial adjoint operator (adB)0 [A] in Eq. (2A.2) and using the power series (2A.3), one finds the explicit formula ˆ + Aˆ + Cˆ = B

∞ X (−1)n n+1 n=1

×

X

pi ,qi ;pi +qi ≥1 q1

(ad A)p1 (adB) p1 ! q1 !

···

1+

1 Pn

i=1

pi

(adA)pn (adB)qn ˆ [A]. pn ! qn !

(2A.5)

The lowest expansion terms are ˆ + A− ˆ 1 1 adA + adB + 1 (ad A)2 + 1 adA adB + 1 (adB)2 +. . . [A] ˆ Cˆ = B 2 6 2 2 2 1 ˆ + 13 (adA)2 + 21 adA adB + 21 adB adA + (adB)2 + . . . [A] 3 ˆ + 1 [A, ˆ B] ˆ + 1 ([A, ˆ [A, ˆ B]] ˆ + [B, ˆ [B, ˆ A]]) ˆ + 1 [A, ˆ [[A, ˆ B], ˆ B]] ˆ ... . = Aˆ + B 2 12 24

(2A.6)

The result can be rearranged to the closely related Zassenhaus formula ˆ

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

eA+B = eA eB eZ2 eZ3 eZ4 · · · ,

(2A.7)

where Zˆ2

=

Zˆ3

=

Zˆ4

= .. .

1 ˆ ˆ [B, A] 2 1 ˆ ˆ ˆ 1 ˆ ˆ ˆ − [B, [B, A]] − [A, [B, A]]) 3 6 1 ˆ ˆ ˆ ˆ ˆ A], ˆ A], ˆ B] ˆ + 1 [[[B, ˆ A], ˆ A], ˆ A] ˆ [[[B, A], B], B] + [[[B, 8 24

(2A.8) (2A.9) (2A.10)

.

To prove the expansion (2A.6), we derive and solve a differential equation for the operator function ˆ B ˆ ˆ = log(eAt C(t) e ). (2A.11) Its value at t = 1 will supply us with the desired result Cˆ in (2A.5). Starting point is the observation ˆ, that for any operator M ˆ ˆ ˆ e−C(t) ˆ ], eC(t) M = ead C(t) [M (2A.12) by definition of adC. Inserting (2A.11), the left-hand side can also be rewritten as ˆ B ˆ ˆ −B ˆ −At ˆ At ˆ ], by definition (2A.4). Hence we have e e Me e , which in turn is equal to ead A t ead B [M ead C(t) = ead A t ead B .

(2A.13)

Differentiation of (2A.11) yields d −C(t) ˆ ˆ e = −A. dt The left-hand side, on the other hand, can be rewritten in general as ˆ

eC(t)

ˆ

eC(t)

d −C(t) ˆ ˆ˙ e = −f (adC(t))[C(t)], dt

where f (z) ≡

ez − 1 . z

(2A.14)

(2A.15)

(2A.16)

202


This will be verified below. It implies that ˆ˙ ˆ f (adC(t))[C(t)] = A.

(2A.17)

We now define the function g(z) as in (2A.3) and see that it satisfies g(ez )f (z) ≡ 1.

(2A.18)

ˆ˙ ˆ˙ C(t) = g(ead C(t) )f (adC(t))[C(t)].

(2A.19)

We therefore have the trivial identity

Using (2A.17) and (2A.13), this turns into the differential equation ˆ˙ ˆ = ead A t ead B [A], ˆ C(t) = g(ead C(t) )[A]

(2A.20)

from which we find directly the result (2A.2). To complete the proof we must verify (2A.15). The expression is not simply equal to ˆ ˆ C(t) ˆ˙ ˆ˙ ˆ˙ −e C(t)M e−C(t) since C(t) does not. in general, commute with C(t). To account for this consider the operator d −C(t)s ˆ ˆ ˆ t) ≡ eC(t)s O(s, e . (2A.21) dt Differentiating this with respect to s gives d ˆ ˆ ˆ ˆ ˆ ˆ t) = eC(t)s ˆ d e−C(t)s ∂s O(s, C(t) − eC(t)s C(t)e−C(t)s dt dt ˆ ˆ ˙ˆ −C(t)s C(t)s = −e C(t)e =

ˆ˙ −ead C(t)s [C(t)].

(2A.22)

Hence ˆ t) − O(0, ˆ t) = O(s,

Z

s

ˆ ′ , t) ds′ ∂s′ O(s

0

= −

∞ X

sn+1 ˙ n ˆ (adC(t)) [C(t)], (n + 1)! n=0

(2A.23)

from which we obtain

d −C(t) ˆ ˆ ˆ t) = eC(t) ˆ˙ O(1, e = −f (adC(t))[C(t)], (2A.24) dt which is what we wanted to prove. Note that the final form of the series for Cˆ in (2A.6) can be rearranged in many different ways, using the Jacobi identity for the commutators. It is a nontrivial task to find a form involving the smallest number of terms.38 The same mathematical technique can be used to derive useful modification of the NeumannLiouville expansion or Dyson series (1.239) and (1.251). This is the so-called Magnus expansion 39 , ˆ (tb , ta ) = eEˆ , and expands the exponent E ˆ as in which one writes one writes U 2 Z tb Z tb Z t2 h i ˆ = −i ˆ 1 ) + 1 −i ˆ 2 ), H(t ˆ 1) E dt1 H(t dt2 dt1 H(t ¯h ta 2 ¯h ta ta 38

For a recent discussion see J.A. Oteo, J. Math. Phys. 32 , 419 (1991). See A. Iserles, A. Marthinsen, and S.P. Nørsett, On the implementation of the method of Magnus series for linear differential equations, BIT 39, 281 (1999) (http://www.damtp.cam.ac.uk/ user/ai/Publications). 39


Appendix 2B 1 + 4

Direct Calculation of Time-Sliced Oscillator Amplitude −i ¯h

3 (

Z

tb

Z

tb

t3

dt3

Z

tb

dt3

Z

ta

1 + 3

t2

dt2

Z

tb

dt2

Z

ta

ta

ta

ta

h h ii ˆ 3 ), H(t ˆ 2 ), H(t ˆ 1) dt1 H(t dt1

ta

hh

i i ˆ 3 ), H(t ˆ 2 ) , H(t ˆ 1) H(t

)

+ ... ,

203

(2A.25)

which converges faster than the Neumann-Liouville expansion.

Appendix 2B

Direct Calculation of Time-Sliced Oscillator Amplitude

After time-slicing the amplitude (2.138), it becomes a multiple integral over short-time amplitudes [using the action (2.185)] 1 i M (xn − xn−1 )2 21 2 2 (xn ǫ|xn−1 0) = p exp − ǫω (xn + xn−1 ) . (2B.26) ¯h 2 ǫ 2 2π¯hiǫ/M We shall write this as

(xn ǫ|xn−1 0) = N1 exp with

i a1 (x2n + x2n−1 ) − 2b1 xn xn−1 , ¯h

ωǫ 2 M 1−2 , 2ǫ 2 1 . N1 = p 2π¯hiǫ/M a1 =

b1 =

(2B.27)

M , 2ǫ (2B.28)

When performing the intermediate integrations in a product of N such amplitudes, the result must have the same general form i 2 2 (xN ǫ|xN −1 0) = NN exp aN (xN + x0 ) − 2bN xN x0 . (2B.29) ¯h Multiplying this by a further short-time amplitude and integrating over the intermediate position gives the recursion relations r iπ¯h NN +1 = N1 NN , (2B.30) aN + a1 a2N − b2N + a1 aN a2 − b21 + a1 aN aN +1 = = 1 , (2B.31) a1 + aN a1 + aN b1 bN bN +1 = . (2B.32) a1 + aN

From (2B.31) we find a2N = b2N + a21 − b21 ,

(2B.33)

and the only nontrivial recursion relation to be solved is that for bN . With (2B.32) it becomes bN +1 = or 1 bN +1

1 = b1

a1 +

b b p 1 N , b2N − (b21 − a21 )

a1 + bN

s

b 2 − a2 1− 1 2 1 bN

!

(2B.34)

.

(2B.35)

204


We now introduce the auxiliary frequency ω ˜ of Eq. (2.156). Then a1 =

M cos ω ˜, 2ǫ

(2B.36)

and the recursion for bN +1 reads 1 bN +1

cos ω ˜ǫ 2ǫ = + bN M

s

1−

M 2 sin2 ω ˜ǫ . 2 2 4ǫ bN

(2B.37)

By introducing the reduced quantities βN ≡

2ǫ bN , M

(2B.38)

with β1 = 1,

(2B.39)

the recursion becomes 1

cos ω ˜ǫ = + βN

βN +1

s

1−

sin2 ω ˜ǫ . 2 βN

(2B.40)

For N = 1, 2, this determines p 1 sin 2˜ ωǫ = cos ω ˜ ǫ + 1 − sin2 ω ˜ǫ = , β2 sin ω ˜ǫ s 1 sin 2˜ ωǫ sin2 2˜ ωǫ sin 3˜ ωǫ = cos ω ˜ǫ + 1 − sin2 ω ˜ǫ = . 2 β3 sin ω ˜ǫ sin ω ˜ ǫ sin ω ˜ǫ

(2B.41)

We therefore expect the general result 1 sin ω ˜ (N + 1)ǫ = . βN +1 sin ω ˜ǫ

(2B.42)

It is easy to verify that this solves the recursion relation (2B.40). From (2B.38) we thus obtain bN +1 =

M sin ω ˜ǫ . 2ǫ sin ω ˜ (N + 1)ǫ

(2B.43)

Inserting this into (2B.30) and (2B.33) yields aN +1 NN +1

cos ω ˜ (N + 1)ǫ M sin ω ˜ǫ , 2ǫ sin ω ˜ (N + 1)ǫ s sin ω ˜ǫ = N1 , sin ω ˜ (N + 1)ǫ =

(2B.44) (2B.45)

such that (2B.29) becomes the time-sliced amplitude (2.192).

Appendix 2C

Derivation of Mehler Formula

Her we briefly sketch the derivation of Mehler’s formula.40 It is based on the observation that the left-hand side of Eq. (2.290), let us call it F (x, x′ ), is the Fourier transform of the function 2 ′2 ′ F˜ (k, k ′ ) = π e−(k +k +akk )/2 ,

(2C.46)

40

See P.M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, Vol. I, p. 781 (1953). H. Kleinert, PATH INTEGRALS

205

Notes and References

as can easily be verified by performing the two Gaussian integrals in the Fourier representation Z ∞Z ∞ dk dk ′ ikx+ik′ x ˜ e F (k, k ′ ). (2C.47) F (x; x′ ) = 2π 2π −∞ −∞ We now consider the right-hand side of (2.290) and form the Fourier transform by recognizing the 2 exponential ek /2−ikx as the generating function of the Hermite polynomials41 ek

2

/2−ikx

=

∞ X (−ik/2)n Hn (x). n! n=0

(2C.48)

This leads to F˜ (k, k ′ ) = ×

Z

∞

Z

∞

−∞ −∞ Z ∞Z ∞ −∞

′

dx dx′ F (x, x′ )e−ikx−ik x = e−(k

2

+k′2 )/2

′ ∞ X ∞ X (−ik/2)n (−ik ′ /2)n dx dx F (x, x ) Hn (x)Hn′ (x). n! n′ ! −∞ n=0 ′

′

′

n =0

Inserting here the expansion on right-hand side of (2.290) and using the orthogonality relation of Hermite polynomials (2.299), we obtain once more (2C.47).

Notes and References The basic observation underlying path integrals for time evolution amplitudes goes back to the historic article P.A.M. Dirac, Physikalische Zeitschrift der Sowjetunion 3, 64 (1933). He observed that the short-time propagator is the exponential of i/¯h times the classical action. See also P.A.M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford, 1947; E.T. Whittaker, Proc. Roy. Soc. Edinb. 61, 1 (1940). Path integrals in configuration space were invented by R. P. Feynman in his 1942 Princeton thesis. The theory was published in 1948 in R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948). The mathematics of path integration had previously been developed by N. Wiener, J. Math. Phys. 2, 131 (1923); Proc. London Math. Soc. 22, 454 (1924); Acta Math. 55, 117 (1930); N. Wiener, Generalized Harmonic Analysis and Tauberian Theorems, MIT Press, Cambridge, Mass., 1964, after some earlier attempts by P.J. Daniell, Ann. Math. 19, 279; 20, 1 (1918); 20, 281 (1919); 21, 203 (1920); discussed in M. Kac, Bull. Am. Math. Soc. 72, Part II, 52 (1966). Note that even the name path integral appears in Wiener’s 1923 paper. Further important papers are I.M. Gelfand and A.M. Yaglom, J. Math. Phys. 1, 48 (1960); S.G. Brush, Rev. Mod. Phys. 33, 79 (1961); E. Nelson, J. Math. Phys. 5, 332 (1964); A.M. Arthurs, ed., Functional Integration and Its Applications, Clarendon Press, Oxford, 1975, 41


206


C. DeWitt-Morette, A. Maheshwari, and B.L. Nelson, Phys. Rep. 50, 255 (1979); D.C. Khandekar and S.V. Lawande, Phys. Rep. 137, 115 (1986). The general harmonic path integral is derived in M.J. Goovaerts, Physica 77, 379 (1974); C.C. Grosjean and M.J. Goovaerts, J. Comput. Appl. Math. 21, 311 (1988); G. Junker and A. Inomata, Phys. Lett. A 110, 195 (1985). The Feynman path integral was applied to thermodynamics by M. Kac, Trans. Am. Math. Soc. 65, 1 (1949); M. Kac, Probability and Related Topics in Physical Science, Interscience, New York, 1959, Chapter IV. A good selection of earlier textbooks on path integrals is R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York 1965, L.S. Schulman, Techniques and Applications of Path Integration, Wiley-Interscience, New York, 1981, F.W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer Science, World Scientific, Singapore, 1986. G. Roepstorff, Path Integral Approach to Quantum Physics, Springer, Berlin, 1994. The path integral in phase space is reviewed by C. Garrod, Rev. Mod. Phys. 38, 483 (1966). The path integral for the most general quadratic action has been studied in various ways by D.C. Khandekar and S.V. Lawande, J. Math. Phys. 16, 384 (1975); 20, 1870 (1979); V.V. Dodonov and V.I. Manko, Nuovo Cimento 44B, 265 (1978); A.D. Janussis, G.N. Brodimas, and A. Streclas, Phys. Lett. A 74, 6 (1979); C.C. Gerry, J. Math. Phys. 25, 1820 (1984); B.K. Cheng, J. Phys. A 17, 2475 (1984); G. Junker and A. Inomata, Phys. Lett. A 110, 195 (1985); H. Kleinert, J. Math. Phys. 27, 3003 (1986) (http://www.physik.fu-berlin.de/~kleinert/144). The caustic phenomena near the singularities of the harmonic oscillator amplitude at tb − ta = integer multiples of π/ω, in particular the phase of the fluctuation factor (2.162), have been discussed by J.M. Souriau, in Group Theoretical Methods in Physics, IVth International Colloquium, Nijmegen, 1975, ed. by A. Janner, Springer Lecture Notes in Physics, 50; P.A. Horvathy, Int. J. Theor. Phys. 18, 245 (1979). See in particular the references therein. The amplitude for the freely falling particle is discussed in G.P. Arrighini, N.L. Durante, C. Guidotti, Am. J. Phys. 64, 1036 (1996); B.R. Holstein, Am. J. Phys. 69, 414 (1997). For the Baker-Campbell-Hausdorff formula see J.E. Campbell, Proc. London Math. Soc. 28, 381 (1897); 29, 14 (1898); H.F. Baker, ibid., 34, 347 (1902); 3, 24 (1905); F. Hausdorff, Berichte Verhandl. Sächs. Akad. Wiss. Leipzig, Math. Naturw. Kl. 58, 19 (1906); W. Magnus, Comm. Pure and Applied Math 7, 649 (1954), Chapter IV; J.A. Oteo, J. Math. Phys. 32, 419 (1991); See also the internet address E.W. Weisstein, http://mathworld.wolfram.com/baker-hausdorffseries.html. The Zassenhaus formula is derived in H. Kleinert, PATH INTEGRALS


207

W. Magnus, Comm. Pure and Appl. Mathematics, 7, 649 (1954); C. Quesne, Disentangling qExponentials, (math-ph/0310038). For Trotter’s formula see the original paper: E. Trotter, Proc. Am. Math. Soc. 10, 545 (1958). The mathematical conditions for its validity are discussed by E. Nelson, J. Math. Phys. 5, 332 (1964); T. Kato, in Topics in Functional Analysis, ed. by I. Gohberg and M. Kac, Academic Press, New York 1987. Faster convergent formulas: M. Suzuki, Comm. Math. Phys. 51, 183 (1976); Physica A 191, 501 (1992); H. De Raedt and B. De Raedt, Phys. Rev. A 28, 3575 (1983); W. Janke and T. Sauer, Phys. Lett. A 165, 199 (1992). See also M. Suzuki, Physica A 191, 501 (1992). The path integral representation of the scattering amplitude is developed in W.B. Campbell, P. Finkler, C.E. Jones, and M.N. Misheloff, Phys. Rev. D 12, 12, 2363 (1975). See also: H.D.I. Abarbanel and C. Itzykson, Phys. Rev. Lett. 23, 53 (1969); R. Rosenfelder, see Footnote 37. The alternative path integral representation in Section 2.18 is due to M. Roncadelli, Europhys. Lett. 16, 609 (1991); J. Phys. A 25, L997 (1992); A. Defendi and M. Roncadelli, Europhys. Lett. 21, 127 (1993).


Aκ´ινητ α κινǫˆις. You stir what should not be stirred. Herodotus

3 External Sources, Correlations, and Perturbation Theory Important information on every quantum-mechanical system is carried by the correlation functions of the path x(t). They are defined as the expectation values of products of path positions at different times, x(t1 ) · · · x(tn ), to be calculated as functional averages. Quantities of this type are observable in simple scattering experiments. The most efficient extraction of correlation functions from a path integral proceeds by adding to the Lagrangian an external time-dependent mechanical force term disturbing the system linearly, and by studying the response to the disturbance. A similar linear term is used extensively in quantum field theory, for instance in quantum electrodynamics where it is no longer a mechanical force, but a source of fields, i.e., a charge or a current density. For this reason we shall call this term generically source or current term. In this chapter, the procedure is developed for the harmonic action, where a linear source term does not destroy the solvability of the path integral. The resulting amplitude is a simple functional of the current. Its functional derivatives will supply all correlation functions of the system, and for this reason it is called the generating functional of the theory. It serves to derive the celebrated Wick rule for calculating the correlation functions of an arbitrary number of x(t). This forms the basis for perturbation expansions of anharmonic theories.

3.1

External Sources

Consider a harmonic oscillator with an action Aω =

Z

tb

ta

dt

M 2 (x˙ − ω 2 x2 ). 2

(3.1)

Let it be disturbed by an external source or current j(t) coupled linearly to the particle coordinate x(t). The source action is Aj =

Z

tb

ta

dt x(t)j(t). 208

(3.2)

209

3.1 External Sources

The total action A = Aω + Aj

(3.3)

is still harmonic in x and x, ˙ which makes it is easy to solve the path integral in the presence of a source term. In particular, the source term does not destroy the factorization property (2.146) of the time evolution amplitude into a classical amplitude eiAj,cl /¯h and a fluctuation factor Fω,j (tb , ta ), (xb tb |xa ta )jω = e(i/¯h)Aj,cl Fω,j (tb , ta ).

(3.4)

Here Aj,cl is the action for the classical orbit xj,cl (t) which minimizes the total action A in the presence of the source term and which obeys the equation of motion x¨j,cl (t) + ω 2 xj,cl (t) = j(t).

(3.5)

In the sequel, we shall first work with the classical orbit xcl (t) extremizing the action without the source term: xcl (t) =

xb sin ω(t − ta ) + xa sin ω(tb − t) . sin ω(tb − ta )

(3.6)

All paths will be written as a sum of the classical orbit xcl (t) and a fluctuation δx(t): x(t) = xcl (t) + δx(t).

(3.7)

Then the action separates into a classical and a fluctuating part, each of which contains a source-free and a source term: A = Aω + Aj ≡ Acl + Afl = (Aω,cl + Aj,cl ) + (Aω,fl + Aj,fl).

(3.8)

The time evolution amplitude can be expressed as (xb tb |xa ta )jω

(i/¯ h)Acl

= e

Z

i Dx exp Afl h ¯

(i/¯ h)(Aω,cl +Aj,cl )

= e

Z

i Dx exp (Aω,fl + Aj,fl) . h ¯

(3.9)

The classical action Aω,cl is known from Eq. (2.152): Aω,cl =

h i Mω (x2b + x2a ) cos ω(tb − ta ) − 2xb xa . 2 sin ω(tb − ta )

(3.10)

The classical source term is known from (3.6): Aj,cl = =

Z

tb

ta

dt xcl (t)j(t)

1 sin ω(tb − ta )

Z

(3.11) tb

ta

dt[xa sin ω(tb − t) + xb sin ω(t − ta )]j(t).

210

3 External Sources, Correlations, and Perturbation Theory

Consider now the fluctuating part of the action, Afl = Aω,fl + Aj,fl. Since xcl (t) extremizes the action without the source, Afl contains a term linear in δx(t). After a partial integration [making use of the vanishing of δx(t) at the ends] it can be written as M Afl = 2

Z

tb

′

′

′

dtdt δx(t)Dω2 (t, t )δx(t ) +

ta

Z

tb

ta

dt δx(t)j(t),

(3.12)

where Dω2 (t, t′ ) is the differential operator Dω2 (t, t′ ) = (−∂t2 − ω 2 )δ(t − t′ ) = δ(t − t′ )(−∂t2′ − ω 2 ),

t, t′ ∈ (ta , tb ).

(3.13)

It may be considered as a functional matrix in the space of the t-dependent functions vanishing at ta , tb . The equality of the two expressions is seen as follows. By partial integrations one has Z

tb

ta

dt f (t)∂t2 g(t) =

Z

tb

ta

dt ∂t2 f (t)g(t),

(3.14)

for any f (t) and g(t) vanishing at the boundaries (or for periodic functions in the R interval). The left-handR side can directly be rewritten as ttab dtdt′ f (t)δ(t−t′ )∂t2′ g(t′), the right-hand side as ttab dtdt′ ∂t2 f (t)δ(t − t′ )g(t′ ), and after further partial integraR tions, as dtdt′ f (t)∂t2 δ(t − t′ )g(t). The inverse Dω−12 (t, t′ ) of the functional matrix (3.13) is formally defined by the relation Z

tb

ta

dt′ Dω2 (t′′ , t′ )Dω−12 (t′ , t) = δ(t′′ − t),

t′′ , t ∈ (ta , tb ),

(3.15)

which shows that it is the standard classical Green function of the harmonic oscillator of frequency ω: Gω2 (t, t′ ) ≡ Dω−12 (t, t′ ) = (−∂t2 − ω 2 )−1 δ(t − t′ ),

t, t′ ∈ (ta , tb ).

(3.16)

This definition is not unique since it leaves room for an additional arbitrary solution R H(t, t′ ) of the homogeneous equation ttab dt′ Dω2 (t′′ , t′ )H(t′ , t) = 0. This freedom will be removed below by imposing appropriate boundary conditions. In the fluctuation action (3.12), we now perform a quadratic completion by a shift of δx(t) to 1 δ˜ x(t) ≡ δx(t) + M

Z

tb

ta

dt′ Gω2 (t, t′ )j(t′ ).

(3.17)

Then the action becomes quadratic in both δ˜ x and j: Afl =

Z

tb

ta

dt

Z

tb

ta

M 1 dt δ˜ x(t)Dω2 (t, t′ )δ˜ x(t′ ) − j(t)Gω2 (t, t′ )j(t′ ) . 2 2M ′

(3.18)


211

3.1 External Sources

The Green function obeys the same boundary condition as the fluctuations δx(t): (

′

Gω2 (t, t ) = 0 for

t = tb , t′ arbitrary, t arbitrary, t′ = ta .

(3.19)

Thus, the shifted fluctuations δ˜ x(t) of (3.17) also vanish at the ends and run through the same functional space as the original δx(t). The measure of path integration R Dδx(t) is obviously unchanged by the simple shift (3.17). Hence the path integral R Dδ˜ x over eiAfl /¯h with the action (3.18) gives, via the first term in Afl , the harmonic fluctuation factor Fω (tb − ta ) calculated in (2.164): 1

Fω (tb − ta ) = q 2πi¯ h/M

s

ω . sin ω(tb − ta )

(3.20)

The source part in (3.18) contributes only a trivial exponential factor Fj,fl = exp

i Aj,fl , h ¯

(3.21)

dt′ j(t)Gω2 (t, t′ )j(t′ ).

(3.22)

whose exponent is quadratic in j(t): Aj,fl

1 =− 2M

Z

tb

ta

dt

Z

tb

ta

The total time evolution amplitude in the presence of a source term can therefore be written as the product (xb tb |xa ta )jω = (xb tb |xa ta )ω Fj,cl Fj,fl,

(3.23)

where (xb tb |xa t)ω is the source-free time evolution amplitude (i/¯ h)Aω,cl

(xb tb |xa ta )ω = e

(

1

Fω (tb − ta ) = q 2πi¯ h/M

s

ω sin ω(tb − ta ) )

i Mω × exp [(x2 + x2a ) cos ω(tb − ta ) − 2xb xa ] , 2¯ h sin ω(tb − ta ) b

(3.24)

and Fj,cl is an amplitude containing the classical action (3.11): Fj,cl = e(i/¯h)Aj,cl ( ) Z tb i 1 = exp dt[xa sin ω(tb − t) + xb sin ω(t − ta )] j(t) . (3.25) h ¯ sin ω(tb − ta ) ta To complete the result we need to know the Green function Gω2 (t, t′ ) explicitly, which will be calculated in the next section.

212

3.2


Green Function of Harmonic Oscillator

According to Eq. (3.16), the Green function in Eq. (3.22) is obtained by inverting the second-order differential operator −∂t2 − ω 2 : Gω2 (t, t′ ) = (−∂t2 − ω 2 )−1 δ(t − t′ ),

t, t′ ∈ (ta , tb ).

(3.26)

As remarked above, this function is defined only up to solutions of the homogeneous differential equation associated with the operator −∂t2 − ω 2 . The boundary conditions removing this ambiguity are the same as for the fluctuations δx(t), i.e., Gω2 (t, t′ ) vanishes if either t or t′ or both hit an endpoint ta or tb (Dirichlet boundary condition). The Green function is symmetric in t and t′ . For the sake of generality, we shall find the Green function also for the more general differential equation with time-dependent frequency, [−∂t2 − Ω2 (t)]GΩ2 (t, t′ ) = δ(t − t′ ),

(3.27)

with the same boundary conditions. There are several ways of calculating this explicitly.

3.2.1

Wronski Construction

The simplest way proceeds via the so-called Wronski construction, which is based on the following observation. For different time arguments, t > t′ or t < t′ , the Green function GΩ2 (t, t′ ) has to solve the homogeneous differential equations (−∂t2 − ω 2 )GΩ2 (t, t′ ) = 0,

(−∂t2′ − ω 2 )GΩ2 (t, t′ ) = 0.

(3.28)

It must therefore be a linear combination of two independent solutions of the homogeneous differential equation in t as well as in t′ , and it must satisfy the Dirichlet boundary condition of vanishing at the respective endpoints. Constant Frequency If Ω2 (t) ≡ ω 2, this implies that for t > t′ , Gω2 (t, t′ ) must be proportional to sin ω(tb − t) as well as to sin ω(t′ − ta ), leaving only the solution Gω2 (t, t′ ) = C sin ω(tb − t) sin ω(t′ − ta ),

t > t′ .

(3.29)

t < t′ .

(3.30)

For t < t′ , we obtain similarly Gω2 (t, t′ ) = C sin ω(tb − t′ ) sin ω(t − ta ), The two cases can be written as a single expression Gω2 (t, t′ ) = C sin ω(tb − t> ) sin ω(t< − ta ),


3.2 Green Function of Harmonic Oscillator

213

where the symbols t> and t< denote the larger and the smaller of the times t and t′ , respectively. The unknown constant C is fixed by considering coincident times t = t′ . There, the time derivative of Gω2 (t, t′ ) must have a discontinuity which gives rise to the δ-function in (3.15). For t > t′ , the derivative of (3.29) is ∂t Gω2 (t, t′ ) = −Cω cos ω(tb − t) sin ω(t′ − ta ),

(3.32)

whereas for t < t′ ∂t Gω2 (t, t′ ) = Cω sin ω(tb − t′ ) cos ω(t − ta ).

(3.33)

At t = t′ we find the discontinuity ∂t Gω2 (t, t′ )|t=t′ +ǫ − ∂t Gω2 (t, t′ )|t=t′ −ǫ = −Cω sin ω(tb − ta ).

(3.34)

Hence −∂t2 Gω2 (t, t′ ) is proportional to a δ-function: −∂t2 Gω2 (t, t′ ) = Cω sin ω(tb − ta )δ(t − t′ ).

(3.35)

By normalizing the prefactor to unity, we fix C and find the desired Green function:

Gω2 (t, t′ ) =

sin ω(tb − t> ) sin ω(t< − ta ) . ω sin ω(tb − ta )

(3.36)

It exists only if tb − ta is not equal to an integer multiple of π/ω. This restriction was encountered before in the amplitude without external sources; its meaning was discussed in the two paragraphs following Eq. (2.161). The constant in the denominator of (3.36) is the Wronski determinant (or Wronskian) of the two solutions ξ(t) = sin ω(tb − t) and η(t) = sin ω(t − ta ) which was introduced in (2.215): ˙ W [ξ(t), η(t)] ≡ ξ(t)η(t) ˙ − ξ(t)η(t).

(3.37)

An alternative expression for (3.36) is Gω2 (t, t′ ) =

− cos ω(tb − ta − |t − t′ |) + cos ω(tb + ta − t − t′ ) . 2ω sin ω(tb − ta )

(3.38)

In the limit ω → 0 we obtain the free-particle Green function 1 (tb − t> )(t< − ta ) (tb − ta ) 1 1 1 ′ ′ ′ = −tt − (tb − ta )|t − t | + (ta + tb )(t + t ) − ta tb . tb − ta 2 2

G0 (t, t′ ) =

(3.39)

214


Time-Dependent Frequency It is just as easy to find the Green functions of the more general differential equation (3.27) with a time-dependent oscillator frequency Ω(t). We construct first a so-called retarded Green function (compare page 38) as a product of a Heaviside function with a smooth function GΩ2 (t, t′ ) = Θ(t − t′ )∆(t, t′ ). (3.40) Inserting this into the differential equation (3.27) we find [−∂t2 − Ω2 (t)]GΩ2 (t, t′ ) = Θ(t − t′ ) [−∂t2 − Ω2 (t)]∆(t, t′ ) ˙ − t′ ) − 2∂t ∆(t, t′ )δ(t − t′ ). − δ(t

(3.41)

Expanding 1 ∆(t, t′ ) = ∆(t, t) + [∂t ∆(t, t′ )]t=t′ (t − t′ ) + [∂t2 ∆(t, t′ )]t=t′ (t − t′ )2 + . . . , 2

(3.42)

and using the fact that ˙ − t′ ) = −δ(t − t′ ), (t − t′ )δ(t

˙ − t′ ) = 0 for n > 1, (t − t′ )n δ(t

(3.43)

the second line in (3.41) can be rewritten as ˙ − t′ )∆(t, t′ ) − δ(t − t′ )∂t ∆(t, t′ ). −δ(t

(3.44)

By choosing the initial conditions ∆(t, t) = 0,

˙ t′ )|t′ =t = −1, ∆(t,

(3.45)

we satisfy the inhomogeneous differential equation (3.27) provided ∆(t, t′ ) obeys the homogeneous differential equation [−∂t2 − Ω2 (t)]∆(t, t′ ) = 0,

for t > t′ .

(3.46)

This equation is solved by a linear combination ∆(t, t′ ) = α(t′ )ξ(t) + β(t′ )η(t)

(3.47)

of any two independent solutions η(t) and ξ(t) of the homogeneous equation [−∂t2 − Ω2 (t)]ξ(t) = 0,

[−∂t2 − Ω2 (t)]η(t) = 0.

(3.48)

˙ Their Wronski determinant W = ξ(t)η(t) ˙ − ξ(t)η(t) is nonzero and, of course, timeindependent, so that we can determine the coefficients in the linear combination (3.47) from (3.45) and find ∆(t, t′ ) =

1 [ξ(t)η(t′ ) − ξ(t′ )η(t)] . W


215


The right-hand side contains the so-called Jacobi commutator of the two functions ξ(t) and η(t). Here we list a few useful algebraic properties of ∆(t, t′ ): ∆(t, t′ ) =

∆(tb , t)∆(t′ , ta ) − ∆(t, ta )∆(tb , t′ ) , ∆(tb , ta )

∆(tb , t)∂tb ∆(tb , ta ) − ∆(t, ta ) = ∆(tb , ta )∂t ∆(tb , t),

(3.50) (3.51)

∆(t, ta )∂tb ∆(tb , ta ) − ∆(tb , t) = ∆(tb , ta )∂t ∆(t, ta ).

(3.52)

GΩ2 (t, t′ ) = Θ(t − t′ )∆(t, t′ ) + a(t′ )ξ(t) + b(t′ )η(t),

(3.53)

The retarded Green function (3.40) is so far not the unique solution of the differential equation (3.27), since one may always add a general solution of the homogeneous differential equation (3.48):

with arbitrary coefficients a(t′ ) and b(t′ ). This ambiguity is removed by the Dirichlet boundary conditions GΩ2 (tb , t) = 0, GΩ2 (t, ta ) = 0,

tb 6= t, t 6= ta .

(3.54)

Imposing these upon (3.53) leads to a simple algebraic pair of equations a(t)ξ(ta ) + b(t)η(ta ) = 0, a(t)ξ(tb ) + b(t)η(tb ) = ∆(t, tb ).

(3.55) (3.56)

Denoting the 2 × 2 -coefficient matrix by Λ=

ξ(ta ) η(ta ) ξ(tb ) η(tb )

!

,

(3.57)

we observe that under the condition det Λ = W ∆(ta , tb ) 6= 0,

(3.58)

the system (3.56) has a unique solution for the coefficients a(t) and b(t) in the Green function (3.53). Inserting this into (3.54) and using the identity (3.50), we obtain from this Wronski’s general formula corresponding to (3.36) GΩ2 (t, t′ ) =

Θ(t − t′ )∆(tb , t)∆(t′ , ta ) + Θ(t′ − t)∆(t, ta )∆(tb , t′ ) . ∆(ta , tb )

(3.59)

At this point it is useful to realize that the functions in the numerator coincide with the two specific linearly independent solutions Da (t) and Db (t) of the homogenous differential equations (3.48) which were introduced in Eqs. (2.221) and (2.222). Comparing the initial conditions of Da (t) and Db (t) with that of the function ∆(t, t′ ) in Eq. (3.45), we readily identify Da (t) ≡ ∆(t, ta ),

Db (t) ≡ ∆(tb , t),

(3.60)

216


and formula (3.59) can be rewritten as GΩ2 (t, t′ ) =

Θ(t − t′ )Db (t)Da (t′ ) + Θ(t′ − t)Da (t)Db (t′ ) . Da (tb )

(3.61)

It should be pointed out that this equation renders a unique and well-defined Green function if the differential equation [−∂t2 − Ω2 (t)]y(t) = 0 has no solutions with Dirichlet boundary conditions y(ta ) = y(tb ) = 0, generally called zero-modes. A zero mode would cause problems since it would certainly be one of the independent solutions of (3.49), say η(t). Due to the property η(ta ) = η(tb ) = 0, however, the determinant of Λ would vanish, thus destroying the condition (3.58) which was necessary to find (3.59). Indeed, the function ∆(t, t′ ) in (3.49) would remain undetermined since the boundary condition η(ta ) = 0 together with (3.55) implies that ˙ also ξ(ta ) = 0, making W = ξ(t)η(t) ˙ − ξ(t)η(t) vanish at the initial time ta , and thus for all times.

3.2.2

Spectral Representation

A second way of specifying the Green function explicitly is via its spectral representation. Constant Frequency For constant frequency Ω(t) ≡ ω, the fluctuations δx(t) which satisfy the differential equation (−∂t2 − ω 2 ) δx(t) = 0, (3.62) and vanish at the ends t = ta and t = tb , are expanded into a complete set of orthonormal functions: xn (t) =

s

2 sin νn (t − ta ), tb − ta

(3.63)

with the frequencies [compare (2.105)] νn =

πn . tb − ta

(3.64)

These functions satisfy the orthonormality relations Z

tb

ta

dt xn (t)xn′ (t) = δnn′ .

(3.65)

Since the operator −∂t2 − ω 2 is diagonal on xn (t), this is also true for the Green function Gω2 (t, t′ ) = (−∂t2 − ω 2 )−1 δ(t − t′ ). Let Gn be its eigenvalues defined by Z

tb

ta

dt Gω2 (t, t′ )xn (t′ ) = Gn xn (t).



217

Then we expand Gω2 (t, t′ ) as follows: Gω2 (t, t′ ) =

∞ X

Gn xn (t)xn (t′ ).

(3.67)

n=1

By definition, the eigenvalues of Gω2 (t, t′ ) are the inverse eigenvalues of the differential operator (−∂t2 − ω 2), which are νn2 − ω 2. Thus Gn = (νn2 − ω 2 )−1 ,

(3.68)

and we arrive at the spectral representation of Gω2 (t, t′ ): Gω2 (t, t′ ) =

∞ 2 X sin νn (t − ta ) sin νn (t′ − ta ) . tb − ta n=1 νn2 − ω 2

(3.69)

We may use the trigonometric relation

sin νn (tb − t) = − sin νn [(t − ta ) − (tb − ta )] = −(−1)n sin νn (t − ta )

to rewrite (3.69) as

Gω2 (t, t′ ) =

∞ 2 X sin νn (tb − t) sin νn (t′ − ta ) (−1)n+1 . tb − ta n=1 νn2 − ω 2

(3.70)

These expressions make sense only if tb − ta is not equal to an integer multiple of π/ω, where one of the denominators in the sums vanishes. This is the same range of tb − ta as in the Wronski expression (3.36). Time-Dependent Frequency The spectral representation can also be written down for the more general Green function with a time-dependent frequency defined by the differential equation (3.27). If yn (t) are the eigenfunctions solving the differential equation with eigenvalue λn K(t)yn (t) = λn yn (t),

(3.71)

and if these eigenfunctions satisfy the orthogonality and completeness relations Z

tb ta

dt yn (t)yn′ (t) = δnn′ ,

X n

yn (t)yn (t′ ) = δ(t − t′ ),

(3.72) (3.73)

and if, moreover, there exists no zero-mode for which λn = 0, then GΩ2 (t, t′ ) has the spectral representation X yn (t)yn (t′ ) GΩ2 (t, t′ ) = . (3.74) λn n This is easily verified by multiplication with K(t) using (3.71) and (3.73). It is instructive to prove the equality between the Wronskian construction and the spectral representations (3.36) and (3.70). It will be useful to do this in several steps. In the present context, some of these may appear redundant. They will, however, yield intermediate results which will be needed in Chapters 7 and 18 when discussing path integrals occurring in quantum field theories.

218

3.3


Green Functions of First-Order Differential Equation

An important quantity of statistical mechanics are the Green functions GpΩ (t, t′ ) which solve the first-order differential equation [i∂t − Ω(t)] GΩ (t, t′ ) = iδ(t − t′ ),

t − t′ ∈ [0, tb − ta ),

(3.75)

or its Euclidean version GpΩ,e (τ, τ ′′ ) which solves the differential equation, obtained from (3.75) for t = −iτ : [∂τ − Ω(τ )] GΩ,e (τ, τ ′ ) = δ(τ − τ ′ ),

τ − τ ′ ∈ [0, h ¯ β).

(3.76)

These can be calculated for an arbitrary function Ω(t).

3.3.1

Time-Independent Frequency

Consider first the simplest case of a Green function Gpω (t, t′ ) with fixed frequency ω which solves the first-order differential equation (i∂t − ω)Gpω (t, t′ ) = iδ(t − t′ ),

t − t′ ∈ [0, tb − ta ).

(3.77)

The equation determines Gpω (t, t′ ) only up to a solution H(t, t′ ) of the homogeneous differential equation (i∂t − ω)H(t, t′ ) = 0. The ambiguity is removed by imposing the periodic boundary condition Gpω (t, t′ ) ≡ Gpω (t − t′ ) = Gpω (t − t′ + tb − ta ),

(3.78)

indicated by the superscript p. With this boundary condition, the Green function Gpω (t, t′ ) is translationally invariant in time. It depends only on the difference between t and t′ and is periodic in it. The spectral representation of Gpω (t, t′ ) can immediately be written down, assuming that tb − ta does not coincide with an even multiple of π/ω: Gpω (t

∞ X 1 i ′ −t)= e−iωm (t−t ) . tb − ta m=−∞ ωm − ω ′

(3.79)

The frequencies ωm are twice as large as the previous νm ’s in (3.64): ωm ≡

2πm , tb − ta

m = 0, ±1, ±2, ±3, . . . .

(3.80)

As for the periodic orbits in Section 2.9, there are “about as many” ωm as νm , since there is an ωm for each positive and negative integer m, whereas the νm are all positive (see the last paragraph in that section). The frequencies (3.80) are the real-time analogs of the Matsubara frequencies (2.373) of quantum statistics with the usual correspondence tb − ta = −i¯ h/kB T of Eq. (2.323). H. Kleinert, PATH INTEGRALS

3.3 Green Functions of First-Order Differential Equation

219

To calculate the spectral sum, we use the Poisson summation formula in the form (1.197): ∞ X

f (m) =

m=−∞

∞

Z

−∞

dµ

∞ X

e2πiµn f (µ).

(3.81)

n=−∞

Accordingly, we rewrite the sum over ωm as an integral over ω ′, followed by an auxiliary sum over n which squeezes the variable ω ′ onto the proper discrete values ωm = 2πm/(tb − ta ): Gpω (t)

=

∞ Z X

∞

n=−∞ −∞

dω ′ −iω′ [t−(tb −ta )n] i e . 2π ω′ − ω

(3.82)

At this point it is useful to introduce another Green function Gω (t−t′ ) associated with the first-order differential equation (3.77) on an infinite time interval: Gω (t) =

Z

∞

−∞

dω ′ −iω′ t i e . 2π ω′ − ω

(3.83)

In terms of this function, the periodic Green function (3.82) can be written as a sum which exhibits in a most obvious way the periodicity under t → t + (tb − ta ): Gpω (t) =

∞ X

n=−∞

Gω (t − (tb − ta )n).

(3.84)

The advantage of using Gω (t − t′ ) is that the integral over ω ′ in (3.83) can easily be done. We merely have to prescribe how to treat the singularity at ω ′ = ω. This also removes the freedom of adding a homogeneous solution H(t, t′ ). To make the integral unique, we replace ω by ω − iη where η is a very small positive number, i.e., by the iη-prescription introduced after Eq. (2.161). This moves the pole in the integrand of (3.83) into the lower half of the complex ω ′-plane, making the integral over ω ′ in Gω (t) fundamentally different for t < 0 and for t > 0. For t < 0, the contour of integration can be closed in the complex ω ′ -plane by a semicircle in the ′ upper half-plane at no extra cost, since e−iω t is exponentially small there (see Fig. 3.1). With the integrand being analytic in the upper half-plane we can contract the contour to zero and find that the integral vanishes. For t > 0, on the other hand, the contour is closed in the lower half-plane containing a pole at ω ′ = ω − iη. When contracting the contour to zero, the integral picks up the residue at this pole and yields a factor −2πi. At the point t = 0, finally, we can close the contour either way. The integral over the semicircles is now nonzero, ∓1/2, which has to be subtracted from the residues 0 and 1, respectively, yielding 1/2. Hence we find dω ′ −iω′ t i e ′ ω − ω + iη −∞ 2π   for t > 0,  1 for t = 0, = e−iωt × 12   0 for t < 0.

Gω (t) =

Z

∞

(3.85)

220


Figure 3.1 Pole in Fourier transform of Green functions Gp,a ω (t), and infinite semicircles in the upper (lower) half-plane which extend the integrals to a closed contour for t < 0 (t > 0).

The vanishing of the Green function for t < 0 is the causality property of Gω (t) discussed in (1.306) and (1.307). It is a general property of functions whose Fourier transforms are analytic in the upper half-plane. The three cases in (3.85) can be collected into a single formula using the Heaviside function Θ(t) of Eq. (1.309): ¯ Gω (t) = e−iωt Θ(t).

(3.86)

The periodic Green function (3.84) can then be written as ∞ X

Gpω (t) =

n=−∞

¯ − (tb − ta )n). e−iω[t−(tb −ta )n] Θ(t

(3.87)

Being periodic in tb − ta , its explicit evaluation can be restricted to the basic interval t ∈ [0, tb − ta ).

(3.88)

Inside the interval (0, tb − ta ), the sum can be performed as follows: Gpω (t)

=

0 X

e−iω[t−(tb −ta )n] =

n=−∞

= −i

e−iω[t−(tb −ta )/2] , 2 sin[ω(tb − ta )/2]

e−iωt 1 − e−iω(tb −ta ) t ∈ (0, tb − ta ).

(3.89)

¯ At the point t = 0, the initial term with Θ(0) contributes only 1/2 so that 1 Gpω (0) = Gpω (0+) − . (3.90) 2 Outside the basic interval (3.88), the Green function is determined by its periodicity. For instance, Gpω (t) = −i

e−iω[t+(tb −ta )/2] , 2 sin[ω(tb − ta )/2]

t ∈ (−(tb − ta ), 0).

(3.91)


221


Note that as t crosses the upper end of the interval [0, tb −ta ), the sum in (3.87) picks up an additional term (the term with n = 1). This causes a jump in Gpω (t) which enforces the periodicity. At the upper point t = tb − ta , there is again a reduction by 1/2 so that Gpω (tb − ta ) lies in the middle of the jump, just as the value 1/2 lies ¯ in the middle of the jump of the Heaviside function Θ(t). The periodic Green function is of great importance in the quantum statistics of Bose particles (see Chapter 7). After a continuation of the time to imaginary values, t → −iτ , tb − ta → −i¯ h/kB T , it takes the form Gpω,e (τ ) =

1 1−

e−¯hω/kB T

e−ωτ ,

τ ∈ (0, h ¯ β),

(3.92)

where the subscript e records the Euclidean character of the time. The prefactor is related to the average boson occupation number of a particle state of energy h ¯ ω, given by the Bose-Einstein distribution function nbω =

1 e¯hω/kB T

In terms of it,

.

(3.93)

τ ∈ (0, h ¯ β).

(3.94)

−1

Gpω,e (τ ) = (1 + nbω )e−ωτ ,

′ The τ -behavior of the subtracted periodic Green function Gpω,e (τ ) ≡ Gpω,e (τ )−1/¯ hβω is shown in Fig. 3.2.

′ Gpω,e (τ )

1

0.8 0.6

Gaω,e (τ )

0.5

0.4 0.2 -1

-0.5

-0.2

0.5

1

1.5

2

τ /¯hβ

-1

-0.4

-0.5

0.5

1

1.5

2

τ /¯hβ

-0.5 -1

′ ≡ Gp (τ )−1/¯ Figure 3.2 Subtracted periodic Green function Gpω,e hβω and antiperiodic ω,e a Green function Gω,e(τ ) for frequencies ω = (0, 5, 10)/¯ hβ (with increasing dash length). The points show the values at the jumps of the three functions (with increasing point size) corresponding to the relation (3.90).

As a next step, we consider a Green function Gpω2 (t) associated with the secondorder differential operator −∂t2 − ω 2 , Gpω2 (t, t′ ) = (−∂t2 − ω 2 )−1 δ(t − t′ ),

t − t′ ∈ [ta , tb ),

(3.95)

Gpω2 (t, t′ ) ≡ Gpω2 (t − t′ ) = Gpω2 (t − t′ + tb − ta ).

(3.96)

which satisfies the periodic boundary condition:

222


Figure 3.3 Two poles in Fourier transform of Green function Gp,a ω 2 (t).

Just like Gpω (t, t′ ), this periodic Green function depends only on the time difference t − t′ . It obviously has the spectral representation Gpω2 (t)

∞ X 1 1 = e−iωm t 2 , tb − ta m=−∞ ωm − ω 2

(3.97)

which makes sense as long as tb − ta is not equal to an even multiple of π/ω. At infinite tb − ta , the sum becomes an integral over ωm with singularities at ±ω which must be avoided by an iη-prescription, which adds a negative imaginary part to the frequency ω [compare the discussion after Eq. (2.161)]. This fixes also the continuation from small tb − ta beyond the multiple values of π/ω. By decomposing 1 1 = ′2 2 ω − ω + iη 2iω

!

i i − ′ , ′ ω − ω + iη ω + ω − iη

(3.98)

the calculation of the Green function (3.97) can be reduced to the previous case. The positions of the two poles of (3.98) in the complex ω ′-plane are illustrated in Fig. 3.3. In this way we find, using (3.89), 1 [Gp (t) − Gp−ω (t)] 2ωi ω 1 cos ω[t − (tb − ta )/2] , = − 2ω sin[ω(tb − ta )/2]

Gpω2 (t) =

t ∈ [0, tb − ta ).

(3.99)

In Gp−ω (t) one must keep the small negative imaginary part attached to the frequency ω. For an infinite time interval tb − ta , this leads to a Green function Gpω2 (t − t′ ): also ¯ G−ω (t) = −e−iωt Θ(−t). (3.100) The directional change in encircling the pole in the ω ′ -integral leads to the exchange ¯ ¯ Θ(t) → −Θ(−t). Outside the basic interval t ∈ [0, tb − ta ), the function is determined by its periodicity. For t ∈ [−(tb − ta ), 0), we may simply replace t by |t|. H. Kleinert, PATH INTEGRALS

223


As a further step we consider another Green function Gaω (t, t′ ). It fulfills the same first-order differential equation i∂t − ω as Gpω (t, t′ ): (i∂t − ω)Gaω (t, t′ ) = iδ(t − t′ ),

t − t′ ∈ [0, tb − ta ),

(3.101)

but in contrast to Gpω (t, t′ ) it satisfies the antiperiodic boundary condition Gaω (t, t′ ) ≡ Gaω (t − t′ ) = −Gaω (t − t′ + tb − ta ).

(3.102)

As for periodic boundary conditions, the Green function Gaω (t, t′ ) depends only on the time difference t − t′ . In contrast to Gpω (t, t′ ), however, Gaω (t, t′ ) changes sign under a shift t → t + (tb − ta ). The Fourier expansion of Gaω (t − t′ ) is Gaω (t) =

∞ X i 1 f e−iωm t f , tb − ta m=−∞ ωm − ω

(3.103)

where the frequency sum covers the odd Matsubara-like frequencies f = ωm

π(2m + 1) . tb − ta

(3.104)

The superscript f stands for fermionic since these frequencies play an important role in the statistical mechanics of particles with Fermi statistics to be explained in Section 7.10 [see Eq. (7.414)]. The antiperiodic Green functions are obtained from a sum similar to (3.82), but modified by an additional phase factor eiπn = (−)n . When inserted into the Poisson summation formula (3.81), such a phase is seen to select the half-integer numbers in the integral instead of the integer ones: ∞ X

f (m + 1/2) =

m=−∞

Z

∞

−∞

dµ

∞ X

(−)n e2πiµn f (µ).

(3.105)

n=−∞

Using this formula, we can expand ∞ Z X

Gaω (t) =

∞

n=−∞ −∞ ∞ X n

=

n=−∞

dω ′ i ′ (−)n e−iω [t−(tb −ta )n] ′ 2π ω − ω + iη

(−) Gω (t − (tb − ta )n),

(3.106)

or, more explicitly, Gaω (t) =

∞ X

n=−∞

¯ − (tb − ta )n). e−iω[t−(tb −ta )n] (−)n Θ(t

(3.107)

For t ∈ [0, tb − ta ), this gives Gaω (t) =

0 X

e−iω[t−(tb −ta )n] (−)n =

n=−∞ −iω[t−(tb −ta )/2]

=

e , 2 cos[ω(tb − ta )/2]

eiωt 1 + e−iω(tb −ta )

t ∈ [0, tb − ta ).

(3.108)

224


Outside the interval t ∈ [0, tb − ta ), the function is defined by its antiperiodicity. The τ -behavior of the antiperiodic Green function Gaω,e (τ ) is also shown in Fig. 3.2. In the limit ω → 0, the right-hand side of (3.108) is equal to 1/2, and the antiperiodicity implies that Ga0 (t) =

1 ǫ(t), 2

t ∈ [−(tb − ta ), (tb − ta )].

(3.109)

Antiperiodic Green functions play an important role in the quantum statistics of Fermi particles. After analytically continuing t to the imaginary time −iτ with tb − ta → −i¯ h/kB T , the expression (3.108) takes the form Gaω,e (τ ) =

1 1+

e−¯hω/kB T

e−ωτ ,

τ ∈ [0, h ¯ β).

(3.110)

The prefactor is related to the average Fermi occupation number of a state of energy h ¯ ω, given by the Fermi-Dirac distribution function nfω =

1 e¯hω/kB T

+1

.

(3.111)

In terms of it, Gaω,e (τ ) = (1 − nfω )e−ωτ ,

τ ∈ [0, h ¯ β).

(3.112)

With the help of Gaω (t), we form the antiperiodic analog of (3.97), (3.99), i.e., the antiperiodic Green function associated with the second-order differential operator −∂t2 − ω 2 : ∞ X 1 1 f e−iωm t f 2 tb − ta m=0 ωm − ω 2 1 = [Ga (t) − Ga−ω (t)] 2ωi ω 1 sin ω[t − (tb − ta )/2] = − , 2ω cos[ω(tb − ta )/2]

Gaω2 (t) =

t ∈ [0, tb − ta ].

(3.113)

Outside the basic interval t ∈ [0, tb − ta ], the Green function is determined by its antiperiodicity. If, for example, t ∈ [−(tb − ta ), 0], one merely has to replace t by |t|. Note that the Matsubara sums Gpω2 ,e (0) =

∞ 1 X 1 , 2 h ¯ β m=−∞ ωm + ω 2

Gpω,e (0) =

∞ 1 X 1 , f 2 h ¯ β m=−∞ ωm + ω 2

(3.114)

can also be calculated from the combinations of the simple Green functions (3.79) and (3.103): i i 1 h p 1 h p 1 Gω,e (η) + Gpω,e (−η) = Gω,e (η) + Gpω,e (¯ hβ −η) = 1 + nbω 1 + e−βω 2ω 2ω 2ω H. Kleinert, PATH INTEGRALS

225


h ¯ ωβ 1 coth , (3.115) 2ω 2 i i 1 h a 1 h a 1 Gω,e (η) + Gaω,e (−η) = Gω,e (η) − Gaω,e (¯ hβ −η) = 1 − nfω 1 − e−βω 2ω 2ω 2ω h ¯ ωβ 1 tanh , (3.116) = 2ω 2 =

where η is an infinitesimal positive number needed to specify on which side of the jump the Green functions Gp,a ω,e (τ ) at τ = 0 have to be evaluated (see Fig. 3.2).

3.3.2

Time-Dependent Frequency

The above results (3.89) and (3.108) for the periodic and antiperiodic Green functions of the first-order differential operator (i∂t − ω) can easily be found also for arbitrary time-dependent frequencies Ω(t), thus solving (3.75). We shall look for the retarded version which vanishes for t < t′ . This property is guaranteed by the ansatz containing the Heaviside function (1.309): ¯ − t′ )g(t, t′ ). GΩ (t, t′ ) = Θ(t

(3.117)

Using the property (1.303) of the Heaviside function, that its time derivative yields the δ-function, and normalizing g(t, t) to be equal to 1, we see that g(t, t′ ) must solve the homogenous differential equation [i∂t − Ω(t)] g(t, t′ ) = 0. The solution is g(t, t′) = K(t′ )e−i The condition g(t, t) = 1 fixes K(t) = ei

Rt c

Rt c

(3.118)

dt′′ Ω(t′′ )

dt′′ Ω(t′′ )

¯ − t′ )e−i GΩ (t, t′ ) = Θ(t

.

(3.119)

, so that we obtain

Rt

t′

dt′′ Ω(t′′ )

.

(3.120)

The most general Green function is a sum of this and an arbitrary solution of the homogeneous equation (3.118): ¯ − t′ ) + C(t′ ) e−i GΩ (t, t ) = Θ(t h

′

i

Rt

t′

dt′′ Ω(t′′ )

.

(3.121)

For a periodic frequency Ω(t) we impose periodic boundary conditions upon the Green function, setting GΩ (ta , t′ ) = GΩ (tb , t′ ). This is ensured if for tb > t > t′ > ta : C(t′ )e−i

R ta t′

dt′′ Ω(t′′ )

= [1 + C(t′ )] e−i

This equation is solved by a t′ -independent C(t′ ): C = npΩ ≡

R tb

1 i

e

R tb ta

dt′′ Ω(t′′ )

t′

. −1

dt′′ Ω(t′′ )

.

(3.122)

(3.123)

226


Hence we obtain the periodic Green function ¯ − t′ ) + npΩ e−i GpΩ (t, t′ ) = Θ(t i

h

Rt

t′

dt′′ Ω(t′′ )

.

(3.124)

For antiperiodic boundary conditions we obtain the same equation with npΩ replaced by −naΩ where 1 naΩ ≡ R tb . (3.125) i t dt Ω(t) a e +1 Note that a sign change in the time derivative of the first-order differential equation (3.75) to [−i∂t − Ω(t)] GΩ (t, t′ ) = iδ(t − t′ ) (3.126)

has the effect of interchanging in the time variable t and t′ of the Green function Eq. (3.120). If the frequency Ω(t) is a matrix, all exponentials have to be replaced by timeordered exponentials [recall (1.252)] i

e

R tb ta

dt Ω(t)

i → Tˆ e

R tb ta

dt Ω(t)

.

(3.127)

As remarked in Subsection 2.15.4, this integral cannot, in general, be calculated explicitly. A simple formula is obtained only if the matrix Ω(t) varies only little around a fixed matrix Ω0 . For imaginary times τ = it we generalize the results (3.92) and (3.110) for the periodic and antiperiodic imaginary-time Green functions of the first-order differential equation (3.76) to time-dependent periodic frequencies Ω(τ ). Here the Green function (3.120) becomes ¯ − τ ′ )e− GΩ (τ, τ ′ ) = Θ(τ

Rτ

τ′

dτ ′′ Ω(τ ′′ )

,

(3.128)

and the periodic Green function (3.124): ¯ − τ ′ ) + nb e− GΩ (τ, τ ′ ) = Θ(τ where

h

i

Rτ

τ′

dτ ′′ Ω(τ ′′ )

,

(3.129)

1 (3.130) nb ≡ R h¯ β ′′ ′′ e 0 dτ Ω(τ ) − 1 is the generalization of the Bose distribution function in Eq. (3.93). For antiperiodic boundary conditions we obtain the same equation, except that the generalized Bose distribution function is replaced by the negative of the generalized Fermi distribution function in Eq. (3.111): 1 nf ≡ R h¯ β ′′ ′′ . (3.131) e 0 dτ Ω(τ ) + 1 For the opposite sign of the time derivative in (3.128), the arguments τ and τ ′ are interchanged. H. Kleinert, PATH INTEGRALS

227


From the Green functions (3.124) or (3.128) we may find directly the trace of the logarithm of the operators [−i∂t + Ω(t)] or [∂τ + Ω(τ )]. At imaginary time, we multiply Ω(τ ) with a strength parameter g, and use the formula Tr log [∂τ + gΩ(τ )] =

Z

g

0

dg ′ Gg′ Ω (τ, τ ).

(3.132)

Inserting on the right-hand side of Eq. (3.129), we find for g = 1: (

"

#)

1 Z ¯hβ ′′ dτ Ω(τ ′′ ) Tr log [∂τ + Ω(τ )] = log 2 sinh 2 0 Z ¯hβ R h¯ β ′′ 1 − dτ Ω(τ ′′ ) ′′ ′′ 0 = dτ Ω(τ ) + log 1 − e , (3.133) 2 0 which reduces at low temperature to 1 Tr log [∂τ + Ω(τ )] = 2

Z

hβ ¯

0

dτ ′′ Ω(τ ′′ ).

(3.134)

The result is the same for the opposite sign of the time derivative and the trace ¯ − τ ′ ) at τ = τ ′ , where it is equal to 1/2. of the logarithm is sensitive only to Θ(τ As an exercise for dealing with distributions it is instructive to rederive this result in the following perturbative way. For a positive Ω(τ ), we introduce an infinitesimal positive quantity η and decompose h

Tr log [±∂τ + Ω(τ )] = Tr log [±∂τ + η] + Tr log 1 + (±∂τ + η)−1 Ω(τ ) h

i

(3.135)

i

= Tr log [±∂τ + η] + Tr log 1 + (±∂τ + η)−1 Ω(τ ) . ∞ The first term Tr log [±∂τ + η] = Tr log [±∂τ + η] = −∞ dω log ω vanishes since −∞ dω log ω = 0 in dimensional regularization by Veltman’s rule [see (2.500)]. Using the Green functions

R

R∞

−1

[±∂τ + η]

′

(τ, τ ) =

(

¯ − τ ′) Θ(τ ¯ ′ − τ) , Θ(τ )

(3.136)

the second term can be expanded in a Taylor series ∞ X

(−1)n+1 n n=1

Z

¯ 1 −τ2 )Ω(τ2 )Θ(τ ¯ 2 −τ3 ) · · · Ω(τn )Θ(τ ¯ n −τ1 ). (3.137) dτ1 · · · dτn Ω(τ1 )Θ(τ

For the lower sign of ±∂τ , the Heaviside functions have reversed arguments τ2 − τ1 , τ3 − τ2 , . . . , τ1 − τn . The integrals over a cyclic product of Heaviside functions in (3.137) are zero since the arguments τ1 , . . . , τn are time-ordered which makes the ¯ n −τ1 ) [or Θ(τ ¯ 1 −τn )] negative and thus Θ(τ ¯ n −τ1 ) = 0 argument of the last factor Θ(τ ¯ 1 − τn )]. Only the first term survives yielding [or Θ(τ Z

¯ 1 − τ1 ) = 1 dτ1 Ω(τ1 )Θ(τ 2

Z

dτ Ω(τ ),

(3.138)

228


such that we re-obtain the result (3.134). This expansion (3.133) can easily be generalized to an arbitrary matrix Ω(τ ) or a ˆ ). Since H(τ ˆ ) and H(τ ˆ ′ ) do not necessarily commute, time-dependent operator, H(τ the generalization is ˆ )] = 1 Tr Tr log[¯ h∂τ + H(τ 2¯ h

"Z

0

hβ ¯

#

ˆ ) − dτ H(τ

∞ X

R h¯ β ′′ 1 ˆ ′′ Tr Tˆ e−n 0 dτ H(τ )/¯h , (3.139) n=1 n

where Tˆ is the time ordering operator (1.241). Each term in the sum contains a power of the time evolution operator (1.255).

3.4

Summing Spectral Representation of Green Function

After these preparations we are ready to perform the spectral sum (3.70) for the Green function of the differential equation of second order with Dirichlet boundary conditions. Setting t2 ≡ tb − t, t1 ≡ t′ − ta , we rewrite (3.70) as Gω2 (t, t′ ) =

∞ 2 X (−1)n+1 (eiνn t2 − e−iνn t2 )(eiνn t1 − e−iνn t1 ) tb − ta n=1 (2i)2 νn2 − ω 2

∞ −iνn (t2 +t1 ) 1 1 X − e−iνn (t2 −t1 ) ) + cc ] n [(e = (−1) 2 tb − ta n=1 νn2 − ω 2

=

∞ X 1 1 e−iνn (t2 +t1 ) − e−iνn (t2 −t1 ) (−1)n . 2 tb − ta n=−∞ νn2 − ω 2

(3.140)

We now separate even and odd frequencies νn and write these as bosonic and f fermionic Matsubara frequencies ωm = ν2m and ωm = ν2m+1 , respectively, recalling the definitions (3.80) and (3.104). In this way we obtain 1 Gω2 (t, t′ ) = 2

(

f ∞ ∞ X X 1 e−iωm (t2 +t1 ) 1 e−iωm (t2 +t1 ) − 2 − ω2 f 2 − ω2 tb − ta m=−∞ ωm tb − ta m=−∞ ωm

f ∞ ∞ X X 1 e−iωm (t2 −t1 ) 1 e−iωm (t2 −t1 ) − + . (3.141) 2 − ω2 f 2 − ω2 tb − ta m=−∞ ωm tb − ta m=−∞ ωm

)

Inserting on the right-hand side the periodic and antiperiodic Green functions (3.99) and (3.108), we obtain the decomposition Gω2 (t, t′ ) =

1 p [G (t2 + t1 ) − Gaω (t2 + t1 ) − Gpω (t2 − t1 ) + Gaω (t2 − t1 )] . (3.142) 2 ω

Using (3.99) and (3.113) we find that Gpω (t2 + t1 ) − Gpω (t2 − t1 ) =

sin ω[t2 − (tb − ta )/2] sin ωt1 , ω sin[ω(tb − ta )/2]

Gaω (t2 + t1 ) − Gaω (t2 − t1 ) = −

cos ω[t2 − (tb − ta )/2] sin ωt1 , ω cos[ω(tb − ta )/2]

(3.143) (3.144)


229

3.4 Summing Spectral Representation of Green Function

such that (3.142) becomes Gω2 (t, t′ ) =

1 sin ωt2 sin ωt1 , ω sin ω(tb − ta )

(3.145)

in agreement with the earlier result (3.36). An important limiting case is ta → −∞,

tb → ∞.

(3.146)

Then the boundary conditions become irrelevant and the Green function reduces to Gω2 (t, t′ ) = −

i −iω|t−t′ | e , 2ω

(3.147)

which obviously satisfies the second-order differential equation (−∂t2 − ω 2 )Gω2 (t, t′ ) = δ(t − t′ ).

(3.148)

The periodic and antiperiodic Green functions Gpω2 (t, t′ ) and Gaω2 (t, t′ ) at finite tb − ta in Eqs. (3.99) and (3.113) are obtained from Gω2 (t, t′ ) by summing over all periodic repetitions [compare (3.106)] Gpω2 (t, t′ ) = Gaω2 (t, t′ ) =

∞ X

n=−∞ ∞ X

G(t + n(tb − ta ), t′ ),

(−1)n Gω2 (t + n(tb − ta ), t′ ).

(3.149)

n=−∞

For completeness let us also sum the spectral representation with the normalized wave functions [compare (3.98)–(3.69)] x0 (t) =

s

1 , tb − ta

xn (t) =

s

2 cos νn (t − ta ), tb − ta

(3.150)

which reads: ′ GN ω 2 (t, t )

∞ X 2 1 cos νn (t − ta ) cos νn (t′ − ta ) = − 2+ . tb − ta 2ω νn2 − ω 2 n=1

"

#

(3.151)

It satisfies the Neumann boundary conditions

′ ∂t GN ω 2 (t, t )

t=tb

= 0,

′ ∂t′ GN ω 2 (t, t ) ′

t =ta

= 0.

(3.152)

The spectral representation (3.151) can be summed by a decomposition (3.140), if that the lowest line has a plus sign between the exponentials, and (3.142) becomes ′ GN ω 2 (t, t ) =

1 p [G (t2 + t1 ) − Gaω (t2 + t1 ) + Gpω (t2 − t1 ) − Gaω (t2 − t1 )] . (3.153) 2 ω

230


Using now (3.99) and (3.113) we find that Gpω (t2 + t1 ) + Gpω (t2 − t1 ) = −

cos ω[t2 − (tb − ta )/2] cos ωt1 , ω sin[ω(tb − ta )/2]

(3.154)

Gaω (t2 + t1 ) + Gaω (t2 − t1 ) = −

sin ω[t2 − (tb − ta )/2] cos ωt1 , ω cos[ω(tb − ta )/2]

(3.155)

and we obtain instead of (3.145): ′ GN ω 2 (t, t ) = −

1 cos ω(tb − t> ) cos ω(t< − ta ), ω sin ω(tb − ta )

(3.156)

which has the small-ω expansion ′ GN ω 2 (t, t ) 2≈ − ω ≈0

3.5

1 tb −ta 1 1 1 ′ ′ 2 ′2 + | − )+ t +t . (3.157) − |t−t (t+t (tb −ta )ω 2 3 2 2 2(tb −ta )

Wronski Construction for Periodic and Antiperiodic Green Functions

The Wronski construction in Subsection 3.2.1 of Green functions with timedependent frequency Ω(t) satisfying the differential equation (3.27) [−∂t2 − Ω2 (t)]GΩ2 (t, t′ ) = δ(t − t′ )

(3.158)

′ can easily be carried over to the Green functions Gp,a Ω2 (t, t ) with periodic and antiperiodic boundary conditions. As in Eq. (3.53) we decompose ′ ′ ′ ′ ′ ¯ Gp,a Ω2 (t, t ) = Θ(t − t )∆(t, t ) + a(t )ξ(t) + b(t )η(t),

(3.159)

with independent solutions of the homogenous equations ξ(t) and η(t), and insert this into (3.27), where δ p,a (t − t′ ) is the periodic version of the δ-function δ

p,a

′

(t − t ) ≡

∞ X

n=−∞

′

δ(t − t − n¯ hβ)

(

1 (−1)n

)

,

(3.160)

and Ω(t) is assumed to be periodic or antiperiodic in tb − ta . This yields again for ∆(t, t′ ) the homogeneous initial-value problem (3.46), (3.45), [−∂t2 − Ω2 (t)]∆(t, t′ ) = 0;

∆(t, t) = 0, ∂t ∆(t, t′ )|t′ =t = −1.

(3.161)

The periodic boundary conditions lead to the system of equations a(t)[ξ(tb ) ∓ ξ(ta )] + b(t)[η(tb ) ∓ η(ta )] = −∆(tb , t), ˙ b ) ∓ ξ(t ˙ a )] + b(t)[η(t a(t)[ξ(t ˙ b ) ∓ η(t ˙ a )] = −∂t ∆(tb , t).

(3.162)


231

3.6 Time Evolution Amplitude in Presence of Source Term

Defining now the constant 2 × 2 -matrices ξ(tb ) ∓ ξ(ta ) η(tb ) ∓ η(ta ) ˙ b ) ∓ ξ(t ˙ a ) η(t ξ(t ˙ b ) ∓ η(t ˙ a)

¯ p,a (ta , tb ) = Λ

!

,

(3.163)

the condition analogous to (3.58),

with

¯ p,a (ta , tb ) = W ∆ ¯ p,a (ta , tb ) 6= 0, det Λ

(3.164)

¯ p,a (ta , tb ) = 2 ± ∂t ∆(ta , tb ) ± ∂t ∆(tb , ta ), ∆

(3.165)

enables us to obtain the unique solution to Eqs. (3.162). After some algebra using the identities (3.51) and (3.52), the expression (3.159) for Green functions with periodic and antiperiodic boundary conditions can be cast into the form ′ ′ Gp,a Ω2 (t, t ) = GΩ2 (t, t ) ∓

[∆(t, ta ) ± ∆(tb , t)][∆(t′ , ta ) ± ∆(tb , t′ )] , ¯ p,a (ta , tb )∆(ta , tb ) ∆

(3.166)

where GΩ2 (t, t′ ) is the Green function (3.59) with Dirichlet boundary conditions. As in (3.59) we may replace the functions on the right-hand side by the solutions Da (t) and Db (t) defined in Eqs. (2.221) and (2.222) with the help of (3.60). The right-hand side of (3.166) is well-defined unless the operator K(t) = −∂t2 − 2 Ω (t) has a zero-mode, say η(t), with periodic or antiperiodic boundary conditions η(tb ) = ±η(ta ), η(t ˙ b ) = ±η(t ˙ a ), which would make the determinant of the 2 × 2 ¯ p,a vanish. -matrix Λ

3.6

Time Evolution Amplitude in Presence of Source Term

Given the Green function Gω2 (t, t′ ), we can write down an explicit expression for the time evolution amplitude. The quadratic source contribution to the fluctuation factor (3.21) is given explicitly by 1 Z tb Z tb ′ dt dt Gω2 (t, t′ ) j(t)j(t′ ) (3.167) 2M ta ta Z tb Z t 1 1 =− dt dt′ sin ω(tb − t) sin ω(t′ − ta )j(t)j(t′ ). M ω sin ω(tb − ta ) ta ta

Aj,fl = −

Altogether, the path integral in the presence of an external source j(t) reads (xb tb |xa ta )jω

=

Z

i Dx exp h ¯

Z

tb

ta

M 2 dt (x˙ − ω 2x2 ) + jx 2

= e(i/¯h)Aj,cl Fω,j (tb , ta ),

with a total classical action h i 1 Mω Aj,cl = (x2b + x2a ) cos ω(tb − ta )−2xb xa 2 sin ω(tb − ta ) Z tb 1 + dt[xa sin ω(tb − t) + xb sin ω(t − ta )]j(t), sin ω(tb − ta ) ta

(3.168)

(3.169)

232


and the fluctuation factor composed of (2.164) and a contribution from the current term eiAj,fl /¯h : iAj,fl /¯ h

Fω,j (tb , ta ) = Fω (tb , ta )e (

i × exp − h ¯ Mω sin ω(tb − ta )

1

=q 2πi¯ h/M tb

Z

ta

dt

Z

t

ta

s

ω sin ω(tb − ta )

′

′

′

)

dt sin ω(tb − t) sin ω(t − ta )j(t)j(t ) . (3.170)

This expression is easily generalized to arbitrary time-dependent frequencies. Using the two independent solutions Da (t) and Db (t) of the homogenous differential equations (3.48), which were introduced in Eqs. (2.221) and (2.222), we find for the action (3.169) theR general expression, composed of the harmonic action (2.261) and the current term ttab dtxcl (t)j(t) with the classical solution (2.241): Aj,cl

i 1 M h 2˙ xb Da (tb )−x2a D˙ b (ta )−2xb xa + = 2Da (tb ) Da (tb )

Z

tb

ta

dt [xb Da (t)+xa Db (t)]j(t). (3.171)

The fluctuation factor is composed of the expression (2.256) for the current-free action, and the generalization of (3.167) with the Green function (3.61): iAj,fl /¯ h

Fω,j (tb , ta ) = Fω (tb , ta )e ×

Z

tb ta

dt

Z

t

ta

1

1

(

i q =q exp − 2¯ hMDa (tb ) 2πi¯ h/M Da (tb )

¯ − t′ )Db (t)Da (t′ ) + Θ(t ¯ ′ − t)Da (t)Db (t′ ) j(t′ ) . (3.172) dt j(t) Θ(t h

′

i

For applications to statistical mechanics which becomes possible after an analytic continuation to imaginary times, it is useful to write (3.169) and (3.170) in another form. We introduce the Fourier transforms of the current tb 1 A(ω) ≡ dte−iω(t−ta ) j (t) , Mω ta Z tb 1 B(ω) ≡ dte−iω(tb −t) j(t) = −e−iω(tb −ta ) A(−ω), Mω ta

Z

(3.173) (3.174)

and see that the classical source term in the exponent of (3.168) can be written as Aj,cl = −i

nh i o Mω xb (eiω(tb −ta ) A − B) + xa (eiω(tb −ta ) B − A) . sin ω(tb − ta )

(3.175)

The source contribution to the quadratic fluctuations in Eq. (3.167), on the other hand, can be rearranged to yield Aj,fl =

i 4Mω

Z

tb

ta

Z

dt

tb

tb

′

dt′ e−iω|t−t | j(t)j(t′ )−

h i Mω eiω(tb −ta ) (A2 +B 2 )−2AB . 2 sin ω(tb −ta ) (3.176) H. Kleinert, PATH INTEGRALS

233

3.6 Time Evolution Amplitude in Presence of Source Term

This is seen as follows: We write the Green function between j(t), j(t′ ) in (3.168) as ¯ − t′ ) + sin ω(tb − t′ ) sin ω(t − ta )Θ(t ¯ ′ − t) − sin ω(tb − t) sin ω(t′ − ta )Θ(t i 1 h iω(tb −ta ) −iω(t−t′ ) ′ ¯ − t′ ) = e e + cc − eiω(tb +ta ) e−iω(t+t ) + cc Θ(t 4 n o + t ↔ t′ . (3.177) h

i

¯ − t′ ) + Θ(t ¯ ′ − t) = 1, this becomes Using Θ(t

1 n iω(tb +ta ) −iω(t′ +t) − e e + cc 4 ′ ¯ − t′ ) + e−iω(t′ −t) Θ(t ¯ ′ − t) +eiω(tb −ta ) e−iω(t−t ) Θ(t

′ ¯ ′ − t)) + eiω(t′ −t) (1 − Θ(t ¯ − t′ )) +e−iω(tb −ta ) eiω(t−t ) (1 − Θ(t

h

(3.178) io

.

A multiplication by j(t), j(t′ ) and an integration over the times t, t′ yield 1h − eiω(tb −ta ) 4M 2 ω 2 (B 2 + A2 ) 4 Z

+ eiω(tb −ta ) − e−iω(tb −ta )

tb

ta

(3.179) dt

Z

tb

tb

i

′

dt′ e−iω|t−t | j(t)j(t′ ) + 4M 2 ω 2 2AB ,

thus leading to (3.176). If the source j(t) is time-independent, the integrals in the current terms of the exponential of (3.169) and (3.170) can be done, yielding the j-dependent exponent i i i Aj = (Aj,cl + Aj,fl) = h ¯ h ¯ h ¯

(

1 [1 − cos ω(tb − ta )](xb + xa )j ω sin ω(tb − ta ) " # ) 1 cos ω(tb − ta ) − 1 2 + ω(tb − ta ) + 2 j . 2Mω 3 sin ω(tb − ta )

(3.180)

Substituting (1−cos α) by sin α tan(α/2), this yields the total source action becomes "

#

1 1 ω(tb − ta ) 2 ω(tb − ta ) Aj = tan (xb + xa )j + ω (tb − ta ) − 2 tan j . (3.181) 3 ω 2 2Mω 2 This result could also have been obtained more directly by taking the potential plus a constant-current term in the action −

Z

tb

ta

dt

M 2 2 ω x − xj , 2

(3.182)

tb − ta 2 j . 2Mω 2

(3.183)

and by completing it quadratically to the form −

Z

tb ta

M j dt ω 2 x − 2 Mω 2

2

+

234


This is a harmonic potential shifted in x by −j/Mω 2 . The time evolution amplitude can thus immediately be written down as (xb tb |xa ta )j=const ω

s

Mω Mω i exp 2πi¯ h sin ω(tb − ta ) 2¯ h sin ω(tb − ta ) (" 2 2 # j j × xb − + xa − cos ω(tb − ta ) (3.184) Mω 2 Mω 2 ! j j i tb − ta 2 xa − + j . −2 xb − Mω 2 Mω 2 h ¯ 2Mω 2

=

In the free-particle limit ω → 0, the result becomes particularly simple: i M (xb − xa )2 = q exp h ¯ 2 tb − ta 2πi¯ h(tb − ta )/M 1 i 1 × exp (xb + xa )(tb − ta )j − (tb − ta )3 j 2 . (3.185) h ¯ 2 24M "

1

(xb ta |xa ta )j=const 0

#

As a cross check, we verify that the total exponent is equal to i/¯ h times the classical action Aj,cl =

Z

tb

ta

dt

M 2 x˙ + jxj,cl , 2 j,cl

(3.186)

calculated for the classical orbit xj,cl (t) connecting xa and xb in the presence of the constant current j. This satisfies the Euler-Lagrange equation x¨j,cl = j/M,

(3.187)

which is solved by

xj,cl (t) = xa + xb − xa −

j t − ta j + (tb − ta )2 (t − ta )2 . 2M tb − ta 2M

(3.188)

Inserting this into the action yields Aj,cl =

M (xb − xa )2 1 (tb − ta )3 j 2 + (xb + xa )(tb − ta )j − , 2 tb − ta 2 24 M

(3.189)

just as in the exponent of (3.185). Let us remark that the calculation of the oscillator amplitude (xa tb |xa t)jω in (3.168) could have proceeded alternatively by using the orbital separation x(t) = xj,cl (t) + δx(t),

(3.190)

where xj,cl (t) satisfies the Euler-Lagrange equations with the time-dependent source term x¨j,cl (t) + ω 2 xj,cl (t) = j(t)/M,


235

3.7 Time Evolution Amplitude at Fixed Path Average

rather than the orbital separation of Eq. (3.7), x(t) = xcl (t) + δx(t), where xcl (t) satisfied the Euler-Lagrange equation with no source. For this inhomogeneous differential equation we would have found the following solution passing through xa at t = ta and xb at t = tb : xj,cl (t) = xa

sin ω(tb − t) sin ω(t − ta ) 1 + xb + sin ω(tb − ta ) sin ω(tb − ta ) M

Z

tb ta

dt′ Gω2 (t, t′ )j(t′ ).

(3.192)

The Green function Gω2 (t, t′ ) appears now at the classical level. The separation (3.190) in the total action would have had the advantage over (3.7) that the source causes no linear term in δx(t). Thus, there would be no need for a quadratic completion; the classical action would be found from a pure surface term plus one half of the source part of the action Acl

tb M 2 M (x˙ cl,j − ω 2 x2j,cl ) + jxj,cl = xj,cl x˙ j,cl = dt ta 2 2 ta Z tb Z tb M j 1 + dt xj,cl −¨ xj,cl − ω 2 xj,cl + + dtxj,cl j 2 M 2 ta ta 1 Z tb M + dtxj,cl (t)j(t). = (xb x˙ b − xa x˙ a ) x=xj,cl 2 2 ta Z

tb

(3.193)

Inserting xj,cl from (3.192) and Gω2 (t, t′ ) from (3.36) leads once more to the exponent in (3.168). The fluctuating action quadratic in δx(t) would have given the same fluctuation factor as in the j = 0 -case, i.e., the prefactor in (3.168) with no further j 2 (due to the absence of a quadratic completion).

3.7

Time Evolution Amplitude at Fixed Path Average

Another interesting quantity to be needed in Chapter 15 is the Fourier transform of the amplitude (3.184): (xb tb |xa ta )xω0 = (tb − ta )

Z

∞

−∞

dj −ij(tb −ta )x0 /¯h e (xb tb |xa ta )jω . 2π¯ h

(3.194)

This is the amplitude for a particle to run from xa to xb along restricted paths whose R −1 tb temporal average x¯ ≡ (tb − ta ) ta dt x(t) is held fixed at x0 : (xb tb |xa ta )xω0

=

Z

i Dx δ(x0 − x¯) exp h ¯

Z

tb

ta

M dt (x˙ 2 − ω 2 x2 ) . 2

(3.195)

This property of the paths follows directly from the fact that the integral over the R time-independent source j (3.194) produces a δ-function δ((tb − ta )x0 − ttab dt x(t)). Restricted amplitudes of this type will turn out to have important applications later in Subsection 3.25.1 and in Chapters 5, 10, and 15.

236


The integral over j in (3.194) is done after a quadratic completion in Aj − j(tb − ta )x0 with Aj of (3.181): "

#

1 ω(tb − ta ) Aj − j(tb − ta )x0 = ω (tb − ta ) − 2 tan (j − j0 )2 + Ax0 , (3.196) 3 2Mω 2 with j0 =

Mω 2 ω(tb − ta ) − 2 tan ω(tb2−ta )

"

#

ω(tb − ta ) ω(tb − ta ) x0 − tan (xb + xa ) , 2

and A

x0

"

ω(tb − ta ) i ω(tb − ta )x0 − tan =− h (xa + xb ) 2 2 ω(tb −ta )−2 tan ω(tb2−ta ) Mω

#2

.

With the completed quadratic exponent (3.196), the Gaussian integral over j in (3.194) can immediately be done, yielding v u u (xb tb |xa ta )t

(xb tb |xa ta )xω0 =

iMω 3 /2π¯ h ω(tb − ta ) − 2 tan ω(tb2−ta )

exp

i x0 A . h ¯

(3.197)

If we set xb = xa and integrate over xb = xa , we find the quantum-mechanical version of the partition function at fixed x0 : Zωx0

1

i ω(tb − ta )/2 =q exp − (tb − ta )Mω 2 x20 . (3.198) 2¯ h 2π¯ h(tb − ta )/Mi sin[ω(tb − ta )/2]

As a check we integrate this over x0 and recover the correct Zω of Eq. (2.404). We may also integrate over both ends independently to obtain the partition function Zωopen,x0 =

v u u t

i ω(tb − ta ) exp − (tb − ta )Mω 2 x20 . sin ω(tb − ta ) 2¯ h

(3.199)

Integrating this over x0 and going to imaginary times leads back to the partition function Zωopen of Eq. (2.405).

3.8

External Source in Quantum-Statistical Path Integral

In the last section we have found the quantum-mechanical time evolution amplitude in the presence of an external source term. Let us now do the same thing for the quantum-statistical case and calculate the path integral (xb h ¯ β|xa 0)jω

=

Z

(

1 Z ¯hβ M 2 Dx(τ ) exp − dτ (x˙ + ω 2 x2 ) − j(τ )x(τ ) h ¯ 0 2

)

. (3.200)

This will be done in two ways. H. Kleinert, PATH INTEGRALS

237

3.8 External Source in Quantum-Statistical Path Integral

3.8.1

Continuation of Real-Time Result

The desired result is obtained most easily by an analytic continuation of the quantum-mechanical results (3.23), (3.168) in the time difference tb − ta to an imaginary time −i¯ h(τb − τa ) = −i¯ hβ. This gives immediately (xb h ¯ β|xa 0)jω

=

s

M 2π¯ h2 β

s

ω¯ hβ 1 exp − Aext [j] , sinh ω¯ hβ h ¯ e

(3.201)

with the extended classical Euclidean oscillator action j j j Aext e [j] = Ae + Ae = Ae + A1,e + A2,e ,

(3.202)

where Ae is the Euclidean action Ae =

i h Mω (x2b + x2a ) cosh ω¯ hβ − 2xb xa , 2 sinh β¯ hω

(3.203)

while the linear and quadratic Euclidean source terms are Aj1,e

1 =− sinh ω¯ hβ

Z

τb

τa

dτ [xa sinh ω(¯ hβ − τ ) + xb sinh ωτ ]j(τ ),

(3.204)

and Aj2,e = −

1 Z ¯hβ Z τ ′ dτ dτ j(τ ) Gω2 ,e (τ, τ ′ )j(τ ′ ), M 0 0

(3.205)

where Gω2 ,e (τ, τ ′ ) is the Euclidean version of the Green function (3.36) with Dirichlet boundary conditions: sinh ω(¯ hβ − τ> ) sinh ωτ< ω sinh ω¯ hβ hβ − τ − τ ′ ) cosh ω(¯ hβ − |τ − τ ′ |) − cosh ω(¯ , = 2ω sinh ω¯ hβ

Gω2 ,e (τ, τ ′ ) =

(3.206)

satisfying the differential equation (−∂τ2 + ω 2 ) Gω2 ,e (τ, τ ′ ) = δ(τ − τ ′ ).

(3.207)

It is related to the real-time Green function (3.36) by Gω2 ,e (τ, τ ′ ) = i Gω2 (−iτ, −iτ ′ ),

(3.208)

the overall factor i accounting for the replacement δ(t − t′ ) → iδ(τ − τ ′ ) on the right-hand side of (3.148) in going to (3.207) when going from the real time t to the Euclidean time −iτ . The symbols τ> and τ< in the first line (3.206) denote the larger and the smaller of the Euclidean times τ and τ ′ , respectively.

238


The source terms (3.204) and (3.205) can be rewritten as follows: Aj1,e = −

i o Mω nh xb (e−β¯hω Ae − Be ) xa (e−β¯hω Be − Ae ) , sinh ω¯ hβ

(3.209)

and Aj2,e

1 =− 4Mω

Z

hβ ¯

0

Z

dτ

hβ ¯ 0

+

′

dτ ′ e−ω|τ −τ | j(τ )j(τ ′ )

i h Mω eβ¯hω (A2e + Be2 ) − 2Ae Be . 2 sinh ω¯ hβ

(3.210)

We have introduced the Euclidean versions of the functions A(ω) and B(ω) in Eqs. (3.173) and (3.174) as hβ ¯ 1 Ae (ω) ≡ iA(ω)|tb −ta =−i¯hβ = dτ e−ωτ j(τ ), (3.211) Mω 0 Z ¯hβ 1 dτ e−ω(¯hβ−τ ) j(τ ) = −e−β¯hω Ae (−ω). (3.212) Be (ω) ≡ iB(ω)|tb −ta =−i¯hβ = Mω 0

Z

From (3.201) we now calculate the quantum-statistical partition function. Setting xb = xa = x, the first term in the action (3.202) becomes Ae =

Mω 2 sinh2 (ω¯ hβ/2)x2 . sinh β¯ hω

(3.213)

If we ignore the second and third action terms in (3.202) and integrate (3.201) over x, we obtain, of course, the free partition function Zω =

1 . 2 sinh(β¯ hω/2)

(3.214)

In the presence of j, we perform a quadratic completion in x and obtain a sourcedependent part of the action (3.202): Aje = Ajfl,e + Ajr,e,

(3.215)

where the additional term Ajr,e is the remainder left by a quadratic completion. It reads Mω eβ¯hω (Ae + Be )2 . 2 sinh ωβ

Ajr,e = −

(3.216)

Combining this with Ajfl,e of (3.210) gives Ajfl,e + Ajr,e = −

1 4Mω

Z

0

hβ ¯

dτ

Z

0

hβ ¯

′

dτ ′ e−ω|τ −τ | j(τ )j(τ ′ ) −

Mω eβ¯hω/2 Ae Be . sinh(β¯ hω/2) (3.217) H. Kleinert, PATH INTEGRALS

239


This can be rearranged to the total source term Aje = −

1 4Mω

Z

hβ ¯

0

Z

dτ

hβ ¯ 0

dτ ′

cosh ω(|τ − τ ′ | − h ¯ β/2) j(τ )j(τ ′ ). sinh(β¯ hω/2)

(3.218)

This is proved by rewriting the latter integrand as nh i 1 ′ ¯ − τ ′) eω(τ −τ ) e−β¯hω/2 + (ω → −ω) Θ(τ 2 sinh(β¯ hω/2) h i o ′ ¯ ′ − τ ) j(τ )j(τ ′ ). + eω(τ −τ ) e−β¯hω/2 + (ω → −ω) Θ(τ

hω/2) and In the second and fourth terms we replace eβ¯hω/2 by e−β¯hω/2 + 2 sinh(β¯ integrate over τ, τ ′ , with the result (3.217). The expression between the currents in (3.218) is recognized as the Euclidean version of the periodic Green function Gpω2 (τ ) in (3.99): Gpω2 ,e (τ ) ≡ iGpω2 (−iτ )|tb −ta =−i¯hβ

1 cosh ω(τ − h ¯ β/2) , 2ω sinh(β¯ hω/2)

=

τ ∈ [0, h ¯ β].

(3.219)

In terms of (3.218), the partition function of an oscillator in the presence of the source term is 1 Zω [j] = Zω exp − Aje . h ¯

(3.220)

For completeness, let us also calculate the partition function of all paths with open ends in the presence of the source j(t), thus generalizing the result (2.405). Integrating (3.201) over initial and final positions xa and xb we obtain Zωopen [j] where

=

s

j 2π¯ h 1 ˜j q e−(A2,e +A2,e )/¯h , Mω sinh[ω(τb − τa )]

1 A˜j2,e = − M

Z

0

hβ ¯

dτ

Z

0

τ

˜ ω2 (τ, τ ′ )j(τ ′ ), dτ ′ j(τ )G

(3.221)

(3.222)

with ˜ ω2(τ, τ ′ )= G

1 {cosh ω¯ hβ[sinh ω(¯ hβ−τ ) sinh ω(¯ hβ−τ ′ )+sinh ωτ sinh ωτ ′ ] 2ω sinh3 ω¯ hβ + sinh ω(¯ hβ −τ ) sinh ωτ ′ + sinh ω(¯ hβ −τ ′ ) sinh ωτ } .(3.223)

By some trigonometric identities, this can be simplified to 1 cosh ω(¯ hβ − τ − τ ′ ) ′ ˜ 2 Gω (τ, τ ) = . ω sinh ω¯ hβ

(3.224)

240


The first step is to rewrite the curly brackets in (3.223) as h

hβ sinh ωτ ′ + sinh ω(¯ hβ −τ ′ ) sinh ωτ cosh ω¯ h

i

i

+ sinh ω(¯ hβ −τ ′ ) cosh ω¯ hβ sinh ω(¯ hβ −τ ) + sinh ω(¯ hβ − ((¯ hβ −τ )) . (3.225) The first bracket is equal to sinh β¯ hω cosh ωτ , the second to sinh β¯ hω cosh ω(¯ hβ −τ ′ ), so that we arrive at h

i

sinh ω¯ hβ sinh ωτ cosh ωτ ′ + sinh ω(¯ hβ −τ ) cosh ω(¯ hβ −τ ′ ) . The bracket is now rewritten as

(3.226)

i 1h sinh ω(τ + τ ′ ) + sinh ω(τ − τ ′ ) + sinh ω(2¯ hβ − τ − τ ′ ) + sinh ω(τ ′ − τ ) , (3.227) 2

which is equal to i 1h sinh ω(¯ hβ + τ + τ ′ − h ¯ β) + sinh ω(¯ hβ + h ¯ β − τ − τ ′) , 2

(3.228)

i 1h 2 sinh ω¯ hβ cosh ω(¯ hβ − τ − τ ′ ) , 2

(3.229)

and thus to

such that we arrive indeed at (3.224). The source action in the exponent in (3.221) is therefore: (Aj2,e + A˜j2,e) = −

1 Z ¯hβ Z τ ′ ′ ′ dτ dτ j(τ )Gopen ω 2 ,e (τ, τ )j(τ ), M 0 0

(3.230)

with (3.205) cosh ω(¯ hβ − |τ − τ ′ |) + cosh ω(¯ hβ − τ − τ ′ ) 2ω sinh ω¯ hβ cosh ω(¯ hβ − τ> ) cosh ωτ< = . ω sinh ω¯ hβ

′ Gopen ω 2 ,e (τ, τ ) =

(3.231)

This Green function coincides precisely with the Euclidean version of Green function ′ GN ω 2 (t, t ) in Eq. (3.151) using the relation (3.208). This coincidence should have been expected after having seen in Section 2.12 that the partition function of all paths with open ends can be calculated, up to a trivial factor le (¯ hβ) of Eq. (2.345), as a sum over all paths satisfying Neumann boundary conditions (2.443), which is calculated using the measure (2.446) for the Fourier components. In the limit of small-ω, the Green function (3.231) reduces to ′ Gopen ω 2 ,e (τ, τ ) ≈ 2

ω ≈0

1 β 1 1 1 2 ′ ′ ′2 + − |τ − τ | − (τ + τ ) + τ + τ , βω 2 3 2 2 2β

(3.232)

which is the imaginary-time version of (3.157). H. Kleinert, PATH INTEGRALS

241


3.8.2

Calculation at Imaginary Time

Let us now see how the partition function with a source term is calculated directly in the imaginary-time formulation, where the periodic boundary condition is used from the outset. Thus we consider Zω [j] =

Z

Dx(τ ) e−Ae [j]/¯h,

(3.233)

with the Euclidean action Ae [j] =

Z

hβ ¯

0

dτ

M 2 (x˙ + ω 2x2 ) − j(τ )x(τ ) . 2

(3.234)

Since x(τ ) satisfies the periodic boundary condition, we can perform a partial integration of the kinetic term without picking up a boundary term xx| ˙ ttba . The action becomes Ae [j] =

Z

hβ ¯

0

M dτ x(τ )(−∂τ2 + ω 2)x(τ ) − j(τ )x(τ ) . 2

(3.235)

Let De (τ, τ ′ ) be the functional matrix Dω2 ,e (τ, τ ′ ) ≡ (−∂τ2 + ω 2 )δ(τ − τ ′ ),

τ − τ ′ ∈ [0, h ¯ β].

(3.236)

Its functional inverse is the Euclidean Green function, Gpω2 ,e (τ, τ ′ ) = Gpω2 ,e (τ − τ ′ ) = Dω−12 ,e (τ, τ ′ ) = (−∂τ2 + ω 2 )−1 δ(τ − τ ′ ),

(3.237)

with the periodic boundary condition. Next we perform a quadratic completion by shifting the path: x → x′ = x −

1 P G 2 j. M ω ,e

(3.238)

This brings the Euclidean action to the form Ae [j] =

Z

hβ ¯ 0

dτ

M ′ 1 x (−∂τ2 + ω 2 )x′ − 2 2M

Z

hβ ¯

0

dτ

Z

0

hβ ¯

dτ ′ j(τ )Gpω2 ,e (τ − τ ′ )j(τ ′ ). (3.239)

The fluctuations over the periodic paths x′ (τ ) can now be integrated out and yield for j(τ ) ≡ 0 −1/2 Zω = Det Dω2 ,e . (3.240) As in Subsection 2.15.2, we find the functional determinant by rewriting the product of eigenvalues as Det Dω2 ,e =

∞ Y

m=−∞

2 (ωm + ω 2 ) = exp

"

∞ X

m=−∞

#

2 log(ωm + ω2) ,

(3.241)

242


and evaluating the sum in the exponent according to the rules of analytic regularization. This leads directly to the partition function of the harmonic oscillator as in Eq. (2.401): 1 Zω = . (3.242) 2 sinh(β¯ hω/2) The generating functional for j(τ ) 6= 0 is therefore 1 Z[j] = Zω exp − Aje [j] , h ¯

(3.243)

1 Z ¯hβ Z ¯hβ ′ dτ dτ j(τ )Gpω2 ,e (τ − τ ′ )j(τ ′ ). 2M 0 0

(3.244)

with the source term: Aje [j] = −

The Green function of imaginary time is calculated as follows. The eigenfunctions 2 of the differential operator −∂τ2 are e−iωm τ with eigenvalues ωm , and the periodic boundary condition forces ωm to be equal to the thermal Matsubara frequencies ωm = 2πm/¯ hβ with m = 0, ±1, ±2, . . . . Hence we have the Fourier expansion Gpω2 ,e (τ ) =

∞ 1 X 1 e−iωm τ . 2 h ¯ β m=−∞ ωm + ω 2

(3.245)

In the zero-temperature limit, the Matsubara sum becomes an integral, yielding Gpω2 ,e (τ ) =

T =0

Z

1 −ω|τ | dωm 1 e−iωm τ = e . 2 2 2π ωm + ω 2ω

(3.246)

The frequency sum in (3.245) may be written as such an integral over ωm , provided the integrand contains an additional Poisson sum (3.81): ∞ X

m=−∞ ¯

δ(m − m) ¯ =

∞ X

i2πnm

e

n=−∞

=

∞ X

einωm ¯hβ .

(3.247)

n=−∞

This implies that the finite-temperature Green function (3.245) is obtained from (3.246) by a periodic repetition: Gpω2 ,e (τ ) = =

∞ X

1 −ω|τ +n¯hβ| e n=−∞ 2ω 1 cosh ω(τ − h ¯ β/2) , 2ω sinh(β¯ hω/2)

τ ∈ [0, h ¯ β].

(3.248)

A comparison with (3.97), (3.99) shows that Gpω2 ,e (τ ) coincides with Gpω2 (t) at imaginary times, as it should. Note that for small ω, the Green function has the expansion Gpω2 ,e (τ )

1 τ2 τ h ¯β = + − + + ... . 2 h ¯ βω 2¯ hβ 2 12

(3.249)


243


The first term diverges in the limit ω → 0. Comparison with the spectral representation (3.245) shows that it stems from the zero Matsubara frequency contribution to the sum. If this term is omitted, the subtracted Green function Gpω2′ ,e (τ ) ≡ Gpω2 ,e (τ ) − has a well-defined ω → 0 limit 1 h ¯β

Gp0,e′ (τ ) =

1 h ¯ βω 2

(3.250)

1 −iωm τ τ2 τ h ¯β e = − + , 2 2¯ hβ 2 12 m=±1,±2,... ωm X

(3.251)

the right-hand side being correct only for τ ∈ [0, h ¯ β]. Outside this interval it must be continued periodically. The subtracted Green function Gpω2′ ,e (τ ) is plotted for different frequencies ω in Fig. 3.4.

0.08

Gpω′2 ,e(τ )

Gaω2 ,e (τ ) 0.2

0.06

0.1

0.04 0.02 -1

-0.5 -0.02

0.5

1

2

1.5

τ /¯hβ

-1

-0.5

0.5

1

1.5

2

τ /¯hβ

-0.1 -0.2

-0.04

Figure 3.4 Subtracted periodic Green function Gpω′2 ,e(τ ) ≡ Gpω2 ,e (τ ) − 1/¯ hβω 2 and antiperiodic Green function Gaω2 ,e (τ ) for frequencies ω = (0, 5, 10)/¯ hβ (with increasing dash length). Compare Fig. 3.2.

The limiting expression (3.251) can, incidentally, be derived using the methods developed in Subsection 2.15.6. We rewrite the sum as 1 h ¯β

(−1)m −iωm (τ −¯hβ/2) e 2 m=±1,±2,... ωm X

(3.252)

and expand 2 − h ¯β

h ¯β 2π

!2

#n

"

1 2π −i (τ − h ¯ β/2) h ¯β n=0,2,4,... n! X

∞ X

(−1)m−1 . 2−n m=1 m

(3.253)

The sum over m on the right-hand side is Riemann’s eta function1 η(z) ≡ 1

∞ X

(−1)m−1 , mz m=1

M. Abramowitz and I. Stegun, op. cit., Formula 23.2.19.

(3.254)

244


which is related to the zeta function (2.513) by η(z) = (1 − 21−z )ζ(z).

(3.255)

Since the zeta functions of negative integers are all zero [recall (2.579)], only the terms with n = 0 and 2 contribute in (3.253). Inserting η(0) = −ζ(0) = 1/2,

η(2) = ζ(2)/2 = π 2 /12,

(3.256)

we obtain 2 − h ¯β

h ¯β 2π

!2 

π2 1  − 12 4

2π h ¯β

!2



(τ − h ¯ β/2)2  =

τ2 τ h ¯β − + , 2¯ hβ 2 12

(3.257)

in agreement with (3.251). It is worth remarking that the Green function (3.251) is directly proportional to the Bernoulli polynomial B2 (z): Gp0,e′ (τ ) =

h ¯β B2 (τ /¯ hβ). 2

(3.258)

These polynomials are defined in terms of the Bernoulli numbers Bk as2 Bn (x) =

n X n

k=0

k

Bk z n−k .

(3.259)

They appear in the expansion of the generating function3 ∞ X ezt tn−1 = B (z) , n et − 1 n=0 n!

(3.260)

and have the expansion B2n (z) = (−1)n−1

∞ 2(2n)! X cos(2πkz) , 2n (2π) k=0 k 2n

(3.261)

B2 (z) = z 2 − z + 1/6, . . . .

(3.262)

with the special cases B1 (z) = z − 1/2,

By analogy with (3.248), the antiperiodic Green function can be obtained from an antiperiodic repetition

2 3

∞ X

Gaω2 ,e (τ ) =

(−1)n −ω|τ +n¯hβ| e n=−∞ 2ω

=

1 sinh ω(τ − h ¯ β/2) , 2ω cosh(β¯hω/2)

τ ∈ [0, h ¯ β],

(3.263)

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 9.620. ibid. Formula 9.621. H. Kleinert, PATH INTEGRALS

245


which is an analytic continuation of (3.113) to imaginary times. In contrast to (3.249), this has a finite ω → 0 limit Gaω2 ,e (τ ) =

τ h ¯β − , 2 4

τ ∈ [0, h ¯ β].

(3.264)

For a plot of the antiperiodic Green function for different frequencies ω see again Fig. 3.4. The limiting expression (3.264) can again be derived using an expansion of the type (3.253). The spectral representation in terms of odd Matsubara frequencies (3.104) ∞ 1 −iωm 1 X f τ e f 2 h ¯ β m=−∞ ωm

Gaω2 ,e (τ ) ≡

(3.265)

is rewritten as ∞ ∞ 1 X 1 (−1)m 1 X f f cos(ω τ ) = sin[ωm (τ − h ¯ β/2)]. m f 2 f 2 h ¯ β m=−∞ ωm h ¯ β m=−∞ ωm

(3.266)

Expanding the sin function yields 2 h ¯β

h ¯β 2π

!2

#n

(−1)(n−1)/2 2π (τ − h ¯ β/2) n! h ¯β n=1,3,5,... "

X

∞ X

m=0

(−1)m

m+

1 2

2−n .

(3.267)

The sum over m at the end is 22−n times Riemann’s beta function4 β(2 − n), which is defined as β(z) ≡

∞ 1 X (−1)m z , 2z m=0 m + 1 2

(3.268)

and is related to Riemann’s zeta function

∞ X

(3.269)

(−1)m 2−z ζ(z, (q + 1)/2), z = ζ(z, q) − 2 (m + q) m=0

(3.270)

ζ(z, q) ≡

1 z. m=0 (m + q)

Indeed, we see immediately that ∞ X

so that

i 1 h 2−z ζ(z, 1/2) − 2 ζ(z, 3/4) . 2z Near z = 1, the function ζ(z, q) behaves like5

β(z) ≡

ζ(z, q) = 4 5

1 − ψ(q) + O(z − 1), z−1

M. Abramowitz and I. Stegun, op. cit., Formula 23.2.21. I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 9.533.2.

(3.271)

(3.272)

246


where ψ(z) is the Digamma function (2.565). Thus we obtain in the limit z → 1:

i 1h π 1 ζ(z, 1/2) − 21−z ζ(z, 3/4) = [−ψ(1/2) + ψ(3/4) + log 2] = . z→1 2 2 4 (3.273) The last result follows from the specific values [compare (2.567)]: π ψ(1/2) = −γ − 2 log 2, ψ(3/4) = −γ − 3 log 2 + . (3.274) 2 For negative odd arguments, the beta function (3.268) vanishes, so that there are no further contributions. Inserting this into (3.267) the only surviving e n = 1 -term yields once more (3.264). Note that the relation (3.273) could also have been found directly from the expansion (2.566) of the Digamma function, which yields

β(1) = lim

1 [ψ(3/2) − ψ(1/4)] , (3.275) 4 and is equal to (3.273) due to ψ(1/4) = −γ − 3 log 2 − π/2. For currents j(τ ), which are periodic in h ¯ β, the source term (3.244) can also be written more simply: β(1) =

Aje [j]

1 Z ¯hβ Z ∞ ′ −ω|τ −τ ′ | =− dτ dτ e j(τ )j(τ ′ ). 4Mω 0 −∞

(3.276)

This follows directly by rewriting (3.276), by analogy with (3.149), as a sum over all periodic repetitions of the zero-temperature Green function (3.246): Gpω2 ,e (τ ) =

∞ 1 X e−ω|τ +n¯hβ| . 2ω n=−∞

(3.277)

When inserted into (3.244), the factors e−nβ¯hω can be removed by an irrelevant periodic temporal shift in the current j(τ ′ ) → j(τ ′ − n¯ hβ) leading to (3.276). For a time-dependent periodic or antiperiodic potential Ω(τ ), the Green function Gp,a ω 2 ,e (τ ) solving the differential equation ′ p,a [∂τ2 − Ω2 (τ )]Gp,a (τ − τ ′ ), ω 2 ,e (τ, τ ) = δ

(3.278)

with the periodic or antiperiodic δ-function δ

p,a

′

(τ − τ ) =

∞ X

n=−∞

′

δ(τ − τ − n¯ hβ)

(

1 (−1)n

)

,

(3.279)

can be expressed6 in terms of two arbitrary solutions ξ(τ ) and η(τ ) of the homogenous differential equation in the same way as the real-time Green functions in Section 3.5: [∆(τ, τa ) ± ∆(τb , τ )][∆(τ ′ , τa ) ± ∆(τb , τ ′ )] ′ ′ 2 ,e (τ, τ ) ∓ Gp,a (τ, τ ) = G , (3.280) 2 ω ω ,e ¯ p,a (τa , τb )∆(τa , τb ) ∆ 6

See H. Kleinert and A. Chervyakov, Phys. Lett. A 245 , 345 (1998) (quant-ph/9803016); J. Math. Phys. B 40 , 6044 (1999) (physics/9712048). H. Kleinert, PATH INTEGRALS

247


where Gω2 ,e (τ, τ ′ ) is the imaginary-time Green function with Dirichlet boundary conditions corresponding to (3.206): Gω2 ,e (τ, τ ′ ) =

¯ − τ ′ )∆(τb , τ ′ )∆(τ, τa ) ¯ − τ ′ )∆(τb , τ )∆(τ ′ , τa ) + Θ(τ Θ(τ , ∆(τa , τb )

(3.281)

with ∆(τ, τ ′ ) =

1 [ξ(τ )η(τ ′ ) − ξ(τ ′ )η(τ )] , W

and

˙ )η(τ ), W = ξ(τ )η(τ ˙ ) − ξ(τ

¯ p,a (τa , τb ) = 2 ± ∂τ ∆(τa , τb ) ± ∂τ ∆(τb , τa ). ∆

(3.282) (3.283)

Let also write down the imaginary-time versions of the periodic or antiperiodic Green functions for time-dependent frequencies. Recall the expressions for constant ¯ β): frequency Gpω (t) and Gaω (t) of Eqs. (3.94) and (3.112) for τ ∈ (0, h Gpω,e (τ ) =

1 1 X −iωm τ −1 e = e−ω(τ −¯hβ/2) h ¯β m iωm − ω 2 sinh(β¯ hω/2)

= (1 + nbω )e−ωτ ,

(3.284)

and Gaω,e (τ ) =

1 X −iωm −1 1 f τ = e−ω(τ −¯hβ/2) e f h ¯β m iωm − ω 2 cosh(β¯ hω/2)

= (1 − nfω )e−ωτ ,

(3.285)

the first sum extending over the even Matsubara frequencies, the second over the odd ones. The Bose and Fermi distribution functions nb,f ω were defined in Eqs. (3.93) and (3.111). For τ < 0, periodicity or antiperiodicity determine p,a Gp,a ¯ β). ω,e (τ ) = ±Gω,e (τ + h

(3.286)

The generalization of these expressions to time-dependent periodic and antiperiodic frequencies Ω(τ ) satisfying the differential equations ′ p,a [−∂τ − Ω(τ )]Gp,a (τ − τ ′ ) Ω,e (τ, τ ) = δ

has for β → ∞ the form ′ Gp,a Ω,e (τ, τ )

¯ − τ ′ )e− = Θ(τ

Rτ 0

dτ ′ Ω(τ ′ )

.

(3.287)

(3.288)

Its periodic superposition yields for finite β a sum analogous to (3.277): ′ Gp,a Ω,e (τ, τ )

=

∞ X

n=0

−

e

R τ +n¯hβ 0

dτ ′ Ω(τ ′ )

(

1 (−1)n

)

,

h ¯ β > τ > τ ′ > 0,

which reduces to (3.284), (3.285) for a constant frequency Ω(τ ) ≡ ω.

(3.289)

248

3.9


Lattice Green Function

As in Chapter 2, it is easy to calculate the above results also on a sliced time axis. This is useful when it comes to comparing analytic results with Monte Carlo lattice simulations. We consider here only the Euclidean versions; the quantum-mechanical ones can be obtained by analytic continuation to real times. The Green function Gω2 (τ, τ ′ ) on an imaginary-time lattice with infinitely many lattice points of spacing ǫ reads [instead of the Euclidean version of (3.147)]: Gω2 (τ, τ ′ ) =

′ ′ ǫ 1 1 e−˜ωe |τ −τ | = e−˜ωe |τ −τ | , 2 sinh ǫ˜ ωe 2ω cosh(ǫ˜ ωe /2)

(3.290)

where ω ˜ e is given, as in (2.398), by

2 ǫω arsinh . ǫ 2 This is derived from the spectral representation Z dω ′ −iω′ (τ −τ ′ ) ǫ2 ′ ′ e Gω2 (τ, τ ) = Gω2 (τ − τ ) = 2π 2(1 − cos ǫω ′ ) + ǫ2 ω 2 ω ˜e =

(3.291)

(3.292)

by rewriting it as Gω2 (τ, τ ′ ) =

Z

∞ 0

ds

Z

dω ′ −iω′ ǫn −s[2(1−cos ǫω′ )+ǫ2 ω2 ]/ǫ2 e e , 2π

(3.293)

with n ≡ (τ ′ − τ )/ǫ, performing the ω ′ -integral which produces a Bessel function I(τ −τ ′ )/ǫ (2s/ǫ2 ), and subsequently the integral over s with the help of formula (2.467). The Green function (3.290) is defined only at discrete τn = n¯hβ/(N + 1). If it is summed over all periodic repetitions n → n + k(N + 1) with k = 0, ±1, ±2, . . . , one obtains the lattice analog of the periodic Green function (3.248): Gpe (τ )

= =

3.10

∞ 1 X ǫ2 e−iωm τ ¯hβ m=−∞ 2(1 − cos ǫωm ) + ǫ2 ω 2

1 cosh ω ˜ (τ − ¯hβ/2) 1 , 2ω cosh(ǫ˜ ω/2) sinh(¯ hω ˜ β/2)

τ ∈ [0, ¯hβ].

(3.294)

Correlation Functions, Generating Functional, and Wick Expansion

Equipped with the path integral of the harmonic oscillator in the presence of an external source it is easy to calculate the correlation functions of any number of position variables x(τ ). We consider here only a system in thermal equilibrium and study the behavior at imaginary times. The real-time correlation functions can be discussed similarly. The precise relation between them will be worked out in Chapter 18. In general, i.e., also for nonharmonic actions, the thermal correlation functions of n-variables x(τ ) are defined as the functional averages (n)

Gω2 (τ1 , . . . , τn ) ≡ hx(τ1 )x(τ2 ) · · · x(τn )i ≡ Z −1

Z

(3.295)

1 Dx x(τ1 )x(τ2 ) · · · x(τn ) exp − Ae . h ¯


249

3.10 Correlation Functions, Generating Functional, and Wick Expansion

They are also referred to as n-point functions. In operator quantum mechanics, the same quantities are obtained from the thermal expectation values of time-ordered products of Heisenberg position operators xˆH (τ ): ˆ (n) Gω2 (τ1 , . . . , τn ) = Z −1 Tr Tˆτ xˆH (τ1 )ˆ xH (τ2 ) · · · xˆH (τn )e−H/kB T

n

where Z is the partition function

h

io

,

ˆ

Z = e−F/kB T = Tr(e−H/kB T )

(3.296)

(3.297)

and Tˆτ is the time-ordering operator. Indeed, by slicing the imaginary-time evolution ˆ operator e−Hτ /¯h at discrete times in such a way that the times τi of the n position (n) operators x(τi ) are among them, we find that Gω2 (τ1 , . . . , τn ) has precisely the path integral representation (3.295). By definition, the path integral with the product of x(τi ) in the integrand is calculated as follows. First we sort the times τi according to their time order, denoting the reordered times by τt(i) . We also set τb ≡ τt(n+1) and τa ≡ τt(0) . Assuming that the times τt(i) are different from one another, we slice the time axis τ ∈ [τa , τb ] into the intervals [τb , τt(n) ], [τt(n) , τt(n−1) ], [τt(n−2) , τt(n−3) ],. . ., [τt(4) , τt(3) ], [τt(2) , τt(1) ], [τt(1) , τa ]. For each of these intervals we calculate the time evolution amplitude (xt(i+1) τt(i+1) |xt(i) τt(i) ) as usual. Finally, we recombine the amplitudes by performing the intermediate x(τt(i) )-integrations, with an extra factor x(τi ) at each τi , i.e., (n) Gω2 (τ1 , . . . , τn )

n+1 Y Z ∞

=

i=1

−∞

dxτt(i) (xt(n+1) τb) |xt(n) τt(n) ) · x(τt(n) ) · . . .

·(xt(i+1) τt(i+1) |xt(i) τt(i) ) · x(τt(i) ) · (xt(i) τt(i) |xt(i−1) τt(i−1) ) · x(τt(i−1) ) · . . . · (xt(2) τt(2) |xt(1) τt(1) ) · x(τt(1) ) · (xt(1) τt(1) |xt(0) τa ). (3.298) We have set xt(n+1) ≡ xb = xa ≡ xt(0) , in accordance with the periodic boundary condition. If two or more of the times τi are equal, the intermediate integrals are accompanied by the corresponding power of x(τi ). Fortunately, this rather complicated-looking expression can be replaced by a much simpler one involving functional derivatives of the thermal partition function Z[j] in the presence of an external current j. From the definition of Z[j] in (3.233) it is easy to see that all correlation functions of the system are obtained by the functional formula (n) Gω2 (τ1 , . . . , τn )

"

#

δ δ ···h ¯ Z[j] = Z[j]−1h ¯ δj(τ1 ) δj(τn )

.

(3.299)

j=0

This is why Z[j] is called the generating functional of the theory. In the present case of a harmonic action, Z[j] has the simple form (3.243), (3.244), and we can write (n) Gω2 (τ1 , . . . , τn )

=

"

h ¯

δ δ ···h ¯ δj(τ1 ) δj(τn )

(3.300)

250

3 External Sources, Correlations, and Perturbation Theory (

1 × exp 2¯ hM

Z

hβ ¯ 0

dτ

Z

hβ ¯

0

dτ

′

j(τ )Gpω2 ,e (τ

′

′

− τ )j(τ )

)#

, j=0

where Gpω2 ,e (τ −τ ′ ) is the Euclidean Green function (3.248). Expanding the exponential into a Taylor series, the differentiations are easy to perform. Obviously, any odd number of derivatives vanishes. Differentiating (3.243) twice yields the two-point function [recall (3.248)] (2)

Gω2 (τ, τ ′ ) = hx(τ )x(τ ′ )i =

h ¯ p G 2 (τ − τ ′ ). M ω ,e

(3.301)

Thus, up to the constant prefactor, the two-point function coincides with the Euclidean Green function (3.248). Inserting (3.301) into (3.300), all n-point func(2) tions are expressed in terms of the two-point function Gω2 (τ, τ ′ ): Expanding the exponential into a power series, the expansion term of order n/2 carries the numeric prefactors 1/(n/2)! · 1/2n/2 and consists of a product of n/2 factors R ¯hβ (2) ′ ′ ′ h2 . The n-point function is obtained by functionally dif0 dτ j(τ )Gω 2 (τ, τ )j(τ )/¯ ferentiating this term n times. The result is a sum over products of n/2 factors (2) Gω2 (τ, τ ′ ) with n! permutations of the n time arguments. Most of these prod(2) ucts coincide, for symmetry reasons. First, Gω2 (τ, τ ′ ) is symmetric in its arguments. Hence 2n/2 of the permutations correspond to identical terms, their num(2) ber canceling one of the prefactors. Second, the n/2 Green functions Gω2 (τ, τ ′ ) in the product are identical. Of the n! permutations, subsets of (n/2)! permutations produce identical terms, their number canceling the other prefactor. Only n!/[(n/2)!2n/2 ] = (n − 1) · (n − 3) · · · 1 = (n − 1)!! terms are different. They all carry a unit prefactor and their sum is given by the so-called Wick rule or Wick expansion: (n)

Gω2 (τ1 , . . . , τn ) =

X

pairs

(2)

(2)

Gω2 (τp(1) , τp(2) ) · · · Gω2 (τp(n−1) , τp(n) ).

(3.302)

Each term is characterized by a different pair configurations of the time arguments in the Green functions. These pair configurations are found most simply by the following rule: Write down all time arguments in the n-point function τ1 τ2 τ3 τ4 . . . τn . Indicate a pair by a common symbol, say τ˙p(i) τ˙p(i+1) , and call it a pair contraction (2) to symbolize a Green function Gω2 (τp(i) , τp(i+1) ). The desired (n − 1)!! pair configurations in the Wick expansion (3.302) are then found iteratively by forming n − 1 single contractions τ˙1 τ˙2 τ3 τ4 . . . τn + τ˙1 τ2 τ˙3 τ4 . . . τn + τ˙1 τ2 τ3 τ˙4 . . . τn + . . . + τ˙1 τ2 τ3 τ4 . . . τ˙n ,

(3.303)

and by treating the remaining n − 2 uncontracted variables in each of these terms likewise, using a different contraction symbol. The procedure is continued until all variables are contracted. H. Kleinert, PATH INTEGRALS

3.10 Correlation Functions, Generating Functional, and Wick Expansion

251

In the literature, one sometimes another shorter formula under the name of Wick’s rule, stating that a single harmonically fluctuating variable satisfies the equality of expectations: 2 2 eKx = eK hx i/2 .

D

E

(3.304)

This follows from the observation that the generating functional (3.233) may also be viewed as Zω times the expectation value of the source exponential D R

Zω [j] = Zω × e

dτ j(τ )x(τ )/¯ h

E

.

(3.305)

Thus we can express the result (3.243) also as D R

e


E

(1/2M ¯ h)

=e

R

dτ

R

dτ ′ j(τ )Gp 2 (τ,τ ′ )j(τ ′ ) ω ,e

.

(3.306)

Since (¯h/M)Gpω2 ,e (τ, τ ′ ) in the exponent is equal to the correlation function (2)

Gω2 (τ, τ ′ ) = hx(τ )x(τ ′ )i by Eq. (3.301), we may also write D R

e


E

R

=e

dτ

R

dτ ′ j(τ )hx(τ )x(τ ′ )ij(τ ′ )/2¯ h2

.

(3.307)

Considering now a discrete time axis sliced at t = tn , and inserting the special source current j(τn ) = Kδn,0 , for instance, we find directly (3.304). The Wick theorem in this form has an important physical application. The intensity of the sharp diffraction peaks observed in Bragg scattering of X-rays on crystal planes is reduced by thermal fluctuations of the atoms in the periodic lattice. The reduction factor is usually written as e−2W and called the Debye-Waller factor . In the Gaussian approximation it is given by e−W ≡ he−∇·u(x) i = e−Σk h |k·u(k)]

2 i/2

,

(3.308)

where u(x) is the atomic displacement field. If the fluctuations take place around hx(τ )i = 6 0, then (3.304) goes obviously over into heP x i = eP hx(τ )i+P

3.10.1

2 hx−hx(τ )ii2 /2

.

(3.309)

Real-Time Correlation Functions

The translation of these results to real times is simple. Consider, for example, the harmonic fluctuations δx(t) with Dirichlet boundary conditions, which vanish at tb and ta . Their correlation functions can be found by using the amplitude (3.23) as a generating functional, if we replace x(t) → δx(t) and xb = xa → 0. Differentiating twice with respect to the external currents j(t) we obtain (2)

Gω2 (t, t′ ) = hx(t)x(t′ )i = i

h ¯ Gω2 (t − t′ ), M

(3.310)

252


with the Green function Gω2 (t − t′ ) of Eq. (3.36), which vanishes if t = tb or t = ta . The correlation function of x(t) ˙ is hx(t) ˙ x(t ˙ ′ )i = i

h ¯ cos ω(tb − t> ) cos ω(t< − ta ) , M ω sin ω(tb − ta )

(3.311)

h ¯ cot ω(tb − ta ). M

(3.312)

and has the value hx(t ˙ b )x(t ˙ b )i = i

As an application, we use this result to calculate once more the time evolution amplitude (xb tb |xa ta ) in a way closely related to the operator method in Section 2.23. We observe that the time derivative of this amplitude has the path integral representation [compare (2.755)] Z

i

D

i¯ h∂tb (xb tb |xa ta ) = − D x L(xb , x˙ b )e

R tb ta

˙ h dt L(x,x)/¯

= −hL(xb , x˙ b )i (xb tb |xa ta ),

(3.313) and calculate the expectation value hL(xb , x˙ b )i as a sum of the classical Lagrangian L(xcl (tb ), x˙ cl (tb )) and the expectation value of the fluctuating part of the Lagrangian hLfl (xb , x˙ b )i ≡ h [L(xb , x˙ b ) − L(xcl (tb ), x˙ cl (tb ))] i. If the Lagrangian has the standard form L = M x˙ 2 /2 − V (x), then only the kinetic term contributes to hLfl (xb , x˙ b )i, so that M hδ x˙ 2b i. hLfl (xb , x˙ b )i = (3.314) 2 There is no contribution from hV (xb ) − V (xcl (tb ))i, due to the Dirichlet boundary conditions. The temporal integral over − [L(xcl (tb ), x˙ cl (tb )) − hLfl (xb , x˙ b )i] agrees with the operator result (2.782), and we obtain the time evolution amplitude from the formula iA(xb ,xa ;tb −ta )/¯ h

(xb tb |xa ta ) = C(xb , xa )e

i exp h ¯

Z

tb

ta

M dtb′ hδ x˙ 2b′ i , 2

(3.315)

where A(xb , xa ; tb − ta ) is the classical action A[xcl ] expressed as a function of the endpoints [recall (4.80)]. The constant of integration C(xb , xa ) is fixed as in (2.768) by solving the differential equation −i¯ h∇b (xb tb |xa ta ) = hpb i(xb tb |xa ta ) = pcl (tb )(xb tb |xa ta ),

(3.316)

and a similar equation for xa [compare (2.769)]. Since the prefactor pcl (tb ) on the right-hand side is obtained from the derivative of the exponential eiA(xb ,xa ;tb −ta )/¯h in (3.315), due to the general relation (4.81), the constant of integration C(xb , xa ) is actually independent of xb and xa . Thus we obtain from (3.315) once more the known result (3.315). As an example, take the harmonic oscillator. The terms linear in δx(t) = x(t) − xcl (t) vanish since they are they are odd in δx(t) while the exponent in (3.313) is H. Kleinert, PATH INTEGRALS

253

3.11 Correlation Functions of Charged Particle in Magnetic Field . . .

even. Inserting on the right-hand side of (3.314) the correlation function (3.311), we obtain in D dimensions hLfl (xb , x˙ b )i =

M h ¯ω hδ x˙ 2b i = i D cot ω(tb − ta ), 2 2

(3.317)

which is precisely the second term in Eq. (2.781), with the appropriate opposite sign.

3.11

Correlation Functions of Charged Particle in Magnetic Field and Harmonic Potential

It is straightforward to find the correlation functions of a charged particle in a magnetic and an extra harmonic potential discussed in Section 2.19. They are obtained by inverting the functional matrix (2.685): (2)

Gω2 ,B (τ, τ ′ ) =

h ¯ −1 D 2 (τ, τ ′ ). M ω ,B

(3.318)

By an ordinary matrix inversion of (2.688), we obtain the Fourier expansion (2)

GB (τ, τ ′ ) =

∞ 1 X ˜ ω2 ,B (ωm )e−iωm (τ −τ ′ ) , G h ¯ β m=−∞

(3.319)

with ¯ 1 ˜ (2)2 (ωm ) = h G ω ,B 2 2 2 + ω2 ) M (ωm + ω+ )(ωm −

2 ωm + ω 2 − ωB2 2ωB ωm 2 −2ωB ωm ωm + ω 2 − ωB2 ,

!

. (3.320)

2 2 2 2 Since ω+ + ω− = 2(ω 2 + ωB2 ) and ω+ − ω− = 4ωωB , the diagonal elements can be written as

h i 1 2 2 2 2 2 (ω + ω ) + (ω + ω ) − 4ω m + m − B 2 + ω 2 )(ω 2 + ω 2 ) 2(ωm + − m # #) (" " 1 1 1 ωB 1 1 = + 2 + − 2 . (3.321) 2 2 2 + ω2 2 + ω2 2 ωm ωm + ω− ω ωm ωm + ω− + +

Recalling the Fourier expansion (3.245), we obtain directly the diagonal periodic correlation function (2) Gω2 ,B,xx=

h ¯ cosh ω+ (|τ −τ ′ |−¯ hβ/2) cosh ω− (|τ −τ ′ | − h ¯ β/2) + , 4Mω sinh(ω+ h ¯ β/2) sinh(ω−h ¯ β/2) "

#

(3.322)

(2)

which is equal to Gω2 ,B,yy . The off-diagonal correlation functions have the Fourier components "

#

2ωB ωm ωm 1 1 = − 2 . 2 2 2 2 2 2 2 (ωm + ω+ )(ωm + ω− ) 2ω ωm + ω+ ωm + ω−

(3.323)

254


Since ωm are the Fourier components of the derivative i∂τ , we can write (2) (2) Gω2 ,B,xy (τ, τ ′ ) = −Gω2 ,B,yx (τ, τ ′ ) =

h ¯ i∂τ 2M

1 cosh ω+ (|τ − τ ′ | − h ¯ β/2) 2ω+ sinh(ω+ h ¯ β/2) # 1 cosh ω− (|τ − τ ′ | − h ¯ β/2) − .(3.324) 2ω− sinh(ω− h ¯ β/2)

"

Performing the derivatives yields (2) (2) Gω2 ,B,xy (τ, τ ′ ) = Gω2 ,B,yx (τ, τ ′ ) =

h ¯ ǫ(τ −τ ′ ) 2Mi

1 sinh ω+ (|τ −τ ′ | − h ¯ β/2) 2ω+ sinh(ω+h ¯ β/2) # 1 sinh ω− (|τ −τ ′ | − h ¯ β/2) − ,(3.325) 2ω− sinh(ω−h ¯ β/2)

"

where ǫ(τ −τ ′ ) is the step function (1.311). For a charged particle in a magnetic field without an extra harmonic oscillator we have to take the limit ω → ωB in these equations. Due to translational invariance of the limiting system, this exists only after removing the zero-mode in the Matsubara sum. This is done most simply in the final expressions by subtracting the hightemperature limits at τ = τ ′ . In the diagonal correlation functions (3.322) this yields 1 (2) ′ (2) ′ (2) Gω2 ,B,xx (τ, τ ′ ) = Gω2 ,B,yy (τ, τ ′ ) = Gω2 ,B,xx − , (3.326) βMω+ ω− where the prime indicates the subtraction. Now one can easily go to the limit ω → ωB with the result (2) ′ Gω2 ,B,xx

′

(τ, τ ) =

(2) ′ Gω2 ,B,yy

h ¯ cosh 2ω(|τ − τ ′ | − h ¯ β/2) 1 (τ, τ ) = − . (3.327) 4Mω sinh(β¯ hω) ω¯ hβ ′

"

#

For the subtracted off-diagonal correlation functions (3.325) we find (2)

(2)

(2)

′ ′ Gω2 ,B,xy (τ, τ ′ ) = −Gω2 ,B,yx (τ, τ ′ ) = Gω2 ,B,xy +

h ¯ ωB ǫ(τ − τ ′ ). 2Miω+ ω−

(3.328)

For more details see the literature.7

3.12

Correlation Functions in Canonical Path Integral

Sometimes it is desirable to know the correlation functions of position and momentum variables (m,n)

Gω2

(τ1 , . . . , τm ; τ1 , . . . , τn ) ≡ hx(τ1 )x(τ2 ) · · · x(τm )p(τ1 )p(τ2 ) · · · p(τn )i (3.329) Z Z Dp(τ ) 1 ≡ Z −1 Dx(τ ) x(τ1 )x(τ2 ) · · · x(τm )p(τ ) p(τ1 )p(τ2 ) · · · p(τn ) exp − Ae . 2π ¯h

These can be obtained from a direct extension of the generating functional (3.233) by another source k(τ ) coupled linearly to the momentum variable p(τ ): Z Z[j, k] = Dx(τ ) e−Ae [j,k]/¯h . (3.330) 7

M. Bachmann, H. Kleinert, and A. Pelster, Phys. Rev. A 62 , 52509 (2000) (quant-ph/0005074); Phys. Lett. A 279 , 23 (2001) (quant-ph/0005100). H. Kleinert, PATH INTEGRALS

255

3.12 Correlation Functions in Canonical Path Integral

3.12.1

Harmonic Correlation Functions

For the harmonic oscillator, the generating functional (3.330) is denoted by Zω [j, k] and its Euclidean action reads Z h¯ β 1 2 M 2 2 Ae [j, k] = dτ −ip(τ )x(τ ˙ )+ p + ω x − j(τ )x(τ ) − k(τ )p(τ ) , (3.331) 2M 2 0 the partition function is denoted by Zω [j, k]. Introducing the vectors in phase space V(τ ) = (p(τ ), x(τ )) and J(τ ) = (j(τ ), k(τ )), this can be written in matrix form as Z h¯ β 1 T V Dω2 ,e V − VT J , (3.332) Ae [J] = dτ 2 0 where Dω2 ,e (τ, τ ′ ) is the functional matrix ′

Dω2 ,e (τ, τ ) ≡

M ω2

i∂τ

−i∂τ

M −1

!

δ(τ − τ ′ ),

τ − τ ′ ∈ [0, ¯hβ].

(3.333)

Its functional inverse is the Euclidean Green function, ′ Gpω2 ,e (τ, τ ′ ) = Gpω2 ,e (τ −τ ′ ) = D−1 ω 2 ,e (τ, τ ) ! M −1 −i∂τ = (−∂τ2 + ω 2 )−1 δ(τ − τ ′ ), i∂τ M ω2

(3.334)

with the periodic boundary condition. After performing a quadratic completion as in (3.238) by shifting the path: V → V′ = V + Gpω2 ,e J,

(3.335)

the Euclidean action takes the form Z Z h¯ β Z h¯ β 1 h¯ β 1 ′T ′ dτ dτ ′ JT (τ ′ )Gpω2 ,e (τ − τ ′ )J(τ ′ ). Ae [J] = dτ V Dω2 ,e V − 2 2 0 0 0

(3.336)

The fluctuations over the periodic paths V′ (τ ) can now be integrated out and yield for J(τ ) ≡ 0 the oscillator partition function −1/2 Zω = Det Dω2 ,e . (3.337) A Fourier decomposition into Matsubara frequencies Dω2 ,e (τ, τ ′ ) =

∞ ′ 1 X Dp 2 (ωm )e−iωm (τ −τ ) , ¯hβ m=−∞ ω ,e

(3.338)

has the components Dpω2 ,e (ωm ) with the determinants

=

M −1

ωm

−ωm

M ω2

!

,

(3.339)

2 det Dpω2 ,e (ωm ) = ωm + ω2,

(3.340)

and the inverses Gpe (ωm )

=

[Dpω2 ,e (ωm )]−1

=

M ω2 ωm

−ωm M

−1

!

2 ωm

1 . + ω2

(3.341)

256


The product of determinants (3.340) for all ωm required in the functional determinant of Eq. (3.337) is calculated with the rules of analytic regularization in Section 2.15, and yields the same partition function as in (3.241), and thus the same partition function (3.242): 1 1 p = . 2 2 2 sinh(β¯hω/2) ωm + ω m=1

Zω = Q∞

(3.342)

We therefore obtain for arbitrary sources J(τ ) = (j(τ ), k(τ )) 6= 0 the generating functional 1 Z[J] = Zω exp − AJ [J] , (3.343) ¯h e with the source term AJ e [J] = −

1 2

Z

0

h ¯β

dτ

Z

0

h ¯β

dτ ′ JT (τ )Gpω2 ,e (τ, τ ′ )J(τ ′ ).

(3.344)

The Green function Gpω2 ,e (τ, τ ′ ) follows immediately from Eq. (3.334) and (3.237): ′ Gpω2 ,e (τ, τ ′ ) = Gpω2 ,e (τ −τ ′ ) = D−1 ω 2 ,e (τ, τ ) =

M −1 i∂τ

−i∂τ M ω2

!

Gpω2 ,e (τ − τ ′ ),

(3.345)

where Gpω2 ,e (τ − τ ′ ) is the simple periodic Green function (3.248). From the functional derivatives of (3.343) with respect to j(τ )/¯h and k(τ )/¯h as in (3.299), we now find the correlation functions ¯h p G 2 (τ − τ ′ ), M ω ,e (2) Gω2 ,e,xp (τ, τ ′ ) ≡ hx(τ )p(τ ′ )i = −i¯hG˙ pω2 ,e (τ − τ ′ ), (2)

Gω2 ,e,xx (τ, τ ′ ) ≡ hx(τ )x(τ ′ )i = (2) Gω2 ,e,px (τ, τ ′ ) (2) Gω2 ,e,pp (τ, τ ′ )

(3.346) (3.347)

≡ hp(τ )x(τ ′ )i = i¯hG˙ pω2 ,e (τ − τ ′ ),

(3.348)

≡ hp(τ )p(τ ′ )i =

(3.349)

h ¯ M ω 2 Gpω2 ,e (τ

− τ ′ ).

The correlation function hx(τ )x(τ ′ )i is the same as in the pure configuration space formulation (3.301). The mixed correlation function hp(τ )x(τ ′ )i is understood immediately by rewriting the current-free part of the action (3.331) as Z h¯ β 1 M 2 2 Ae [0, 0] = dτ (p − iM x) ˙ + x˙ + ω 2 x2 , (3.350) 2M 2 0 which shows that p(τ ) fluctuates harmonically around the classical momentum for imaginary time iM x(τ ˙ ). It is therefore not surprising that the correlation function hp(τ )x(τ ′ )i comes out to be the same as that of iM hx(τ ˙ )x(τ ′ )i. Such an analogy is no longer true for the correlation function ′ hp(τ )p(τ )i. In fact, the correlation function hx(τ ˙ )x(τ ˙ ′ )i is equal to hx(τ ˙ )x(τ ˙ ′ )i = −¯hM ∂τ2 Gpω2 ,e (τ − τ ′ ).

(3.351)

Comparison with (3.349) reveals the relation hp(τ )p(τ ′ )i

¯ h −∂τ2 + ω 2 Gpω2 ,e (τ − τ ′ ) M ¯h ′ = hx(τ ˙ )x(τ ˙ )i + δ(τ − τ ′ ). M = hx(τ ˙ )x(τ ˙ ′ )i +

(3.352)

The additional δ-function on the right-hand side is the consequence of the fact that p(τ ) is not equal to iM x, ˙ but fluctuates around it harmonically. H. Kleinert, PATH INTEGRALS

257


For the canonical path integral of a particle in a uniform magnetic field solved in Section 2.18, there are analogous relations. Here we write the canonical action (2.635) with a vector potential (2.632) in the Euclidean form as Ae [p, x] =

Z

h ¯β

dτ

0

i2 M 1 h e 2 2 p − B × x − iM x˙ + ω x , 2M c 2

(3.353)

showing that p(τ ) fluctuates harmonically around the classical momentum pcl (τ ) = (e/c)B × x − ˙ For a magnetic field pointing in the z-direction we obtain, with the frequency ωB = ωL /2 iM x. of Eq. (2.640), the following relations between the correlation functions involving momenta and those involving only coordinates given in (3.322), (3.324), (3.324): (2)

(2)

(2)

(3.354)

Gω2 ,B,xpy (τ, τ ′ ) ≡ hx(τ )py (τ ′ )i = iM ∂τ ′ Gω2 ,B,xy (τ, τ ′ ) + M ωB Gω2 ,B,xx (τ, τ ′ ),

(2)

(2)

(2)

(3.355)

Gω2 ,B,zpz (τ, τ ′ ) ≡ hz(τ )pz (τ ′ )i = iM ∂τ ′ Gω2 ,B,zz (τ, τ ′ ),

(2)

(2)

(3.356)

Gω2 ,B,xpx (τ, τ ′ ) ≡ hx(τ )px (τ ′ )i = iM ∂τ ′ Gω2 ,B,xx (τ, τ ′ ) − M ωB Gω2 ,B,xy (τ, τ ′ ),

(2) Gω2 ,B,px px (τ, τ ′ )

(2) Gω2 ,B,px py (τ, τ ′ )

≡

(2) (2) hpx (τ )px (τ ′ )i = −M 2 ∂τ ∂τ ′ Gω2 ,B,xx (τ, τ ′ )−2iM 2ωB ∂τ Gω2 ,B,xy (τ, τ ′ ) (2)

2 ¯ M δ(τ − τ ′ ), + M 2 ωB Gω2 ,B,xx (τ, τ ′ ) + h ′

≡ hpx (τ )py (τ )i = −M

2

(2) ∂τ ∂τ ′ Gω2 ,B,xy (τ, τ ′ )

+ iM

2

(2)

2 Gω2 ,B,xy (τ, τ ′ ), + M 2 ωB (2)

(3.357)

(2) ∂τ Gω2 ,B,xx (τ, τ ′ )

(2)

Gω2 ,B,pz pz (τ, τ ′ ) ≡ hpz (τ )pz (τ ′ )i = −M 2 ∂τ ∂τ ′ Gω2 ,B,zz (τ, τ ′ ) + h ¯ M δ(τ − τ ′ ).

(3.358) (3.359)

Only diagonal correlations between momenta contain the extra δ-function on the right-hand side (2) (2) according to the rule (3.352). Note that ∂τ ∂τ ′ Gω2 ,B,ab (τ, τ ′ ) = −∂τ2 Gω2 ,B,ab (τ, τ ′ ). Each correlation function is, of course, invariant under time translations, depending only on the time difference τ − τ ′. The correlation functions hx(τ )x(τ ′ )i and hx(τ )y(τ ′ )i are the same as before in Eqs. (3.324) and (3.325).

3.12.2

Relations between Various Amplitudes

A slight generalization of the generating functional (3.330) contains paths with fixed endpoints rather than all periodic paths. If the endpoints are held fixed in configuration space, one defines (xb ¯hβ|xa 0)[j, k] =

Z

x(¯ hβ)=xb

Dx

x(0)=xa

Dp 1 exp − Ae [j, k] . 2π¯h ¯ h

(3.360)

If the endpoints are held fixed in momentum space, one defines (pb ¯hβ|pa 0)[j, k] =

Z

p(¯ hβ)=pb

Dp 1 exp − Ae [j, k] . 2π¯h ¯ h

(3.361)

dxb e−i(pb xb −pa xa )/¯h (xb ¯hβ|xa 0)[j, k] .

(3.362)

Dx

p(0)=pa

The two are related by a Fourier transformation (pb ¯hβ|pa 0)[j, k] =

Z

+∞

−∞

dxa

Z

+∞

−∞

We now observe that in the canonical path integral, the amplitudes (3.360) and (3.361) with fixed endpoints can be reduced to those with vanishing endpoints with modified sources. The modification consists in shifting the current k(τ ) in the action by the source term ixb δ(τb − τ ) −

258


ixa δ(τ − τa ) and observe that this produces in (3.361) an overall phase factor in the limit τb ↑ ¯hβ and τa ↓ 0: lim lim (pb ¯hβ|pa 0)[j(τ ), k(τ ) + ixb δ(τb − τ ) − ixa δ(τ − τa )] i (pb xb − pa xa ) (pb ¯hβ|pa 0)[j(τ ), k(τ )] . = exp ¯h

τb ↑¯ hβ τa ↓0

(3.363)

By inserting (3.363) into the inverse of the Fourier transformation (3.362), Z +∞ Z dpa +∞ dpb i(pb xb −pa xa )/¯h (xb ¯hβ|xa 0)[j, k] = e (pb ¯hβ|pa 0)[j, k], h −∞ 2π¯h −∞ 2π¯

(3.364)

we obtain (xb ¯hβ|xa 0)[j, k] = lim lim (0 h ¯ β|0 0)[j(τ ), k(τ ) + ixb δ(τb −τ ) − ixa δ(τ −τa )] . τb ↑¯ hβ τa ↓0

(3.365)

In this way, the fixed-endpoint path integral (3.360) can be reduced to a path integral with vanishing endpoints but additional δ-terms in the current k(τ ) coupled to the momentum p(τ ). There is also a simple relation between path integrals with fixed equal endpoints and periodic path integrals. The measures of integration are related by Z

x(¯ hβ)=x x(0)=x

DxDp = 2π¯h

I

DxDp δ(x(0) − x) . 2π¯h

(3.366)

Using the Fourier decomposition of the delta function, we rewrite (3.366) as Z

x(¯ hβ)=x

x(0)=x

DxDp = lim τa′ ↓0 2π¯h

Z

+∞

dpa ipa x/¯h e 2π¯h

I

DxDp −i e 2π¯h

R h¯ β

dτ pa δ(τ −τa′ )x(τ )/¯ h

.

(3.367)

×Z [j(τ ) − ipa δ(τ − τa′ ), k(τ ) + ixb δ(τb − τ ) − ixa δ(τ − τa )] ,

(3.368)

−∞

0

Inserting now (3.367) into (3.365) leads to the announced desired relation Z +∞ dpa (xb ¯hβ|xa 0)[k, j] = lim lim lim ′ τb ↑¯ hβ τa ↓0 τa ↓0 −∞ 2π¯ h

where Z[j, k] is the thermodynamic partition function (3.330) summing all periodic paths. When using (3.368) we must be careful in evaluating the three limits. The limit τa′ ↓ 0 has to be evaluated prior to the other limits τb ↑ ¯hβ and τa ↓ 0.

3.12.3

Harmonic Generating Functionals

Here we write down explicitly the harmonic generating functionals with the above shifted source terms: ˜ ) = k(τ ) + ixb δ(τb − τ ) − ixa δ(τ − τa ) , k(τ

˜j(τ ) = j(τ ) − ipδ(τ − τa′ ),

(3.369)

leading to the factorized generating functional ˜ ˜j] = Z (0) [0, 0]Z (1) [k, j]Z p [k, j] . Zω [k, ω ω ω

(3.370)

The respective terms on the right-hand side of (3.370) read in detail 1 2 p ′ ′ (0) p ′ p ′ Zω [0, 0] = Zω exp 2 −p Gxx (τa , τa ) − 2p xa Gxp (τa , τa ) + xb Gxp (τa , τb ) 2¯ h H. Kleinert, PATH INTEGRALS

259


Zω(1) [k, j] =

Zωp [k, j] =

−x2a Gppp (τa , τa ) − x2b Gppp (τb , τb ) + 2xa xb Gppp (τa , τb ) , (3.371) Z h¯ β 1 exp dτ j(τ )[−ipGpxx (τ, τa′ ) + ixb Gpxp (τ, τb ) − ixa Gpxp (τ, τa )] 2 ¯h 0 +k(τ )[−ipGpxp (τ, τa′ ) + ixb Gppp (τ, τb ) − ixa Gppp (τ, τa )] , (3.372) ( Z h¯ β Z h¯ β 1 exp dτ dτ2 [(j(τ1 ), k(τ2 )) 1 2¯ h2 0 0 p Gxx (τ1 , τ1 ) Gpxp (τ1 , τ2 ) j(τ2 ) × , (3.373) Gppx (τ1 , τ2 ) Gppp (τ1 , τ2 ) k(τ2 )

where Zω is given by (3.342) and Gpxp (τ1 , τ2 ) etc. are the periodic Euclidean Green functions (2)

Gω2 ,e,ab (τ1 , τ2 ) defined in Eqs. (3.346)–(3.349) in an abbreviated notation. Inserting (3.370) into (0)

(3.368) and performing the Gaussian momentum integration, over the exponentials in Zω [0, 0] (1) and Zω [k, j], the result is ( Z ) 1 h¯ β (xb ¯hβ|xa 0)[k, j] = (xb ¯hβ|xa 0)[0, 0] × exp dτ [xcl (τ )j(τ ) + pcl (τ )k(τ )] ¯h 0 ( ! #) Z h¯ β Z h¯ β (D) (D) 1 Gxx (τ1 , τ2 ) Gxp (τ1 , τ2 ) j(τ2 ) × exp dτ1 dτ2 [(j(τ1 ), k(τ2 )) , (D) (D) k(τ2 ) 2¯ h2 0 Gpx (τ1 , τ2 ) Gpp (τ1 , τ2 ) 0 (3.374) (D)

where the Green functions Gab (τ1 , τ2 ) have now Dirichlet boundary conditions. In particular, the (D) Green function Gab (τ1 , τ2 ) is equal to (3.36) continued to imaginary time. The Green functions (D) (D) Gxp (τ1 , τ2 ) and Gpp (τ1 , τ2 ) are Dirichlet versions of Eqs. (3.346)–(3.349) which arise from the above Gaussian momentum integrals. After performing the integrals, the first factor without currents is s Zω 2π¯h2 (xb ¯hβ|xa 0)[0, 0] = lim lim lim p τb ↑¯ hβ τa ↓0 τa′ ↓0 2π¯ h Gxx (τa′ , τa′ ) ) ( ) ( Gpxp 2 (τa′ , τa ) Gpxp 2 (τa′ , τb ) 1 p 2 p 2 × exp xa − Gpp (τa , τa ) + xb − Gpp (τb , τb ) Gpxx (τa′ , τa′ ) Gpxx (τa′ , τa′ ) 2¯h2 p ′ Gxp (τa , τa )Gpxp (τa′ , τb ) p −2xa xb − G (τ , τ ) . (3.375) pp a b Gpxx (τa′ , τa′ ) Performing the limits using ¯h lim lim Gpxp (τa′ , τa ) = −i , 2

τa ↓0 τa′ ↓0

(3.376)

where the order of the respective limits turns out to be important, we obtain the amplitude (2.403): s Mω (xb ¯hβ|xa 0)[0, 0] = 2π¯h sinh h ¯ βω 2 Mω × exp − (xa + x2b ) cosh ¯hβω − 2xa xb . (3.377) 2¯h sinh h ¯ βω The first exponential in (3.374) contains a complicated representation of the classical path p ′ Gxp (τa , τa )Gpxx (τ, τa′ ) i p xcl (τ ) = lim lim lim xa + Gxp (τa , τ ) τb ↑¯ hβ τa ↓0 τa′ ↓0 ¯ h Gpxx (τa′ , τa′ )

260


−xb

Gpxp (τa′ , τb )Gpxx (τ, τa′ ) p + G (τ , τ ) , xp b Gpxx (τa′ , τa′ )

(3.378)

and of the classical momentum pcl (τ )

i = lim lim lim τb ↑¯ hβ τa ↓0 τa′ ↓0 ¯ h

Gpxp (τa′ , τa )Gpxp (τa′ , τ ) p xa − Gpp (τa , τ ) Gpxx (τa′ , τa′ ) p ′ Gxp (τa , τb )Gpxp (τa′ , τ ) p − Gpp (τb , τ ) . −xb Gpxx (τa′ , τa′ )

(3.379)

Indeed, inserting the explicit periodic Green functions (3.346)–(3.349) and going to the limits we obtain xcl (τ )

=

xa sinh ω(¯ hβ − τ ) + xb sinh ωτ sinh h ¯ βω

(3.380)

−xa cosh ω(¯ hβ − τ ) + xb cosh ωτ , sinh h ¯ βω

(3.381)

and pcl (τ )

= iM ω

the first being the imaginary-time version of the classical path (3.6), the second being related to it by the classical relation pcl (τ ) = iM dxcl (τ )/dτ . The second exponential in (3.374) quadratic in the currents contains the Green functions with Dirichlet boundary conditions Gpxx (τ1 , 0)Gpxx (τ2 , 0) , Gpxx (τ1 , τ1 ) Gpxx (τ1 , 0)Gpxp (τ2 , 0) p G(D) , xp (τ1 , τ2 ) = Gxp (τ1 , τ2 ) + Gpxx (τ1 , τ1 ) Gpxp (τ1 , 0)Gpxx (τ2 , 0) p G(D) , px (τ1 , τ2 ) = Gpx (τ1 , τ2 ) + Gpxx (τ1 , τ1 ) Gpxp (τ1 , 0)Gpxp (τ2 , 0) p G(D) (τ , τ ) = G (τ , τ ) − . 1 2 1 2 pp pp Gpxx (τ1 , τ1 ) p G(D) xx (τ1 , τ2 ) = Gxx (τ1 , τ2 ) −

(3.382) (3.383) (3.384) (3.385)

After applying some trigonometric identities, these take the form ¯h [cosh ω(¯ hβ −|τ1 −τ2 |)−cosh ω(¯ hβ −τ1 −τ2 )], 2M ω sinh h ¯ βω i¯h G(D) {θ(τ1 −τ2 ) sinh ω(¯ hβ −|τ1 −τ2 |) xp (τ1 , τ2 ) = 2 sinh h ¯ βω −θ(τ2 −τ1 ) sinh ω(¯ hβ − |τ2 −τ1 |)+sinh ω(¯ hβ − τ1 −τ2 )}, i¯h G(D) {θ(τ1 −τ2 ) sinh ω(¯ hβ − |τ1 −τ2 |) px (τ1 , τ2 ) = − 2 sinh h ¯ βω −θ(τ2 −τ1 ) sinh ω(¯ hβ − |τ2 −τ1 |)−sinh ω(¯ hβ − τ1 −τ2 )}, M ¯hω G(D) [cosh ω(¯ hβ − |τ1 −τ2 |) + cosh ω(¯ hβ −τ1 −τ2 )]. pp (τ1 , τ2 ) = 2 sinh h ¯ βω G(D) xx (τ1 , τ2 ) =

(3.386)

(3.387)

(3.388) (3.389)

The first correlation function is, of course, the imaginary-time version of the Green function (3.206). Observe the symmetry properties under interchange of the time arguments: (D) G(D) xx (τ1 , τ2 ) = Gxx (τ2 , τ1 ) ,

G(D) px (τ1 , τ2 )

=

−G(D) px (τ2 , τ1 ) ,

(D) G(D) xp (τ1 , τ2 ) = −Gxp (τ2 , τ1 ) ,

G(D) pp (τ1 , τ2 )

=

G(D) pp (τ2 , τ1 ) ,

(3.390) (3.391)


261

3.13 Particle in Heat Bath and the identity (D) G(D) xp (τ1 , τ2 ) = Gpx (τ2 , τ1 ).

(3.392)

In addition, there are the following derivative relations between the Green functions with Dirichlet boundary conditions: G(D) xp (τ1 , τ2 ) = G(D) px (τ1 , τ2 ) = G(D) pp (τ1 , τ2 ) =

∂ (D) ∂ (D) Gxx (τ1 , τ2 ) = iM G (τ1 , τ2 ) , ∂τ1 ∂τ2 xx ∂ (D) ∂ (D) G (τ1 , τ2 ) = −iM G (τ1 , τ2 ) , iM ∂τ1 xx ∂τ2 xx ∂2 h ¯ M δ(τ1 −τ2 ) − M 2 G(D) (τ1 −τ2 ) . ∂τ1 ∂τ2 xx

(3.393)

−iM

(3.394) (3.395)

Note that Eq. (3.382) is a nonlinear alternative to the additive decomposition (3.142) of a Green function with Dirichlet boundary conditions: into Green functions with periodic boundary conditions.

3.13

Particle in Heat Bath

The results of Section 3.8 are the key to understanding the behavior of a quantummechanical particle moving through a dissipative medium at a fixed temperature T . We imagine the coordinate x(t) a particle of mass M to be coupled linearly to a heat bath consisting of a great number of harmonic oscillators Xi (τ ) (i = 1, 2, 3, . . .) with various masses Mi and frequencies Ωi . The imaginary-time path integral in this heat bath is given by YI

(xb h ¯ β|xa 0) =

i

DXi (τ )

(

Z

(

Z

1 × exp − h ¯ 1 × exp − h ¯

Z

hβ ¯ 0

x(0)=xa

dτ

Dx(τ )

X Mi

2

i

hβ ¯ 0

x(¯ hβ)=xb

2 (X˙ i + Ω2i Xi2 )

)

(3.396)

"

X M 2 dτ x˙ + V (x(τ )) − ci Xi (τ )x(τ ) 2 i

#)

1 , i Zi

×Q

where we have allowed for an arbitrary potential V (x). The partition functions of the individual bath oscillators ( ) I Z 1 ¯hβ Mi ˙ 2 2 2 Zi ≡ DXi (τ ) exp − dτ (Xi + Ωi Xi ) h ¯ 0 2 1 = (3.397) 2 sinh(¯ hβΩi /2) have been divided out, since their thermal behavior is trivial and will be of no interest in the sequel. The path integrals over Xi (τ ) can be performed as in Section 3.1 leading for each oscillator label i to a source expression like (3.243), in which ci x(τ ) plays the role of a current j(τ ). The result can be written as (xb h ¯ β|xa 0) =

Z

x(¯ hβ)=xb

x(0)=xa

(

1 Dx(τ ) exp − h ¯

Z

hβ ¯ 0

)

M 2 1 dτ x˙ + V (x(τ )) − Abath [x] , 2 h ¯ (3.398)

262


where Abath [x] is a nonlocal action for the particle motion generated by the bath 1 Z ¯hβ Z ¯hβ ′ Abath [x] = − dτ dτ x(τ )α(τ − τ ′ )x(τ ′ ). 2 0 0

(3.399)

The function α(τ − τ ′ ) is the weighted periodic correlation function (3.248): α(τ − τ ′ ) =

X

=

X

c2i

i

i

1 p G 2 (τ − τ ′ ) Mi Ωi ,e

c2i cosh Ωi (|τ − τ ′ | − h ¯ β/2) . 2Mi Ωi sinh(Ωih ¯ β/2)

(3.400)

Its Fourier expansion has the Matsubara frequencies ωm = 2πkB T /¯ h α(τ − τ ′ ) =

∞ 1 X ′ αm e−iωm (τ −τ ) , h ¯ β m=−∞

(3.401)

with the coefficients αm =

c2i 1 . 2 + ω2 Mi ωm i

X i

(3.402)

Alternatively, we can write the bath action in the form corresponding to (3.276) as Abath [x] = −

1 Z ¯hβ Z ∞ ′ dτ dτ x(τ )α0 (τ − τ ′ )x(τ ′ ), 2 0 −∞

(3.403)

with the weighted nonperiodic correlation function [recall (3.277)] α0 (τ − τ ′ ) =

X i

c2i ′ e−Ωi |τ −τ | . 2Mi Ωi

(3.404)

The bath properties are conveniently summarized by the spectral density of the bath X c2i ′ δ(ω ′ − Ωi ). (3.405) ρb (ω ) ≡ 2π i 2Mi Ωi The frequencies Ωi are by definition positive numbers. The spectral density allows us to express α0 (τ − τ ′ ) as the spectral integral ′

α0 (τ − τ ) =

Z

0

∞

dω ′ ′ ′ ρb (ω ′ )e−ω |τ −τ | , 2π

(3.406)

and similarly α(τ − τ ′ ) =

Z

0

∞

dω ′ cosh ω ′ (|τ − τ ′ | − h ¯ β/2) ρb (ω ′ ) . 2π sinh(ω ′h ¯ β/2)

(3.407)


263

3.13 Particle in Heat Bath

For the Fourier coefficients (3.402), the spectral integral reads Z

αm =

∞

0

dω ′ 2ω ′ ρb (ω ′ ) 2 . 2π ωm + ω ′2

(3.408)

It is useful to subtract from these coefficients the first term α0 , and to invert the sign of the remainder making it positive definite. Thus we split αm = 2

Z

∞

0

2 dω ′ ρb (ω ′) ωm 1 − 2 + ω ′2 2π ω ′ ωm

!

= α0 − gm .

(3.409)

Then the Fourier expansion (3.401) separates as α(τ − τ ′ ) = α0 δ p (τ − τ ′ ) − g(τ − τ ′ ),

(3.410)

where δ p (τ − τ ′ ) is the periodic δ-function (3.279): δ p (τ − τ ′ ) =

∞ ∞ X 1 X ′ e−iωm (τ −τ ) = δ(τ − τ ′ − n¯ hβ), h ¯ β m=−∞ n=−∞

(3.411)

the right-hand sum following from Poisson’s summation formula (1.197). The subtracted correlation function ∞ 1 X ′ g(ωm )e−iωm (τ −τ ) , g(τ − τ ) = h ¯ β m=−∞ ′

(3.412)

has the coefficients gm =

X i

Z ∞ 2 2 c2i ωm dω ′ ρb (ω ′ ) 2ωm = . 2 + Ω2 2 + ω ′2 Mi ωm 2π ω ′ ωm 0 i

(3.413)

The corresponding decomposition of the bath action (3.399) is Abath [x] = Aloc + A′bath [x],

(3.414)

where A′bath [x]

1 = 2

Z

0

hβ ¯

dτ

hβ ¯

Z

0

and

dτ ′ x(τ )g(τ − τ ′ )x(τ ′ ),

(3.415)

α0 ¯hβ dτ x2 (τ ), (3.416) Aloc = − 2 0 is a local action which can be added to the original action in Eq. (3.398), changing merely the curvature of the potential V (x). Because of this effect, it is useful to introduce a frequency shift ∆ω 2 via the equation Z

2

M∆ω ≡ −α0 = −2

Z

0

∞

X c2i dω ′ ρb (ω ′ ) = − . 2 2π ω ′ i Mi Ωi

(3.417)

264


Then the local action (3.416) becomes Z ¯hβ M dτ x2 (τ ). ∆ω 2 2 0

Aloc =

(3.418)

This can be absorbed into the potential of the path integral (3.398), yielding a renormalized potential M (3.419) Vren (x) = V (x) + ∆ω 2 x2 . 2 With the decomposition (3.414), the path integral (3.398) acquires the form (xb h ¯ β|xa 0) =

Z

x(¯ hβ)=xb

x(0)=xa

(

1 Dx(τ ) exp − h ¯

Z

hβ ¯

0

)

M 2 1 dτ x˙ + Vren (x(τ )) − A′bath [x] . 2 h ¯ (3.420)

The subtracted correlation function (3.412) has the property Z

0

hβ ¯

dτ g(τ − τ ′ ) = 0.

(3.421)

Thus, if we rewrite in (3.415) 1 x(τ )x(τ ′ ) = {x2 (τ ) + x2 (τ ′ ) − [x(τ ) − x(τ ′ )]2 }, 2

(3.422)

the first two terms do not contribute, and we remain with A′bath [x]

1 =− 4

Z

0

hβ ¯

dτ

Z

0

hβ ¯

dτ ′ g(τ − τ ′ )[x(τ ) − x(τ ′ )]2 .

(3.423)

If the oscillator frequencies Ωi are densely distributed, the function ρb (ω ′ ) is continuous. As will be shown later in Eqs. (18.208) and (18.311), an oscillator bath introduces in general a friction force into classical equations of motion. If this is to have the usual form −Mγ x(t), ˙ the spectral density of the bath must have the approximation (3.424) ρb (ω ′) ≈ 2Mγω ′ [see Eqs. (18.208), (18.311)]. This approximation is characteristic for Ohmic dissipation. In general, a typical friction force increases with ω only for small frequencies; for larger ω, it decreases again. An often applicable phenomenological approximation is the so-called Drude form ρb (ω ′ ) ≈ 2Mγω ′

2 ωD , 2 ωD + ω ′2

(3.425)

where 1/ωD ≡ τD is Drude’s relaxation time. For times much shorter than the Drude time τD , there is no dissipation. In the limit of large ωD , the Drude form describes again Ohmic dissipation. H. Kleinert, PATH INTEGRALS

265

3.14 Heat Bath of Photons

Inserting (3.425) into (3.413), we obtain the Fourier coefficients for Drude dissipation 2 gm = 2MγωD

Z

0

∞

2 dω 1 2ωm ωD = M|ω |γ . m 2 2 + ω2 2π ωD + ω 2 ωm |ωm | + ωD

(3.426)

It is customary, to factorize gm ≡ M|ωm |γm,

(3.427)

so that Drude dissipation corresponds to γm = γ

ωD , |ωm | + ωD

(3.428)

and Ohmic dissipation to γm ≡ γ. The Drude form of the spectral density gives rise to a frequency shift (3.417) ∆ω 2 = −γωD ,

(3.429)

which goes to infinity in the Ohmic limit ωD → ∞.

3.14

Heat Bath of Photons

The heat bath in the last section was a convenient phenomenological tool to reproduce the Ohmic friction observed in many physical systems. In nature, there can be various different sources of dissipation. The most elementary of these is the deexcitation of atoms by radiation, which at zero temperature gives rise to the natural line width of atoms. The photons may form a thermally equilibrated gas, the most famous example being the cosmic black-body radiation which is a gas of the photons of 3 K left over from the big bang 15 billion years ago (and which create a sizable fraction of the blips on our television screens). The theoretical description is quite simple. We decompose the vector potential A(x, t) of electromagnetism into Fourier components of wave vector k A(x, t) =

X k

ck (x)Xk (t),

ck = √

eikx , 2Ωk V

X k

=

Z

d3 kV . (2π)3

(3.430)

The Fourier components Xk (t) can be considered as a sum of harmonic oscillators of frequency Ωk = c|k|, where c is the light velocity. A photon of wave vector k is a quantum of Xk (t). A certain number N of photons with the same wave vector can be described as the Nth excited state of the oscillator Xk (t). The statistical sum of these harmonic oscillators led Planck to his famous formula for the energy of black-body radiation for photons in an otherwise empty cavity whose walls have a temperature T . These will form the bath, and we shall now study its effect on the quantum mechanics of a charged point particle. Its coupling to the vector potential is given by the interaction (2.626). Comparison with the coupling to the

266


heat bath in Eq. (3.396) shows that we simply have to replace − P ˙ ). The bath action (3.399) takes then the form − k ck Xk (τ )x(τ Abath [x] = −

1 2

Z

hβ ¯

0

dτ

Z

hβ ¯

0

P

i ci Xi (τ )x(τ )

dτ ′ x˙ i (τ )αij (x(τ ), τ ; x(τ ′ ), τ ′ )x˙ j (τ ′ ),

by

(3.431)

where αij (x, τ ; x′ , τ ′ ) is a 3 × 3 matrix generalization of the correlation function (3.400): αij (x, τ ; x′ , τ ′ ) =

e2 X i c−k (x)ck (x′ )hX−k (τ )Xkj (τ ′ )i. h ¯ c2 k

(3.432)

We now have to account for the fact that there are two polarization states for each photon, which are transverse to the momentum direction. We therefore introduce a transverse Kronecker symbol T ij δk

≡ (δ ij − k i k j /k2 )

(3.433)

i and write the correlation function of a single oscillator X−k (τ ) as ′ î ˆj ′ Gij ¯ Tδkij δkk′ Gpω2 ,e k (τ − τ ′ ), −k′ k (τ − τ ) = hX−k′ (τ )Xk′ (τ )i = h

(3.434)

with Gpω2 ,e k (τ − τ ′ ) ≡

1 cosh Ωk (|τ − τ ′ | − h ¯ β/2) . 2Ωk sinh(Ωk h ¯ β/2)

(3.435)

Thus we find e2 α (x, τ ; x , τ ) = 2 c ij

′

′

′

Z

d3 k T ij eik(x−x ) cosh Ωk (|τ − τ ′ | − h ¯ β/2) . δk 3 (2π) 2Ωk sinh(Ωk h ¯ β/2)

(3.436)

At zero temperature, and expressing Ωk = c|k|, this simplifies to e2 α (x, τ ; x , τ ) = 3 c ij

′

′

′

Z

′

d3 k T ij eik(x−x )−c|k||τ −τ | δ . (2π)3 k 2|k|

(3.437)

Forgetting for a moment the transverse Kronecker symbol and the prefactor e2 /c2 , the integral yields ′ ′ GR e (x, τ ; x , τ ) =

1 1 , 2 2 ′ 2 4π c (τ − τ ) + (x − x′ )2 /c2

(3.438)

which is the imaginary-time version of the well-known retarded Green function used in electromagnetism. If the system is small compared to the average wavelengths in the bath we can neglect the retardation and omit the term (x − x′ )2 /c2 . In the finite-temperature expression (3.437) this amounts to neglecting the x-dependence. H. Kleinert, PATH INTEGRALS

267

3.15 Harmonic Oscillator in Ohmic Heat Bath

The transverse Kronecker symbol can then be averaged over all directions of the wave vector and yields simply 2δ ij /3, and we obtain the approximate function 2e2 1 α (x, τ ; x , τ ) = 2 δ ij 3c 2πc2 ij

′

′

Z

dω cosh ω(|τ − τ ′ | − h ¯ β/2) ω . 2π sinh(ω¯ hβ/2)

(3.439)

This has the generic form (3.407) with the spectral function of the photon bath ρpb (ω ′) =

e2 ′ ω. 3c2 π

(3.440)

This has precisely the Ohmic form (3.424), but there is now an important difference: the bath action (3.431) contains now the time derivatives of the paths x(τ ). This gives rise to an extra factor ω ′2 in (3.424), so that we may define a spectral density for the photon bath: e2 ρpb (ω ′) ≈ 2Mγω ′3 , γ = 2 . (3.441) 6c πM In contrast to the usual friction constant γ in the previous section, this has the dimension 1/frequency.

3.15

Harmonic Oscillator in Ohmic Heat Bath

For a harmonic oscillator in an Ohmic heat bath, the partition function can be calculated as follows. Setting Vren (x) =

M 2 2 ω x, 2

(3.442)

the Fourier decomposition of the action (3.420) reads M¯ h Ae = kB T

(

∞ h i ω2 2 X 2 x0 + ωm + ω 2 + ωm γm |xm |2 . 2 m=1

)

(3.443)

The harmonic potential is the full renormalized potential (3.419). Performing the Gaussian integrals using the measure (2.439), we obtain the partition function for the damped harmonic oscillator of frequency ω [compare (2.400)] Zωdamp

kB T = h ¯ω

(

∞ Y

m=1

"

2 ωm + ω 2 + ωm γ m 2 ωm

#)−1

.

(3.444)

For the Drude dissipation (3.426), this can be written as Zωdamp =

∞ 2 kB T Y ωm (ωm + ωD ) . 3 2 h ¯ ω m=1 ωm + ωm ωD + ωm (ω 2 + γωD ) + ωD ω 2

(3.445)

Let w1 , w2 , w3 be the roots of the cubic equation w 3 − w 2 ωD + w(ω 2 + γωD ) − ω 2 ωD = 0.

(3.446)

268


Then we can rewrite (3.445) as Zωdamp =

∞ kB T Y ωm ωm ωm ωm + ωD . h ¯ ω m=1 ωm + w1 ωm + w2 ωm + w3 ωm

(3.447)

Using the product representation of the Gamma function8 n nz Y m n→∞ z m=1 m + z

Γ(z) = lim

(3.448)

and the fact that w1 + w2 + w3 − ωD = 0,

w1 w2 w3 = ω 2 ωD ,

(3.449)

the partition function (3.447) becomes Zωdamp =

1 ω Γ(w1 /ω1 )Γ(w2 /ω1 )Γ(w3 /ω1) , 2π ω1 Γ(ωD /ω1 )

(3.450)

where ω1 = 2πkB T /¯ h is the first Matsubara frequency, such that wi /ω1 = wi β/2π. In the Ohmic limit ωD → ∞, the roots w1 , w2 , w3 reduce to w1 = γ/2 + iδ,

w1 = γ/2 − iδ,

with δ≡ and (3.450) simplifies further to Zωdamp =

w3 = ωD − γ,

q

ω 2 − γ 2 /4,

(3.451) (3.452)

1 ω Γ(w1 /ω1 )Γ(w2 /ω1 ). 2π ω1

(3.453)

For vanishing friction, the roots w1 and w2 become simply w1 = iω, w2 = −iω, and the formula9 π Γ(1 − z)Γ(z) = (3.454) sin πz can be used to calculate ω1 π ω1 π Γ(iω/ω1 )Γ(−iω/ω1 ) = = , (3.455) ω sinh(πω/ω1) ω sinh(ω¯ h/2kB T ) showing that (3.450) goes properly over into the partition function (3.214) of the undamped harmonic oscillator. The free energy of the system is F (T ) = −kB T [log(ω/2πω1) − log Γ(ωD /ω1 ) + log Γ(w1 /ω1 ) + log Γ(w2 /ω1 ) + log Γ(w3 /ω1 )] . 8 9

(3.456)

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.322. ibid., Formula 8.334.3. H. Kleinert, PATH INTEGRALS

269

3.15 Harmonic Oscillator in Ohmic Heat Bath

Using the large-z behavior of log Γ(z)10 1 1 1 1 log Γ(z) = z − log z − z + log 2π + − − O(1/z 5 ), 2 2 12z 360z 3

(3.457)

we find the free energy at low temperature 1 1 1 ω2 + + − F (T ) ∼ E0 − w1 w2 w1 w1 w2 w3

!

π γπ (kB T )2 = E0 − 2 (kB T )2 , (3.458) 6¯ h 6ω h ¯

where E0 = −

h ¯ [w1 log(w1 /ωD ) + w2 log(w2 /ωD ) + w3 log(w3 /ωD )] 2π

(3.459)

is the ground state energy. For small friction, this reduces to h ¯ω γ ωD γ2 4ω E0 = + log − 1+ + O(γ 3 ). 2 2π ω 16ω πωD

(3.460)

The T 2 -behavior of F (T ) in Eq. (3.458) is typical for Ohmic dissipation. At zero temperature, the Matsubara frequencies ωm = 2πmkB T /¯ h move arbitrarily close together, so that Matsubara sums become integrals according to the rule Z ∞ 1 X dωm − −−→ . (3.461) T →0 h ¯β m 2π 0

Applying this limiting procedure to the logarithm of the product formula (3.445), the ground state energy can also be written as an integral h ¯ E0 = 2π

Z

0

∞

3 2 ωm + ωm ωD + ωm (ω 2 + γωD ) + ωD ω 2 dωm log , 2 (ω + ω ) ωm m D

"

#

(3.462)

which shows that the energy E0 increases with the friction coefficient γ. It is instructive to calculate the density of states defined in (1.578). Inverting the Laplace transform (1.577), we have to evaluate 1 ρ(ε) = 2πi

Z

η+i∞ η−i∞

dβ eiεβ Zωdamp (β),

(3.463)

where η is an infinitesimally small positive number. In the absence of friction, the P −β¯ hω(n+1/2) integral over Zω (β) = ∞ yields n=0 e ρ(ε) =

∞ X

n=0

δ(ε − (n + 1/2)¯ hω).

(3.464)

In the presence of friction, we expect the sharp δ-function spikes to be broadened. The calculation is done as follows: The vertical line of integration in the complex β-plane in (3.463) is moved all the way to the left, thereby picking up the poles

270


Figure 3.5 Poles in complex β-plane of Fourier integral (3.463) coming from the Gamma functions of (3.450)

of the Gamma functions which lie at negative integer values of wi β/2π. From the representation of the Gamma function11 Γ(z) =

Z

∞

1

dt tz−1e−t +

∞ X

(−1)n n=0 n!(z + n)

(3.465)

we see the size of the residues. Thus we obtain the sum

Rn,1 =

∞ X 3 1 X ρ(ε) = Rn,i e−2πnε/wi , ω n=1 i=1

(3.466)

ω (−1)n−1 Γ(−nw2 /w1)Γ(−nw3 /w1 ) , w12 (n − 1)! Γ(−nωD /w1 )

(3.467)

with analogous expressions for Rn,2 and Rn,3 . The sum can be done numerically and yields the curves shown in Fig. 3.6 for typical underdamped and overdamped situations. There is an isolated δ-function at the ground state energy E0 of (3.459) which is not widened by the friction. Right above E0 , the curve continues from a finite value ρ(E0 + 0) = γπ/6ω 2 determined by the first expansion term in (3.458).

3.16

Harmonic Oscillator in Photon Heat Bath

It is straightforward to extend this result to a photon bath where the spectral density is given by (3.441) and (3.468) becomes Zωdamp 10 11

kB T = h ¯ω

(

∞ Y

m=1

"

2 3 ωm + ω 2 + ωm γ 2 ωm

#)−1

,

(3.468)

ibid., Formula 8.327. ibid., Formula 8.314. H. Kleinert, PATH INTEGRALS

271

3.17 Perturbation Expansion of Anharmonic Systems

3.5 3 ρ(ε) 2.5 2 1.5 1 0.5 0

δ(ε − E0 )

1

2

3

4

5

6

7

8

δ(ε − E0 )

3.5 3 ρ(ε) 2.5 2 1.5 1 0.5 0 ε/ω

1

2

3

4

5

6

7

8

ε/ω

Figure 3.6 Density of states for weak and strong damping in natural units. On the left, the parameters are γ/ω = 0.2, ωD /ω = 10, on the right γ/ω = 5, ωD /ω = 10. For more details see Hanke and Zwerger in Notes and References.

with γ = e2 /6c2 πM. The power of ωm accompanying the friction constant is increased by two units. Adding a Drude correction for the high-frequency behavior we replace γ by ωm /(ωm + ωD ) and obtain instead of (3.445) Zωdamp =

∞ 2 kB T Y ωm (ωm + ωD )(1 + γωD ) . 3 2 ω + ω ω2 + ω ω2 h ¯ ω m=1 ωm (1 + γωD ) + ωm D m D

(3.469)

The resulting partition function has again the form (3.450), except that w123 are the solutions of the cubic equation w 3 (1 + γωD ) − w 2 ωD + wω 2 − ω 2ωD = 0.

(3.470)

Since the electromagnetic coupling is small, we can solve this equation to lowest order in γ. If we also assume ωD to be large compared to ω, we find the roots eff /2 + iω, w1 ≈ γpb

eff w1 ≈ γpb /2 − iω,

eff w3 ≈ ωD /(1 + γpb ωD /ω 2),

(3.471)

where we have introduced an effective friction constant of the photon bath eff γpb =

e2 ω2, 2 6c πM

(3.472)

which has the dimension of a frequency, just as the usual friction constant γ in the previous heat bath equations (3.451).

3.17

Perturbation Expansion of Anharmonic Systems

If a harmonic system is disturbed by an additional anharmonic potential V (x), to be called interaction, the path integral can be solved exactly only in exceptional cases. These will be treated in Chapters 8, 13, and 14. For sufficiently smooth and small potentials V (x), it is always possible to expand the full partition in powers of the interaction strength. The result is the so-called perturbation series. Unfortunately, it only renders reliable numerical results for very small V (x) since, as we shall prove

272


in Chapter 17, the expansion coefficients grow for large orders k like k!, making the series strongly divergent. The can only be used for extremely small perturbations. Such expansions are called asymptotic (more in Subsection 17.10.1). For this reason we are forced to develop a more powerful technique of studying anharmonic systems in Chapter 5. It combines the perturbation series with a variational approach and will yield very accurate energy levels up to arbitrarily large interaction strengths. It is therefore worthwhile to find the formal expansion in spite of its divergence. Consider the quantum-mechanical amplitude (xb tb |xa ta ) =

x(tb )=xb

Z

x(ta )=xa

(

i Dx exp h ¯

Z

M 2 ω2 dt x˙ − M x2 − V (x) 2 2 "

tb

ta

#)

,

(3.473)

and expand the integrand in powers of V (x), which leads to the series Z

(xb tb |xa ta ) =

x(tb )=xb

x(ta )=xa

Dx 1 −

i h ¯

Z

tb ta

dtV (x(t))

tb tb 1 − 2 dt2 V (x(t2 )) dt1 V (x(t1 )) ta 2!¯ h ta Z tb Z tb Z tb i + 3 dt3 V (x(t3 )) dt2 V (x(t2 )) dt1 V (x(t1 )) + . . . ta ta 3!¯ h ta Z t i b M 2 2 2 dt x˙ − ω x . (3.474) × exp h ¯ ta 2

Z

Z

If we decompose the path integral in the nth term into a product (2.4), the expansion can be rewritten as Z i Z tb dt1 dx1 (xb tb |x1 t1 )V (x1 )(xb tb |xa ta ) (3.475) (xb tb |xa ta ) = (xb tb |xa ta ) − h ¯ ta Z tb Z tb Z 1 − 2 dt2 dt1 dx1 dx2 (xb tb |x2 t2 )V (x2 )(x2 t2 |x1 t1 )V (x1 )(x1 t1 |xa ta ) + . . . . ta 2!¯ h ta A similar expansion can be given for the Euclidean path integral of a partition function Z=

I

(

1 Dx exp − h ¯

Z

hβ ¯ 0

M 2 dτ (x˙ + ω 2 x2 ) + V (x) 2

)

,

(3.476)

where we obtain Z =

Z

Dx 1 −

1 h ¯ −

Z

hβ ¯ 0

1 3 3!¯ (h

dτ V (x(τ )) + Z

0

hβ ¯

1 2!¯ h2

dτ3 V (x(τ3 ))

Z

Z

0

hβ ¯

0 hβ ¯

dτ2 V (x(τ2 ))

dτ2 V (x(τ2 ))

1 Z ¯hβ M 2 ω2 × exp − dt x˙ + M x2 h ¯ 0 2 2 "

#)

.

Z

Z

0

hβ ¯

0 hβ ¯

dτ1 V (x(τ1 ))

dτ1 V (x(τ1 )) + . . .

(3.477)

The individual terms are obviously expectation values of powers of the Euclidean interaction Aint,e ≡

Z

0

hβ ¯

dτ V (x(τ )),


273

3.17 Perturbation Expansion of Anharmonic Systems

calculated within the harmonic-oscillator partition function Zω . The expectation values are defined by h. . .iω ≡

Zω−1

Z

(

1 Dx . . . exp − h ¯

Z

hβ ¯

M 2 dτ (x˙ + ω 2x2 ) 2

)

.

(3.479)

1 1 1 2 3 Z = 1 − hAint,e iω + 2 hAint,e iω − 3 hAint,e iω + . . . Zω . h ¯ 2!¯ h 3!¯ h

(3.480)

0

With these, the perturbation series can be written in the form

As we shall see immediately, it is preferable to resum the prefactor into an exponential of a series 1 1 1 2 3 1 − hAint,eiω + 2 hAint,e iω − 3 hAint,e iω + . . . h ¯ 2!¯ h 3!¯ h 1 1 1 2 3 = exp − hAint,e iω + hA iω,c − hA iω,c + . . . . h ¯ 2!¯ h2 int,e 3!¯ h3 int,e

(3.481)

The expectation values hA3int,e iω,c are called cumulants. They are related to the original expectation values by the cumulant expansion:12 hA2int,eiω,c ≡ hA2int,e iω − hAint,e i2ω

(3.482)

hA3int,eiω,c

(3.483)

= h[Aint,e − hAint,e iω ]2 iω , ≡ hA3int,e iω − 3hA2int,e iω hAint,e iω + 2hAint,e i3ω = h[Aint,e − hAint,e iω ]3 iω , .. . .

The cumulants contribute directly to the free energy F = −(1/β) log Z. From (3.481) and (3.480) we conclude that the anharmonic potential V (x) shifts the free hβω/2)] by energy of the harmonic oscillator Fω = (1/β) log[2 sinh(¯ 1 ∆F = β

1 1 1 2 hAint,e iω − hA3 iω,c + . . . . 2 hAint,e iω,c + h ¯ 2!¯ h 3!¯ h3 int,e

D

(3.484)

E

grow for large β with the nth Whereas the original expectation values Anint,e ω power of β, due to contributions of n disconnected diagrams of first D orderE in g which are integrated independently over τ from 0 to h ¯ β, the cumulants Anint,,e are ω proportional to β, thus ensuring that the free energy F has a finite limit, the ground state energy E0 . In comparison with the ground state energy of the unperturbed harmonic system, the energy E0 is shifted by 1 ∆E0 = lim β→∞ β 12

1 1 1 2 hAint,e iω − hA3 iω,c + . . . . 2 hAint,e iω,c + h ¯ 2!¯ h 3!¯ h3 int,e

(3.485)

Note that the subtracted expressions in the second lines of these equations are particularly simple only for the lowest two cumulants given here.

274


There exists a simple functional formula for the perturbation expansion of the partition function in terms of the generating functional Zω [j] of the unperturbed harmonic system. Adding a source term into the action of the path integral (3.476), we define the generating functional of the interacting theory: Z[j] =

(

1 Dx exp − h ¯

I

Z

hβ ¯

0

)

M 2 dτ x˙ + ω 2 x2 + V (x) − jx . 2

(3.486)

The interaction can be brought outside the path integral in the form 1 −h ¯

Z[j] = e

R h¯ β 0

dτ V (δ/δj(τ ))

Zω [j] .

(3.487)

The interacting partition function is obviously Z = Z[0].

(3.488)

Indeed, after inserting on the right-hand side the explicit path integral expression for Z[j] from (3.233): Zω [j] =

Z

(

1 Dx exp − h ¯

Z

0

hβ ¯

)

M 2 (x˙ + ω 2 x2 ) − jx dτ 2

,

(3.489)

and expanding the exponential in the prefactor 1 −h ¯

e

R h¯ β 0

1 = 1− h ¯

dτ V (δ/δj(τ ))

Z

0

hβ ¯

dτ V (δ/δj(τ ))

hβ ¯ hβ ¯ 1 dτ2 V (δ/δj(τ2 )) dτ1 V (δ/δj(τ1 )) (3.490) 2 0 2!¯ h 0 Z ¯hβ Z ¯hβ 1 Z ¯hβ − 3 dτ3 V (δ/δj(τ3 )) dτ2 V (δ/δ(τ2 )) dτ1 V (δ/δ(τ1 )) + . . . , 0 0 3!¯ h 0 the functional derivatives of Z[j] with respect to the source j(τ ) generate inside the path integral precisely the expansion (3.480), whose cumulants lead to formula (3.484) for the shift in the free energy. Before continuing, let us mention that the partition function (3.476) can, of course, be viewed as a generating functional for the calculation of the expectation values of the action and its powers. We simply have to form the derivatives with respect to h ¯ −1 : n n −1 ∂ hA i = Z (3.491) −1 n Zω [j] −1 . ∂¯ h h =0 ¯ For a harmonic oscillator where Z is given by (3.242), this yields

Z

+

hAi = lim Zω−1 h ¯2 h→∞ ¯

Z

∂ h ¯ ωβ Zω = lim h ¯ = 0. h→∞ 2 sinh h ¯ ∂¯ h ¯ ωβ/2

(3.492)

The same result is, incidentally, obtained by calculating the expectation value of the action with analytic regularization: dω ′ ω ′2 dω ′ ω 2 x˙ (τ ) + ω x (τ ) = + = ω ω 2π ω ′2 + ω 2 2π ω ′2 + ω 2 The integral vanishes by Veltman’s rule (2.500). D

2

E

2

D

2

E

Z

Z

Z

dω ′ = 0. (3.493) 2π


3.18 Rayleigh-Schr¨ odinger and Brillouin-Wigner Perturbation Expansion

3.18

275

Rayleigh-Schr¨ odinger and Brillouin-Wigner Perturbation Expansion

The expectation values in formula (3.484) can be evaluated by means of the socalled Rayleigh-Schrödinger perturbation expansion, also referred to as old-fashioned perturbation expansion. This expansion is particularly useful if the potential V (x) is not a polynomial in x. Examples are V (x) = δ(x) and V (x) = 1/x. In these two cases the perturbation expansions can be summed to all orders, as will be shown for the first example in Section 9.5. For the second example the reader is referred to the literature.13 We shall explicitly demonstrate the procedure for the ground state and the excited energies of an anharmonic oscillator. Later we shall also give expansions for scattering amplitudes. To calculate the free-energy shift ∆F in Eq. (3.484) to first order in V (x), we need the expectation hAint,e iω ≡ Zω−1

Z

hβ ¯ 0

dτ1

Z

dxdx1 (x h ¯ β|x1 τ1 )ω V (x1 )(x1 τ1 |x 0)ω .

(3.494)

The time evolution amplitude on the right describes the temporal development of the harmonic oscillator located initially at the point x, from the imaginary time 0 up to τ1 . At the time τ1 , the state is subject to the interaction depending on its position x1 = x(τ1 ) with the amplitude V (x1 ). After that, the state is carried to the final state at the point x by the other time evolution amplitude. To second order we have to calculate the expectation in V (x): 1 2 hA iω ≡ Zω−1 2 int,e

Z

0

hβ ¯

dτ2

Z

0

hβ ¯

dτ1

Z

dxdx2 dx1 (x h ¯ β|x2 τ2 )ω V (x2 )

×(x2 τ2 |x1 τ1 )ω V (x1 )(x1 τ1 |x 0)ω .

(3.495)

The integration over τ1 is taken only up to τ2 since the contribution with τ1 > τ2 would merely render a factor 2. The explicit evaluation of the integrals is facilitated by the spectral expansion (2.293). The time evolution amplitude at imaginary times is given in terms of the eigenstates ψn (x) of the harmonic oscillator with the energy En = h ¯ ω(n + 1/2): (xb τb |xa τa )ω =

∞ X

ψn (xb )ψn∗ (xa )e−En (τb −τa )/¯h .

(3.496)

n=0

The same type of expansion exists also for the real-time evolution amplitude. This leads to the Rayleigh-Schrödinger perturbation expansion for the energy shifts of all excited states, as we now show. The amplitude can be projected onto the eigenstates of the harmonic oscillator. For this, the two sides are multiplied by the harmonic wave functions ψn∗ (xb ) and 13

M.J. Goovaerts and J.T. Devreese, J. Math. Phys. 13, 1070 (1972).

276


ψn (xa ) of quantum number n and integrated over xb and xa , respectively, resulting in the expansion Z

dxb dxa ψn∗ (xb )(xb tb |xa ta )ψn (xa ) =

Z

dxb dxa ψn∗ (xb )(xb tb |xa ta )ω ψn (xa )

i 1 i 2 3 × 1 + hn|Aint|niω − 2 hn|Aint |niω − 3 hn|Aint |niω + . . . , h ¯ 2!¯ h 3!¯ h with the interaction

Aint ≡ −

Z

tb

ta

dt V (x(t)).

(3.497)

(3.498)

The expectation values are defined by hn| . . . |niω ≡

−1 ZQM,ω,n

Z

Z

dxb dxa ψn∗ (xb )

x(tb )=xb

x(ta )=xa

iAω /¯ h

Dx . . . e

!

ψn (xa ), (3.499)

where ZQM,ω,n ≡ e−iω(n+1/2)(tb −ta )

(3.500)

is the projection of the quantum-mechanical partition function of the harmonic oscillator ∞ ZQM,ω =

X

e−iω(n+1/2)(tb −ta )

n=0

[see (2.40)] onto the nth excited state. The expectation values are calculated as in (3.494), (3.495). To first order in V (x), one has hn|Aint |niω ≡

−1 −ZQM,ω,n

Z

tb

ta

dt1

Z

dxb dxa dx1 ψn∗ (xb )(xb tb |x1 t1 )ω × V (x1 )(x1 t1 |xa ta )ω ψn (xa ).

(3.501)

The time evolution amplitude on the right-hand side describes the temporal development of the initial state ψn (xa ) from the time ta to the time t1 , where the interaction takes place with an amplitude −V (x1 ). After that, the time evolution amplitude on the left-hand side carries the state to ψn∗ (xb ). To second order in V (x), the expectation value is given by the double integral tb t2 1 −1 hn|A2int|niω ≡ ZQM,ω,n dt2 dt1 dxb dxa dx2 dx1 2 ta ta ×ψn∗ (xb )(xb tb |x2 t2 )ω V (x2 )(x2 t2 |x1 t1 )ω V (x1 )(x1 t1 |xa ta )ω ψn (xa ).

Z

Z

Z

(3.502)

As in (3.495), the integral over t1 ends at t2 . By analogy with (3.481), we resum the corrections in (3.497) to bring them into the exponent: 1 i i 2 3 1 + hn|Aint |niω − (3.503) 2 hn|Aint |niω − 3 hn|Aint |niω + . . . h ¯ 2!¯ h 3!¯ h i 1 i 2 3 = exp hn|Aint|niω − hn|Aint|niω,c − hn|Aint |niω,c + . . . . h ¯ 2!¯ h2 3!¯ h3 H. Kleinert, PATH INTEGRALS

3.18 Rayleigh-Schr¨ odinger and Brillouin-Wigner Perturbation Expansion

277

The cumulants in the exponent are hn|A2int |niω,c ≡ = 3 hn|Aint |niω,c ≡ = .. . .

hn|A2int |niω − hn|Aint |ni2ω hn|[Aint − hn|Aint |niω ]2 |niω , (3.504) 3 2 3 hn|Aint |niω − 3hn|Aint|niω hn|Aint |niω + 2hn|Aint |niω hn|[Aint − hn|Aint |niω ]3 |niω , (3.505)

From (3.503), we obtain the energy shift of the nth oscillator energy i¯ h ∆En = lim tb −ta →∞ tb − ta

i 1 2 hn|Aint |niω − 2 hn|Aint |niω,c h ¯ 2!¯ h i 3 − 3 hn|Aint|niω,c + . . . , 3!¯ h

(3.506)

which is a generalization of formula (3.485) which was valid only for the ground state energy. At n = 0, the new formula goes over into (3.485), after the usual analytic continuation of the time variable. The cumulants can be evaluated further with the help of the real-time version of the spectral expansion (3.496): (xb tb |xa ta )ω =

∞ X

ψn (xb )ψn∗ (xa )e−iEn (tb −ta )/¯h .

(3.507)

n=0

To first order in V (x), it leads to hn|Aint |niω ≡ −

Z

tb

ta

dt

Z

dxψn∗ (x)V (x)ψn (x) ≡ −(tb − ta )Vnn .

(3.508)

To second order in V (x), it yields tb t2 1 −1 hn|A2int|niω ≡ ZQM,ω,n dt2 dt1 2 ta ta X × e−iEn (tb −t2 )/¯h−iEk (t2 −t1 )/¯h−iEn (t1 −ta )/¯h Vnk Vkn .

Z

Z

(3.509)

k

The right-hand side can also be written as Z

tb

ta

dt2

Z

t2

ta

dt1

X

ei(En −Ek )t2 /¯h+i(Ek −En )t1 /¯h Vnk Vkn

(3.510)

k

and becomes, after the time integrations, −

X k

h i Vnk Vkn h ¯2 i¯ h(tb − ta )− ei(En −Ek )(tb −ta )/¯h −1 . Ek − En En − Ek (

)

(3.511)

278


As it stands, the sum makes sense only for the Ek 6= En -terms. In these, the second term in the curly brackets can be neglected in the limit of large time differences tb − ta . The term with Ek = En must be treated separately by doing the integral directly in (3.510). This yields Vnn Vnn

(tb − ta )2 , 2

(3.512)

so that X Vnm Vmn 1 (tb − ta )2 i¯ h(tb − ta ) + Vnn Vnn hn|A2int |niω = − . 2 2 m6=n Em − En

(3.513)

The same result could have been obtained without the special treatment of the Ek = En -term by introducing artificially an infinitesimal energy difference Ek − En = ǫ in (3.511), and by expanding the curly brackets in powers of tb − ta . When going over to the cumulants 12 hn|A2int |niω,c according to (3.504), the k = n term is eliminated and we obtain X Vnk Vkn 1 hn|A2int |niω,c = − i¯ h(tb − ta ). 2 k6=n Ek − En

(3.514)

For the energy shifts up to second order in V (x), we thus arrive at the simple formula ∆1 En + ∆2 En = Vnn −

Vnk Vkn . k6=n Ek − En X

(3.515)

The higher expansion coefficients become rapidly complicated. The correction of third order in V (x), for example, is ∆3 En =

X Vnk Vkl Vln Vnk Vkn − Vnn . 2 k6=n l6=n (Ek − En )(El − En ) k6=n (Ek − En ) XX

(3.516)

For comparison, we recall the well-known formula of Brillouin-Wigner equation 14 ¯ nn (En + ∆En ), ∆En = R

(3.517)

ˆ¯ ¯ nn (E) are the diagonal matrix elements hn|R(E)|ni where R of the level shift operator ˆ ¯ R(E) which solves the integral equation 1 − Pˆn ˆ¯ ˆ¯ R(E) = Vˆ + Vˆ R(E). ˆω E−H

(3.518)

The operator Pˆn ≡ |nihn| is the projection operator onto the state |ni. The factors 1 − Pˆn ensure that the sums over the intermediate states exclude the quantum 14

L. Brillouin and E.P. Wigner, J. Phys. Radium 4, 1 (1933); M.L. Goldberger and K.M. Watson, Collision Theory, John Wiley & Sons, New York, 1964, pp. 425–430. H. Kleinert, PATH INTEGRALS

3.19 Level-Shifts and Perturbed Wave Functions from Schr¨ odinger Equation

279

number n of the state under consideration. The integral equation is solved by the series expansion in powers of Vˆ : 1 − Pˆn ˆ 1 − Pˆn ˆ 1 − Pˆn ˆ ˆ¯ R(E) = Vˆ + Vˆ V + Vˆ V V + ... . ˆω ˆω E − H ˆω E−H E −H

(3.519)

Up to the third order in Vˆ , Eq. (3.517) leads to the Brillouin-Wigner perturbation expansion E − En = Rnn (E) = Vnn +

XX Vnk Vkn Vnk Vkl Vln + + . . . , (3.520) E − Ek k6=n l6=n (E − Ek )(E − El ) k6=n X

which is an implicit equation for ∆En = E − En . The Brillouin-Wigner equation (3.517) may be converted into an explicit equation for the level shift ∆En : ′ ′ 2 ′′ ∆En = Rnn (En ) + Rnn (En )Rnn (En )+[Rnn (En )Rnn (En )2 + 12 Rnn (En )Rnn (En )] ′ 3 2 ′ ′′ 3 ′′′ 3 1 +[Rnn (En )Rnn (En ) + 2 Rnn (En )Rnn (En )Rnn (En )+ 6 Rnn (En )Rnn (En )]+ . . . .(3.521)

Inserting (3.520) on the right-hand side, we recover the standard RayleighSchrödinger perturbation expansion of quantum mechanics,which coincides precisely with the above perturbation expansion of the path integral whose first three terms were given in (3.515) and (3.516). Note that starting from the third order, the explicit solution (3.521) for the level shift introduces more and more extra disconnected terms with respect to the simple systematics in the Brillouin-Wigner expansion (3.520). For arbitrary potentials, the calculation of the matrix elements Vnk can become quite tedious. A simple technique to find them is presented in Appendix 3A. The calculation of the energy shifts for the particular interaction V (x) = gx4 /4 is described in Appendix 3B. Up to order g 3 , the result is h ¯ω g (2n + 1) + 3(2n2 + 2n + 1)a4 2 4 2 g 1 − 2(34n3 + 51n2 + 59n + 21)a8 4 h ¯ω 3 g 1 + 4 · 3(125n4 + 250n3 + 472n2 + 347n + 111)a12 2 2 . 4 h ¯ ω

∆En =

(3.522)

The perturbation series for this as well as arbitrary polynomial potentials can be carried out to high orders via recursion relations for the expansion coefficients. This is done in Appendix 3C.

3.19

Level-Shifts and Perturbed Wave Functions from Schr¨ odinger Equation

It is instructive to rederive the perturbation expansion from ordinary operator Schrödinger theory. This derivation provides us also with the perturbed eigenstates to any desired order.

280


ˆ is split into a free and an interacting part The Hamiltonian operator H ˆ =H ˆ 0 + Vˆ . H

(3.523)

ˆ 0 and |ψ (n) i those of H: ˆ Let |ni be the eigenstates of H ˆ 0 |ni = E (n) |ni, H 0

ˆ (n) i = E (n) |ψ (n) i. H|ψ

(3.524)

We shall assume that the two sets of states |ni and |ψ (n) i are orthogonal sets, the first with unit norm, the latter normalized by scalar products (n) a(n) i = 1. n ≡ hn|ψ

(3.525)

Due to the completeness of the states |ni, the states |ψ (n) i can be expanded as X |ψ (n) i = |ni + a(n) m |mi,

(3.526)

m6=n

where

(n) a(n) i m ≡ hm|ψ

(3.527)

are the components of the interacting states in the free basis. Projecting the right-hand Schrödinger equation in (3.524) onto hm| and using (3.527), we obtain (m) (n) am

E0

+ hm|Vˆ |ψ (n) i = E (n) a(n) m .

(3.528)

Inserting here (3.526), this becomes (m) (n) am

E0

+ hm|Vˆ |ni +

X

(n)

k6=n

ak hm|Vˆ |ki = E (n) a(n) m ,

and for m = n, due to the special normalization (3.525), X (n) (n) E0 + hn|Vˆ |ni + ak hn|Vˆ |ki = E (n) .

(3.529)

(3.530)

k6=n

(n)

Multiplying this equation with am and subtracting it from (3.529), we eliminate the unknown (n) exact energy E (n) , and obtain a set of coupled algebraic equations for am :   X (n) 1 ˆ ˆ  hm − a(n) ak hm − a(n) (3.531) a(n) m n|V |ki , m n|V |ni + m = (n) (m) E0 − E0 k6=n (n)

(n)

where we have introduced the notation hm − am n| for the combination of states hm| − am hn| , for brevity. This equation can now easily be solved perturbatively order by order in powers of the inter(n) action strength. To count these, we replace Vˆ by g Vˆ and expand am as well as the energies E (n) in powers of g as: a(n) m (g) =

∞ X

(n)

am,l (−g)l

l=1

(m 6= n),

(3.532)

(n)

(3.533)

and (n)

E (n) = E0

−

∞ X

(−g)l El .

l=1


281

3.20 Calculation of Perturbation Series via Feynman Diagrams

Inserting these expansions into (3.530), and equating the coefficients of g, we immediately find the perturbation expansion of the energy of the nth level (n)

E1

(n)

El

= hn|Vˆ |ni, X (n) = ak,l−1 hn|Vˆ |ki

(3.534) l > 1.

(3.535)

k6=n

(n)

The expansion coefficients am,l are now determined by inserting the ansatz (3.532) into (3.531). This yields (n)

am,1 = and for l > 1: (n) am,l =

hm|Vˆ |ni

(m)

E0

(n)

− E0

,

(3.536)





l−2 X (n) X X (n) (n) ˆ |ki− −a(n) hn|Vˆ |ni+ a hm| V a ak,l−1−l′ hn|Vˆ |ki. ′ m,l−1 k,l−1 m,l (m) (n) E0 −E0 ′ k6=n l =1 k6=n

1

(3.537)

Using (3.534) and (3.535), this can be simplified to   l−1 X (n) X 1 (n) (n) (n)  am,l = (m) ak,l−1 hm|Vˆ |ki − am,l′ El−l′  . (n) E0 −E0 ′ k6=n l =1

(3.538)

Together with (3.534), (3.535), and (3.536), this is a set of recursion relations for the coefficients (n) (n) am,l and El . The recursion relations allow us to recover the perturbation expansions (3.515) and (3.516) for the energy shift. The second-order result (3.515), for example, follows directly from (3.537) and (3.538), the latter giving (n)

E2

=

X

k6=n

(n)

ak,1 hn|Vˆ |ki =

X hk|Vˆ |nihn|Vˆ |ki (k)

k6=n

(n)

E0 − E0

.

(3.539)

If the potential Vˆ = V (ˆ x) is a polynomial in x ˆ, its matrix elements hn|Vˆ |ki are nonzero only for n in a finite neighborhood of k, and the recursion relations consist of finite sums which can be solved exactly.

3.20

Calculation of Perturbation Series via Feynman Diagrams

The expectation values in formula (3.484) can be evaluated also in another way which can be applied to all potentials which are simple polynomials pf x. Then the partition function can be expanded into a sum of integrals associated with certain Feynman diagrams. The procedure is rooted in the Wick expansion of correlation functions in Section 3.10. To be specific, we assume the anharmonic potential to have the form g V (x) = x4 . 4

(3.540)

The graphical expansion terms to be found will be typical for all so-called ϕ4 theories of quantum field theory.

282


To calculate the free energy shift (3.484) to first order in g, we have to evaluate the harmonic expectation of Aint,e . This is written as g hAint,e iω = 4

hβ ¯

Z

0

dτ hx4 (τ )iω .

(3.541)

The integrand contains the correlation function (4)

hx(τ1 )x(τ2 )x(τ3 )x(τ4 )iω = Gω2 (τ1 , τ2 , τ3 , τ4 ) at identical time arguments. According to the Wick rule (3.302), this can be expanded into the sum of three pair terms (2)

(2)

(2)

(2)

(2)

(2)

Gω2 (τ1 , τ2 )Gω2 (τ3 , τ4 ) + Gω2 (τ1 , τ3 )Gω2 (τ2 , τ4 ) + Gω2 (τ1 , τ4 )Gω2 (τ2 , τ3 ), (2)

where Gω2 (τ, τ ′ ) are the periodic Euclidean Green functions of the harmonic oscillator [see (3.301) and (3.248)]. The expectation (3.541) is therefore equal to the integral Z g ¯hβ (2) dτ Gω2 (τ, τ )2 . (3.542) hAint,e iω = 3 4 0 The right-hand side is pictured by the Feynman diagram . 3

Because of its shape this is called a two-loop diagram. In general, a Feynman diagram consists of lines meeting at points called vertices. A line connecting two points (2) represents the Green function Gω2 (τ1 , τ2 ). A vertex indicates a factor g/4¯ h and a variable τ to be integrated over the interval (0, h ¯ β). The present simple diagram has only one point, and the τ -arguments of the Green functions coincide. The number underneath counts how often the integral occurs. It is called the multiplicity of the diagram. To second order in V (x), the harmonic expectation to be evaluated is hA2int,e iω

2 Z

g = 4

0

hβ ¯

dτ2

Z

hβ ¯

0

dτ1 hx4 (τ2 )x4 (τ1 )iω .

(3.543)

(8)

The integral now contains the correlation function Gω2 (τ1 , . . . , τ8 ) with eight time arguments. According to the Wick rule, it decomposes into a sum of 7!! = 105 (2) products of four Green functions Gω2 (τ, τ ′ ). Due to the coincidence of the time arguments, there are only three different types of contributions to the integral (3.543): hA2int,e iω

2 Z

g = 4

0

hβ ¯

dτ2

Z

0

hβ ¯

h

(2)

(2)

(2)

dτ1 72Gω2 (τ2 , τ2 )Gω2 (τ2 , τ1 )2 Gω2 (τ1 , τ1 )

(2)

(2)

(2)

i

+24Gω2 (τ2 , τ1 )4 + 9Gω2 (τ2 , τ2 )2 Gω2 (τ1 , τ1 )2 .

(3.544)

The integrals are pictured by the following Feynman diagrams composed of three loops: H. Kleinert, PATH INTEGRALS

283


. 72

24

9

They contain two vertices indicating two integration variables τ1 , τ2 . The first two diagrams with the shape of three bubbles in a chain and of a watermelon, respectively, are connected diagrams, the third is disconnected . When going over to the cumulant hA2int,e iω,c , the disconnected diagram is eliminated. To higher orders, the counting becomes increasingly tedious and it is worth developing computer-algebraic techniques for this purpose. Figure 3.7 shows the diagrams for the free-energy shift up to four loops. The cumulants eliminate precisely all disconnected diagrams. This diagram-rearranging property of the logarithm is very general and happens to every order in g, as can be shown with the help of functional differential equations.

βF = βFω + 3

1 + 3!

1 − 2!

!

+ 72

24

+ 2592

+ 1728

!

+ 3456

+ ...

1728

Figure 3.7 Perturbation expansion of free energy up to order g3 (four loops).

The lowest-order term βFω containing the free energy of the harmonic oscillator [recall Eqs. (3.242) and (2.518)] 1 β¯ hω Fω = log 2 sinh β 2

!

(3.545)

is often represented by the one-loop diagram 1 1 (2) βFω = − Tr log Gω2 = − 2 2¯ hβ

Z

hβ ¯ 0

h

(2)

i

dτ log Gω2 (τ, τ ) = −

1 2

.

(3.546)

With it, the graphical expansion in Fig. 3.7 starts more systematically with one loop rather than two. The systematics is, however, not perfect since the line in the one-loop diagram does not show that integrand contains a logarithm. In addition, the line is not connected to any vertex. All τ -variables in the diagrams are integrated out. The diagrams have no open lines and are called vacuum diagrams. The calculation of the diagrams in Fig. 3.7 is simplified with the help of a factorization property: If a diagram consists of two subdiagrams touching each other

284


at a single vertex, its Feynman integral factorizes into those of the subdiagrams. Thanks to this property, we only have to evaluate the following integrals (omitting the factors g/4¯ h for each vertex) =

Z

hβ ¯

(2)

dτ Gω2 (τ, τ ) = h ¯ βa2 ,

0

=

Z

¯ βZ ¯ h hβ

=

Z

hβZ ¯

0

0

0

0

≡ h ¯β

1 (2) dτ1 dτ2 Gω2 (τ1 , τ2 )2 ≡ h ¯ β a42 , ω

¯ βZ ¯ h hβ

1 ω

0

2

hβZ ¯ ¯ hβ

=

Z

¯ βZ h h ¯ βZ ¯ hβ

0

0

≡ h ¯β =

Z

0

1 ω

0

2

≡ h ¯β

0

1 ω

0

2

(2)

(2)

(2)

dτ1 dτ2 dτ3 Gω2 (τ1 , τ2 )Gω2 (τ2 , τ3 )Gω2 (τ3 , τ1 )3

a10 3 ,

¯ βZ h h ¯ βZ ¯ hβ

(2)

1 (2) dτ1 dτ2 Gω2 (τ1 , τ2 )4 ≡ h ¯ β a82 , ω

Z

0

(2)

a63 ,

=

0

(2)

dτ1 dτ2 dτ3 Gω2 (τ1 , τ2 )Gω2 (τ2 , τ3 )Gω2 (τ3 , τ1 )

(2)

(2)

(2)

dτ1 dτ2 dτ3 Gω2 (τ1 , τ2 )2 Gω2 (τ2 , τ3 )2 Gω2 (τ3 , τ1 )2

a12 3 .

(3.547)

Note that in each expression, the last τ -integral yields an overall factor h ¯ β, due to the translational invariance along the τ -axis. The others give rise to a factor 1/ω, for dimensional reasons. The temperature-dependent quantities a2L V are labeled by the number of vertices V and lines L of the associated diagrams. Their dimension is length to the nth power [corresponding to the dimension of the n x(τ )-variables in the diagram]. For more than four loops, there can be more than one diagram for each V and L, such that one needs an additional label in a2L V to specify the diagram may be written as a product of the basic length scale (¯ h/Mω)L uniquely. Each a2L V multiplied by a function of the dimensionless variable x ≡ β¯ hω: a2L V

=

h ¯ Mω

!L

αV2L (x).

(3.548)

The functions αV2L (x) are listed in Appendix 3D. As an example for the application of the factorization property, take the Feynman integral of the second third-order diagram in Fig. 3.7 (called a “daisy” diagram H. Kleinert, PATH INTEGRALS

285


because of its shape):

=

Z

0

¯ βZ h h ¯ βZ ¯ hβ 0

0

(2)

(2)

(2)

dτ1 dτ2 dτ3 Gω2 (τ1 , τ2 )Gω2 (τ2 , τ3 )Gω2 (τ3 , τ1 ) (2)

(2)

(2)

× Gω2 (τ1 , τ1 )Gω2 (τ2 , τ2 )Gω2 (τ3 , τ3 ). It decomposes into a product between the third integral in (3.547) and three powers of the first integral: →

×

3

.

Thus we can immediately write = h ¯β

1 ω

2

a63 (a2 )3 .

In terms of a2L V , the free energy becomes g 4 1 g 2 2 4 2 72a a2 a + 24a82 (3.549) 3a − 4 2!¯ hω 4 3 h i 1 g 2 12 + 2 2 2592a2 (a42 )2 a2 + 1728a63 (a2 )3 + 3456a10 a + 1728a + ... . 3 3 3!¯ hω 4

F = Fω +

In the limit T → 0, the integrals (3.547) behave like a42

→ a4 ,

a82

→

a12 3

a63

1 8 a, 2 3 12 → a , 8

a10 3

3 6 a, 2 5 10 → a , 8 →

(3.550)

and the free energy reduces to F =

2

h ¯ω g 4 g + 3a − 2 4 4

42a8

3

1 g + h ¯ω 4

4 · 333a12

1 h ¯ω

2

+ ... .

(3.551)

In this limit, it is simpler to calculate the integrals (3.547) directly with the zero′ (2) temperature limit of the Green function (3.301), which is Gω2 (τ, τ ′ ) = a2 e−ω|τ −τ | with a2 = h ¯ /2ωM [see (3.248)]. The limits of integration must, however, be shifted R ¯hβ/2 R∞ by half a period to −¯hβ/2 dτ before going to the limit, so that one evaluates −∞ dτ R∞ rather than 0 dτ (the latter would give the wrong limit since it misses the left-hand side of the peak at τ = 0). Before integration, the integrals are conveniently split as in Eq. (3D.1).

286

3.21


Perturbative Definition of Interacting Path Integrals

In Section 2.15 we have seen that it is possible to define a harmonic path integral without time slicing by dimensional regularization. With the techniques developed so far, this definition can trivially be extended to path integrals with interactions, if these can be treated perturbatively. We recall that in Eq. (3.480), the partition function of an interacting system can be expanded in a series of harmonic expectation values of powers of the interaction. The procedure is formulated most conveniently in terms of the generating functional (3.486) using formula (3.487) for the generating functional with interactions and Eq. (3.488) for the associated partition. The harmonic generating functional on the right-hand side of (3.487), Zω [j] =

I

(

1 Dx exp − h ¯

Z

0

hβ ¯

)

M 2 dτ (x˙ + ω 2x2 ) − jx 2

,

(3.552)

can be evaluated with analytic regularization as described in Section (2.15) and yields, after a quadratic completion [recall (3.243), (3.244)]: (

)

1 1 Z ¯hβ Z ¯hβ ′ Zω [j] = exp dτ dτ j(τ )Gpω2 ,e (τ − τ ′ )j(τ ′ ) , (3.553) 2 sin(ω¯ hβ/2) 2M¯ h 0 0 where Gpω2 ,e (τ ) is the periodic Green function (3.248) Gpω2 ,e (τ ) =

1 cosh ω(τ − h ¯ β/2) , 2ω sinh(β¯ hω/2)

τ ∈ [0, h ¯ β].

(3.554)

As a consequence, Formula (3.487) for the generating functional of an interacting theory 1

Z[j] = e− h¯

R h¯ β 0

dτ V (¯ hδ/δj(τ ))

Zω [j] ,

(3.555)

is completely defined by analytic regularization. By expanding the exponential prefactor as in Eq. (3.490), the full generating functional is obtained from the harmonic one without any further path integration. Only functional differentiations are required to find the generating functional of all interacting interacting correlation functions Z[j] from the harmonic one Zω [j]. This procedure yields the perturbative definition of arbitrary path integrals. It is widely used in the quantum field theory of particle physics15 and critical phenomena16 It is also the basis for an important extension of the theory of distributions to be discussed in detail in Sections 10.6–10.11. It must be realized, however, that the perturbative definition is not a complete definition. Important contributions to the path integral may be missing: all those 15

C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill (1985). H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories, World Scientific, Singapore 2001, pp. 1–487 (www.physik.fu-berlin.de/~kleinert/re.html#b8). 16


287

3.22 Generating Functional of Connected Correlation Functions

which are not expandable in powers of the interaction strength g. Such contributions are essential in understanding many physical phenomena, for example, tunneling, to be discussed in Chapter 17. Interestingly, however, information on such phenomena can, with appropriate resummation techniques to be developed in Chapter 5, also be extracted from the large-order behavior of the perturbation expansions, as will be shown in Subsection 17.10.4.

3.22

Generating Functional of Connected Correlation Functions

In Section 3.10 we have seen that the correlation functions obtained from the functional derivatives of Z[j] via relation (3.295) contain many disconnected parts. The physically relevant free energy F [j] = −kB T log Z[j], on the other hand, contains only in the connected parts of Z[j]. In fact, from statistical mechanics we know that meaningful description of a very large thermodynamic system can only be given in terms of the free energy which is directly proportional to the total volume V . The partition function Z = e−F/kB T has no meaningful infinite-volume limit, also called the thermodynamic limit, since it contains a power series in V . Only the free energy density f ≡ F/V has an infinite-volume limit. The expansion of Z[j] diverges therefore for V → ∞. This is why in thermodynamics we always go over to the free energy density by taking the logarithm of the partition function. This is calculated entirely from the connected diagrams. Due to this thermodynamic experience we expect the logarithm of Z[j] to provide us with a generating functional for all connected correlation functions. To avoid factors kB T we define this functional as W [j] = log Z[j],

(3.556)

and shall now prove that the functional derivatives of W [j] produce precisely the connected parts of the Feynman diagrams for each correlation function. Consider the connected correlation functions G(n) c (τ1 , . . . , τn ) defined by the functional derivatives δ δ G(n) ··· W [j] . (3.557) c (τ1 , . . . , τn ) = δj(τ1 ) δj(τn ) Ultimately, we shall be interested only in these functions with zero external current, where they reduce to the physically relevant connected correlation functions. For the general development in this section, however, we shall consider them as functionals of j(τ ), and set j = 0 only at the end. Of course, given all connected correlation functions G(n) c (τ1 , . . . , τn ), the full (n) correlation functions G (τ1 , . . . , τn ) in Eq. (3.295) can be recovered via simple composition laws from the connected ones. In order to see this clearly, we shall derive the general relationship between the two types of correlation functions in Section 3.22.2. First, we shall prove the connectedness property of the derivatives (3.557).

288

3.22.1


Connectedness Structure of Correlation Functions

We first prove that the generating functional W [j] collects only connected diagrams in its Taylor coefficients δ n W/δj(τ1 ) . . . δj(τn ). Later, after Eq. (3.585), we shall see that these functional derivatives comprise all connected diagrams in G(n) (τ1 , . . . , τn ). Let us write the path integral for the generating functional Z[j] as follows (here we use natural units with h ¯ = 1): Z[j] =

Z

Dx e−Ae [x,j]/¯h,

(3.558)

with the action Ae [x, j] =

Z

hβ ¯

0

M 2 2 2 dτ x˙ + ω x + V (x) − j(τ )x(τ ) . 2

(3.559)

In the following structural considerations we shall use natural physical units in which h ¯ = 1, for simplicity of the formulas. By analogy with the integral identity Z

dx

d −F (x) = 0, e dx

which holds by partial integration for any function F (x) which goes to infinity for x → ±∞, the functional integral satisfies the identity Z

Dx

δ e−Ae [x,j] = 0, δx(τ )

(3.560)

since the action Ae [x, j] goes to infinity for x → ±∞. Performing the functional derivative, we obtain Z

Dx

δAe [x, j] −Ae [x,j] e = 0. δx(τ )

(3.561)

To be specific, let us consider the anharmonic oscillator with potential V (x) = λx4 /4!. We have chosen a coupling constant λ/4! instead of the previous g in (3.540) since this will lead to more systematic numeric factors. The functional derivative of the action yields the classical equation of motion δAe [x, j] λ = M(−¨ x + ω 2x) + x3 − j = 0, δx(τ ) 3!

(3.562)

which we shall write as δAe [x, j] λ 3 = G−1 x − j = 0, 0 x+ δx(τ ) 3!

(3.563)

where we have set G0 (τ, τ ′ ) ≡ G(2) to get free space for upper indices. With this notation, Eq. (3.561) becomes Z

Dx

(

G−1 0 x(τ )

)

λ + x3 (τ ) − j(τ ) e−Ae [x,j] = 0. 3!

(3.564)



289

We now express the paths x(τ ) as functional derivatives with respect to the source current j(τ ), such that we can pull the curly brackets in front of the integral. This leads to the functional differential equation for the generating functional Z[j]:  

"

 

(3.565)

δ δ δ ··· Z[j], δj(τ1 ) δj(τ2 ) δj(τn )

(3.566)

λ δ δ G−1 +  0 δj(τ ) 3! δj(τ )

#3

− j(τ ) Z[j] = 0.

With the short-hand notation

Zj(τ1 )j(τ2 )...j(τn ) [j] ≡



where the arguments of the currents will eventually be suppressed, this can be written as G−1 0 Zj(τ ) +

λ Zj(τ )j(τ )j(τ ) − j(τ ) = 0. 3!

(3.567)

Inserting here (3.556), we obtain a functional differential equation for W [j]: G−1 0 Wj +

λ Wjjj + 3Wjj Wj + Wj3 − j = 0. 3!

(3.568)

We have employed the same short-hand notation for the functional derivatives of W [j] as in (3.566) for Z[j], Wj(τ1 )j(τ2 )...j(τn ) [j] ≡

δ δ δ ··· W [j], δj(τ1 ) δj(τ2 ) δj(τn )

(3.569)

suppressing the arguments τ1 , . . . , τn of the currents, for brevity. Multiplying (3.568) functionally by G0 gives λ Wj = − G0 Wjjj + 3Wjj Wj + Wj3 + G0 j. 3!

(3.570)

We have omitted the integral over the intermediate τ ’s, for brevity. More specifically, R we have written G0 j for dτ ′ G0 (τ, τ ′ )j(τ ′ ). Similar expressions abbreviate all functional products. This corresponds to a functional version of Einstein’s summation convention. Equation (3.570) may now be expressed in terms of the one-point correlation function G(1) c = Wj ,

(3.571)

defined in (3.557), as G(1) c

h i λ (1) (1) (1) 3 = − G0 Gc jj + 3Gc j G(1) + G + G0 j. c c 3!

(3.572)

290


The solution to this equation is conveniently found by a diagrammatic procedure displayed in Fig. 3.8. To lowest, zeroth, order in λ we have G(1) c = G0 j.

(3.573)

From this we find by functional integration the zeroth order generating functional W0 [j] W0 [j] =

Z

1 Dj G(1) c = jG0 j, 2

(3.574)

up to a j-independent constant. Subscripts of W [j] indicate the order in the interaction strength λ. Reinserting (3.573) on the right-hand side of (3.572) gives the first-order expression G(1) = −G0 c

i λh 3G0 G0 j + (G0 j)3 + G0 j, 3!

(3.575)

represented diagrammatically in the second line of Fig. 3.8. Equation (3.575) can be integrated functionally in j to obtain W [j] up to first order in λ. Diagrammatically, this process amounts to multiplying each open lines in a diagram by a current j, and dividing the arising j n s by n. Thus we arrive at W0 [j] + W1 [j] =

1 λ λ jG0 j − G0 (G0 j)2 − (G0 j)4 , 2 4 24

(3.576)

Figure 3.8 Diagrammatic solution of recursion relation (3.570) for the generating functional W [j] of all connected correlation functions. First line represents Eq. (3.572), second (3.575), third (3.576). The remaining lines define the diagrammatic symbols. H. Kleinert, PATH INTEGRALS

291


as illustrated in the third line of Fig. 3.8. This procedure can be continued to any order in λ. The same procedure allows us to prove that the generating functional W [j] collects only connected diagrams in its Taylor coefficients δ n W/δj(x1 ) . . . δj(xn ). For the lowest two orders we can verify the connectedness by inspecting the third line in Fig. 3.8. The diagrammatic form of the recursion relation shows that this topological property remains true for all orders in λ, by induction. Indeed, if we suppose it to be true for some n, then all G(1) c inserted on the right-hand side are connected, and so are the diagrams constructed from these when forming G(1) c to the next, (n + 1)st, order. Note that this calculation is unable to recover the value of W [j] at j = 0 which is an unknown integration constant of the functional differential equation. For the purpose of generating correlation functions, this constant is irrelevant. We have seen in Fig. 3.7 that W [0], which is equal to −F/kB T , consists of the sum of all connected vacuum diagrams contained in Z[0].

3.22.2

Correlation Functions versus Connected Correlation Functions

Using the logarithmic relation (3.556) between W [j] and Z[j] we can now derive general relations between the n-point functions and their connected parts. For the one-point function we find G(1) (τ ) = Z −1 [j]

δ δ Z[j] = W [j] = G(1) c (τ ). δj(τ ) δj(τ )

(3.577)

This equation implies that the one-point function representing the ground state expectation value of the path x(τ ) is always connected: hx(τ )i ≡ G(1) (τ ) = G(1) c (τ ) = X.

(3.578)

Consider now the two-point function, which decomposes as follows: δ δ Z[j] δj(τ1 ) δj(τ2 ) ( ! ) δ δ −1 = Z [j] W [j] Z[j] δj(τ1 ) δj(τ2 )

G(2) (τ1 , τ2 ) = Z −1 [j]

n

o

= Z −1 [j] Wj(τ1 )j(τ2 ) + Wj(τ1 ) Wj(τ2 ) Z[j] (1) (1) = G(2) c (τ1 , τ2 ) + Gc (τ1 ) Gc (τ2 ) .

(3.579)

In addition to the connected diagrams with two ends there are two connected diagrams ending in a single line. These are absent in a x4 -theory at j = 0 because of the symmetry of the potential, which makes all odd correlation functions vanish. In that case, the two-point function is automatically connected.

292


For the three-point function we find δ δ δ Z[j] δj(τ1 ) δj(τ2 ) δj(τ3 ) (" # ) δ δ δ W [j] Z[j] = Z −1 [j] δj(τ1 ) δj(τ2 ) δj(τ3 ) i o δ nh = Z −1 [j] Wj(τ3 )j(τ2 ) + Wj(τ2 ) Wj(τ3 ) Z[j] δj(τ1 )

G(3) (τ1 , τ2 , τ3 ) = Z −1 [j]

n

(3.580)

= Z −1 [j] Wj(τ1 )j(τ2 )j(τ3 ) + Wj(τ1 ) Wj(τ2 )j(τ3 ) + Wj(τ2 ) Wj(τ1 )j(τ3 )

o

+ Wj(τ3 ) Wj(τ1 )j(τ2 ) + Wj(τ1 ) Wj(τ2 ) Wj(τ3 ) Z[j]

h

i

(1) (2) (1) (1) (1) = G(3) c (τ1 , τ2 , τ3 ) + Gc (τ1 )Gc (τ2 , τ3 ) + 2 perm + Gc (τ1 )Gc (τ2 )Gc (τ3 ),

and for the four-point function h

(3) (1) G(4) (τ1 , . . . , τ4 ) = G(4) c (τ1 , . . . , τ4 ) + Gc (τ1 , τ2 , τ3 ) Gc (τ4 ) + 3 perm

h

(2) + G(2) c (τ1 , τ2 ) Gc (τ3 , τ4 ) + 2 perm

h

i

(1) (1) + G(2) c (τ1 , τ2 ) Gc (τ3 )Gc (τ4 ) + 5 perm (1) + G(1) c (τ1 ) · · · Gc (τ4 ).

i

i

(3.581)

In the pure x4 -theory there are no odd correlation functions, because of the symmetry of the potential. For the general correlation function G(n) , the total number of terms is most easily retrieved by dropping all indices and differentiating with respect to j (the arguments τ1 , . . . , τn of the currents are again suppressed):

G(1) = e−W eW

G(2) = e−W eW

G(3) = e−W eW

G(4) = e−W eW

j

= Wj = G(1) c

jj

(1) 2 = Wjj + Wj 2 = G(2) c + Gc

jjj

(2) (1) (1)3 = Wjjj + 3Wjj Wj + Wj 3 = G(3) c + 3Gc Gc + Gc

jjjj

= Wjjjj + 4Wjjj Wj + 3Wjj 2 + 6Wjj Wj 2 + Wj 4 (3) (1) (2)2 (1)2 = G(4) + 6G(2) + G(1)4 c + 4Gc Gc + 3Gc c Gc c .

(3.582)

All equations follow from the recursion relation (n−1)

G(n) = G j

+ G(n−1) G(1) c ,

n ≥ 2,

(3.583)

(n−1)

if one uses Gc j = G(n) and the initial relation G(1) = G(1) c c . By comparing the first four relations with the explicit expressions (3.579)–(3.581) we see that the numerical factors on the right-hand side of (3.582) refer to the permutations of the arguments τ1 , τ2 , τ3 , . . . of otherwise equal expressions. Since there is no problem in H. Kleinert, PATH INTEGRALS


293

reconstructing the explicit permutations we shall henceforth write all composition laws in the short-hand notation (3.582). The formula (3.582) and its generalization is often referred to as cluster decomposition, or also as the cumulant expansion, of the correlation functions. We can now prove that the connected correlation functions collect precisely all connected diagrams in the n-point functions. For this we observe that the decomposition rules can be inverted by repeatedly differentiating both sides of the equation W [j] = log Z[j] functionally with respect to the current j: G(1) c G(2) c G(3) c G(4) c

= = = =

G(1) G(2) − G(1) G(1) G(3) − 3G(2) G(1) + 2G(1)3 G(4) − 4G(3) G(1) + 12G(2) G(1)2 − 3G(2)2 − 6G(1)4 .

(3.584)

Each equation follows from the previous one by one more derivative with respect to j, and by replacing the derivatives on the right-hand side according to the rule (n)

G j = G(n+1) − G(n) G(1) .

(3.585)

Again the numerical factors imply different permutations of the arguments and the subscript j denotes functional differentiations with respect to j. Note that Eqs. (3.584) for the connected correlation functions are valid for symmetric as well as asymmetric potentials V (x). For symmetric potentials, the equations simplify, since all terms involving G(1) = X = hxi vanish. It is obvious that any connected diagram contained in G(n) must also be contained (n) in G(n) c , since all the terms added or subtracted in (3.584) are products of G j s, and thus necessarily disconnected. Together with the proof in Section 3.22.1 that the contain only the connected parts of G(n) , we can now be correlation functions G(n) c sure that G(n) contains precisely the connected parts of G(n) . c

3.22.3

Functional Generation of Vacuum Diagrams

The functional differential equation (3.570) for W [j] contains all information on the connected correlation functions of the system. However, it does not tell us anything about the vacuum diagrams of the theory. These are contained in W [0], which remains an undetermined constant of functional integration of these equations. In order to gain information on the vacuum diagrams, we consider a modification of the generating functional (3.558), in which we set the external source j equal to zero, but generalize the source j(τ ) in (3.558) coupled linearly to x(τ ) to a bilocal form K(τ, τ ′ ) coupled linearly to x(τ )x(τ ′ ): Z[K] =

Z

where Ae [x, K] is the Euclidean action Ae [x, K] ≡ A0 [x] + Aint [x] +

1 2

Dx(τ ) e−Ae [x,K],

(3.586)

Z

(3.587)

dτ

Z

dτ ′ x(τ )K(τ, τ ′ )x(τ ′ ).

294


When forming the functional derivative with respect to K(τ, τ ′ ) we obtain the correlation function in the presence of K(τ, τ ′ ): G(2) (τ, τ ′ ) = −2Z −1 [K]

δZ . δK(τ, τ ′ )

(3.588)

At the end we shall set K(τ, τ ′ ) = 0, just as previously the source j. When differentiating Z[K] twice, we obtain the four-point function G(4) (τ1 , τ2 , τ3 , τ4 ) = 4Z −1 [K]

δ2Z . δK(τ1 , τ2 )δK(τ3 , τ4 )

(3.589)

As before, we introduce the functional W [K] ≡ log Z[K]. Inserting this into (3.588) and (3.589), we find δW , (3.590) δK(τ, τ ′ ) " # δ2W δW δW (4) + . (3.591) G (τ1 , τ2 , τ3 , τ4 ) = 4 δK(τ1 , τ2 )δK(τ3 , τ4 ) δK(τ1 , τ2 ) δK(τ3 , τ4 ) G(2) (τ, τ ′ ) = 2

With the same short notation as before, we shall use again a subscript K to denote functional differentiation with respect to K, and write G(2) = 2WK ,

G(4) = 4 [WKK + WK WK ] = 4WKK + G(2) G(2) .

(3.592)

From Eq. (3.582) we know that in the absence of a source j and for a symmetric potential, G(4) has the connectedness structure (2) (2) G(4) = G(4) c + 3Gc Gc .

(3.593)

This shows that in contrast to Wjjjj , the derivative WKK does not directly yield a connected four-point function, but two disconnected parts: (2) (2) 4WKK = G(4) c + 2Gc Gc ,

(3.594)

the two-point functions being automatically connected for a symmetric potential. More explicitly, (3.594) reads 4δ 2 W δK(τ1 , τ2 )δK(τ3 , τ4 ) (2) (2) (2) (2) = G(4) c (τ1 , τ2 , τ3 , τ4 ) + Gc (τ1 , τ3 )Gc (τ2 , τ4 ) + Gc (τ1 , τ4 )Gc (τ2 , τ3 ). (3.595) Let us derive functional differential equations for Z[K] and W [K]. By analogy with (3.560) we start out with the trivial functional differential equation Z

Dx x(τ )

δ e−Ae [x,K] = −δ(τ − τ ′ )Z[K], δx(τ ′ )



295

which is immediately verified by a functional integration by parts. Performing the functional derivative yields Z

Dx x(τ )

δAe [x, K] −Ae [x,K] e = δ(τ − τ ′ )Z[K], δx(τ ′ )

(3.597)

or Z

Dx

Z

dτ

Z

dτ

′

(

′ ′ x(τ )G−1 0 (τ, τ )x(τ )

)

λ + x(τ )x3 (τ ′ ) e−Ae [x,K] = δ(τ − τ ′ )Z[K]. 3! (3.598)

For brevity, we have absorbed the source in the free-field correlation function G0 : −1 G0 → [G−1 0 − K] .

(3.599)

The left-hand side of (3.598) can obviously be expressed in terms of functional derivatives of Z[K], and we obtain the functional differential equation whose short form reads λ 1 G−1 (3.600) 0 ZK + ZKK = Z. 3 2 Inserting Z[K] = eW [K], this becomes λ 1 G−1 0 WK + (WKK + WK WK ) = . 3 2 It is useful to reconsider the functional W [K] as a functional W [G0 ]. δG0 /δK = G20 , and the derivatives of W [K] become WK = G20 WG0 ,

WKK = 2G30 WG0 + G40 WG0 G0 ,

(3.601) Then (3.602)

and (3.601) takes the form λ 1 G0 WG0 + (G40 WG0 G0 + 2G30 WG0 + G40 WG0 WG0 ) = . 3 2

(3.603)

This equation is represented diagrammatically in Fig. 3.9. The zeroth-order solution

G0 WG0 = 8

1 −1 4 λG0 WG0 G0 + 2G0 λG20 WG0 + WG0 G20 λG20 WG0 + 4! 2

Figure 3.9 Diagrammatic representation of functional differential equation (3.603). For the purpose of finding the multiplicities of the diagrams, it is convenient to represent here by a vertex the coupling strength −λ/4! rather than g/4 in Section 3.20.

296


to this equation is obtained by setting λ = 0: 1 W (0) [G0 ] = Tr log(G0 ). (3.604) 2 Explicitly, the right-hand side is equal to the one-loop contribution to the free energy in Eq. (3.546), apart from a factor −β. The corrections are found by iteration. For systematic treatment, we write W [G0 ] as a sum of a free and an interacting part, W [G0 ] = W (0) [G0 ] + W int [G0 ],

(3.605)

insert this into Eq. (3.603), and find the differential equation for the interacting part: λ −λ 2 G0 WGint0 + (G40 WGint0 G0 + 3G30 WGint0 + G40 WGint0 WGint0 ) = 6 G. 3 4! 0

(3.606)

This equation is solved iteratively. Setting W int [G0 ] = 0 in all terms proportional to λ, we obtain the first-order contribution to W int [G0 ]: −λ 2 (3.607) G. 4! 0 This is precisely the contribution (3.542) of the two-loop Feynman diagram (apart from the different normalization of g). In order to see how the iteration of Eq. (3.606) may be solved systematically, let us ignore for the moment the functional nature of Eq. (3.606), and treat G0 as an ordinary real variable rather than a functional matrix. We expand W [G0 ] in a Taylor series: !p ∞ X 1 −λ int Wp W [G0 ] = (G0 )2p , (3.608) p! 4! p=1 W int [G0 ] = 3

and find for the expansion coefficients the recursion relation  

Wp+1 = 4 [2p (2p − 1) + 3(2p)] Wp + 

p−1 X q=1

p q

!

 

2q Wq × 2(p − q)Wp−q . (3.609) 

Solving this with the initial number W1 = 3, we obtain the multiplicities of the connected vacuum diagrams of pth order: 3, 96, 9504, 1880064, 616108032, 301093355520, 205062331760640, 185587468924354560, 215430701800551874560, 312052349085504377978880.(3.610) To check these numbers, we go over to Z[G] = eW [G0 ] , and find the expansion: 

∞ X 1 1 −λ Z[G0 ] = exp  Tr log G0 + Wp 2 4! p=1 p!



∞ X

1 −λ = Det1/2 [G0 ] 1 + zp 4! p=1 p!

!p

!p



(G0 )2p  

(G0 )2p  .

(3.611)


297


The expansion coefficients zp count the total number of vacuum diagrams of order p. The exponentiation (3.611) yields zp = (4p − 1)!!, which is the correct number of Wick contractions of p interactions x4 . In fact, by comparing coefficients in the two expansions in (3.611), we may derive another recursion relation for Wp : !

!

p−1 p−1 p−1 Wp + 3 Wp−1 + 7·5 ·3 + . . . + (4p −5)!! 1 2 p−1

!

= (4p −1)!!, (3.612)

which is fulfilled by the solutions of (3.609). In order to find the associated Feynman diagrams, we must perform the differentiations in Eq. (3.606) functionally. The numbers Wp become then a sum of diagrams, for which the recursion relation (3.609) reads Wp+1=



4G40

p−1 X d d2 3 W + 3 · G W + p p 0 d ∩2 d∩ q=1

p q

!

!

!

d d Wq G20 · G20 Wp−q  , d∩ d∩

(3.613)

where the differentiation d/d∩ removes one line connecting two vertices in all possible ways. This equation is solved diagrammatically, as shown in Fig. 3.10.

Wp+1 = 4

"

G40

# p−1 X p d d2 d d 3 2 2 Wp + 3 · G0 Wp + Wq G0 · G0 Wp−q q d ∩2 d∩ d∩ d∩ q=1

Figure 3.10 Diagrammatic representation of recursion relation (3.609). A vertex represents the coupling strength −λ.

Starting the iteration with W1 = 3 , we have dWp /d ∩ = 6 and 2 d Wp /d ∩ = 6 . Proceeding to order five loops and going back to the usual vertex notation −λ, we find the vacuum diagrams with their weight factors as shown in Fig. 3.11. For more than five loops, the reader is referred to the paper quoted in Notes and References, and to the internet address from which Mathematica programs can be downloaded which solve the recursion relations and plot all diagrams of W [0] and the resulting two- and four-point functions. 2

3.22.4

Correlation Functions from Vacuum Diagrams

The vacuum diagrams contain information on all correlation functions of the theory. One may rightly say that the vacuum is the world. The two- and four-point functions are given by the functional derivatives (3.592) of the vacuum functional W [K]. Diagrammatically, a derivative with respect to K corresponds to cutting one line of a vacuum diagram in all possible ways. Thus, all diagrams of the two-point function G(2) can be derived from such cuts, multiplied by a factor 2. As an example, consider

298


diagrams and multiplicities g1 g

1 2!

2

g3

g

3

4

1 3!

gg q

q + 72 24 q l

g q q q q l 1728 Tm + 3456 q + 2592 q

ggg q q

!

g q gggg q q q + 1728 gq gq g

!

i g g q q gq q q ql q qg q qg 1 q m g q q q q q l q 62208 q q + 66296 q gq + 248832 q + 497664 T + 165888 + 248832 l q q 4! g ! qg g qg qgq g qg ggggg q q q q ggg q q g q + 248832 165888 q l qg + 62208 q + 124416 qg

Figure 3.11 Vacuum diagrams up to five loops and their multiplicities. The total numbers to orders gn are 3, 96, 9504, 1880064, respectively. In contrast to Fig. 3.10, and to the previous diagrammatic notation in Fig. 3.7, a vertex stands here for −λ/4! for brevity. For more than five loops see the tables on the internet (http://www.physik.fu-berlin/~kleinert/b3/programs).

the first-order vacuum diagram of W [K] in Fig. 3.11. Cutting one line, which is possible in two ways, and recalling that in Fig. 3.11 a vertex stands for −λ/4! rather than −λ, as in the other diagrams, we find W1 [0] =

1 8

− − −→

(2)

G1 (τ1 , τ2 ) = 2 ×

1 2 8

.

(3.614)

The second equation in (3.592) tells us that all connected contributions to the four-point function G(4) may be obtained by cutting two lines in all combinations, and multiplying the result by a factor 4. As an example, take the second-order vacuum diagrams of W [0] with the proper translation of vertices by a factor 4!, which are 1 1 W2 [0] = + . (3.615) 16 48 Cutting two lines in all possible ways yields the following contributions to the connected diagrams of the two-point function: G

(4)

1 1 =4× 2·1· +4·3· 16 48

.

(3.616)

It is also possible to find all diagrams of the four-point function from the vacuum diagrams by forming a derivative of W [0] with respect to the coupling constant −λ, H. Kleinert, PATH INTEGRALS

299


and multiplying the result by a factor 4!. This follows directly from the fact that R this differentiation applied to Z[0] yields the correlation function dτ hx4 i. As an example, take the first diagram of order g 3 in Table 3.11 (with the same vertex convention as in Fig. 3.11): 1 W2 [0] = . (3.617) 48 Removing one vertex in the three possible ways and multiplying by a factor 4! yields G(4) = 4! ×

3.22.5

1 3 48

.

(3.618)

Generating Functional for Vertex Functions. Effective Action

Apart from the connectedness structure, the most important step in economizing the calculation of Feynman diagrams consists in the decomposition of higher connected correlation functions into one-particle irreducible vertex functions and one-particle irreducible two-particle correlation functions, from which the full amplitudes can easily be reconstructed. A diagram is called one-particle irreducible if it cannot be decomposed into two disconnected pieces by cutting a single line. There is, in fact, a simple algorithm which supplies us in general with such a decomposition. For this purpose let us introduce a new generating functional Γ[X], to be called the effective action of the theory. It is defined via a Legendre transformation of W [j]: −Γ[X] ≡ W [j] − Wj j.

(3.619)

Here and in theRfollowing, we use a short-hand notation for the functional multiplication, Wj j = dτ Wj (τ )j(τ ), which considers fields as vectors with a continuous index τ . The new variable X is the functional derivative of W [j] with respect to j(τ ) [recall (3.569)]: X(τ ) ≡

δW [j] ≡ Wj(τ ) = hxij(τ ) , δj(τ )

(3.620)

and thus gives the ground state expectation of the field operator in the presence of the current j. When rewriting (3.619) as −Γ[X] ≡ W [j] − X j,

(3.621)

and functionally differentiating this with respect to X, we obtain the equation ΓX [X] = j.

(3.622)

This equation shows that the physical path expectation X(τ ) = hx(τ )i, where the external current is zero, extremizes the effective action: ΓX [X] = 0.

(3.623)

300


We shall study here only physical systems for which the path expectation value is a constant X(τ ) ≡ X0 . Thus we shall not consider systems which possess a timedependent X0 (τ ), although such systems can also be described by x4 -theories by admitting more general types of gradient terms, for instance x(∂ 2 − k02 )2 x. The ensuing τ -dependence of X0 (τ ) may be oscillatory.17 Thus we shall assume a constant X0 = hxi|j=0 ,

(3.624)

which may be zero or non-zero, depending on the phase of the system. Let us now demonstrate that the effective action contains all the information on the proper vertex functions of the theory. These can be found directly from the functional derivatives: δ δ ... Γ[X] . (3.625) Γ(n) (τ1 , . . . , τn ) ≡ δX(τ1 ) δX(τn ) We shall see that the proper vertex functions are obtained from these functions P by a Fourier transform and a simple removal of an overall factor (2π)D δ ( ni=1 ωi ) to ensure momentum conservation. The functions Γ(n) (τ1 , . . . , τn ) will therefore be called vertex functions, without the adjective proper which indicates the absence of the δ-function. In particular, the Fourier transforms of the vertex functions Γ(2) (τ1 , τ2 ) and Γ(4) (τ1 , τ2 , τ3 , τ4 ) are related to their proper versions by ¯ (2) (ω1 ), Γ(2) (ω1 , ω2 ) = 2πδ (ω1 + ω2 ) Γ Γ(4) (ω1 , ω2 , ω3 , ω4 ) = 2πδ

4 X i=1

(3.626)

!

¯ (4) (ω1 , ω2 , ω3 , ω4 ). ωi Γ

(3.627)

For the functional derivatives (3.625) we shall use the same short-hand notation as for the functional derivatives (3.569) of W [j], setting ΓX(τ1 )...X(τn ) ≡

δ δ ... Γ[X] . δX(τ1 ) δX(τn )

(3.628)

The arguments τ1 , . . . , τn will usually be suppressed. In order to derive relations between the derivatives of the effective action and the connected correlation functions, we first observe that the connected one-point function G(1) c at a nonzero source j is simply the path expectation X [recall (3.578)]: G(1) c = X.

(3.629)

Second, we see that the connected two-point function at a nonzero source j is given by G(2) c

=

(1) Gj

δX = Wjj = = δj

δj δX

!−1

= Γ−1 XX .

(3.630)

17

In higher dimensions there can be crystal- or quasicrystal-like modulations. See, for example, H. Kleinert and K. Maki, Fortschr. Phys. 29, 1 (1981) (http://www.physik.fu-berlin.de/~kleinert/75). This paper was the first to investigate in detail icosahedral quasicrystalline structures discovered later in aluminum. H. Kleinert, PATH INTEGRALS


301

The inverse symbols on the right-hand side are to be understood in the functional sense, i.e., Γ−1 XX denotes the functional matrix: −1 ΓX(τ )X(y)

δ2Γ ≡ δX(τ )δX(τ ′ ) "

#−1

,

(3.631)

which satisfies Z

−1 ′′ dτ ′ ΓX(τ )X(τ ′ ) ΓX(τ ′ )X(τ ′′ = δ(τ − τ ).

(3.632)

Relation (3.630) states that the second derivative of the effective action determines directly the connected correlation function G(2) c (ω) of the interacting theory in the presence of the external source j. Since j is an auxiliary quantity, which eventually be set equal to zero thus making X equal to X0 , the actual physical propagator is given by

G(2) c

j=0

−1 = ΓXX

X=X0

.

(3.633)

By Fourier-transforming this relation and removing a δ-function for the overall momentum conservation, the full propagator Gω2 (ω) is related to the vertex function Γ(2) (ω), defined in (3.626) by 1 ¯ (2) (k) = . Gω2 (ω) ≡ G (2) ¯ Γ (ω)

(3.634)

The third derivative of the generating functional W [j] is obtained by functionally differentiating Wjj in Eq. (3.630) once more with respect to j, and applying the chain rule: Wjjj = −Γ−2 XX ΓXXX

δX (2) 3 = −Γ−3 ΓXXX . XX ΓXXX = −Gc δj

(3.635)

This equation has a simple physical meaning. The third derivative of W [j] on the left-hand side is the full three-point function at a nonzero source j, so that 3

(2) ΓXXX . G(3) c = Wjjj = −Gc

(3.636)

This equation states that the full three-point function arises from a third derivative of Γ[X] by attaching to each derivation a full propagator, apart from a minus sign. We shall express Eq. (3.636) diagrammatically as follows:

where

302


denotes the connected n-point function, and

the negative n-point vertex function. For the general analysis of the diagrammatic content of the effective action, we observe that according to Eq. (3.635), the functional derivative of the correlation function G with respect to the current j satisfies 3

(3) (2) G(2) ΓXXX . c j = Wjjj = Gc = −Gc

(3.637)

This is pictured diagrammatically as follows:

(3.638)

This equation may be differentiated further with respect to j in a diagrammatic way. From the definition (3.557) we deduce the trivial recursion relation G(n) c (τ1 , . . . , τn ) =

δ G(n−1) (τ1 , . . . , τn−1 ) , δj(τn ) c

(3.639)

which is represented diagrammatically as

By applying δ/δj repeatedly to the left-hand side of Eq. (3.637), we generate all higher connected correlation functions. On the right-hand side of (3.637), the chain rule leads to a derivative of all correlation functions G = G(2) with respect to j, c thereby changing a line into a line with an extra three-point vertex as indicated in the diagrammatic equation (3.638). On the other hand, the vertex function ΓXXX must be differentiated with respect to j. Using the chain rule, we obtain for any n-point vertex function: ΓX...Xj = ΓX...XX

δX = ΓX...XX G(2) c , δj

(3.640)

which may be represented diagrammatically as



303

With these diagrammatic rules, we can differentiate (3.635) any number of times, and derive the diagrammatic structure of the connected correlation functions with an arbitrary number of external legs. The result up to n = 5 is shown in Fig. 3.12.

Figure 3.12 Diagrammatic differentiations for deriving tree decomposition of connected correlation functions. The last term in each decomposition yields, after amputation and removal of an overall δ-function of momentum conservation, precisely all one-particle irreducible diagrams.

The diagrams generated in this way have a tree-like structure, and for this reason they are called tree diagrams. The tree decomposition reduces all diagrams to their one-particle irreducible contents. The effective action Γ[X] can be used to prove an important composition theorem: The full propagator G can be expressed as a geometric series involving the so-called self-energy. Let us decompose the vertex function as ¯ (2) = G−1 + Γ ¯ int , Γ 0 XX

(3.641)

304


such that the full propagator (3.633) can be rewritten as ¯ int G = 1 + G0 Γ XX

−1

G0 .

(3.642)

Expanding the denominator, this can also be expressed in the form of an integral equation: ¯ int G0 + G0 Γ ¯ int G0 Γ ¯ int G0 − . . . . G = G0 − G0 Γ XX XX XX

(3.643)

¯ int is called the self-energy, commonly denoted by Σ: The quantity −Γ XX ¯ int , Σ ≡ −Γ XX

(3.644)

i.e., the self-energy is given by the interacting part of the second functional derivative of the effective action, except for the opposite sign. According to Eq. (3.643), all diagrams in G can be obtained from a repetition of self-energy diagrams connected by a single line. In terms of Σ, the full propagator reads, according to Eq. (3.642): −1 G ≡ [G−1 0 − Σ] .

(3.645)

This equation can, incidentally, be rewritten in the form of an integral equation for the correlation function G: G = G0 + G0 ΣG.

3.22.6

(3.646)

Ginzburg-Landau Approximation to Generating Functional

Since the vertex functions are the functional derivatives of the effective action [see (3.625)], we can expand the effective action into a functional Taylor series Γ[X] =

∞ X

1 n=0 n!

Z

dτ1 . . . dτn Γ(n) (τ1 , . . . , τn )X(τ1 ) . . . X(τn ).

(3.647)

The expansion in the number of loops of the generating functional Γ[X] collects systematically the contributions of fluctuations. To zeroth order, all fluctuations are neglected, and the effective action reduces to the initial action, which is the mean-field approximation to the effective action. In fact, in the absence of loop diagrams, the vertex functions contain only the lowest-order terms in Γ(2) and Γ(4) : (2)

Γ0 (τ1 , τ2 ) = M −∂τ21 + ω 2 δ(τ1 −τ2 ),

(3.648)

Γ0 (τ1 , τ2 , τ3 , τ4 ) = λ δ(τ1 −τ2 )δ(τ1 − τ3 )δ(τ1 − τ4 ).

(3.649)

(4)

Inserted into (3.647), this yields the zero-loop approximation to Γ[X]: Γ0 [X] =

M 2!

Z

dτ [(∂τ X)2 + ω 2 X 2 ] +

λ 4!

Z

dτ X 4 .

(3.650)


305


This is precisely the original action functional (3.559). By generalizing X(τ ) to be a magnetization vector field, X(τ ) → M(x), which depends on the three-dimensional space variables x rather than the Euclidean time, the functional (3.650) coincides with the phenomenological energy functional set up by Ginzburg and Landau to describe the behavior of magnetic materials near the Curie point, which they wrote as18 Γ[M] =

Z

3 1X m2 2 λ 4 d3 x (∂i M)2 + M + M . 2 i=1 2! 4!

"

#

(3.651)

The use of this functional is also referred to as mean-field theory or mean-field approximation to the full theory.

3.22.7

Composite Fields

Sometimes it is of interest to study also correlation functions in which two fields coincide at one point, for instance 1 G(1,n) (τ, τ1 , . . . , τn ) = hx2 (τ )x(τ1 ) · · · x(τn )i. 2

(3.652)

If multiplied by a factor Mω 2 , the composite operator Mω 2 x2 (τ )/2 is precisely the frequency term in the action energy functional (3.559). For this reason one speaks of a frequency insertion, or, since in the Ginzburg-Landau action (3.651) the frequency ω is denoted by the mass symbol m, one speaks of a mass insertion into the correlation function G(n) (τ1 , . . . , τn ). Actually, we shall never make use of the full correlation function (3.652), but only of the integral over τ in (3.652). This can be obtained directly from the generating functional Z[j] of all correlation functions by differentiation with respect to the square mass in addition to the source terms Z

dτ G

(1,n)

(τ, τ1 , . . . , τn ) = − Z

−1

∂ δ δ · · · Z[j] . M∂ω 2 δj(τ1 ) δj(τn ) j=0

(3.653)

By going over to the generating functional W [j], we obtain in a similar way the connected parts: Z

dτ

Gc(1,n) (τ, τ1 , . . . , τn )

∂ δ δ =− · · · W [j] . 2 M∂ω δj(τ1 ) δj(τn ) j=0

(3.654)

The right-hand side can be rewritten as Z

dτ G(1,n) (τ, τ1 , . . . , τn ) = − c

∂ G(n) (τ1 , . . . , τn ). M∂ω 2 c

(3.655)

The connected correlation functions G(1,n) (τ, τ1 , . . . , τn ) can be decomposed into c tree diagrams consisting of lines and one-particle irreducible vertex functions 18

L.D. Landau, J.E.T.P. 7 , 627 (1937).

306


Γ(1,n) (τ, τ1 , . . . , τn ). If integrated over τ , these are defined from Legendre transform (3.619) by a further differentiation with respect to Mω 2 : Z

dτ Γ

(1,n)

δ δ ∂ (τ, τ1 , . . . , τn ) = − · · · Γ[X] , M∂ω 2 δX(τ1 ) δX(τn ) X0

(3.656)

implying the relation Z

3.23

dτ Γ(1,n) (τ, τ1 , . . . , τn ) = −

∂ Γ(n) (τ1 , . . . , τn ). M∂ω 2

(3.657)

Path Integral Calculation of Effective Action by Loop Expansion

Path integrals give the most direct access to the effective action of a theory avoiding the cumbersome Legendre transforms. The derivation will proceed diagrammatically loop by loop, which will turn out to be organized by the powers of the Planck constant h ¯ . This will now be kept explicit in all formulas. For later applications to quantum mechanics we shall work with real time.

3.23.1

General Formalism

Consider the generating functional of all Green functions Z[j] = eiW [j]/¯h ,

(3.658)

where W [j] is the generating functional of all connected Green functions. The vacuum expectation of the field, the average X(t) ≡ hx(t)i,

(3.659)

is given by the first functional derivative X(t) = δW [j]/δj(t).

(3.660)

This can be inverted to yield j(t) as a functional of X(t): j(t) = j[X](t),

(3.661)

which leads to the Legendre transform of W [j]: Γ[X] ≡ W [j] −

Z

dt j(t)X(t),

(3.662)

where the right-hand side is replaced by (3.661). This is the effective action of the theory. The effective action for time independent X(t) ≡ X defines the effective potential 1 V eff (X) ≡ − Γ[X]. (3.663) tb − ta H. Kleinert, PATH INTEGRALS

307

3.23 Path Integral Calculation of Effective Action by Loop Expansion

The first functional derivative of the effective action gives back the current δΓ[X] = −j(t). δX(t)

(3.664)

The generating functional of all connected Green functions can be recovered from the effective action by the inverse Legendre transform W [j] = Γ[X] +

Z

dt j(t)X(t).

(3.665)

We now calculate these quantities from the path integral formula (3.558) for the generating functional Z[j]: Z[j] =

Z

Dx(t)e(i/¯h){A[x]+

R

dt j(t)x(t)}

.

(3.666)

With (3.658), this amounts to the path integral formula for Γ[X]: e h¯ {Γ[X]+ i

R

dt j(t)X(t)}

=

Z

Dx(t)e(i/¯h){A[x]+

dt j(t)x(t)}

R

.

(3.667)

The action quantum h ¯ is a measure for the size of quantum fluctuations. Under many physical circumstances, quantum fluctuations are small, which makes it desirable to develop a method of evaluating (3.667) as an expansion in powers of h ¯.

3.23.2

Mean-Field Approximation

For h ¯ → 0, the path integral over the path x(t) in (3.666) is dominated by the classical solution xcl (t) which extremizes the exponent

δA[x] = −j(t), δx(t) x=xcl (t)

(3.668)

and is a functional of j(t) which may be written, more explicitly, as xcl (t)[j]. At this level we can identify W [j] = Γ[X] +

Z

dt j(t)X(t) ≈ A[xcl [j]] +

Z

dt j(t)xcl (t)[j].

(3.669)

By differentiating W [j] with respect to j, we have from the general first part of Eq. (3.659): X=

δW δΓ δX δX = +X +j . δj δX δj δj

(3.670)

Inserting the classical equation of motion (3.668), this becomes X=

δA δxcl δxcl + xcl + j = xcl . δxcl δj δj

(3.671)

308


Thus, to this approximation, X(t) coincides with the classical path xcl (t). Replacing xcl (t) → X(t) on the right-hand side of Eq. (3.669), we obtain the lowest-order result, which is of zeroth order in h ¯ , the classical approximation to the effective action: Γ0 [X] = A[X].

(3.672)

For an anharmonic oscillator in N dimensions with unit mass and an interaction x4 , where x = (x1 , . . . , xN ), which is symmetric under N-dimensional rotations O(N), the lowest-order effective action reads Γ0 [X] =

Z

1 ˙2 g 2 2 dt Xa − ω 2 Xa2 − Xa , 2 4!

(3.673)

where repeated indices a, b, . . . are summed from 1 to N following Einstein’s summation convention. The effective potential (3.663) is simply the initial potential V0eff (X) = V (X) =

ω 2 2 g 2 2 X + Xa . 2 a 4!

(3.674)

For ω 2 > 0, this has a minimum at X ≡ 0, and there are only two non-vanishing vertex functions Γ(n) (t1 , . . . , tn ): For n = 2: Γ(2) (t1 , t2 )ab ≡

δ2Γ δ2A = δXa (t1 )δXb (t2 ) Xa =0 xa (t1 )xb (t2 ) xa =Xa =0

= (−∂t2 − ω 2)δab δ(t1 − t2 ).

(3.675)

This determines the inverse of the propagator: Γ(2) (t1 , t2 )ab = [i¯ hG−1 ]ab (t1 , t2 ).

(3.676)

Thus we find to this zeroth-order approximation that Gab (t1 , t2 ) is equal to the free propagator: Gab (t1 , t2 ) = G0ab (t1 , t2 ). (3.677) For n = 4: Γ(4) (t1 , t2 , t3 , t4 )abcd ≡

δ4Γ = gTabcd , δXa (t1 )δXb (t2 )δXc (t3 )δXd (t4 )

(3.678)

with

1 Tabcd = (δab δcd + δac δbd + δad δbc ). (3.679) 3 According to the definition of the effective action, all diagrams of the theory can be composed from the propagator Gab (t1 , t2 ) and this vertex via tree diagrams. Thus we see that in this lowest approximation, we recover precisely the subset of H. Kleinert, PATH INTEGRALS

309


all original Feynman diagrams with a tree-like topology. These are all diagrams which do not involve any loops. Since the limit h ¯ → 0 corresponds to the classical equations of motion with no quantum fluctuations we conclude: Classical theory corresponds to tree diagrams. For ω 2 < 0 the discussion is more involved since the minimum of the potential (3.674) lies no longer at X = 0, but at a nonzero vector X0 with an arbitrary direction, and a length q (3.680) |X0| = −6ω 2 /g. The second functional derivative (3.675) at X is anisotropic and reads ω2 > 0

V eff (X)

ω2 < 0

V eff (X)

X2

X2 X1

X1

Figure 3.13 Effective potential for ω 2 > 0 and ω 2 < 0 in mean-field approximation, pictured for the case of two components X1 , X2 . The right-hand figure looks like a Mexican hat or a champaign bottle..

(2)

Γ (t1 , t2 )ab

δ2Γ δ2A ≡ = δXa (t1 )δXb (t2 ) Xa 6=0 xa (t1 )xb (t2 ) xa =Xa 6=0 g 2 2 2 = −∂t − ω − δab Xc + 2Xa Xb δab δ(t1 − t2 ). 6

(3.681)

This is conveniently separated into longitudinal and transversal derivatives with ˆ = X/|X|. We introduce associated projection matrices: respect to the direction X ˆ =X âX ˆb, PLab (X)

ˆ = δab − X âX ˆb , PT ab (X)

(3.682)

and decompose (2) ˆ + Γ(2) (t1 , t2 )ab PT ab (X), ˆ Γ(2) (t1 , t2 )ab = ΓL (t1 , t2 )ab PLab (X) T

where (2) ΓL (t1 , t2 )ab

=

−∂t2

g − ω + X2 6

2

δ(t1 − t2 ),

(3.683) (3.684)

310


and

g (2) ΓL (t1 , t2 )ab = −∂t2 − ω 2 + 3 X2 6 This can easily be inverted to find the propagator

h

G(t1 , t2 )ab = i¯ h Γ(2) (t1 , t2 )

i−1 ab

δ(t1 − t2 ).

ˆ ˆ = GL (t1 , t2 )ab PLab (X)+G T (t1 , t2 )ab PT ab (X),

(3.685)

(3.686)

where i¯ h i¯ h = , 2 ΓL (t1 , t2 ) −∂t − ωL2 (X) i¯ h i¯ h = = 2 (2) −∂t − ωT2 (X) ΓT (t1 , t2 )

GL (t1 , t2 )ab =

(3.687)

GT (t1 , t2 )ab

(3.688)

are the longitudinal and transversal parts of the Green function. For convenience, we have introduced the X-dependent frequencies of the longitudinal and transversal Green functions: g g ωL2 (X) ≡ ω 2 + 3 X2 , ωT2 (X) ≡ ω 2 + X2 . 6 6

(3.689)

To emphasize the fact that this propagator is a functional of X we represent it by the calligraphic letter G. For ω 2 > 0, we perform the fluctuation expansion around the minimum of the potential (3.663) at X = 0, where the two Green functions coincide, both having the same frequency ω: GL (t1 , t2 )ab |X=0 = GT (t1 , t2 )ab |X=0 = G(t1 , t2 )ab |X=0 =

i¯ h , − ω2

−∂t2

(3.690)

For ω 2 < 0, however, where the minimum lies at the vector X0 of length (3.680), they are different: GL (t1 , t2 )ab |X=X0 =

i¯ h i¯h , G (t , t ) | = . T 1 2 ab X=X 0 −∂t2 + 2ω 2 −∂t2

(3.691)

Since the curvature of the potential at the minimum in radial direction of X is positive at the minimum, the longitudinal part has now the positive frequency −2ω 2 . The movement along the valley of the minimum, on the other hand, does not increase the energy. For this reason, the transverse part has zero frequency. This feature, observed here in lowest order of the fluctuation expansion, is a very general one, and can be found in the effective action to any loop order. In quantum field theory, there exists a theorem asserting this called Nambu-Goldstone theorem. It states that if a quantum field theory without long-range interactions has a continuous symmetry which is broken by a nonzero expectation value of the field corresponding to the present X [recall (3.659)], then the fluctuations transverse to it have a zero mass. They are called Nambu-Goldstone modes or, because of their bosonic nature, Nambu-Goldstone bosons. The exclusion of long-range interactions is necessary, H. Kleinert, PATH INTEGRALS

311


since these can mix with the zero-mass modes and make it massive. This happens, for example, in a superconductor where they make the magnetic field massive, giving it a finite penetration depth, the famous Meissner effect. One expresses this pictorially by saying that the long-range mode can eat up the Nambu-Goldstone modes and become massive. The same mechanism is used in elementary particle physics to explain the mass of the W ± and Z 0 vector bosons as a consequence of having eaten up a would be Nambu-Goldstone boson of an auxiliary Higgs-field theory. In quantum-mechanical systems, however, a nonzero expectation value with the associated zero frequency mode in the transverse direction is found only as an artifact of perturbation theory. If all fluctuation corrections are summed, the minimum of the effective potential lies always at the origin. For example, it is well known, that the ground state wave functions of a particle in a double-well potential is symmetric, implying a zero expectation value of the particle position. This symmetry is caused by quantum-mechanical tunneling, a phenomenon which will be discussed in detail in Chapter 17. This phenomenon is of a nonperturbative nature which cannot be described by an effective potential calculated order by order in the fluctuation expansion. Such a potential does, in general, posses a nonzero minimum at some X0 somewhere near the zero-order minimum (3.680). Due to this shortcoming, it is possible to derive the Nambu-Goldstone theorem from the quantum-mechanical effective action in the loop expansion, even though the nonzero expectation value X0 assumed in the derivation of the zero-frequency mode does not really exist in quantum mechanics. The derivation will be given in Section 3.24. The use of the initial action to approximate the effective action neglecting corrections caused by the fluctuations is referred to as mean-field approximation.

3.23.3

Corrections from Quadratic Fluctuations

In order to find the first h ¯ -correction to the mean-field approximation we expand the action in powers of the fluctuations of the paths around the classical solution δx(t) ≡ x(t) − xcl (t),

(3.692)

and perform a perturbation expansion. The quadratic term in δx(t) is taken to be the free-particle action, the higher powers in δx(t) are the interactions. Up to second order in the fluctuations δx(t), the action is expanded as follows: A[xcl + δx] +

Z

dt j(t) [xcl (t) + δx(t)]

= A[xcl ] +

Z

dt j(t) xcl (t) +

Z

dt

(



 δA δx(t) j(t) + δx(t) x=xcl 

δ2A 1Z ′ 3 + dt dt′ δx(t) δx(t ) + O (δx) . 2 δx(t)δx(t′ ) x=xcl

(3.693)

The curly bracket multiplying the linear terms in the variation δx(t) vanish due to the extremality property of the classical path xcl expressed by the equation of

312


motion (3.668). Inserting this expansion into (3.667), we obtain the approximate expression  i Z



 δ2A ′ Z[j] ≈ e(i/¯h){A[xcl ]+ dt j(t)xcl (t)} Dδx exp dt dt′ δx(t) δx(t ) . h  ¯ δx(t)δx(t′ ) x=xcl (3.694) √ ¯ due to the We now observe that the fluctuations δx(t) will be of average size h n h ¯ -denominator in the Fresnel exponent. Thus the fluctuations (δx) are of average √ n size h ¯ . The approximate path integral (3.694) is of the Fresnel type and my be integrated to yield Z

R

(i/¯ h){A[xcl ]+

e

R

dt j(t)xcl (t)}

= e(i/¯h){A[xcl ]+

R

δ2A det δx(t)δx(t′ )

"

#−1/2

(3.695)

x=xcl

dt j(t)xcl (t)+i(¯ h/2)Tr log[δ2 A/δx(t)δx(t′ )|x=xcl }

.

Comparing this with the left-hand side of (3.667), we find that to first order in h ¯, the effective action may be recovered by equating Γ[X] +

Z

dt j(t)X(t) = A[xcl [j]] +

Z

dt j(t) xcl (t)[j] +

i¯ h δ 2 A [xcl [j]] Tr log . (3.696) 2 δx(t)δx(t′ )

In the limit h ¯ → 0, the tracelog term disappears and (3.696) reduces to the classical expression (3.669). To include the h ¯ -correction into Γ[X], we expand W [j] as W [j] = W0 [j] + h ¯ W1 [j] + O(¯ h2 ).

(3.697)

¯: Correspondingly, the path X differs from Xcl by a correction term of order h X = xcl + h ¯ X1 + O(¯ h2 ).

(3.698)

Inserting this into (3.696), we find Z

Γ[X] + dt jX = A [X −¯ hX1 ] + i + h ¯ Tr log 2

Z

dt jX − h ¯

δ 2 A δxa δxb

Z

x=X−¯ hX1

dt jX1

+O h ¯2 .

(3.699)

Expanding the action up to the same order in h ¯ gives Γ[X] = A[X] − h ¯

Z

δA[X] i δ 2 A dt + j X1 + h ¯ Tr log + O h ¯ 2 . (3.700) δX 2 δxa δxb x=X (

)

Due to (3.668), the curly-bracket term is only of order h ¯ 2 , so that we find the oneloop form of the effective action

1 ˙ 2 ω 2 2 g 2 2 Γ[X] = Γ0 [X] + h ¯ Γ1 [X] = dt X − Xa − Xa 2 2 4! i g 2 2 2 + h ¯ Tr log −∂t − ω − δab Xc + 2Xa Xb . (3.701) 2 6 Z

"

#



313

Using the decomposition (3.683), the tracelog term can be written as a sum of transversal and longitudinal parts i i (2) (2) h ¯ Tr log ΓL (t1 , t2 )ab + (N − 1)¯ h Tr log ΓT (t1 , t2 )ab (3.702) 2 2 i i = h ¯ Tr log −∂t2 − ωL2 (X) + (N − 1)¯ h Tr log −∂t2 − ωT2 (X) . 2 2

h ¯ Γ1 [X] =

What is the graphical content in the Green functions at this level of approxima¯ = 0, as in tion? Assuming ω 2 > 0, we find for j = 0 that the minimum lies at X the mean-field approximation. Around this minimum, we may expand the tracelog in powers of X. For the simplest case of a single X-variable, we obtain i g i i i X2 h ¯ Tr log −∂t2 −ω 2 − X 2 = h ¯ Tr log −∂t2 −ω 2 + h ¯ Tr log 1+ 2 ig 2 2 2 2 −∂t −ω 2 2 ! n ∞ i h ¯ h ¯X g n1 Tr = i Tr log −∂t2 − ω 2 − i −i X2 . (3.703) 2 2 2 n=1 2 n −∂t − ω 2

!

If we insert G0 =

−∂t2

i , − ω2

(3.704)

this can be written as ∞ h ¯X g h ¯ −i i Tr log −∂t2 − ω 2 − i 2 2 n=1 2

n

n 1 Tr G0 X 2 . n

(3.705)

More explicitly, the terms with n = 1 and n = 2 read: h ¯ − g dt dt′ δ(t − t′ )G0 (t, t′ )X 2 (t′ ) 4 Z g2 +i¯ h dt dt′ dt′′ δ 4 (t − t′′ )G0 (t, t′ )X 2 (t′ )G0 (t′ , t′′ )X 2 (t′′ ) + . . . . 16 Z

(3.706)

The expansion terms of (3.705) for n ≥ 1 correspond obviously to the Feynman diagrams (omitting multiplicity factors) (3.707)

A[xcl ] =

The series (3.705) is therefore a sum of all diagrams with one loop and any number of fundamental X 4 -vertices To systematize the entire expansion (3.705), the tracelog term is [compare (3.546)] pictured by a single-loop diagram h ¯ 1 i Tr log −∂t2 − ω 2 = 2 2

.

(3.708)

314


The first two diagrams in (3.707) contribute corrections to the vertices Γ(2) and Γ(4) . The remaining diagrams produce higher vertex functions and lead to more involved tree diagrams. In Fourier space we find from (3.706) i g dk (3.709) 2 2 2π k − ω 2 + iη "Z # dk g2 i i (4) + 2 perm . Γ (qi ) = g − i 2 2π k 2 − ω 2 + iη (q1 + q2 − k)2 − ω 2 + iη (3.710) Γ(2) (q) = q 2 − ω 2 − h ¯

Z

We may write (3.709) in Euclidean form as 1 dk 2 2π k + ω 2 g 1 = − q2 + ω2 + h ¯ , 2 2ω g2 Γ(4) (qi ) = g − h ¯ [I (q1 + q2 ) + 2 perm] , 2 Γ(2) (q) = −q 2 − ω 2 − h ¯

g 2

Z

(3.711) (3.712)

with the Euclidean two-loop integral I(q1 + q2 ) =

dk i 1 , 2 2 2π k + ω (q1 + q2 − k)2 + ω 2

Z

(3.713)

to be calculated explicitly in Chapter 10. It is equal to J((q1 + q2 )2 )/2π with the functions J(z) of Eq. (10.258). ¯ 6= 0, the expansion For ω 2 < 0 where the minimum of the effective action lies at X of the trace of the logarithm in (3.701) must distinguish longitudinal and transverse parts.

3.23.4

-2 Effective Action to Order h

Let us now find the next correction to the effective action.19 Instead of truncating the expansion (3.693), we keep all terms, reorganizing only the linear and quadratic terms as in (3.694). This yields e(i/¯h){Γ[X]+jX} = ei(¯h/2)W [j] = e(i/¯h){(A[xcl ]+jxcl)+(i¯h/2)Tr log Axx [xcl ]} e(i/¯h)¯h

2

W2 [xcl ]

. (3.714)

The functional W2 [xcl ] is defined by the path integral over the fluctuations e(i/¯h)¯h 19

2

W2 [xcl ]

=

R

Dx exp ¯hi R

n

1 δxD[xcl ]δx 2

Dδx exp ¯hi

n

o

+ R[xcl , δx]

1 δxAxx [xcl ]δx 2

o

,

(3.715)

R. Jackiw, Phys. Rev. D 9 , 1687 (1976) H. Kleinert, PATH INTEGRALS


315

where D[xcl ] ≡ Axx [xcl ] is the second functional derivative of the action at x = xcl . The subscripts x of Axx denote functional differentiation. For the anharmonic oscillator: g D[xcl ] ≡ Axx [xcl ] = −∂t2 − ω 2 − x2cl . (3.716) 2 The functional R collects all unharmonic terms: R [xcl , δx] = A [xcl + δx] − A[xcl ] − −

Z

dt Ax [xcl ](t)δx(t)

1 2

Z

dtdt′ δx(t)Axx [xcl ](t, t′ )δx(t′ ).

(3.717)

In condensed functional vector notation, we shall write expressions like the last term as Z 1 1 dtdt′ δx(t)Axx [xcl ](t, t′ )δx(t′ ) → δxAxx [xcl ]δx. (3.718) 2 2 By construction, R is at least cubic in δx. The path integral (3.715) may thus be considered as the generating functional Z fl of a fluctuating variable δx(τ ) with a propagator G[xcl ] = i¯ h{Axx [xcl ]}−1 ≡ i¯ hD −1 [xcl ], and an interaction R[xcl , x], ˙ both depending on j via xcl . We know from the previous sections, and will immediately see this explicitly, that h ¯ 2 W2 [xcl ] is of order h ¯ 2 . Let us write the full generating functional W [j] in the form ¯ ∆1 [xcl ], W [j] = A[xcl ] + xcl j + h

(3.719)

where the last term collects one- and two-loop corrections (in higher-order calculations, of course, also higher loops): i ∆1 [xcl ] = Tr log D[xcl ] + h ¯ W2 [xcl ]. 2

(3.720)

From (3.719) we find the vacuum expectation value X = hxi as the functional derivative X=

δxcl δW [j] = xcl + h ¯ ∆1xcl [xcl ] , δj δj

(3.721)

implying the correction term X1 : X1 = ∆1xcl [xcl ]

δxcl . δj

(3.722)

The only explicit dependence of W [j] on j comes from the second term in (3.719). In all others, the j-dependence is due to xcl [j]. We may use this fact to express j as a function of xcl . For this we consider W [j] for a moment as a functional of xcl : W [xcl ] = A[xcl ] + xcl j[xcl ] + h ¯ ∆1 [xcl ].

(3.723)

316


The combination W [xcl ] − jX gives us the effective action Γ[X] [recall (3.662)]. We therefore express xcl in (3.723) as X − h ¯ X1 − O(¯ h2 ) from (3.698), and re-expand everything around X rather than xcl , yields 1 2 Γ[X] = A[X] − h ¯ AX [X]X1 − h ¯ X1 j[X] + h ¯ 2 X1 jX [X]X1 + h ¯ X1 D[X]X1 2 + h ¯ ∆1 [X] − h ¯ 2 ∆1X [X]X1 + O(¯ h3 ). (3.724) Since the action is extremal at xcl , we have AX [X − h ¯ X1 ] = −j[X] + O(¯ h2 ),

(3.725)

and thus AX [X] = −j[X] + h ¯ AXX [X]X1 + O(¯ h2 ) = −j[X] + h ¯ D[X]X1 + O(¯ h2 ),

(3.726)

and therefore: ¯ Γ[X] = A[X] + h ¯ ∆1 [X] + h

2

1 − X1 D[X]X1 + X1 jX [X]X1 − ∆1X X1 . (3.727) 2

From (3.722) we see that δj X1 = ∆1xcl [xcl ]. δxcl

(3.728)

Replacing xcl → X with an error of order h ¯ , this implies δj X = ∆1X [X] + O(¯ h). δX

(3.729)

Inserting this into (3.727), the last two terms in the curly brackets cancel, and the only remaining h ¯ 2 -terms are −

h ¯2 X1 D[X]X1 + h ¯ 2 W2 [X] + O(¯ h3 ). 2

(3.730)

From the classical equation of motion (3.668) one has a further equation for δj/δxcl : δj = −Axx [xcl ] = −D[xcl ]. δxcl

(3.731)

Inserting this into (3.722) and replacing again xcl → X, we find X1 = −D −1 [X]∆1X [X] + O(¯ h).

(3.732)

We now express ∆1X [X] via (3.720). This yields !

i δ ∆1X [X] = Tr D −1 [X] D[X] + h ¯ W2X [X] + O(¯ h2 ). 2 δX

(3.733)



317

Inserting this into (3.732) and further into (3.727), we find for the effective action the expansion up to the order h ¯ 2: Γ[X] = A[X] + h ¯ Γ1 [X] + h ¯ 2 Γ2 [X] h ¯ = A[X] + i Tr log D[X] + h ¯ 2 W2 [X] 2 ! ! h ¯2 1 δ 1 δ −1 −1 −1 + Tr D [X] D[X] D [X] Tr D [X] D[X] . (3.734) 2 2 δX 2 δX ¯ . The remainder R[X; x] in (3.717) has We now calculate W2 [X] to lowest order in h the expansion R[X; δx] =

1 1 AXXX [X]δx δx δx + AXXXX [X]δx δx δx δx + . . . . (3.735) 3! 4!

Being interested only in the h ¯ 2 -corrections, we have simply replaced xcl by X. In order to obtain W2 [X], we have to calculate all connected vacuum diagrams for the interaction terms in R[X; δx] with a δx(t)-propagator G[X] = i¯ h{AXX [X]}−1 ≡ i¯ hD −1[X]. Since every contraction brings in a factor h ¯ , we can truncate the expansion (3.735) after δx4 . Thus, the only contributions to i¯ hW2 [X] come from the connected vacuum diagrams (3.736) 1 + 1 +1 , 8

12

8

where a line stands now for G[X], a four-vertex for (i/¯ h)AXXXX [X] = (i/¯ h)DXX [X],

(3.737)

(i/¯ h)AXXX [X] = (i/¯ h)DX [X].

(3.738)

and a three-vertex for

Only the first two diagrams are one-particle irreducible. As a pleasant result, the third diagram which is one-particle reducible cancels with the last term in (3.734). To see this we write that term more explicitly as h ¯ 2 −1 −1 D AX X X D −1 AX3′ X1′ X2′ DX , 1′ X2′ 8 X1 X2 1 2 3 X3 X3′

(3.739)

which corresponds precisely to the third diagram in Γ2 [X], except for an opposite sign. Note that the diagram has a multiplicity 9. Thus, at the end, only the one-particle irreducible vacuum diagrams contribute to the h ¯ 2 -correction to Γ[X]: 3 −1 1 −1 −1 −1 −1 iΓ2 [X] = i D12 AX1 X2 X3 X4 D34 + i 2 AX1 X2 X3 DX DX DX AX1 X2 X3 . 1 X1′ 2 X2′ 3 X3′ 4! 4! (3.740)

318


Their diagrammatic representation is i 2 h ¯ Γ2 [X] = 18 h ¯

+ 121 (3.741)

The one-particle irreducible nature of the diagrams is found to all orders in h ¯.

3.23.5

Finite-Temperature Two-Loop Effective Action

At finite temperature, and in D dimensions, the expansion proceeds with the imaginary-time versions of the X-dependent Green functions (3.687) and (3.688) GL (τ1 , τ2 ) =

¯ h cosh(ωL |τ1 − τ2 | − ¯hβωL /2) , 2M ωL sinh(¯ hβωL /2)

(3.742)

GT (τ1 , τ2 ) =

¯h cosh(ωT |τ1 − τ2 | − ¯hβωT /2) , 2M ωT sinh(¯ hβωT /2)

(3.743)

and

where we have omitted the argument X in √ ωL (X) and ωT (X). Treating here the general rotationally symmetric potential V (x) = v(x), x = x2 , the two frequencies are 2 ωL (X) ≡

1 ′′ 1 ′ v (X), ωT2 (X) ≡ v (X). M MX

(3.744)

We also decompose the vertex functions into longitudinal and transverse parts. The three-point vertex is a sum ′′ ∂ 3 v(X) v (X) v ′ (X) L ′′′ T = Pijk v (X) + Pijk − , (3.745) ∂Xi ∂Xj ∂Xk X X2 with the symmetric tensors L Pijk ≡

Xi Xj Xk X3

and

T Pijk ≡ δij

Xk Xj Xi L + δik + δjk − 3Pijk . X X X

(3.746)

The four-point vertex reads ′′ ′′′ ∂ 4 v(X) v (X) v ′ (X) L T v (X) S = Pijkl v (4) (X) + Pijkl + Pijkl − , ∂Xi ∂Xj ∂Xk ∂Xl X X2 X3

(3.747)

with the symmetric tensors L Pijkl =

Xi Xj Xk Xl , X4

T Pijkl = δij

(3.748)

Xk Xl Xj Xl Xj Xk Xi Xl Xi Xk Xi Xk L +δik +δil +δjk +δjl +δkl −6Pijkl , 2 2 2 2 2 X X X X X X2

S L T Pijkl = δij δkl + δik δjl + δil δjk − 3Pijkl − 3Pijkl .

(3.749) (3.750)

The tensors obey the following relations: Xi L L P = Pjk , X ijk

Xi T T P = Pjk , X ijk

(3.751)


3.23 Path Integral Calculation of Effective Action by Loop Expansion Xk T Xl T Xj T T L Pjl + Pjk , PijL Pikl = P , PijT Pikl = 0, X X X kl

(3.752)

T T T L T T L T T Phij Phkl = PijT Pkl + Pik Pjl + PilL Pjk + Pjk Pil + PjlL Pik ,

(3.753)

L L T PijL Pikl = Pjkl , PijT Pikl = L L L Phij Phkl = Pijkl ,

L T T T L L L L T L Phij Phkl = PijL Pkl , Phij Phkl = PijT Pkl , PijL Pijkl = Pkl , PijT Pijkl = (D−1)Pkl , T T PijL Pijkl = Pkl ,

319

L PijT Pijkl = 0,

S T S T L PijL Pijkl = −2Pkl , PijT Pijkl = (D + 1)Pkl − 2(D − 1)Pkl .

(3.754) (3.755) (3.756)

Instead of the effective action, the diagrammatic expansion (3.741) yields now the free energy (i/¯h)Γ[X] → −βF (X).

(3.757)

Using the above formulas we obtain immediately the mean field contribution to the free energy Z h¯ β M ˙2 −βFMF = − dτ X + v(X) , (3.758) 2 0 and the one-loop contribution [from the trace-log term in Eq. (3.734)]: −βF1−loop = − log [2 sinh(¯ hβωL /2)] − (D − 1) log [2 sinh(¯ hβωT /2)] . The first of the two-loop diagrams in (3.741) yields the contribution to the free energy ′′ v (X) v ′ (X) 2 (4) 2 2 −β∆1 F2−loop = −β GL (τ, τ )v (X) + (D − 1) GT (τ, τ ) − X2 X3 ′′′ ′′ ′ v (X) 2v (X) 2v (X) + 2(D−1)GL(τ, τ )GT (τ, τ ) − + . X X2 X3 From the second diagram we obtain the contribution Z h¯ β Z h¯ β 1 2 −β∆2 F2−loop = 2 dτ1 dτ2 GL3 (τ1 , τ2 ) [v ′′′ (X)] ¯h 0 0 2 ′′ v (X) v ′ (X) + 3(D − 1)GL (τ1 , τ2 )GT2 (τ1 , τ2 ) − . X X2

(3.759)

(3.760)

(3.761)

The explicit evaluation yields ¯ 2β h (2M )2

1 2 hβωL /2)v (4) (X) (3.762) 2 coth (¯ ωL ′′ D2 − 1 v (X) v ′ (X) 2 + coth (¯ h βω /2) − T ωT2 X2 X3 ′′′ 2(D − 1) v (X) 2v ′′ (X) 2v ′ (X) + coth(¯ hβωL /2) coth(¯ hβωT /2) − + . ωL ωT X X2 X3

−β∆1 F2−loop = −

and 2¯h2 β 1 1 ′′′ 2 1 −β∆2 F2−loop = [v (X)] + ωL (2M ωL )3 3 sinh2 (¯ hβωL /2) ′′ 2 2 6¯h β(D − 1) 1 1 v (X) v ′ (X) + − (3.763) 2ωT + ωL 2M ωL (2M ωT )2 X X2 1 ωT sinh[¯ hβ(2ωT −ωL )/2] ωT × coth2 (¯ hβωT /2) + + . ωL sinh2 (¯ 2ω −ω hβωT /2) hβωL /2) sinh2 (¯ hβωT /2) T L sinh(¯

320


In the limit of zero temperature, the effective potential in the free energy becomes ( ¯hωL ¯hωT ¯h2 1 (4) + (D − 1) + (X) Veff (X) = v(X) + 2v T →0 2 2 8(2M )2 ωL ) D2 − 1 v ′′ (X) v ′ (X) 2(D − 1) v ′′′ (X) 2v ′′ (X) 2v ′ (X) + − + − + ωT2 X2 X3 ωL ωT X X2 X3 ( ) ′′ 2 ¯h2 1 3(D − 1) 1 v (X) v ′ (X) ′′′ 2 − [v (X)] + − + O(¯ h3 ). 4 6(2M )3 3ωL 2ωT + ωL ωL ωT2 X X2

(3.764)

For the one-dimensional potential V (x) =

M 2 2 g3 3 g4 4 ω x + x + x , 2 3! 4!

(3.765)

the effective potential becomes, up to two loops, Veff (X) =

g4 M 2 2 1 1 ω X +g3 X 3 +g4 X 4 + log (2 sinh h ¯ βω/2)+¯ h2 2 β 8(2M ω)2 tanh2 (¯ hβω/2) 2 2 1 ¯h (g3 + g4 X) 1 + O(¯ h3 ) , (3.766) − + 2 3 6ω (2M ω) 3 sinh (¯ hβω/2)

whose T → 0 limit is Veff (X)

=

T →0

M 2 2 g3 3 g4 4 ¯hω g4 ω X + X + X + +h ¯2 2 3! 4! 2 8(2M ω)2 −

¯ 2 (g3 + g4 X)2 h + O(¯ h3 ) . 18ω (2M ω)3

(3.767)

If the potential is a polynomial in X, the effective potential at zero temperature can be solved more efficiently than here and to much higher loop orders with the help of recursion relations. This will be shown in Appendix 3C.5.

3.23.6

Background Field Method for Effective Action

In order to find the rules for the loop expansion to any order, let us separate the total effective action into a sum of the classical action A[X] and a term Γfl [X] which collects the contribution of all quantum fluctuations: Γ[X] = A[X] + Γfl [X].

(3.768)

To calculate the fluctuation part Γfl [X], we expand the paths x(t) around some arbitrarily chosen background path X(t):20 x(t) = X(t) + δx(t),

(3.769)

and calculate the generating functional W [j] by performing the path integral over the fluctuations: Z i i exp W [j] = Dδx exp A [X + δx] + j[X](X + δx) . (3.770) h ¯ h ¯ 20

In the theory of fluctuating fields, this is replaced by a more general background field which explains the name of the method. H. Kleinert, PATH INTEGRALS

321


From W [j] we find a j-dependent expectation value Xj = hxij as Xj = δW [j]/δj, and the Legendre transform Γ[X] = W [j] − jXj . In terms of Xj , Eq. (3.770) can be rewritten as i exp Γ[Xj ] + j[Xj ] Xj = h ¯

Z

Dδx exp

i

h ¯

A [X + δx] + j[X](X + δx)

. (3.771)

The expectation value Xj has the property of extremizing Γ[X], i.e., it satisfies the equation δΓ[X] = −ΓX [Xj ]. (3.772) j=− ∂X X=Xj We now choose j in such a way that Xj equals the initially chosen X, and find o i in exp A [X+δx] − ΓX [X]δx . Γ[X] = Dδx exp h ¯ h ¯

Z

(3.773)

This is a functional integro-differential equation for the effective action Γ[X] which we can solve perturbatively order by order in h ¯ . This is done diagrammatically. The diagrammatic elements are lines representing the propagator (3.686) δ 2 A[X] = Gab [X] ≡ i¯ h δXa δXb "

#−1

,

(3.774)

ab

and vertices n

6

1

5 2

=

δ n A[X] . δXa1 δXa2 . . . δXan

(3.775)

4 3

From the explicit calculations in the last two subsections we expect the effective action to be the sum of all one-particle irreducible vacuum diagrams formed with these propagators and vertices. This will now be proved to all orders in perturbation theory. h i ˜ X, ˜j which governs the corWe introduce an auxiliary generating functional W relation functions of the fluctuations δx around the above fixed backgound X: ˜ X, ˜j /¯ exp iW h ≡ n

h

i

o

Z

i ˜ Dδx exp A [X, δx] + h ¯

Z

dt ˜j(t) δx(t)

,

(3.776)

with the action of fluctuations ˜ A[X, δx] = A[X + δx] − A[X] − AX [X]δx,

(3.777)

322


whose expansion in powers of δx(t) starts out with a quadratic term. A source ˜j(t) is coupled to the fluctuations δx(t). By comparing (3.776) with (3.773) we see that for the special choice of the current ˜j = −ΓX [X] + AX[X] = −Γ ˜ X [X],

(3.778)

˜ [X, ˜j] contains prethe right-hand sides coincide, such that the auxiliary functional W fl cisely the diagrams in Γ [X] which we want to calculate. We now form the Legendre ˜ [X, ˜j], which is an auxiliary effective action with two arguments: transform of W ˜ X, X ˜ ≡W ˜ [X, ˜j] − Γ h

i

Z

˜ dt ˜j X,

(3.779)

with the auxiliary conjugate variable ˜ ˜ ˜j]. ˜ = δ W [X, j] = X[X, ˜ X δ˜j

(3.780)

This is the expectation value of the fluctuations hδxi in the path integral (3.776). If ˜j has the value (3.778), this expectation vanishes, i.e. X ˜ = 0. The auxiliary action fl ˜ Γ [X, 0] coincides with the fluctuating part Γ [X] of the effective action which we want to calculate. ˜ [X, ˜j] with respect to ˜j yield all connected corThe functional derivatives of W relation h ifunctions of the fluctuating variables δx(t). The functional derivatives of ˜ ˜ ˜ select from these the one-particle irreducible correlation Γ X, X with respect to X ˜ = 0, only vacuum diagrams survive. functions. For X Thus we have proved that the full effective action is obtained from the sum of the classical action Γ0 [X] = A[X], the one-loop contribution Γ1 [X] given by the trace of the logarithm in Eq. (3.702), the two-loop contribution Γ2 [X] in (3.741), and the sum of all connected one-particle irreducible vacuum diagrams with more than two loops i X n i¯h Γn [X] = h ¯ n≥3 (3.781) Observe that in the expansion of Γ[X]/¯ h, each line carries a factor h ¯ , whereas each n-point vertex contributes a factor h ¯ −1 . The contribution of an n-loop diagram to Γ[X] is therefore of order h ¯ n . The higher-loop diagrams are most easily generated by a recursive treatment of the type developed in Subsection 3.22.3. For a harmonic oscillator, the expansion stops after the trace of the logarithm (3.702), and reads simply, in one dimension: i Γ[X] = A[X] + h ¯ Tr log Γ(2) (tb , ta ) "2 # Z tb M ˙ 2 Mω 2 2 i = dt X − X + h ¯ Tr log −∂t2 − ω 2 . 2 2 2 ta

(3.782)


323

3.24 Nambu-Goldstone Theorem

Evaluating the trace of the logarithm we find for a constant X the effective potential (3.663): V eff (X) = V (X) −

i log{2πi sin[ω(tb − ta )] /Mω}. 2(tb − ta )

(3.783)

If the boundary conditions are periodic, so that the analytic continuation of the result can be used for quantum statistical calculations, the result is V eff (X) = V (X) −

i log{2i sin[ω(tb − ta )/2]}. (tb − ta )

(3.784)

It is important to keep in mind that a line in the above diagrams contains an infinite series of fundamental Feynman diagrams of the original perturbation expansion, as can be seen by expanding the denominators in the propagator Gab in Eqs. (3.686)–(3.688) in powers of X2 . This expansion produces a sum of diagrams which can be obtained from the loop diagrams in the expansion of the trace of the logarithm in (3.707) by cutting the loop. If the potential is a polynomial in X, the effective potential at zero temperature can be solved most efficiently to high loop orders with the help of recursion relations. This is shown in detail in Appendix 3C.5.

3.24

Nambu-Goldstone Theorem

The appearance of a zero-frequency mode as a consequence of a nonzero expectation value X can easily be proved for any continuous symmetry and to all orders in perturbation theory by using the full effective action. To be more specific we consider as before the case of O(N)-symmetry, and perform infinitesimal symmetry transformations on the currents j in the generating functional W [j]: ja → ja − iǫcd (Lcd )ab jb ,

(3.785)

where Lcd are the N(N −1)/2 generators of O(N)-rotations with the matrix elements (Lcd )ab = i (δca δdb − δda δcb ) ,

(3.786)

and ǫab are the infinitesimal angles of the rotations. Under these, the generating functional is assumed to be invariant: δW [j] = 0 =

Z

dt

δW [j] i (Lcd )ab jb ǫcd = 0. δja (x)

(3.787)

Expressing the integrand in terms of Legendre-transformed quantities via Eqs. (3.620) and (3.622), we obtain Z

dtXa (t)i (Lcd )ab

δΓ[X] ǫcd = 0. δXb (t)

(3.788)

324


This expresses the infinitesimal invariance of the effective action Γ[X] under infinitesimal rotations Xa → Xa − iǫcd (Lcd )ab Xb . The invariance property (3.788) is called the Ward-Takakashi identity for the functional Γ[X]. It can be used to find an infinite set of equally named identities for all vertex functions by forming all Γ[X] functional derivatives of Γ[X] and setting X equal to the expectation value at the minimum of Γ[X]. The first derivative of Γ[X] gives directly from (3.788) (dropping the infinitesimal parameter ǫcd ) (Lcd )ab jb (t) = (Lcd )ab = −

Z

δΓ[X] δX(t)b

dt′ Xa′ (t′ ) (Lcd )a′ b

δ 2 Γ[X] . δXb (t′ )δXn (t)

(3.789)

¯ this Denoting the expectation value at the minimum of the effective potential by X, yields Z

¯ a′ (t ) (Lcd ) ′ dt X ab ′

′

δ 2 Γ[X] = 0. δXb (t′ )δXa (t) X(t)=X¯

(3.790)

Now the second derivative is simply the vertex function Γ(2) (t′ , t) which is the functional inverse of the correlation function G(2) (t′ , t). The integral over t selects the zero-frequency component of the Fourier transform ˜ (2) (ω ′) ≡ Γ

Z

′

dt′ eiω t Γ(2) (t′ , t).

(3.791)

If we define the Fourier components of Γ(2) (t′ , t) accordingly, we can write (3.790) in Fourier space as ˜ −1 (ω ′ = 0) = 0. Xa0′ (Lcd )a′ b G ba

(3.792)

Inserting the matrix elements (3.786) of the generators of the rotations, this equation ¯ 6= 0, the fully interacting transverse propagator has to possess a shows that for X singularity at ω ′ = 0. In quantum field theory, this implies the existence of N − 1 massless particles, the Nambu-Goldstone boson. The conclusion may be drawn only if there are no massless particles in the theory from the outset, which may be “eaten up” by the Nambu-Goldstone boson, as explained earlier in the context of Eq. (3.688). As mentioned before at the end of Subsection 3.23.1, the Nambu-Goldstone theorem does not have any consequences for quantum mechanics since fluctuations are too violent to allow for the existence of a nonzero expectation value X. The effective action calculated to any finite order in perturbation theory, however, is incapable of reproducing this physical property and does have a nonzero extremum and ensuing transverse zero-frequency modes. H. Kleinert, PATH INTEGRALS

325

3.25 Effective Classical Potential

3.25

Effective Classical Potential

The loop expansion of the effective action Γ[X] in (3.768), consisting of the trace of the logarithm (3.702) and the one-particle irreducible diagrams (3.741), (3.781) and the associated effective potential V (X) in Eq. (3.663), can be continued in a hβ to form the Euclidean straightforward way to imaginary times setting tb −ta → −i¯ effective potential Γe [X]. For the harmonic oscillator, where the expansion stops after the trace of the logarithm and the effective potential reduces to the simple expression (3.782), we find the imaginary-time version V

eff

!

1 β¯ hω (X) = V (X) + log 2 sinh . β 2

(3.793)

Since the effective action contains the effect of all fluctuations, the minimum of the effective potential V (X) should yield directly the full quantum statistical partition function of a system: Z = exp[−βV (X) ]. (3.794) min

Inserting the harmonic oscillator expression (3.793) we find indeed the correct result (2.399). For anharmonic systems, we expect the loop expansion to be able to approximate V (X) rather well to yield a good approximation for the partition function via Eq. (3.794). It is easy to realize that this cannot be true. We have shown in Section 2.9 that for high temperatures, the partition function is given by the integral [recall (2.345)] Zcl =

Z

∞

−∞

dx −V (x)/kB T e . le (¯ hβ)

(3.795)

This integral can in principle be treated by the same background field method as the path integral, albeit in a much simpler way. We may write x = X + δx and find a loop expansion for an effective potential. This expansion evaluated at the extremum will yield a good approximation to the integral (3.795) only if the potential is very close to a harmonic one. For any more complicated shape, the integral at small β will cover the entire range of x and can therefore only be evaluated numerically. Thus we can never expect a good result for the partition function of anharmonic systems at high temperatures, if it is calculated from Eq. (3.794). It is easy to find the culprit for this problem. In a one-dimensional system, the correlation functions of the fluctuations around X are given by the correlation function [compare (3.301), (3.248), and (3.687)] h ¯ p G 2 (τ − τ ′ ) M Ω (X),e h ¯ 1 cosh Ω(X)(|τ − τ ′ | − h ¯ β/2) = , M 2Ω(X) sinh[Ω(X)¯ hβ/2] (2)

hδx(τ )δx(τ ′ )i = GΩ2 (X) (τ, τ ′ ) =

|τ − τ ′ | ∈ [0, h ¯ β], (3.796)

326


with the X-dependent frequency given by g Ω2 (X) = ω 2 + 3 X 2 . 6

(3.797)

At equal times τ = τ ′ , this specifies the square width of the fluctuations δx(τ ): D

E

[δx(τ )]2 =

1 Ω(X)¯ hβ h ¯ coth . M 2Ω(X) 2

(3.798)

The point is now that for large temperatures T , this width grows linearly in T D

[δx(τ )]2

E

T →∞

− −−→

kB T . MΩ2

(3.799)

The linear behavior follows the historic Dulong-Petit law for the classical fluctuation width of a harmonic oscillator [compare with the Dulong-Petit law (2.595) for the thermodynamic quantities]. It is a direct consequence of the equipartition theorem for purely thermal fluctuations, according to which the potential energy has an average kB T /2: MΩ2 D 2 E kB T x = . (3.800) 2 2 If we consider the spectral representation (3.245) of the correlation function, GpΩ2 ,e (τ

∞ 1 X 1 ′ −τ )= e−iωm (τ −τ ) , 2 2 h ¯ β m=−∞ ωm + Ω ′

(3.801)

we see that the linear growth is entirely due to term with zero Matsubara frequency. The important observation is now that if we remove this zero frequency term from the correlation function and form the subtracted correlation function [recall (3.250)] GpΩ′2 ,e (τ ) ≡ GpΩ2 ,e (τ )−

1 1 cosh Ω(|τ |−¯ hβ/2) 1 = , − 2 h ¯ βΩ 2Ω sinh[Ω¯hβ/2] h ¯ βΩ2

(3.802)

we see that the subtracted square width a2Ω ≡ GpΩ′2 ,e (0) =

1 Ω¯hβ 1 coth − 2Ω 2 h ¯ βΩ2

(3.803)

decrease for large T . This is shown in Fig. 3.14. Due to this decrease, there exists a method to substantially improve perturbation expansions with the help of the so-called effective classical potential.

3.25.1

Effective Classical Boltzmann Factor

The above considerations lead us to the conclusion that a useful approximation for partition function can be obtained only by expanding the path integral in powers of H. Kleinert, PATH INTEGRALS

327


Figure 3.14 Local fluctuation width compared with the unrestricted fluctuation width of harmonic oscillator and its linear Dulong-Petit approximation. The vertical axis shows units of h ¯ /M Ω, a quantity of dimension length2 .

the subtracted fluctuations δ ′ x(τ ) which possess no zero Matsubara frequency. The quantity which is closely related to the effective potential V eff (X) in Eq. (3.663) but allows for a more accurate evaluation of the partition function is the effective classical potential V eff cl (x0 ). Just as V eff (X), it contains the effects of all quantum fluctuations, but it keeps separate track of the thermal fluctuations which makes it a convenient tool for numerical treatment of the partition function. The definition starts out similar to the background method in Subsection 3.23.6 in Eq. (3.769). We split the paths as in Eq. (2.435) into a time-independent constant background x0 and a fluctuation η(τ ) with zero temporal average η¯ = 0: x(τ ) = x0 + η(τ ) ≡ x0 +

∞ X

xm eiωm τ + cc ,

m=1

x0 = real,

x−m ≡ x∗m ,

(3.804)

and write the partition function using the measure (2.440) as Z=

I

Dx e−Ae /¯h =

Z

∞

−∞

dx0 le (¯ hβ)

I

D ′ x e−Ae /¯h ,

(3.805)

where I

′

−Ae /¯ h

D xe

=

∞ Y

m=1

"Z

∞

−∞

Z

∞

−∞

#

d Re xm d Im xm −Ae /¯h e . 2 πkB T /Mωm

(3.806)

Comparison of (2.439) with the integral expression (2.344) for the classical partition function Zcl suggests writing the path integral over the components with nonzero Matsubara frequencies as a Boltzmann factor B(x0 ) ≡ e−V

eff cl (x

0 )/kB T

(3.807)

and defined the quantity V eff cl (x0 ) as the effective classical potential. The full partition function is then given by the integral Z=

Z

∞

−∞

dx0 −V eff cl (x0 )/kB T e , le (¯ hβ)

(3.808)

328


where the effective classical Boltzmann factor B(x0 ) contains all information on the quantum fluctuations of the system and allows to calculate the full quantum statistical partition function from a single classically looking integral. At hightemperature, the partition function (3.808) takes the classical limit (2.454). Thus, by construction, the effective classical potential V eff cl (x0 ) will approach the initial potential V (x0 ): T →∞

−−→ V (x0 ). V eff cl (x0 ) −

(3.809)

This is a direct consequence of the shrinking fluctuation width (3.803) for growing temperature. The path integral representation of the effective classical Boltzmann factor B(x0 ) ≡

I

D ′x e−Ae /¯h

(3.810)

can also be written as a path integral in which one has inserted a δ-function to ensure the path average 1 Z ¯hβ dτ x(τ ). (3.811) x¯ ≡ h ¯β 0 Let us introduce the slightly modified δ-function [recall (2.345)] ˜ x − x0 ) ≡ le (¯ δ(¯ hβ)δ(¯ x − x0 ) =

s

2π¯ h2 β δ(¯ x − x0 ). M

(3.812)

Then we can write −V eff cl (x0 )/kB T

B(x0 ) ≡ e

= =

I I

′

−Ae /¯ h

D xe

=

I

˜ x − x0 ) e−Ae /¯h Dx δ(¯

˜ η ) e−Ae /¯h . Dη δ(¯

(3.813)

As a check we evaluate the effective classical Boltzmann factor for the harmonic action (2.437). With the path splitting (3.804), it reads h i Mω 2 2 M Z ¯hβ Ae [x0 + η] = h ¯β x0 + dτ η˙ 2 (τ ) + ω 2η 2 (τ ) . 2 2 0

(3.814)

After representing the δ function by a Fourier integral ˜ η) = le (¯ δ(¯ hβ)

Z

i∞

−i∞

dλ 1 exp λ 2πi h ¯β

Z

!

dτ η(τ ) ,

(3.815)

we find the path integral Bω (x0 ) =

I

dλ −i∞ 2πi ( " #) I Z ¯hβ 1 M 2 λ × Dη exp − dτ η˙ (τ ) − η(τ ) . (3.816) h ¯ 0 2 β

˜ η) e−Ae /¯h = e−βM ω Dη δ(¯

2 x2 /2 0

le (¯ hβ)

Z

i∞


329


The path integral over η(τ ) in the second line can now be performed without the restriction h ¯ β = 0 and yields, recalling (3.552), (3.553), and inserting there j(τ ) = λ/β, we obtain for the path integral over η(τ ) in the second line of (3.816): 1 λ2 exp 2 sinh(β¯ hω/2) 2M¯ hβ 2 (

Z

hβ ¯ 0

dτ

Z

hβ ¯

0

dτ

′

Gpω2 ,e (τ

′

)

−τ ) .

(3.817)

The integrals over τ, τ ′ are most easily performed on the spectral representation (3.245) of the correlation function: Z

0

hβ ¯

dτ

Z

0

hβ ¯

dτ

′

Gpω2 ,e (τ

′

− τ )=

Z

hβ ¯ 0

dτ

Z

hβ ¯

0

∞ h ¯β 1 X 1 ′ e−iωm (τ −τ ) = 2 . dτ 2 2 h ¯ β m=−∞ ωm + ω ω ′

(3.818)

The expression (3.817) has to be integrated over λ and yields 1 2 sinh(β¯ hω/2)

Z

i∞

−i∞

dλ λ2 exp 2πi 2Mω 2 β

!

=

1 1 ω¯hβ. 2 sinh(β¯ hω/2) le (¯ hβ)

(3.819)

Inserting this into (3.816) we obtain the local Boltzmann factor eff cl (x

Bω (x0 ) ≡ e−Vω

0 )/kB T

=

I

˜ η ) e−Ae /¯h = Dη δ(¯

β¯ hω/2 2 2 e−βM ω x0 . sinh(β¯ hω/2)

(3.820)

The final integral over x0 in (3.805) reproduces the correct partition function (2.401) of the harmonic oscillator.

3.25.2

Effective Classical Hamiltonian

It is easy to generalize the expression (3.813) to phase space, where we define the effective classical Hamiltonian H eff cl (p0 , x0 ) and the associated Boltzmann factor B(p0 , x0 ) by the path integral Dp hδ(p0 − p) e−Ae [p,x]/¯h, δ(x0 − x)2π¯ 2π¯ h (3.821) R ¯hβ R ¯hβ where x = 0 dτ x(τ )/¯ hβ and p = 0 dτ p(τ )/¯ hβ are the temporal averages of position and momentum, and Ae [p, x] is the Euclidean action in phase space h

B(p0 , x0 ) ≡ exp −βH

eff cl

i

(p0 , x0 ) ≡

Ae [p, x] =

Z

hβ ¯ 0

I

Dx

I

dτ [−ip(τ )x(τ ˙ ) + H(p(τ ), x(τ ))].

(3.822)

The full quantum-mechanical partition function is obtained from the classicallooking expression [recall (2.338)] Z=

Z

∞ −∞

dx0

Z

∞

−∞

dp0 −βH eff cl (p0 ,x0 ) e . 2π¯ h

(3.823)

330


The definition is such that in the classical limit, H eff cl (p0 , x0 )) becomes the ordinary Hamiltonian H(p0 , x0 ). For a harmonic oscillator, the effective classical Hamiltonian can be directly deduced from Eq. (3.820) by “undoing” the p0 -integration: eff cl (p

Bω (p0 , x0 ) ≡ e−Hω

0 ,x0 )/kB T

= le (¯ hβ)

β¯ hω/2 2 2 2 e−β(p0 /2M +M ω x0 ) . sinh(β¯ hω/2)

(3.824)

Indeed, inserting this into (3.823), we recover the harmonic partition function (2.401). Consider a particle in three dimensions moving in a constant magnetic field B along the z-axis. For the sake of generality, we allow for an additional harmonic oscillator centered at the origin with frequencies ωk in z-direction and ω⊥ in the xy-plane (as in Section 2.19). It is then easy to calculate the effective classical Boltzmann factor for the Hamiltonian [recall (2.681)] H(p, x) =

1 2 M 2 2 M p + ω⊥ x⊥ (τ ) + ωk2 z 2 (τ ) + ωB lz (p(τ ), x(τ )), 2M 2 2

(3.825)

where lz (p, x) is the z-component of the angular momentum defined in Eq. (2.639). We have shifted the center of momentum integration to p0 , for later convenience (see Subsection 5.11.2). The vector x⊥ = (x, y) denotes the orthogonal part of x. As in the generalized magnetic field action (2.681), we have chosen different frequencies in front of the harmonic oscillator potential and of the term proportional to lz , for generality. The effective classical Boltzmann factor follows immediately from (2.695) by “undoing” the momentum integrations in px , py , and using (3.824) for the motion in the z-direction: h ¯ βωk /2 −βH(p0 ,x0 ) h ¯ βω+ /2 h ¯ βω− /2 , e sinh h ¯ βω+ /2 sinh h ¯ βω− /2 sinh h ¯ βωk /2 (3.826) where ω± ≡ ωB ± ω⊥ , as in (2.691). As in Eq. (3.820), the restrictions of the path ¯ and p0 = p ¯ give rise to the extra integrals over x and p to the fixed averages x0 = x numerators in comparison to (2.695). B(p0 , x0 )= e−βH

3.25.3

eff cl (p

0 ,x0 )

= le3 (¯ hβ)

High- and Low-Temperature Behavior

We have remarked before in Eq. (3.809) that in the limit T → ∞, the effective classical potential V eff cl (x0 ) converges by construction against the initial potential V (x0 ). There exists, in fact, a well-defined power series in h ¯ ω/kB T which describes this approach. Let us study this limit explicitly for the effective classical potential of the harmonic oscillator calculated in (3.820), after rewriting it as sinh(¯ hω/2kB T ) M 2 2 + ω x0 (3.827) h ¯ ω/2kB T 2 " # M 2 2 h ¯ω h ¯ω −¯ hω/kB T = ω x0 + + kB T log(1 − e ) − log . 2 2 kB T

Vωeff cl (x0 ) = kB T log


331


Due to the subtracted logarithm of ω in the brackets, the effective classical potential has a power series 

¯ω 1 1 h M 2 2 ω x0 + h − Vωeff cl (x0 ) = ¯ω  2 24 kB T 2880

h ¯ω kB T

!3



+ ... .

(3.828)

This pleasant high-temperature behavior is in contrast to that of the effective potential which reads for the harmonic oscillator hω/2kB T )] + Vωeff (x0 ) = kB T log [2 sinh(¯ =

M 2 2 ω x0 2

M 2 2 h ¯ω ω x0 + + kB T log(1 − e−¯hω/kB T ), 2 2

(3.829)

as we can see from (3.793). The logarithm of ω prevents this from having a power series expansion in h ¯ ω/kB T , reflecting the increasing width of the unsubtracted fluctuations. Consider now the opposite limit T → 0, where the final integral over the Boltzmann factor B(x0 ) can be calculated exactly by the saddle-point method. In this limit, the effective classical potential V eff cl (x0 ) coincides with the Euclidean version of the effective potential:

V eff cl (x0 ) → V eff (x0 ) ≡ Γe [X]/β

X=x0

T →0

,

(3.830)

whose real-time definition was given in Eq. (3.663). Let us study this limit again explicitly for the harmonic oscillator, where it becomes T →0 h ¯ω M 2 2 h ¯ω Vωeff cl (x0 ) − −−→ + ω x0 − kB T log , (3.831) 2 2 kB T i.e., the additional constant tends to h ¯ ω/2. This is just the quantum-mechanical zero-point energy which guarantees the correct low-temperature limit Zω

h ¯ω − −−→ e kB T −¯ hω/2kB T = e . T →0

−¯ hω/2kB T

Z

∞

−∞

dx0 −M ω2 x20 /2kB T e le (¯ hβ) (3.832)

The limiting partition function is equal to the Boltzmann factor with the zero-point energy h ¯ ω/2.

3.25.4

Alternative Candidate for Effective Classical Potential

It is instructive to compare this potential with a related expression which can be defined in terms of the partition function density defined in Eq. (2.325): V˜ωeff cl (x) ≡ kB T log [le (¯ hβ) z(x)] .

(3.833)

332


This quantity shares with Vωeff cl (x0 ) the property that it also yields the partition function by forming the integral [compare (2.324)]: Z=

Z

∞

−∞

dx0 −V˜ eff cl (x0 )/kB T e . le (¯ hβ)

(3.834)

It may therefore be considered as an alternative candidate for an effective classical potential. For the harmonic oscillator, we find from Eq. (2.326) the explicit form "

#

Mω kB T 2¯ hω h ¯ω h ¯ω 2 V˜ωeff cl (x) = − log + +kB T log 1−e−2¯hω/kB T + tanh x .(3.835) 2 kB T 2 h ¯ kB T

This shares with the effective potential V eff (X) in Eq. (3.829) the unpleasant property of possessing no power series representation in the high-temperature limit. The low-temperature limit of V˜ωeff cl (x) looks at first sight quite similar to (3.831): T →0 h ¯ω Mω 2 kB T 2¯ hω V˜ eff cl (x0 ) − −−→ + kB T x − log , 2 h ¯ 2 kB T

(3.836)

and the integration leads to the same result (3.832) in only a slightly different way: Zω

s

2¯ hω − −−→ e kB T −¯ hω/2kB T =e . T →0

−¯ hω/2kB T

Z

∞

−∞

dx −M ωx2 /¯h e le (¯ hβ) (3.837)

There is, however, an important difference of (3.836) with respect to (3.831). The width of a local Boltzmann factor formed from the partition function density (2.325): ˜ eff cl ˜ hβ) z(x) = e−V (x)/kB T B(x) ≡ le (¯

(3.838)

is much wider than that of the effective classical Boltzmann factor B(x0 ) = eff cl e−V (x0 )/kB T . Whereas B(x0 ) has a finite width for T → 0, the Boltzmann fac˜ tor B(x) has a width growing to infinity in this limit. Thus the integral over x in (3.837) converges much more slowly than that over x0 in (3.832). This is the principal reason for introducing V eff cl (x0 ) as an effective classical potential rather than V˜ eff cl (x0 ).

3.25.5

Harmonic Correlation Function without Zero Mode

By construction, the correlation functions of η(τ ) have the desired subtracted form (3.802): hη(τ )η(τ ′ )iω =

h ¯ p′ h ¯ cosh ω(|τ − τ ′ |−¯ hβ/2) 1 Gω2 ,e (τ −τ ′ ) = − , M 2Mω sinh(β¯ hω/2) h ¯ βω 2

(3.839)


333


with the square width as in (3.803): hη 2 (τ )iω ≡ a2ω = Gpω′2 ,e (0) =

β¯ hω 1 1 coth − , 2ω 2 h ¯ βω 2

(3.840)

which decreases with increasing temperature. This can be seen explicitly by adding R a current term − dτ j(τ )η(τ ) to the action (3.814) which winds up in the exponent of (3.816), replacing λ/β by j(τ ) + λ/β and multiplies the exponential in (3.817) by a factor 1 2M¯ hβ 2

(Z

hβ ¯ 0

dτ

Z

0

hβ ¯

′

h

2

′

dτ λ + λβj(τ ) + λβj(τ )

(

i

Gpω2 ,e (τ

′

−τ ) )

1 Z ¯hβ Z ¯hβ ′ × exp dτ dτ j(τ )Gpω2 ,e (τ − τ ′ )j(τ ′ ) . 2M¯ h 0 0

)

(3.841)

In the first exponent, one of the τ -integrals over Gpω2 ,e (τ − τ ′ ), say τ ′ , produces a factor 1/ω 2 as in (3.818), so that the first exponent becomes (

1 h ¯β λβ λ2 2 + 2 2 2 2M¯ hβ ω ω

hβ ¯

Z

0

)

dτ j(τ ) .

(3.842)

If we now perform the integral over λ, the linear term in λ yields, after a quadratic completion, a factor (

1 exp − 2Mβ¯ h2 ω 2

Z

hβ ¯

0

dτ

Z

hβ ¯ 0

′

′

)

dτ j(τ )j(τ ) .

(3.843)

Combined with the second exponential in (3.841) this leads to a generating functional for the subtracted correlation functions (3.839): Zωx0 [j]

(

β¯ hω/2 1 2 2 = e−βM ω x0 /2 exp sin(β¯ hω/2) 2M¯ h

Z

0

hβ ¯

dτ

Z

0

hβ ¯

dτ

′

j(τ )Gpω′2 ,e (τ −τ ′ )j(τ ′ )

)

.

(3.844) For j(τ ) ≡ 0, this reduces to the local Boltzmann factor (3.820).

3.25.6

Perturbation Expansion

We can now apply the perturbation expansion (3.480) to the path integral over η(τ ) in Eq. (3.813) for the effective classical Boltzmann factor B(x0 ). We take the action Ae [x] =

Z

0

hβ ¯

M 2 dτ x˙ + V (x) , 2

(3.845)

and rewrite it as Ae = h ¯ βV (x0 ) + A(0) e [η] + Aint,e [x0 ; η],

(3.846)

334


with an unperturbed action A(0) e [η]

=

Z

hβ ¯

0

M 2 M dτ η˙ (τ ) + Ω2 (x0 )η 2 (τ ) , Ω2 (x0 ) ≡ V ′′ (x0 )/M, 2 2

and an interaction Aint,e [x0 ; η] =

Z

hβ ¯

0

dτ V int (x0 ; η(τ )),

(3.847)

(3.848)

containing the subtracted potential 1 V int (x0 ; η(τ )) = V (x0 + η(τ )) − V (x0 ) − V ′ (x0 )η(τ ) − V ′′ (x0 )η 2 (τ ). 2

(3.849)

This has a Taylor expansion starting with the cubic term V int (x0 ; η) =

1 ′′′ 1 V (x0 )η 3 + V (4) (x0 )η 4 + . . . . 3! 4!

(3.850)

Since η(τ ) has a zero temporal average, the linear term 0¯hβ dτ V ′ (x0 )η(τ ) is absent in (3.847). The effective classical Boltzmann factor B(x0 ) in (3.813) has then the perturbation expansion [compare (3.480)] R

B(x0 ) =

1−

1 1 D 2 Ex0 1 D 3 Ex0 hAint,e ixΩ0 + A − A + . . . BΩ (x0 ). (3.851) int,e Ω int,e Ω h ¯ 2!¯ h2 3!¯ h3

The harmonic expectation values are defined with respect to the harmonic path integral BΩ (x0 ) =

Z

(0)

˜ η ) e−Ae Dη δ(¯

[η]//¯ h

.

(3.852)

For an arbitrary functional F [x] one has to calculate hF [x]ixΩ0 = BΩ−1 (x0 )

Z

(0)

˜ η ) F [x] e−Ae Dη δ(¯

[η]/¯ h

.

(3.853)

Some calculations of local expectation values are conveniently done with the explicit Fourier components of the path integral. Recalling (3.806) and expanding the action (3.814) in its Fourier components using (3.804), they are given by the product of integrals hF [x]ixΩ0

=

[ZΩx0 ]−1

∞ Y

m=1

"Z

im dxre − M Σ∞ [ω 2 +Ω2 (x0 )]|xm |2 m dxm e kB T m=1 m F [x]. 2 πkB T /Mωm

#

(3.854)

This implies the correlation functions for the Fourier components D

xm x∗m′

Ex0 Ω

= δmm′

kB T 1 . 2 + Ω2 (x ) M ωm 0


335


From these we can calculate once more the correlation functions of the fluctuations η(τ ) as follows: hη(τ )η(τ

′

x )iΩ0

*

=

∞ X

xm x∗m′ e−i(ωm −ωm′ )τ

m,m′ 6=0

+x0 Ω

=2

1 X 1 . (3.856) 2 Mβ m=1 ωm + Ω2 (x0 )

Performing the sum gives once more the subtracted correlation function Eq. (3.839), whose generating functional was calculated in (3.844). The calculation of the harmonic averages in (3.851) leads to a similar loop expansion as for the effective potential in Subsection 3.23.6 using the background field method. The path average x0 takes over the role of the background X and the nonzero Matsubara frequency part of the paths η(τ ) corresponds to the fluctuations. The only difference with respect to the earlier calculations is that the correlation functions of η(τ ) contain no zero-frequency contribution. Thus they are obtained from the subtracted Green functions GpΩ2′ (x0 ),e(τ ) defined in Eq. (3.802). All Feynman diagrams in the loop expansion are one-particle irreducible, just as in the loop expansion of the effective potential. The reducible diagrams are absent since there is no linear term in the interaction (3.850). This trivial absence is an advantage with respect to the somewhat involved proof required for the effective action in Subsection 3.23.6. The diagrams in the two expansions are therefore precisely the same and can be read off from Eqs. (3.741) and (3.781). The only difference lies in the replacement X → x0 in the analytic expressions for the lines and vertices. In addition, there is the final integral over x0 to obtain the partition function Z in Eq. (3.808). This is in contrast to the partition function expressed in terms of the effective potential V eff (X), where only the extremum has to be taken.

3.25.7

Effective Potential and Magnetization Curves

The effective classical potential V eff cl (x0 ) in the Boltzmann factor (3.807) allows us to estimate the effective potential defined in Eq. (3.663). It can be derived from the generating functional Z[j] restricted to time-independent external source j(τ ) ≡ j, in which case Z[j] reduces to a mere function of j: Z(j) =

Z

(

Dx(τ ) exp −

Z

0

β

)

1 dτ x˙ 2 + V (x(τ )) + βj x¯ , 2

(3.857)

where x¯ is the path average of x(τ ). The function Z(j) is obtained from the effective classical potential by a simple integral over x0 : Z(j) =

Z

∞

−∞

dx eff cl √ 0 e−β[V (x0 )−jx0 ] . 2πβ

(3.858)

The effective potential V eff (X) is equal to the Legendre transform of W (j) = log Z(j): 1 V eff (X) = − W (j) + Xj, (3.859) β

336


Figure 3.15 Magnetization curves in double-well potential V (x) = −x2 /2 + gx4 /4 with g = 0.4, at various inverse temperatures β. The integral over these curves returns the effective potential V eff (X). The curves arising from the approximate effective potential W1 (x0 ) are labeled by β1 (- - -) and the exact curves (found by solving the Schr¨ odinger equation numerically) by βex (—–). For comparison we have also drawn the classical curves (· · ·) obtained by using the potential V (x0 ) in Eqs. (3.861) and (3.858) rather than W1 (x0 ). They are labeled by βV . Our approximation W1 (x0 ) is seen to render good magnetization curves for all temperatures above T = 1/β ∼ 1/10. The label β carries several subscripts if the corresponding curves are indistinguishable on the plot. Note that all approximations are monotonous, as they should be (except for the mean field, of course).

where the right-hand side is to be expressed in terms of X using X = X(j) =

1 d W (j). β dj

(3.860)

To picture the effective potential, we calculate the average value of x(τ ) from the integral Z ∞ n o dx √ 0 x0 exp −β[V eff cl (x0 ) − jx0 ] X = Z(j)−1 (3.861) 2πβ −∞ and plot X = X(j). By exchanging the axes we display the inverse j = j(X) which is the slope of the effective potential: j(X) =

dV eff (X) . dX


337


The curves j(X) are shown in Fig. 3.15 for the double-well potential with a coupling strength g = 0.4 at various temperatures. Note that the x0 -integration makes j(X) necessarily a monotonous function of X. The effective potential is therefore always a convex function of X, no matter what the classical potential looks like. This is in contrast to j(X) before fluctuations are taken into account, the mean-field approximation to (3.862) [recall the discussion in Subsection 3.23.1], which is given by j = dV (X)/dX.

(3.863)

For the double-well potential, this becomes j = −X + gX 3.

(3.864)

Thus, the mean-field effective potential coincides with the classical potential V (X), which is obviously not convex. In magnetic systems, j is a constant magnetic field and X its associated magnetization. For this reason, plots of j(X) are referred to as magnetization curves.

3.25.8

First-Order Perturbative Result

To first order in the interaction V int (x0 ; η), the perturbation expansion (3.851) becomes 1 (3.865) B(x0 ) = 1 − hAint,e ixΩ0 + . . . BΩ (x0 ), h ¯ and we have to calculate the harmonic expectation value of Aint,e . Let us assume that the interaction potential possesses a Fourier transform V int (x0 ; η(τ )) =

Z

∞

−∞

dk ik(x0 +η(τ )) ˜ int V (k). e 2π

(3.866)

Then we can write the expectation of (3.848) as hAint,e [x0 ; η]ixΩ0

=

Z

hβ ¯

dτ

0

Z

D Ex0 dk ˜ int V (k)eikx0 eikη(τ ) . Ω 2π

∞

−∞

(3.867)

We now use Wick’s rule in the form (3.304) to calculate D

eikη(τ )

Ex0 Ω

x

0 2 2 = e−k hη (τ )iΩ /2 .

(3.868)

We now use Eq. (3.840) to write this as D

eikη(τ )

Ex0 Ω

−k 2 a2Ω(x ) /2

=e

0

.

(3.869)

Thus we find for the expectation value (3.867): hAint,e [x0 ; η]ixΩ0

=

Z

0

hβ ¯

dτ

Z

∞ −∞

dk ˜ int ikx −k 2 a2Ω(x ) /2 0 V (k)e 0 . 2π

(3.870)

338


Due to the periodic boundary conditions satisfied by the correlation function and the associated invariance under time translations, this result is independent of τ , so that the τ -integral can be performed trivially, yielding simply a factor h ¯ β. We now reinsert the Fourier coefficients of the potential Z

V˜ int (k) =

∞

−∞

dx V int (x0 ; η) e−ik(x0+η) ,

(3.871)

perform the integral over k via a quadratic completion, and obtain D

V int (x(τ ))

E x0 Ω

int

≡ Va2 (x0 ) = Ω

Ex0

D

Z

dx′0

∞

−∞

−η2 /2a2Ω(x

q

2πa2Ω(x0 )

e

0)

V int (x0 ; η).

(3.872)

int

The expectation V int (x(τ )) ≡ Va2 (x0 ) of the potential arises therefore from a Ω Ω convolution integral of the original potential with a Gaussian distribution of square width a2Ω(x0 ) . The convolution integral smears the original interaction potential int

Va2 (x0 ) out over a length scale aΩ(x0 ) . In this way, the approximation accounts for Ω the quantum-statistical path fluctuations of the particle. As a result, we can write the first-order Boltzmann factor (3.865) as follows: B(x0 ) ≈

n o Ω(x0 )¯ hβ int exp −βMΩ(x0 )2 x20 /2 − βVa2 (x0 ) . Ω 2 sin[Ω(x0 )¯ hβ/2]

(3.873)

Recalling the harmonic effective classical potential (3.831), this may be written as a Boltzmann factor associated with the first-order effective classical potential int

eff cl V eff cl (x0 ) ≈ VΩ(x (x0 ) + Va2 (x0 ). 0)

(3.874)

Ω

Given the power series expansion (3.850) of the interaction potential V

int

(x0 ; η) =

∞ X

1 (k) V (x0 )¯ hk , k=3 k!

(3.875)

we may use the integral formula Z

∞

−∞

dη −η2 /2a2 k √ e η = 2πa2

(

(k − 1)!! ak 0

)

(

)

even for k = , odd

(3.876)

we find the explicit smeared potential Vaint 2 (x0 ) =

∞ X

(k − 1)!! (k) V (x0 )ak (x0 ). k! k=4,6,...

(3.877)


3.26 Perturbative Approach to Scattering Amplitude

3.26

339

Perturbative Approach to Scattering Amplitude

In Eq. (2.739) we have derived a path integral representation for the scattering amplitude. It involves calculating a path integral of the general form Z tb Z Z Z Z i M 2 d3 ya d3 za D3 y D3 z exp dt y˙ − z˙ 2 F [y(t) − z(0)], (3.878) ¯h ta 2 where the paths y(t) and z(t) vanish at the final time t = tb whereas the initial positions are integrated out. In lowest approximation, we may neglect the fluctuations in y(t) and z(0) and obtain the eikonal approximation (2.742). In order to calculate higher-order corrections to path integrals of the form (3.878) we find the generating functional of all correlation functions of y(t) − z(0).

3.26.1

Generating Functional

For the sake of generality we calculate the harmonic path integral over y: Z tb Z Z i M 2 2 2 3 3 dt y˙ − ω y − jy y . Z[jy ] ≡ d ya D y exp ¯h ta 2

(3.879)

This differs from the amplitude calculated in (3.168) only by an extra Fresnel integral over the initial point and a trivial extension to three dimensions. This yields Z j Z[jy ] = d3 ya (yb tb |ya ta )ωy Z tb i 1 dt yb (sin [ω(t − ta )] + sin [ω(tb − t)]) jy = exp ¯h ta sin ω(tb − ta ) Z tb Z t i ¯h ′ ′ ′ ¯ × exp − 2 dt jy (t)Gω2 (t, t )jy (t ) , (3.880) dt ¯h M ta ta ¯ ω2 (t, t′ ) is obtained from the Green function (3.36) with Dirichlet boundary conditions by where G adding the result of the quadratic completion in the variable yb − ya preceding the evaluation of the integral over d3 ya : ¯ ω2 (t, t′ ) = G

1 sin ω(tb − t> ) [sin ω(t< − ta ) + sin ω(tb − t< )] . ω sin ω(tb − ta )

(3.881)

We need the special case ω = 0 where ¯ ω2 (t, t′ ) = tb − t> . G

(3.882)

In contrast to Gω2 (t, t′ ) of (3.36), this Green function vanishes only at the final time. This reflects the fact that the path integral (3.878) is evaluated for paths y(t) which vanish at the final time t = tb . A similar generating functional for z(t) leads to the same result with opposite sign in the exponent. Since the variable z(t) appears only with time argument zero in (3.878), the relevant generating functional is Z Z Z Z 3 3 3 Z[j] ≡ d ya d za D y D 3 z Z tb M 2 i 2 2 2 2 × exp dt y˙ − z˙ − ω y − z − j yz , (3.883) ¯h ta 2

with yb = zb = 0, where we have introduced the subtracted variable yz (t) ≡ y(t) − z(0),

(3.884)

340


for brevity. From the above calculations we can immediately write down the result Z tb Z tb 1 i ¯h ′ ′ ′ ′ ¯ Z[j] = exp − 2 dt dt j(t)Gω2 (t, t )j(t ) , Dω ¯h 2M ta ta

(3.885)

where Dω is the functional determinant associated with the Green function (3.881) which is obtained by integrating (3.881) over t ∈ (tb , ta ) and over ω 2 : Dω =

1 exp cos2 [ω(tb − ta )]

Z

tb −ta

0

dt (cos ωt − 1) , t

(3.886)

¯ ′ 2 (t, t′ ) is the subtracted Green function (3.881): and G ω ¯ ′ 2 (t, t′ ) ≡ G ¯ ω2 (t, t′ ) − G ¯ ω2 (0, 0). G ω

(3.887)

For ω = 0 where Dω = 1, this is simply ¯ ′0 (t, t′ ) ≡ −t> , G

(3.888)

where t> denotes the larger of the times t and t′ . It is important to realize that thanks to the subtraction in the Green function (3.882) caused by the z(0)-fluctuations, the limits ta → −∞ and tb → ∞ can be taken in (3.885) without any problems.

3.26.2

Application to Scattering Amplitude

We can now apply this result to the path integral (2.739). With the abbreviation (3.884) we write it as Z p d2 b e−iqb/¯h fpb pa = 2πi¯h Z ∞ h Z i i M ¯ ′ (t, t′ )]−1 yz eiχb,p [yz ] − 1 , (3.889) × D3 yz exp dt yz [G 0 ¯h −∞ 2 ¯ ′0 (t, t′ )]−1 is the functional inverse of the subtracted Green function (3.888), and χb,p [yz ] where [G the integral over the interaction potential V (x): Z 1 ∞ p χb,p [yz ] ≡ − dt V b + t + yz (t) . (3.890) ¯h −∞ M

3.26.3

First Correction to Eikonal Approximation

The first correction to the eikonal approximation (2.742) is obtained by expanding (3.890) to first order in yz (t). This yields Z 1 ∞ p dt ∇V b + t yz (t). (3.891) χb,p [y] = χei − b ,p ¯h −∞ M The additional terms can be considered as an interaction Z 1 ∞ − dt yz (t) j(t), ¯h −∞

(3.892)

with the current j(t)

=

p ∇V b + t . M

(3.893)


3.26 Perturbative Approach to Scattering Amplitude

341

Using the generating functional (3.885), this is seen to yield an additional scattering phase Z ∞ Z ∞ 1 p p dt dt ∇V b + t ∇V b + t2 t> . (3.894) ∆1 χei = 1 2 1 b,p 2M ¯h −∞ M M −∞ To evaluate this we shall always change, as in (2.744), the time variables t1,2 to length variables z1,2 ≡ p1,2 t/M along the direction of p. √ For spherically symmetric potentials V (r) with r ≡ |x| = b2 + z 2 , we may express the derivatives parallel and orthogonal to the incoming particle momentum p as follows: ∇k V = z V ′ /r,

∇⊥ V = b V ′ /r.

(3.895)

Then (3.894) reduces to ∆1 χei b ,p

M2 = 2¯ hp 3

Z

∞ −∞

dz1

Z

∞

V ′ (r1 ) V ′ (r2 ) 2 b + z1 z2 z1 . r1 r2

dz2

−∞

(3.896)

The part of the integrand before the bracket is obviously symmetric under z → −z and under the exchange z1 ↔ z2 . For this reason we can rewrite Z Z M2 ∞ V ′ (r1 ) ∞ V ′ (r2 ) 2 (3.897) ∆1 χei = dz z dz b − z22 . 1 1 2 b,p 3 ¯hp −∞ r1 r2 −∞ Now we use the relations (3.895) in the opposite direction as zV ′ /r = ∂z V,

bV ′ /r = ∂b V,

(3.898)

and performing a partial integration in z1 to obtain21 ∆1 χei b,p

M2 = − 3 (1 + b∂b ) ¯hp

Z

∞

−∞

dz V 2

p b2 + z 2 .

(3.899)

Compared to the leading eikonal phase (2.745), this √ is suppressed by a factor V (0)M/p2 . 2 Note that for the Coulomb potential where V ( b2 + z 2 ) ∝ 1/(b2 + z 2 ), the integral is proportional to 1/b which is annihilated by the factor 1 + b∂b . Thus there is no first correction to the eikonal approximation (1.503).

3.26.4

Rayleigh-Schr¨ odinger Expansion of Scattering Amplitude

In Section 1.16 we have introduced the scattering amplitude as the limiting matrix element [see (1.513)] ˆ ai ≡ hpb |S|p

lim

tb −ta →∞

ei(Eb −Ea )tb /¯h (pb 0|pa ta )e−iEa ta /¯h .

(3.900)

A perturbation expansion for these quantities can be found via a Fourier transformation of the expansion (3.474). We only have to set the oscillator frequency of the harmonic part of the action equal to zero, since the particles in a scattering process are free far away from the scattering center. Since scattering takes usually place in three dimensions, all formulas will be written down in such a space. 21

This agrees with results from Schrödinger theory by S.J. Wallace, Ann. Phys. 78 , 190 (1973); S. Sarkar, Phys. Rev. D 21 , 3437 (1980). It differs from R. Rosenfelder’s result (see Footnote 37 on p. 192) who derives a prefactor p cos(θ/2) instead of the incoming momentum p.

342


We shall thus consider the perturbation expansion of the amplitude Z

(pb 0|pa ta ) =

d3 xb d3 xa e−ipb xb (xb 0|xa ta )eipa xa ,

(3.901)

where (xb 0|xa ta ) is expanded as in (3.474). The immediate result looks as in the expansion (3.497), if we replace the external oscillator wave functions ψn (xb ) and ψa (xb ) by free-particle plane waves e−ipb xb and eipa xa : (pb 0|pa ta ) = (pb 0|pa ta )0 i 1 i 2 + hpb |Aint |pa i0 − hpb |A3int|pa i0 + . . . . 2 hpb |Aint |pa i0 − h ¯ 2!¯ h 3!¯ h3 Here

2

(pb 0|pa ta )0 = (2π¯ h)3 δ (3) (pb − pa )eipb ta /2M ¯h

(3.902)

(3.903)

is the free-particle time evolution amplitude in momentum space [recall (2.133)] and the matrix elements are defined by hpb | . . . |pa i0 ≡

Z

3

3

−ipb xb

d xb d xa e

Z

3

iA0 /¯ h

D x...e

eipa xa .

(3.904)

In contrast to (3.497) we have not divided out the free-particle amplitude (3.903) in this definition since it is too singular. Let us calculate the successive terms in the expansion (3.902). First hpb |Aint |pa i0 = −

Z

0 ta

dt1

Z

d3 xb d3 xa d3 x1 e−ipb xb (xb 0|x1 t1 )0 × V (x1 )(x1 t1 |xa ta )0 eipa xa .

(3.905)

Since Z

Z

2

d3 xb e−ipb xb (xb tb |x1 t1 )0 = e−ipb x1 e−ipb (tb −t1 )/2M ¯h , 2

d3 xa (x1 t1 |xa ta )0 e−ipb xb = e−ipa x1 eipa (t1 −ta )/2M ¯h ,

(3.906)

this becomes hpb |Aint|pa i0 = −

Z

0 ta

2

2

2

dt1 ei(pb −pa )t1 /2M ¯h Vpb pa eipa ta /2M ¯h ,

(3.907)

Z

(3.908)

where Vpb pa

≡ hpb |Vˆ |pa i =

d3 xei(pb −pa )x/¯h V (x) = V˜ (pb − pa )

[recall (1.491)]. Inserting a damping factor eηt1 into the time integral, and replacing p2 /2M by the corresponding energy E, we obtain i 1 hpb |Aint |pa i0 = − Vpb pa eiEa ta . h ¯ Eb − Ea − iη


343

3.27 Functional Determinants from Green Functions

Inserting this together with (3.903) into the expansion (3.902), we find for the scattering amplitude (3.900) the first-order approximation ˆ ai ≡ hpb |S|p

lim

tb −ta →∞

i(Eb −Ea )tb /¯ h

e

"

#

1 Vp p (2π¯ h) δ (pb − pa ) − Eb − Ea − iη b a (3.910) 3 (3)

corresponding precisely to the first-order approximation of the operator expression (1.516), the Born approximation. Continuing the evaluation of the expansion (3.902) we find that Vpb pa in (3.910) is replaced by the T -matrix [recall (1.474)] d 3 pc 1 Vpb pc Vp p (3.911) 3 (2π¯ h) Ec − Ea − iη c a Z Z d 3 pc d 3 pd 1 1 + Vpb pc Vpc pd Vp p + . . . . 3 3 (2π¯ h) (2π¯ h) Ec − Ea − iη Ed − Ea − iη d a

Tpb pa = Vpb pa −

Z

This amounts to an integral equation Tpb pa = Vpb pa −

Z

d 3 pc 1 Vpb pc Tp p , 3 (2π¯ h) Ec − Ea − iη c a

(3.912)

which is recognized as the Lippmann-Schwinger equation (1.522) for the T -matrix.

3.27

Functional Determinants from Green Functions

In Subsection 3.2.1 we have seen that there exists a simple method, due to Wronski, for constructing Green functions of the differential equation (3.27), O(t)Gω2 (t, t′ ) ≡ [−∂t2 − Ω2 (t)]Gω2 (t, t′ ) = δ(t − t′ ),

(3.913)

with Dirichlet boundary conditions. That method did not require any knowledge of the spectrum and the eigenstates of the differential operator O(t), except for the condition that zero-modes are absent. The question arises whether this method can be used to find also functional determinants.22 The answer is positive, and we shall now demonstrate that Gelfand and Yaglom’s initial-value problem (2.206), (2.207), (2.208) with the Wronski construction (2.218) for its solution represents the most concise formula for the functional determinant of the operator O(t). Starting point is the observation that a functional determinant of an operator O can be written as Det O = eTr log O ,

(3.914)

and that a Green function of a harmonic oscillator with an arbitrary time-dependent frequency has the integral Tr

22

Z

0

1

dg Ω2 (t)[−∂t2 − gΩ2 (t)]−1 δ(t − t′ )

See the reference in Footnote 6 on p. 246.

= −Tr {log[−∂t2 − Ω2 (t)]δ(t − t′ )} +Tr {log[−∂t2 ]δ(t − t′ )}.

(3.915)

344


If we therefore introduce a strength parameter g ∈ [0, 1] and an auxiliary Green function Gg (t, t′ ) satisfying the differential equation Og (t)Gg (t, t′ ) ≡ [−∂t2 − gΩ2 (t)]Gg (t, t′ ) = δ(t − t′ ),

(3.916)

we can express the ratio of functional determinants Det O1 /Det O0 as Det (O0−1 O1 ) = e−

R1 0

dg Tr [Ω2 (t)Gg (t,t′ )]

.

(3.917)

Knowing of the existence of Gelfand-Yaglom’s elegant method for calculating functional determinants in Section 2.4, we now try to relate the right-hand side in (3.917) to the solution of the Gelfand-Yaglom’s equations (2.208), (2.206), and (2.207): Og (t)Dg (t) = 0; Dg (ta ) = 0, D˙ g (ta ) = 1.

(3.918)

By differentiating these equations with respect to the parameter g, we obtain for the g-derivative Dg′ (t) ≡ ∂g Dg (t) the inhomogeneous initial-value problem Og (t)Dg′ (t) = Ω2 (t)Dg (t); Dg′ (ta ) = 0, D˙ g′ (ta ) = 0.

(3.919)

The unique solution of equations (3.918) can be expressed as in Eq. (2.214) in terms of an arbitrary set of solutions ηg (t) and ξg (t) as follows ξg (ta )ηg (t) − ξg (t)ηg (ta ) = ∆g (t, ta ), Wg

Dg (t) =

(3.920)

where Wg is the constant Wronski determinant Wg = ξg (t)η˙ g (t) − ηg (t)ξ˙g (t).

(3.921)

DetΛg = ∆g (tb , ta ), Wg

(3.922)

We may also write Dg (tb ) =

where Λg is the constant 2 × 2 -matrix Λg =

ξg (ta ) ηg (ta ) ξg (tb ) ηg (tb )

!

.

(3.923)

With the help of the solution ∆g (t, t′ ) of the homogenous initial-value problem (3.918) we can easily construct a solution of the inhomogeneous initial-value problem (3.919) by superposition: Dg′ (t)

=

Z

t

ta

dt′ Ω2 (t′ )∆g (t, t′ )∆g (t′ , ta ).

(3.924)

Comparison with (3.59) shows that at the final point t = tb Dg′ (tb )

= ∆g (tb , ta )

Z

tb

ta

dt′ Ω2 (t′ )Gg (t′ , t′ ).


345


Together with (3.922), this implies the following equation for the integral over the Green function which solves (3.913) with Dirichlet’s boundary conditions: det Λg Tr [Ω (t)Gg (t, t )] = −∂g log Wg 2

′

!

= −∂g log Dg (tb ).

(3.926)

Inserting this into (3.915), we find for the ratio of functional determinants the simple formula Det (O0−1 Og ) = C(tb , ta )Dg (tb ).

(3.927)

The constant of g-integration, which still depends in general on initial and final times, is fixed by applying (3.927) to the trivial case g = 0, where O0 = −∂t2 and the solution to the initial-value problem (3.918) is D0 (t) = t − ta .

(3.928)

At g = 0, the left-hand side of (3.927) is unity, determining C(tb , ta ) = (tb − ta )−1 and the final result for g = 1: Det (O0−1 O1 )

det Λ1 = W1

,

D1 (tb ) DetΛ0 = , W0 tb − ta

(3.929)

in agreement with the result of Section 2.7. The same method permits us to find the Green function Gω2 (τ, τ ′ ) governing quantum statistical harmonic fluctuations which satisfies the differential equation ′ 2 2 p,a ′ p,a Og (τ )Gp,a (τ − τ ′ ), g (τ, τ ) ≡ [∂τ − gΩ (τ )]Gg (τ, τ ) = δ

(3.930)

with periodic and antiperiodic boundary conditions, frequency Ω(τ ), and δ-function. The imaginary-time analog of (3.915) for the ratio of functional determinants reads Det (O0−1 O1 ) = e−

R1 0

dgTr [Ω2 (τ )Gg (τ,τ ′ )]

.

(3.931)

′ The boundary conditions satisfied by the Green function Gp,a g (τ, τ ) are ′ p,a ′ Gp,a g (τb , τ ) = ±Gg (τa , τ ), ′ ′ ˙ p,a G˙ p,a g (τb , τ ) = ±Gg (τa , τ ).

(3.932)

According to Eq. (3.166), the Green functions are given by ′ ′ Gp,a g (τ, τ ) = Gg (τ, τ ) ∓

[∆g (τ, τa ) ± ∆g (τb , τ )][∆g (τ ′ , τa ) ± ∆g (τb , τ ′ )] , (3.933) ¯ p,a ∆ g (τa , τb ) · ∆g (τa , τb )

where [compare (3.49)] ∆(τ, τ ′ ) =

1 [ξ(τ )η(τ ′ ) − ξ(τ ′ )η(τ )] , W

(3.934)

346


˙ )η(τ ), and [compare (3.165)] with the Wronski determinant W = ξ(τ )η(τ ˙ ) − ξ(τ ¯ p,a (τa , τb ) = 2 ± ∂τ ∆g (τa , τb ) ± ∂τ ∆g (τb , τa ). ∆ g

(3.935)

The solution is unique provided that ¯ p,a (τa , τb ) 6= 0. ¯ p,a = Wg ∆ det Λ g g

(3.936)

The right-hand side is well-defined unless the operator Og (t) has a zero-mode with ηg (tb ) = ±ηg (ta ), η˙g (tb ) = ±η˙g (ta ), which would make the determinant of the 2 × 2 ¯ p,a vanish. -matrix Λ g We are now in a position to rederive the functional determinant of the operator O(τ ) = ∂τ2 −Ω2 (τ ) with periodic or antiperiodic boundary conditions more elegantly than in Section 2.11. For this we formulate again a homogeneous initial-value problem, but with boundary conditions dual to Gelfand and Yaglom’s in Eq. (3.918): ¯ g (τ ) = 0; D ¯ g (τa ) = 1, D ¯˙ g (τa ) = 0. Og (τ )D

(3.937)

In terms of the previous arbitrary set ηg (t) and ξg (t) of solutions of the homogeneous differential equation, the unique solution of (3.937) reads ˙ ¯ g (τ ) = ξg (τ )η˙g (τa ) − ξg (τa )ηg (τ ) . D Wg

(3.938)

This can be combined with the time derivative of (3.920) at τ = τb to yield ¯ g (τb ) = ±[2 − ∆ ¯ p,a (τa , τb )]. D˙ g (τb ) + D g

(3.939)

By differentiating Eqs. (3.937) with respect to g, we obtain the following inhomoge¯ g′ (τ ) = ∂g D ¯ g (τ ): neous initial-value problem for D ¯ ′ (τ ) = Ω2 (τ )D ¯ ′ (τ ); D ¯ ′ (τa ) = 1, D ¯˙ ′ g (τa ) = 0, Og (τ )D g g g

(3.940)

whose general solution reads by analogy with (3.924) ¯ ′ (τ ) = − D g

Z

τ

τa

˙ g (τa , τ ′ ), dτ ′ Ω2 (τ ′ )∆g (τ, τ ′ )∆

(3.941)

˙ g (τa , τ ′ ) acts on the first imaginary-time argument. With the where the dot on ∆ ¯ ′ (τ ) at τ = τb can help of identities (3.939) and (3.940), the combination D˙ ′ (τ ) + D g now be expressed in terms of the periodic and antiperiodic Green functions (3.166), by analogy with (3.925), ¯ ′ (τb ) = ±∆ ¯ p,a (τa , τb ) D˙ g′ (τb ) + D g g

Z

τb

τa

dτ Ω2 (τ )Gp,a g (τ, τ ).

(3.942)


347


Together with (3.939), this gives for the temporal integral on the right-hand side of (3.917) the simple expression analogous to (3.926) 2

Tr [Ω

¯ p,a det Λ g = −∂g log Wg h i ¯ g (τb ) , = −∂g log 2 ∓ D˙ g (τb ) ∓ D !

′ (τ )Gp,a g (τ, τ )]

(3.943)

so that we obtain the ratio of functional determinants with periodic and antiperiodic boundary conditions ˜ −1 Og ) = C(tb , ta ) 2 ∓ D˙ g (τb ) ∓ D ¯ g (τb ) , Det (O h

i

(3.944)

˜ = O0 − ω 2 = ∂ 2 − ω 2 . The constant of integration C(tb , ta ) is fixed in where O τ the way described after Eq. (3.915). We go to g = 1 and set Ω2 (τ ) ≡ ω 2 . For the operator O1ω ≡ −∂τ2 − ω 2 , we can easily solve the Gelfand-Yaglom initial-value problem (3.918) as well as the dual one (3.937) by D1ω (τ ) =

1 sin ω(τ − τa ), ω

¯ ω (τ ) = cos ω(τ − τa ), D 1

(3.945)

so that (3.944) determines C(tb , ta ) by 1 = C(tb , ta )

(

4 sin2 [ω(τb − τa )/2] 4cos2 [ω(τb − τa )/2]

periodic case, antiperiodic case.

(3.946)

Hence we find the final results for periodic boundary conditions ¯ p1 det Λ Det (O O1 ) = W1 ˜ −1

,

¯ ω1 p ¯ 1 (τb ) DetΛ 2 − D˙ 1 (τb ) − D = , 2 W1ω 4 sin [ω(τb − τa )/2]

(3.947)

and for antiperiodic boundary conditions ¯a ˜ −1 O1 ) = det Λ1 Det (O W1

,

¯ω a ¯ 1 (τb ) DetΛ 2 + D˙ 1 (τb ) + D 1 . = W1ω 4 cos2 [ω(τb − τa )/2]

(3.948)

The intermediate expressions in (3.929), (3.947), and (3.948) show that the ratios of functional determinants are ordinary determinants of two arbitrary independent solutions ξ and η of the homogeneous differential equation O1 (t)y(t) = 0 or O1 (τ )y(τ ) = 0. As such, the results are manifestly invariant under arbitrary linear transformations of these functions (ξ, η) → (ξ ′ , η ′). It is useful to express the above formulas for the ratio of functional determinants (3.929), (3.947), and (3.948) in yet another form. We rewrite the two independent solutions of the homogenous differential equation [−∂t2 − Ω2 (t)]y(t) = 0 as follows ξ(t) = q(t) cos φ(t),

η(t) = q(t) sin φ(t).

(3.949)

348


The two functions q(t) and φ(t) parametrizing ξ(t) and η(t) satisfy the constraint 2 ˙ φ(t)q (t) = W,

(3.950)

where W is the constant Wronski determinant. The function q(t) is a soliton of the Ermankov-Pinney equation23 q¨ + Ω2 (t)q − W 2 q −3 = 0.

(3.951)

For Dirichlet boundary conditions we insert (3.949) into (3.929), and obtain the ratio of fluctuation determinants in the form Det (O0−1 O1 ) =

1 q(ta )q(tb ) sin[φ(tb ) − φ(ta )] . W tb − ta

(3.952)

For periodic or antiperiodic boundary conditions with a corresponding frequency Ω(t), the functions q(t) and φ(t) in Eq. (3.949) have the same periodicity. The initial value φ(ta ) may always be assumed to vanish, since otherwise ξ(t) and η(t) could be combined linearly to that effect. Substituting (3.949) into (3.947) and (3.948), the function q(t) drops out, and we obtain the ratios of functional determinants for periodic boundary conditions ˜ −1 O1 ) = 4 sin2 Det (O

φ(tb ) ω(tb − ta ) 4 sin2 , 2 2

(3.953)

and for antiperiodic boundary conditions ˜ −1 O1 ) = 4 cos2 Det (O

φ(tb ) ω(tb − ta ) . 4 cos2 2 2

(3.954)

For a harmonic oscillator with Ω(t) ≡ ω, Eq. (3.951) is solved by q(t) ≡

s

W , ω

(3.955)

and Eq. (3.950) yields φ(t) = ω(t − ta ).

(3.956)

Inserted into (3.952), (3.953), and (3.954) we reproduce the known results: Det (O0−1 O1 ) = 23

sin ω(tb − ta ) , ω(tb − ta )

˜ −1 O1 ) = 1. Det (O

For more details see J. Rezende, J. Math. Phys. 25, 3264 (1984). H. Kleinert, PATH INTEGRALS

Appendix 3A

349

Matrix Elements for General Potential

Appendix 3A

Matrix Elements for General Potential

The matrix elements hn|Vˆ |mi can be calculated for an arbitrary potential Vˆ = V (ˆ x) as follows: We represent V (ˆ x) by a Fourier integral as a superposition of exponentials i∞

Z

V (ˆ x) =

−i∞

dk V (k) exp(kˆ x), 2πi

(3A.1)

√ and express x) in terms of creation and annihilation operators as exp(kˆ x) = exp[k(ˆ a+ˆ a† )/ 2], √ exp(kˆ set k ≡ 2ǫ, and write down the obvious equation √ 1 ∂n ∂m ǫ 2ˆ x αˆ a ǫ(ˆ a+ˆ a† ) βˆ a† hn|e |mi = √ h0|e e e |0i . (3A.2) n!m! ∂αn ∂β m α=β=0

We now make use of the Baker-Campbell-Hausdorff Formula (2A.1) with (2A.6), and rewrite ˆ ˆ

ˆ

ˆ

ˆ ˆ

1

ˆ ˆ ˆ

1

ˆ

ˆ ˆ

eA eB = eA+B+ 2 [A,B]+ 12 ([A,[A,B]]+[B,[B,A]])+.... .

(3A.3)

ˆ with a Identifying Aˆ and B ˆ and a ˆ† , the property [ˆ a, a ˆ† ] = 1 makes this relation very simple: †

eǫ(â+â

)

†

= eǫâ eǫâ e−ǫ

2

/2

,

(3A.4)

and the matrix elements (3A.2) become †

†

†

h0|eαâ eǫ(â+â ) eβâ |0i = h0|e(α+ǫ)â e(β+ǫ)â |0ie−ǫ

2

/2

.

(3A.5)

The bra and ket states on the right-hand side are now eigenstates of the annihilation operator a ˆ with eigenvalues α + ǫ and β + ǫ, respectively. Such states are known as coherent states. Using once more (3A.3), we obtain †

h0|e(α+ǫ)â e(β+ǫ)â |0i = e(ǫ+α)(ǫ+β),

(3A.6)

and (3A.2) becomes simply hn|e

√ x ǫ 2ˆ

1 ∂ n ∂ m (α+ǫ)(β+ǫ) −ǫ2 /2 |mi = √ . e e n!m! ∂αn ∂β m α=β=0

(3A.7)

We now calculate the derivatives

∂ n ∂ m (ǫ+α)(ǫ+β) e ∂αn ∂β m

α=β=0

∂n m ǫ(ǫ+α) = (ǫ + α) e ∂αn

.

(3A.8)

α=0

Using the chain rule of differentiation for products f (x) = g(x) h(x): f

(n)

(x) =

n X n l=0

l

g (l) (x)h(n−l) (x),

(3A.9)

the right-hand side becomes ∂n m ǫ(ǫ+α) (ǫ + α) e ∂αn

= α=0

=

n l n−l X n ∂ ∂ m ǫ(ǫ+α) (ǫ + α) e l ∂αl ∂αn−l l=0 α=0 n X 2 n m(m − 1) · · · (m − l + 1)ǫn+m−2l eǫ . l l=0

(3A.10)

350


Hence we find hn|eǫ

√ 2ˆ x

n 2 1 X n m |mi = √ l!ǫn+m−2l eǫ /2 . l n!m! l=0 l

(3A.11)

From this we obtain the matrix elements of single powers x ˆp by forming, with the help of (3A.9) q q ǫ2 /2 and (∂ /∂ǫ )e |ǫ=0 = q!!, the derivatives ∂ p n+m−2l ǫ2 /2 p p! ǫ e [2l − (n + m − p)]!! = l−(n+m−p)/2 . = ∂ǫp n+m−2l 2 [l − p − (n+m−p)/2]! ǫ=0 (3A.12) The result is min(n,m) X n m p! 1 . hn|ˆ xp |mi = √ l! l+p−(n+m)/2 l l 2 [l − (n + m − p)/2]! n!m! l=(n+m−p)/2

For the special case of a pure fourth-order interaction, this becomes √ √ √ √ hn|ˆ x4 |n − 4i = n − 3 n − 2 n − 1 n, √ √ hn|ˆ x4 |n − 2i = (4n − 2) n − 1 n, hn|ˆ x4 |ni = hn|ˆ x4 |n + 2i =

hn|ˆ x4 |n + 4i =

6n2 + 6n + 3, √ √ (4n + 6) n + 1 n + 2, √ √ √ √ n + 1 n + 2 n + 3 n + 4.

(3A.13)

(3A.14)

For a general potential (3A.1) we find Z i∞ n 2 1 X n m 1 dk √ hn|V (ˆ x)|mi = l! l−(n+m)/2 V (k)k n+m−2l ek /4 . l l 2πi 2 n!m! l=0 −i∞

Appendix 3B

(3A.15)

Energy Shifts for gx4 /4 -Interaction

For the specific polynomial interaction V (x) = gx4 /4, the shift of the energy E (n) to any desired order is calculated most simply as follows. Consider the expectations of powers x ˆ4 (z1 )ˆ x4 (z2 ) · · · xˆ4 (zn ) † −1 of the operator x ˆ(z) = (ˆ a z+a ˆz ) between the excited oscillator states hn| and |ni. Here a ˆ and a ˆ† are the usual creation and annihilation operators of the harmonic oscillator, and √ |ni = (a† )n |0i/ n! . To evaluate these expectations, we make repeated use of the commutation rules [ˆ a, a ˆ† ] = 1 and of the ground state property a ˆ|0i = 0. For n = 0 this gives hx4 (z)iω = 3,

hx4 (z1 )x4 (z2 )iω = 72z1−2z22 + 24z1−4z24 + 9, hx4 (z1 )x4 (z2 )x4 (z3 )iω = 27 · 8z1−2 z22 + 63 · 32z1−2z2−2 z34

(3B.1)

+ 351 · 8z1−2 z32 + 9 · 8z1−4 z24 + 63 · 32z1−4z22 z32 + 369 · 8z1−4 z34 + 27 · 8z2−2 z32 + 9 · 8z2−4 z34 + 27.

The cumulants are hx4 (z1 )x4 (z2 )iω,c = 72z1−2z22 + 24z1−4z24 , 4

4

4

hx (z1 )x (z2 )x (z3 )iω,c =

288(7z1−2z2−2 z34

(3B.2) +

9z1−2 z32

+

7z1−4 z22 z32

+

10z1−4 z34 ).

The powers of z show by how many steps the intermediate states have been excited. They determine the energy denominators in the formulas (3.515) and (3.516). Apart from a factor (g/4)n and a H. Kleinert, PATH INTEGRALS

Energy Shifts for gx4 /4-Interaction

Appendix 3B

351

factor 1/(2ω)2n which carries the correct length scale of x(z), the energy shifts ∆E = ∆1 E0 + ∆2 E0 + ∆3 E0 are thus found to be given by ∆1 E0 = 3, 1 1 ∆2 E0 = − 72 · + 24 · , 2 4 1 1 1 1 1 1 1 1 = 333 · 4. ∆3 E0 = 288 7 · · + 9 · · + 7 · · + 10 · · 2 4 2 2 4 2 4 4

(3B.3)

Between excited states, the calculation is somewhat more tedious and yields

4

hx4 (z)iω = 6n2 + 6n + 3,

4

hx (z1 )x (z2 )iω,c = + + + 4

4

4

(3B.4)

(16n + 96n + 212n + 204n + 72)z1−2 z22 (n4 + 10n3 + 35n2 + 50n + 24)z1−4 z24 (n4 − 6n3 + 11n2 − 6n)z14 z2−4 (16n4 − 32n3 + 20n2 − 4n)z12 z2−2 , 6 5 4 3 4

3

2

(3B.5) 2

hx (z1 )x (z2 )x (z3 )iω,c = [(16n + 240n + 1444n + 4440n + 7324n + 6120n + 2016)

× (z1−2 z2−2 z34 +z1−4 z22 z32 ) + (384n5 + 2880n4 + 8544n3 + 12528n2 + 9072n + 2592)z1−2z32

+ (48n5 + 600n4 + 2880n3 + 6600n2 + 7152n + 2880)z1−4z34 + (16n6 − 144n5 + 484n4 − 744n3 + 508n2 − 120n)z14z2−2 z3−2

+ (−48n5 + 360n4 − 960n3 + 1080n2 − 432n)z14z3−4 + (16n6 + 48n5 + 4n4 − 72n3 − 20n2 + 24n)z12 z2−4 z32

+ (−384n5 + 960n4 − 864n3 + 336n2 − 48n)z12z3−2 + (16n6 − 144n5 + 484n4 − 744n3 + 508n2 − 120n)z12z22 z3−4 + (16n6 + 48n5 + 4n4 − 72n3 − 20n2 + 24n)z1−2 z24 z3−2 ].

(3B.6)

From these we obtain the reduced energy shifts: ∆1 E0 = 6n2 + 6n + 3, ∆2 E0 = −(16n4 + 96n3 + 212n2 + 204n + 72) · −(n4 + 10n3 + 35n2 + 50n + 24) · −(n4 − 6n3 + 11n2 − 6n) · −1 4

−(16n4 − 32n3 + 20n2 − 4n) · = 2 · (34n3 + 51n2 + 59n + 21), 6

5

4

(3B.7) 1 2

1 4

−1 2

(3B.8) 3

2

1 2 1 2

∆3 E0 = [(16n + 240n + 1444n + 4440n + 7324n + 6120n + 2016) · ( · +(384n5 + 2880n4 + 8544n3 + 12528n2 + 9072n + 2592) · 12 · +(48n5 + 600n4 + 2880n3 + 6600n2 + 7152n + 2880) ·

1 4

·

+(16n6 − 144n5 + 484n4 − 744n3 + 508n2 − 120n) · +(−48n5 + 360n4 − 960n3 + 1080n2 − 432n) · 14 · 14

1 4

·

1 2

+(16n6 − 144n5 + 484n4 − 744n3 + 508n2 − 120n) · +(16n6 + 48n5 + 4n4 − 72n3 − 20n2 + 24n) · 12 · −1 2 ]

1 2

·

1 4

+(16n6 + 48n5 + 4n4 − 72n3 − 20n2 + 24n) · +(−384n5 + 960n4 − 864n3 + 336n2 − 48n) ·

= 4 · 3 · (125n4 + 250n3 + 472n2 + 347n + 111).

1 2 1 2

· ·

1 4

+

1 4

1 2

· )

1 4

1 4 1 2

(3B.9)

352


Appendix 3C

Recursion Relations for Perturbation Coefficients of Anharmonic Oscillator

Bender and Wu24 were the first to solve to high orders recursion relations for the perturbation coefficients of the ground state energy of an anharmonic oscillator with a potential x2 /2 + gx4 /4. Their relations are similar to Eqs. (3.534), (3.535), and (3.536), but not the same. Extending their method, we derive here a recursion relation for the perturbation coefficients of all energy levels of the anharmonic oscillator in any number of dimensions D, where the radial potential is l(l + D − 2)/2r2 + r2 /2 + (g/2)(a4 r4 + a6 r6 + . . . + a2q x2q ), where the first term is the centrifugal barrier of angular momentum l in D dimensions. We shall do this in several steps.

3C.1

One-Dimensional Interaction x4

In natural physical units with h ¯ = 1, ω = 1, M = 1, the Schrödinger equation to be solved reads 1 d2 1 2 4 + x + gx ψ (n) (x) = E (n) ψ (n) (x). (3C.1) − 2 dx2 2 At g = 0, this is solved by the harmonic oscillator wave functions ψ (n) (x, g = 0) = N n e−x

2

/2

Hn (x),

(3C.2)

with proper normalization constant N n , where Hn (x) are the Hermite polynomial of nth degree n X

Hn (x) =

hpn xp .

(3C.3)

p=0

Generalizing this to the anharmonic case, we solve the Schr¨ odinger equation (3C.1) with the power series ansatz ψ (n) (x)

=

e−x

2

/2

∞ X

(n)

(−g)k Φk (x),

(3C.4)

k=0

E (n)

=

∞ X

(n)

g k Ek .

(3C.5)

k=0 (n)

To make room for derivative symbols, the superscript of Φk (x) is now dropped. Inserting (3C.4) and (3C.5) into (3C.1) and equating the coefficients of equal powers of g, we obtain the equations xΦ′k (x) − nΦk (x) =

k X ′ 1 ′′ (n) Φk (x) − x4 Φk−1 (x) + (−1)k Ek′ Φk−k′ (x), 2 ′

(3C.6)

k =1

where we have inserted the unperturbed energy (n)

E0

= n + 1/2,

(3C.7)

and defined Φk (x) ≡ 0 for k < 0. The functions Φk (x) are anharmonic versions of the Hermite polynomials. They turn out to be polynomials of (4k + n)th degree: Φk (x) =

4k+n X

Apk xp .

(3C.8)

p=0

24

C.M. Bender and T.T. Wu, Phys. Rev. 184 , 1231 (1969); Phys. Rev. D 7 , 1620 (1973). H. Kleinert, PATH INTEGRALS

Appendix 3C

Recursion Relations for Perturbation Coefficients

353

In a more explicit notation, the expansion coefficients Apk would of course carry the dropped (n) superscript of Φk . All higher coefficients vanish: Apk ≡ 0

for p ≥ 4k + n + 1.

(3C.9)

From the harmonic wave functions (3C.2), Φ0 (x) = N n Hn (x) = N n

n X

hpn xp ,

(3C.10)

p=0

we see that the recursion starts with Ap0 = hpn N n .

(3C.11)

For levels with an even principal quantum number n, the functions Φk (x) are symmetric. It is convenient to choose the normalization ψ (n) (0) = 1, such that N n = 1/h0n and A0k = δ0k .

(3C.12)

For odd values of n, the wave functions Φk (x) are antisymmetric. Here we choose the normalization ψ (n)′ (0) = 3, so that N n = 3/h1n and A1k = 3δ0k .

(3C.13)

Defining Apk ≡ 0

for p < 0 or k < 0,

(3C.14)

we find from (3C.6), by comparing coefficients of xp , (p − n)Apk =

k X ′ 1 (n) p−4 (−1)k Ek′ Apk−k′ . (p + 2)(p + 1)Ap+2 + A + k k−1 2 ′

(3C.15)

k =1

The last term on the right-hand side arises after exchanging the order of summation as follows: k X

4(k−k′ )+n k′

(−1)

(n) Ek′

X

Apk−k′ xp =

p=0

k′ =1

4k+n X

xp

p=0

k X

′

(n)

(−1)k Ek′ Apk−k′ .

(3C.16)

k′ =1

For even n, Eq. (3C.15) with p = 0 and k > 0 yields [using (3C.14) and (3C.12)] the desired expansion coefficients of the energies (n)

Ek

= −(−1)k A2k .

(3C.17)

For odd n, we take Eq. (3C.15) with p = 1 and odd k > 0 and find [using (3C.13) and (3C.14)] the expansion coefficients of the energies: (n)

Ek

= −(−1)k A3k .

(3C.18)

For even n, the recursion relations (3C.15) obviously relate only coefficients carrying even indices with each other. It is therefore useful to set ′

′

p n = 2n′ , p = 2p′ , A2p k = Ck ,

(3C.19)

leading to ′

′

′

p −2 2(p′ − n′ )Ckp = (2p′ + 1)(p′ + 1)Ckp +1 + Ck−1 −

k X

k′ =1

′

p Ck1′ Ck−k ′.

(3C.20)

354


For odd n, the substitution ′

′

+1 n = 2n′ + 1 , p = 2p′ + 1 , A2p = Ckp , k

(3C.21)

leads to ′

′

′

p −2 2(p′ − n′ )Ckp = (2p′ + 3)(p′ + 1)Ckp +1 + Ck−1 −

k X

′

p Ck1′ Ck−k ′.

(3C.22)

k′ =1

The rewritten recursion relations (3C.20) and (3C.22) are the same for even and odd n, except ′ for the prefactor of the coefficient Ckp +1 . The common initial values are 2p′ 0 ′ hn /hn for 0 ≤ p′ ≤ n′ , C0p = (3C.23) 0 otherwise. The energy expansion coefficients are given in either case by (n)

Ek

= −(−1)k Ck1 .

(3C.24)

The solution of the recursion relations proceeds in three steps as follows. Suppose we have p′ calculated for some value of k all coefficients Ck−1 for an upper index in the range 1 ≤ p′ ≤ ′ 2(k − 1) + n . ′ In a first step, we find Ckp for 1 ≤ p′ ≤ 2k + n′ by solving Eq. (3C.20) or (3C.22), starting with p′ = 2k + n′ and lowering p′ down to p′ = n′ + 1. Note that the knowledge of the coefficients Ck1 (which determine the yet unknown energies and are contained in the last term of the recursion ′ relations) is not required for p′ > n′ , since they are accompanied by factors C0p which vanish due to (3C.23). Next we use the recursion relation with p′ = n′ to find equations for the coefficients Ck1 contained in the last term. The result is, for even k, " # k−1 X 1 n′ +1 n′ −2 1 ′ ′ 1 n′ Ck = (2n + 1)(n + 1)Ck + Ck−1 − Ck′ Ck−k′ . (3C.25) n′ C 0 k′ =1

For odd k, the factor (2n′ + 1) is replaced by (2n′ + 3). These equations contain once more the ′ coefficients Ckn . ′ Finally, we take the recursion relations for p′ < n′ , and relate the coefficients Ckn −1 , . . . , Ck1 to ′ (n) Ckn . Combining the results we determine from Eq. (3C.24) all expansion coefficients Ek . The relations can easily be extended to interactions which are an arbitrary linear combination V (x) =

∞ X

a2n ǫn x2n .

(3C.26)

n=2

A short Mathematica program solving the relations can be downloaded from the internet.25 The expansion coefficients have the remarkable property of growing, for large order k, like r 1 6 12 (n) Ek − − −→ − (−3)k Γ(k + n + 1/2). (3C.27) π π n! This will be shown in Eq. (17.323). Such a factorial growth implies the perturbation expansion to have a zero radius of convergence. The reason for this will be explained in Section 17.10. At the expansion point g = 0, the energies possess an essential singularity. In order to extract meaningful numbers from a Taylor series expansion around such a singularity, it will be necessary to find a convergent resummation method. This will be provided by the variational perturbation theory to be developed in Section 5.14. 25

See http://www.physik.fu-berlin/~kleinert/b3/programs. H. Kleinert, PATH INTEGRALS

Appendix 3C

3C.2


355

General One-Dimensional Interaction

Consider now an arbirary interaction which is expandable in a power series v(x) =

∞ X

g k vk+2 xk+2 .

(3C.28)

k=1

Note that the coupling constant corresponds now to the square root of the previous one, the lowest interaction terms being gv3 x3 + g 2 v4 x4 + . . . . The powers of g count the number of loops of the associated Feynman diagrams. Then Eqs. (3C.6) and (3C.15) become xΦ′k (x) − nΦk (x) =

k k X X ′ ′ ′ 1 ′′ (n) Φk (x) − (−1)k vk′ +2 xk +2 Φk−k′ + (−1)k Ek′ Φk−k′ (x), 2 ′ ′ k =1

(3C.29)

k =1

and (p − n)Apk =

k k X X ′ 1 (n) p−j−2 k′ ′ (p + 2)(p + 1)Ap+2 − (−1) v A + (−1)k Ek′ Apk−k′ . (3C.30) k +2 k−k′ k 2 ′ ′ k =1

k =1

The expansion coefficients of the energies are, as before, given by (3C.17) and (3C.18) for even and odd n, respectively, but the recursion relation (3C.30) has to be solved now in full.

3C.3

Cumulative Treatment of Interactions x4 and x3

There exists a slightly different recursive treatment which we shall illustrate for the simplest mixed interaction potential V (x) =

M 2 2 ω x + gv3 x3 + g 2 v4 x4 . 2

(3C.31)

Instead of the ansatz (3C.4) we shall now factorize the wave function of the ground state as follows: ψ

(n)

(x) =

Mω π¯h

1/4

Mω 2 (n) exp − x + φ (x) , 2¯h

(3C.32)

i.e., we allow for powers series expansion in the exponent: φ(n) (x) =

∞ X

(n)

g k φk (x) .

(3C.33)

k=1

We shall find that this expansion contains fewer terms than in the Bender-Wu expansion of the correction factor in Eq. (3C.4). For completeness, we keep here physical dimensions with explicit constants ¯h, ω, M . Inserting (3C.32) into the Schrödinger equation ¯h2 d2 M 2 2 3 2 4 (n) − + ω x + gv3 x + g v4 x − E ψ (n) (x) = 0, (3C.34) 2M dx2 2 we obtain, after dropping everywhere the superscript (n), the differential equation for φ(n) (x): −

¯ 2 ′′ h ¯2 ′ h 2 φ (x) + h ¯ ω x φ′ (x) − [φ (x)] + gv3 x3 + g 2 v4 x4 = n¯hω + ǫ, 2M 2M

where ǫ denotes the correction to the harmonic energy 1 E=h ¯ω n + +ǫ. 2

(3C.35)

(3C.36)

356


We shall calculate ǫ as a power series in g: ǫ=

∞ X

g k ǫk .

(3C.37)

k=1

From now on we shall consuder only the ground state with n = 0. Inserting expansion (3C.33) into (3C.35), and comparing coefficients, we obtain the infinite set of differential equations for φk (x): −

k−1 ¯ 2 ′′ h ¯2 X ′ h φk (x) + h ¯ ω x φ′k (x) − φk−l (x) φ′l (x) + δk,1 v3 x3 + δk,2 v4 x4 = ǫk . 2M 2M

(3C.38)

l=1

Assuming that φk (x) is a polynomial, we can show by induction that its degree cannot be greater than k + 2, i.e., φk (x) =

∞ X

m c(k) m x ,

m=1

with c(k) m ≡0

for m > k + 2 ,

(3C.39)

(k)

The lowest terms c0 have been omitted since they will be determined at the end the normalization of the wave function ψ(x). Inserting (3C.39) into (3C.38) for k = 1, we find (1)

c1 = −

v3 , M ω2

(1)

c2 = 0,

(1)

c3 = −

v3 , 3¯ hω

ǫ1 = 0 .

(3C.40)

For k = 2, we obtain 3v4 v32 v4 7v32 (2) (2) − , c = 0, c = − , 3 4 8M 2 ω 4 4M ω 2 8M ¯hω 3 4¯hω 11v32 h ¯2 3v4 ¯h2 ǫ2 = − + . 3 4 8M ω 4M 2 ω 2

(2)

c1

(2)

= 0, c2 =

(3C.41) (3C.42)

For the higher-order terms we must solve the recursion relations c(k) m =

k−1 X m+1 X ¯h (m + 2)(m + 1)¯ h (k) (k−l) cm+2 + n(m + 2 − n) c(l) n cm+2−n , 2mM ω 2mM ω n=1

(3C.43)

l=1

ǫk = −

k−1 ¯ 2 (k) h ¯ 2 X (l) (k−l) h c2 − c1 c1 . M 2M

(3C.44)

l=1

Evaluating this for k = 3 yields (3)

c1 = −

5v33 ¯h 6v3 v4 ¯h 13v33 3v3 v4 (3) (3) + , c = 0, c = − + , 2 3 4 7 3 5 3 6 M ω M ω 12M ω 2M 2 ω 4 v33 v3 v4 (3) (3) c4 = 0, c5 = − + , ǫ3 = 0, 10M 2 ¯hω 5 5M ¯hω 3

(3C.45)

and for k = 4: 305v34 ¯h 123v32 v4 ¯h 21v42 ¯h 99v34 47v32 v4 11v42 (4) (4) − + , c3 = 0, c4 = − + , 5 9 4 7 3 5 4 8 6 32M ω 8M ω 8M ω 64M ω 16M ω 16M 2 ω 4 5v34 v32 v4 v42 (4) (4) − + , (3C.46) c5 = 0, c6 = 48M 3 ¯hω 7 4M 2 ¯hω 5 12M ¯hω 3 465v34 ¯h3 171v32 v4 ¯h3 21v42 ¯h3 ǫ4 = − + − . (3C.47) 6 9 5 7 32M ω 8M ω 8M 4 ω 5 (4)

(4)

c1 = 0, c2 =


Appendix 3C


357

The general form of the coefficients is, now in natural units with h ¯ = 1, M = 1,: c(k) m

=

⌊k/2⌋

X

λ=0

ǫk =

⌊k/2⌋

X

λ=0

v3k−2λ v4λ

ω

(k) c , 5k/2−m/2−2λ m,λ

v3k−2λ v4λ ω 5k/2−1−2λ

with

(k) cm,λ

k ≡ 0 for m > k + 2, or λ > , 2

ǫk,λ .

(3C.48)

(3C.49)

This leads to the recursion relations (k)

cm,λ =

k−1 m+1 λ (m+2)(m+1) (k) 1 XX X (l) (k−l) cm+2,λ + n(m + 2 − n)cn,λ−λ′ cm+2−n,λ′ , 2m 2m n=1 ′ l=1

(3C.50)

λ =0

(k)

with cm,λ ≡ 0 for m > k + 2 or λ > ⌊k/2⌋. The starting values follow by comparing (3C.40) and (3C.41) with (3C.48): 1 (1) (1) (1) c1,0 = −1, c2,0 = 0, c3,0 = − , 3 7 3 (2) (2) (2) (2) c1,0 = 0, c1,1 = 0, c2,0 = , c2,1 = − , 8 4 1 1 (2) (2) (2) (2) c3,0 = 0, c3,1 = 0, c4,0 = , c4,1 = − . 8 4

(3C.51)

(3C.52)

The expansion coefficients ǫk,λ for the energy corrections ǫk are obtained by inserting (3C.49) and (3C.48) into (3C.44) and going to natural units: (k)

ǫk,λ = −c2,λ −

k−1 λ 1 X X (l) (k−l) c1,λ−λ′ c1,λ′ . 2 ′

(3C.53)

l=1 λ =0

Table 3.1 shows the nonzero even energy corrections ǫk up to the tenth order.

3C.4

Ground-State Energy with External Current

In the presence of a constant external current j, the time-independent Schrödinger reads ¯ 2 ′′ h − ψ (x) + 2M

M 2 2 3 2 4 ω x + gv3 x + g v4 x − jx ψ(x) = Eψ(x). 2

(3C.54)

For zero coupling constant g = 0, we may simply introduce the new variables x′ and E ′ : x′ = x −

j j2 and E ′ = E + , 2 Mω 2M ω 2

(3C.55)

and the system becomes a harmonic oscillator in x′ with energy E ′ = h ¯ ω/2. Thus we make the ansatz for the wave function ψ(x) ∝ e

φ(x)

∞

j Mω 2 X k , with φ(x) = x− x + g φk (x), ¯hω 2¯ h

(3C.56)

k=1

and for the energy E(j) =

∞

X ¯ω h j2 − + g k ǫk . 2 2 2M ω k=1

(3C.57)

358


k

ǫk

2

−11v32 + 6v4 ω 2 8ω 4

4

−

465v34 − 684v32 v4 ω 2 + 84v42 ω 4 32ω 9

−39709v36 + 91014v34v4 ω 2 − 47308v32v42 ω 4 + 2664v43 ω 6 128ω 14

6 8

−3(6416935v38 − 19945048v36v4 ω 2 + 18373480v34v42 ω 4 +4962400v32v43 ω 6 164720v44ω 8 )/(2048ω 19)

10

(−2944491879v310 + 11565716526v38v4 ω 2 − 15341262168v36v42 ω 4 +7905514480v34v43 ω 6 − 1320414512v32v44 ω 8 + 29335392v45ω 10 )/(8192ω 24)

Table 3.1 Expansion coefficients for the ground-state energy of the anharmonic oscillator (3C.31) up to the 10th order.

The equations (3C.38) become now −

k−1 ¯ 2 ′′ h ¯h2 X ′ j¯h φk (x)− φk−l (x)φ′l (x) + ¯hωx − φ′k (x) + δk,1 v3 x3 + δk,2 v4 x4 = ǫk . 2M 2M Mω l=1

(3C.58)

The results are now for k = 1 (1)

c1 = −

j 2 v3 jv3 v3 3¯ hjv3 j 3 v3 v3 (1) (1) − , c = − , c = − , ǫ = + , 1 2 3 M ω2 M 2 ¯hω 5 2M ¯hω 3 3¯ hω 2M ω 3 M 3 ω 6 (3C.59)

and for k = 2: 17jv32 4j 3 v32 5jv4 j 3 v4 + − − , 4M 3 ω 6 M 4 ¯hω 9 2M 2 ω 4 M 3 ¯hω 7 2 2 2 2 7v3 3j v3 3v4 j v4 (2) c2 = + − − , 2 4 3 7 2 8M ω 2M ¯hω 4M ω 2M 2 ¯hω 5 jv32 jv4 v32 v4 (2) (2) c3 = − , c4 = − , 2 5 3 2M ¯hω 3M ¯hω 8M ¯hω 3 4¯ hω 11¯h2 v32 27¯hj 2 v32 9j 4 v32 3¯h2 v4 3¯hj 2 v4 j 4 v4 ǫ2 = − − − + + + . 8M 3 ω 4 4M 4 ω 7 2M 5 ω 10 4M 2 ω 2 M 3ω5 M 4ω8 (2)

c1

=

(3C.60)

The recursive equations (3C.43) and (3C.44) become c(k) m

=

k−1 X m+1 X (m + 2)(m + 1)¯ h (k) ¯h (k−l) cm+2 + n(m + 2 − n)c(l) n cm+2−n 2mM ω 2mM ω n=1 l=1

j(m + 1) (k) + c , M mω 2 m+1

(3C.61)


Appendix 3C

ǫk

359


=

−

k−1 j¯h (k) h ¯ 2 (k) ¯ 2 X (l) (k−l) h c1 − c2 − c1 c1 . Mω M 2M

(3C.62)

l=1

Table 3.2 shows the energy corrections ǫk in the presence of an external current up to the sixth order using natural units, h ¯ = 1, M = 1.

k

ǫk

1

v3 j(2j 2 + 3ω 3 ) 2ω 6

2

2v4 ω 2 (4j 4 + 12j 2 ω 3 + 3ω 6 ) − v32 (36j 4 + 54j 2 ω 3 + 11ω 6 ) 8ω 10

3

v3 j[3v32 (36j 4 + 63j 2 ω 3 + 22ω 6 ) − 2v4 ω 2 (24j 4 + 66j 2 ω 3 + 31ω 6 )] 4ω 14

4

[36v32 v4 ω 2 (112j 6 + 324j 4 ω 3 + 212j 2ω 6 + 19ω 9 ) −4v42 ω 4 (64j 6 + 264j 4 ω 3 + 248j 2 ω 6 + 21ω 9 ) 4 −3v3 (2016j 6 + 4158j 4ω 3 + 2112j 2 ω 6 + 155ω 9)]/(32ω 10 )

5

v3 j[27v34 (1728j 6 + 4158j 4 ω 3 + 2816j 2ω 6 + 465ω 9) +4v42 ω 4 (1536j 6 + 6408j 4 ω 3 + 7072j 2ω 6 + 1683ω 9) 2 −12v3 v4 ω 2 (3456j 6 + 10908j 4ω 3 + 9176j 2ω 6 + 1817ω 9)]/(32ω 22 )

6

[8v43 ω 6 (1536j 8 + 8544j 6 ω 3 + 14144j 4ω 6 + 6732j 2ω 9 + 333ω 12) 2 2 4 −4v3 v4 ω (103680j 8 + 454032j 6ω 3 + 584928j 4ω 6 + 221706j 2ω 9 + 11827ω 12) +6v34 v4 ω 2 (285120j 8 + 991224j 6ω 3 + 1024224j 4ω 6 + 323544j 2ω 9 + 15169ω 12) 6 −v3 (1539648j 8 + 4266108j 6ω 3 + 3649536j 4ω 6 + 979290j 2ω 9 + 39709ω 12)]/(128ω 26 )

Table 3.2 Expansion coefficients for the ground-state energy of the anharmonic oscillator (3C.31) in the presence of an external current up to the 6th order.

3C.5

Recursion Relation for Effective Potential

It is possible to derive a recursion relation directly for the zero-temperature effective potential (5.259). To this we observe that according to Eq. (3.771), the fluctuating part of the effective potential is given by the Euclidean path integral Z o fl 1n fl e−βVeff (X) = Dδx exp − A [X +δx]−A [X] − AX [X]δx−Veff (X)δx . (3C.63) X ¯h This can be rewritten as [recall (3.768)] I Z 1 h¯ β −β[Veff (X)−V (X)] e = Dδx exp − dτ ¯h 0 M 2 ′ × δ x˙ (τ ) + V (X + δx) − V (X) − Veff (X)δx . 2

(3C.64)

360


Going back to the integration variable x = X + δx, and taking all terms depending only on X to the left-hand side, this becomes I Z ′ 1 h¯ β M 2 ′ e−β[Veff (X)−Veff (X)X] = Dx exp − dτ x˙ (τ ) + V (x) − Veff (X)x . (3C.65) ¯h 0 2 (0)

In the limit of zero temperature, the right-hand side is equal to e−βE (X) , where E (0) (X) is the ground state of the Schrödinger equation associated with the path integral. Hence we obtain −

¯ 2 ′′ h ′ ′ ψ (x) + [V (x) − Veff (X)x]ψ(x) = [Veff (X) − Veff (X)X]ψ(x). 2M

For the mixed interaction of the previous subsection, this reads ¯h2 ′′ M 2 2 ′ − ψ (x) + ω x + gv3 x3 + g 2 v4 x4 − Veff (X)x ψ(x) 2M 2 ′ = [Veff (X) − Veff (X)X] ψ(x),

(3C.66)

(3C.67)

and may be solved recursively. We expand the effective potential in powers of is expanded in the coupling constant g: ∞ X

Veff (X) =

g k Vk (X) ,

(3C.68)

(k) m Cm X .

(3C.69)

k=0

and assume Vk (X) to be a polynomial in X: Vk (X) =

k+2 X

m=0 ′ ′ Comparison with Eq. (3C.54) shows that we may set j = Veff (X) and calculate Veff (X) − Veff (X)X by analogy to the energy in (3C.57). Inserting the ansatz (3C.68), (3C.69) into (3C.67) we find all equations for Vk (X) by comparing coefficients of g k and X m . It turns out that for even or odd k, also Vk (X) is even or odd in X, respectively. Table 3.3 shows the first six orders of the effective potential, which have been obtained in this way. The equations for Vk (X) are obtained as follows. We insert into (3C.67) the ansatz for the wave function ψ(x) ∝ eφ(x) with ∞

M ωX Mω 2 X k φ(x) = x− x + g φk (x), ¯h 2¯ h

(3C.70)

k=1

and expand Veff (X) =

∞

¯ω M 2 2 X k h + ω X + g Vk (X), 2 2

(3C.71)

k=1

to obtain the set of equations −

k−1 ¯ 2 ′′ h ¯2 X ′ h φk (x) − φk−l (x)φ′l (x) + h ¯ ω(x−X)φ′k (x)−xVk′ (X) + δk,1 v3 x3 + δk,2 v4 x4 2M 2M l=1

= Vk (X) − Vk′ (X)X.

(3C.72)

From these we find for k = 1: (1)

c1 =

v3 2v3 X 2 (1) v3 X (1) v3 3v3 ¯h + , c2 = − , c =− , V1 (X) = + v3 X 3 , 2 2M ω ¯hω 2¯ hω 3 3¯ hω 2M ω

(3C.73)


Appendix 3C


k

Vk (X)

0

ω ω2 2 + X 2 2 v3 X 3 +

1

361

3v3 X 2ω

v32 (1 + 9ωX 2) + v4 ω 2 (3 + 12ωX 2) 4ω 4

2

v4 X 4 −

3

v3 X[3v32 (4 + 9ωX 2) − 2v4 ω 2 (13 + 18ωX 2)] 4ω 6

4

−[4v42 ω 4 (21 + 104ωX 2 + 72ω 2 X 4 ) − 12v32 v4 ω 2 (13 + 152ωX 2 + 108ω 2 X 4 ) +v34 (51 + 864ωX 2 + 810ω 2X 4 )]/(32ω 9 )

5

3v3 X[9v34 (51 + 256ωX 2 + 126ω 2X 4 ) + 4v42 ω 4 (209 + 544ωX 2 + 216ω 2X 4 ) −4v32 v4 ω 2 (341 + 1296ωX 2 + 540ω 2 X 4 )]/(32ω 11 )

6

[24v43 ω 6 (111 + 836ωX 2 + 1088ω 2X 4 + 288ω 3 X 6 ) −36v32 v42 ω 4 (365 + 5654ωX 2 + 8448ω 2X 4 + 2160ω 3X 6 ) +6v34 v4 ω 2 (2129 + 46008ωX 2 + 85248ω 2X 4 + 22680ω 3X 6 ) −v36 (3331 + 90882ωX 2 + 207360ω 2X 4 + 61236ω 3X 6 )]/(128ω 14 )

Table 3.3 Effective potential of the anharmonic oscillator (3C.31) up to the 6th order, expanded in the coupling constant g (in natural units with h ¯ = 1 and M = 1). The lowest terms agree, of course, with the two-loop result (3.767).

and for k = 2: 13v32 X 2v 2 X 3 7v4 X 3v4 X 3 v32 3v4 v4 X 2 (2) − 3 3 + + , c2 = − − , 2 4 2 2 4 2 4M ω M ¯hω 2M ω ¯hω 8M ω 4M ω 2¯hω v32 X v4 X (2) v32 v4 (2) c3 = − ,c = − , 4 2M ¯hω 3 3¯hω 8M ¯hω 3 4¯hω 2 2 2 ¯h v3 9¯hv32 X 2 3¯ h v4 3¯hv4 X 2 V2 (X) = − − + + + v4 X 4 . 3 4 2 3 2 2 4M ω 4M ω 4M ω Mω (2)

c1

= −

(3C.74) (3C.75)

For k ≥ 3, we must solve recursively c(k) m

=

k−1 m+1 X X (m + 2)(m + 1)¯ h (k) ¯h (k−l) cm+2 + n(m + 2 − n)c(l) n cm+2−n 2mM ω 2mM ω n=1 l=1

(k)

c1

=

X(m + 1) (k) cm+1 for m ≥ 2 and with c(k) + m ≡ 0 for m > k + 2, m k−1 3¯h (k) ¯h X (k−l) (l) 1 ′ (k) (k−l) (l) c3 + 2Xc2 + c2 c1 + c1 c2 + V (X), Mω Mω ¯hω k l=1

(3C.76) (3C.77)

362

3 External Sources, Correlations, and Perturbation Theory k−1

Vk (X) = − −

¯ 2 (k) 3¯h2 h ¯ 2 X (k−l) (l) h (k) (k) (k−l) (l) c2 − Xc3 − 2¯hωX 2 c2 − X c2 c1 + c1 c2 M M M l=1

2 k−1 X

¯ h 2M

(l) (k−l)

c1 c1

.

(3C.78)

l=1

The results are listed in Table 3.3.

Interaction r4 in D -Dimensional Radial Oscillator

3C.6

It is easy to generalize these relations further to find the perturbation expansions for the eigenvalues of the radial Schrödinger equation of an anharmonic oscillator in D dimensions 1 d2 1D−1 d l(l + D − 2) 1 2 g 4 − − + + r + r Rn (r) = E (n) Rn (r). (3C.79) 2 dr2 2 r dr 2r2 2 4 The case g = 0 will be solved in Section 9.2, with the energy eigenvalues E (n) = 2n′ + l + D/2 = n + D/2,

n = 0, 1, 2, 3, . . .

, l = 0, 1, 2, 3, . . . .

(3C.80)

For a fixed principal quantum number n = 2nr + l, the angular momentum runs through l = 0, 2, . . . , n for even, and l = 1, 3, . . . , n for odd n. There are (n + 1)(n + 2)/2 degenerate levels. Removing a factor rl from Rn (r), and defining Rn (r) = rl wn (r), the Schrödinger equation becomes 1 d2 1 2l + D − 1 d 1 2 g 4 − − + r + r wn (r) = E (n) wn (r). (3C.81) 2 dr2 2 r dr 2 4 The second term modifies the differential equation (3C.6) to k

X ′ (2l + D − 1) ′ 1 (n) rΦ′k (r)−2n′ Φk (r) = Φ′′k (r)+ Φk (r)+r4 Φk−1 (r)+ (−1)k Ek′ Φk−k′ (r). (3C.82) 2 2r ′ k =1

The extra terms change the recursion relation (3C.15) into k

(p − 2n′ )Apk =

X ′ 1 (n) (−1)k Ek′ Apk−k′ .(3C.83) [(p + 2)(p + 1) + (p + 2)(2l + D−1)]Ap+2 + Ap−4 k k−1 + 2 ′ k =1

′

For even n = 2n + l with l = 0, 2, 4, . . . , n, we normalize the wave functions by setting Ck0 = (2l + D)δ0k ,

(3C.84)

rather than (3C.12), and obtain ′

2(p − n

′

′ )Ckp

′

′

= [(2p + 1)(p + 1) + (p

′

′ + 1)(l + D/2 − 1/2)]Ckp +1

+

p′ −2 Ck−1

−

k X

′

p Ck1′ Ck−k ′ , (3C.85)

k′ =1

instead of (3C.20). For odd n = 2n′ + l with l = 1, 3, 5, . . . , n, the equations analogous to (3C.13) and (3C.22) are Ck1 = 3(2l + D)δ0k

(3C.86)

and ′

′

′

p −2 2(p′ − n′ )Ckp = [(2p′ + 3)(p′ + 1) + (p′ + 3/2)(l + D/2−1/2)]Ckp +1 + Ck−1 −

In either case, the expansion coefficients of the energy are given by (n)

Ek

=−

(−1)k 2l + D + 1 1 Ck . 2 2l + D

k X

′

p Ck1′ Ck−k ′ . (3C.87)

k′ =1

(3C.88)


Feynman Integrals for T 6= 0

Appendix 3D

3C.7

363

Interaction r2q in D Dimensions

A further extension of the recursion relation applies to interactions gx2q /4. Then Eqs. (3C.20) and ′ (3C.22) are changed in the second terms on the right-hand side which become Ckp −q . In a first ′ step, these equations are now solved for Ckp for 1 ≤ p ≤ qk + n, starting with p′ = qk + n′ and lowering p′ down to p′ = n′ + 1. As before, the knowledge of the coefficients Ck1 (which determine the yet unknown energies and are contained in the last term of the recursion relations) is not required for p′ > n′ . The second and third steps are completely analogous to the case q = 2. The same generalization applies to the D-dimensional case.

3C.8

Polynomial Interaction in D Dimensions

If the Schrödinger equation has the general form 1 d2 1D−1 d l(l + D − 2) 1 2 − − + + r 2 2 dr 2 r dr 2r2 2 i g 4 6 + (a4 r + a6 r + . . . + a2q x2q ) Rn (r) = E (n) Rn (r), 4

(3C.89)

we simply have to replace in the recursion relations (3C.85) and (3C.87) the second term on the right-hand side as follows ′

′

′

′

p −2 p −2 p −3 p −q Ck−1 → a4 Ck−1 + a6 Ck−1 + . . . + a2q Ck−1 ,

(3C.90)

and perform otherwise the same steps as for the potential gr2q /4 alone.

Appendix 3D

Feynman Integrals for T = /0

The calculation of the Feynman integrals (3.547) can be done straightforwardly with the help of the symbolic program Mathematica. The first integral in Eqs. (3.547) is trivial. The second and forth integrals are simple, since one overall integration over, say, τ3 yields merely a factor ¯hβ, due to translational invariance of the integrand along the τ -axis. The triple integrals can then be split as Z h¯ βZ h¯ βZ h¯ β dτ1 dτ2 dτ3 f (|τ1 − τ2 |, |τ2 − τ3 |, |τ3 − τ1 |) 0

0

=h ¯β

Z

0 h ¯ βZ h ¯β

0

=h ¯β

Z

0

0 h ¯β

dτ1 dτ2 f (|τ1 − τ2 |, |τ2 |, |τ1 |)

dτ2

Z

0

τ2

dτ1 f (τ2 − τ1 , τ2 , τ1 ) +

(3D.1) Z

0

h ¯β

dτ2

Z

h ¯β

τ2

!

dτ1 f (τ1 − τ2 , τ2 , τ1 ) ,

to ensure that the arguments of the Green function have the same sign in each term. The lines represent the thermal correlation function G(2) (τ, τ ′ ) =

¯ cosh ω[|τ − τ ′ | − ¯hβ/2] h . 2M ω sinh(ω¯hβ/2)

(3D.2)

With the dimensionless variable x ≡ ω¯hβ, the result for the quantities α2L V defined in (3.547) in the Feynman diagrams with L lines and V vertices is 1 x coth , 2 2 1 1 = (x + sinh x) , 8 sinh2 x2

a2 =

(3D.3)

α42

(3D.4)

364

3 External Sources, Correlations, and Perturbation Theory 1 x 1 x 3x x 2 −3 cosh + 2 x cosh + 3 cosh + 6 x sinh , 64 sinh3 x2 2 2 2 2 1 1 = (6 x + 8 sinh x + sinh 2x) , 256 sinh4 x2 1 x 1 x 3x = −40 cosh + 24 x2 cosh + 35 cosh 4096 sinh5 x2 2 2 2 x 3x 5x + 72 x sinh + 12 x sinh , + 5 cosh 2 2 2 1 1 = − 48 + 32 x2 − 3 cosh x + 8 x2 cosh x 16384 sinh6 x2 + 48 cosh 2 x + 3 cosh 3 x + 108 x sinhx , 1 1 = (5 +24 cosh x) , 24 sinh2 x2 1 x 1 x 3x = 3 x cosh + 9 sinh + sinh , 72 sinh3 x2 2 2 2 1 1 (30 x + 104 sinhx + 5 sinh2x) . = 2304 sinh4 x2

α63 =

(3D.5)

α82

(3D.6)

α10 3

α12 3

α62 α83 α10 3′

(3D.7)

(3D.8) (3D.9) (3D.10) (3D.11)

For completeness, we have also listed the integrals α62 , α83 , and α10 3′ , corresponding to the three diagrams ,

,

,

(3D.12)

respectively, which occur in perturbation expansions with a cubic interaction potential x3 . These will appear in a modified version in Chapter 5. In the low-temperature limit where x = ω¯hβ → ∞, the x-dependent factors α2L V in Eqs. (3D.3)– (3D.11) converge towards the constants 1/2, 1/4, 3/16, 1/32, 5/(8 · 25 ),

3/(8 · 26 ),

1/12, 1/18, 5/(9 · 25 ),

(3D.13)

respectively. From these numbers we deduce the relations (3.550) and, in addition, a62 →

2 6 a , 3

a83 →

8 8 a , 9

a10 3′ →

5 10 a . 9

(3D.14)

In the high-temperature limit x → 0, the Feynman integrals h ¯ β(1/ω)V −1 a2L V with L lines and V V L vertices diverge like β (1/β) . The first V factors are due to the V -integrals over τ , the second are the consequence of the product of n/2 factors a2 . Thus, a2L V behaves for x → 0 like a2L V

∝

¯ h Mω

L

xV −1−L .

(3D.15)

Indeed, the x-dependent factors α2L V in (3D.3)–(3D.11) grow like α2 α42 α63 α82 α10 3

≈

1/x + x/12 + . . . ,

≈ ≈

1/x + x3 /720, 1/x + x5 /30240 + . . . ,

≈

1/x3 + x/240 − x3 /15120 + x7 /6652800 + . . . ,

≈

1/x3 + x/120 − x3 /3780 + x5 /80640 + . . . ,


Feynman Integrals for T 6= 0

Appendix 3D α12 3 α62 α83 α10 3′

≈ ≈

≈ ≈

365

1/x4 + 1/240 + x2 /15120 − x6 /4989600 + 701 x8 /34871316480 + . . . , 1/x2 + x2 /240 − x4 /6048 + . . . ,

1/x2 + x2 /720 − x6 /518400 + . . . , 1/x3 + x/360 − x5 /1209600 + 629 x9 /261534873600 + . . . .

(3D.16)

For the temperature behavior of these Feynman integrals see Fig. 3.16. We have plotted the reduced Feynman integrals a ˆ2L V (x) in which the low-temperature behaviors (3.550) and (3D.14) 2L have been divided out of aV .

1.2

a2

1 0.8 0.6 0.4 0.2

a ˆ2L V L/x

0.1 0.2 0.3 0.5 0.4 2L Figure 3.16 Plot of reduced Feynman integrals a ˆV (x) as a function of L/x = LkB T /¯ hω. The integrals (3D.4)–(3D.11) are indicated by decreasing dash-lengths. The integrals (3D.4) and (3D.5) for a42 and a63 can be obtained from the integral (3D.3) for a2 by the operation n n ∂ ¯hn 1 ∂ ¯hn − 2 = − , (3D.17) n!M n ∂ω n!M n 2ω ∂ω with n = 1 and n = 2, respectively. This follows immediately from the fact that the Green function ′ G(2) ω (τ, τ ) =

∞ X ′ 1 ¯h e−iωm (τ −τ ) 2 , ¯hM β m=−∞ ωm + ω 2

(3D.18)

with ω 2 shifted to ω 2 + δω 2 can be expanded into a geometric series ∞ X ′ 1 ¯h δω 2 ¯h2 (2) ′ −iω (τ −τ ) m pace−1.9cmG√ω2 +δω2 (τ, τ ) = e − 2 + ω2 2 + ω 2 )2 ¯hM β m=−∞ ωm ¯ (ωm h # 2 2 δω ¯h3 pace1.0cm + +. . . , (3D.19) 2 ¯h (ωm + ω 2 )3 which corresponds to a series of convoluted τ -integrals Z M δω 2 h¯ β (2) ′ (2) ′ G√ω2 +δω2 (τ, τ ′ ) = G(2) (τ, τ ) − dτ1 G(2) ω ω (τ, τ1 )Gω (τ1 , τ ) ¯h 0 2 Z h¯ β Z h¯ β M δω 2 (2) (2) ′ + dτ1 dτ2 G(2) ω (τ, τ1 )Gω (τ1 , τ2 )Gω (τ2 , τ ) + . . . . ¯h 0 0

(3D.20)

In the diagrammatic representation, the derivatives (3D.17) insert n points into a line. In quantum field theory, this operation is called a mass insertion. Similarly, the Feynman integral (3D.7) is obtained from (3D.6) via a differentiation (3D.17) with n = 1 [see the corresponding diagrams in (3.547)]. A factor 4 must be removed, since the differentiation inserts a point into each of the four

366


lines which are indistinguishable. Note that from these rules, we obtain directly the relations 1, 2, and 4 of (3.550). Note that the same type of expansion allows us to derive the three integrals from the one-loop diagram (3.546). After inserting (3D.20) into (3.546) and re-expanding the logarithm we find the series of Feynman integrals ω 2 +δω 2

− −−→

M δω 2 + ¯h

−

M δω 2 ¯h

2

1 2

+

M δω 2 ¯h

3

1 3

−... ,

from which the integrals (3D.3)–(3D.5) can be extracted. As an example, consider the Feynman integral 1 = h ¯ β a42 . ω It is obtained from the second-order Taylor expansion term of the tracelog as follows: 2 1 1 ¯h2 ∂ [−2βVω ]. (3D.21) − ¯hβ a41 = 2 ω 2!M 2 ∂ω 2 A straightforward calculation, on the other hand, yields once more a42 of Eq. (3D.5).

Notes and References The theory of generating functionals in quantum field theory is elaborated by J. Rzewuski, Field Theory, Hafner, New York, 1969. For the usual operator derivation of the Wick expansion, see S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper and Row, New York, 1962, p. 435. The derivation of the recursion relation in Fig. 3.10 was given in H. Kleinert, Fortschr. Physik. 30, 187 (1986) (http://www.physik.fu-berlin.de/~kleinert/82), Fortschr. Physik. 30, 351 (1986) (ibid.http/84). See in particular Eqs. (51)–(61). Its efficient graphical evaluation is given in H. Kleinert, A. Pelster, B. Kastening, M. Bachmann, Recursive Graphical Construction of Feynman Diagrams and Their Multiplicities in x4 - and in x2 A-Theory, Phys. Rev. D 61, 085017 (2000) (hep-th/9907044). This paper develops a Mathematica program for a fast calculation of diagrams beyond five loops, which can be downloaded from the internet at ibid.http/b3/programs. The Mathematica program solving the Bender-Wu-like recursion relations for the general anharmonic potential (3C.26) is found in the same directory. This program was written in collaboration with W. Janke. The path integral calculation of the effective action in Section 3.23 can be found in R. Jackiw, Phys. Rev. D 9, 1686 (1974). See also C. De Dominicis, J. Math. Phys. 3, 983 (1962), C. De Dominicis and P.C. Martin, ibid. 5, 16, 31 (1964), B.S. DeWitt, in Dynamical Theory of Groups and Fields, Gordon and Breach, N.Y., 1965, A.N. Vassiliev and A.K. Kazanskii, Teor. Math. Phys. 12, 875 (1972), J.M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D 10, 1428 (1974), and the above papers by the author in Fortschr. Physik 30. H. Kleinert, PATH INTEGRALS


367

The path integral of a particle in a dissipative medium is discussed in A.O. Caldeira and A.J. Leggett, Ann. Phys. 149, 374 (1983), 153; 445(E) (1984). See also A.J. Leggett, Phys. Rev. B 30, 1208 (1984); A.I. Larkin, and Y.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 86, 719 (1984) [Sov. Phys. JETP 59, 420 (1984)]; J. Stat. Phys. 41, 425 (1985); H. Grabert and U. Weiss, Z. Phys. B 56, 171 (1984); L.-D. Chang and S. Chakravarty, Phys. Rev. B 29, 130 (1984); D. Waxman and A.J. Leggett, Phys. Rev. B 32, 4450 (1985); P. Hänggi, H. Grabert, G.-L. Ingold, and U. Weiss, Phys. Rev. Lett. 55, 761 (1985); D. Esteve, M.H. Devoret, and J.M. Martinis, Phys. Rev. B 34, 158 (1986); E. Freidkin, P. Riseborough, and P. Hänggi, Phys. Rev. B 34, 1952 (1986); H. Grabert, P. Olschowski and U. Weiss, Phys. Rev. B 36, 1931 (1987), and in the textbook U. Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 1993. See also Notes and References in Chapter 18. For alternative approaches to the damped oscillator see F. Haake and R. Reibold, Phys. Rev. A 32, 2462 (1985), A. Hanke and W. Zwerger, Phys. Rev. E 52, 6875 (1995); S. Kehrein and A. Mielke, Ann. Phys. (Leipzig) 6, 90 (1997) (cond-mat/9701123). X.L. Li, G.W. Ford, and R.F. O’Connell, Phys. Rev. A 42, 4519 (1990). The effective potential (5.259) was derived in D dimensions by H. Kleinert and B. Van den Bossche, Nucl. Phys. B 632, 51 (2002) (http://arxiv.org/ abs/cond-mat/0104102">cond-mat/0104102). By inserting D = 1 and changing the notation appropriately, one finds (5.259). The finite-temperature expressions (3.760)–(3.763) are taken from S.F. Brandt, Beyond Effective Potential via Variational Perturbation Theory, M.S. thesis, FU-Berlin 2004 (http://hbar.wustl. edu/~sbrandt/diplomarbeit.pdf). See also S.F. Brandt, H. Kleinert, and A. Pelster, J. Math. Phys. 46, 032101 (2005) (quant-ph/0406206) .


Take the gentle path. George Herbert (1593-1633), Discipline

4 Semiclassical Time Evolution Amplitude The path integral approach renders a clear intuitive understanding of quantummechanical effects in terms of quantum fluctuations, exhibiting precisely how the laws of classical mechanics are modified by these fluctuations. In some limiting situations, the modifications may be small, for instance, if an electron in an atom is highly excited. Its wave packet encircles the nucleus in almost the same way as a point particle in classical mechanics. Then it is relatively easy to calculate quite accurate quantum-mechanical amplitudes by expanding them around classical expressions in powers of the fluctuation width.

4.1

Wentzel-Kramers-Brillouin (WKB) Approximation

In Schrödinger’s theory, an important step towards understanding the relation between classical and quantum mechanics consists in proving that the center of a Schrödinger wave packet moves like a classical particle. The approach to the classical limit is described by the so-called eikonal approximation, or the Wentzel-KramersBrillouin approximation (short: WKB approximation), which proceeds as follows: First, one rewrites the time-independent Schrödinger equation of a point particle (

in the form

h ¯2 2 − ∇ − [E − V (x)] ψ(x) = 0 2M )

h

i

−¯ h2 ∇2 − p2 (x) ψ(x) = 0,

where

p(x) ≡

q

2M[E − V (x)]

(4.1)

(4.2) (4.3)

is the local classical momentum of the particle. In a second step, one re-expresses the wave function as an exponential ψ(x) = eiS(x)/¯h . 368

(4.4)

4.1 Wentzel-Kramers-Brillouin (WKB) Approximation

369

For the exponent S(x), called the eikonal , the Schrödinger equation amounts to the a differential equation : −i¯ h∇2 S(x) + [∇S(x)]2 − p2 (x) = 0.

(4.5)

To solve this equation approximately, one assumes that the function p(x) shows little relative change over the de Broglie wavelength λ≡

2π 2π¯ h ≡ , k(x) p(x)

i.e.,

2π¯ h ∇p(x) ε≡ ≪ 1. p(x) p(x)

(4.6)

(4.7)

This condition is called the WKB condition. In the extreme case of p(x) being a constant the condition is certainly fulfilled and the Riccati equation is solved by the trivial eikonal S = px + const , (4.8) which makes (4.4) a plain wave. For slow variations, the first term in the Riccati equation is much smaller than the others and can be treated systematically in a “smoothness expansion”. Since the small ratio (4.7) carries a prefactor h ¯ , the Planck constant may be used to count the powers of the smallness parameter ε, i.e., it may formally be considered as a small expansion parameter. The limit h ¯ → 0 of the equation determines the lowest-order approximation to the eikonal, S0 (x), by [∇S0 (x)]2 − p2 (x) = 0. (4.9) Being independent of h ¯ , this is a classical equation. Indeed, it is equivalent to the Hamilton-Jacobi differential equation of classical mechanics: For time-independent systems, the action can be written as A(x, t) = S0 (x) − tE, where S0 (x) is defined by S0 (x) ≡

Z

˙ dt p(t)x(t),

(4.10)

(4.11)

and E is the constant energy of the orbit under consideration. The action solves the Hamilton-Jacobi equation (1.64). In three dimensions, it is a function A(x, t) of the orbital endpoints. According to Eq. (1.62), the derivative of A(x, t) is equal to the momentum p. Since E is a constant, the same thing holds for S0 (x, t). Hence p ≡ ∇A(x) = ∇S0 (x).

(4.12)

In terms of the action A, the Hamilton-Jacobi equation reads 1 (∇A)2 + V (x) = −∂t A. 2M

(4.13)

370

4 Semiclassical Time Evolution Amplitude

By inserting Eqs. (4.10) and (4.12), we recover Eq. (4.9). This is why S0 (x) is also called the classical eikonal . The corrections to the classical eikonal are calculated most systematically by imagining h ¯ 6= 0 to be a small quantity and expanding the eikonal around S0 (x) in a power series in h ¯: S = S0 − i¯ hS1 + (−i¯ h)2 S2 + (−i¯ h)3 S3 + . . . .

(4.14)

This is called the semiclassical expansion of the eikonal. Inserting it into the Riccati equation, we find the sequence of WKB equations (∇S0 )2 − p2 ∇2 S0 + 2∇S0 · ∇S1 ∇2 S1 + (∇S1 )2 + 2∇S0 · ∇S2 ∇2 S2 + 2∇S1 · ∇S2 + 2∇S0 · ∇S3 .. . 2

∇ Sn +

n+1 X

m=0

= = = =

0, 0, 0, 0,

∇Sm · ∇Sn+1−m = 0, .. .

(4.15)

.

Note that these equations involve only the vectors qn = ∇Sn

(4.16)

and allow for a successive determination of S0 , S1 , S2 , . . . , giving higher and higher corrections to the eikonal S(x). In one dimension we recognize in (4.5) the √ Riccati differential equation (2.539) fulfilled by ∂τ S(x), if we identify x with τ 2M, and v(τ ) with E − V (τ ), where (4.5) reads −i¯ h∂τ [∂τ S(τ )] + [∂τ S(τ )]2 = p2 (τ ) = v(τ ). (4.17) If we re-express the expansion terms (2.544) in terms of w(τ ) = replace w(τ ), w ′(τ ), . . . by p(x), p′ (x), . . . , and find directly 1 p′ (x) , 2 p(x) " # 1 p′′ (x) 3 p′2 (x) q2 = ± − ≡ ∓p(x)ǫ(x), 4 p2 (x) 8 p3 (x)

q0 = ±p(x),

q

v(τ ), we may

q1 = −

1 q3 = ǫ′ (x), . . . . 2

(4.18)

The equation for q2 (x) defines also the quantity g(x) used in subsequent equations. The eikonal has the expansion S(x) = ±

Z

dx p(x)[1 + h ¯ 2 g(x)]

i +¯ h [log p(x) + h ¯ 2 g(x)] ± . . . . 2


4.1 Wentzel-Kramers-Brillouin (WKB) Approximation

371

Keeping only terms up to the order h ¯ , which is possible if h ¯ 2 |ǫ(x)| ≪ 1, we find the (as yet unnormalized) WKB wave function Rx ′ ′ 1 ψWKB (x) = q e±(i/¯h) dx p(x ) . p(x)

(4.20)

In the classically accessible regime V (x) ≤ E, this is an oscillating wave function; in the inaccessible regime V (x) ≥ E, it decreases or increases exponentially. The transition from one to the other is nontrivial since for V (x) ≈ E, the WKB approximation breaks down. After some analytic work1 , however, it is possible to derive simple connection rules for the linearly independent solutions. Let k(x) ≡ p(x)/¯ h in the oscillating regime and κ(x) ≡ |p(x)|/¯ h in the exponential regime. Suppose that there is a crossover at x = a connecting an inaccessible regime on the left of x = a with an accessible one on the right. Then the connection rules are V (x) > E 1 − R a dx′ κ √ e x κ 1 R a dx′ κ √ ex κ

V (x) < E x 2 √ cos ←− − −− −→ −→ dx′ k − a k Z x 1 ←− ←− − −−→ − √ sin dx′ k − a k

Z

π , 4 π . 4

(4.21) (4.22)

In the opposite situation at the point x = b, they turn into V (x) < E b 2 √ cos dx′ k − x k Z b 1 − √ sin dx′ k − x k

Z

!

π 4 ! π 4

←− ←− − −−→ ←− − −− −→ −→

V (x) > E 1 − R x dx′ κ √ e b , κ 2 Rx ′ √ e b dx κ . κ

(4.23) (4.24)

The connection rules can be used safely only along the direction of the double arrows. For their derivation one solves the Schrödinger equation exactly in the neighborhood of the turning points where the potential rises or falls approximately linearly. These solutions are connected with adjacent WKB wave functions. The connection formulas can also be found directly by a formal trick: When approaching the dangerous turning points, one escapes into the complex x-plane and passes around the singularities at a finite distance. This has to be sufficiently large to preserve the WKB condition (4.7), but small enough to allow for the linear approximation of the potential near the turning point. Take for example the rule (4.23). When approaching the turning √ point at x = b from the right, the function κ(x) is approximately proportional to x − b. Going around this zero in the upper complex half-plane takes 1

R.E. Langer, Phys. Rev. 51 , 669 (1937). See also W.H. Furry, Phys. Rev. 71 , 360 (1947), and the textbooks S. Fl¨ ugge, Practical Quantum Mechanics, Springer, Berlin, 1974; L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1955; N. Fröman and P.O. Fröman, JWKBApproximation, North-Holland, Amsterdam, 1965.

372


√ −1 −i R x dx′ k κ(x) into ik(x) and the wave function κ e becomes e k e Rb . √ −1 i x dx′ k iπ/4 Going around the turning point in the lower half-plane produces e k e b . √ −1 Rb ′ The sum of the two terms is 2 k cos( x dx k − π/4). The argument does not show why one should use the sum rather than the average. This becomes clear only after a more detailed discussion found in quantum-mechanical textbooks2 . The simplest derivation of the connection formulas is based on the large-distance behaviors (2.617) and (2.622) of wave the function to the right and left of a linearly rising potential and applying this to the linearly rising section of the general potential near the turning point. In a simple potential well, the function p(x) has two zeros, say one at x = a and one at x = b. The bound-state wave functions must satisfy the boundary condition to vanish exponentially fast at x = ±∞. Imposing these, the connection formulas lead to the semiclassical or Bohr-Sommerfeld quantization rule √

Z

b a

dx k(x) = (n + 1/2)π,

−1 −

Rx b

dx′ κ

−iπ/4

n = 0, ±1, ±2, . . . .

(4.25)

For the harmonic oscillator, the semiclassical quantization rule (4.25) gives the exact energy levels. Indeed, for an energy E, the classical crossover points with V (xE ) = E are s 2E , (4.26) xc = ± Mω 2 to be identified in (4.25) with a and b, respectively. Inserting further p(x) k(x) = = h ¯

s

1 2M E − Mω 2 x2 , 2 2 h ¯

(4.27)

we obtain the WKB approximation for the energy levels Z

xE

−xE

dx′ k(x) =

E π = (n + 1/2)π, h ¯ω

(4.28)

which indeed coincides with the exact ones. Only nonnegative values n = 0, 1, 2, . . . lead to oscillatory waves. As an example consider the quartic potential V (x) = gx4 /2 for which the Schrödinger equation cannot be solved exactly. Inserting this into equation (4.28), we obtain q 1 Z xE dx 2M(E − gx4 /4) ≡ ν(E)π = (n + 1/2)π, (4.29) h ¯ −xE with the turning points ±xE = ±(4E/g)1/4 . The integral is done using the formula3 Z

0

1

dt tµ−1 (1 − tλ )ν−1 =

1 B(µ/λ, ν) λ

(4.30)

2

See for example E. Merzbacher, Quantum Theory, John Wiley & Sons, New York, 1970, p. 122; M.L. Goldberger and K.M. Watson, Collision Theory, John Wiley & Sons, New York, 1964, p. 324. The analytical argument is given in L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965, p. 158. 3 I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 3.251.1. H. Kleinert, PATH INTEGRALS

4.2 Saddle Point Approximation

373

which for µ = 1, λ = 4, and ν = 3/2 yields (1/4)B(1/4, 3/2), so√ that the left-hand side of Eq.(4.29) can be written with Γ(1/4) = 2π/Γ(3/4) 2 as ν(E)π ≡ E 3/4 2M 1/2 π 3/2 /3¯ hg 1/4 Γ2 (3/4), and (4.29) determines the energy with principal quantum numebr n in the Bohr-Sommerfeld approximation by the consition ν(E) = n + 1/2 resulting in (n) EBS

=

g¯ h 4M 2 ω 3

(n) h ¯ ωκBS

!1/3

,

(4.31)

with (n)

κBS =

1 π 2/3

4/3

3 2

Γ8/3 (3/4) (n + 12 )4/3 ≈ 0.688 253 702 × 2(n + 12 )4/3 .

(4.32)

This large-n result may be compared with the precise of κ(0) = 0.667986 . . . to be derived in Section 5.19 (see Table 5.9).

4.2

Saddle Point Approximation

Let us now look at the semiclassical expansion within the path integral approach to quantum mechanics. Consider the time evolution amplitude (xb tb |xa ta ) =

Z

Dx eiA[x]/¯h ,

(4.33)

imagining Planck’s constant h ¯ to be again a free parameter which is very small compared to the typical fluctuations of the action. With h ¯ appearing in the denominator of an imaginary exponent we see that in the limit h ¯ → 0, the path integral becomes a sum of rapidly oscillating terms which will approximately cancel each other. This phenomenon is known from ordinary integrals Z

√

dx ia(x)/¯h e , 2πi¯ h

(4.34)

which converge to zero for h ¯ → 0 according to the Riemann-Lebesgue lemma. The precise behavior is given by the saddle point expansion of integrals which we shall first recapitulate.

4.2.1

Ordinary Integrals

The evaluation of an integral of the type (4.34) proceeds for small h ¯ via the so-called saddle point approximation. In the limit h ¯ → 0, the integral is dominated by the extremum of the function a(x) with the smallest absolute value, call it xcl (assuming it to be unique, for simplicity), where a′ (xcl ) = 0.

(4.35)

374


In the path integral, the point xcl in this example corresponds to the classical orbit for which the functional derivative vanishes. This is the reason for using the subscript cl. For x near the extremum, the oscillations of the integrand are weakest. The leading oscillatory behavior of the integral is given by Z

∞

−∞

√

dx ia(x)/¯h ¯h→0 e − −−→ const × eia(xcl )/¯h , 2πi¯ h

(4.36)

with a constant proportionality factor independent of h ¯ . This can be calculated by expanding a(x) around its extremum as 1 1 a(x) = a(xcl ) + a′′ (xcl )(δx)2 + a(3) (xcl )(δx)3 + . . . , 2 3!

(4.37)

where δx ≡ x − xcl is the deviation from xcl . It is the analog of the quantum fluctuation introduced in Section 2.2. Due to (4.35), the linear term in δx is absent. If a′′ (xcl ) 6= 0 and the higher derivatives are neglected, the integral is of the Fresnel type and can be done, yielding Z

∞

−∞

dx ia(x)/¯h √ e → eia(xcl )/¯h 2πi¯ h

Z

∞

−∞

dδx ia′′ (xcl )(δx)2 /2¯h eia(xcl )/¯h √ e =q . 2πi¯ h a′′ (xcl )

(4.38)

The right-hand side is the saddle point approximation to the integral (4.34). The saddle point approximation may be viewed as the consequence of the classical limit of the exponential function: √ h→0 2πi¯ h ia(x)/¯ h ¯ e − −−→ q δ(x − xcl ) . (4.39) ′′ a (xcl )

Corrections can be calculated perturbatively by expanding the integral in powers of h ¯ , leading to what is called the saddle point expansion. For this we expand the remaining exponent in powers of δx: i 1 (3) 1 exp a (xcl )(δx)3 + a(4) (xcl )(δx)4 + . . . h ¯ 3! 4! i 1 (3) 1 =1+ a (xcl )(δx)3 + a(4) (xcl )(δx)4 + . . . h ¯ 3! 4! 1 1 (3) − 2 a (xcl )2 (δx)6 + . . . + . . . h ¯ 72

(4.40)

and perform the resulting integrals of the type Z

∞

−∞

(n − 1)!! dδx ia′′ (xcl )(δx)2 /2¯h (i¯ h)n/2 , n = even, n ′′ (1+n)/2 √ e (δx) = [a (x )] cl   2πi¯ h 0, n = odd.   

(4.41)

q

Each factor δx in (4.40) introduces a power h ¯ /a′′ (xcl ). This is the average relative size of the quantum fluctuations. The increasing powers of h ¯ ensure the decreasing H. Kleinert, PATH INTEGRALS


375

importance of the higher terms for small h ¯ . For instance, the fourth-order term (4) 4 a (xcl )(δx) /4! is accompanied by h ¯ , and the lowest correction amounts to a factor 3!! h ¯ 1 − ia(4) (xcl ) . (4.42) 4! [a′′ (xcl )]2 The cubic term a(3) (xcl )(δx)3 /3! yields a factor h ¯ 5!! 1 + i[a(3) (xcl )]2 . ′′ 72 [a (xcl )]3 Thus we obtain the saddle point expansion to the integral is

(4.43)

dx ia(x)/¯h 1 a(4) (xcl ) eia(xcl )/¯h 5 [a(3) (xcl )]2 √ e =q 1 − i¯ h − + O(¯ h2 ) . ′′ (x )]2 ′′ (x )]3 ′′ 8 [a 24 [a −∞ 2πi¯ h cl cl a (xcl ) (4.44) Expectation values in this integral can also be expanded in powers of h ¯ , for instance hxi = xcl + hδxi where Z

(

∞

"

#

1 a(3) (xcl ) 2 [a′′ (xcl )]2 " # (3) (4) 5 [a(3) (xcl )]3 1 a(5) (xcl ) 2 2 a (xcl )a (xcl ) −h ¯ − − + O(¯ h3 ). 3 [a′′ (xcl )]4 8 [a′′ (xcl )]5 8 [a′′ (xcl )]3

)

hδxi = −i¯ h

(4.45)

Since the saddle point expansion is organized in powers of h ¯ , it corresponds precisely to the semiclassical expansion of the eikonal in the previous section. The saddle point expansion can be used for very small h ¯ to calculate an integral with increasing accuracy. It is impossible, however, to achieve arbitrary accuracy since the resulting series is divergent for all physically interesting systems. It is merely an asymptotic series whose usefulness decreases rapidly with an increasing size of the expansion parameter. A variational expansion must be used to achieve convergence. For more details, see Sections 5.15 and 17.9. An important property of the semiclassical approximation is that Fourier transformations become very simple. Consider the Fourier integral Z

∞

−∞

dx e−ipx/¯h eia(x)/¯h .

(4.46)

For small h ¯ , this can be done in the saddle point approximation according to the rule (4.39), and obtain Z ∞ √ ei[a(xcl )−pxcl ]/¯h −ipx/¯ h ia(x)/¯ h , (4.47) dx e e → 2πi¯ h q −∞ a′′ (xcl )

where xcl is now the extremum of the action with a source term p, i.e., it is determined by the equation p = a′ (xcl ). Note that the formula holds also if the exponential carries an x-dependent prefactor, since the x-dependence gives only corrections of the order of h ¯ in the exponent: Z ∞ √ ei[a(xcl )−pxcl ]/¯h dx e−ipx/¯h c(x)eia(x)/¯h → 2πi¯ h c(xcl ) q . (4.48) −∞ a′′ (xcl )

376


If the equation p = a′ (xcl ) is inverted to find xcl as a function xcl (p), the exponent a(xcl ) − pxcl may be considered as a function of p: b(p) = a(xcl ) − pxcl ,

p = a′ (xcl ).

(4.49)

This function is recognized as being the Legendre transform of the function a(x) [recall (1.9)]. The original function a(x) can be recovered from b(p) via an inverse Legendre transformation a(x) = b(pcl ) + xpcl , x = −b′ (pcl ). (4.50)

This formalism is the basis for many thermodynamic calculations. For large statistical systems, fluctuations of global properties such as the volume and the total internal energy are very small so that the saddle point approximation is very good. In this chapter, the formalism will be applied on many occasions.

4.2.2

Path Integrals

A similar saddle point expansion exists for the path integral (4.33). For small h ¯ , the amplitudes eiA/¯h from the various paths will mostly cancel each other by interference. The dominant contribution comes from the functional regime where the oscillations are weakest, which is the extremum of the action δA[x] = 0.

(4.51)

This gives the classical Euler-Lagrange equation of motion. For a point particle with the action Z tb M 2 x˙ − V (x) , (4.52) A[x] = dt 2 ta it reads M x¨ = −V ′ (x). (4.53)

Let xcl (t) denote the classical orbit. After multiplying (4.53) by x, ˙ an integration in t yields the law of energy conservation M 2 x˙ + V (xcl ) = const . 2 cl This implies that the classical momentum E=

(4.54)

pcl (t) ≡ M x˙ cl (t)

(4.55)

pcl (t) = p(xcl (t)),

(4.56)

can be written as

where p(x) is the local classical momentum defined in (4.3). From (4.54), the time dependence of the classical orbit xcl (t) is given by t − t0 =

Z

0

xcl

M dx = p(x)

Z

0

xcl

M dx q . 2M[E − V (x)]

(4.57)



377

When solving the integral on the right-hand side we find for a given time interval t = tb − ta the energy for which a pair of positions xa , xb can be connected by a classical orbit: E = E(xb , xa ; tb − ta ). (4.58) The classical action is given by Z

A[xcl ] =

tb

ta tb

Z

=

M 2 x˙cl − V (xcl ) 2

t

dt [pcl (t)x˙ cl − H(pcl , xcl )]

xa

dxp(x) − (tb − ta )E.

Z axb

=

dt

(4.59)

Just like E, the classical action is a function of xb , xa and tb − ta , to be denoted by A(xb , xa ; tb − ta ), for which (4.59) reads more explicitly A(xb , xa ; tb − ta ) ≡

Z

xb

xa

dx p(x) − (tb − ta )E(xb , xa ; tb − ta ).

(4.60)

Recalling (4.11), the first term on the right-hand side is seen to be the classical eikonal Z xb S(xb , xa ; E) = dx p(x), (4.61) xa

where E is the energy function (4.58) and p(x) is given by (4.3), The eikonal may be viewed as a functional SE [x] of paths x(t) of a fixed energy, in which case it is extremal on the classical orbits. This was observed as early as 1744 by Maupertius. The proof for this is quite simple: We insert the classical momentum (4.3) into SE [x] and write SE [x] ≡

Z

p(x)dx =

Z

dt p(x)x˙ =

Z

dt LE (x, x) ˙ =

Z

q

dt 2M[E − V (x)]x, ˙ (4.62)

thus introducing a Lagrangian LE (x, x) ˙ for this problem. Lagrange equation reads

The associated Euler-

d ∂LE ∂LE = . dt ∂ x˙ ∂x

(4.63)

Inserting LE (x, x) ˙ = p(x)x˙ we find the correct equation of motion p˙ = −V ′ (x). There is an interesting geometrical aspect to this variational procedure. In order to see this let us go to D dimensions and write the eikonal (4.62) as SE [x] =

Z

˙ = dt LE (x, x)

Z

q

dt gij (x)x˙ i (t)x˙ j (t),

(4.64)

with an energy-dependent metric gij (x) = p2E (x)δij .

(4.65)

378


Then the Euler-Lagrange equations for x(t) coincides with the equation (1.72) for the geodesics in a Riemannian space with a metric gij (x). In this way, the dynamical problem has been reduced to a geometric problem. The metric gij (x) may be called dynamical metric of the space with respect to the potential V (x). This geometric view is further enhanced by the fact that the eikonal (4.62) is, in fact, independent of the parametrization of the trajectory. Instead of the time t we could have used any parameter τ to describe x(τ ) and write the eikonal (4.62) as SE [x] =

Z

dτ

q

gij (x)x˙ i (τ )x˙ j (τ ).

(4.66)

Einstein has certainly been inspired by this ancient description of classical trajectories when geometrizing the relativistic Kepler motion by attributing a dynamical Riemannian geometry to spacetime. It is worth pointing out a subtlety in this variational principle, in view of a closely related situation to be encountered later in Chapter 10. The variations are supposed to be carried out at a fixed energy M 2 (4.67) x˙ + V (x). 2 This is a nonholonomic constraint which destroys the independence of the variation δx(t) and δ x. ˙ They are related by E=

1 ∇V (x)δx. (4.68) M It is, however, possible to regain the independence by allowing for a simultaneous variation of the time argument in x(t) when varying x(t). As a consequence, we can no longer employ the standard equality δ x˙ = dδx/dt which is necessary for the derivation of the Euler-Lagrange equation (4.63). Instead, we calculate x˙ δ x˙ = −

δ x˙ =

dx + dδx d d − x˙ = δx − x˙ δt, dt + dδt dt dt

(4.69)

which shows that variation and time derivatives no longer commute with each other. Combining this with the relation (4.68) we see that the variations of x and x˙ can be made independent if we vary t along the orbit according to the relation x˙ 2

d d 1 δt = x˙ δx + ∇V (x)δx. dt dt M

(4.70)

With (4.69), the variations of the eikonal (4.64) are δSE [x] =

Z

dt

!

∂LE d ∂LE δx + δx + ∂ x˙ dt ∂x

Z

!

∂LE d dt LE − x˙ δt , ∂ x˙ dt

(4.71)

where we have kept the usual commutativity of variation and time derivative of the time itself. In the second integral, we may set −LE +

∂LE x˙ ≡ HE . ∂ x˙



379

˙ and The function HE arises from LE by the same combination of LE (x, x) ˙ ˙ as in a Legendre transformation which brings a Lagrangian to the ∂LE /∂ x(x, x) associated Hamiltonian [recall (1.13)]. But in contrast to the usual procedure we do not eliminate x˙ in favor of a canonical momentum variable ∂LE /∂ x˙ [recall (1.14)], ˙ Note that it is not equal to the energy. i.e., the HE is a function HE (x, x). The variation (4.71) shows that the extra variation δt of the time does not change the Euler-Lagrange equations for the above Lagrangian in Eq. (4.64), LE = q ˙ the associated HE vanishes identically, so 2M[E − V (x)]x˙ 2 . Being linear in x, that the second term disappears and we recover the ordinary equation of motion ∂LE d ∂LE = . dt ∂ x˙ ∂x

(4.73)

In general, however, we must keep the second term. Expressing dδt/dt via (4.70), we find "

#

x˙ d ∂LE dt − HE 2 δx δSE [x] = ∂ x˙ x˙ dt " # Z ∂LE 1 1 + dt − HE 2 ∇V (x) δx, ∂x x˙ M Z

(4.74)

and the general equation of motion becomes "

#

x˙ ∂LE d ∂LE 1 1 − HE 2 = − HE 2 ∇V (x), x˙ x˙ M dt ∂ x˙ ∂x

(4.75)

rather than (4.73). Let us illustrate this by rewriting the eikonal as a functional SE [x] =

Z

˙ =M dt L′E (x, x)

Z

dt x˙ 2 (t),

(4.76)

which is the same functional as (4.62) as long as the energy E is kept fixed. If we insert the new Lagrangian L′E into (4.75), we obtain the correct equation of motion ¨ = −∇V (x). Mx

(4.77)

In this case, the equation of motion can actually be found more directly. We vary the eikonal (4.76) as follows: δSE [x] = M

Z

δdt x˙ 2 + M

Z

dt x˙ δ x˙ + M

Z

˙ x. ˙ dt xδ

(4.78)

In the last term we insert the relation (4.69) and write δSE [x] = M

"Z

2

δdt x˙ +

Z

dt x˙ δ x˙ +

Z

#

Z d d dt x˙ δx − dt x˙ 2 δt . dt dt

(4.79)

The two terms containing δt cancel each other, so that relation (4.70) is no longer needed. Using now (4.68), we obtain directly the equation of motion (4.77).

380


With the help of the eikonal (4.61), we write the classical action (4.59) as A(xb , xa ; tb − ta ) ≡ S(xb , xa ; E) − (tb − ta )E,

(4.80)

where E is given by (4.58). The action has the property that its derivatives with respect to the endpoints xb , xa at a fixed tb − ta yield the initial and final classical momenta: ∂ A(xb , xa ; tb − ta ) = ±p(xb,a ). ∂xb,a

(4.81)

Indeed, the differentiation gives ∂A = p(xb ) + ∂xb

"Z

xb

xa

#

∂p(x) ∂E dx − (tb − ta ) , ∂E ∂xb

(4.82)

and using ∂p(x) M 1 = = , ∂E p(x) x˙

(4.83)

∂p(x) = ∂E

(4.84)

we see that Z

xb

xa

dx

Z

tb

ta

dt = tb − ta ,

so that the bracket in (4.82) vanishes, and (4.81) is indeed fulfilled [compare also (4.12)]. The relation (4.84) implies that the eikonal (4.61) has the energy derivative ∂ S(xb , xa ; E) = tb − ta . ∂E

(4.85)

As a conjugate relation, the derivative of the action with respect to the time tb at fixed xb gives the energy with a minus sign [compare (4.10)]: ∂ A(xb , xa ; tb − ta ) = −E(xb , xa ; tb − ta ). ∂tb

(4.86)

This is easily verified: "Z # xb ∂ ∂p ∂E A= dx − (tb − ta ) − E = −E. ∂tb ∂E ∂tb xa

(4.87)

Thus, the classical action function A(xb , xa ; tb − ta ) and the eikonal S(xb , xa ; E) are Legendre transforms of each other. The equation 1 (∂x A)2 + V (x) = ∂t A 2M



381

is, of course, the Hamilton-Jacobi equation (4.13) of classical mechanics. We have therefore found the leading term in the semiclassical approximation to the amplitude [corresponding to the approximation (4.36)]: h→0 ¯

(xb tb |xa ta ) − −−→ const × eiA(xb ,xa ;tb −ta )/¯h .

(4.89)

In general, this leading term will be multiplied by a fluctuation factor (xb tb |xa ta ) = eiA(xb ,xa ;tb −ta )/¯h F (xb , xa ; tb − ta ).

(4.90)

In contrast to the purely harmonic case in Eq. (2.146) this will depend on the initial and final coordinates xa and xb . The calculation of the leading contribution to the fluctuation factor is the next step in the saddle point expansion of the path integral (4.33). For this we expand the action (4.52) in the neighborhood of the classical orbit in powers of the fluctuations δx(t) = x(t) − xcl (t).

(4.91)

This yields the fluctuation expansion δA δx(t) δx(t) ta Z 1 tb δ2A ′ + dtdt δx(t)δx(t′ ) (4.92) ′ 2 ta δx(t)δx(t ) Z δ3A 1 tb + dtdt′ dt′′ δx(t)δx(t′ )δx(t′′ ) + . . . , 3! ta δx(t)δx(t′ )δx(t′′ )

A[x, x] ˙ = A[xcl ] +

Z

tb

dt

where all functional derivatives on the right-hand side are evaluated along the classical orbit x(t) = xcl (t). The linear term in the quantum fluctuation δx(t) is absent since A[x, x] ˙ is extremal at xcl (t). For a point particle, the quadratic term is 1 2

Z

tb ta

δ2A dtdt δx(t)δx(t′ ) = δx(t)δx(t′ ) ′

Z

tb

ta

M 1 dt (δ x) ˙ 2 + V ′′ (xcl (t))(δx)2 . 2 2 (4.93)

Thus the fluctuations behave like those of a harmonic oscillator with a timedependent frequency Ω2 (t) =

1 ′′ V (xcl (t)). M

(4.94)

By definition, the fluctuations vanish at the endpoints: δx(ta ) = 0, δx(tb ) = 0.

(4.95)

If we include only the quadratic terms in the fluctuation expansion (4.92), we can integrate out the fluctuations in the path integral (4.33). Since x(t) and δx(t)

382


differ only by a fixed additive function xcl (t), the measure of the path integral over x(t) transforms trivially into that over δx(t). Thus we conclude that the leading semiclassical limit of the amplitude is given by the product (xb tb |xa ta )sc = eiA(xb ,xa ;tb −ta )/¯h Fsc (xb , xa ; tb − ta ),

(4.96)

with the semiclassical fluctuation factor [compare (2.193)] Fsc (xb , xa ; tb − ta ) =

Z

Dδx(t) exp

i h ¯

Z

ta

tb

dt

M [δ x˙ 2 − Ω2 (t)δx2 ] 2

1 = q det (−∇∇ − Ω2 (t))−1/2 2πiǫ¯ h/M =

1

q

2πi¯ h(tb − ta )/M

v u u t

det (−∂t2 ) . det (−∂t2 − Ω2 (t))

(4.97)

In principle, we would now have to solve the differential equation [−∂t2 − Ω2 (t)]yn (t) = [−∂t2 − V ′′ (xcl (t))/M]yn (t) = λn yn (t),

(4.98)

and find the energies of the eigenmodes yn (t) of the fluctuations. The ratio of fluctuation determinants D0 det (−∂t2 ) = (4.99) D det (−∂t2 − Ω2 (t)) in the second line of (4.97) would then be found from the product of ratios of eigenvalues, λn /λ0n , where λ0n are the eigenvalues of the differential equation −∂t2 yn (t) = λ0n yn (t).

(4.100)

Fortunately, we can save ourselves all this work using the Gelfand-Yaglom method of Section 2.4 which provides a much simpler and more direct way of calculating fluctuation determinants with a time-dependent frequency without the knowledge of the eigenvalues λn .

4.3

Van Vleck-Pauli-Morette Determinant

According to the Gelfand-Yaglom method of Section 2.4, a functional determinant of the form det (−∂t2 − Ω2 (t)) is found by solving the differential equation (4.98) at zero eigenvalue [−∂t2 − Ω2 (t)]Da (t) = 0,

(4.101)

with the initial conditions Da (ta ) = 0,

D˙ a (ta ) = 1.

(4.102)


4.3 Van Vleck-Pauli-Morette Determinant

383

Then Da (tb ) is the desired fluctuation determinant. In Eq. (2.233), we have constructed the solution to these equations in terms of an arbitrary solution ξ(t) of the homogenous equation [−∂t2 − Ω2 (t)]ξ(t) = 0 as

(4.103)

dt′ . (4.104) ta ξ 2 (t′ ) In general, it is difficult to find an analytic solution to Eq. (4.103). In the present fluctuation problem, however, the time-dependent frequency Ω(t) has a special form Ω2 (t) = V ′′ (xcl (t))/M of (4.94). We shall now prove that, just as in the purely harmonic action in Section 2.5, all information on the fluctuation determinant is contained in the classical orbit xcl (t), and ultimately in the mixed spatial derivatives of the classical action A(xb , xa ; tb − ta ). In fact, the solution ξ(t) is simply equal to the velocity Dren = ξ(t)ξ(ta)

Z

tb

ξ(t) = x˙ cl (t).

(4.105)

This is seen directly by differentiating the equation of motion (4.53) with respect to t, yielding ∂t [M x¨cl + V ′ (xcl (t))] = [M∂t2 + V ′′ (xcl (t))]x˙ cl (t) = 0,

(4.106)

which is precisely the homogenous differential equation (4.103) for x˙ cl (t). There is a simple symmetry argument to understand (4.105) as a completely general consequence of the time translation invariance of the system. The fluctuation δx(t) ∝ x˙ cl (t) describes an infinitesimal translation of the classical solution xcl (t) in time, xcl (t) → xcl (t + ǫ) = xcl + ǫx˙ cl + . . . . Interpreted as a translational fluctuation of the solution xcl (t) along the time axis it cannot carry any energy λn and y0 (t) ∝ x˙ cl (t) must therefore solve Eq. (4.98) with λ0 = 0. With the special solution (4.105), the functional determinant (4.104) becomes Dren = x˙ cl (tb )x˙ cl (ta )

Z

tb

ta

dt x˙ 2cl (t)

.

(4.107)

Note that also the Green-function of the quadratic fluctuations associated with Eq. (4.103) can be given explicitly in terms of the classical solution xcl (t). For Dirichlet boundary conditions, it is equal to the combination (3.61) of the solutions Da (t) and Db (t) of the homogeneous differential equation (4.103) satisfying the boundary conditions (2.221) and (2.222), whose d’Alembert construction (2.232) becomes here Da (t) = x˙ cl (t)x˙ cl (ta )

Z

t

ta

dt , 2 x˙ cl (t)

Db (t) = x˙ cl (tb )x˙ cl (t)

Z

t

tb

dt x˙ 2cl (t)

.

(4.108)

In Eqs. (2.245) and (2.262) we have found two simple expressions for the fluctuation determinant in terms of the classical action Dren

∂ x˙ b =− ∂xa

!−1

∂2 = −M Acl ∂xb ∂xa "

#−1

.

(4.109)

384


These were derived for purely quadratic actions with an arbitrary time-dependent frequency Ω2 (t). But they hold for any action. First, the equality between the second and third expression is a consequence of the general relation (4.81). Second, we may consider the semiclassical approximation to the path integral as an exact path integral associated with the lowest quadratic approximation to the action in (4.92), (4.93): Aqu [x, x] ˙ = A[xcl ] +

Z

tb

ta

dt

M (δ x) ˙ 2 + Ω2 (t)(δx)2 , 2

(4.110)

with Ω2 (t) = V ′′ (xcl (t))/M of (4.94). Then, since the classical orbit running from xa to xb satisfies the equation of motion (4.106), also a slightly different orbit (xcl + δxcl )(t) from x′a = xa + δxa to x′b = xa + δxb satisfies (4.106). Although the small change of the classical orbit gives rise to a slightly different frequency Ω2 (t) = V ′′ ((xcl + δxcl )(t))/M, this contributes only to second order in δxa and δxb . As a consequence, the derivative Da (t) = −∂ x˙ b (t)/∂xa satisfies Eq. (4.106) as well. Also the boundary conditions of Da (t) are the same as those of Da (t) in Eqs. (2.221). Hence the quantity Da (tb ) is the correct fluctuation determinant also for the general action in the semiclassical approximation under study. Another way to derive this formula makes use of the general relation (4.81), from which we find ∂ ∂ ∂ M ∂E A(xb , xa ; tb − ta ) = p(xb ) = . ∂xb ∂xa ∂xa p(xb ) ∂xa

(4.111)

On the right-hand side we have suppressed the arguments of the function E(xb , xa ; tb − ta ). After rewriting ∂E ∂ ∂A ∂ ∂A = − =− ∂xa ∂xa ∂tb ∂tb ∂xa ∂ M ∂E = p(xa ) = , ∂tb p(xa ) ∂tb

(4.112)

we see that ∂ ∂ 1 ∂E A(xb , xa ; tb − ta ) = . ∂xb ∂xa x(t ˙ b )x(t ˙ a ) ∂tb

(4.113)

From (4.57) we calculate ∂E ∂tb

∂tb ∂E

= =

"

−

Z

!−1 xb

xa

"

= −

M2 dx 3 p

Z

xb

xa

#−1

M ∂p dx 2 p ∂E "

= −M

Z

#−1

tb

ta

dt p2

(4.114) #−1

"

1 = − M

Z

tb

ta

dt 2 x˙ cl (t)

#−1

.

Inserting this into (4.113), we obtain once more formula (4.109) for the fluctuation determinant. H. Kleinert, PATH INTEGRALS

4.3 Van Vleck-Pauli-Morette Determinant

385

A relation following from (4.85): ∂2S ∂E 2

∂E = ∂tb

!−1

,

(4.115)

∂2S . ∂E 2

(4.116)

leads to an alternative expression Dren = −M x˙ cl (tb )x˙ cl (ta )

The fluctuation factor is therefore also here [recall the normalization from Eqs. (2.195), (2.197), and (2.205)] "

1

∂ x˙ b F (xb , ta ; tb −ta ) = q 2πi¯ h/M ∂xa

#1/2

=√

1 [−∂xb ∂xa A(xb , xa ; tb −ta )]1/2. (4.117) 2πi¯ h

Its D-dimensional generalization of (4.117) is F (xb , xa ; tb − ta ) = √

1

o1/2

n

detD [−∂xib ∂xja A(xb , xa ; tb − ta )]

D

2πi¯ h

,

(4.118)

and the semiclassical time evolution amplitude reads (xb tb |xa ta ) = √

1 D

2πi¯ h

n

o1/2

detD [−∂xib ∂xja A(xb , xa ; tb − ta )]

eiA(xb ,xa ;tb −ta )/¯h . (4.119)

The D × D -determinant in the curly brackets is the so-called Van Vleck-PauliMorette determinant.4 It is the analog of the determinant in the right-hand part of Eq. (2.263). As discussed there, the result is initially valid only as long as the fluctuation determinant is regular. Otherwise we must replace the determinant by its absolute value, and multiply the fluctuation factor by the phase factor e−iν/2 with the Maslov-Morse index ν [see Eq. (2.264)]. Using the relation (4.81) in D dimensions ∂pib ∂xib ∂xja A(xb , xa ; tb − ta ) = j , (4.120) ∂xa we shall often write (4.119) as (xb tb |xa ta ) = √

1 D

2πi¯ h

"

detD

∂pb − ∂xa

!#1/2

eiA(xb ,xa ;tb −ta )/¯h ,

(4.121)

where the subscripts a and b can be interchanged in the determinant, if the sign is changed [recall (2.245)]. This concludes the calculation of the semiclassical approximation to the time evolution amplitude. 4

J.H. Van Vleck, Proc. Nat. Acad. Sci. (USA) 14 , 178 (1928); W. Pauli, Selected Topics in Field Quantization, MIT Press, Cambridge, Mass. (1973); C. DeWitt-Morette, Phys. Rev. 81 , 848 (1951).

386


As a simple application, we use this formula to write down the semiclassical amplitude for a free particle and a harmonic oscillator. The first has the classical action A(xb , xa ; tb − ta ) =

M (xb − xa )2 , 2 tb − ta

(4.122)

and Eq. (4.109) gives Dren = tb − ta ,

(4.123)

as it should. The harmonic-oscillator action is A(xb , xa ; tb − ta ) =

i h Mω (x2b + x2a ) cos ω(tb − ta ) − 2xb xa , (4.124) 2 sin ω(tb − ta )

and has the second derivative −∂xb ∂xa A =

Mω , sin ω(tb − ta )

(4.125)

so that (4.117) coincides with fluctuation factor (2.209).

4.4

Fundamental Composition Law for Semiclassical Time Evolution Amplitude

The determinant ensures that the semiclassical approximation for the time evolution amplitude satisfies the fundamental composition law (2.4) in D dimensions (xb tb |xa ta ) =

N Z Y

n=1

∞

−∞

dD xn

NY +1 n=1

(xn tn |xn−1 tn−1 ),

(4.126)

if the intermediate x-integrals are evaluated in the saddle point approximation. To leading order in h ¯ , only those intermediate x-values contribute which lie on the classical trajectory determined by the endpoints of the combined amplitude. To next order in h ¯ , the quadratic correction to the intermediate integrals renders an inverse square root of the fluctuation determinant. If two such amplitudes are connected with each other by an intermediate integration according to the composition law (4.126), the product of the two fluctuation factors turns into the correct fluctuation factor of the combined time interval. This is seen after rewriting the matrix ∂xib ∂xja A(xb , xa ; tb −ta ) with the help of (4.12) as ∂pb /∂xa . The intermediate integral over x in the product of two amplitudes receives a contribution only from continuous paths since, at the saddle point, the adjacent momenta have to be equal: ∂ ∂ A(xb , x; tb − t) + A(x, xa ; t − ta ) = −p′ (xb , x; tb − t) + p(x, xa ; t − ta ) = 0. ∂x ∂x (4.127) H. Kleinert, PATH INTEGRALS

4.4 Fundamental Composition Law for Semiclassical Time Evolution Amplitude

387

To obtain the combined amplitude, we obviously need the relation 

detD −

   ∂pb   detD − ∂x xb  

∂p ∂x ′

+ xb

∂p ∂x

xa

 

p′ =p

 ∂pb  , ∂xa



= detD −

−1  

detD

 

!

∂p − ∂xa x

(4.128)

xb

where we have indicated explicitly the variables kept fixed in p′ (xb , x; tb − t) and p(x, xa ; t − ta ) when forming the partial derivatives. To prove (4.128), we use the product rule for determinants 











∂pb  ∂pb  ∂x  − −  det = det det−1 D D D ∂x xb ∂xa xb ∂xa xb

(4.129)

to rewrite (4.128) as detD −

! ∂p ∂xa x



= detD −

∂p ∂x ′

xb

+

∂p ∂x

xa



 ∂x  detD  . ∂xa 



p′ =p

(4.130)

xb

This equation is true due to the chain rule of differentiation applied to the momentum p′ (xb , x; tb − t)= p(x, xa ; t − ta ), after expressing p(x, xa ; t − ta ) explicitly in terms of the variables xb and xa as p(x(xb , xa ; tb − ta ), xa ; t − ta ), to enable us to hold xb fixed in the second partial derivative:

∂p′ ∂p ∂p(x(xb , xa ; tb − ta ), xa ; t − ta ) ∂p ∂xa ∂p = = = + . ∂x xb ∂x xb ∂x ∂x x ∂x xb ∂x xa xb (4.131) It may be expected, and can indeed be proved, that it is possible to proceed in the opposite direction and derive the semiclassical expressions (4.119) and (4.121) with the Van Vleck-Pauli-Morette determinant from the fundamental composition law (4.126).5 In the semiclassical approximation, the composition law (4.126) can also be written as a temporal integral (in D dimensions) (xb tb |xa ta ) =

Z

dt(xb tb |xcl (t)t) x˙ cl (t) (xcl (t)t|xa ta )

(4.132)

over a classical orbit xcl (t), where the t-integration is done in the saddle point approximation, assuming that the fluctuation determinant does not happen to be degenerate. Just as in the saddle point expansion of ordinary integrals, it is possible to calculate higher corrections in h ¯ . The result is a saddle point expansion of the path 5

H. Kleinert and B. Van den Bossche, Berlin preprint 2000 (http://www.physik.fu-berlin.de/~kleinert/301).

388


integral which is again a semiclassical expansion. The counting of the h ¯ -powers is the same as for the integral. The lowest approximation is of the exponential form eiAcl /¯h . Thus, in the exponent, the leading term is of order 1/¯ h.6 The fluctuation factor F contributes to this an additive term log F , which is of order h ¯ 0 . To first order in h ¯ , one finds expressions containing the third and fourth functional derivative of the action in the expansion (4.92), corresponding to the expressions (4.42) and (4.43) in the integral. Unfortunately, the functional case offers little opportunity for further analytic corrections, so we shall not dwell on this more academic possibility.

4.5

Semiclassical Fixed-Energy Amplitude

As pointed out at the end of Subsection 4.2.1, we have observed that the semiclassical approximation allows for a simple evaluation of Fourier integrals. As an application of the rules presented there, let us evaluate the Fourier transform of the time evolution amplitude, the fixed-energy amplitude introduced in (1.307). It is given by the temporal integral (xb |xa )E = √

1 2πi¯ h

Z

∞

ta

dtb [−∂xb ∂xa A(xb , xa ; tb − ta )]1/2 × ei[A(xb ,xa ;tb −ta )+(tb −ta )E]/¯h ,

(4.133)

which may be evaluated in the same saddle point approximation as the path integral. The extremum lies at ∂ A(xb , xa ; tb − ta ) = −E. (4.134) ∂t Because of (4.86), the left-hand side is the function −E(xb , xa ; tb − ta ). At the extremum, the time interval tb − ta is some function of the endpoints and the energy E: (4.135) tb − ta = t(xb , xa ; E). The exponent is equal to the eikonal function S(xb , xa ; E) of Eq. (4.80), whose derivative with respect to the energy gives [recalling (4.85)]

∂ (4.136) S(xb , xa ; E) = t(xb , xa ; E). ∂E The expansion of the exponent around the extremum has the quadratic term i ∂ 2 A(xb , xa ; tb − ta ) [tb − ta − t(xb , xa ; E)]2 . h ¯ ∂t2b

(4.137)

The time integral over tb yields a factor √

∂ 2 A(xb , xa ; tb − ta ) 2πi¯ h ∂t2b "

#−1/2

.

(4.138)

6

Since h ¯ has the dimension of an action, the dimensionless number h ¯ /Acl should really be used as an appropriate dimensionless expansion parameter, but it has become customary to count directly the orders in h ¯. H. Kleinert, PATH INTEGRALS

4.5 Semiclassical Fixed-Energy Amplitude

389

With this, the fixed-energy amplitude has precisely the form (4.48): h

(xb |xa )E = −∂xb ∂xa A(xb , xa ; t)/∂t2 A(xb , xa ; t)

i1/2

eiS(xb ,xa ;E)/¯h .

(4.139)

Since the fluctuation factor has to be evaluated at a fixed energy E, it is advantageous to express it in terms of S(xb , xa ; E). For ∂t2 A, the evaluation is simple since ∂2A ∂E ∂t =− =− 2 ∂t ∂t ∂E

!−1

∂2S =− ∂E 2

!−1

.

(4.140)

For ∂xb ∂xa A, we observe that the spatial derivatives of the action must be performed at a fixed time, so that a variation of xb implies also a change of the energy E(xb , xa ; t). This is found from the condition ∂t = 0, ∂xb

(4.141)

which after inserting (4.136), goes over into

We now use the relation

∂2S ∂ 2 S ∂E ∂ 2 S = + = 0. ∂E∂xb t ∂E∂xb ∂E 2 ∂xb

(4.142)

∂A ∂S ∂E ∂S ∂S ∂E ∂E ∂S + = − t= − t= ∂xb t ∂xb t ∂xb t ∂xb ∂E ∂xb t ∂xb t ∂xb E

(4.143)

and find from it

∂ 2 A ∂2S ∂ 2 S ∂E + = ∂xb ∂xa t ∂xb ∂xa ∂xb ∂E ∂xa

(4.144)

∂2S ∂2S ∂2S = − ∂xb ∂xa ∂xa ∂E ∂xb ∂E

,

∂2S . ∂E 2

Thus the fixed-energy amplitude (4.133) takes the simple form 1/2

(xb |xa )E = DS eiS(xb ,xa ;E)/¯h , with the 2 × 2-determinant DS =

∂2S ∂xb ∂xa ∂2S ∂xb ∂E

∂2S ∂E∂xa ∂2S ∂E 2

.

(4.145)

(4.146)

The determinant can be simplified by the fact that a differentiation of the HamiltonJacobi equation ! ∂S H , xb = E (4.147) ∂xb

390


with respect to xa leads to the equation ∂H ∂ 2 S ∂2S = x˙ b = 0. ∂pb ∂xb ∂xa ∂xb ∂xa

(4.148)

It implies the vanishing of the upper left element in (4.146), reducing DS to DS = −

∂2S ∂2S . ∂xb ∂E ∂xa ∂E

(4.149)

Since ∂S/∂xb,a = ±pb,a and ∂p/∂E = 1/x, ˙ one arrives at DS =

1 . x˙ b x˙ a

(4.150)

Let us calculate the semiclassical fixed-energy amplitude for a free particle. The classical action function is M (xb − xa )2 A(xb , xa ; tb − ta ) = , 2 tb − ta

(4.151)

so that the function E(xb , xa ; tb − ta ) is given by E(xb , xa ; tb − ta ) = −

∂ M (xb − xa )2 M (xb − xa )2 = . ∂tb 2 tb − ta 2 (tb − ta )2

(4.152)

By a Legendre transformation, or directly from the defining equation (4.61), we calculate √ (4.153) S(xb , xa ; E) = 2ME|xb − xa |. From this we calculate the determinant (4.150) as Ds =

M , 2E

(4.154)

and the fixed-energy amplitude (4.145) becomes (xb |xa )E =

4.6

s

M i√2M E|xb −xa |/¯h e . 2E

(4.155)

Semiclassical Amplitude in Momentum Space

The simple way of finding Fourier transforms in the semiclassical approximation can be used to derive easily amplitudes in momentum space. Consider first the time evolution amplitude (xb tb |xa ta )sc . The momentum space version is given by the two-dimensional Fourier integral [recall (2.37) and insert (4.96)] (pb tb |pa ta )sc =

Z

dxb dxa e−i(pb xb −pa xa )/¯h eiA(xb ,xa ;tb −ta )/¯h F (xb , xa ; tb − ta ).

(4.156)


4.6 Semiclassical Amplitude in Momentum Space

391

The semiclassical evaluation according to the general rule (4.48) yields √ 2πi¯ h [−∂xb ∂xa A(xb , xa ; tb − ta )]1/2 ei[A(xb ,xa ;tb −ta )−pb xb +pa xa ]/¯h,(4.157) (pb tb |pa ta )sc = √ det H where H is the matrix 

H=

∂x2b A(xb , xa ; tb − ta ) ∂xb ∂xa A(xb , xa ; tb − ta ) ∂x2a A(xb , xa ; tb − ta )

∂xa ∂xb A(xb , xa ; tb − ta )



.

(4.158)

The exponent must be evaluated at the extremum with respect to xb and xa , which lies at pb = ∂xb A(xb , xa ; tb − ta ),

pa = −∂xb A(xb , xa ; tb − ta ).

(4.159)

The exponent contains then the Legendre transform of the action A(xb , xa ; tb − ta ) which depends naturally on pb and pa : A(pb , pa ; tb − ta ) = A(xb , xa ; tb − ta ) − pb xb + pa xa .

(4.160)

The inverse Legendre transformation to (4.159) is xb = −∂pb A(xb , xa ; tb − ta ),

xa = ∂xb A(xb , xa ; tb − ta ).

(4.161)

The important observation which greatly simplifies the result is that for a 2 × 2 matrix Hab with (a, b = 1, 2), the matrix element −H12 /det H is equal to H12 . By writing the matrix H and its inverse as

H=

    

∂pb ∂pb ∂xb ∂xa ∂pa ∂pa − − ∂xb ∂xa



  , 

H

−1

=

     

∂xb ∂xb − ∂pb ∂pa ∂xa ∂xa − ∂pb ∂pa



  ,  

(4.162)

we see that, just as in the Eqs. (2.271) and (2.272): −1 H12 =

∂xa ∂ 2 A(pb , pa ; tb − ta ) = . ∂pb ∂pb ∂pa

(4.163)

As a result, the semiclassical time evolution amplitude in momentum space (4.157) takes the simple form 2π¯ h (pb tb |pa ta )sc = √ [−∂pb ∂pa A(pb , pa ; tb − ta )]1/2 eiA(pb ,pa ;tb −ta )/¯h. 2πi¯ h

(4.164)

In D dimensions, this becomes (pb tb |pa ta ) = √

1 D

2πi¯ h

n

o1/2

detD [−∂pib ∂pja A(pb , pa ; tb − ta )]

eiA(pb ,pa ;tb −ta )/¯h , (4.165)

392


or (pb tb |pp ta ) = √

1 D

2πi¯ h

(

detD

"

∂pb − ∂xa

#)1/2

eiA(pb ,pa ;tb −ta )/¯h ,

(4.166)

these results being completely analogous to the x-space expression (4.119) and (4.121), respectively. As before, the subscripts a and b can be interchanged in the determinant. If we apply these formulas to the harmonic oscillator with a time-dependent frequency, we obtain precisely the amplitude (2.278). Thus in this case, the semiclassical time evolution amplitude (pb tb |pa ta )sc happens to coincide with the exact one. For a free particle with the action A(xb , xa ; tb − ta ) = M(xb − xa )2 /2(tb − ta ), the formula (4.157) cannot be applied since determinant of H vanishes, so that the saddle point approximation is inapplicable. The formal infinity one obtains when trying to apply Eq. (4.157) is a reflection of the δ-function in the exact expression (2.133), which has no semiclassical approximation. The Legendre transform of the action can, however, be calculated correctly and yields via the derivatives pa = pb ≡ p = A(xb , xa ; tb − ta ) = M(xb − xa )/2(tb − ta ) the expression A(pb , pa ; tb − ta ) = −

p2 (tb − ta ), 2

(4.167)

which agrees with the exponent of (2.133).

4.7

Semiclassical Quantum-Mechanical Partition Function

From the result (4.96) we can easily derive the quantum-mechanical partition function (1.537) in semiclassical approximation: sc ZQM (tb − ta ) =

Z

dxa (xa tb |xa ta )sc =

Z

dxa F (xa , xa ; tb − ta )eiA(xa ,xa ;tb −ta )/¯h . (4.168)

Within the semiclassical approximation the path integral, as the final trace integral may be performed using the saddle point approximation. At the saddle point one has [as in (4.127)]

∂ ∂ ∂ A(xa , xa ; tb − ta ) = A(xb , xa ; tb − ta ) + A(xb , xa ; tb − ta ) ∂xa ∂xb ∂xa xb =xa xb =xa

= pb − pa = 0,

(4.169)

i.e., only classical orbits contribute whose momenta are equal at the coinciding endpoints. This restricts the orbits to periodic solutions of the equations of motion. The semiclassical limit selects, among all paths with xa = xb , the paths solving the equation of motion, ensuring the continuity of the internal momenta along these paths. The integration in (4.168) enforces the equality of the initial and final momenta on these paths and permits a continuation of the equations of motion beyond H. Kleinert, PATH INTEGRALS

4.7 Semiclassical Quantum-Mechanical Partition Function

393

the final time tb in a periodic fashion, leading to periodic orbits. Along each of these orbits, the energy E(xa , xa , tb − ta ) and the action A(xa , xa , tb − ta ) do not depend on the choice of xa . The phase factor eiA/¯h in the integral (4.168) is therefore a constant. The integral must be performed over a full period between the turning points of each orbit in the forward and backward direction. It contains a nontrivial xa -dependence only in the fluctuation factor. Thus, (4.168) can be written as sc ZQM (tb − ta ) =

Z

dxa F (xa , xa ; tb − ta ) eiA(xa ,xa ;tb −ta )/¯h .

(4.170)

For the integration over the fluctuation factor we use the expression (4.117) and the equation 1 ∂2A ∂ ∂ A(xb , xa ; tb − ta ) = − , ∂xb ∂xa x˙ b x˙ a ∂t2b

(4.171)

following from (4.113) and (4.86), and have 1 1 ∂2A F (xb , xa ; tb − ta ) = √ ˙ b )x(t ˙ a ) ∂t2b 2πi¯ h x(t "

#1/2

.

(4.172)

Inserting xa = xb leads to 1 1 ∂2A F (xa , xa ; tb − ta ) = √ 2πi¯ h x˙ a ∂t2b "

#1/2

.

(4.173)

The action of a periodic path does not depend on xa ,so that the xb -integration in (4.168) requires only integrating 1/x˙ a forward and back, which produces the total period: tb − ta = 2

Z

x+

x−

1 dxa =2 x˙ a

Z

x+

x−

M dx q . 2M[E − V (x)]

(4.174)

Hence we obtain from (4.168): sc ZQM (tb

1/2

tb − ta ∂ 2 A − ta ) = √ 2πi¯ h ∂t2b

eiA(tb −ta )/¯h−iπ .

(4.175)

There is a phase factor e−iπ associated with a Maslov-Morse index ν = 2, first introduced in the fluctuation factor (2.264). In the present context, this phase factor arises from the fact that when doing the integral (4.170), the periodic orbit passes through the turning points x− and x+ where the integrand of (4.174) becomes singular, even though the integral remains finite. Near the turning points, the semiclassical approximation breaks down, as discussed in Section 4.1 in the context of the WKB approximation to the Schrödinger equation. This breakdown required special attention in the derivation of the connection formulas relating the wave functions on

394


one side of the turning points to those on the other side. There, the breakdown was circumvented by escaping into the complex x-plane. When going around the singu1/2

q

larity in the clockwise sense, the prefactor 1/p(x) = 1/ 2M(E − V (x)) acquired a phase factor e−iπ/2 . For a periodic orbit, both turning points had to be encircled producing twice this phase factor, which is precisely the phase e−iπ given in (4.175). The result (4.175) takes an especially simple form after a Fourier transform action: sc Z˜QM (E) =

Z

∞

ta

sc dtb eiE(tb −ta )/¯h ZQM (tb − ta )

1/2

∂2A 1 Z∞ = √ dtb (tb − ta ) 2 ∂t t 2πi¯ h a b

ei[A(tb −ta )+(tb −ta )E]/¯h−iπ .

(4.176)

In the semiclassical approximation, the main contribution to the integral at a given energy E comes from the time where tb − ta is equal to the period of the particle orbit with this energy. It is determined as in (4.133) by the extremum of A(tb − ta ) + (tb − ta )E. Thus it satisfies −

(4.177)

∂ A(tb − ta ) = E. ∂tb

(4.178)

As in (4.134), the extremum determines the period tb − ta of the orbit with an energy E. It will be denoted by t(E). The second derivative of the exponent is (i/¯ h)∂ 2 A(tb − ta )/∂t2b . For this reason, the quadratic correction in the saddle point approximation to the integral over tb cancels the corresponding prefactor in (4.176) and leads to the simple expression sc Z˜QM (E) = t(E)ei[A(t)+t(E)E]/¯h−iπ .

(4.179)

The exponent contains again the eikonal S(E) = A(t) + t(E)E, the Legendre transform of the action A(t) defined by S(E) = A(t) − t

∂A(t) , ∂t

(4.180)

where the variable t has to be replaced by E(t) = −∂A(t)/∂t. Via the inverse Legendre transformation, the derivative ∂S(E)/∂E = t leads back to A(t) = S(E) −

∂S(E) E. ∂E

(4.181)

Explicitly, S(E) is given by the integral (4.61): S(E) = 2

Z

x+

x−

dx p(x) = 2

Z

x+

x−

q

dx 2M[E − V (x)].

(4.182)


4.7 Semiclassical Quantum-Mechanical Partition Function

395

Finally, we have to take into account that the periodic orbit is repeatedly traversed for an arbitrary number of times. Each period yields a phase factor eiS(E)/¯h−iπ . The sum is sc Z˜QM (E) =

∞ X

n=1

t(E)ein[S(E)/¯h−π] = −t(E)

eiS(E)/¯h . 1 + eiS(E)/¯h

(4.183)

This expression possesses poles in the complex energy plane at points where the eikonal satisfies the condition S(En ) = 2π¯ h(n + 1/2),

n = 0, ±1, ±2, . . . .

(4.184)

This condition agrees precisely with the Bohr-Sommerfeld rule (4.25) for semiclassical quantization. At the poles, one has sc Z˜QM (E) ≈ t(E)

i¯ h S ′ (En )(E

− En )

.

(4.185)

Due to (4.85), the pole terms acquire the simple form sc Z˜QM (E) ≈

i¯ h . E − En

(4.186)

From (4.183) we derive the density of states defined in (1.578). For this we use the general formula 1 ρ(E) = disc Z˜QM (E), (4.187) 2π¯ h where disc ZQM (E) is the discontinuity ZQM (E + iη) − ZQM (E − iη) across the singularities defined in Eq. (1.324). If we equip the energies En in (4.186) with the usual small imaginary part −iη, we can also write (4.187) as ρ(E) =

1 ReZ˜QM (E). π¯ h

(4.188)

Inserting here the sum (4.183), we obtain the semiclassical approximation ρ¯sc (E) = or

∞ t(E) X cos{n[S(E)/¯ h − π]} π¯ h n=1

(4.189)

∞ X t(E) ∆ρsc (E) = −1 + ein[S(E)/¯h−π] . 2π¯ h n=−∞

!

(4.190)

We have added a ∆-symbol to this quantity since it is really the semiclasscial correction to the classical density of states ρ(E), as we sall see in a moment. With the help of Poisson’s summation formula (1.197), this goes over into ∆ρsc (E) = −

∞ t(E) t(E) X + δ[S(E)/¯ h − 2π(n + 1/2)]. 2π¯ h h ¯ n=−∞

(4.191)

396


The right-hand side contains δ-functions which are singular at the semiclassical energy values (4.184). Using once more the relation (4.85), the formula δ(ax) = a−1 δ(x) leads to the simple expression ∆ρsc (E) = −

∞ X t(E) δ(E − En ). + 2π¯ h n=−∞

(4.192)

This result has a surprising property: Consider the spacing between the energy levels ∆En = En − En−1 = 2π¯ h

∆En ∆Sn

(4.193)

and average the sum in (4.192) over a small energy interval ∆E containing several energy levels. Then we obtain an average density of states: S ′ (E) t(E) ρav (E) = = . 2π¯ h 2π¯ h

(4.194)

It cancels precisely the first term in (4.192). Thus, the semiclassical formula (4.183) possesses a vanishing average density of states. This cannot be correct and we conclude that in the derivation of the formula, a contribution must have been overlooked. This contribution comes from the classical partition function. Within the above analysis of periodic orbits, there are also those which return to the point of departure after an infinitesimally small time (which leaves them with no time to fluctuate). The expansion (4.183) does not contain them, since the saddle point approximation to the time integration (4.176) used for its derivation fails at short times. The reason for this failure is the singular behavior of the fluctuation factor ∝ 1/(tb − ta )1/2 in (4.96). In order to recover the classical contribution, one simply uses the short-time amplitude in the form (2.341) to calculate the purely classical contribution to Z(E): Zcl (E) ≡

Z

dx

Z

dp i¯ h . 2π¯ h E − H(p, x)

(4.195)

This implies a classical contribution to the density of states ρcl (E) ≡

Z

dx ρcl (E; x),

(4.196)

which is a spatial integral over the classical local density of states ρcl (E; x) ≡

Z

dp δ[E − H(p, x)]. 2π¯ h

(4.197)

The δ-function in the integrand can be rewritten as δ(E − H(p, x)) =

M [δ(p − p(E; x)) + δ(p + p(E; x))] , p(E; x)

(4.198)


4.8 Multi-Dimensional Systems

397

where p(E; x) is the local momentum associated with the energy E p(E; x) =

q

2M[E − V (x)],

(4.199)

which was defined in (4.3), except that we have now added the energy to the argument, to have a more explicit notation. It is then trivial to evaluate the integral (4.197) and (4.196) yielding the classical local density of states M 1 , π¯ h p(E; x)

ρcl (E; x) =

(4.200)

and its integral

1 M 1 (4.201) = t(E) = ρav (E), π¯ h p(E; x) 2π¯ h which coincides with the average classical density of states in (4.194). Thus the full semiclassical density of states consists of the sum of (4.192) and (4.201): (4.202) ρsc (E) = ρcl (E) + ∆ρsc (E). ρcl (E) =

Z

dx

This has, on the average, the correct classical value. Note that by Eq. (4.194), the eikonal S(E) is related to the integral over the classical density of states ρcl (E) by a factor 2π¯ h: S(E) = 2π¯ h

Z

E

−∞

dE ρcl (E).

(4.203)

Recalling the definition (1.582) of the number of states up to the energy E we see that S(E) = 2π¯ hN(E), (4.204) which shows that the Bohr-Sommerfeld quantization condition (4.184) is the semiclassical version of the completely general equation (1.583).

4.8

Multi-Dimensional Systems

The D-dimensional generalization of the classical partition function (4.195) reads Zcl (E) ≡

Z

D

d x

Z

i¯ h dD p , 2π¯ h E − H(p, x)

(4.205)

and of the density of states (4.197): ρcl (E; x) ≡

Z

dD p δ[E − H(p, x)]. (2π¯ h)D

(4.206)

The Hamiltonian of the standard form H(p, x) = p2 /2M +V (x) allows us to perform the momentum integration by separating it into radial and angular parts, Z

dD p = (2π¯ h)D

Z

dp pD−1

Z

dˆ p.

(4.207)

398


The angular integral yields the surface of a unit sphere in D dimensions: Z

dˆ p = SD =

2π D/2 . Γ(D/2)

(4.208)

The δ-function δ(E − H(p, x)) can again be rewritten as in (4.198), which selects the momenta of magnitude p(E; x) =

q

2M[E − V (x)].

(4.209)

ρcl (E) ≡

Z

(4.210)

Thus we find dD x ρcl (E; x),

where ρcl (E; x) is the classical local density of states. ρcl (E; x) = SD

M pD (E; x) 1 2M = {2M[E − V (x)]}D/2−1, (4.211) 2 D/2 2 D p (E; x) (2π¯ h) Γ(D/2) (4π¯ h)

generalizing expression (4.201). The number of states with energies between E and E + dE in the volume element dD x is dEd3xρcl (E; x). For completeness we state some features of the semiclassical results which appear when generalizing the theory to D dimensions. For a detailed derivation see the rich literature on this subject quoted at the end of the chapter. For an arbitrary number D of dimensions. the Van Vleck-Pauli-Morette determinant (4.118) takes the form F (xb , xa ; tb − ta ) = √

1

detD [−∂xi ∂xj A(xb , xa ; tb

D

b

2πi¯ h

a

1/2

− ta )]

e−iπν/2 , (4.212)

where ν is the Maslov-Morse index. The fixed-energy amplitude becomes the sum over all periodic orbits:7 (xb |xa )E = √

1 D−1

2πi¯ h

X p

′

|DS |1/2 eiS(xb ,xa ;E)/¯h−iπν /2 ,

(4.213)

where S(xb , xa ; E) is the D-dimensional generalization of (4.61) and DS the (D + 1) × (D + 1)-determinant: D+1

DS = (−1)

7

det

     

∂2S ∂xb ∂xa ∂2S ∂xb ∂E

∂2S ∂E∂xa ∂2S ∂E∂E



  .  

(4.214)

M.C. Gutzwiller, J. Math. Phys. 8 , 1979 (1967); 11 , 1791 (1970); 12 , 343 (1971). H. Kleinert, PATH INTEGRALS


399

The factor (−1)D+1 makes the determinant positive for short trajectories. The index ν ′ differs from ν by one unit if ∂ 2 S/∂E 2 = ∂t(E)/∂E is negative. In D dimensions, the Hamilton-Jacobi equation leads to ∂H ∂2S ∂2S ˙ · = xb · = 0, ∂pb ∂xb ∂xa ∂xb ∂xa

(4.215)

instead of (4.148). Only the longitudinal projection of the D×D-matrix ∂ 2 S/∂xb ∂xa along the direction of motion vanishes now. In this direction x˙b ·

∂2S = 1, ∂xb ∂E

(4.216)

so that the determinant (4.214) can be reduced to 1 ∂2S DS = det − ⊥ ⊥ , |x˙ b ||x˙ a | ∂xb ∂xa !

(4.217)

instead of (4.150). Here x⊥ b,a denotes the deviations from the orbit orthogonal to x˙ b,a , and we have used (2.283) to arrive at (4.217). As an example, let us write down the D-dimensional generalization of the freeparticle amplitude (4.155). The eikonal is obviously √ S(xa , xb ; E) = 2ME|xb − xa |, (4.218) and the determinant (4.217) becomes DS =

M (2ME)(D−1)/2 . 2E |xa − xb |D−1

(4.219)

Thus we find (xb |xa )E =

s

M 1 (2ME)(D−1)/4 i√2M E|xb −xa |/¯h e . 2E (2πi¯ h)(D−1)/2 |xa − xb |(D−1)/2

(4.220)

For D = 1, this reduces to (4.155). Note that the semiclassical result coincides with the large-distance behavior (1.355) of the exact result (1.351), since the semiclassical limit implies a large momenta k in the Bessel function (1.351). When calculating the partition function, one has to perform a D-dimensional integral over all xb = xa . This is best decomposed into a one-dimensional integral along the orbit and a D − 1 -dimensional one orthogonal to it. The eikonal function S(xa , xa ; E) is constant along the orbit, as in the one-dimensional case. When leaving the orbit, however, this is no longer true. The quadratic deviation of S orthogonal to the orbit is 1 ∂ 2 S(x, x; E) (x − x⊥ )T (x − x⊥ ), 2 ∂x⊥ ∂x⊥

(4.221)

400


where the superscript T denotes the transposed vector to be multiplied from the left with the matrix in the middle. After the exact trace integration along the orbit and a quadratic approximation in the transversal direction for each primitive orbit, which is not repeated, we obtain the contribution to the partition function

Zsc =

∂ 2 S(x , x ; E) 1/2 b a ⊥ ⊥ ∞ ∂xb ∂xa X xb =xa =x t(E) ein[S(E)/¯h−iπν/2] , 1/2 ∂ 2 S(x, x; E) n=1 ∂x⊥ ∂x⊥

(4.222)

where ν is the Maslov-Morse index of the orbit. The ratio of the determinants is conveniently expressed in terms of the determinant of the so-called stability matrix M in phase space, which is introduced in classical mechanics as follows: Consider a classical orbit in phase space and vary slightly the initial point, mov⊥ ing it orthogonally away from the orbit by δx⊥ a , δpa . This produces variations at ⊥ the final point δx⊥ b , δpb , related to those at the initial point by the linear equation δx⊥ b δp⊥ b

!

=

A B C D

!

δx⊥ a δp⊥ a

!

≡M

δx⊥ a δp⊥ a

!

.

(4.223)

The 2(D − 1) × 2(D − 1)-dimensional matrix is the stability matrix M. It can be expressed in terms of the second derivatives of S(xb , xa ; E). These appear in the relation ! ! ! δp⊥ −a −b δx⊥ a a = , (4.224) bT δp⊥ c δx⊥ b b where a, b, and c are the (D − 1) × (D − 1)-dimensional matrices ∂2S , ⊥ ∂x⊥ a ∂xa

a=

b=

∂2S , ⊥ ∂x⊥ a ∂xb

c=

∂2S . ⊥ ∂x⊥ b ∂xb

(4.225)

From this one calculates the matrix elements of the stability matrix (4.223): A = −b−1 a,

B = −b−1 ,

C = bT − cb−1 a,

D = −cb−1 .

(4.226)

The stability properties of the classical orbits are classified by the eigenvalues of the stability matrix (4.223). In three dimensions, the eigenvalues are given by the zeros of the characteristic polynomial of the 4 × 4 -matrix M:

A−λ B P (λ) = |M − λ| = C D−λ

− b−1 a − λ −b−1 = T b − cb−1 a −cb−1 − λ

The usual manipulations bring this to the form

.

(4.227)

1 −a − λb −b−1 a − λ −b−1 −1 P (λ) = = bT + λ −λ |b| bT + (a + c)λ + λ2 b 0 1 T b + (a + c)λ + λ2 b . = |b|

(4.228)



401

Precisely this expression appears, with xb = xa , in the prefactor of (4.222) if this is rewritten as ∂ 2 S 1/2 ∂x⊥ ∂x⊥ a xb ≈xa =x b . (4.229) 1/2 2 ∂2S ∂S ∂S ∂ S ⊥ ⊥ +2 + ⊥ ⊥ ⊥ ⊥ ∂x ∂x ∂x ∂x ∂x ∂x a a a xb =xa =x b b b Due to (4.226), this coincides with P (1)−1/2 . The semiclassical limit to the quantum-mechanical partition function takes therefore the simple form referred to as Gutzwiller’s trace formula Zsc (E) = t(E)

eiS(E)−iπν/2 1 . P (1)1/2 1 − eiS(E)−iπν/2

(4.230)

The energy eigenvalues lie at the poles and satisfy the quantization rules [compare (4.25), (4.184)] h(n + ν/4). (4.231) S(En ) = 2π¯ The eigenvalues of the stability matrix come always in pairs λ, 1/λ, as is obvious from (4.228). For this reason, one has to classify only two eigenvalues. These must be either both real or mutually complex-conjugate. One distinguishes the following cases: 1. elliptic,

if

λ = eiχ , e−iχ , with a real phase χ 6= 0,

2. direct parabolic, inverse parabolic,

if

λ = 1, λ = −1,

3.

direct hyperbolic, inverse hyperbolic,

λ = e±χ , λ = −e±χ ,

if

4. loxodromic,

λ = eu±v .

if

In these cases, P (1) =

2 Y

i=1

(λi − 1)(1/λi − 1)

has the values 1.

4 sin2 (χ/2),

2.

0

3.

− 4 sinh2 (χ/2)

4.

4 sin[(u + v)/2] sin[(u − v)/2].

or

4, or

4 cosh2 (χ/2),

(4.232)

402


Only in the parabolic case are the equations of motion integrable, this being obviously an exception rather than a rule, since it requires the fulfillment of the equation a + c = ±2b. Actually, since the transverse part of the trace integration in the partition function results in a singular determinant in the denominator of (4.230), this case requires a careful treatment to arrive at the correct result.8 In general, a system will show a mixture of elliptic and hyperbolic behavior, and the particle orbits exhibit what is called a smooth chaos. In the case of a purely hyperbolic behavior one speaks of a hard chaos, which is simpler to understand. The semiclassical approximation is based precisely on those orbits of a system which are exceptional in a chaotic system, namely, the periodic orbits. The expression (4.230) also serves to obtain the semiclassical density of states in D-dimensional systems via Eq. (4.187). In D dimensions the paths, with vanishing length contribute to the partition function the classical expression [compare (4.205)]. Application of semiclassical formulas has led to surprisingly simple explanations of extremely complex experimental data on highly excited atomic spectra which classically behave in a chaotic manner. For completeness, let us also state the momentum space representation of the semiclassical fixed-energy amplitude (4.139). It is given by the momentum space analog of (4.213): (2π¯ h)D X ˜ 1/2 iS(pb ,pa ;E)/¯h−iπν ′ /2 | DS | e , (pb |pa )E = √ D−1 p 2πi¯ h

(4.233)

where S(pb , pa ; E) is the Legendre transform of the eikonal S(pb , pa ; E) = S(pb , pa ; E) − pb xb + pa xa ,

(4.234)

evaluated at the classical momenta pb = ∂pb S(pb , pa ; E) and pa = ∂pa S(pb , pa ; E). The determinant can be brought to the form: 1 ∂2S DS = det − ⊥ ⊥ , |p˙ b ||p˙ a | ∂pb ∂pa !

(4.235)

˙ a. where p⊥ a is the momentum orthogonal to p This formula cannot be applied to the free particle fixed-energy amplitude (3.216) for the same degeneracy reason as before. Higher h ¯ -corrections to the trace formula (4.230) have also been derived, but the resulting expressions are very complicated to handle. See the citations at the end of this chapter.

4.9

Quantum Corrections to Classical Density of States

There exists a simple way of calculating quantum corrections to the semiclassical expressions (4.201) and its D-dimensional generalization (4.210) for the density of 8

M.V. Berry and M. Tabor, J. Phys. A 10 , 371 (1977), Proc. Roy. Soc. A 356 , 375 (1977). H. Kleinert, PATH INTEGRALS

4.9 Quantum Corrections to Classical Density of States

403

ˆ via the spectral states. To derive them we introduce an operator δ-function δ(E − H) representation X ˆ ≡ δ(E − H) δ(E − En )|nihn|, (4.236) n

ˆ The δ-function (4.236) where |ni are the eigenstates of the Hamiltonian operator H. has the Fourier representation [recall (1.193)] ˆ = δ(E − H)

Z

∞

−∞

dt −i(H−E)t/¯ ˆ h . e 2π¯ h

(4.237)

Its matrix elements between eigenstates |xi of the position operator, ˆ ρ(E; x) = hx|δ(E − H)|xi =

Z

∞

−∞

dt iEt/¯h ˆ e hx|e−iHt/¯h |xi, 2π¯ h

(4.238)

define the quantum-mechanical local density of states. The amplitude on the righthand side is the time evolution amplitude ˆ

hx|e−iHt/¯h |xi = (x t|x 0),

(4.239)

which can be represented by a path integral as described in Chapter 2. In the semiclassical limit, only the short-time behavior of (x t|x 0) is relevant.

4.9.1

One-Dimensional Case

For a one-dimensional harmonic oscillator, the short-time expansion of (4.239) can easily be written down. For short times t ≡ tb − ta compared to the period 1/ω, we expand the amplitude (2.168) at equal initial and final space points x = xa = xb as follows in a power series of t: 1

ω 2 x2 t/¯ h −i M 2

e (x tb |x ta ) = q 2πi¯ ht/M

(

t2 i t3 Mω 4 x2 + . . . , 1 + ω2 − 12 h ¯ 24 )

(4.240)

This expansion is valid for an arbitrary smooth potential V (x) if the exponential prefactor containing the harmonic potential is replaced by e−iV (x)t/¯h ,

(4.241)

whereas ω 2 and Mω 4 x2 are substituted as follows: 1 ′′ V (x), M 1 ′ → [V (x)]2 . M

ω2 → Mω 4 x2

(4.242) (4.243)

Hence: 1

−iV (x)t/¯ h

(x tb |x ta ) = q e 2πi¯ ht/M

(

t2 i t3 1+ V ′′ (x) − [V ′ (x)]2 + . . . . (4.244) 12M h ¯ 24M )

404


Inserting this into (4.238) yields the local density of states dt 1 q e−i[V (x)−E]t/¯h 2π¯ h 2πi¯ ht/M

ρ(E; x) =

Z

∞

×

(

t2 i t3 1+ V ′′ (x) − [V ′ (x)]2 + . . . . 12M h ¯ 24M

−∞

)

(4.245)

For positive E − V (x), the integration along the real axis can be deformed into the upper complex plane to enclose the square-root cut along the positive imaginary t-axis in the anti-clockwise sense. Setting t = iτ and using the fact that the discontinuity across a square root cut produces a factor two, we have ρ(E; x) =2

Z

∞

0

dτ 1 τ 2 ′′ 1 τ3 q e−[E−V (x)]τ /¯h 1− V (x)− [V ′ (x)]2 + . . . . 2π¯ h 2π¯ 12M h ¯ 24M hτ /M (

)

(4.246)

The first term can easily be integrated for E > V (x), and yields the classical local density of states (4.201): ρcl (E; x) =

1 M M 1 q = . π¯ h 2M[E − V (x)] π¯ h p(E; x)

(4.247)

In order to calculate the effect of the correction terms in the expansion (4.246), we observe that a factor τ in the integrand is the same as a derivative h ¯ d/dV applied to the exponential. Thus we find directly the semiclassical expansion for the density of states (4.246), valid for E > V (x): h ¯ 2 ′′ d2 h ¯2 d3 ρ(E; x) = 1 − V (x) 2 − [V ′ (x)]2 3 + . . . ρcl (E; x). 12M dV 24M dV (

)

(4.248)

Inserting (4.247) and performing the differentiations with respect to V we obtain 1 ρ(E; x) = π¯ h

s

M 1 h ¯ 2 ′′ 3 1 V (x) − 1/2 2 [E − V (x)] 12M 4 [E − V (x)]5/2 (

h ¯2 15 1 − [V ′ (x)]2 + . . . .(4.249) 24M 8 [E − V (x)]7/2 )

Note that the proceeds in powers of higher gradients of the potential; it is a gradient expansion. The integral over (4.249) yields a gradient expansion for ρ(E) [1]. The second term can be integrated by parts which, under the assumption that V (x) vanishes at the boundaries, simply changes the sign of the third term, so that we find 1 ρ(E) = π¯ h

s

MZ 1 h ¯2 15 1 dx + [V ′ (x)]2 + ... 1/2 2 [E − V (x)] 24M 8 [E − V (x)]7/2 (

)

.(4.250)



4.9.2

405

Arbitrary Dimensions

In D dimensions, the short-time expansion of the time evolution amplitude (4.244) takes the form 1

−iV (x)t/¯ h

(x tb |x ta ) = q De 2πi¯ ht/M

(

t2 i t3 1+ ∇2 V (x) − [∇V (x)]2 + . . . . 12M h ¯ 24M )

(4.251) Recalling the iη-prescription on page 114, according to which the singularity at t = 0 has to be shifted slightly into the upper half plane by replacing t → t − iη, we use the formula9 Z

∞

−∞

dt 1 aν−1 −aη ita e = Θ(a) e , 2π (it + η)ν Γ(ν)

(4.252)

and obtain the obvious generalization of (4.248): 3 h ¯2 d2 h ¯2 2 2 d + . . . ρcl (E; x), (4.253) ρ(E; x) = 1 − ∇ V (x) 2 − [∇V (x)] 12M dV 24M dV 3

(

)

where ρcl (E; x) is the classical D-dimensional local density of states (4.211). The way this appears here is quite different from that in the earlier classical calculation (4.206), which may be expressed with the help of the local momentum (4.209) as an integral ρcl (E; x) =

Z

dD p δ[E − H(p, x)] = (2π¯ h)D

Z

dD p M δ[p − p(E; x)]. (4.254) D (2π¯ h) p(E; x)

In order to see the relation to the appearance in (4.253) we insert the Fourier decomposition of the leading term of the short-time expansion of the time evolution amplitude Z dD p −i[p2 /2M +V (x)]t/¯h (x t|x 0)cl = e (4.255) (2π¯ h)D into the integral representation (4.238) which takes the form ρ(E; x) =

Z

∞

−∞

dt 2π¯ h

Z

dD p −i[p2 −p2 (E;x)]t/2M ¯h e . (2π¯ h)D

(4.256)

By doing the integral over the time first, the size of the momentum is fixed to the local momentum p2 (E; x) resulting in the original representation (4.254). The expression (4.211) for the density of states, on the other hand, corresponds to first integrating over all momenta. The time integration selects from the result of this the correct local momenta p2 (E; x). 9

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 3.382.6. The formula is easily derived by R∞ expressing (it + η)−ν = Γ−1 (ν) 0 dτ τ ν−1 e−τ (it+η) .

406


This generalizes (4.249) to M D/2 1 ρ(E; x) = [E − V (x)]D/2−1 2 Γ(D/2) 2π¯ h i 1 h ¯2 h 2 [E − V (x)]D/2−3 − ∇ V (x) 12M Γ(D/2 − 2) ) 1 h ¯2 2 D/2−4 + [∇V (x)] [E − V (x)] + ... . 24M Γ(D/2 − 3)

(

(4.257)

When integrating the density (4.257) over all x, the second term in the curly brackets can again be converted into the third term changing its sign, as in (4.250). The right-hand side can easily be integrated for all pure power potentials. This will be done in Appendix 4A.

4.9.3

Bilocal Density of States

It is useful to generalize the local density of states (4.238) and introduce a bilocal density of states: ˆ ai = ρ(E; xb , xa ) = hxb |δ(E − H)|x =

Z

∞

−∞

Z

∞

−∞

dt iEt/¯h ˆ e hxb |e−iHt/¯h |xa i 2π¯ h

dt iEt/¯h (xb t|xa 0). e 2π¯ h

(4.258)

The semiclassical expansion requires now the nondiagonal version of the short-time expansions (4.251). For the one-dimensional harmonic oscillator, the expansion (4.240) is generalized to 1 2 2 2 (xb tb |xa ta ) = q eiM (xb −xa ) /2t¯h e−iM ω x¯ t/2¯h 2πi¯ ht/M

t2 i t3 i t × 1 + ω2 − Mω 4 x¯2 − (xb − xa )2 Mω 2 + . . . , 12 h ¯ 24 h ¯ 24 (

)

(4.259)

where x¯ = (xb +xa )/2 is the mean position of the two endpoints. In this expansion we have included all terms whose size is of the order t3 , keeping in mind that (xb − xa )2 is of the order h ¯ in a finite amplitude. Going to D dimensions and performing the substitutions (4.242) and (4.243), this expansion is generalized to 1 iM (xb −xa )2 /2t¯ h−iV (¯ x)t/¯ h (xb tb |xa ta ) = q De 2πi¯ ht/M

(4.260)

t2 i t3 i t × 1+ ∇2 V (¯ x)− [∇V (¯ x)]2 − [(xb −xa )∇]2 V (¯ x)+. . . . 12M h ¯ 24M h ¯ 24 (

)

For a derivation without substitution trick in (4.242) and (4.243) see Appendix 4B. H. Kleinert, PATH INTEGRALS


407

Inserting this amplitude into the integral in Eq. (4.258), we obtain the bilocal density of states ρ(E; xb , xa ) =

Z

∞

−∞

dt 1 iM (xb −xa )2 /2t¯ h −i[V (¯ x)−E]t/¯ h e q De 2π¯ h 2πi¯ ht/M

t2 i t3 i t × 1+ x) − x) + . . . . (4.261) ∇2 V (¯ [∇V (¯ x)]2 − [(xb −xa )∇]2 V (¯ 12M h ¯ 24M h ¯ 24 (

)

The first term in the integrand is simply the time evolution amplitude of the freeparticle in a constant potential V (¯ x) which has the Fourier decomposition [recall (1.331)]: Z dD p ip(xb −xa )/¯h −iH(p,¯x)t/¯h (xb tb |xa ta )cl = e e . (4.262) (2π¯ h)D Indeed, inserting this into (4.258), and performing the integration over time, we find ρcl (E; xb , xa ) =

Z

dD p ¯ ))eip(xb −xa )/¯h . δ(E − H(p, x (2π¯ h)D

(4.263)

Decomposing the momentum integral into radial and angular parts as in (4.207), we can integrate out the radial part as in (4.197), whereas the angular integral yields the following function of R = |xb − xa |: Z

dˆ p eip(xb −xa )/¯h = SD (pR/¯ h),

(4.264)

which is a direct generalization of the surface of a sphere in D dimensions (4.208). It reduces to it for p = 0. This integral will be calculated in Section 9.1. The result is SD (z) = (2π)D/2 JD/2−1 (z)/z D/2−1 ,

(4.265)

where Jν (z) are Bessel functions. For small z, these behave like10 Jν (z) ≈

(z/2)ν , Γ(ν + 1)

(4.266)

thus ensuring that SD (kR) is indeed equal to SD at R = 0. Altogether, the classical limit of the bilocal density of states is 1 ρcl (E; xb , xa ) = 2π¯ h2

D/2

M

¯ )R/¯ JD/2−1 (p(E; x h) . D/2−1 (R/¯ h)

At xb = xa , this reduces to the density (4.206). 10


(4.267)

408


In three dimensions, the Bessel function becomes J1/2 (z) =

s

2 sin z, πz

(4.268)

and (4.267) yields 1 ρcl (E; xb , xa ) = 2π¯ h2

3/2

¯ )R/¯ 1 M sin[p(E; x h] √ . Γ(3/2) 2 R/¯ h

(4.269)

From the D-dimensional version of the short-time expansion (4.261) we obtain, after using once more the equivalence of t and i¯ hd/dV , (

ρ(E; xb , xa ) = 1 −

4.9.4

3 i d2 h ¯2 h 2 h ¯2 2 d ∇ V (¯ x) − [∇V (¯ x )] 12M dV 2 24M dV 3 ) 1 d 2 + [(xb −xa )∇] V (¯ x) + . . . ρcl (E; xb , xa ).(4.270) 24 dV

Gradient Expansion of Tracelog of Hamiltonian Operator

Starting point is formula (1.585) for the tracelog of the Hamiltonian operator. By performing the trace in the local basis |xi, we arrive at the useful formula involving the density of states (4.238) Z Z ∞ D ˆ Tr log H = d x dE ρ(E; x) log E. (4.271) −∞

Inserting here the classical density of states (4.211), and integrating over the classical spectrum E ∈ (E0 , ∞), where E0 is the bottom of the potential V (x), we obtain the classical limit of the tracelog: D/2 Z Z Z ∞ M ˆ cl = dD x [Tr log H] dE ρcl (E; x) log E = dD x ID/2 (V (x)), (4.272) 2π¯h2 E0 where 1 Iα (V ) ≡ Γ(α)

Z

∞

V

dE (E − V )α−1 log E.

(4.273)

The integrals ID/2 (V (x)) diverge, but can be calculated with the techniques explained in Section 2.15 from the analytically regularized integrals11 Z ∞ 1 Γ(−α + η) dE (E − V )α−1 E −η = V α−η . (4.274) Iαη (V ) ≡ Γ(α) V Γ(η) Since E −η = 1 − η log E + O(η 2 ), the coefficient of −η in the Taylor series of Iαη (V ) will yield the desired integral. Since 1/Γ(η) ≈ η, we obtain directly Iα (V ) = −Γ(−α)V α ,

(4.275)

so that (4.272) becomes ˆ cl = − [Tr log H] 11

M 2π¯h2

D/2

Γ(−D/2)

Z

dD x [V (x)]D/2 .

(4.276)



409

The same result can be obtained with the help of formulas (4.238), (4.251), and (2.498) as Z Z ∞ Z ∞ Z ∞ ′ dt eit[E−V (x)]/¯h dt −iEt′ /¯h ˆ cl = − dD x e . (4.277) [Tr log H] dE D/2 2π¯ h t′ (2πi¯ h t/M ) 0 −∞ −∞ Integrating over the energy yields ˆ cl = − [Tr log H]

Z

D

d x

Z

∞

0

dt 1 e−itV (x)/¯h . t (2πi¯ht/M )D/2

(4.278)

Deforming the contour of integration by the substitution t = −iτ , we arrive at the integral representation of the Gamma function (2.490) which reproduces immediately the result (4.276). The quantum corrections are obtained by multiplying this with the prefactor in curly brackets in the expansion (4.253): ˆ Tr log H

D/2 M = − Γ(−D/2) 2π¯h2 Z 3 ¯h2 d2 ¯h2 2 D 2 d × d x 1− ∇ V (x) 2 − [∇V (x)] + . . . [V (x)]D/2 . 12M dV 24M dV 3

(4.279)

The second term can be integrated by parts, which replaces ∇2 V (x) → −[∇V (x)]2 d/dV , so that we obtain the gradient expansion ˆ =− Tr log H

M 2π¯h2

D/2

Γ(−D/2)

Z

dD x

¯2 h d3 1+ [∇V (x)]2 + . . . [V (x)]D/2 . (4.280) 24M dV 3

The curly brackets can obviously be replaced by ¯h2 Γ(3 − D/2) [∇V (x)]2 1− + ... . 24M Γ(−D/2) [V (x)]3 In one dimension and with M = 1/2, this amounts to the formula Z p 1 ¯h2 [V ′ (x)]2 2 2 Tr log[−¯h ∂x + V (x)] = dx V (x) 1 + + ... . ¯h 32 V 3 (x)

(4.281)

(4.282)

It is a useful exercise to rederive this with the help of the Gelfand-Yaglom method in Section 2.4. There exists another method for deriving the gradient expansion (4.279). We split V (x) into a constant term V and a small x-dependent term δV (x), and rewrite ¯h2 2 ¯h2 2 Tr log − ∇ + V (x) = Tr log − ∇ + V + δV (x) 2M 2M ¯h2 2 = Tr log − ∇ + V +Tr log (1 + ∆V δV ) , (4.283) 2M

where ∆V denotes the functional matrix ∆V (x, x′ ) =

−1 Z ′ ¯2 2 h dD p eip(x−x )/¯h − ∇ +V = ≡ ∆V (x − x′ ). 2M (2π¯h)D p2 /2M + V

(4.284)

This coincides with the fixed-energy amplitude (i/¯h)(x|x′ )E at E = −V [recall Eq. (1.344)]. The first term in (4.283) is equal to (4.276) if we replace V (x) in that expression by the constant V , so that we may write ˆ = [Tr log H] ˆ cl Tr log H + Tr log (1 + ∆V δV ) . (4.285) V (x)→V

410


We now expand the remainder 1 Tr log (1 + ∆V δV ) = Tr ∆V δV − Tr (∆V δV )2 + . . . , 2 and evaluate the expansion terms. The first term is simply Z Z D Tr ∆V δV = d x ∆V (x, x)δV (x) = ∆V (0) dD x δV (x).

(4.286)

(4.287)

where ∆V (0) =

Z

1 dD p ˆ cl = ∂V [Tr log H] . D 2 V (x)→V (2π¯h) p /2M + V

The result of the integration was given in Eq. (1.348). The second term in the remainder (4.286) reads explicitly Z Z 1 1 2 − Tr (∆V δV ) = − dD x dD x′ ∆V (x, x′ )δV (x′ )∆V (x′ , x)δV (x). 2 2

(4.288)

(4.289)

We now make use of the operator relation ˆ B] ˆ = f ′ (A)[ ˆ A, ˆ B] ˆ − 1 f ′′ (A)[ ˆ A, ˆ [A, ˆ B]] ˆ + ... , [f (A), 2

(4.290)

to expand h i h i δV ∆V = ∆V δV + ∆2V Tˆ, δV + ∆3V Tˆ, [Tˆ, δV ] + . . . ,

(4.291)

ˆ 2 /2M . It commutes with any function f (x) as where Tˆ is the operator of the kinetic energy p follows: ¯2 2 h ∇ f + 2 (∇f ) · ∇ , 2M ¯h4 2 2 (∇ ) f + 4[∇∇2 f ] · ∇ + 4[∇i ∇i f ]∇i ∇j , 2 4M

[Tˆ, f ] = − [Tˆ, [Tˆ, f ]] = .. .

(4.292) (4.293)

.

Inserting this into (4.289), we obtain a first contribution Z Z 1 − dD x dD x′ ∆V (x, x′ )∆V (x′ , x)[δV (x)]2 . 2

(4.294)

The spatial integrals are performed by going to momentum space, where we derive the general formula Z Z Z dD x dD x1 · · · dD xn ∆V (x, x1 )∆V (x1 , x2 ) · · · ∆V (xn−1 , xn )∆V (xn , x) Z 1 (−1)n n dD p = = ∂V ∆V (0). (4.295) n+1 D (2π¯h) (p2 /2M +V ) n! This simplifies (4.294) to 1 ∂V ∆V (0) 2

Z

dD x [δV (x)]2 .

(4.296)

We may now combine the non-gradient terms of δV (x) consisting of the first term in (4.285), of (4.287), and of (4.296), and replace in the latter ∆V (0) according to (4.288), to obtain the first ˆ cl with the full x-dependent V (x) in Eq. (4.276). three expansion terms of [Tr log H] H. Kleinert, PATH INTEGRALS

4.9 Quantum Corrections to Classical Density of States The next contribution to (4.289) coming from (4.292) is Z Z Z ¯h2 dD x dD x1 dD x2 ∆V (x, x1 )∆V (x1 , x2 )∆V (x2 , x) 4M n o 2 × ∇2 δV (x) δV (x) + 2 [∇δV (x)] + 2[∇δV (x)]δV (x)∇ ,

411

(4.297)

where the last ∇ acts on the first x in ∆V (x, x1 ), due to the trace. It does not contribute to the integral since it is odd in x − x1 . We now perform the integrals over x1 and x2 using formula (4.295) and find Z n o ¯h2 2 dD x ∇2 δV (x) [δV (x)]+2 [∇δV (x)] ∂V2 ∆V (0). (4.298) 8M

The first term can be integrated by parts, after which it removes half of the second term. A third contribution to (4.289) which contains only the lowest gradients of δV (x) comes from the third term in (4.293): Z Z Z Z h4 ¯ D D D − d x d x d x dD x3 ∆V (x, x1 )∆V (x1 , x2 )∆V (x2 , x3 )∆V (x3 , x) 1 2 8M 2 × 4 [∇i ∇j δV (x)] δV (x)∇i ∇j , (4.299) where the last ∇i ∇j acts again on the first x in ∆V (x, x1 ), as a consequence of the trace. In momentum space, we encounter the integral # " Z Z 8M δij V dD p −4pi pj /¯h2 dD p 1 = − 2 − (2π¯h)D (p2 /2M + V )4 (2π¯h)D (p2 /2M +V )3 (p2 /2M +V )4 ¯h D 8M δij 1 2 1 = − 2 ∂V + V ∂V3 ∆V (0). (4.300) 6 ¯h D 2 so that the third contribution to (4.289) reads, after an integration by parts, Z ¯h2 1 2 1 2 δij D 3 ∂ + V ∂V ∆V (0). − d x [∇δV (x)] M D 2 V 6

(4.301)

Combining all gradient terms in [∇V (x)]2 and replacing ∆V (0) according to (4.288), we recover the previous result (4.280) with the curly brackets (4.281). For the one-dimensional tracelog, this leads to the formula Z p ¯h2 [V ′ (x)]2 2 2 Tr log[−¯h ∂x + V (x)] = dx V (x) 1 + + ... . (4.302) 32 V 3 (x)

It is a useful exercise to rederive this with the help of the Gelfand-Yaglom method in Section 2.4. This expansion can actually be deduced, and carried to much higher order, with the help of the gradient expansion of the trace of the logarithm of the operator −¯h2 ∂τ2 + w2 (τ ) derived in Subsection 2.15.4. If we replace τ by x, v(τ ) by V (x), h ¯ by 1, we obtain from (2.544): ( ! ! Z 2 3 p V′ 5V ′ V ′′ 15V ′ 9V ′ V ′′ V (3) 1 2 3 dx V (x) 1 − ¯h 3/2 − ¯h − − ¯h − + ¯h 32V 3 8V 2 4V 64V 9/2 32V 7/2 16V 5/2 !) 4 2 2 1105V ′ 221V ′ V ′′ 19V ′′ 7V ′ V (3) V (4) − ¯h4 − + + − . (4.303) 6 5 4 4 2048V 256V 128V 32V 32V 3 The ¯h-term in the curly brackets vanishes if V (x) is the same at the boundaries, and the h ¯ 2 term 2 goes over into the ¯h -term in (4.302).

412

4.9.5


Local Density of States on Circle

For future use, let us also calculate this determinant for x on a circle x = (0, b), so that, as a side result, we obtain also the gradient expansion of the tracelog of the operator (−∂τ2 + ω 2 (τ )) at a finite temperature. For this we recall that for a τ -independent frequency, the starting point is Eq. (2.550), according to which the tracelog of the operator (−∂τ2 + ω 2 ) with periodic boundary conditions in τ ∈ (0, ¯hβ) is given by " # Z ∞ ∞ X 2 ¯h dτ −1/2 −(n¯ hβ)2 /4τ τ 1+2 e e−τ ω . (4.304) Fω = − √ π 0 τ n=1 The first term is the zero-temperature expression, the second comes from the Poisson summation formula and gives p the finite-temperature effects. In the first (classical) term of the density (4.246), the factor 1/ 2π¯hτ /M came from the integral over the Boltzmann factor involving the kinetic R∞ 2 energy −∞ (dk/2π)e−τ h¯ k /2M . For periodic boundary conditions in x ∈ (0, b), this is changed to P −τ h¯ k2 /2M m (1/b) m e , where km = 2πm/b. By Poisson’s formula (1.205), this can be replaced by the integral and an auxiliary sum ∞ Z ∞ ∞ X X 2 2 2 1 X −τ h¯ km dk −τ h¯ k2 /2M+ibkn 1 /2M e−n Mb /2¯hτ. (4.305) e = e =p b m 2π 2π¯hτ /M n=−∞ n=−∞ −∞ If the sum is inserted into the integral (4.246), we obtain the density ρ(E; x) on a circle of circumference b, with the classical contribution Z ∞ 2 2 ∞ dτ X e−n Mb /2¯hτ −τ [E−V (x)]/¯h ρcl (b, E; x) = 2 . (4.306) 2π¯h n=−∞ (2π¯hτ /M )1/2 0

The n = 0 -term in the sum leads back to the original expression (4.246) on an infinite x-axis. The τ -integrals are now done with the help of formula (2.551) which yields, due to Kν (z) = K−ν (z), ν Z ∞ dτ ν −n2 Mb2 /2¯hτ −[E−V (x)]τ /¯h nM b τ e =2 Kν (np(E; x)b/¯h), (4.307) τ p(E; x) 0 and we obtain, instead of (4.247), ρcl (b, E; x) = Inserting K1/2 (z) =

1/2 ∞ X 1 1 nM b p 2 K1/2 (np(E; x)b/¯h). π¯h 2π¯h/M n=0 p(E; x)

p π/2z e−z [recall (2.553)], this becomes 1 M ρcl (b, E; x) = π¯h p(E; x)

The sum

P∞

n=1

1+2

∞ X

e

−np(E;x)b/¯ h

n=1

!

.

(4.308)

(4.309)

αn is equal to α/(1 − α), so that we obtain

p M coth[p(E; x)b/2¯h] M coth 2M [E − V (x)] b/2¯h p ρcl (b, E; x) = = . π¯h p(E; x) π¯h 2M [E − V (x)]

(4.310)

For b → ∞, this reduces to the previous density (4.247). If we include the higher powers of τ in (4.246), we obtain the generalization of expression (4.248): 3 ¯h2 ′′ d2 ¯h2 ′ 2 d ρ(b, E; x) = 1 − V (x) 2 − [V (x)] + . . . ρcl (b, E; x). (4.311) 12M dV 24M dV 3 H. Kleinert, PATH INTEGRALS


413

The tracelog is obtained by integrating this over dE log E from V (x) to infinity. The integral diverges, and we must employ analytic regularization. We proceed by using the R ∞ as in (4.277), ′ real-time version of (4.306) and rewriting log E as an integral − 0 (dt′ /t′ )e−iEt /¯h , so that the leading term in (4.311) is given by ˆ cl = − [Tr log H]

Z

0

b

Z dx

∞

dE

−∞

Z

∞

−∞

The integral over E leads now to ∞ Z X ˆ cl = − [Tr log H] n=−∞

Z 2 2 ∞ dt X e−in Mb /2¯ht+it[E−V (x)]/¯h ∞ dt′ −iEt′ /¯h e . (4.312) 2π¯h n=−∞ t′ (2πi¯ht/M )1/2 0

b

dx 0

Z

∞

0

2 2 dt 1 e−in Mb /2¯ht−itV (x)/¯h . 1/2 t (2πi¯ht/M )

(4.313)

Deforming again the contour of integration by the substitution t = −iτ , creating the τ −1 in the denominator by an integration over V (x)/¯h, we see that Z b Z ∞ ˆ cl = −π [Tr log H] dx dV ρcl (b, E; x) 0

= π

Z

b

dx

0

E−V (x)→V

V (x)

Z

∞

V (x)

1 2

p V /Eb dV 1 coth p , 1 Eb 4πb V /Eb 2

(4.314)

where Eb ≡ ¯h/2M b2 is the energy associated with the length b. The integration over V produces a factor h ¯ /τ in the integrand. Thus we obtain   s  Z b   2 3 h ¯ d 2 1 V (x) ˆ=   . Tr log H dx 1 + [V ′ (x)]2 + . . . log 2 sinh (4.315)  24M dV 3 b 2 Eb  0

For M = h ¯ 2 /2, this give us the finite-b correction to formula (4.302). Replacing x by the Euclidean time τ , b by h ¯ β, and V (x) by the time-dependent square frequency ω 2 (τ ), we obtain the gradient expansion Z h¯ β 2 ¯hβω(τ ) [∂τ ω 2 (τ )]2 3 2 2 Tr log[−∂τ + ω (τ )] = dτ 1 + ∂ω2 +. . . log 2 sinh . (4.316) 12 ¯hβ 2 0

4.9.6

Quantum Corrections to Bohr-Sommerfeld Approximation

The expansion (4.303) can be used to obtain a higher-order expansion of the density of states ρ(E), thereby extending Eq. (4.250). For this we recall Eq. (1.590) according to which we can calculate the exact density of states from the formula 1 ρ(E) = − ∂E Im Tr log −∂x2 + [V (x) − E] . π

(4.317)

Integrating this over the energy yields, according to Eq. (1.591), the number of states times π, and thus the simple exact quantization condition for a nondegenerate one-dimensional system: −Im Tr log −∂x2 + [V (x) − E] = π(n + 1/2). (4.318) By comparison with Eq. (4.203) we may define a fullly quantum corrected version of the classical eikonal: (4.319) Sqc (E) = −2¯h Im Tr log −∂x2 + [V (x) − E] .

414


The semiclassical expansionpof this can p be obtained from our earlier result (4.303) by replacing V (x) → V (x) − E, so that V (x) → −i E − V (x), yielding # ( " Z 2 p 5V ′ V ′′ 2 ¯ Sqc (E) = 2 dx E − V (x) 1 + h 3 + 2 32(E −V ) 8(E −V ) " #) 4 2 2 1105V ′ 221V ′ V ′′ 19V ′′ 7V ′ V (3) V (4) 4 −¯h + + + + + . . . . (4.320) 2048(E −V )6 256(E −V )5 128(E −V )4 32(E −V )4 32(E −V )3 The first term in the expansion corresponds to the Bohr-Sommerfeld approximation, the remaining ones yield the quantum corrections. The integrand agrees, of course, with the WKB expansion of the eikonal (4.14) with the expansion terms (4.16), (4.18). Using Eq. (4.317) we obtain from Sqc (E) the density of states # ( " Z 2 1 1 3 V ′′ 1 25 V ′ 2 Sqc (E) = dx √ + ρ(E) = 1 − ¯h 2π¯h 2π¯h E−V 32 (E − V )3 8 (E − V )2 " #) 12155 V ′ 4 1989 V ′ 2 V ′′ 133 V ′′ 2 49 V ′ V (3) 5 V (4) 3 +¯ h . (4.321) 6 + 5 + 4 + 4 + 3 2048 (E − V ) 256 (E − V ) 128 (E − V ) 32 (E − V ) 32 (E − V ) Let us calculate the quantum corrections to the semiclassical energies for a purely quartic potential V (x) = gx4 /4, where the integral over the first term in (4.320) between the turning points ±xE = ±(4E/g)1/4 gave the Bohr-Sommerfeld approximation (4.29). The integrals of the higher terms in (4.320) are divergent, but can be calculated in analytically regularized form using once more the integral formula (4.30). This extends the Bohr-Sommerfeld equation ν(E) = n+ 1/2 to the exact equation N (E) ≡ Sqc /2π¯h = n + 1/2 [recall (1.591)]. If we express N (E) in terms of ν(E) defined in Eq. (4.29) rather than E, we obtain the expansion N (ν) = ν − −

1 11π 2 4697π 390065π 4 + − + 12πν 10368Γ8( 34 )ν 3 1866240Γ8( 34 )ν 5 501645312Γ16( 34 )ν 7

53352893π 3 + . . . = n + 1/2. 7739670528Γ16( 34 )ν 9

(4.322)

The function is plotted in Fig. 4.322 for increasing orders in y. Given a solution ν (n) of this equation, we obtain the energy E (n) from Eq. (4.31) with √ κ(n) = [ν (n) 3Γ(3/2)2 /2 π]4/3 . (4.323) For large n, where ν (n) → n, we recover the Bohr-Sommerfeld result (4.32). We can invert the series (4.322) and obtain 0.026525823 0.002762954 0.001299177 ν (n) = (n + 12 ) 1 + − − (n + 21 )2 (n + 21 )4 (n + 21 )6 0.003140091 0.007594497 + + + . . . . (4.324) (n + 12 )8 (n + 21 )10 The results are compared with the exact ones in Table 4.1, which approach rapidly the BohrSommerfeld limit 0.688 253 702 . . . The approach is illustrated in the right-hand part of Fig. 4.1 (n) where log[ν (n) /(n + 21 )4/3 − 1] = log[E (n) /EBS − 1] is plotted once for the exact values and once for the semiclassical expansion in Fig. 4.1. The second excited states is very well represented by the series. For a detailed study of the convergence of the semiclassical expansion see Ref. [5]. Having obtained the quantum-corrected eikonal Sqc (E) we can write down a quantum-corrected partition function replacing the classical eikonal S(E) in Eq. (4.230): Zqc (E) = tqc (E)

1 eiSqc (E)−iπν/2 . Pqc (1)1/2 1 − eiSqc (E)−iπν/2

(4.325)



2

415

2

N (ν)

n + 1/2

1.5 2 1

4

6

8

10

-2

0.5 -4 (n)

0.5

1

1.5

ν

-0.5

2

2.5

3 -6

log[E (n) /EBS − 1]

-8 -1

Figure 4.1 Determination of energy eigenvalues E (n) for purely quartic potential gx4 /4 in semiclassical expansion. The intersections of N (ν) with the horizontal lines yield ν (n) , from which E (n) is obtained via Eqs. (4.323) and (4.31). The increasing dash lengths show the expansions of N (ν) to increasing orders in ν. For the ground state with n = 0, the expansion is too divergent to give improvements to the lowest approximation without resumming the series; Right: Comparison between exact and semiclassical energies. The (n) plot is for log[E (n) /EBS − 1].

Table 4.1 Particle energies in purely anharmonic potential gx4 /4 for n = 0, 2, 4, 6, 8, 10. n 0 2 4 6 8 10

E (n) /(g¯h/4M 2 )1/3 0.667 986 259 155 777 108 3 4.696 795 386 863 646 196 2 10.244 308 455 438 771 076 0 16.711 890 073 897 950 947 1 23.889 993 634 572 505 935 5 31.659 456 477 221 552 442 8

κ(n) /2(n + 1/2)4/3 0.841 609 948 950 895 526 0.692 125 685 914 981 314 0.689 449 772 359 340 765 0.688 828 486 600 234 466 0.688 590 146 947 993 676 0.688 474 290 179 981 433

416


′ where tqc (E) ≡ Sqc (E) [compare (4.192)], and Pqc (1)1/2 is the quantum-corrected determinant (4.228) whose calculation will require extra work.

4.10

Thomas-Fermi Model of Neutral Atoms

The density of states calculated in the last section forms the basis for the ThomasFermi model of neutral atoms. If an atom has a large nuclear charge Z, most of the electrons move in orbits with large quantum numbers. For Z → ∞, we expect them to be described by semiclassical limiting formulas, which for decreasing values of Z require quantum corrections. The largest quantum correction is expected for electrons near the nucleus which must be calculated separately.

4.10.1

Semiclassical Limit

Filling up all negative energy states with electrons of both spin directions produces some local particle density n(x), which is easily calculated from the classical local density (4.211) over all negative energies, yielding the Thomas-Fermi density of states (−) ρcl (x)

M = dE ρcl (E; x) = V (x) 2π¯ h2 Z

0

D/2

1 [−V (x)]D/2 . Γ(D/2 + 1)

(4.326)

This expression can also be obtained directly from the phase space integral over the accessible free-particle energies. At each point x, the electrons occupy all levels up to a Fermi energy pF (x)2 EF = + V (x). (4.327) 2M The associated local Fermi momentum is equal to the local momentum function (4.209) at E = EF : pF (x) = p(EF ; x) =

q

2M[EF − V (x)].

(4.328)

The electrons fill up the entire Fermi sphere |p| ≤ pF (x): 1 2π D/2 pD F (x) . D (2π¯ h) Γ(D/2) D |p|≤pF (x) 0 (4.329) For neutral atoms, the Fermi energy is zero and we recover the density (4.326). By occupying each state of negative energy twice, we find the classical electron density (−) n(x) = 2ρcl (x). (4.330) (−) ρcl (x)

=

Z

1 dD p = SD D (2π¯ h) (2π¯ h)D

Z

pF (x)

dp pD−1 =

The potential energy density associated with the levels of negative energy is obviously (−)

Epot TF (x) = V (x)ρ(−) (x) = −

M 2π¯ h2

D/2

1 [−V (x)]D/2+1 . (4.331) Γ(D/2 + 1) H. Kleinert, PATH INTEGRALS

4.10 Thomas-Fermi Model of Neutral Atoms

417

To find the kinetic energy density we integrate (−) Ekin TF (x)

Z

=

0 V (x)

dE [E − V (x)]ρcl (E; x)

D/2 M = D/2 + 1 2π¯ h2

D/2

1 [−V (x)]D/2+1 . Γ(D/2 + 1)

(4.332)

As in the case of the density of states (4.329), this expression can be obtained directly from the phase space integral over the free-particle energies (−) Ekin TF (x)

=

Z

|p|≤pF (x)

d D p p2 . (2π¯ h)D 2M

(4.333)

Performing the momentum integral on the right-hand side yields the energy density (−) Ekin TF (x)

1 1 = SD D (2π¯ h) 2M

Z

pF (x)

0

dp pD+1 =

1 SD pD+2 (x) F , (4.334) D (2π¯ h) D + 2 2M

in agreement with (4.332). The sum of the two is the Thomas-Fermi energy density (−) ETF (x)

=

Z

0

V (x)

dE E ρcl (E; x)

1 M = − D/2 + 1 2π¯ h2

D/2

1 [−V (x)]D/2+1 . Γ(D/2 + 1)

(4.335)

The three energies are related by (−)

ETF (x) = −

1 1 (−) (−) Ekin TF (x) = E (x). D/2 D/2 + 1 pot TF

(4.336)

Note that the Thomas-Fermi model can also be applied to ions. Then the energy levels are filled up to a nonzero Fermi energy EF , so that the density of states (4.326) and the kinetic energy (4.332) have −V replaced by EF − V . This follows immediately from the representations (4.329) and (4.333) where the right-hand sides q depend only on pF (x) = 2M[EF − V (x)]. In the potential energy (4.331), the expression (−V )D/2+1 is replaced by (−V )(EF − V )D/2 , whereas in the ThomasFermi energy density (4.335) it becomes (1 − EF ∂/EF )(EF − V )D/2+1 .

4.10.2

Self-Consistent Field Equation

The total electrostatic potential energy V (x) caused by the combined charges of the nucleus and the electron cloud is found by solving the Poisson equation ∇2 V (x) = 4πe2 [Zδ (3) (x) − n(x)] ≡ 4πe2 [nC (x) − n(x)].

(4.337)

The nucleus is treated as a point charge which by itself gives rise to the Coulomb potential Ze2 VC (x) = − . (4.338) r

418


Recall that in these units e2 = α¯ hc, where α is the dimensionless fine-structure constant (1.502). A single electron near the ground state of this potential has orbits with diameters of the order naH /Z, where n is the principal quantum number and aH the Bohr radius of the hydrogen atom, which will be discussed in detail in Chapter 13. The latter is expressed in terms of the electron charge e and mass M as h ¯2 1 C aH = = λ . (4.339) Me2 α M This equation implies that aH is about 137 times larger than the Compton wavelength of the electron λC ¯ /Mc ≈ 3.861 593 23 × 10−13 cm. (4.340) M ≡h

It is convenient to describe the screening effect of the electron cloud upon the Coulomb potential (4.338) by a multiplicative dimensionless function f (x). Restricting our attention to the ground state, which is rotationally symmetric, we shall write the solution of the Poisson equation (4.337) as V (x) = −

Ze2 f (r). r

(4.341)

At the origin the function f (r) is normalized to unity, f (0) = 1,

(4.342)

to ensure that the nuclear charge is not changed by the electrons. It is useful to introduce a length scale of the electron cloud aTF

1 2π¯ h2 Γ(5/2) = 2 1/3 eZ M 2 · 4π "

#2/3

=

1 3π 2 4

2/3

aH aH ≈ 0.8853 1/3 , 1/3 Z Z

(4.343)

which is larger than the smallest orbit aH /Z by roughly a factor Z 2/3 . All length scales will now be specified in units of aTF , i.e., we set r = aTF ξ.

(4.344)

In these units, the electron density (4.330) becomes simply (2Ze2 M) n(x) = − 3π 2h ¯3

3/2

"

f (ξ) aTF ξ

#3/2

"

Z f (ξ) = a3TF 3 4πaTF ξ

#3/2

.

(4.345)

The left-hand side of the Poisson equation (4.337) reads ∇2 V (x) =

1 d Ze2 1 ′′ rV (x) = − f (ξ), r dr 2 a3TF ξ

(4.346)

so that we obtain the self-consistent Thomas-Fermi equation 1 f ′′ (ξ) = √ f 3/2 (ξ), ξ

ξ > 0.



419

f (ξ)

ξ

Figure 4.2 Solution for screening function f (ξ) in Thomas-Fermi model.

The condition ξ > 0 excludes the nuclear charge from the equation, whose correct size is incorporated by the initial condition (4.342). Equation (4.347) is solved by the function shown in Fig. 4.2. Near the origin, it starts out like f (ξ) = 1 − sξ + . . . , (4.348) with a slope s ≈ 1.58807.

(4.349)

For large ξ, it goes to zero like f (ξ) ≈

144 . ξ3

(4.350)

This power falloff is a weakness of the model since the true screened potential should fall off exponentially fast. The right-hand side by itself happens to be an exact solution of (4.347), but does not satisfy the desired boundary condition f (0) = 1.

4.10.3

Energy Functional of Thomas-Fermi Atom

Let us derive an energy functional whose functional extremization yields the Thomas-Fermi equation (4.347). First, there is the kinetic energy of the spin-up and spin-down electrons in a potential V (x). It is given by the volume integral over twice the Thomas-Fermi expression (4.332): (−) Ekin

3 M =2 5 2π¯ h2

3/2

1 Γ(5/2)

Z

d3 x [−V (x)]5/2 .

(4.351)

This can be expressed in terms of the electron density (4.330) as (−) Ekin

3 Z 3 5/3 = κ d x n (x), 5

(4.352)

h ¯ 2 2/3 3π . 2M

(4.353)

where κ≡

420


The potential energy (−) Epot

=

Z

d3 x V (x)n(x)

(4.354)

(−)

is related to Ekin via relation (4.336) as 5 (−) (−) Epot = − Ekin , 3

(4.355)

and the total electron energy in the potential V (x) is (−)

(−)

Ee(−) = Ekin + Epot =

2 (−) E . 5 pot

(4.356)

We now observe that if we consider the energy as a functional of an arbitrary density n(x), Z Z 3 Ee(−) = Ee [n] ≡ κ d3 x n5/3 (x) + d3 x V (x)n(x), (4.357) 5 the physical particle density (4.330) constitutes a minimum of the functional, which satisfies κn2/3 (x) = −V (x). In the Thomas-Fermi atom, V (x) on the right-hand side of (4.357) is, of course, the nuclear Coulomb potential, i.e., (−) Epot

=

(−) EC

≡

Z

d3 x VC (x) n(x).

(4.358)

The energy Ee(−) has to be supplemented by the energy due to the Coulomb repulsion between the electrons (−) Eee

e2 Z 3 3 ′ 1 n(x′ ). = Eee [n] = d xd x n(x) ′ 2 |x − x |

(4.359)

The physical energy density should now be obtained from the minimum of the combined energy functional Z 3 Z 3 5/3 e2 Z 3 3 ′ 1 3 Etot [n] = κ d x n (x) + d x VC (x) n(x) + d xd x n(x) n(x′ ). 5 2 |x − x′ | (4.360) Since we are not very familiar with extremizing nonlocal functionals, it will be convenient to turn this into a local functional. This is done as follows. We introduce an auxiliary local field ϕ(x) and rewrite the interaction term as

Eee [n, ϕ] = Eϕ [n, ϕ] − Eϕϕ [ϕ] ≡

Z

d3 x ϕ(x)n(x) −

1 8πe2

Z

d3 x ∇ϕ(x)∇ϕ(x). (4.361)

Extremizing this in ϕ(x), under the assumption of a vanishing ϕ(x) at spatial infinity, yields the electric potential of the electron cloud ∇2 ϕ(x) = −4πe2 n(x),



421

which is the same as (4.337), but without the nuclear point charge at the origin. Inserting this into (4.361) we reobtain precisely to the repulsive electron-electron interaction energy (4.359). Replacing the last term in (4.360) by the functional (4.361), we obtain a total energy functional Etot [n, ϕ], for which it is easy to find the extremum with respect to n(x). This lies at κ n2/3 (x) = −V (x), (4.363) where V (x) = VC (x) + ϕ(x)

(4.364)

is the combined Thomas-Fermi potential of the nucleus and the electron cloud solving the Poisson equation (4.337). If the extremal density (4.363) is inserted into the total energy functional Etot [n, ϕ], we may use the relation (4.355) to derive the following functional of ϕ(x): 2 Etot [ϕ] = − 5

Z

1 d x V (x)n(x) + 8πe2 3

Z

d3 x ϕ(x)∇2 ϕ(x).

(4.365)

When extremizing this expression with respect to ϕ(x) we must remember that V (x) = VC (x) + ϕ(x) is also present in n(x) with a power 3/2. The extremum lies therefore again at a field satisfying the Poisson equation (4.362).

4.10.4

Calculation of Energies

We now proceed to calculate explicitly the energies occuring in Eq. (4.365). They turn out to depend only on the slope of the screening function f (ξ) at the origin. (−) Consider first Epot . The common prefactor appearing in all energy expressions can be expressed in terms of the Thomas-Fermi length scale aTF of Eq. (4.343) as 2

M 3/2 4π 3 Z −1/2 e = 3/2 . (2π¯ h)3/2 Γ(5/2) aTF

(4.366)

We therefore obtain the simple energy integral involving the screening function f (ξ): (−)

Epot = −

Z 2 e2 Z ∞ 1 dξ √ f 5/2 (ξ). a 0 ξ

(4.367)

The interaction energy between the electrons at the extremal ϕ(x) satisfying (4.362) becomes simply Z 1 (−) Eee = d3 x n(x)ϕ(x), (4.368) 2 which can be rewritten as (−) Eee =

1 2

Z

1 (−) 1 (−) d3 x n(x)[V (x) − VC (x)] = Epot − EC . 2 2

(4.369)

422


Inserting this into (4.365) we find the alternative expression for the total energy 2 (−) 1 (−) 1 (−) (−) (−) = − Epot + EC . Etot = Epot − Eee 5 10 2

(4.370)

(−)

The energy EC of the electrons in the Coulomb potential is evaluated as follows. Replacing n(x) by −∇2 ϕ(x)/4πe2 , we have, after two partial integrations with vanishing boundary terms and recalling (4.337), (−) EC

Z 1 Z 3 2 =− d x ϕ(x)∇ VC (x) = − d3 x ϕ(x)nC (x). 4πe2

(4.371)

Now, since ϕ(x) = V (x) − VC (x) = −

Ze2 [f (ξ) − 1], r

nC = Zδ (3) (x),

(4.372)

(−)

we see that the Coulomb energy EC depends only on ϕ(0), which can be expressed in terms of the negative slope (4.349) of the function f (ξ) as: ϕ(0) =

Ze2 s. a

(4.373)

Thus we obtain

Z 2 e2 s. (4.374) a We now turn to the integral associated with the potential energy in Eq. (4.367): (−)

EC = − Z

I[f ] =

∞

0

1 dξ √ f 5/2 (ξ). ξ

(4.375)

By a trick it can again be expressed in terms of the slope parameter s. We express the energy functional (4.365) in terms of the screening function f (ξ) as Z 2 e2 Etot [ϕ] = − ε[f ], aTF

(4.376)

with the dimensionless functional 2 1 ε[f ] ≡ I[f ] + J[f ] = 5 2

Z

0

∞

(

)

2 1 5/2 1 √ f (ξ) − [f (ξ) − 1]f ′′ (ξ) . dξ 5 ξ 2

(4.377)

The second integral can also be rewritten as J[f ] =

Z

∞ 0

dξ [f ′ (ξ)]2.

(4.378)

This follows from a partial integration J[f ] = −

Z

0

∞

dξ [f (ξ) − 1]f ′′ (ξ) =

Z

0

∞

dξ [f ′ (ξ)]2 − [f (ξ) − 1]f ′ (ξ)|∞ 0 ,

(4.379)



423

inserting the boundary condition f (ξ) − 1 = 0 at ξ = 0 and f ′ (ξ) = 0 at ξ = ∞. We easily verify that the Euler-Lagrange equation following from ε[f ] is the Thomas-Fermi differential equation (4.347). As a next step in calculating the integrals I and J, we make use of the fact that under a scaling transformation f (ξ) → f¯(ξ) = f (λξ), (4.380) the functional ε[f ] goes over into

2 1 ελ [f ] = λ−1/2 I[f ] + λJ[f ]. (4.381) 5 2 This must be extremal at λ = 1, from which we deduce that for f (ξ) satisfying the differential equation (4.347): 5 I[f ] = J[f ]. (4.382) 2 This relation permits us to express the integral J in terms of the slope of f (ξ) at the origin. For this we separate the two terms in (4.379), and replace f ′′ (ξ) via the Thomas-Fermi differential equation (4.347) to obtain Z ∞ Z ∞ Z ∞ 1 ′′ ′′ J =− dξ f (ξ)f (ξ) + dξ f (ξ) = − dξ √ f 5/2 (ξ) −f ′ (0) = −I + s. (4.383) ξ 0 0 0 Together with (4.382), this implies 5 2 I = s, J = s. 7 7 Thus we obtain for the various energies: 3 Z 2 e2 5 s, 5 aTF 7 Z 2 e2 (−) EC = − s, aTF and the total energy is (−)

Ekin =

(−)

Etot =

(−)

Z 2 e2 5 s, aTF 7 Z 2 e2 1 = s, aTF 7

Epot = − (−) Eee

(4.384)

Ee(−) = −

2 Z 2 e2 5 s, 5 aTF 7

(4.385)

2 (−) 1 (−) 1 (−) 3 Z 2 e2 e2 (−) Epot −Eee = − Epot + EC = − s ≈ −0.7687 Z 7/3 . (4.386) 5 10 2 7 aTF aH

The energy increases with the nuclear charge Z like Z 2 /aTF ∝ Z 7/3 . At the extremum, we may express the energy functional ε[f ] with the help of (4.383) as Z 1 ∞ 1 1 ε¯[f ] = − dξ √ f 5/2 (ξ) − f (0)f ′ (0), (4.387) 10 0 2 ξ or in a form corresponding to (4.370): 1 1 1 Z∞ 1 1Z ∞ 1 ε¯[f ] ≡ − I[f ] + JC [f ] = − dξ √ f 5/2 (ξ) + dξ √ f 3/2 (ξ). (4.388) 10 2 10 0 2 0 ξ ξ Using the Thomas-Fermi equation (4.347), the second integral corresponding to the Coulomb energy can be reduced to a surface term yielding JC [f ] = s, so that (4.388) gives the same total energy as (4.377).

424

4.10.5


Virial Theorem

Note that the total energy is equal in magnitude and opposite in sign to the kinetic energy. This is a general consequence of the so-called viral theorem for Coulomb systems. The kinetic energy of the many-electron Schrödinger equation contains the Laplace differential operator proportional to ∇2 , whereas the Coulomb potentials are proportional to 1/r. For this reason, a rescaling x → λx changes the sum of kinetic and total potential energies Etot = Ekin + Epot

(4.389)

λ2 Ekin + λEpot .

(4.390)

into Since this must be extremal at λ = 1, one has the relation 2Ekin + Epot = 0,

(4.391)

Etot = −Ekin .

(4.392)

which proves the virial theorem

In the Thomas-Fermi model, the role of total potential energy is played by the (−) (−) , and Eq. (4.385) shows that the theorem is satisfied. combination Epot − Eee

4.10.6

Exchange Energy

In many-body theory it is shown that due to the Fermi statistics of the electronic wave functions, there exists an additional electron-electron exchange interaction which we shall now take into account. For this purpose we introduce the bilocal density of all states of negative energy by analogy with (4.326): (−) ρcl (xb , xa )

=

Z

0

V (¯ x)

(−)

dEρcl (E; xb , xa ).

(4.393)

In three dimensions we insert (4.269) and rewrite the energy integral as Z

0

V (¯ x)

dE =

1 Z pF (¯x) dp p , M 0

(4.394)

¯ [see (4.328)]. with the Fermi momentum pF (¯ x) of the neutral atom at the point x In this way we find (−)

ρcl (xb , xa ) =

p3F (¯ x) 1 (sin z − z cos z), 2 2π h ¯ 3 z3

(4.395)

z ≡ pF (¯ x)R/¯ h.

(4.396)

where



425

This expression can, incidentally, be obtained alternatively by analogy with the local expression (4.329) from a momentum integral over free wavefunctions d3 p ip(xb −xa )/¯h e . (2π¯ h)D

Z

(−)

ρcl (xb , xa ) =

|p|≤pF (¯ x)

(4.397)

The simplest way to derive the exchange energy is to re-express the density of states ρ(−) (E; x) as the diagonal elements of the bilocal density ρ(−) (x) = ρ(−) (xb , xa )

(4.398)

and rewrite the electron-electron energy (4.359) as (−) Eee =4×

e2 2

Z

d3 xd3 x′ ρ(−) (x, x)

1 ρ(−) (x′ , x′ ). 4π|x − x′ |

(4.399)

The factor 4 accounts for the four different spin pairs in the first and the second bilocal density. ↑ ↑; ↑ ↑;

↑ ↑; ↓ ↓;

↓ ↓; ↑ ↑;

↓ ↓; ↓ ↓ .

In the first and last case, there exists an exchange interaction which is obtained by interchanging the second arguments of the bilocal densities and changing the sign. This yields (−)

Eexch = −2 ×

e2 2

Z

d3 xd3 x′ ρ(−) (x, x′ )

1 ρ(−) (x′ , x). 4π|x − x′ |

(4.400)

The integral over x − x′ may be performed using the formula Z

0

∞

dzz 2

1 1 (sin z − z cos z) z z3

2

1 = , 4

(4.401)

and we obtain the exchange energy (−) Eexch

e2 =− 3 4π

Z

"

pF (¯ x) ¯ dx h ¯ 3

#4

.

(4.402)

Inserting s

2Z f (ξ) , aH aTF ξ

(4.403)

2 4 aTF Z 2 5/3 e I ≈ −0.3588 Z I2 , 2 π 2 aH aH aH

(4.404)

pF (x) = the exchange energy becomes (−)

Eexch = − where I2 is the integral

I2 ≡

Z

∞ 0

dξ f 2 (ξ) ≈ 0.6154.

(4.405)

426


Hence we obtain (−)

Eexch ≈ −0.2208Z 5/3

e2 , aH

(4.406)

giving rise to a correction factor Cexch (Z) = 1 + 0.2872 Z −2/3

(4.407)

to the Thomas-Fermi energy (4.386).

4.10.7

Quantum Correction Near Origin

The Thomas-Fermi energy with exchange corrections calculated so far would be reliable for large-Z only if the potential was smooth so that the semiclassical approximation is applicable. Near the origin, however, the Coulomb potential is singular and this condition is no longer satisfied. Some more calculational effort is necessary to account for the quantum effects near the singularity, based on the following observation [3]. For levels with an energy smaller than some value ε < 0, which is large compared to the ground state energy Z 2 e2 /aH , but much smaller than the average Thomas Fermi energy per particle Z 2 e2 /aZ ∼ Z 2 e2 /aH Z 2/3 , i.e., for Z 2 e2 1 Z 2 e2 ≪ −ε ≪ , aH Z 2/3 aH

(4.408)

we have to recalculate the energy. Let us define a parameter ν by −ε ≡

Z2 , 2aH ν 2

(4.409)

which satisfies 1 ≪ ν 2 ≪ Z 2/3 .

(4.410)

The contribution of the levels with energy Ze2 p2 − < −ε 2M r

(4.411)

(−)

to Ekin,TF is given by an integral like (4.334), where the momentum runs from 0 q

to p−ε (x) = 2M[−ε − V (x)]. In the kinetic energy (4.335), the potential V (x) is simply replaced by −ε − V (x), and the spatial integral covers the small sphere of radius rmax , where −ε − V (x) > 0. For the screening function f (ξ) = V (r)r/Ze2 this implies the replacement f (ξ) → [−ε − V (r)] where ξm =

r = f (ξ) − ξ/ξm, Ze2

2ν 2 aH Z a

(4.412)



427

is small of the order Z −2/3 . Using relation (4.392) between total and kinetic energies we find the additional total energy (−) ∆Etot

3 Z 2 e2 =− 5 a

Z

ξmax

0

1 dξ √ [f (ξ) − ξ/ξm ]5/2 , ξ

(4.414)

where ξmax ≡ rmax /a is the place at which the integrand vanishes, i.e., where ξmax = Ze2 εf (ξmax)/a,

(4.415)

this being the dimensionless version of −ε − V (rmax ) = 0.

(4.416)

Under the condition (4.410), the slope of f (ξ) may be ignored and we can use the approximation (4.417) ξmax ≈ ξm corresponding to rmax = Ze2 ε, with an error of relative order Z −2/3 . After this, the integral Z

0

1/c

1 1 1 5 dλ √ (1 − cλ)5/2 = √ B (1/2, 7/2) = √ π, c c8 λ

(4.418)

yielding a Beta function B(x, y) ≡ Γ(x)Γ(y)/Γ(x + y), leads to an energy (−) ∆Etot

3 Z 2 e2 5 π =− 5 a 8M

r

aH ν Z 2 e2 ν, = − 2a Z 1/3 a

(4.419)

showing that the correction to the energy will be of relative order 1/Z 1/3 . Expressing a in terms of aH via (4.343), we find (−)

∆Etot = −

Z 2 e2 ν. aH

(4.420)

The point is now that this energy can easily be calculated more precisely. Since the slope of the screening function can be ignored in the small selected radius, the potential is Coulomb-like and we may simply sum all occupied exact quantummechanical energies En in a Coulomb potential −Ze2 /r which lie below the total energy −ε. They depend on the principal quantum number n in the well-known way: Z 2 e2 1 En = − . (4.421) aH 2n2 Each level occurs with angular momentum l = 0, . . . , n − 1, and with two spin directions so that the total degeneracy is 2n2 . By Eq. (4.409), the maximal energy

428


−ε corresponds to a maximal quantum number nmax = ν. The sum of all energies En up to the energy ε is therefore given by ν Z 2 e2 1 X = −2 1 aH 2 n=0

(−) ∆QM Etot

= −

Z 2 e2 [ν] , aH

(4.422)

where [ν] is the largest integer number smaller than ν. The difference between the semiclassical energy (4.419) and the true quantum-mechanical one (4.422) yields the desired quantum correction (−) ∆Ecorr =−

Z 2 e2 ([ν] − ν). aH

(4.423)

For large ν, we must average over the step function [ν], and find 1 h[ν]i = ν − , 2

(4.424)

and therefore

Z 2 e2 1 . (4.425) aH 2 This is the correction to the energy of the atom due to the failure of the quasiclassical expansion near the singularity of the Coulomb potential. With respect to the Thomas-Fermi energy (4.386) which grows with increasing nuclear charge Z like −0.7687 Z 7/3 e2 /aH , this produces a correction factor (−) ∆Ecorr =

7aTF ≈ 1 − 0.6504 Z −1/3 6aH s

Csing (Z) = 1 −

(4.426)

to the Thomas-Fermi energy (4.386).

4.10.8

Systematic Quantum Corrections to Thomas-Fermi Energies

Just as for the density of states in Section 4.9, we can derive the quantum corrections to the energies in the Thomas-Fermi atom. The electrons fill up all negative-energy levels in the combined potential V (x). The density of states in these levels can be selected by a Heaviside function of the negative Hamiltonian operator as follows: ˆ ρ(−) (x) = hx|Θ(−H)|xi.

(4.427)

Using the Fourier representation (1.308) for the Heaviside function we write Z ∞ dt ˆ ˆ Θ(−H) = e−iHt/¯h , 2πi(t − iη) −∞

(4.428)

and obtain the integral representation ρ

(−)

(x) =

Z

∞

−∞

dt (x t|x 0). 2πi(t − iη)

(4.429)



429

Inserting the short-time expansion (4.251), and the correspondence t → i¯hd/dV in the time integral, we find 3 ¯2 h d2 ¯h2 2 2 d ρ(x) = 1 − ∇ V (x) 2 − [∇V (x)] + . . . ρTF (x). 12M dV 24M dV 3

(4.430)

The potential energy is simply given by (−) Epot (x)

ˆ = hx|V (x)Θ(−H)|xi =

Z

d3 xV (x)ρ(−) (x).

(4.431)

With (4.430), this becomes ¯2 h d2 ¯h2 d3 (−) Epot (x) = V (x) 1 − ∇2 V (x) 2 − [∇V (x)]2 + . . . ρTF (x). 12M dV 24M dV 3

(4.432)

For the energy of all negative-energy states we may introduce a density function ˆ ˆ E (−) (x) = hx|HΘ(− H)|xi.

(4.433)

The derivative of this with respect to V (x) is equal to the density of states (4.427): ∂ E (−) (x) = ρ(−) (x). ∂V (x)

(4.434)

ˆ This follows right-away from ∂ H/∂V (x) = 1 and ∂[xΘ(x)]/δx = Θ(x). ˆ can be obtained by applying the differential Inserting the representation (4.428), the factor H operator i¯h∂t to the exponential function. After a partial integration, we arrive at the integral representation Z ∞ dt E (−) (x) = i¯h (x t|x 0). (4.435) 2 −∞ 2πi(t − iη) Inserting on the right-hand side the expansion (4.251), the leading term produces the local ThomasFermi energy density (4.335): (−)

ETF (x) = −

M 2π¯h

D/2

1 [−V (x)]D/2+1 . Γ(D/2 + 2)

(4.436)

The short-time expansion terms yield, with the correspondence t → i¯hd/dV , the energy including the quantum corrections E

(−)

(x) =

3 ¯2 h d2 ¯h2 (−) 2 2 d 1− ∇ V (x) 2 − [∇V (x)] + . . . ETF (x). 12M dV 24M dV 3

(4.437)

One may also calculate selectively the kinetic energy density from the expression E (−) (x) =

1 ˆ hx|ˆ p2 Θ(−H)|xi. 2M

(4.438)

This can obviously be extracted from (4.435) by a differentiation with respect to the mass: Z ∞ ∂ dt ˆ (−) Ekin (x) = −M i¯h hx|e−iHt/¯h |xi 2 ∂M 2πi(t − iη) −∞ ∂ (−) = −M E (x). (4.439) ∂M

430

4 Semiclassical Time Evolution Amplitude (−)

According to Eq. (4.437), the first quantum correction to the energy Ee is Z 3 ¯h2 d2 ¯h2 2 (−) 3 2 d ∇ V − (∇V ) ∆Ee = − d x − 12M dV 2 24M dV 3 3/2 2 M 1 × 2 (−V )5/2 5 2π¯h Γ(5/2) √ Z 2M 1 3 1/2 2 −3/2 2 = d x (−V ) ∇ V − (−V ) (∇V ) . 12¯hπ 2 4

(4.440)

It is useful to bring the second term to a more convenient form. For this we note that by the chain rule of differentiation i 3h i 3 h ∇2 V 3/2 = ∇ V 1/2 ∇V = V −1/2 (∇V )2 + 2V 1/2 ∇2 V . (4.441) 2 4 As a consequence we find ∆Ee(−)

=

√ Z 2M 2 2 3 1/2 2 3/2 d x (−V ) ∇ V − ∇ (−V ) . 24¯hπ 2 3

(4.442)

This energy evaluated with the potential V determined above describes directly the lowest correction to the total energy. To prove this, consider the new total energy [recall (4.361)] (−)

Etot = Ee(−) + ∆Ee(−) − Eϕϕ [ϕ].

(4.443)

Extremizing this in the field ϕ(x) and denoting the new extremal field by ϕ(x) + ∆ϕ(x), we obtain for ∆ϕ(x) the field equation: δ 1 [Ee(−) + ∆Ee(−) ] + 2 ∇2 [ϕ(x) + ∆ϕ(x)] = 0. δV (x) e

(4.444)

Taking advantage of the initial extremality condition (4.362), we derive the field equation (−)

1 2 δ∆Ee ∇ ∆ϕ(x) = − . e2 δV (x)

(4.445) (−)

If we now expand the corrected energy up to first order in ∆Ee (−)

Etot = Ee(−) + ∆Ee(−) +

Z

(−)

d3 x

1 δEe ∆ϕ(x) − Eϕϕ [ϕ] − δV (x) 4πe2

Z

and ∆ϕ(x), we obtain d3 x∇ϕ(x)∇∆ϕ(x).

(4.446)

Due to the extremality property (4.362) of the uncorrected energy at the original field ϕ(x) , the second and fourth terms cancel each other, and the correction to the total energy is indeed given by (4.442). Actually, the statement that the energy (4.442) is the next quantum correction is not quite true. When calculating the first quantum correction in Subsection 4.10.7, we subtracted the contribution of all orbits with total energies E < −ε. (4.447) After that we calculated in (4.422) the exact quantum corrections coming from the neighborhood r < rmax = Ze2 ε

(4.448)

of the origin. Thus we have to omit this neighborhood from all successive terms in the semiclassical (−) expansion, in particular from (4.442). According to the remarks after Eq. (4.335), the energy Ee D/2+1 of electrons filling all levels up to a total energy EF is found from (4.331) by replacing (−V ) H. Kleinert, PATH INTEGRALS


431

by (1 − EF ∂/EF )(EF − V )D/2+1 . The energy level satisfying (4.447) correspond to a Fermi level EF , so that the energy of the electrons in these levels is Ee(−) = 2

2 (−) 2 Epot TF = −2 5 5

M 2π¯h

3/2

Z 1 (1−EF ∂EF ) d3 x[−EF − V (x)]5/2. Γ(5/2)

We therefore have to subtract from the correction (4.440) a term √ Z 2M 1 (−) 3 1/2 2 −3/2 2 ∆sub Ee = (1−EF ∂EF ) d x (−E − V ) ∇ V − (−E − V ) (∇V ) . F F 12¯hπ 2 4

(4.449)

(4.450)

The true correction can then be decomposed into a contribution from the finite region outside the small sphere √ Z 2M (−) d3 x(−V )1/2 ∇2 V, (4.451) ∆Eoutside = 24¯hπ 2 r≥rmax plus a subtracted contribution from the inside √ Z h i 2M (−) 3 1/2 1/2 ∆Einside = d x (−V ) − (1 − E ∂ )(−E − V ) ∇2 V, F E F F 24¯hπ 2 r 0) in attractive and repulsive cases.

For positive energy, we define a momentum √ pE = pE (∞) = 2ME,

(4.488)

and the parameters e2 Me2 l 2 p2 l2 l2 va2 = 2 , ǫ2 ≡ 1 + 2 E4 = 1 + a≡ = 1 + , va = 2|E| pE M e aMe2 e4

s

e2 pE = . aM M (4.489) The eccentricity ǫ is now larger than unity. Apart from this, the solutions to the equations of motion are the same as before, and the orbits are hyperbolas as shown in Fig. 4.3. The y-coordinate above the focus is now l2 l2 h = a(ǫ − 1) = = 2 . Me2 pE a 2

(4.490)

For a repulsive interaction, we change the sign of e2 in the above equations. The equation (4.485) for r becomes now h = −1 + ǫ cos(φ − φ0 ), r

(4.491)

and yields the right-hand hyperbola shown in Fig. 4.3. For later discussions in Chapter 13, where we shall solve the path integral of the Coulomb system exactly, we also note that by introducing a new variable in terms of the variable ξ, to so-called eccentric anomaly r = a(1 − ǫ cos ξ),

(4.492)

we can immediately perform the integral (4.481) to find a t= va

Z

dξ (1 − ǫ cos ξ) =

1 (ξ − ǫ sin ξ), ω

(4.493)

436


where we have chosen the integration constant to zero. Using Eq. (4.491) we see that h−r = a(cos ξ − ǫ), ǫ √ √ y = r sin φ = r 2 − x2 = a 1 − ǫ2 sin ξ = b sin ξ.

x = r cos φ =

(4.494) (4.495)

Equations (4.492) and (4.493) represent a parametric representation of the orbit. From Eqs. (4.493) and (4.492) we see that 1 dt = d(ξ/ω), r a

(4.496)

exhibiting ξ/ω is a path-dependent pseudotime. As a function of this pseudotime, the coordinates x and y oscillate harmonically. The pseudotime facilitates the calculation of the classical action A(xb , xa ; tb −ta ), as done first in the eighteenth century by Lambert.12 If we denote the derivative with respect to ξ by a prime, the classical action reads A=

Z

ξb

ξa

e2 M dξ aω x′2 + y ′2 + . 2r aω "

#

(4.497)

For an elliptic orbit with principal axes a, b, this becomes A=

Z

ξb

ξa

M a2 sin2 ξ + b2 cos2 ξ dξ ω + Mωa2 (ξb − ξa ). 2 1 − ǫ cos ξ "

#

After performing the integral using the formula √ Z dξ 1 − ǫ2 tan(ξ/2) 2 arctan = , 1 − ǫ cos ξ 1 − ǫ2 1−ǫ

(4.498)

(4.499)

we find A =

M 2 a ω [3(ξb − ξa ) + ǫ(sin ξb − sin ξa )] . 2

(4.500)

Introducing the parameters α, β, γ, and δ by the relations cos α ≡ ǫ cos[(ξb + ξa )/2],

β ≡ (ξb − ξa )/2,

γ ≡ α + β,

δ ≡ α − β,

(4.501)

the action becomes A =

M 2 a ω [(3γ + sin γ) − (3δ + sin δ)] . 2

(4.502)

12

Johann Heinrich Lambert (1728–1777) was an ingenious autodidactic taylor’s son who with 16 years found Lambert’s law for the apparent motion of comets (and planets) on the sky: If the sun lies on the concave (convex) side of the apparent orbit, comet is closer to (farther from) the sun than the earth. In addition, he laid the foundations to photometry. H. Kleinert, PATH INTEGRALS

4.11 Classical Action of Coulomb System

437

Using (4.493), we find for the elapsed time tb − ta the relation tb − ta =

1 [(γ − sin γ) ∓ (δ − sin δ)] . ω

(4.503)

The ∓-signs apply to an ellipse whose short or long arc connects the two endpoints, respectively. The parameters γ and δ in the action and in the elapsed time are related to the endpoints xb and xa by rb + ra + R = 4a sin2 (γ/2) ≡ 4aρ+ ,

rb + ra − R = 4a sin2 (δ/2) ≡ 4aρ− , (4.504)

where rb ≡ |xb |, ra ≡ |xa |, R ≡ |xb − xa |, and ρ± ∈ [0, 1]. Expressing the semimajor axis in (4.502) in terms of ω as a = (e2 /Mω 2 )1/3 , and ω in terms of tb − ta we obtain the desired classical action A(xb , xa ; tb − ta ). The elapsed time depends on the endpoints via a transcendental equation which can only be solved by a convergent power series (Lambert’s series) ∞ 1 X (2j)! 1 j+1/2 j+1/2 j−1 ǫ ρ ± ρ . tb − ta = √ + − 2ω j=1 22j j!2 j 2 − 1/4

(4.505)

In the limit of a parabolic orbit the series has only the first term, yielding √ 2 3/2 3/2 tb − ta = (4.506) (ρ+ ± ρ− ). 3ω We can also express a and ω in terms of the energy E as e2 /(−2E) and −2E/Ma2 , respectively, and go over to the eikonal S(xb , xa ; E) via the Legendre transformation (4.80). Substituting for tb − ta the relation (4.503), we obtain for the short arc of the ellipse q

S(xb , xa ; E) = A(xb , xa ; E) + (tb − ta ) E =

M 2 a ω 4(γ − δ). 2

(4.507)

For a complete orbit, the expression (4.500) yields a total action A =

M 2 M 2 2π p¯E 2π a ω 2π(1 + 2) = va (1 + 2) = (1 + 2), 2 2 ω 2M ω

(4.508)

where the numbers 1 and 2 indicate the source of the contributions from the kinetic and potential parts of the action (4.497), respectively. Since the action is the difference between kinetic and potential energy, the average potential energy is minus twice as big as the kinetic energy, which is the single-particle version of the virial theorem observed in the Thomas-Fermi approximation (4.391). For positive energies E, the eccentricity is ǫ > 1 and the orbit is a hyperbola (see Fig. 4.3). Then equations (4.492)–(4.495) become r = a(ǫ cosh ξ − 1), x = −aǫ(cosh ξ − ǫ),

a (ξ − ǫ sin ξ), va √ y = a ǫ2 − 1 sinh ξ. t=

(4.509) (4.510)

438


The orbits take a simple form in momentum space. Using Eq. (4.469), we find from (4.487): l px = − [ǫ + sin(φ − φ0 )] , h

py =

l cos(φ − φ0 ). h

(4.511)

As a function of time, the momenta describe a circle of radius p0 =

Me2 p¯E l = =√ h l 1 − ǫ2

(4.512)

around a center on the py -axis with (see Fig. 4.4) ǫ l pcx = − ǫ = − √ . h 1 − ǫ2

(4.513)

py √ 1 1−ǫ2

py √ 1 ǫ2 −1

1

1

px

E0 √ ǫ 1−ǫ2

√ ǫ ǫ2 −1

Figure 4.4 Circular orbits in momentum space in units p¯E for E < 0 and pE for E > 0.

For positive energies, the above solutions can be used to describe the scattering of electrons or ions on a central atom. For helium nuclei obtained by α-decay of radioactive atoms, this is the famous Rutherford scattering process. The potential is then repulsive, and e2 in the potential (4.467) must be replaced by −2Ze2 , where 2e is the charge of the projectiles and Ze the charge of the central atom. As we can see on Fig. 4.5, the trajectories in an attractive potential are simply related to those in a repulsive potential. The momentum p¯E is the asymptotic momentum of the projectile, and may be called p∞ . The impact parameter b of the projectile fixes the angular momentum via l = bp∞ = b¯ pE . (4.514) Inserting l and p¯E we see that b coincides with the previous parameter b. The relation of the impact parameter b with the scattering angle θ may be taken from Fig. 4.5, which shows that (see also Fig. 4.4) tan Thus we have

θ p0 1 = = 2 . 2 p∞ ǫ −1 θ b = a cot . 2

(4.515)


4.11 Classical Action of Coulomb System

439

The particles impinging into a circular annulus of radii b and b + db come out between the angles θ and θ + dθ, with db/dθ = −a/2 sin2 (θ/2). The area of the annulus dσ = 2πbdb is the differential cross section for this scattering process. The absolute ratio with respect to the associated solid angle dΩ ≡ 2π sin θdθ is then a2 dσ Z 2 α2 M 2 = = . dΩ 4 sin4 (θ/2) 4p4∞ sin4 (θ/2)

(4.517)

The right-hand side is the famous Rutherford formula, which arises after expressing a in terms of the incoming momentum p∞ [recall (4.489)] as a = Ze2 /2E = Zα¯ hcM/p2∞ .

Let us also calculate the classical eikonal in momentum space. We shall do this for the attractive interaction at positive energy. Inserting (4.487) and (4.511) we have S(pb , pa ; E) = −l

Z

φb

φa

dφ . 1 + ǫ(φ − φ0 )

(4.518)

pb pb θ

p0

pa (pb )

pb (pa )

φa b pa

scattering center

p∞ γa

E = const. p∞

pa

Figure 4.5 Geometry of scattering in momentum space. The solid curves are for attractive Coulomb potential, the dashed curves for repulsive (Rutherford scattering). The right-hand part of the figure shows the circle on which the momentum moves from pa to pb as the angle φ runs from φa to φb . The distance b is the impact parameter, and θ is the scattering angle.

440


Using the formula13 Z

√ 2 dξ 1 + ǫ + ǫ2 − 1 tan(ξ/2) √ =√2 log . 1 + ǫ cos ξ ǫ −1 1 − ǫ − ǫ2 − 1 tan(ξ/2)

(4.519)

√ Inserting l/ 1 − ǫ2 = Mva a = Me2 /p∞ , we find after some algebra S(pb , pa ; E) = − with ζ≡

Me2 ζ +1 log , p∞ ζ −1

v u u t1 +

p∞ =

√

(p2b − p2∞ )(p2b − p2∞ ) . p2∞ |pb − pa |2

2ME,

(4.520)

(4.521)

The expression (4.520) has no definite limit if the impinging particle comes in from spatial infinity where pa becomes equal to p∞ . There is a logarithmic divergence which is due to the infinite range of the Coulomb potential. In nature, the charges are always screened at some finite radius R, after which the logarithmic divergence disappears. This was discussed before when deriving the eikonal approximation (1.499) to Coulomb scattering. There is a simple geometric meaning to the quantity ζ. Since the force is central, the change in momentum along a classical orbit is always in the direction of the center, so that (4.518) can also be written as S(pb , pa ; E) = −

Z

pb

pa

r dp.

(4.522)

Expressing r in terms of momentum and total energy, this becomes for an attractive (repulsive) potential S(pb , pa ; E) = ±2Me2

Z

pb

p2

pa

dp . − 2ME

(4.523)

Now we observe, that for E < 0, the integrand is the arc length on a sphere of radius 1/¯ pE in a four-dimensional momentum space. Indeed, the three-dimensional momentum space can be mapped onto the surface of a four-dimensional unit sphere by the following transformation n4 ≡

p2 − p¯2E , p2 + p¯2E

Then we find that p2 13

n≡

2¯ pE p . 2 p + p¯2E

dp dϑ = , 2 + p¯E 2¯ pE

(4.524)

(4.525)

See the previous footnote. H. Kleinert, PATH INTEGRALS

4.12 Semiclassical Scattering

441

where dϑ is the infinitesimal arc length on the unit sphere. But then the eikonal becomes simply 2Me2 S(pb , pa ; E) = ± ϑba , (4.526) p¯E where ϑ is the angular difference between the images of the momenta pb and pa . This is easily calculated. From (4.524) we find directly cos ϑba =

p¯2E (pb − pa )2 4¯ p2E pb · pa + (p2b − p¯2E )(p2a − p¯2E ) = 1 − 2 . (p2b + p¯2E )(p2a + p¯2E ) (p2b + p¯2E )(p2a + p¯2E )

(4.527)

Continuing E analytically to positive energies, we may replace p¯E by ip∞ = 2ME, and obtain p2 (pb − pa )2 cos ϑba = 1 + 2 2 ∞ 2 . (4.528) (pb − p∞ )(p2a − p2∞ ) ¯ with Hence ϑ becomes imaginary, ϑ = iϑ, sin

ϑ¯ p2 (pb − pa )2 = 2 ∞ 2 , 2 (pb − p∞ )(p2a − p2∞ )

(4.529)

and the eikonal function (4.530) takes the form S(pb , pa ; E) = ±

2Me2 ¯ ϑba . p∞

(4.530)

This is precisely the expression (4.520) with ζ = 1/ tanh(ϑ¯ab /2).

4.12

Semiclassical Scattering

Let us also derive the semiclassical limit for the scattering amplitude.

4.12.1

General Formulation

Consider a particle impinging with a momentum pa and energy E = Ea = p2a /2M upon a nonzero potential concentrated around the origin. After a long time, it will be found far from the potential with some momentum pb and the same energy E = Eb = p2b /2M. Let us derive the scattering amplitude for such a process from the heuristic formula (1.507):

fpb pa

pb = M

q

3

2π¯ hM/i (2π¯ h)3

lim

1

tb →∞ t1/2 b

eiEb (tb −ta )/¯h [(pb tb |pa ta )−hpb |pa i] .

(4.531)

In the semiclassical approximation we replace the exact propagator in the momentum representation by a sum over all classical trajectories and associated phases,

442


connecting pa to pb in the time tb − ta . According to formula (4.166) we have in three dimensions (2π¯ h)3 (pb tb |pa ta )−hpb |pa i = (2π¯ h/i)3/2

X′

class. traj.

!

∂xa 1/2 iA(pb ,pa ;tb −ta )/¯h−iνπ¯h/2 − e . ∂pb

det

(4.532)

The sum carries a prime to indicate that unscattered trajectories are omitted. The classical action in momentum space is A(pb , pa ; tb − ta ) =

Z

pa

pb

x · p˙ −

Z

tb ta

Hdt = S(pb , pa ; E) − E(tb − ta ),

(4.533)

where S(pb , pa ; E) is the eikonal function introduced in Eqs. (4.234) and (4.61). Inserting (4.532) into (1.507) we obtain the semiclassical scattering transition amplitude fpb pa = lim

tb →∞

p ∂x 1/2 √ b det a eiS(pb ,pa ;E)/¯h−iνπ/2 . ∂pb traj. tb M

X′

class.

(4.534)

The determinant has a simple physical meaning. To see this we rewrite

so that (4.536) becomes fpb pa = lim

tb →∞

−1

∂pb ∂xa = , ∂pb p ∂xa p

(4.535)

a

p ∂pb −1/2 iS(pb ,pa ;E)/¯h−iνπ/2 √ det e . ∂xa pa tb M traj.

X′

class.

a

(4.536)

We now note that for large tb

pb = p(tb ) = Mxb (tb )/tb

(4.537)

along any trajectory. Thus we find fpb pa = lim

tb →∞

X′

class. traj.

rb det

∂xb −1/2 iS(pb ,pa ;E)/¯h−iνπ/2 e , ∂xa pb

(4.538)

where rb = |xb |. From the definition of the scattering amplitude (1.494) we expect the prefactor of the exponential to be equal to the square root of the classical differential cross section dσcl /dΩ. Let us choose convenient coordinates in which the particle trajectories start out at a point with cartesian coordinates xa = (xa , ya , za ) with a large negative za and a momentum pa ≈ pa ˆz, where zˆ is the direction of the z-axis. The final ˆb = points xb of the trajectories will be described in spherical coordinates. If p H. Kleinert, PATH INTEGRALS


443

(sin θb cos φb , sin θ sin φb , cos θb ) denotes the direction of the final momentum pb = ˆ b , then xb = rb p ˆ . Let us introduce an auxiliary triplet of spherical coordinates pb p sb ≡ (rb , θb , φb ). Then we factorize the determinant in (4.538) as det

∂xb ∂sb ∂sb ∂xb = det × det = rb2 det . ∂xa ∂sa ∂xa ∂xa

We further calculate

∂rb ∂rb ∂rb    ∂xa ∂ya ∂za     ∂θ ∂θb ∂θb  ∂sb b   det = . (4.539)  ∂xa  ∂xa ∂ya ∂za    ∂φ ∂φb ∂φb    b ∂xa ∂ya ∂za Long after the collision, for tb → ∞, a small change of the starting point along the trajectory dza will not affect the scattering angle. Thus we may approximate the matrix elements in the third column by ∂za ≈ ∂φ/∂za ≈ 0. After the same amount of time the particle will only wind up at a slightly more distant rb , where drb ≈ dzb . Thus we may replace the matrix element in the right upper corner by 1, so that the determinant (4.539) becomes in the limit

lim det

tb →∞





∂sb ≈ det ∂xa

    

∂θb ∂xa ∂φb ∂xa

∂θb ∂ya ∂φb ∂ya

    

=

dθb dφb dΩ = , dxb dyb dσ

(4.540)

where dΩ = sin θb dθdφb is the element of the solid angle of the emerging trajectories, and dσ the area element in the x − y -plane, for which the trajectories arrive in an element of the final solid angle dΩ. Thus we obtain "

∂xb det ∂xa

#−1

≈

1 dσ . rb2 dΩ

(4.541)

The ratio dσ/dΩ is precisely the classical differential cross section of the scattering process. Combining (4.538) and (4.540), we see that the contribution of an individual trajectory to the semiclassical amplitude is of the expected form [4] fpb pa =

s

dσcl dΩ

×

X′

eiS(pb ,pa ;E)/¯h−iνπ/2 .

(4.542)

class. traj.

Note that this equation is also valid for some potentials which are not restricted to a finite regime around the origin, such as the Coulomb potentials. In the operator theory of quantum-mechanical scattering processes, such potentials always cause considerable problems since the outgoing wave functions remain distorted even at large distances from the scattering center.

444


Usually, there are only a few trajectories contributing to a process with a given scattering angle. If the actions of these trajectories differ by less than h ¯ , the semiclassical approximation fails since the fluctuation integrals overlap. Examples are the light scattering causing the ordinary rainbow in nature, and glory effects seen at night around the moonlight. We now turn to a derivation of the amplitude (4.542) from the more reliable formula (1.528) for the interacting wave function Î (0, ta )|pa i = hxb |U

−2πi¯ hta M

lim

ta →−∞

!3/2

2

(xb tb |xa ta )ei(pa xa −pa ta /2M )/¯h

xa =pa ta /M

,

by isolating the factor of eipa rb /rb for large rb , as discussed at the end of Section 1.16. On the right-hand side we now insert the x-space form (4.121) of the semiclassical amplitude, and use (4.80) to write Î (0, ta )|pa i = hxb |U

lim

ta →−∞

−ta M

3/2 "

det3

∂pa − ∂xb

!#1/2

× ei[S(xb ,xa ;Ea )+ipa xa −iνπ/2]/¯h

xa =pa ta /M

Now we observe that

−ta M

3/2 "

det3

∂pa − ∂xb

!#1/2

"

= det3

∂xa ∂xb

!#1/2

In Eq. (4.541) we have found that this determinant is equal to Eq. (4.543) to the form 1 hxb |UÎ (0, ta )|pa i = lim ta →−∞ rb

s

.

.

(4.543)

(4.544)

q

dσ/dΩ/rb , bringing

dσcl i[S(xb ,xa ;Ea )+ipa xa −iνπ/2]/¯h e dΩ

. (4.545) xa =pa ta /M

For large xb in the direction of the final momentum pb , we can rewrite the exponent as [recalling (4.234)] S(xb , xa ; Ea ) + ipa xa = pb rb + S(pb , pa ; Ea )

(4.546)

so that (4.545) consists of an outgoing spherical wave function eipb rb /¯h /rb multiplied by the scattering amplitude fpb pa =

s

dσcl dΩ

×

X′

eiS(pb ,pa ;E)/¯h−iνπ/2 ,

(4.547)

class. traj.

the same as in (4.542). H. Kleinert, PATH INTEGRALS


4.12.2

445

Semiclassical Cross Section of Mott Scattering

If the scattering particle is distinguishable from the target particles, the extra phase in the semiclassical formula (4.547) does not change the classical result (4.517). A quantum-mechanical effect becomes visible only if we consider electron-electron scattering, also referred to as Mott scattering. The potential is repulsive, and the above Coulomb potential holds for the relative motion of the two identical particles in their center-of-mass frame. Moreover, the identity of particles requires us to add the amplitudes for the trajectories going to pb and to −pb [see Fig. 4.6], so that the differential cross section is dσsc = |fpb pa − fpb ,−pa |2 . dΩ

(4.548)

The minus sign accounts for the Fermi statistics of the two electrons. For two identical bosons, we have to use a plus sign instead. Now the eikonal function S(p, p, E) enters into the result. According to Eq. (4.520), this is given by pb

pa

−pb

Figure 4.6 Classical trajectories in Coulomb potential plotted in the center-of-mass frame. For identical particles, trajectories which merge with a scattering angle θ and π − θ are indistinguishable. Their amplitudes must be subtracted from each other, yielding the differential cross section (4.548).

where p∞ =

√

√ M¯ hcα 1+∆+1 S(pb , pa ; E) = − log √ , p∞ 1+∆−1

(4.549)

2ME is the impinging momentum at infinite distance, and ∆≡

(p2b − p2∞ )(pa 2 − p2∞ ) . p2∞ |pb − pa |2

(4.550)

The eikonal function is needed only for momenta pb , pa in the asymptotic regime where pb , pa ≈ p∞ , so that ∆ is small and S(pb , pa ; E) ≈

M¯ hcα log ∆, p∞

(4.551)

446


which may be rewritten as M¯ hcα log(sin2 θ/2), p∞

(4.552)

M¯ hcα (p2b − p2∞ )(pa 2 − p2∞ ) σ0 = log , 2p∞ p4∞

(4.553)

S(p, pa E) ≈ 2σ0 − with "

#

and the scattering angle determined by cos θ = [pb · pa /pb pa ]. The logarithmically diverging constant σ0 for pa = pb p∞ does , fortunately, not depend on the scattering angle, and is therefore the semiclassical approximation for the phase shift at angular momentum l = 0. It therefore drops, fortunately, out of the difference of the amplitudes in Eq. (4.548). Inserting (4.552) with (4.553) into (4.547) and (4.548), we obtain the differential cross section for Mott scattering (see Fig. 4.7 for a plot) h ¯ cα 4E

!2 (

"

#)

1 1 1 2αMc ± + 2 cos log(cot θ/2) . 4 2 4 2 p∞ sin θ/2 cos θ/2 sin θ/2 cos θ/2 (4.554) This semiclassical result happens to be identical to the exact result. The exactness is caused by two properties of the Coulomb motion: First there is only one trajectory for each scattering angle, second the motion can be mapped onto that of a harmonic oscillator in four dimensions, as we shall see in Chapter 13. dσ = dΩ

dσ dΩ

dσ dΩ

fermions θ

bosons θ

Figure 4.7 Oscillations in differential Mott scattering cross section caused by statistics. For scattering angle θ = 900 , the cross section vanishes due to the Pauli exclusion principle. The right-hand plot shows the situation for identical bosons.

Appendix 4A

Semiclassical Quantization for Pure Power Potentials

Let us calculate the local density of states (4.257) for the general pure power potential V (x) = gxp /p in D dimensions. For D = 1 and p = 2, we shall recover the exact spectrum of the harmonic oscillator, for p = −1 that of the one-dimensional hydrogen atom. For p = 4, we shall find the energies of the purely quartic potential which can be compared with the strong-coupling limit of H. Kleinert, PATH INTEGRALS

Appendix 4A

Semiclassical Quantization for Pure Power Potentials

447

the anharmonic oscillator with V (x) = ω 2 x2 /2 + gx4 /4 to be calculated in Section 5.16. The integrals on the right-hand side of Eq. (4.257) can be calculated using the formula ν Z Z rE g p D µ ν D−1+µ d x r [E −V (x)] = SD dr r E− r p 0 −(D+µ)/p Γ(1 + ν)Γ((D + µ)/p) g E ν+(D+µ)/p , p > 0, (4A.1) = SD p Γ(1 + ν + (D + µ)/p) p where rE = (pE/g)1/p . For p < 0, the right-hand side must be replaced by −(D+µ)/p Γ(1 + ν)Γ(−ν − (D + µ)/p) g = SD − (−E)ν+(D+µ)/p , p < 0. p Γ(1 − (D + µ)/p) p Recalling (4.208), we find the total density of states for p > 0: ( D/2 −D/p 2/p Γ D p 2 M g ¯h2 g ρ(E) = 1 − 2 D D p 24M p Γ 2 p 2¯ h Γ D 2 + p) ) D D p2 Γ D−2 p +2 Γ 2 + p) −1−2/p E + . . . E (D/2)(1+2/p)−1, p > 0, × 2 D Γ D−2 (1 + ) Γ 2 p p

(4A.2)

(4A.3)

and for p < 0:

D/2 −D/p Γ 1 − D − D ( 2/p 2 p g ¯h2 g − 1+ ρ(E) = − D − p 24M p Γ( 2 ) p Γ 1− D p ) 2 D p2 Γ 1− D−2 2 (1+ p ) Γ 1− p (−E)−1−2/p + . . . (−E)(D/2)(1+2/p)−1, p < 0. × D D Γ −1− D−2 Γ 1− − ) p 2 p 2

M 2¯h2

For a harmonic oscillator with p = 2 and g = M ω 2 , we obtain 1 1 ¯h2 ω 2 D D−1 D−3 ρ(E) = E − E + ... . D Γ(D) 24 Γ(D−2) (¯ hω)

(4A.4)

(4A.5)

In one two dimension, only the first term survives and ρ(E) = 1/¯hω or ρ(E) = E/(¯ hω)2 . Inserting RE ′ ′ this into Eq. (1.581), we find the number of states N (E) = 0 dE ρ(E ) = E/¯hω or E 2 /2(¯ hω)2 . According to the exact quantization condition (1.583), we set N (E) = n + 1/2 the exact energies En = (n + 1/2)¯ hω. Since the semiclassical expansion (4A.3) contains only the first term, the exact quantization condition (1.583) agrees with the Bohr-Sommerfeld quantization condition (4.184). In two dimensions we obtain ρ(E) = E/(¯ hω)2 and N (E) = E 2 /2(¯ hω)2 . Here the exact quantization condition N (E) = n + 1/2 cannot be used to find the energies En , due to the degeneracies of the energy eigenvalues En . In order to see that it is nevertheless a true equation, let us expanding the partition function of the two-dimensional oscillator [recall (2.399)] Z=

∞ X 1 1 −¯ hβω = e = (n + 1)e−(n+1)¯hβω . [2 sinh(¯ hβω/2)]2 (1 − e−¯hβω )2 n=0

(4A.6)

This shows that the the energies are En = (n + 1)¯ hω withy n + 1 -fold degeneracy. The density of states is found from this by the Fourier transform (1.580): ρ(E) =

∞ X

(n′ + 1)δ(E − ¯hω(n′ + 1)).

n′ =0

(4A.7)

448


Integrating this over E yields the number of states Z E ∞ X ′ ′ N (E) = dE ρ(E ) = (n′ + 1)Θ(E − En ). 0

(4A.8)

n′ =0

For E = En this becomes [recall (1.309)] N (En ) =

n−1 X

(n′ + 1) + (n + 1)

n′ =0

1 1 = (n + 1)2 . 2 2

(4A.9)

This shows that the exact energies En = (n + 1)¯ hω of the two-dimensional oscillator satisfy the quantization condition N (En ) = (n + 1)2 /2 rather than (1.583). For a quartic potential gx4 /4, Eq. (4A.3) becomes   D 4 12  3D D 4 − 4  3 D 3D − 32 Γ( 4 ) g¯h Γ( 2 + 4 )Γ( 4 )E 1 g¯h + . . . E 4 −1. (4A.10) ρ(E) = 1− D 3D D 3 2 2   M 2 Γ( 2 ) Γ( 4 ) M 3Γ( 4 )Γ 4 (D − 2) Integrating this over E yields N (E). Setting N (E) = n + 1/2 in one dimension, we obtain the Bohr-Sommerfeld energies (4.31) plus a first quantum correction. Since we have studied these corrections to high order in Subsection 4.9.6 (see Fig. 4.1), we do not write the result down here. A physically important case is p = −1, g = h ¯ cα, with α of Eq. (1.502), where V (x) = −¯hcα/r becomes the Coulomb potential. Here we obtain from (4A.10): D/2 Γ 1+ D 2 M g2 2 ρ(E) = Γ (1 + D) Γ( D ) 2π¯h2 ( 2 ) 2 2 Γ (1 + D) p Γ D−2 ¯h 2 E + . . . (−E)(D/2)(1+2/p)−1 . × 1− (4A.11) 24M g −2 Γ (D − 3) Γ 1 + D ) 2

For d = 1, only the leading term survives and r M g2 (−E)−3/2 , ρ(E) = 2¯ h2

(4A.12)

implying r

M g2 (−E)−1/2 . (4A.13) 2¯ h2 In order to find the bound-state energies, we must watch out for a subtlety in one dimension: only the positive half-space is accessible to the particle in a Coulomb potential, due to the strong singularity at the origin. For this reason, the “surface of a sphere” SD for D = 1 , which is equal to 2, must be replaced by 1, so that we must equate N (E)/2 to n + 1/2. This yields the spectrum En = −α2 M c2 /2(n + 1/2)2 . As in the harmonic oscillator, the Bohr-Sommerfeld approximation gives the exact energies. N (E) = 2

Appendix 4B

Derivation of Semiclassical Time Evolution Amplitude

Here we derive the semiclassical approximation to the time evolution amplitude (4.260). We shall do this for imaginary times τ = it. Decomposing the path x(τ ) into path average of the ends points x = (xb + xa )/2 and fluctuations (τ ), we calculate the imaginary-time amplitude Z (τb )=∆x/2 Z 1 τb M 2 ˙ (τ ) + V x + (τ ) (xb τb |xa τa ) = dτ D exp − , (4B.1) (τa )=−∆x/2 ¯h τa 2 H. Kleinert, PATH INTEGRALS

Appendix 4B


where ∆x ≡ xb − xa . For smooth potentials we expand 1 V x + (τ ) = V (x) + ∂i V (x) ηi (τ ) + ∂i ∂j V (x) ηi (τ ) ηj (τ ) + . . . , 2

where Vij... (x) ≡ ∂i ∂j · · · V (x), and rewrite the path integral (4B.1) as Z (τb )=∆x/2 Z 1 τb M 2 −βV (x) ˙ (τ ) (xb τb |xa τa ) = e D exp − dτ (τa )=−∆x/2 ¯h τa 2 ( Z i 1 τb h 1 × 1− dτ Vi (x) ηi (τ ) + Vij (x) ηi (τ ) ηj (τ ) + . . . ¯h τa 2 ) Z τb Z τb h i 1 ′ ′ + 2 dτ dτ Vi (x)Vj (x) ηi (τ )ηj (τ )+. . . +. . . . 2¯h τa τa

At this point it is useful to introduce an auxiliary harmonic imaginary-time amplitude Z (τb )=∆x/2 Z 1 τb M 2 ˙ (τ ) (∆x/2 τb |−∆x/2 τa ) = D exp − dτ (τa )=−∆x/2 ¯h τa 2 and the harmonic expectation values Z τb Z (τb )=∆x/2 1 1 M 2 ˙ (τ ) , D F [ ] exp − hF [ ]i ≡ dτ (∆x/2 τb |−∆x/2 τa ) (τa )=−∆x/2 ¯h τa 2

449

(4B.2)

(4B.3)

(4B.4)

(4B.5)

which allows us to rewrite (4B.3) more concisely as Z 1 τb (xb τb |xa τa ) = e (∆x/2 τb |−∆x/2 τa ) 1 − dτ Vi (x) hηi (τ )i ¯h τ Z τb Z τb Za τb 1 1 ′ ′ − Vij (x) dτ hηi (τ )ηj (τ )i+ 2 Vi (x)Vj (x) dτ dτ hηi (τ )ηj (τ )i+. . . . 2¯ h 2¯ h τa τa τa −βV (x)

(4B.6)

The amplitudes (4B.4) reads explicitly, with ∆τ ≡ τb − τa : (∆x/2 τb | − ∆x/2 τa ) =

M 2π¯h∆τ

D/2

M 2 exp − (∆x) , 2¯h∆τ

(4B.7)

and (4B.5) can be calculated from the generating functional Z τb Z (τb )=∆x/2 1 M 2 ˙ (τ ) − j(τ ) (τ ) , (∆x/2 τb |−∆x/2 τa )[j] = D − dτ (τa )=−∆x/2 ¯h τa 2

(4B.8)

whose explicit solution is D/2 Z τb M M 1 (∆x/2 τb |−∆x/2 τa )[j] = exp − (∆x)2 + dτ (τ − τ¯) ∆x j(τ ) 2π¯h∆τ 2¯ h∆τ ¯h∆τ τa Z τb Z τb ′ ′ ′ ′ 1 ′ Θ(τ − τ )(∆τ − τ )τ + Θ(τ − τ )(∆τ − τ )τ ′ + dτ dτ j(τ ) j(τ ) . 2¯ h τa M ∆τ τa The expectation values in (4B.6) and (4B.5) are obtained from the functional derivatives 1 ¯hδ hηi (τ )i = (∆x/2 τb |−∆x/2 τa)[j] , (4B.9) (∆x/2 τb | − ∆x/2 τa ) δji (τ ) j=0 1 ¯hδ ¯hδ . hηi (τ )ηj (τ ′ )i = (∆x/2 τ |−∆x/2 τ )[j] (4B.10) b a ′ (∆x/2 τb |−∆x/2 τa ) δji (τ ) δjj (τ ) j=0

450


This yields the ∆x-dependent expectation value hηi (τ )i = (τ − τ¯)

∆xi , ∆τ

(4B.11)

and the ∆x-dependent correlation function ¯h [Θ(τ − τ ′ )(∆τ − τ )τ ′ + Θ(τ ′ − τ )(∆τ − τ ′ )τ ] δij M ∆τ ∆xi ∆xj + (τ − τ¯) (τ ′ − τ¯) ∆τ ∆τ

hηi (τ )ηj (τ ′ )i =

≡

A2 (τ, τ ′ )δij + B 2 (τ, τ ′ )∆xi ∆xj ≡ Gij (τ, τ ′ ),

suppressing the argument ∆x in Gij (τ, τ ′ ), for brevity. Note that Z τb dτ hηi (τ )i = 0,

(4B.12)

(4B.13)

τa

and τb

τb

τb

τb

¯h dτ ′ Gij (τ, τ ′ ) = ∆τ (∆τ 2 − τa2 )δij , 12M τa τa τa τa Z τb Z τb ∆τ ¯h ∆τ 2 2 dτ hηi (τ )ηj (τ )i = − τa δij + ∆xi ∆xj . dτ Gij (τ, τ ) = M 6 12 τa τa Z

Z dτ

dτ ′ hηi (τ )ηj (τ ′ )i =

Z

dτ

Z

(4B.14) (4B.15)

Thus we obtain the semiclassical imaginary-time amplitude D/2 M M ∆τ 2 (xb τb |xa τa ) = exp − ∆x − V (x) (4B.16) 2π¯h∆τ 2¯ h∆τ ¯h ∆τ ∆τ 3 ∆τ 2 2 2 2 × 1− ∇ V (x) − (∆x∇) V (x) + [∇V (x)] + . . . . 12M 24¯h 24M ¯h

This agrees precisely with the real-time amplitude (4.260). For the partition function at inverse temperature β = (τb − τa )/¯h, this implies the semiclassical approximation Z Z = dD x (x ¯hβ|x 0) ≈

M 2π¯h2 β

D/2 Z

dD x¯

¯ 2β2 2 h ¯ 2β3 h 1− ∇ V (x) + [∇V (x)]2 e−βV (x) . 12M 24M

(4B.17)

A partial integration simplifies this to Z

≈ ≈

D/2 Z M ¯h2 β 2 2 D ∇ V (x) e−βV (x) d x ¯ 1 − 24M 2π¯h2 β D/2 Z M ¯h2 β 2 2 dD x¯ exp −βV (x) − ∇ V (x) . 24M 2π¯h2 β

(4B.18)

Actually, it is easy to calculate all terms in (4B.16) proportional to V (x) and its derivatives. Instead of the expansion (4B.6), we evaluate Z 1 τb −βV (x) (xb τb |xa τa ) = e (∆x/2 τb |−∆x/2 τa ) 1 − dτ hV (x + (τ )) − V (x)i . ¯h τa (4B.19) H. Kleinert, PATH INTEGRALS

Appendix 4B


By rewriting V (x + ) as a Fourier integral Z dD k ˜ V (k) exp [ik (x + )] , V (x + ) = (2π)D

451

(4B.20)

we obtain (xb τb |xa τa )

= (∆x/2 τb |−∆x/2 τa) Z Z D E 1 τb dD k ˜ ik (τ ) ikx × e−βV (x) 1 − dτ V (k)e e − 1 . ¯h τa (2π)D

(4B.21)

The expectation value can be calculated using Wick’s theorem (3.307) as Z τb Z τb D E Z τb 2 2 ik (τ ) −ki kj hηi (τ )ηj (τ )i/2 dτ e = dτ e = dτ e−ki kj [A (τ,τ )δij +B (τ,τ )∆xi∆xj ]/2 . τa

τa

τa

(4B.22)

where A2 (τ, τ ), B 2 (τ, τ ) are from (4B.12): A2 (τ, τ ) =

¯ h (∆τ − τ )τ, M ∆τ

2

B 2 (τ, τ ) =

(τ − τ¯) . ∆τ 2

Inserting the inverse of the Fourier decomposition (4B.20), Z V˜ (k) = dD η V (x + ) exp [−ik (x + )] , where

is now a time-independent variable of integration, we find Z τb D Z τb Z Z D d k −(1/2)ki Gij (τ,τ )kj −iki ηi (τ ) E dτ eik (τ ) = dτ dDη V (x + ) e . (2π)D τa τa

(4B.23)

(4B.24)

(4B.25)

After a quadratic completion of the exponent, the momentum integral can be performed and yields Z Z τb Z τb D E −1/2 −(1/2)ηi G−1 (τ,τ )ηj ij = dD η V (x + ) dτ [det G(τ, τ )] e . (4B.26) dτ eik (τ ) τa

τa

Using the transverse and longitudinal projection matrices PijT = δij − 2

∆xi ∆xj , (∆x)2

PijL =

∆xi ∆xj , (∆x)2

(4B.27)

2

satisfying P T = P T , P L = P L , we can decompose Gij (τ, τ ′ ) as Gij ≡ A2 (τ, τ ′ )PijT + A2 (τ, τ ′ ) + B 2 (τ, τ ′ )(∆x)2 PijL .

(4B.28)

It is then easy to find the determinant

det G(τ, τ ′ ) = [A2 (τ, τ ′ )]D−1 [A2 (τ, τ ′ ) + B 2 (τ, τ ′ )(∆x)2 ],

(4B.29)

and the inverse matrix G−1 ij (τ )

1 ∆xi ∆xj 1 ∆xi ∆xj = 2 δij − + 2 . ′ 2 ′ 2 ′ 2 A (τ, τ ) (∆x) A (τ, τ ) + B (τ, τ )(∆x) (∆x)2

(4B.30)

Inserting (4B.26) back into (4B.21), and taking the correction into the exponent, we arrive at R τb −(1/¯ h) dτ Vsm (x,τ ) τa (xb τb |xa τa ) = (∆x/2 τb |−∆x/2 τa) e , (4B.31)

452


where Vsm (x, τ ) is the harmonically smeared potential Z −1 −1/2 dD η V (x + )e−(1/2)ηi Gij (τ )ηj . Vsm (x, τ ) ≡ [det G(τ, τ )]

(4B.32)

By expanding V (x + ) to second order in , the exponent in (4B.31) becomes Z τb 1 −βV (x) − Vij (x) dτ Gij (τ, τ ) + . . . . 2¯ h τa

(4B.33)

According to Eq. (4B.15), we have Z τb ¯ ∆τ 2 h ∆τ δij + ∆xi ∆xj , dτ Gij (τ, τ ) = M 6 12 τa

(4B.34)

so that we reobtain the first two correction terms in the curly brackets of (4B.16) The calculation of the higher-order corrections becomes quite tedious. One rewrites the ex ηi (τ )∂i pansion (4B.2) as V x + (τ ) = e V (x) and (4B.19) as ∞ X (−1)n (xb τb |xa τa ) = (∆x/2 τb |−∆x/2 τa ) × n! n=0

*

∞ Z Y

n=0

τb

+

dτn eηi (τn )∂i V (x) .

τa

(4B.35)

Now we apply Wick’s rule (3.307) for harmonically fluctuating variables, to re-express D E eη(τ )∂i = ehηi (τ )ηj (τ )i/2 = eGij (τ,τ )∂i ∂j /2 , D E ′ ′ ′ ′ eηi (τ )∂i eηi (τ )∂i = e[hηi (τ )ηj (τ )i∂i ∂j +hηi (τ )ηj (τ )i∂i ∂j +2hηi (τ )ηj (τ )i∂i ∂j ]/2 ′ ′ ′ = e[Gij (τ,τ )∂i ∂j +Gij (τ ,τ )∂i ∂j +2Gij (τ,τ )∂i ∂j ]/2 .. . .

(4B.36)

Expanding the exponentials and performing the τ -integrals in (4B.35) yields all desired higherorder corrections to (4B.16). For ∆x = 0, the expansion has been driven to high orders in Ref. [2] (including a minimal interaction with a vector potential).

Notes and References For the eikonal expansion, see the original works by G. Wentzel, Z. Physik 38, 518 (1926); H.A. Kramers, Z. Physik 39, 828 (1926); L. Brillouin, C. R. Acad. Sci. Paris 183, 24 (1926); V.P. Maslov and M.V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics, Reidel, Dordrecht, 1982; J.B. Delos, Semiclassical Calculation of Quantum Mechanical Wave Functions, Adv. Chem. Phys. 65, 161 (1986); M.V. Berry and K.E. Mount, Semiclassical Wave Mechanics, Rep. Prog. Phys. 35, 315 (1972); and the references quoted in the footnotes. For the semiclassical expansion of path integrals see R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D 10, 4114, 4130 (1974), R. Rajaraman, Phys. Rep. 21C, 227 (1975); S. Coleman, Phys. Rev. D 15, 2929 (1977); and in The Whys of Subnuclear Physics, Erice Lectures H. Kleinert, PATH INTEGRALS


453

1977, Plenum Press, 1979, ed. by A. Zichichi. Recent semiclassical treatments of atomic systems are given in R.S. Manning and G.S. Ezra, Phys. Rev. 50, 954 (1994). Chaos 2, 19 (1992). Semiclassical scattering is treated in J.M. Rost and E.J. Heller, J. Phys. B 27, 1387 (1994). For the semiclassical approach to chaotic systems see the textbook M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, Berlin, 1990, where the trace formula (4.230) is derived. In Section 12.4 of that book, the action (4.502) and the eikonal (4.520) of the Coulomb system are calculated. Sections 6.3 and 6.4 discuss the properties of the stability matrix in (4.223). Applications to complex highly excited atomic spectra are described by H. Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989); P. Cvitanović and B. Eckhardt, Phys. Rev. Lett. 63, 823 (1991); G. Tanner, P. Scherer, E.B. Bogomonly, B. Eckhardt, and D. Wintgen, Phys. Rev. Lett. 67, 2410 (1991); G.S. Ezra, K. Richter, G. Tanner, and D. Wintgen, J. Phys. B 24, L413 (1991); B. Eckhardt and D. Wintgen, J. Phys. A 24, 4335 (1991); D. Wintgen, K. Richter, and G. Tanner, Chaos 2, 19 (1992). P. Gaspard, D. Alonso, and I. Burghardt, Adv. Chem. Phys. XC 105 (1995); B. Grémaud, Phys. Rev. E 65, 056207 (2002); E 72, 046208 (2005). The individual citations refer to the following works: [1] C.M. Fraser, Z. Phys. C 28, 101 (1985); J.Iliopoulos, C. Itzykson, A. Martin, Rev. Mod. Phys. 47, 165 (1975); K. Kikkawa, Prog. Theor. Phys. 56, 947 (1976); H. Kleinert, Fortschr. Phys. 26, 565 (1978); R. MacKenzie, F. Wilczek, and A. Zee, Phys. Rev. Lett. 53, 2203 (1984); I.J.R. Aitchison and C.M. Fraser, Phys. Lett. B 146, 63 (1984). [2] D. Fliegner, M.G. Schmidt, and C. Schubert, Z. Phys. C64, 111 (1994) (hep-ph/9401221); D. Fliegner, P. Haberl, M.G. Schmidt, and C. Schubert, Ann. Phys. (N.Y.) 264, 51 (1998) (hep-th/9707189). [3] J. Schwinger, Phys. Phys. A 22, 1827 (1980), A24, 2353 (1981). [4] The form (4.542) of the scattering amplitude was first derived by P. Pechukas, Phys. Rev. 181, 166 (1969). See also J.M. Rost and E.J. Heller, J. Phys. B 27, 1387 (1994). For rainbow and glory scattering see the paper by Pechukas and by K.W. Ford and J.A. Wheeler, Ann. Phys. 7, 529 (1959). For the semiclassical treatment of the Coulomb problem see A. Northcliffe and I.C. Percival, J. Phys. B 1, 774, 784 (1968); A. Northcliffe, I.C. Percival, and M.J. Roberts, J. Phys. B 2, 590, 578 (1968). For an alternative path integral formula for the scattering matrix see W.B. Campbell, P. Finkler, C.E. Jones, and M.N. Misheloff, Phys. Rev. D 12, 2363 (1975). [5] C.M. Bender, K. Olaussen, and P.S. Wang, Phys. Rev. D 16, 1740 (1977).


Who can believe what varies every day, Nor ever was, nor will be at a stay? John Dryden (1631-1700), Hind and the Panther (1687)

5 Variational Perturbation Theory Most path integrals cannot be performed exactly. It is therefore necessary to develop approximation procedures which allow us to approach the exact result with any desired accuracy, at least in principle. The perturbation expansion of Chapter 3 does not serve this purpose since it diverges for any coupling strength. Similar divergencies appear in the semiclassical expansion of Chapter 4. The present chapter develops a convergent approximation procedure to calculate Euclidean path integrals at a finite temperature. The basis for this procedure is a variational approach due to Feynman and Kleinert, which was recently extended to a systematic and uniformly convergent variational perturbation expansion [1].

5.1

Variational Approach to Effective Classical Partition Function

Starting point for the variational approach will be the path integral representation (3.813) for the effective classical potential introduced in Section 3.25. Explicitly, the effective classical Boltzmann factor B(x0 ) for a quantum system with an action Ae =

Z

hβ ¯

0

M 2 dτ x˙ (τ ) + V (x(τ )) 2

(5.1)

has the path integral representation [recall (3.813) and (3.806)] −V eff cl (x0 )/kB T

B(x0 ) ≡ e

=

I

′

−Ae /¯ h

D xe

=

∞ Y

m=1

Z ∞ M X 1 ¯h/kB T 2 × exp − ωm |xm |2 − dτ V kB T m=1 h ¯ 0 "

"Z

im dxre m dxm × 2 πkB T /Mωm

x0 +

#

∞ X

m=1

−iωm τ

(xm e

(5.2) + c.c.)

!#

,

im with the notation xre m = Re xm , xm = Im xm . To make room for later subscripts, we shall in this chapter write A instead of Ae . In Section 3.25 we have derived a eff cl perturbation expansion for B(x0 ) = e−βV (x0 ) . Here we shall find a simple but quite accurate approximation for B(x0 ) whose effective classical potential V eff cl (x0 ) approaches the exact expression always from above.

368

369

5.2 Local Harmonic Trial Partition Function

5.2

Local Harmonic Trial Partition Function

The desired approximation is obtained by comparing the path integral in question with a solvable trial path integral. The trial path integral consists of a suitable superposition of local harmonic oscillator path integrals centered at arbitrary average positions x0 , each with an own frequency Ω2 (x0 ). The coefficients of the superposition and the frequencies are chosen in such a way that the effective classical potential of the trial system is an optimal upper bound to the true effective classical potential. In systems with a smooth or at least not too singular potential, the accuracy of the approximation will be very good. In Section 5.13 we show how to use this approximation as a starting point of a systematic variational perturbation expansion which permits improving the result to any desired accuracy. As a local trial action we shall take the harmonic action (3.847), which may also be considered as the action of a harmonic oscillator centered around some point x0 : AxΩ0

=

Z

h/kB T ¯

0

(x − x0 )2 x˙ 2 + Ω2 (x0 ) . dτ M 2 2 #

"

(5.3)

However, instead of using the specific frequency Ω2 (x0 ) ≡ V ′′ (x0 )/M in (3.847), we shall choose Ω(x0 ) to be an as yet unknown local trial frequency. The effective classical Boltzmann factor B(x0 ) associated with this trial action can be taken directly from (3.820). We simply replace the harmonic potential Mω 2 x2 /2 in the defining expression (3.813) by the local trial potential Ω(x0 )(x − x0 )2 /2. Then the first term in the fluctuation expansion (3.814) of the action vanishes, and we obtain, instead of (3.820), the local Boltzmann factor BΩ (x0 ) ≡ e−V

eff cl (x

0 )/kB T

=

h ¯ Ω(x0 )/2kB T ≡ ZΩx0 . sinh[¯hΩ(x0 )/2kB T ]

(5.4)

The exponential exp (−βMω 2 x20 /2) in (3.820) is absent since the local trial potential vanishes at x = x0 . The local Boltzmann factor BΩ (x0 ) is a local partition function of paths whose temporal average is restricted to x0 , and this fact will be emphasized by the alternative notation ZΩx0 which we shall now find convenient to use. The effective classical potential of the harmonic oscillators may also be viewed as a local free energy associated with the local partition function ZΩx0 which we defined as FΩx0 ≡ −kB T log ZΩx0 ,

such that we may identify VΩeff cl (x0 )

=

FΩx0

(

(5.5) )

sinh[¯hΩ(x0 )/2kB T ] = kB T log . h ¯ Ω(x0 )/2kB T

(5.6)

We now define the local expectation values of an arbitrary functional F [x(τ )] within the harmonic path integral ZΩx0 = =

Z

x0

˜ x − x0 )e−AΩ Dx(τ )δ(¯

∞ Y

m=1

"Z

/¯ h

# P∞ im dxre − kMT [ω 2 +Ω2 (x0 )]|xm |2 m dxm m=1 m B e , 2 πkB T /Mωm

(5.7)

370

5 Variational Perturbation Theory

˜ x − x0 ) is the slightly modified δ-function introduced in Eq. (3.812). The where δ(¯ definition is hF [x(τ )]ixΩ0

≡

[ZΩx0 ]−1

Z

x0

˜ x − x0 )e−AΩ Dx δ(¯

/¯ h

F [x(τ )].

(5.8)

The effective classical potential can then be re-expressed as a path integral e−V

eff cl (x

0 )/kB T

= Z x0 = Z

≡

Z

˜ x − x0 )e−A/¯h Dx δ(¯ x0

˜ x − x0 )e−AΩ Dx δ(¯ D

x0

= ZΩx0 e−(A−AΩ

E )/¯ h x0 Ω

x

/¯ h −(A−AΩ0 )/¯ h

e

.

(5.9)

We now take advantage of the fact that the expectation value on the right-hand side possesses an easily calculable bound given by the Jensen-Peierls inequality: D

x0

e−(A−AΩ

E )/¯ h x0 Ω

x0

≥ e−hA/¯h−AΩ

x0

/¯ hi

Ω

.

(5.10)

This implies that the effective classical potential has an upper bound V eff cl (x0 ) ≤ FΩx0 (x0 ) + kB T hA/¯ h − AxΩ0 /¯ hixΩ0 .

(5.11)

The Jensen-Peierls inequality is a consequence of the convexity of the exponential function: The average of two exponentials is always larger than the exponential at the average point (see Fig. 5.1): x1 +x2 e−x1 + e−x2 ≥ e− 2 . 2

(5.12)

This convexity property of the exponential function can be generalized to an exponential functional. Let O[x] be an arbitrary functional in the space of paths x(τ ), and Z hO[x]i ≡ Dµ[x]O[x] (5.13)

Figure 5.1 Illustration of convexity of exponential function e−x , satisfying he−x i ≥ e−hxi everywhere. H. Kleinert, PATH INTEGRALS

371


an expectation value in this space. The measure of integration µ[x] is supposed to be normalized so that h1i = 1. Then (5.12) generalizes to D

E

e−O ≥ e−hOi .

(5.14)

To prove this we first observe that the inequality (5.12) remains valid if x1 , x2 are replaced by the values of an arbitrary function O(x): O(x1 )+O(x2 ) e−O(x1 ) + e−O(x2 ) 2 ≥ e− . 2

(5.15)

This inequality is then generalized with the help of any positive measure µ(x), with R unit normalization, dµ(x) = 1, to and further to

Z

dµ(x)e−O(x) ≥ e

Z

Dµ[x]e−O[x] ≥ e

R

dµ(x)e−O(x)

,

(5.16)

R

Dµ[x]e−O[x]

,

(5.17) R

where µ[x] is any positive functional measure with the normalization Dµ[x] = 1. This shows that Eq. (5.14) is true, and hence also the Jensen-Peierls inequality (5.10). Since the kinetic energies in the two actions A and AxΩ0 in (5.11) are equal, the inequality (5.11) can also be written as V

eff cl

(x0 ) ≤

FΩx0

kB T + h ¯

Z

h/kB T ¯

dτ

0

*

+x 0

i Ω2 (x0 ) V (x(τ )) − M (x(τ ) − x0 )2 2

h

. (5.18)

Ω

The local expectation value on the right-hand side is easily calculated. Recalling the definition (3.853) we have to use the correlation functions of η(τ ) without the zero frequency in Eq. (3.839): hη(τ )η(τ

′

x )iΩ0

≡

a2τ τ ′ (x0 )

h ¯ 1 cosh Ω(x0 )(|τ −τ ′ | − h ¯ β/2) 1 − = , (5.19) M 2Ω(x0 ) sinh(Ω(x0 )¯ hβ/2) h ¯ βΩ2 "

#

valid for arguments τ, τ ′ ∈ [0, h ¯ β]. By analogy with (3.803) we may denote this 2 ′ quantity by aΩ(x0 ) (τ, τ ), which we have shortened to a2τ τ ′ (x0 ), to avoid a pile-up of indices. All subtracted correlation functions can be obtained from the functional derivatives of the local generating functional ZΩx0 [j]

≡

Z

˜ x − x0 ) e−(1/¯h) Dx(τ )δ(¯

R h¯ β 0

dτ [ M x˙ 2 (τ )+Ω2 (x0 )[x(τ )−x0 ]2 ]−j(τ )[x(τ )−x0 ]} 2 {

,(5.20)

whose explicit form was calculated in (3.844): ZΩx0 [j]

=

ZΩx0

(

1 exp 2M¯ h

Z

0

hβ ¯

dτ

Z

0

hβ ¯

dτ

′

j(τ )GpΩ′2 (x0 ),e(τ −τ ′ )j(τ ′ )

)

.

(5.21)

372


If desired, the Green function can be continued analytically to the real-time retarded Green function ′ GR Ω2 (t, t ) =

M Θ(t − t′ ) hx(t)x(t′ )i h ¯

(5.22)

in a way to be explained later in Section 18.2. Using the spectral decomposition of these correlation functions to be derived in Section 18.2, it is possible to show that the low-temperature value of Ω(x0 ) at the potential minimum gives an approximation to the energy difference between ground and first excited states. In Table 5.1 we see that for the anharmonic oscillator, this approximation is quite good. As shown in Fig. 3.14, the local fluctuation square width D

η 2 (τ )

Ex0 Ω

D

= (x(τ ) − x0 )2

Ex0 Ω

= a2τ τ (x0 ) ≡ a2Ω(x0 ) ,

(5.23)

with the explicit form (3.803), a2Ω(x0 )

∞ 2 X 1 1 h ¯ βΩ(x0 ) h ¯ βΩ(x0 ) = coth − 1 , (5.24) = 2 2 2 Mβ m=1 ωm + Ω (x0 ) MβΩ (x0 ) 2 2

"

#

goes to zero for high temperature like a2Ω(x0 ) − −−→ h ¯ 2 /12MkB T.

(5.25)

T →∞

This is in contrast to the unrestricted expectation h(x(τ ) − x0 )2 iΩ(x0 ) of the harmonic oscillator which includes the ωm = 0 -term in the spectral decomposition (3.800): D

a2tot ≡ (x(τ ) − x0 )2

E

Ω(x0 )

= a2Ω(x0 ) +

kB T . MΩ2 (x0 )

(5.26)

This grows linearly with T following the equipartition theorem (3.800). As discussed in Section 3.25, this difference is essential for the reliability of a perturbation expansion of the effective classical potential. It will also be essential for the quality of the variational approach in this chapter. The local fluctuation square width a2Ω(x0 ) measures the importance of quantum fluctuations at nonzero temperatures. These decrease with increasing temperatures. In contrast, the square width of the ω0 = 0 -term grows with the temperature showing the growing importance of classical fluctuations. This behavior of the fluctuation width is in accordance with our previous observation after Eq. (3.837) on the finite width of the Boltzmann factor B(x0 ) = eff cl e−V (x0 )/kB T for low temperatures in comparison to the diverging width of the alternative Boltzmann factor (3.838) formed from the partition function density ˜ eff cl ˜ 0 ) ≡ le (¯ B(x hβ) z(x) = e−V (x0 )/kB T . Since a2Ω(x0 ) is finite at all temperatures, the quantum fluctuations can be treated approximately. The approximation improves with growing temperatures where a2Ω(x0 ) tends to zero. The thermal fluctuations, on the other hand, diverge at high H. Kleinert, PATH INTEGRALS

373


temperatures. Their evaluation requires a numeric integration over x0 in the final effective classical partition function (3.808). Having determined a2Ω(x0 ) , the calculation of the local expectation value hV (x(τ ))ixΩ0 is quite easy following the steps in Subsection 3.25.8. The result is the smearing formula analogous to (3.872): We write V (x(τ )) as a Fourier integral dk ikx(τ ) ˜ e V (k), −∞ 2π and obtain with the help of Wick’s rule (3.304) the expectation value ∞

Z

V (x(τ )) =

hV (x(τ ))ixΩ0 = Va2 (x0 ) ≡

Z

∞

−∞

dk ˜ 2 2 V (k)eikx0 −a (x0 )k /2 . 2π

(5.27)

(5.28)

For brevity, we have used the shorter notation a2 (x0 ) for a2Ω(x0 ) , and shall do so in the remainder of this chapter. Reinsert the Fourier coefficients of the potential V˜ (k) =

Z

∞

−∞

dx V (x)e−ikx ,

(5.29)

we may perform the integral over k and obtain the convolution integral hV

(x(τ ))ixΩ0

= Va2 (x0 ) ≡

Z

dx′0

∞

−∞

q

2πa2 (x0 )

′

2 /2a2 (x

e−(x0 −x0 )

0)

V (x′0 ).

(5.30)

As in (3.872), the convolution integral smears the original potential V (x0 ) out over a length scale a(x0 ), thus accounting for the effects of quantum-statistical path fluctuations. x The expectation value h(x(τ ) − x0 )2 iΩ0 in Eq. (5.26) is, of course, a special case of this general smearing rule: (x − x0 )2a2 =

Z

∞

−∞

dx′ −(1/2a2 )(x′ −x0 )2 ′ 2 √ e (x − x0 ) = a2 (x0 ). 2 2πa

(5.31)

Hence we obtain for the effective classical potential the approximation M 2 Ω (x0 )a2 (x0 ), 2 which by the Jensen-Peierls inequality lies always above the true result: W1Ω (x0 ) ≡ FΩx0 + Va2 (x0 ) −

W1Ω (x0 ) ≥ V eff cl (x0 ).

(5.32)

(5.33)

A minimization of W1Ω (x0 ) in Ω(x0 ) produces an optimal variational approximation to be denoted by W1 (x0 ). For the harmonic potential V (x) = Mω 2 x2 /2, the smearing process leads to Va2 (x0 ) = Mω 2 (x20 + a2 )/2. The extremum of W1Ω (x0 ) lies at Ω(x0 ) ≡ ω, so that the optimal upper bound is x2 W1 (x0 ) = Fωx0 + Mω 2 0 . (5.34) 2 Thus, for the harmonic oscillator, W1Ω (x0 ) happens to coincide with the exact effective classical potential Vωeff cl (x0 ) found in (3.827).

374

5.3


The Optimal Upper Bound

We now determine the frequency Ω(x0 ) of the local trial oscillator which optimizes the upper bound in Eq. (5.33). The derivative of W1Ω (x0 ) with respect to Ω2 (x0 ) has two terms: ∂W1Ω (x0 ) ∂W1Ω (x0 ) dW1Ω (x0 ) ∂a2 (x0 ) = + . dΩ2 (x0 ) ∂Ω2 (x0 ) ∂a2 (x0 ) Ω(x0 ) ∂Ω2 (x0 ) The first term is ( " ! # ) ∂W1 (x0 ) M kB T h ¯Ω h ¯Ω 2 = coth − 1 − a (x0 ) . ∂Ω2 (x0 ) 2 MΩ2 (x0 ) 2kB T 2kB T

(5.35)

It vanishes automatically due to (5.24). Thus we only have to minimize W1Ω (x0 ) with respect to a2 (x0 ) by satisfying the condition ∂W1Ω (x0 ) = 0. ∂a2 (x0 )

(5.36)

Inserting (5.32), this determines the trial frequency Ω2 (x0 ) =

2 ∂Va2 (x0 ) . M ∂a2 (x0 )

(5.37)

In the Fourier integral (5.28) for Va2 (x0 ), the derivative 2(∂/∂a2 )Va2 is represented by a factor −k 2 which, in turn, is equivalent to ∂/∂x20 . This leads to the alternative equation: " # 1 ∂2 2 Ω (x0 ) = Va2 (x0 ) . (5.38) M ∂x20 a2 =a2 (x0 ) Note that the partial derivatives must be taken at fixed a2 which is to be set equal to a2 (x0 ) at the end. The potential W1Ω (x0 ) with the extremal Ω2 (x0 ) and the associated a2 (x0 ) of (5.24) constitutes the Feynman-Kleinert approximation W1 (x0 ) to the effective classical potential V eff cl (x0 ). It is worth noting that due to the vanishing of the partial derivative ∂W1Ω (x0 )/∂Ω2 (x0 ) in (5.35) we may consider Ω2 (x0 ) and a2 (x0 ) as arbitrary variational parameters in the expression (5.32) for W1Ω (x0 ). Then the independent variation of W1Ω (x0 ) with respect to these two parameters yields both (5.24) and the minimization condition (5.37) for Ω2 (x0 ). From the extremal W1Ω (x0 ) we obtain the approximation for the partition function and the free energy [recall (3.834)] −F1 /kB T

Z1 = e

=

Z

∞

−∞

dx0 −W1 (x0 )/kB T e ≤ Z, le (¯ hβ)

(5.39)

where le (¯ hβ) is the thermal de Broglie length defined in Eq. (2.345). We leave it to the reader to calculate the second derivative of W1Ω (x0 ) with respect to Ω2 (x0 ) and to prove that it is nonnegative, implying that the above extremal solution is a local minimum. H. Kleinert, PATH INTEGRALS

375

5.4 Accuracy of Variational Approximation

5.4

Accuracy of Variational Approximation

The accuracy of the approximate effective classical potential W1 (x0 ) can be estimated by the following observation: In the limit of high temperatures, the approximation is perfect by construction, due the shrinking width (5.25) of the nonzero frequency fluctuations. This makes W1 (x0 ) in (5.31) converge against V (x0 ), just as the exact effective classical potential in Eq. (3.809). In the opposite limit of low temperatures, the integral over x0 in the general expression (3.808) is dominated by the minimum of the effective classical potential. If its position is denoted by xm , we have the saddle point approximation (see Section 4.2) Z dx0 −[V eff cl (xm )]′′ (x0 −xm )2 /2kB T −V eff cl (xm )/kB T e . (5.40) Z− −−→ e T →0 le (¯ hβ) The exponential of the prefactor yields the leading low-temperature behavior of the free energy: (5.41) F− −−→ V eff cl (xm ). T →0

The Gaussian integral over x0 contributes a term  

h ¯ ∆F = kB T log  kB T

s



[V eff cl (xm )]′′  ,  M

(5.42)

which accounts for the entropy of x0 fluctuations around xm [recall Eq. (1.563)]. Moreover, at zero temperature, the free energy F converges against the ground state energy E (0) of the system, so that E (0) = V eff cl (xm ).

(5.43)

The minimum of the approximate effective classical potential, W1Ω (x0 ) with respect to Ω(x0 ) supplies us with a variational approximation to the free energy F1 , which in the limit T → 0 yields a variational approximation to the ground state energy (0)

E1 = F1 |T =0 ≡ W1 (xm )|T =0 .

(5.44)

By taking the T → 0 limit in (5.32) we see that lim W1Ω (x0 ) =

T →0

i 1h h ¯ Ω(x0 ) − MΩ2 (x0 )a2 (x0 ) + Va2 (x0 ). 2

(5.45)

In the same limit, Eq. (5.24) gives lim a2 (x0 ) =

T →0

so that

h ¯ , 2MΩ(x0 )

1 1 h ¯2 lim W1Ω (x0 ) = h ¯ Ω(x0 ) + Va2 (x0 ) = + Va2 (x0 ) . T →0 4 8 Ma2 (x0 )

(5.46)

(5.47)

376


The right-hand side is recognized as the expectation value of the Hamiltonian operator 2 ˆ = pˆ + V (x) H (5.48) 2M in a normalized Gaussian wave packet of width a centered at x0 : 1 1 ψ(x) = exp − 2 (x − x0 )2 . 2 1/4 (2πa ) 4a

(5.49)

Indeed,

1 h ¯2 ˆ ≡ dxψ (x)Hψ(x) = + Va2 (x0 ). (5.50) ψ 8 Ma2 −∞ Let E1 be the minimum of this expectation under the variation of x0 and a2 : D

ˆ H

E

Z

∞

∗

D

ˆ E1 = minx0 ,a2 H

E

ψ

.

(5.51)

This is the variational approximation to the ground state energy provided by the Rayleigh-Ritz method . In the low temperature limit, the approximation F1 to the free energy converges toward E1 : lim F1 = E1 . (5.52) T →0

The approximate effective classical potential W1 (x0 ) is for all temperatures and x0 more accurate than the estimate of the ground state energy E0 by the minimal expectation value (5.51) of the Hamiltonian operator in a Gaussian wave packet. For potentials with a pronounced unique minimum of quadratic shape, this estimate is known to be excellent. In Table 5.1 we list the energies E1 = W1 (0) for a particle in an anharmonic oscillator potential. Its action will be specified in Section 5.7, where the approximation W1 (x0 ) will be calculated and discussed in detail. The table shows that this approximation promises to be quite good [2]. With the effective classical potential having good high- and low-temperature limits, it is no surprise that the approximation is quite reliable at all temperatures. Even if the potential minimum is not smooth, the low-temperature limit can be of acceptable accuracy. An example is the three-dimensional Coulomb system for which the limit (5.51) becomes (with the obvious optimal choice x0 = 0) E1 = mina

¯2 2 e2 3 h √ √ − 8 Ma2 π 2a2

!

=−

3 h ¯2 . 8 Ma2min

(5.53)

q

The minimal value of a is amin = 9π/32aH where aH = h ¯ 2 /Me2 is the Bohr radius (4.339) of the hydrogen atom. In terms of it, the minimal energy has the value E1 = −(4/3π)e2 /aH . This is only 15% percent different from the true ground (0) state energy of the Coulomb system Eex = −(1/2)e2 /aH . Such a high degree of accuracy may seem somewhat surprising since the exact Coulomb wave function ψ(x) = (πa3H )−1/2 exp(−r/aH ) is far from being a Gaussian. H. Kleinert, PATH INTEGRALS

377

5.5 Weakly Bound Ground State Energy in Finite-Range Potential Well

Table 5.1 Comparison of variational energy E1 = limT →0 F1 , obtained from Gaussian (0) trial wave function, with exact ground state energy Eex . The energies of the first two (1) (2) (0) (1) (0) excited states Eex and Eex are listed as well. The level splitting ∆Eex = Eex − Eex to the first excited state is shown in column 6. We see that it is well approximated by the value of Ω(0), as it should (see the discussion after Eq. (5.21). g/4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 10 50 100 500 1000

E1 0.5603 0.6049 0.6416 0.6734 0.7017 0.7273 0.7509 0.7721 0.7932 0.8125 1.5313 2.5476 3.1924 5.4258 6.8279

(0)

Eex 0.559146 0.602405 0.637992 0.668773 0.696176 0.721039 0.743904 0.765144 0.785032 0.803771 1.50497 2.49971 3.13138 5.31989 6.69422

(1)

Eex 1.76950 1.95054 2.09464 2.21693 2.32441 2.42102 2.50923 2.59070 2.66663 2.73789 5.32161 8.91510 11.1873 19.0434 23.9722

(2)

Eex 3.13862 3.53630 3.84478 4.10284 4.32752 4.52812 4.71033 4.87793 5.03360 5.17929 10.3471 17.4370 21.9069 37.3407 47.0173

(0)

∆Eex 1.21035 1.34810 1.45665 1.54816 1.62823 1.69998 1.76533 1.82556 1.86286 1.93412 3.81694 6.41339 8.05590 13.7235 17.2780

Ω(0) 1.222 1.370 1.487 1.585 1.627 1.749 1.819 1.884 1.944 2.000 4.000 6.744 8.474 14.446 18.190

a2 (0) 0.4094 0.3650 0.3363 0.3154 0.2991 0.2859 0.2749 0.2654 0.2572 0.2500 0.1250 0.0741 0.0590 0.0346 0.0275

The partition function of the Coulomb system can be calculated only after subtracting the free-particle partition function and screening the 1/r -behavior down to a finite range. The effective classical free energy F1 of the Coulomb potential obtained by this method is, at any temperature, more accurate than the difference (0) between E1 and Eex . More details will be given in Section 5.10.

5.5

Weakly Bound Ground State Energy in Finite-Range Potential Well

The variational approach allows us to derive a simple approximation for the boundstate energy in an arbitrarily shaped potential of finite range, for which the binding energy is very weak. Precisely speaking, the falloff of the ground state wave function has to lie outside the range of the potential. A typical example for this situation is the binding of electrons to Cooper pairs in a superconductor. The attractive force comes from the electron-phonon interaction which is weakened by the Coulomb repulsion. The potential has a complicated shape, but the binding energy is so weak that the wave function of a Cooper pair reaches out to several thousand lattice spacings, which is much larger than the range of the potential, which extends only over maximally a hundred lattice spacings. In this case one may practically replace the potential by an equivalent δ-function potential.

378


The present considerations apply to this situation. Let us assume the absolute minimum of the potential to lie at the origin. The first-order variational energy at the origin is given by Ω W1 (0) = + Va2 (0), (5.54) 2 where by (5.30) Va2 (0) =

Z

∞

−∞

dx′ ′2 2 √ 0 e−x0 /2a V (x′0 ). 2πa2

(5.55)

By assumption, the binding energy is so small that the ground state wave function does not fall off within the range of V (x0 ). Hence we can approximate Va2 (0) ≈

s

Ω π

Z

∞

−∞

dx′0 V (x′0 ),

(5.56)

where we have inserted a2 = 1/2Ω. Extremizing this in Ω yields the approximate ground state energy Z ∞ 2 1 (0) E1 ≈ − − dx′0 V (x′0 ) . (5.57) 2π −∞ By applying this result to a simple δ-function potential at the origin, V (x) = −gδ(x),

g > 0,

(5.58)

we find an approximate ground state energy (0)

E1 = −

1 2 g . 2π

(5.59)

The exact value is

1 E (0) = − g 2 . (5.60) 2 The failure of the variational approximation is due to the fact that outside √ the range −k|x| of the potential, the wave function is a simple exponential e with k = −2E (0) , and not a Gaussian. In fact, if we consider the expectation value of the Hamiltonian operator 1 d2 H=− − gδ(x) (5.61) 2 dx2 for a normalized trial wave function √ ψ(x) = Ke−K|x|, (5.62) we obtain a variational energy W1 =

K2 − gK, 2


379

5.6 Possible Direct Generalizations

whose minimum gives the exact ground state energy (5.60). Thus, problems of the present type call for the development of a variational perturbation theory for which Eq. (5.54) and (5.55) read K2 (5.64) W (0) = + VK (0), 2 where Z ∞ dx′0 −K|x0| VK (0) = K V (x0 ). (5.65) e −∞ a For an arbitrary attractive potential whose range is much shorter than a, this leads to the correct energy for a weakly bound ground state (0)

E1 ≈ −

5.6

1 − 2

Z

∞

−∞

dx′0 V (x0 )

2

.

(5.66)

Possible Direct Generalizations

Let us remark that there is a possible immediate generalization of the above variational procedure. One may treat higher components xm with m > 0 accurately, say up to m = m ¯ − 1, where m ¯ is some integer > 1, using the ansatz ( Z Z 1 h¯ /kB T M x˙ 2 (τ ) Zm Dx(τ ) exp − + Ω2 (x0 , . . . , xm ¯ ≡ ¯) ¯h 0 2 2 !2   m−1 ¯  X × x(τ ) − x0 − (xm e−iωm τ + c.c.)  e−(1/kB T )Lm¯ (x0 ,...,xm¯ )  m=1

=

Z

∞

−∞

dx0 le (¯ hβ)

m−1 ¯Y Z n=1

im 2 dxre ωm + Ω2 (x0 ) m dxm 2 2 πkB T /M ωm ωm

¯ Ω(x0 )/2kB T h e−Lm¯ (x0 ,...,xm¯ )/kB T , sinh(¯ hΩ(x0 )/2kB T )

×

(5.67)

with the trial function Lm ¯: Lm ¯ (x0 , . . . , xm ¯)

=

kB T ¯h

Z

0

h ¯ /kB T

dτ Va2m¯

x0 +

m−1 ¯ X

!

(xm e−iωm τ + c.c.)

m=1

−

M 2 2 Ω (x0 , . . . , xm ¯ )am ¯, 2 (5.68)

and a smearing square width of the potential a2m ¯

= =

∞ 2kB T X 1 2 M m=m ω m + Ω2 ¯ m−1 ¯ kB T ¯hΩ ¯hΩ 2kB T X 1 coth − 1 − . 2 + Ω2 M Ω2 2kB T 2kB T M m=1 ωm

(5.69)

For the partition function alone the additional work turns out to be not very rewarding since it renders only small improvements. It turns out that in the low-temperature limit T → 0, the free energy is still equal to the optimal expectation of the Hamiltonian operator in the Gaussian wave packet (5.49).

380


Note that the ansatz (5.7) [as well as (5.67)] cannot be improved by allowing the trial frequency Ω(x0 ) to be a matrix Ωmm′ (x0 ) in the space of Fourier components xm [i.e., by using P P ∗ 2 m,m′ Ωmm′ (x0 )xm xm′ instead of Ω(x0 ) m |xm | ]. This would also lead to an exactly integrable trial partition function. However, after going through the minimization procedure one would fall back to the diagonal solution Ωmm′ (x0 ) = δmm′ Ω(x0 ).

5.7

Effective Classical Potential for Anharmonic Oscillator and Double-Well Potential

For a typical application of the approximation method consider the Euclidean action A[x] =

Z

0

h/kB T ¯

M 2 g 4 2 2 dτ x˙ + ω x + x . 2 4

(5.70)

¯ = kB = 1. We have Let us write 1/kB T as β and use natural units with M = 1, h to distinguish two cases: a) Case ω 2 > 0, Anharmonic Oscillator Setting ω 2 = 1, the smeared potential (5.30) is according to formula (3.876): Va2 (x0 ) =

x20 g 4 a2 3 2 2 3g 4 + x0 + + gx0 a + a . 2 4 2 2 4

(5.71)

Differentiating this with respect to a2 /2 gives, via (5.37), Ω2 (x0 ) = [1 + 3gx20 + 3ga2 (x0 )].

(5.72)

This equation is solved at each x0 by iteration together with (5.24), "

#

1 βΩ(x0 ) βΩ(x0 ) a (x0 ) = coth −1 . βΩ2 (x0 ) 2 2 2

(5.73)

An initial approximation such as Ω(x0 ) = 0 is inserted into (5.73) to find a2 (x0 ) ≡ β/12, which serves to calculate from (5.72) an improved Ω2 (x0 ), and so on. The iteration converges rapidly. Inserting the final a2 (x0 ), Ω2 (x0 ) into (5.71) and (5.32), we obtain the desired approximation W1 (x0 ) to the effective classical potential V eff cl (x0 ). By performing the integral (5.39) in x0 we find the approximate free energy F1 plotted as a function of β in Fig. 5.2. The exact free-energy values are obtained from the known energy eigenvalues of the anharmonic oscillator. They are seen to lie closely below the approximate F1 curve. For comparison, we have also plotted the classical approximation Fcl = −(1/β) log Zcl which does not satisfy the Jensen-Peierls inequality and lies below the exact curve. In his book on statistical mechanics [3], Feynman gives another approximation, called here F0 , which can be obtained from the present W1 (x0 ) by ending the iteration of (5.72), (5.73) after the first step, i.e., by using the constant nonminimal H. Kleinert, PATH INTEGRALS

381

5.7 Effective Classical Potential for Anharmonic Oscillator

Figure 5.2 Approximate free energy F1 of anharmonic oscillator as compared with the R∞ √ exact energy Fex , the classical limit Fcl = −(1/β) log −∞ (dx/ 2πβ)e−βV (x) , as well as an earlier approximation F0 = −(1/β) log Z0 of Feynman’s corresponding to F1 for the nonoptimal choice Ω = 0, a2 = β/12. Note that F0 , F1 satisfy the inequality F0,1 ≥ F , while Fcl does not.

variational parameters Ω(x0 ) ≡ 0, a2 (x0 ) ≡ h ¯ 2 β/12M. This leads to the approximation 1 h ¯ 2 β ′′ 1 V (x0 ) + Va2 (x0 ) ≈ V (x0 ) + 2 12M 8

h ¯ 2β 12M

!2

V (4) (x0 ) + . . . ,

(5.74)

referred to as Wigner’s expansion [4]. The approximation F0 is good only at higher temperatures, as seen in Fig. 5.2. Just like F1 , the curve F0 lies also above the exact curve since it is subject to the Jensen-Peierls inequality. Indeed, the inequality holds for the potential W1 (x0 ) in the general form (5.32), i.e., irrespective of the minimization in a2 (x0 ). Thus it is valid for arbitrary Ω2 (x0 ), in particular for Ω2 (x0 ) ≡ 0. In the limit T → 0, the free energy F1 yields the following approximation for the ground state energy E (0) of the anharmonic oscillator: (0)

E1 =

1 Ω 3 g + + . 4Ω 4 4 4Ω2

(5.75)

382


This approximation is very good for all coupling strengths, including the strongcoupling limit. In this limit, the optimal frequency and energy have the expansions Ω1 =

1/3 "

g 4

1 1 61/3 + 1/3 4/3 + ... 2 3 (g/4)2/3

#

,

(5.76)

and 1/3 " 4/3

g 4

3 4

#

1 1 + 7/3 1/3 + ... 2 3 (g/4)2/3 # 1/3 " g 1 ≈ 0.681420 + 0.13758 + ... . 4 (g/4)2/3

(0) E1 (g) ≈

(5.77)

The coefficients are quite close to the precise limiting expression to be calculated in Section 5.15 (listed in Table 5.9). b) Case ω 2 < 0: The Double-Well Potential For ω 2 = −1, we slightly modify the potential by adding a constant 1/4g, so that it becomes x2 g 4 1 (5.78) V (x) = − + x + . 2 4 4g The additional constant ensures a smooth behavior of the energies in the limit g → 0. Since the potential possesses now two symmetric minima, it is called the double-well potential . Its smeared version Va2 (x0 ) can be taken from (5.71), after a sign change in the first and third terms (and after adding the constant 1/4g). Now the trial frequency Ω2 (x0 ) = −1 + 3gx20 + 3ga2(x0 )

(5.79)

can become negative, although it turns out to remain always larger than −4π 2 /β 2, since the solution is incapable of crossing the first singularity in the sum (5.24) from the right. Hence the smearing square width a2 (x0 ) is always positive. For Ω2 ∈ (−4π 2 /β 2 , 0), the sum (5.24) gives a2 (x0 ) =

∞ 2 X 1 2 β m=1 ωm + Ω2 (x0 )

1 = 2 βΩ (x0 )

!

β|Ω(x0 )| β|Ω(x0 )| cot −1 , 2 2

(5.80)

which is the expression (5.73), continued analytically to imaginary Ω(x0 ). The above procedure for finding a2 (x0 ) and Ω2 (x0 ) by iteration of (5.79) and (5.80) is not applicable near the central peak of the double well, where it does not converge. There one finds the solution by searching for the zero of the function of Ω2 (x0 ) 1 f (Ω2 (x0 )) ≡ a2 (x0 ) − [1 + Ω2 (x0 ) − 3gx20 ], (5.81) 3g H. Kleinert, PATH INTEGRALS


383

with a2 (x0 ) calculated from (5.80) or (5.73). At T = 0, the curves have for g ≤ gc two symmetric nontrivial minima at ±xm with xm = where Eq. (5.79) becomes These disappear for

s

1 − 3ga2 , g

Ω2 (xm ) = 2 − 6ga2 (xm ).

(5.82) (5.83)

s

4 2 ≈ 0.3629 . (5.84) 9 3 The resulting effective classical potentials and the free energies are plotted in Figs. 5.3 and 5.4. g > gc =

Figure 5.3 Effective classical potential of double well V (x) = −x2 /2 + gx4 /4 + 1/4g at various g for T = 0 and T = ∞ [where it is equal to the potential V (x) itself]. The > 0.4, but not if quantum fluctuations at T = 0 smear out the double well completely if g ∼ g = 0.2.

It is useful to compare the approximate effective classical potential W1 (x) with the true one V eff cl (x) in Fig. 5.5. The latter was obtained by Monte Carlo simulations of theR path integral of the double-well potential, holding the path average x¯ = (1/β) 0β dτ x(τ ) fixed at x0 . The coupling strength is chosen as g = 0.4, where the worst agreement is expected. (0) In the limit T → 0, the approximation F1 yields an approximation E1 for the ground state energy. In the strong-coupling limit, the leading behavior is the same as in Eq. (5.77) for the anharmonic oscillator.

384


Figure 5.4 Free energy F1 in double-well potential (5.78), compared with the exact free energy Fex , the classical limit Fcl , and Feynman’s approximation F0 (which coincides with F1 for the nonminimal values Ω = 0, a2 = β/12).

Let us end this section with the following remark. The entire approximation procedure can certainly also be applied to a time-sliced path integral in which the time axis contains N + 1 discrete points τn = nǫ, n = 0, 1, . . . N. The only change in the above treatment consists in the replacement 2 ¯ m = 1 [2 − 2 cos(ǫωm )]. ωm → Ωm Ω ǫ2

(5.85)

Hence the expression for the smearing square width parameter a2 (x0 ) of (5.24) is replaced by mY max h i 2kB T ∂ 2 log Ω Ω + Ω (x ) m m 0 M ∂Ω2 (x0 ) m=1 kB T 1 ∂ sinh(¯ hΩN (x0 )/2kB T ) = log (5.86) M Ω ∂Ω h ¯ Ω(x0 )/2kB T " # h ¯ Ω(x0 ) h ¯ ΩN (x0 ) 1 kB T = coth −1 , MΩ2 (x0 ) 2kB T 2kB T cosh(ǫΩN (x0 )/2)

a2 (x0 ) =



385

Figure 5.5 Comparison of approximate effective classical potential W1 (x0 ) (dashed curves) and W3 (x0 ) (solid curves) with exact V eff cl (x0 ) (dots) at various inverse temperatures β = 1/T . The data dots are obtained from Monte Carlo simulations using 105 configurations [W. Janke and H. Kleinert, Chem. Phys. Lett. 137 , 162 (1987) (http://www.physik.fu-berlin.de/~kleinert/154)]. We have picked the worst case, g = 0.4. The solid lines represent the higher approximation W3 (x0 ), to be calculated in Section 5.13.

where mmax = N/2 for even and (N − 1)/2 for odd N [recall (2.383)], and ΩN (x0 ) is defined by sinh[ǫΩN (x0 )/2] ≡ ǫΩ(x0 )/2 (5.87) [see Eq. (2.391)]. The trial potential W1 (x0 ) now reads W1 (x0 ) ≡ kB T log

sinh h ¯ ΩN (x0 )/2kB T M + Va2 (x0 ) (x0 ) − Ω2 (x0 )a2 (x0 ), h ¯ Ω(x0 )/2kB T 2

(5.88)

rather than (5.32). Minimizing this in a2 (x0 ) gives again (5.37) and (5.38) for Ω2 (x0 ). In Fig. 5.6 we have plotted the resulting approximate effective classical potential W1 (x0 ) of the double-well potential (5.78) with g = 0.4 at a fixed large value β = 20 for various numbers of lattice points N + 1. It is interesting to compare these plots with the exact curves, obtained again from Monte Carlo simulations. For N = 1, the agreement is exact. For small N, the agreement is good near and outside the potential minima. For larger N, the exact effective classical potential has oscillations which are not reproduced by the approximation.

386

5 Variational Perturbation Theory 1.0 g =0.4, β =20 0.8

0.6 W1 0.4

0.2

0.0

-2

-1

0 x0

1

2

Figure 5.6 Effective classical potential W1 (x0 ) for double-well potential (5.78) with g = 0.4 at fixed low temperature T = 1/β = 1/20, for various numbers of time slices N + 1 = 2 (×t), 4 (△), 8 (▽), 16 (3), 32 (+), 64 ( ). The dashed line represents the original potential V (x0 ). For the source of the data points, see the previous figure caption.

5.8

Particle Densities

It is possible to find approximate particle densities from the optimal effective classical potential W1 (x0 ) [5, 6]. Certainly, the results cannot be as accurate as those for the free energies. In Schrödinger quantum mechanics, it is well known that variational methods can give quite accurate energies even if the trial wave functions are only of moderate quality. This has also been seen in the Eq. (5.53) estimate to the ground state energy of the Coulomb system by a Gaussian wave packet. The energy is a rather global property of the system. For physical quantities such as particle densities which contain local information on the wave functions, the approximation is expected to be much worse. Let us nevertheless calculate particle densities of a quantum-mechanical system. For this we tie down the periodic particle orbit in the trial partition function Z1 for an arbitrary time at a particular position, say xa . Mathematically, this is enforced with the help of a δ-function: δ(xa − x(τ )) = δ(xa − x0 − =

Z

∞

−∞

∞ X

(xm e−iωm τ + c.c.))

m=1

∞ X dk exp ik xa − x0 − (xm e−iωm τ + c.c.) 2π m=1

(

"

#)

.

(5.89)

With this, we write the path integral for the particle density [compare (2.345)] ρ(xa ) = Z −1

I

Dx δ(xa − x(τ ))e−A/¯h


387

5.8 Particle Densities

and decompose ρ(xa ) = Z −1 ×

Z

Z

∞

−∞

dx0 le (¯ hβ)

Z

dk ik(xa −x0 ) e 2π ∞

˜ x − x0 )e−ikΣm=1 (xm +c.c.) e−A/¯h . Dxδ(¯

(5.91)

The approximation W1 (x0 ) is based on a quasiharmonic treatment of the xm -

Figure 5.7 Approximate particle density (5.94) of anharmonic oscillator for g = 40, as P compared with the exact density ρ(x) = Z −1 n |ψn (x)|2 e−βEn , obtained by integrating the Schr¨ odinger equation numerically. The curves are labeled by their β values with the subscripts 1, ex, cl indicating the approximation.

fluctuations for m > 0. For harmonic fluctuations we use Wick’s rule of Section 3.10 to evaluate D

∞

e−ikΣm=1 (xm e

−iωm τ +c.c.)

Ex0 Ω

≈ e−k

2 Σ∞ m=1

x

h|xm |2 iΩ0 ≡ e−k2 a2 /2 ,

(5.92)

which is true for any τ . Thus we could have chosen any τ in the δ-function (5.89) to find the distribution function. Inserting (5.92) into (5.91) we can integrate out k and find the approximation to the particle density ρ(xa ) ≈ Z

−1

Z

∞

−∞

2

2

dx0 e−(xa −x0 ) /2a (x0 ) −V eff cl (x0 )/kB T q e . le (¯ hβ) 2πa2 (x0 )

(5.93)

388


Figure 5.8 Particle density (5.94) in double-well potential (5.78) for the worst choice of the coupling constant, g = 0.4. Comparison is made with the exact density ρ(x) = P Z −1 n |ψn (x)|2 e−βEn obtained by integrating the Schr¨ odinger equation numerically. The curves are labeled by their β values with the subscripts 1, ex, cl indicating the approxima2 /2a2 √ −x 2 tion. For β → ∞, the distribution tends to the Gaussian e / 2πa with a2 = 1.030 (see Table 5.1).

By inserting for V eff cl (x0 ) the approximation W1 (x0 ), which for Z yields the approximation Z1 , we arrive at the corresponding approximation for the particle distribution function: 2 2 Z ∞ dx0 e−(xa −x0 ) /2a (x0 ) −W1 (x0 )/kB T q e ρ1 (xa ) = Z1−1 . (5.94) hβ) −∞ le (¯ 2πa2 (x0 ) ∞ This has obviously the correct normalization −∞ dxa ρ1 (xa ) = 1. Figure 5.7 shows a comparison of the approximate particle distribution functions of the anharmonic oscillator with the exact ones. Both agree reasonably well with each other. In Fig. 5.8, the same plot is given for the double-well potential at a coupling g = 0.4. Here the agreement at very low temperature is not as good as in Fig. 5.7. Compare, for example, the zero-temperature curve ∞1 with the exact curve ∞ex . The first has only a single central peak, the second a double peak. The reason for this discrepancy is the correspondence of the approximate distribution to an optimal Gaussian wave function which happens to be centered at the origin, in spite of the double-well shape of the potential. In Fig. 5.3 we see the reason for this: The approximate effective classical potential W1 (x0 ) has, at small temperatures up to T ∼ 1/10, only one minimum at the origin, and this becomes the center of the optimal Gaussian wave function. For larger temperatures, there are two minima and the approximate distribution function ρ1 (x) corresponds roughly to two Gaussian wave packets centered around these minima. Then, the agreement with the exact

R


5.9 Extension to D Dimensions

389

distribution becomes better. We have intentionally chosen the coupling g = 0.4, where the result would be about the worst. For g ≫ 0.4, both the true and the approximate distributions have a single central peak. For g ≪ 0.4, both have two peaks at small temperatures. In both limits, the approximation is acceptable.

5.9

Extension to D Dimensions

The method can easily be extended to approximate the path integral of a particle moving in a D-dimensional x-space. Let xi be the D components of x. Then the trial frequency Ω2ij (x0 ) in (5.7) must be taken as a D × D -matrix. In the special √ case of V (x) being rotationally symmetric and depending only on r = x2 , we may introduce, as in the discussion of the effective action in Eqs. (3.681)–(3.686), longitudinal and transverse parts of Ω2ij (x0 ) via the decomposition Ω2ij (x0 ) = Ω2L (r0 )PLij (x0 ) + Ω2T (r0 )PT ij (x0 ),

(5.95)

where PLij (ˆ x0 ) = x0i x0j /r02 ,

PT ij (ˆ x0 ) = δij − x0i x0j /r02 ,

(5.96)

are projection matrices into the longitudinal and transverse directions of x0 . Analogous projections of a vector are defined by xL ≡ PL (x0 ) x, xT ≡ PT (x0 ) x. Then the anisotropic generalization of W1Ω (x0 ) becomes "

#

sinh h ¯ ΩL /2kB T sinh h ¯ ΩT /2kB T W1 (r0 ) = kB T log + (D − 1) log h ¯ ΩL /2kB T h ¯ ΩT /2kB T i Mh 2 2 − ΩL aL + (D − 1)Ω2T a2T + VaL ,aT , (5.97) 2 with all functions on the right-hand side depending only on r0 . The bold-face su perscript indicates the presence of two variational frequencies ≡ (ΩL , ΩT ). The smeared potential is now given by 1

1

VaL ,aT (r0 ) = q q D−1 2πa2L 2πa2T

D Z Y

i=1

∞

−∞

dδxi

1 2 −2 2 × exp − a−2 V (x0 + δx), (5.98) L δxL + aT δxT 2 which can also be written as " # Z Z 1 δxL2 δxT2 D−1 VaL ,aT (r0 ) = q dδxL d δxT exp − 2 − 2 V (x0 +δx). (5.99) 2aL 2aT (2π)D a2 a2D−2

L T

For higher temperatures where the smearing widths a2L , a2T are small, we set V (x) ≡ v(r 2 ), so that V (x) = v(r02 + 2r0 δxL + δx2L + δx2T ) = =

v(r02

λ(2r0 δxL +δx2L +δx2T )

+ ∂λ )e

∞ X

n 1 (n) 2 v (r0 ) 2r0 δxL + δx2L + δx2T n=0 n!

. λ=0

(5.100)

390


Inserting this into the right-hand side of (5.98), we find 



1 1 2 2 2 2   VaL ,aT (r0 ) = v(r02 + ∂λ ) q e2r0 λ aL /(1−2aL λ)  q D−1 2 1 − 2aL λ 1 − 2a2T λ

,

(5.101)

λ=0

which has the expansion

VaL ,aT (r0 ) = v(r02 ) + v ′ (r02 )[a2L + (D − 1)a2T ] 1 + v ′′ (r02 )[3a4L + 2(D − 1)a2L a2T + (D 2 − 1)a4T + 4r02 a2L ] + . . . . 2

(5.102)

The prime abbreviates the derivative with respect to r02 . In general it is useful to insert into (5.98) the Fourier representation for the potential Z dD k ikx V (x) = e V (k), (5.103) (2π)D which makes the x-integration Gaussian, so that (5.99) becomes Va2L ,a2T (r0 ) =

Z

a2L 2 a2T 2 dD k V (k) exp − k − kT − ir0 kL . (2π)D 2 L 2 !

(5.104)

Exploiting the rotational symmetry of the potential by writing V (k) ≡ v(k 2 ), we decompose the measure of integration as Z

SD−1 dD k = D (2π) (2π)D

Z

where SD =

∞ −∞

dkL

Z

0

∞

dkT kTD−2 ,

(5.105)

2π D/2 Γ(D/2)

(5.106)

is the surface of a sphere in D dimensions, and further with kL = k cos φ, kT = k sin φ: Z Z ∞ dD k SD−1 Z 1 = d cos φ dk k D−1 . (5.107) D D (2π) (2π) −1 0 This brings (5.104) to the form Va2L ,a2T (r0 ) =

SD−1 (2π)D

Z

1 −1

du

Z

∞

−1

dk k D−1 v(k 2 )

i 1 h 2 2 × exp − aL u + a2T (1 − u2 ) k 2 − ir0 u . 2

(5.108)

The final effective classical potential is found by minimizing W1 (r0 ) at each r0 in aL , aT , ΩL , ΩT . To gain a rough idea about the solution, it is usually of advantage to study first the isotropic approximation obtained by assuming a2T (r0 ) = a2L (r0 ), and to proceed later to the anisotropic approximation. H. Kleinert, PATH INTEGRALS

391

5.10 Application to Coulomb and Yukawa Potentials

5.10

Application to Coulomb and Yukawa Potentials

The effective classical potential can be useful also for singular potentials as long as the smearing procedure makes sense. An example is the Yukawa potential V (r) = −(e2 /r)e−mr ,

(5.109)

which reduces to the Coulomb potential for m ≡ 0. Using the Fourier representation V (r) = 4π

Z

d3 k eikx = 4π (2π)3 k 2 + m2

Z

∞

0

dτ

Z

d3 k −τ (k2 +m2 ) e , (2π)3

(5.110)

we easily calculate the isotropically smeared potential d3 x

1 (x0 − x)2 V (r) 3 2 2 2a 2πa Z ∞ 1 2 2 2 2 ′2 2 ′2 = −e2 2πem a /2 da′ √ exp −r /2a − m a /2 0 3 a2 ′2 2πa √ Z m2 a2 /2 1 r0 / 2a2 2 2 2 2 2 2e √ = −e dte−(t +m r0 /4t ) . (5.111) π r0 0

Va2 (r0 ) = −e2

Z

√

exp −

In the Coulomb limit m → 0, the smeared potential becomes equal to the Coulomb potential multiplied by an error function, Va2 (r0 ) = −

√ e2 erf(r0 / 2a2 ), r0

(5.112)

where the error function is defined by 2 Zz 2 erf(z) ≡ √ dxe−x . π 0

(5.113)

The smeared potential is no longer singular at the origin, e2 Va2 (0) = − a

s

2 m2 a2 /2 e . π

(5.114)

The singularity has been removed by quantum fluctuations. In this way the effective classical potential explains the stability of matter in quantum physics, i.e., the fact that atomic electrons do not fall into the origin. The effective classical potential of the Coulomb system is then by the isotropic version of (5.97) 



3 sinh[¯hβΩ(x0 )/2] e2 |x0 |  3 W1Ω (x0 ) = ln − erf  q − MΩ2 (x0 )a2 (x0 ). (5.115) 2 β h ¯ βΩ(x0 )/2 |x0 | 2 2a (x0 )

Minimizing W1 (r0 ) with respect to a2 (r0 ) gives an equation analogous to Eq. (5.37) determining the frequency Ω2 (r0 ) to be 2 1 1 −r02 /2a2 Ω2 (r0 ) = e2 √ e . 3 2π (a2 )3/2

(5.116)

392


Figure 5.9 Approximate effective classical potential W1 (r0 ) of Coulomb system at various temperatures (in multiples of 104 K). It is calculated once in the isotropic (dashed curves) and once in the anisotropic approximation. The improvement is visible in the insert which shows W1 /V . The inverse temperature values of the different curves are β = 31.58, 15.78, 7.89, 3.945, 1.9725, 0.9863, 0 atomic units, respectively.

We have gone to atomic units in which e = M= h ¯ = kB = 1, so that energies and 4 temperatures are measured in units of E0 = Me /M¯ h2 ≈ 4.36 × 10−11 erg ≈ 27.21eV and T0 = E0 /kB ≈ 31575K, respectively. Solving (5.116) together with (5.24), we find a2 (r0 ) and the approximate effective classical potential (5.32). The result is shown in Fig. 5.9 as a dashed curve. The above approximation may now be improved by treating the fluctuations anisotropically, as described in the previous section, with different Ω2 (x0 ) for radial and tangential fluctuations δxL and δxT , and the effective potential following Eqs. (5.97) and (5.98). For the anisotropically smeared Coulomb potential we calculate from (5.108): s

1 Va2L ,a2T (r0 ) = − 2π

r02 u2 du q exp − . 2 [a2L u2 + a2T (1 − u2 )] −1 a2L u2 + a2T (1 − u2 )

Z

e2

1

(

)

(5.117)

q

q

Introducing the variable λ = a2L u/ a2L λ2 + a2T (1 − λ2 ), which runs through the same interval [−1, 1] as u, we rewrite this as Va2L ,a2T (r0 ) = −e2

s

a2L 2π

Z

1

−1

dλ

exp[−(r02 /2a2L )λ2 ] . a2L (1 − λ2 ) + a2T λ2

(5.118)

Extremization of W1 (r0 ) in Eq. (5.97) yields the equations for the trial frequencies s

∂ a2L 2 2 2 Ω2T (r0 ) ≡ V = e ∂a2T aT ,aL 2π

Z

1

−1

dλ

λ2 exp[−(r02 /2a2L )λ2 ] , [a2L (1 − λ2 ) + a2T λ2 ]2

(5.119)


393

5.10 Application to Coulomb and Yukawa Potentials

√ 3 Figure 5.10 Particle distribution g(r) ≡ 2πβ ρ(r) in Coulomb potential at different temperatures T (the same as in Fig. 5.9), calculated once in the isotropic and once in the anisotropic approximation. The dotted curves show the classical distribution. For low and intermediate temperatures the exact distributions of R.G. Storer, J. Math. Phys. 9 , 964 (1968) are well represented by the two lowest energy levels for which ρ(r)=π −1 e−2r eβ/2 +(1/8π)(1 − r + r 2 /2)e−r eβ/8 +(1/38 π)(243 − 324r + 216r 2 −48r 3 +4r 4 )e−2r/3 eβ/18 . "

#

1 2 1 ∂ ∂ Ω2 (r0 ) ≡ [ΩL + 2Ω2T ] = + 2 2 Va2T ,a2L 2 3 3 ∂aL ∂aT 2 1 1 = e2 √ q exp(−r02 /2a2L ). 3 2π a2 a4 L T

(5.120)

These equations have to be solved together with a2L,T (r0 )

"

1

#

βΩL,T (r0 ) βΩL,T (r0 ) = coth −1 . 2 βΩL,T (r0 ) 2 2

(5.121)

Upon inserting the solutions into (5.97), we find the approximate effective classical potential plotted in Fig. 5.9 as a solid curve. Let us calculate the approximate particle distribution functions using a three-dimensional anisotropic version of Eq. (5.94). With the potential W1 (r0 ), we arrive at the integral [6] ρ1 (r) =

Z

2 /2a2 L

2

2

2

e−(x0 +y0 )/2aT e−βW1 (r0 ) d x0 q 2 3/2 2πa2L (r0 ) 2πaT (r0 ) (2πβ)

= 2π

e−(z0 −r)

3

Z

0

∞

dr0

2 /2a2 L

1

e−(λr0 −r)

2

2

2

e−r0 (1−λ )/2aT e−βW1 (r0 ) dλ q . 2πa2T (r0 ) (2πβ)3/2 −1 2πa2L (r0 )

Z

(5.122)

The resulting curves to different temperatures are plotted in Fig. 5.10 and compared with the exact distribution as given by Storer. The distribution obtained from the earlier isotropic approximation (5.116) to the trial frequency Ω2 (r0 ) is also shown.

394

5.11


Hydrogen Atom in Strong Magnetic Field

The recent discovery of magnetars [7] has renewed interest in the behavior of charged particle systems in the presence of extremely strong external magnetic fields. In this new type of neutron stars, electrons and protons from decaying neutrons produce magnetic fields B reaching up to 1015 G, much larger than those in neutron stars and white dwarfs, where B is of order 1010 − 1012 G and 106 − 108 G, respectively. Analytic treatments of the strong-field properties of an atomic system are difficult, even in the zero-temperature limit. The reason is a logarithmic asymptotic behavior of the ground state energy to be derived in Eq. (5.132). In the weak-field limit, on the other hand, perturbative approaches yield well-known series expansions in powers of B 2 up to B 60 [8]. These are useful, however, only for B ≪ B0 , where B0 is the atomic magnetic field strength B0 = e3 M 2 /¯ h3 ≈ 2.35 × 105 T = 2.35 × 109 G. So far, the most reliable values for strong uniform fields were obtained by numerical calculations [9]. The variational approach can be used to derive a single analytic expression for the effective classical potential applicable to all field strengths and temperatures [10]. The Hamiltonian of the electron in a hydrogen atom in a uniform external magnetic field pointing along the positive z-axis is the obvious extension of the expression in Eq. (2.638) by a Coulomb potential: H(p, x) =

e2 1 2 M 2 2 p + ωB x − ωB lz (p, x) − , 2M 2 |x|

(5.123)

where ωB denotes the B-dependent magnetic frequency ωL /2 = eB/2Mc of Eq. (2.643), i.e., half the Landau or cyclotron frequency. The magnetic vector potential has been chosen in the symmetric gauge (2.632). Recall that lz is the z-component of the orbital angular momentum lz (p, x) = (x × p)z [see (2.639)]. At first, we restrict ourselves here to zero temperature. From the imaginary-time version of the classical action (2.632) we see that the particle distribution function in the orthogonal direction of the magnetic field is, for xb = 0, yb = 0, proportional to M (5.124) exp − ωB x2a . 2 This is the same distribution as for a transverse harmonic oscillator with frequency ωB . Being at zero temperature, the first-order variational energy requires knowing the smeared potential at the origin. Allowing for a different smearing width a2k and a2⊥ along an orthogonal to the magnetic field, we may use Eq. (5.118) to write Va2k ,a2⊥ (0) =

v u 2u −e t

1 2πa2k

Z

1

dλ

−1

(1 −

λ2 )

1 . + λ2 a2⊥ /a2k

(5.125)

Performing the integral yields

Va2k ,a2⊥ (0) =

v u u −e2 t

ak 1 2 arccosh . 2 − a⊥ ) a⊥

2π(a2k

(5.126)


395

5.11 Hydrogen Atom in Strong Magnetic Field

Since the ground state energies of the parallel and orthogonal oscillators are Ωk /2 and 2 × Ω⊥ /2, we obtain immediately the first-order variational energy W1 (0) = Ω⊥ +

Ωk M + Va2k ,a2⊥ (0) + (ωk2 − Ω2k )a2k + M(ωB2 − Ω2⊥ )a2⊥ , 2 2

(5.127)

with ωk = 0 and a2k,⊥ = 1/2Ωk,⊥ . In this expression we have ignored the second term in the Hamiltonian (5.123), since the angular momentum lz of the ground state must have a zero expectation value. For very strong magnetic fields, the transverse variational frequency ΩT will become equal to ωB , such that in this limit s

(5.128)

4e4 2πωB log2 4 , π e

(5.129)

Ωk Ωk 8ωB W (0) ≈ ωB + − e2 log . 4 π Ωk Extremizing this in Ωk yields Ωk ≈

and thus an approximate ground state energy (0)

E1 ≈ ωB −

e4 2πωB log2 4 . π e

(5.130)

The approach to very strong fields can be found by extremizing the energy (5.127) also in Ω⊥ . Going over to atomic units with e = 1 and m = 1, where energies are measured in units of ǫ0 = Me4 /¯ h2 ≡ 2 Ryd ≈ 27.21 eV, temperatures in ǫ0 /kB ≈ 3.16 × 105 K, distances in Bohr radii aB = h ¯ 2 /Me2 ≈ 0.53 × 10−8 cm, and magnetic 3 field strengths in B0 = e3 M 2 /¯ h ≈ 2.35 × 105 T = 2.35 × 109 G, the extremization yields 2 2a a2 √ Ω⊥ = ln B − 2lnln B + + 2 + b + O(ln−3 B), π ln B ln B (5.131) with abbreviations a = 2 − ln 2 ≈ 1.307 and b = ln(π/2) − 2 ≈ −1.548. The associated optimized ground state energy is, up to terms of order ln−2 B, B Ωk ≈ , 2

E

(0)

!

q

B 1 − (B) = 2 π

ln2 B − 4 ln B lnln B + 4 ln2 ln B − 4b lnln B + 2(b + 2) ln B + b2 −

i 1 h 2 8 ln ln B − 8b lnln B + 2b2 + O(ln−2 B). ln B

(5.132)

The prefactor 1/π of the leading ln2 B-term using a variational ansatz of the type (5.64), (5.65) for the transverse degree of freedom is in contrast to the value 1/2 calculated in the textbook by Landau and Lifshitz [11]. The calculation of higher orders in variational perturbation theory would drive our value towards 1/2.

396


Table 5.2 Example for competing leading six terms in large-B expansion (5.132) at B = 105 B0 ≈ 2.35 × 1014 G. (1/π)ln2 B 42.1912

−(4/π)ln B lnln B

(4/π) ln2 ln B

−35.8181

7.6019

−(4b/π) lnln B

[2(b + 2)/π] ln B

b2 /π

3.3098

0.7632

4.8173

30 Landau: − log2 B 2 /2

ǫ(B)/2Ryd

10

20

8 6

10

4 2

0

0

1000

500

2000

1000

1500

2000

B/B0

Figure 5.11 First-order variational result for binding energy (5.133) as a function of the strength of the magnetic field. The dots indicate the values derived in the reference given in Ref. [12]. The long-dashed curve on the left-hand side shows the simple estimate 0.5 ln2 B of the textbook by Landau and Lifshitz [11]. The right-hand side shows the successive approximations from the strong-field expansion (5.134) for N = 0, 1, 2, 3, 4, with decreasing dash length. Fat curve is our variational approximation.

The convergence of the expansion (5.132) is quite slow. At a magnetic field strength B = 105 B0 , which corresponds to 2.35 × 1010 T = 2.35 × 1014 G, the contribution from the first six terms is 22.87 [2 Ryd]. The next three terms suppressed by a factor ln−1 B contribute −2.29 [2 Ryd], while an estimate for the ln−2 B-terms yields nearly −0.3 [2 Ryd]. Thus we find ε(1) (105 ) = 20.58 ± 0.3 [2 Ryd]. Table 5.2 lists the values of the first six terms of Eq. (5.132). This shows in particular the significance of the second term in (5.132), which is of the same order of the leading first term, but with an opposite sign. The field dependence of the binding energy

ε(B) ≡

B − E (0) 2

(5.133)

is plotted in Fig. 5.11, where it is compared with the results of other authors who used completely different methods, with satisfactory agreement [12]. On the strongcoupling side we have plotted successive orders of a strong-field expansion [24]. The H. Kleinert, PATH INTEGRALS

397


curves result fromqan iterative solution of the sequence of implicit equations for the quantity w(B) = ǫ(4B)/2 for N = 1, 2, 3, 4: w=

N X 1 B log 2 + an (B, w), 2 w n=1

(5.134)

where s

1 π2 2 B √ log − πw , a4 = a1 ≡ − (γ +log 2), a2 ≡ , a3 ≡ − 2 12w B 2w 2

s

√ 2 π D, B 2 (5.135)

√ R and D denotes the integral D ≡ γ −2 0∞ dy (y/ y 2 + 1−1) log y ≈ −0.03648, where γ ≈ 0.5773 is the Euler-Mascheroni constant (2.461). Our results are of similar accuracy as those of other first-order calculations based on an operator optimization method [25]. The advantage of our variational approach is that it yields good results for all magnetic field strengths and temperatures, and that it can be improved systematically by methods to be developed in Section 5.13, with rapid convergence. The figure shows also the energy of Landau and Lifshitz which grossly overestimates the binding energies even at very large magnetic fields, such as 2000B0 ∝ 1012 G. Obviously, the nonleading terms in Eq. (5.132) give important contributions to the asymptotic behavior even at such large magnetic fields. As an peculiar property of the asymptotic behavior, the absolute value of the difference between the Landau-Lifshitz result and our approximation (5.132) diverges with increasing magnetic field strengths B. Only the relative difference decreases.

5.11.1

Weak-Field Behavior

Let us also calculate the weak-field behavior of the variational energy (5.127). Setting η ≡ Ωk /Ω⊥ , we rewrite W1 (0) as s √ Ω⊥ η B2 ηΩ⊥ 1 1− 1−η √ √ W1 (0) = 1+ + + ln . (5.136) 2 2 8Ω⊥ π 1−η 1+ 1−η This is minimized in η and Ω⊥ by expanding η(B) and Ω(B) in powers of B 2 with unknown coefficients, and inserting these expansions into extremality equations. The expansion coefficients are then determined order by order. The optimal expansions are inserted into (5.136), yielding the optimized binding energy ε(1) (B) as a power series ∞ W1 (0) =

X

εn B 2n .

(5.137)

n=0

The coefficients εn are listed in Table 5.3 and compared with the exact ones. Of course, the higher-order coefficients of this first-order variational approximation become rapidly inaccurate, but the results can be improved, if desired, by going to higher orders in variational perturbation theory of Section 5.13.

398


Table 5.3 Perturbation coefficients up to order B 6 in weak-field expansions of variational parameters, and binding energy in comparison to exact ones (from J.E. Avron et al. and B.G. Adams et al. quoted in Notes and References). n

0

ηn Ωn εn

1.0 32 ≈ 1.1318 9π 4 − ≈ −0.4244 3π

εex n

5.11.2

1

2

3

405π 2 − ≈ −0.5576 7168 99π ≈ 1.3885 224 9π ≈ 0.2209 128

16828965π 4 ≈ 1.3023 1258815488 1293975π 3 − ≈ −2.03982 19668992 8019π 3 − ≈ −0.1355 1835008 53 − ≈ −0.2760 192

3886999332075π 6 − ≈ −4.2260 884272562962432 524431667187π 5 ≈ 5.8077 27633517592576 256449807π 5 ≈ 0.2435 322256764928 5581 ≈ 1.2112 4608

−0.5

0.25

Effective Classical Hamiltonian

The quantum statistical properties of the system at an arbitrary temperature are contained in the effective classical potential H eff cl (p0 , x0 ) defined by the threedimensional version of Eq. (3.821): D 3 p (3) δ (x0 − x)(2π¯ h)3 δ(p0 − p) e−A[p,x]/¯h , (2π¯ h)3 (5.138) where Ae [p, x] is the Euclidean action B(p0 , x0 ) ≡ e−βH

≡

I

D3x

Ae [p, x] =

Z

hβ ¯

eff cl (p

0 ,x0 )

0

I

˙ ) + H(p(τ ), x(τ ))], dτ [−ip(τ )x(τ

(5.139)

and x = 0¯hβ dτ x(τ )/¯ hβ and p = 0¯hβ dτ p(τ )/¯ hβ are the temporal averages of position and momentum. Note that the deviations of p(τ ) from the average p0 share with x(τ )−x0 the property that the averages of the squares go to zero with increasing temperatures like 1/T , and remains finite for T → 0. while the expectation of p2 grows linearly with T (Dulong-Petit law). For T → 0, the averages of the squares of p(τ ) remain finite. This property is the basis for a reliable accuracy of the variational treatment. Thus we separate the action (5.139) (omitting the subscript e) as R

R

Ae [p, x] = βH(p0 , x0 ) + ApΩ0 ,x0 [p, x] + Aint [p, x],

(5.140)

where ApΩ0 ,x0 [p, x] is the most general harmonic trial action containing the magnetic field. It has the form (3.825), except that we use capital frequencies to emphasize that they are now variational parameters: 1 [p(τ ) − p0 ]2 2M 0 o M 2 M +ΩB lz (p(τ )−p0 , x(τ )−x0 ) + Ω⊥ [x⊥ (τ )−x0⊥ ]2 + Ω2k [z(τ ) − z0 ]2 . (5.141) 2 2

ApΩ0 ,x0 [p, x]

=

Z

hβ ¯

dτ

n

˙ )+ − i[p(τ ) − p0 ] · x(τ

The vector x⊥ = (x, y) is the projection of x orthogonal to B. H. Kleinert, PATH INTEGRALS

399


The trial frequencies Ω = (ΩB , Ω⊥ , Ωk ) are arbitrary functions of p0 , x0 , and B. Inserting the decomposition (5.140) into (5.138), we expand the exponential of the interaction, exp {−Aint [p, x]/¯ h}, and obtain a series of expectation values of powers p0 ,x0 of the interaction h Anint [p, x] iΩ , defined in general by the path integral 1

h O[p, x] ipΩ0 ,x0 =

p0 ,x0 ZΩ

I

D3x

D3p O[p, x] δ (3) (x0 − x)(2π¯ h)3 δ(p0 − p) (2π¯ h)3

1 p0 ,x0 × exp − AΩ [p, x] , h ¯

(5.142)

p0 ,x0 where the local partition function in phase space ZΩ is the normalization factor p0 ,x0 which ensures that h 1 iΩ = 1. From Eq. (3.826) we know that p0 ,x0 ZΩ ≡ e−βF

p0 ,x0

= le3 (¯ hβ)

h ¯ βΩk /2 h ¯ βΩ− /2 h ¯ βΩ+ /2 , sinh h ¯ βΩ+ /2 sinh h ¯ βΩ− /2 sinh h ¯ βΩk /2

(5.143)

where Ω± ≡ ΩB ± Ω⊥ . In comparison to (3.826), the classical Boltzmann factor e−βH(p0 ,x0 ) is absent due to the shift of the integration variables in the action (5.141). Note that the fluctuations p(τ ) − p0 decouple from p0 just as x(τ ) − x0 decoupled from x0 due to the absence of zero frequencies in the fluctuations. Rewriting the perturbation series as a cumulant expansion, evaluating the expectation values, and integrating out the momenta on the right-hand side of Eq. (5.138) leads to a series representation for the effective classical potential Veff (x0 ). Since it is impossible to sum up the series, the perturbation expansion (N ) must be truncated, leading to an Nth-order approximation WΩ (x0 ) for the ef(N ) fective classical potential. Since the parameters Ω are arbitrary, WΩ (x0 ) should depend minimally on Ω. This determines the optimal values of Ω to be equal (N ) (N ) (N ) to Ω(N ) (x0 ) = (ΩB (x0 ), Ω⊥ (x0 ), Ωk (x0 )) of Nth order. Reinserting these into (N )

(N )

WΩ (x0 ) yields the optimal approximation W (N ) (x0 ) ≡ WΩ(N) (x0 ). The first-order approximation to the effective classical potential is then, with ωk = 0, ω⊥ = ωB , (1)

M ΩB (x0 )[ωB − ΩB (x0 )] b2⊥ (x0 ) a2⊥ (x0 ) 2 * +p0 ,x0 i Mh 2 1 2 2 e2 2 + ωB − Ω⊥ (x0 ) − Ωk ak (x0 ) − . 2 2 |x| Ω

WΩ (x0 ) = FΩp0 ,x0 −

(5.144)

The smearing of the Coulomb potential is performed as in Section 5.10. This yields the result (5.117). with the longitudinal width a2k (x0 )

=

x0 G(2) (τ, τ ) zz

"

#

h ¯ βΩk (x0 ) h ¯ βΩk (x0 ) 1 = coth −1 , 2 βMΩk (x0 ) 2 2

and an analog transverse width.

(5.145)

400


3

2

1

0

−1

0

5

10

15

20

Figure 5.12 Effective classical potential of atom in strong q magnetic field plotted along two directions: once as a function of the coordinate ρ0 = x20 + y02 perpendicular to the field lines at z0 = 0 (solid curves), and once parallel to the magnetic field as a function of z0 at ρ0 = 0 (dashed curves). The inverse temperature is fixed at β = 100, and the strengths of the magnetic field B are varied (all in natural units).

The quantity b2⊥ (x0 ) is new in this discussion based on the canonical path integral. It denotes the expectation value associated with the z-component of the angular momentum 1 b2⊥ (x0 ) ≡ h lz ipΩ0 ,x0 , (5.146) MΩB which can also be written as b2⊥ (x0 ) =

2 h x(τ )py (τ ) ipΩ0 ,x0 . MΩT 1

(5.147)

According to Eq. (3.355), the correlation function h x(τ )py (τ ) ipΩ0 ,x0 is given by p ,x0

h x(τ )py (τ ′ ) iΩ0

(2)

(2)

= iM∂τ ′ Gω2 ,B,xx (τ, τ ′ ) − MωB Gω2 ,B,xy (τ, τ ′ ),

(5.148)

where the expressions on the right-hand side are those of Eqs. (3.326) and (3.328), with ω replaced by Ω. The variational energy (5.144) is minimized at each x0 , and the resulting (N ) W (x0 ) is displayed for a low temperature and different magnetic fields in Fig. 5.12. The plots show the anisotropy with respect to the magnetic field direction. The anisotropy grows when lowering the temperature and increasing the field strength. Far away from the proton at the origin, the potential becomes isotropic, due to the decreasing influence of the Coulomb interaction. Analytically, this is seen by going H. Kleinert, PATH INTEGRALS

401

5.12 Variational Approach to Excitation Energies

to the limits ρ0 → ∞ or z0 → ∞, where the expectation value of the Coulomb potential tends to zero, leaving an effective classical potential M (1) 2 WΩ (x0 ) → FΩp0 ,x0 − MΩB (ω⊥ − ΩB ) b2⊥ + M ω⊥ − Ω2⊥ a2⊥ − Ω2k a2k . (5.149) 2 (1)

(1)

This is x0 -independent, and optimization yields the constants ΩB = Ω⊥ = ωB and (1) Ωk = 0, with the asymptotic energy 1 β¯ hωB W (1) (x0 ) → − log . (5.150) β sinh β¯ hωB The B = 0 -curves agree, of course, with those obtained from the previous variational perturbation theory of the hydrogen atom [26]. For large temperatures, the anisotropy decreases since the violent thermal fluctuations have a smaller preference of the z-direction.

5.12

Variational Approach to Excitation Energies

As explained in Section 5.4, the success of the above variational treatment is rooted in the fact that for smooth potentials, the ground state energy can be approximated quite well by the optimal expectation value of the Hamiltonian operators in a Gaussian wave packet. The question arises as to whether the energies of excited states can also be obtained by calculating an optimized expectation value between excited oscillator wave functions. If the potential shape has only a rough similarity with that of a harmonic oscillator, when there are no multiple minima, the answer is positive. Consider again the anharmonic oscillator with the action Z ¯hβ M 2 1 4 2 2 A[x] = dτ x˙ + ω x + gx . (5.151) 2 4 0 As for the ground state, we replace the action A by AxΩ0 +Axint0 with the trial oscillator action centered around the arbitrary point x0 AxΩ0

=

1 Ω2 dτ M x˙ 2 + (x − x0 )2 , 2 2 #

(5.152)

ω 2 − Ω2 g dτ M (x − x0 )2 + x4 2 4

(5.153)

Z

hβ ¯

0

"

and a remainder Axint0

=

Z

0

β

"

#

(n)

to be treated as an interaction. Let ψΩ (x − x0 ) be the wave functions of the trial oscillator.1 With these, we form the projections ZΩ (x0 ) ≡

Z

dxb dxa ψΩ (xb )(xb τb |xa τa )ψΩ (xa )

=

Z

(n)∗ dxb dxa ψΩ (xb )

(n)

1

(n)∗

(n)

Z

(xa ,0);(xb ,¯ hβ)

!

(n)

Dxe−A/¯h ψΩ (xa ).

(5.154)

In contrast to the earlier notation, we now use superscripts in parentheses to indicate the principal quantum numbers. The subscripts specify the level of approximation.

402


If the temperature tends to zero, an optimization in the parameters x0 and Ω(x0 ) should yield information on the energy E (n) of this state by containing an exponentially decreasing function (n) Z (n) ≈ e−βE . (5.155) (n)

Since the trial wave functions ψΩ (x−x0 ) are not the true ones, the behavior (5.155) (n′ ) contains an admixture of Boltzmann factors e−βE with n′ 6= n, which have to be eliminated. They are easily recognized by the powers of g which they carry. The calculation of (5.154) is done in the same approximation that rendered good results for the ground state, i.e., we approximate the part of Z (n) behaving like (5.155) as follows: x0

(n)

Z (n) ≈ ZΩ e−hAint /¯hiΩ , (n)

(5.156)

(n)

where ZΩ denotes the contribution of the nth excited state to the oscillator partition function (n) (5.157) ZΩ = e−β¯hΩ(n+1/2) , (n)

x0 and hAxint0 /¯ hi(n) h in the state ψΩ (x − x0 ). This Ω stands for the expectation of Aint /¯ approximation corresponds precisely to the first term in the perturbation expansion (3.506) (after continuing to imaginary times). Note that the approximation on the right-hand side of (5.156) is not necessarily smaller than the left-hand side, as in the Jensen-Peierls inequality (5.10), since the measure of integration in (5.154) is no longer positive. For the action (5.151), the best value of x0 lies at the coordinate origin. This 2k simplifies the calculation of the expectation hAxint0 /¯ hi(n) in Ω . The expectation of x (n) the state ψΩ (x) is given by

D

x2k

E(n) Ω

=

h ¯k n2k , (MΩ)k

(5.158)

where 3 5 n2 = (n + 1/2), n4 = (n2 + n + 1/2), n6 = (2n3 + 3n2 + 4n + 3/2), 2 4 1 n8 = (70n4 + 140n3 + 344n2 + 280n + 105), . . . . (5.159) 16 After inserting (5.153) into (5.156), the expectations (5.158) yield the approximation Z

(n)

h 1 2 ¯ n2 g h ¯ 2 n4 ≈ exp −β Ω + ω2 + 2 Ω 4 M 2 Ω2 (

"

#)

.

(5.160)

h/M 2 ω 3, this corresponds to the With the dimensionless coupling constant g ′ ≡ g¯ variational energies W

(n)

1 ω2 g′ ω3 =h ¯ Ω+ n2 + n4 . 2 Ω 4 Ω2 "

!

#

(5.161)


403

5.12 Variational Approach to Excitation Energies

They are optimized by the extremal Ω-values (n)

Ω = Ω1 =

    

where

√2 3

ω cosh

√2 3

ω cos

h

1 3

h

arcosh (g/g (n))

1 3

arccos(g/g (n))

i i

for

g > g (n) , g < g (n) ,

2 n2 M 2 ω 3 . g (n) ≡ √ ¯ 3 3 n4 h

(5.162)

(5.163)

(n) The optimized W (n) yield the desired approximations Eapp for the excited energies of the anharmonic oscillator. For large g, the trial frequency grows like

(n)

1/3

g¯ h 4M 2 ω 3

!1/3

Ω

≡6

n ˜,

(5.164)

n ˜≡

n2 + n + 1/2 4n4 /3 = , 2n2 n + 1/2

(5.165)

where

(n) grow like making the energy Eapp

(n) EBS

(n)

→h ¯ ωκ

g¯ h 4M 2 ω 3

!1/3

,

for large g,

(5.166)

with κ(n) =

3 · 61/3 ˜ 1/3 . (2n + 1) n ˜ 1/3 ≈ 0.68142 × (2n + 1) n 8

(5.167)

For n = 0, this is in good agreement with the precise growth behavior to be calculated in Section 5.16 where we shall find κ(0) ex = 0.667 986 . . . (see Table 5.9). In the limit of large g and n, this can be compared with the exact behavior obtained from the semiclassical approximation of Bohr and Sommerfeld, which Table 5.4 Approach of variational energies of nth excited state to Bohr-Sommerfeld approximation with increasing n. Values in the last column converge rapidly towards the Bohr-Sommerfeld value 0.688 253 702 . . . in Eq. (4.32). n 0 2 4 6 8 10

E (n) /(g¯h/4M 2 ) 0.667 986 259 155 777 108 3 4.696 795 386 863 646 196 2 10.244 308 455 438 771 076 0 16.711 890 073 897 950 947 1 23.889 993 634 572 505 935 5 31.659 456 477 221 552 442 8

κ(n) /(n + 1/2)4/3 0.841 609 948 112 105 001 0.692 125 685 914 981 314 0.689 449 772 359 340 765 0.688 828 486 600 234 466 0.688 590 146 947 993 676 0.688 474 290 179 981 433 1 2

404


Table 5.5 Energies of the nth excited states of anharmonic oscillator ω 2 x2 /2 + gx4 /4 for various coupling strengths g (in natural units). In each entry, the upper number shows the energies obtained from a numerical integration of the Schr¨ odinger equation, whereas the lower number is our variational result. E (0)

g/4

E (1)

E (2)

E (3)

E (4)

E (5)

E (6)

E (7)

E (8)

0.1

0.559 146 0.560 307

1.769 50 1.773 39

3.138 62 3.138 24

4.628 88 4.621 93

6.220 30 6.205 19

7.899 77 7.875 22

9.657 84 9.622 76

11.4873 11.4407

13.3790 13.3235

0.2

0.602 405 0.604 901

1.950 54 1.958 04

3.536 30 3.534 89

5.291 27 5.278 55

7.184 46 7.158 70

9.196 34 9.156 13

11.313 2 11.257 3

13.5249 13.4522

15.8222 15.7328

0.3

0.637 992 0.641 630

2.094 64 2.104 98

3.844 78 3.842 40

5.796 57 5.779 48

7.911 75 7.878 23

10.1665 10.1151

12.5443 12.4736

15.0328 14.9417

17.6224 17.5099

0.4

0.668 773 0.673 394

2.216 93 2.229 62

4.102 84 4.099 59

6.215 59 6.194 95

8.511 41 8.471 69

10.9631 10.9028

13.5520 13.4698

16.2642 16.1588

19.0889 18.9591

0.5

0.696 176 0.701 667

2.324 41 2.339 19

4.327 52 4.323 52

6.578 40 6.554 75

9.028 78 8.983 83

11.6487 11.5809

14.4177 14.3257

17.3220 17.2029

20.3452 20.2009

0.6

0.721 039 0.727 296

2.421 02 2.437 50

4.528 12 4.523 43

6.901 05 6.874 77

9.487 73 9.438 25

12.2557 12.1816

15.1832 15.0828

18.2535 18.1256

21.4542 21.2974

0.7

0.743 904 0.750 859

2.509 23 2.527 29

4.710 33 4.705 01

7.193 27 7.164 64

9.902 61 9.849 11

12.8039 12.7240

15.8737 15.7658

19.0945 18.9573

22.4530 22.2852

0.8

0.765 144 0.772 736

2.590 70 2.610 21

4.877 93 4.872 04

7.461 45 7.430 71

10.2828 10.2257

13.3057 13.2206

16.5053 16.3907

19.8634 19.7179

23.3658 23.1880

0.9

0.785 032 0.793 213

2.666 63 2.687 45

5.033 60 5.027 18

7.710 07 7.677 39

10.6349 10.5744

13.7700 13.6801

17.0894 16.9687

20.5740 20.4209

24.2091 24.0221

1

0.803 771 0.812 500

2.737 89 2.759 94

5.179 29 5.172 37

7.942 40 7.907 93

10.9636 10.9000

14.2031 14.1090

17.6340 17.5076

21.2364 21.0763

24.9950 24.7996

10

1.504 97 1.531 25

5.321 61 5.382 13

10.3471 10.3244

16.0901 15.9993

22.4088 22.2484

29.2115 28.9793

36.4369 36.1301

44.0401 43.6559

51.9865 51.5221

50

2.499 71 2.547 58

8.915 10 9.023 38

17.4370 17.3952

27.1926 27.0314

37.9385 37.6562

49.5164 49.1094

61.8203 61.2842

74.7728 74.1029

88.3143 87.5059

100

3.131 38 3.192 44

11.1873 11.3249

21.9069 21.8535

34.1825 33.9779

47.7072 47.3495

62.2812 61.7660

77.7708 77.0924

94.0780 93.2307

111.128 110.106

500

5.319 89 5.425 76

19.0434 19.2811

37.3407 37.2477

58.3016 57.9489

81.4012 80.7856

106.297 105.411

132.760 131.595

160.622 159.167

189.756 188.001

1000

6.694 22 6.827 95

23.9722 24.2721

47.0173 46.9000

73.4191 72.9741

102.516 101.740

133.877 132.760

167.212 165.743

202.311 200.476

239.012 236.799

gives the same leading powers in g and n as in (5.166), but a 3% larger prefactor 0.688 253 702 × 2 [recall (4.32)]. The exact values of E (n) /(g¯ h/4M 2 ) and 1 (n) /(n + 1/2)4/3 are for n = 0, 2, 4, 6, 8, 10 are shown in Table 5.4. 2κ (n) A comparison of the approximate energies Eapp with the precise numerical solutions of the Schrödinger equation in natural units h ¯ = 1, M = 1 in Table 5.5 shows an excellent agreement for all coupling strengths. (n) Near the strong-coupling limit, the optimal frequencies Ω1 and the approximate (n) energies E1 behave as follows [in natural units; compare (5.76) and (5.77)]: (n) Ω1

= 6

1/3

n ˜g 4

1/3 "

1/3

1 3 1+ 9 4

#

1 + ... , (˜ ng/4)2/3 H. Kleinert, PATH INTEGRALS

405

5.13 Systematic Improvement of Feynman-Kleinert Approximation . . .

(n) E1

4/3

3 4

=

n ˜g (2n + 1) 4

1/3 "

1 61/3 1+ + ... . 9 (˜ ng/4)2/3 #

(5.168)

The tabulated energies can be used to calculate an approximate partition function at all temperatures: Z≈

∞ X

(n)

e−βEapp .

(5.169)

n=0

The resulting free energies F1′ = − log Z/β agree well with the previous variational results of the Feynman-Kleinert approximation — in the plots of Fig. 5.2, the curves are indistinguishable. This is not astonishing since both approximations are dom(0) inated near zero temperature by the same optimal energy Eapp , while approaching the semiclassical behavior at high temperatures. The previous free energy F1 does so exactly, the free energy F1′ to a very good approximation. (n) By combining the Boltzmann factors with the oscillator wave functions ψΩ (x), we also calculate the density matrix (xb τb |xa τa ) and the distribution functions ρ(x) = (x h ¯ β|x 0) in this approximation: (xb τb |xa τa ) ≈

∞ X

(n)

ψ (n) (xb )ψ (n)∗ (xa )e−βEapp .

(5.170)

n=0

They are in general less accurate than the earlier calculated particle density ρ1 (x0 ) of (5.94).

5.13

Systematic Improvement of Feynman-Kleinert Approximation. Variational Perturbation Theory

A systematic improvement of the variational approach leads to a convergent variational perturbation expansion for the effective classical potential of a quantummechanical system [27]. To derive it, we expand the action in powers of the deviations of the path from its average x0 = x¯: δx(τ ) ≡ x(τ ) − x0 .

(5.171)

A = V (x0 ) + AxΩ0 + A′xint0 ,

(5.172)

The expansion reads where AxΩ0 is the quadratic action of the deviations δx(τ ) AxΩ0

=

Z

h/kB T ¯ 0

dτ

o Mn [δ x(τ ˙ )]2 + Ω2 (x0 )[δx(τ )]2 , 2

(5.173)

and A′xint0 contains all higher powers in δx(τ ): A′xint0

=

Z

0

hβ ¯

g2 g3 g4 dτ [δx(τ )]2 + [δx(τ )]3 + [δx(τ )]4 + . . . . 2! 3! 4!

(5.174)

406


The coupling constants are, in general, x0 -dependent: gi (x0 ) = V (i) (x0 ) − Ω2 δi2 ,

V (i) (x0 ) ≡

di V (x0 ) . dxi0

(5.175)

For the anharmonic oscillator, they take the values g2 (x0 ) = M[ω 2 − Ω2 (x0 )] + 3gx20 , g3 (x0 ) = 6gx0 , g4 (x0 ) = 6g.

(5.176)

Introducing the parameter r 2 = 2M(ω 2 − Ω2 )/g,

(5.177)

which has the dimension length square, we write g2 (x0 ) as g2 (x0 ) = g(r 2/2 + 3x20 ).

(5.178)

With the decomposition (5.172), the Feynman-Kleinert approximation (5.32) to the effective classical potential can be written as W1Ω (x0 ) = V (x0 ) + FΩx0 +

1 hAxint0 ixΩ0 . h ¯β

(5.179)

To generalize this, we replace the local free energy FΩx0 by FΩx0 +∆F x0 , where ∆F x0 denotes the local analog of the cumulant expansion (3.484). This leads to the variational perturbation expansion for the effective classical potential 1 hAxint0 ixΩ0 h ¯β Ex0 1 D 1 D x0 3 Ex0 − 2 Axint0 2 + Aint + ... Ω,c Ω,c 2!¯ hβ 3!¯ h3 β

V eff cl (x0 ) = V (x0 ) + FΩx0 +

(5.180)

in terms of the connected expectation values of powers of the interaction: D

Axint0 2

Ex0

Ω,c D E x0 3 x0 Aint Ω,c

≡ ≡ .. .

D

Axint0 2

Ex0

Ω D E x0 3 x0 Aint Ω

− hAxint0 ixΩ0 2 , D

− 3 Axint0 2

Ex0 Ω

(5.181) hAxint0 ixΩ0 + 2 hAxint0 ixΩ0 3 ,

(5.182)

.

By construction, the infinite sum (5.180) is independent of the choice of the trial frequency Ω(x0 ). When truncating (5.180) after the Nth order, we obtain the approximation WNΩ (x0 ) to the effective classical potential V eff cl (x0 ). In many applications, it is sufficient to work with the third-order approximation 1 x hA′xint0 iΩ0 h ¯β Ex0 1 D ′x0 2 Ex0 1 D − 2 A int + 3 A′xint0 3 . Ω,c Ω,c 2¯ hβ 6¯ hβ

W3Ω (x0 ) = V (x0 ) + FΩx0 +

(5.183)


5.13 Systematic Improvement of Feynman-Kleinert Approximation . . .

407

In contrast to (5.180), the truncated sums WN (x0 ) do depend on Ω(x0 ). Since the infinite sum is Ω(x0 )-independent, the best truncated sum WN (x0 ) should lie at the frequency ΩN (x0 ) where WNΩ (x0 ) depends minimally on it. The optimal ΩN (x0 ) is called the frequency of least dependence. Thus we require for (5.183) ∂W3Ω (x0 ) = 0. ∂Ω(x0 )

(5.184)

At the frequency Ω3 (x0 ) fixed by this condition, Eq. (5.183) yields the desired thirdorder approximation W3 (x0 ) to the effective classical potential. The explicit calculation of the expectation values on the right-hand side of (5.183) proceeds according to the rules of Section 3.18. We have to evaluate the Feynman integrals associated with all vacuum diagrams. These are composed of p-particle vertices carrying the coupling constant gp /p!¯ h, and of lines representing the correlation function of the fluctuations introduced in (5.23): h ¯ ′x0 G (τ − τ ′ ) M Ω = a2 (τ − τ ′ , x0 ).

x

hδx(τ )δx(τ ′ )iΩ0 =

(5.185)

Since this correlation function contains no zero frequency, diagrams of the type shown in Fig. 5.13 do not contribute. Their characteristic property is to fall apart when cutting a single line. They are called one-particle reducible diagrams. A vacuum subdiagram connected with the remainder by a single line is called a tadpole diagram, alluding to its biological shape. Tadpole diagrams do not contribute to the variational perturbation expansion since they vanish as a consequence of energy conservation: the connecting line ending in the vacuum, must have a vanishing frequency where the spectral representation of the correlation function hδx(τ )δx(τ ′ )ixΩ0 has no support, by construction. The number of Feynman diagrams to be evaluated is reduced by ignoring at first all diagrams containing the vertices g2 . The omitted diagrams can be recovered from the diagrams without g2 -vertices by calculating the latter at the initial frequency ω, and by replacing ω by a modified local trial frequency, ˜ 0) ≡ ω → Ω(x

q

Ω2 (x0 ) + g2 (x0 )/M.

(5.186)

Figure 5.13 Structure of a one-particle reducible vacuum diagram. The dashed box encloses a so-called tadpole diagram. Such diagrams vanish in the present expansion since an ending line cannot carry any energy and since the correlation function hδx(τ )δx(τ ′ )i contains no zero frequency.

408


After this replacement, which will be referred to as the square-root trick , all diagrams are re-expanded in powers of g [remembering that g2 (x0 ) is by (5.178) proportional to g] up to the maximal power g 3 .

5.14

Applications of Variational Perturbation Expansion

The third-order approximation W3 (x0 ) is far more accurate than W1 (x0 ). This will now be illustrated by performing the variational perturbation expansion for the anharmonic oscillator and the double-well potential. The reason for the great increase in accuracy will become clear in Section 5.15, where we shall demonstrate that this expansion converges rapidly towards the exact result at all coupling strengths, in contrast to ordinary perturbation expansions which diverges even for arbitrarily small values of g.

5.14.1

Anharmonic Oscillator at T = 0

Consider first the case of zero temperature, where the calculation is simplest and the approximation should be the worst. At T = 0, only the point x0 = 0 contributes, and δx(τ ) coincides with the path itself ≡ x(τ ). Thus we may omit the superscript x0 in all equations, so that the interaction in (5.182) becomes writing it as Aint

g = 4

Z

hβ ¯

0

dτ (r 2 x2 + x4 ).

(5.187)

The effect of the r 2 -term is found by replacing the frequency ω in the original perturbation expansion for the anharmonic oscillator according to the square-root trick (5.186), which for x0 = 0 is simply ˜≡ ω→Ω

q

Ω2 + (ω 2 − Ω2 ) =

q

Ω2 + gr 2 /2M.

(5.188)

After this replacement, all terms are re-expanded in powers of g. Finally, r 2 is again replaced by 2M 2 (ω − Ω2 ). (5.189) r2 → g Since the interaction is even in x, the zero-temperature expansion is automatically free of tadpole diagrams. The perturbation expansion to third order was given in Eq. (3.551). With the above replacement it leads to the free energy F =

˜ h ¯Ω g g + 3a4 + 2 4 4

2

42a8

3

1 g + ˜ 4 h ¯Ω

with a2 =

h ¯ . ˜ 2M Ω

4 · 333a12

1 ˜ h ¯Ω

2

,

(5.190)


409

5.14 Applications of Variational Perturbation Expansion

Table 5.6 Second- and third-order approximations to ground state energy, in units of hω, of anharmonic oscillator at various coupling constants g in comparison with exact ¯ (0) (0) values Eex (g) and the Feynman-Kleinert approximation E1 (g) of previous section. g/4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 50 100 500 1000

(0)

Eex (g) 0.559146 0.602405 0.637992 0.668773 0.696176 0.721039 0.743904 0.765144 0.785032 0.803771 1.50497 2.49971 3.13138 5.31989 6.69422

(0)

E1 (g) 0.560307371 0.604900748 0.641629862 0.673394715 0.701661643 0.727295668 0.750857818 0.772736359 0.793213066 0.812500000 1.53125000 2.54758040 3.19244404 5.42575605 6.82795331

(0)

E2 (g) 0.559152139 0.602450713 0.638088735 0.668922455 0.696376950 0.721288789 0.744199436 0.765483301 0.785412037 0.804190095 1.50674000 2.50312133 3.13578530 5.32761969 6.70400326

(0)

E3 (g) 0.559154219 0.602430621 0.638035760 0.668834137 0.696253632 0.721131776 0.744010317 0.765263697 0.785163494 0.803914053 1.50549750 2.50069963 3.13265656 5.32211709 6.69703286

The higher orders can most easily be calculated with the help of the Bender-Wu recursion relations derived in Appendix 3C.2 By expanding F in powers of g up to g 3 we obtain h ¯Ω + g(3a4 − r 2 a2 )/4 − g 2(21a8 /8 + 3a6 r 2 /4 + a4 r 4 /16)/¯hΩ 2 +g 3 (333a12 /16 + 105a10 r 2 /16 + 3a8 r 4 /4 + a6 r 6 /32)/(¯ hΩ)2 .

W3Ω =

(5.192)

After the replacement (5.189), we minimize W3 in Ω, and obtain the third-order (0) approximation E3 (g) to the ground state energy. Its accuracy at various coupling strengths is seen in Table 5.6 where it is compared with the exact values obtained from numerical solutions of the Schrödinger equation. The improvement with respect to the earlier approximation W1 is roughly a factor 50. The maximal error is now smaller than 0.05%. We shall see in the last subsection that up to rather high orders N, the minimum happens to be unique. Observe that when truncating the expansion (5.183) after the second order and working with the approximation W2Ω (x0 ), there exists no minimum in Ω, as can be seen in Fig. 5.14. The reason for this is the alternating sign of the cumulants in (5.183). This gives an alternating sign to the highest power a and thus to the highest power of 1/Ω in the g n -terms of Eq. (5.192), causing the trial energy of order N to diverge for Ω → 0 like (−1)N −1 g N × (1/Ω)3N −1 . Since the trial energy goes for 2

The Mathematica program is available on the internet under http://www.physik.fu-berlin/ .de~kleinert/294/programs.

410


0.8 0.75

N =1

0.7

N =2

0.65 N =3

0.6

W

0.55 0.5 0.45 0.4

1

2

1.5

2.5

3

Ω

Figure 5.14 Typical Ω-dependence of approximations W1,2,3 at T = 0. The coupling constant has the value g = 0.4. The second-order approximation W2Ω has no extremum. Here the minimal Ω-dependence lies at the turning point, and the condition ∂ 2 W2Ω /∂Ω2 renders the best approximation to the energy (short dashes).

large Ω to positive infinity, only the odd approximations are guaranteed to possess a minimum. The second-order approximation W2 can nevertheless be used to find an improved energy value. As shown by Fig. 5.14, the frequency of least dependence Ω2 is well defined. It is the frequency where the Ω-dependence of W2 has its minimal absolute value. Thus we optimize Ω with the condition ∂ 2 W2Ω = 0. ∂Ω2

(5.193)

(0)

This leads to the energy values E2 (g) listed in Table 5.6. They are more accurate (0) than the values E1 (g) by an order of magnitude.

5.14.2

Anharmonic Oscillator for T > 0

Consider now the anharmonic oscillator at a finite temperature, where the expansion (5.183) consists of the sum of one-particle irreducible vacuum diagrams ˜ W3Ω

1 = β

(

1 − 2

1 − 2!

˜ Ω

˜ Ω

+ 3 ˜ Ω

˜ Ω

+ 72

˜ Ω

+ 24

6

!

(5.194)


411


1 + 3!

+

+

2592

˜ Ω

˜ Ω

˜ Ω

˜ Ω

1728

˜ Ω

+

˜ Ω

+

3456

+

.

648

648

1728

!)

The vertices represent the couplings g3 (x0 )/3!¯ h, g4 (x0 )/4!¯ h, whereas the lines stand for the correlation function G′ xΩ˜0 (τ, τ ′ ). The numbers under each diagram are their multiplicities acting as factors. Only the five integrals associated with the diagrams ;

;

;

;

˜ or by factorneed to be evaluated explicitly; all others arise by the expansion of Ω ization. The explicit form of three of these integrals can be found in the first, forth, and sixth of Eqs. (3.547). The results of the integrations are listed in Appendix 5A. Only the second and fourth diagrams are new since they involve vertices with three legs. They can be found in Eqs. (3D.7) and (3D.8). In quantum field theory one usually calculates Feynman integrals in momentum space. At finite temperatures, this requires the evaluation of multiple sums over Matsubara frequencies. The present quantum-mechanical example corresponds to a D = 1 -dimensional quantum field theory. Here it is more convenient to evaluate the integrals in τ -space. The diagrams ;

,

for example, are found by performing the integrals 2

h ¯ βa

≡

Z

hβ ¯

0

dτ G2 (τ, τ ),

hβ ¯ ¯ hβ ¯ hβ 1 2 12 a3 ≡ G2 (τ1 , τ2 )G2 (τ2 , τ3 )G2 (τ3 , τ1 ) dτ1 dτ2 dτ3 . ω 0 0 0 The factor h ¯ β on the left-hand side is due to an overall τ -integral and reflects the temporal translation symmetry of the system; the factors 1/ω arise from the remaining τ -integrations whose range is limited by the correlation time 1/ω. In general, the β- and x0 -dependent parameters a2L V have the dimension of a length to the nth power and the associated diagrams consist of m vertices and n/2 lines (defining a21 ≡ a2 ). ˜ and expand everything in powers We now use the rule (5.188) to replace ω by Ω of g2 up to the third order. The expansion can be performed diagrammatically in each Feynman diagram. Letting a dot on a line indicate the coupling g2 /2¯ h, the one-loop diagram is expanded as follows:

h ¯β

Z

Z

Z

˜ Ω

=

−2

+

−

1 3

.

(5.195)

412


The other diagrams are expanded likewise: ˜ Ω

=

−

˜ Ω

1 2

+

1 6

+

=

1 2

−

1 6

,

=

1 2

−

1 6

,

=

1 2

1 6

,

˜ Ω

1 2 ˜ Ω

1 2

−

1 6

−

1 6

.

In this way, we obtain from (5.194) the complete graphical expansion for W3Ω (x0 ) including all vertices associated with the coupling g2 (x0 ): βW3Ω (x0 )

1 + 2 1 + 2

1 =− 2 1 1 − 2! 2 1 + 3!

1 + 8

"

1 + 8

1 +3 4

1 + 2



+3

1 4

1 + 24

+



3 +  16

1 4 +

1 + 6

+

1 6

+

1 2

+

1 4

+

1 8

1 8 +

3 4

+

 

#

3 4



.

(5.196)

In the latter diagrams, the vertices represent directly the couplings gn /¯ h. The denominators n! of the previous vertices gn /n!¯ h have been combined with the multiplicities of the diagrams yielding the indicated prefactors. The corresponding analytic expression for W3Ω (x0 ) is g2 2 g4 4 = V (x0 ) + + a + a 2 8 " # 2 2 1 g2 4 g2 g4 4 2 g4 4 4 g42 8 g32 6 − a + a a + a2 a + a2 + + a2 2!¯ hΩ 2 2 2 2 8 24 6 " ! 2 2 1 g g4 g g4 + 2 2 g23a63 + 3 2 (a42 )2 + 2 a63 a2 4 2 3!¯ hΩ

W3Ω (x0 )

FΩx0

(5.197)


413


g2 g42 4 2 2 g2 g42 6 4 g2 g42 10 g2 g32 8 +3 (a2 ) a + aa + a + a 4 4 3 6 3 2 3 !# 3g43 4 2 2 g43 6 6 g43 10 2 g43 12 3g32g4 10 3g32 g4 8 2 + (a a ) + a3 a + a3 a + a3 + a′ + aa . 16 2 8 4 8 4 3 4 3 !

The quantities a2L V are ordered in the same way as the associated diagrams in (5.196). As before, we have omitted the variable x0 in all but the first three terms, for brevity. The optimal trial frequency Ω(x0 ) is found numerically by searching, at each value of x0 , for the real roots of the first derivate of W3Ω (x0 ) with respect to Ω(x0 ). Just as for zero temperature, the solution happens to be unique. By calculating the integral Z3 =

Z

∞ −∞

dx0 −W3 (x0 )/kB T , e le (¯ hβ)

(5.198)

we obtain the approximate free energy 1 F3 = − ln Z3 . β

(5.199)

The results are listed in Table 5.7 for various coupling constants g and temperatures. They are compared with the exact free energy X 1 exp(−βEn ), Fex = − log β n

(5.200)

whose energies En were obtained by numerically solving the Schrödinger equation. We see that to third order, the new approximation yields energies which are better than those of F1 by a factor of 30 to 50. The remaining difference with respect to the exact energies lies in the fourth digit. In the high-temperature limit, all approximations WN (x0 ) tend to the classical result V (x0 ), as they should. Thus, for small β, the approximations W3 (x0 ) and W1 (x0 ) are practically indistinguishable. The accuracy is worst at zero temperature. Using the T → 0 -limits of the Ω Feynman integrals a2L V given in (3.550) and (3D.14), the approximation W3 takes the simple form h ¯ Ω(x0 ) g2 2 g4 4 + a + a 2 8 " 2 # 1 g22 4 g2 g4 6 g32 2 4 g42 7 8 − a + a + a + a (5.201) 2¯ hΩ 2 2 6 3 24 2 1 13 10 33 6 24 8 2 8 2 2 35 10 3 37 12 + g + g g + g g 3a + g g + g g + g , a a a a a 2 3 2 4 2 4 3 4 4 6¯ hΩ2 2 2 3 3 16 64 W3Ω (x0 ) = V (x0 ) +

where a2 = h ¯ /2MΩ. As in (5.197), we have omitted the arguments x0 in Ω and gi , for brevity. At zero temperature, the remaining integral over x0 in the partition function (5.198) receives its only contribution from the point x0 = 0, where W3 (0) is minimal. There it reduces to the energy W3 of Eq. (5.192).

414


g 0.002 0.4 2.0

4.0 20

200 2000

80000

β 2.0 1.0 5.0 1.0 5.0 10.0 1.0 5.0 1.0 5.0 10.0 5.0 0.1 1.0 10.0 0.1 3.0

F1 0.427937 0.226084 0.559155 0.492685 0.699431 0.700934 0.657396 0.809835 1.18102 1.24158 1.24353 2.54587 2.6997 5.40827 5.4525 18.1517 18.501

F3 0.427937 0.226075 0.558678 0.492578 0.696180 0.696285 0.6571051 0.803911 1.17864 1.22516 1.22515 2.50117 2.69834 5.32319 5.3225 18.0470 18.146

Fex 0.427741 0.226074 0.558675 0.492579 0.696118 0.696176 0.6571049 0.803758 1.17863 1.22459 1.22459 2.49971 2.69834 5.31989 5.3199 18.0451 18.137

Table 5.7 Free energy of anharmonic oscillator with potential V (x) = x2 /2 + gx4 /4 for various coupling strengths g and β = 1/kT .

5.15

Convergence of Variational Perturbation Expansion

For a single interaction xp , the approximation WN at zero temperature can easily be carried to high orders [13, 14]. The perturbation coefficients are available exactly from recursion relations, which were derived for the anharmonic oscillator with p = 4 in Appendix 3C. The starting point is the ordinary perturbation expansion for the energy levels of the anharmonic oscillator E

(n)

=ω

∞ X

k=0

(n) Ek

g 4ω 3

k

.

(5.202)

It was remarked in (3C.27) and will be proved in Section 17.10 [see Eq. (17.323)] (n) that the coefficients Ek grow for large k like 1 (n) Ek − − −→ − π

s

6 12n (−3)k Γ(k + n + 1/2). π n!

(5.203)

Using Stirling’s formula3 n! ≈ (2π)1/2 nn−1/2 e−n , this amounts to (n) Ek 3

(5.204)

√ " #n+k 2 3 (−4)n −3(k + n) − − −→ − . π n! e

(5.205)

M. Abramowitz and I. Stegun, op. cit., Formulas 6.1.37 and 6.1.38. H. Kleinert, PATH INTEGRALS

415

5.15 Convergence of Variational Perturbation Expansion 0.8 0.75

N =1 N =3

0.7 W

N =5

0.65 0.6

N =11

1

2

1.5

3

2.5

3.5

4

Ω 0.59 0.58

N =2

N =4

N =6

0.57

N =10

W 0.56 0.55

1

1.5

2

2.5 Ω

3

3.5

4

Figure 5.15 Typical Ω-dependence of N th approximations WN at T = 0 for increasing orders N . The coupling constant has the value g/4 = 0.1. The dashed horizontal line indicates the exact energy.

(n)

Thus, Ek grows faster than any power in k. Such a strong growth implies that the expansion has a zero radius of convergence. It is a manifestation of the fact that the energy possesses an essential singularity in the complex g-plane at the expansion point g = 0. The series is a so-called asymptotic series. The precise form of the singularity will be calculated in Section 17.10 with the help of the semiclassical approximation. If we want to extract meaningful numbers from a divergent perturbation series such as (5.202), it is necessary to find a convergent resummation procedure. Such a procedure is supplied by the variational perturbation expansion, as we now demonstrate for the ground state energy of the anharmonic oscillator. Truncating the infinite sum (5.202) after the Nth term, the replacement (5.188) followed by a re-expansion in powers of g up to order N leads to the approximation WN at zero temperature:

416


W

0.55914635

N =21

N =15 0.55914625

N =11 1.6

1.4

2

1.8

2.2

Ω

Figure 5.16 New plateaus in WN developing for higher orders N ≥ 15 in addition to the minimum which now gives worse results. For N = 11 the new plateau is not yet extremal, but it is the proper region of least Ω-dependence yielding the best approximation to the exact energy indicated by the dashed horizontal line. The minimum has fallen far below this value and is no longer useful. The figure looks similar for all couplings (in the plot, g = 0.4). The reason is the scaling property (5.215) proved in Appendix 5B.

WNΩ = Ω

N X l=0

(0)

εl

g 4Ω3

l

,

(5.206)

with the re-expansion coefficients (0) εl

=

l X

j=0

(0) Ej

(1 − 3j)/2 l−j

!

(−4σ)l−j .

(5.207)

Here σ denotes the dimensionless function of Ω 1 Ω(Ω2 − 1) σ ≡ − Ωr 2 = . 2 g

(5.208)

In Fig. 5.15 we have plotted the Ω-dependence of WN for increasing N at the coupling constant g/4 = 0.1. For odd and even N, an increasingly flat plateau develops at the optimal energy. At larger orders N ≥ 15, the initially flat plateau is deformed into a minimum with a larger curvature and is no longer a good approximation. However, a new plateau has developed yielding the best energy. This is seen on the high-resolution plot in Fig. 5.16. At N = 11, the new plateau is not yet extremal but close to the correct energy. The worsening extrema in Fig. 5.16 correspond here to points leaving the optimal dashed into the upward direction. The newly forming plateaus lie always on the dashed curve. H. Kleinert, PATH INTEGRALS

417

5.15 Convergence of Variational Perturbation Expansion 10 g/4 =1 8

6 ΩN 4

2

0

0

20

40

60

80

100

N

Figure 5.17 Trial frequencies ΩN extremizing the variational approximation WN at T = 0 for odd N ≤ 91. The coupling is g/4 = 1. The dashed curve corresponds to the approximation (5.211) [related to ΩN via (5.208)]. The frequencies on this curve produce the fastest convergence. The worsening extrema in Fig. 5.16 correspond here to points leaving the optimal dashed into the upward direction. The newly forming plateaus lie always on the dashed curve. 6

g/4 =1 ΩN

4

2 0

10

N

20

30

Figure 5.18 Extremal and turning point frequencies ΩN in variational approximation WN at T = 0 for even and odd N ≤ 30. The coupling is g/4 = 1. The dashed curve corresponds to the approximation (5.211) [related to ΩN via (5.208)].

The set of all extremal ΩN -values for odd N up to N = 91 is shown in Fig. 5.17. The optimal frequencies with smallest curvature are marked by a fat dot. In Subsection 17.10.5. we shall derive that ΩN (Ω2N − 1) σN = g

(5.209)

418

5 Variational Perturbation Theory 100 g/4 =1 10-5 |E-Eex| 10-10

10-15

10-20

0

20

40

N

60

80

100

Figure 5.19 Difference between approximate ground state energies E = WN and exact energies Eex for odd N corresponding to the ΩN -values shown in Fig. 5.17. The coupling is g/4 = 1. The lower curve follows roughly the error estimate to be derived in Eq. (17.409). The extrema in Fig. (5.17) which move away from the dashed curve lie here on horizontal curves whose accuracy does not increase.

grows for large N like σN ≈ cN ,

c = 0.186047 . . . ,

(5.210)

so that ΩN grows like ΩN ≈ (cNg)1/3 . For smaller N, the best ΩN -values in Fig. 5.17 can be fitted with the help of the corrected formula (5.210): 6.85 σN ≈ cN 1 + 2/3 . N

(5.211)

The associated ΩN -curve is shown as a dashed line. It is the lower envelope of the extremal frequencies. The set of extremal and turning point frequencies ΩN is shown in Fig. 5.18 for even and odd N up to N = 30. The optimal extrema with smallest curvature are again marked by a fat dot. The theoretical curve for an optimal convergence calculated from (5.211) and (5.209) is again plotted as a dashed line. In Table 5.8, we illustrate the precision reached for large orders N at various coupling constants g by a comparison with accurate energies derived from numerical solutions of the Schrödinger equation. The approach to the exact energy values is illustrated in Fig. 5.19 which shows that a good convergence is achieved by using the lowest of all extremal frequencies, which lie roughly on the dashed theoretical curve in Fig. 5.17 and specify the position of the plateaus. The frequencies ΩN on the higher branches leaving the dashed curve in that figure, on the other hand, do not yield converging energy values. The sharper minima in Fig. 5.16 correspond precisely to those branches which no longer determine the region of weakest Ω-dependence. H. Kleinert, PATH INTEGRALS

419

5.15 Convergence of Variational Perturbation Expansion N 1 2 3 4 5 10 15 20 25 exact N 1 2 3 4 5 10 15 20 25 exact

g/4 =0.1 0.5603073711 0.5591521393 0.5591542188 0.5591457408 0.5591461596 0.5591463266 0.5591463272 0.5591463272 0.5591463272 0.5591463272 g/4 =50 2.5475803996 2.5031213253 2.5006996279 2.4995980125 2.4996213227 2.4997071960 2.4997089403 2.4997089079 2.4997087731 2.4997087726

g/4 = 0.3 0.6416298621 0.6380887347 0.6380357598 0.6379878713 0.6379899084 0.6379917677 0.6379917838 0.6379917836 0.6379917832 0.6379917832 g/4 =200 4.0084608812 3.9365586048 3.9325538203 3.9307488127 3.9307857892 3.9309286743 3.9309316283 3.9309315732 3.9309313396 3.9309313391

g/4 =0.5 0.7016616429 0.6963769499 0.6962536326 0.6961684978 0.6961717475 0.6961757782 0.6961758231 0.6961743059 0.6961758208 0.6961758208 g/4 =1000 6.8279533136 6.7040032606 6.6970328638 6.6939036178 6.6939667971 6.6942161680 6.6942213631 6.6942212659 6.6942208522 6.6942208505

g/4 =1.0 0.8125000000 0.8041900946 0.8039140528 0.8037563457 0.8037615232 0.8037705329 0.8037706596 0.8037706575 0.8037706513 0.8037706514 g/4 =8000 13.635282593 13.386598486 13.372561189 13.366269038 13.366395347 13.366898079 13.366908583 13.366908387 13.366907551 13.366907544

g/4 =2.0 0.9644035598 0.9522936298 0.9517997694 0.9515444198 0.9515517450 0.9515682249 0.9515684933 0.9515684887 0.9515584121 0.9515684727 g/4 =20000 18.501658712 18.163979967 18.144908389 18.136361642 18.136533060 18.137216200 18.137230481 18.137230214 18.137230022 18.137229073

Table 5.8 Comparison of the variational approximations WN at T = 0 for increasing N with the exact ground state energy at various coupling constants g.

The anharmonic oscillator has the remarkable property that a plot of the ΩN values in the N, σN -plane is universal in the coupling strengths g; the plots do not depend on g. To see the reason for this, we reinsert explicitly the frequency ω (which was earlier set equal to unity). Then the re-expanded energy WN in Eq. (5.206) has the general scaling form g, ω ˆ 2 ), WNΩ = ΩwN (ˆ

(5.212)

where wN is a dimensionless function of the reduced coupling constant and frequency gˆ ≡

g , Ω3

ω ˆ≡

ω , Ω

(5.213)

respectively. When differentiating (5.212), "

#

d d d WN = 1 − 3ˆ g − 2ˆ ω 2 2 wN (ˆ g, ω ˆ 2), dΩ dˆ g dˆ ω

(5.214)

we discover that the right-hand side can be written as a product of gˆN and a dimensionless polynomial of order N depending only on σ = Ω(Ω2 − ω 2 )/g: d W Ω = gˆN pN (σ). dΩ N

(5.215)

A proof of this will be given in Appendix 5B for any interaction xp . The universal optimal σN -values are obtained from the zeros of pN (σ). It is possible to achieve the same universality for the optimal frequencies of the even approximations WN by determining them from the extrema of pN (σ) rather than from the turning points of WN as a function of Ω.

420


The universal functions pN (σ) are found most easily by replacing the variable σ (0) in the coefficients εl of the re-expansion (5.206) by its ω = 0 -limit σ|ω=0 = Ω3 /g = 1/ˆ g . This yields the simpler expression WNΩ = ΩwN (ˆ g , 0) = Ω

N X

gˆ 4

(0) εl

l=0

with (0) εl

=

l X

!

(1 − 3j)/2 l−j

(0) Ej

j=0

!l

,

(5.216)

(−4/ˆ g )l−j .

(5.217)

The derivative of WN with respect to Ω yields −N

pN (σ) = gˆ

"

#

d 1 − 3ˆ g wN (ˆ g , 0) . dˆ g gˆ=1/σ

(5.218) (0)

In Section 17.10, we show the re-expansion coefficients εk in (5.207) to be for (0) large k proportional to Ek : (0)

(0)

εk ≈ e−2σN Ek ,

σN =

ΩN (Ω2N − 1) g

(5.219)

[see Eq. (17.396)]. Thus, at any fixed Ω, the re-expanded series has the same asymptotic growth as the original series with the same vanishing radius of convergence. The behavior (5.219) can be seen in Fig. 5.20(a) where we have plotted the logarithm log Sk

log Sk

30

10 N= 1

20

0

10

N= 5

-10

30

k

40

N =10 -10

10 0

20

N= 9 10

20

30

40

N =20

50

-20

k

N =30

-30

a)

b)

Figure 5.20 Logarithmic plot of kth terms in re-expanded perturbation series at a coupling constant g/4 = 1: (a) Frequencies ΩN extremizing the approximation WN . The dashed curves indicate the theoretical asymptotic behavior (5.219). (b) Frequencies ΩN corresponding to the dashed curve in Fig. 5.17. The minima lie for each N precisely at k = N , producing the fastest convergence. The curves labeled Ω = ω indicate the kth term in the original perturbation series.

of the absolute value of the kth term Sk =

(0) εk

gˆ 4

!k

(5.220)


421

5.16 Variational Perturbation Theory for Strong-Coupling Expansion

of the re-expanded perturbation series (5.202) for various optimal values ΩN and g = 40. All curves show a growth ∝ k k . The terms in the original series start growing immediately (precocious growth). Those in the re-expanded series, on the other hand, decrease initially and go through a minimum before they start growing (retarded growth). The dashed curves indicate the analytically calculated asymptotic behavior (5.219). The increasingly retarded growth is the reason why energies obtained from the variational expansion converge towards the exact result. Consider the terms Sk of the resummed series with frequencies ΩN taken from the theoretical curve of optimal convergence in Fig. 5.17 (or 5.18). In Fig. 5.20(b) we see that the terms Sk are minimal at k = N, i.e., at the last term contained in the approximation WN . In general, a divergent series yields an optimal result if it is truncated after the smallest term Sk . The size of the last term gives the order of magnitude of the error in the truncated evaluation. The re-expansion makes it possible to find, for every N, a frequency ΩN which makes the truncation optimal in this sense.

5.16

Variational Perturbation Theory for Strong-Coupling Expansion

From the ω → 0 -limit of (5.206), we obtain directly the strong-coupling behavior of WN . Since Ω = (g/ˆ g)1/3 , we can write WN = (g/ˆ g)1/3 wN (ˆ g , 0),

(5.221)

and evaluate this at the optimal value gˆ = 1/σN . The large-g behavior of WN is therefore WN − −−→(g/4)1/3b0 ,

(5.222)

b0 = (4/ˆ g )1/3 wN (ˆ g , 0)|gˆ=1/σN .

(5.223)

g→∞

with the coefficient

The higher corrections to the leading behavior (5.222) are found just as easily. By expanding wN (ˆ g, ω ˆ 2) in powers of ω ˆ 2, WN = and inserting

g gˆ

!1/3  1 + ω ˆ2

d ω ˆ4 + dˆ ω2 2!

ω ˆ2 =

d dˆ ω2

!2

2  + . . . wN (ˆ g, ω ˆ ) 

,

(5.224)

ω ˆ 2 =0

gˆ2/3 , (g/ω 3)2/3

(5.225)

we obtain the expansion 1/3 "

g WN = 4

b0 + b1

g 4ω 3

−2/3

+ b2

g 4ω 3

−4/3

#

+ ... ,

(5.226)

422


log |α0 − αex 0 | 81/3

Figure 5.21 b0 and b1 .

log |α1 − αex 1 |

271/3 641/3 N 1/3

1251/3

2161/3

81/3

271/3 641/3 N 1/3

1251/3

2161/3

Logarithmic plot of N -behavior of strong-coupling expansion coefficients

Figure 5.22 Oscillations of approximate strong-coupling expansion coefficient b0 as a function of N when approaching exponentially fast the exact limit. The exponential behavior has been factored out. The upper and lower points show the odd-N and even-N approximations, respectively.

with the coefficients 1 bn = n!

gˆ 4

!(2n−1)/3

d dˆ ω2

!n

2 wN (ˆ g, ω ˆ )

.

gˆ=1/σN

(5.227)

,ˆ ω 2 =0

The derivatives on the right-hand side have the expansions d dˆ ω2

!n

wN (ˆ g, ω ˆ 2) =

N X l=0

(0) εn l

gˆ 4

!l

,

(5.228)


5.16 Variational Perturbation Theory for Strong-Coupling Expansion

423

Table 5.9 Coefficients bn of strong-coupling expansion of ground state energy of anharmonic oscillator obtained from a perturbation expansion of order 251. An extremely ˇ ıˇzek, J. Math. Phys. 32, 3392 (1991): precise value for b0 was given by F. Vinette and J. C´ b0 = 0.667 986 259 155 777 108 270 962 016 198 601 994 304 049 36 . . . . n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

bn 0.667 986 259 155 777 108 270 96 0.143 668 783 380 864 910 020 3 −0.008 627 565 680 802 279 128 0.000 818 208 905 756 349 543 −0.000 082 429 217 130 077 221 0.000 008 069 494 235 040 966 −0.000 000 727 977 005 945 775 0.000 000 056 145 997 222 354 −0.000 000 002 949 562 732 712 −0.000 000 000 064 215 331 954 0.000 000 000 048 214 263 787 −0.000 000 000 008 940 319 867 0.000 000 000 001 205 637 215 −0.000 000 000 000 130 347 650 0.000 000 000 000 010 760 089 −0.000 000 000 000 000 445 890 1 −0.000 000 000 000 000 058 989 8 0.000 000 000 000 000 019 196 00 −0.000 000 000 000 000 003 288 13 0.000 000 000 000 000 000 429 62 −0.000 000 000 000 000 000 044 438 0.000 000 000 000 000 000 003 230 5 −0.000 000 000 000 000 000 000 031 4

with the re-expansion coefficients l−n 1 (0) X (0) εn l = Ej n! j=0

(1 − 3j)/2 l−j

!

l−j n

!

(−1)l−j−n (4/ˆ g )l−j .

(5.229)

For increasing N, the coefficients b0 , b1 , . . . converge rapidly against the values shown in Table 5.9. From the logarithmic plot in Fig. 5.21 we extract a convergence 8.2−9.7N |b0 − bex 0 | ∼ e

1/3

,

6.4−9.1N |b1 − bex 1 | ∼ e

1/3

.

(5.230)

This behavior will be derived in Subsection 17.10.5, where we shall find that for any −2/3 1/3 g > 0, the error decreases at large N roughly like e−[9.7+(cg) ] N [see Eq. (17.409)]. The approach to this limiting behavior is oscillatory, as seen in Fig. 5.22 where 1/3 we have removed the exponential falloff and plotted e−6.5+9.42N (b0 − bex 0 ) against N [16].

424


For the proof of the convergence of the variational perturbation expansion in Subsection 17.10.5. it will be important to know that the strong-coupling expansion for the ground state energy E

(0)

1/3 "

g = 4

b0 + b1

g 4ω 3

−2/3

+ b2

g 4ω 3

−4/3

#

+ ... ,

(5.231)

converges for large enough g > gs .

(5.232)

The same is true for the excited energies.

5.17

General Strong-Coupling Expansions

The coefficients of the strong-coupling expansion can be derived for any divergent perturbation series EN (g) =

N X

an g n ,

(5.233)

n=0

for which we know that it behaves at large couplings g like M X

E(g) = g p/q

bm (g −2/q )m .

(5.234)

m=0

The series (5.233) can trivially be rewritten as EN (g) = ω

N X

p

an

n=0

g ωq

n

,

(5.235)

with ω = 1. We now apply the square-root trick (5.188) and replace ω by the identical expression q ω ≡ Ω2 + (ω 2 − Ω2 ), (5.236)

containing a dummy scaling parameter Ω. The series (5.235) is then re-expanded in powers of g up to the order N, thereby treating ω 2 − Ω2 as a quantity of order g. The result is most conveniently expressed in terms of dimensionless parameters gˆ ≡ g/Ωq and σ ≡ (1 − ω ˆ 2 )/ˆ g , where ω ˆ ≡ ω/Ω. Then the replacement (5.236) amounts to ω− − −→ Ω(1 − σˆ g )1/2 , (5.237) so that the re-expanded series reads explicitly p

WN (ˆ g , σ) = Ω

N X

εn (σ) (ˆ g )n ,

(5.238)

!

(5.239)

n=0

with the coefficients: εn (σ) =

n X

j=0

aj

(p − qj)/2 n−j

(−σ)n−j .


425

5.17 General Strong-Coupling Expansions

For any fixed g, we form the first and second derivatives of WNΩ (g) with respect to Ω, calculate the Ω-values of the extrema and the turning points, and select the smallest of these as the optimal scaling parameter ΩN . The function WN (g) ≡ WN (g, ΩN ) constitutes the Nth variational approximation EN (g) to the function E(g). We now take this approximation to the strong-coupling limit g → ∞. For this we observe that (5.238) has the general scaling form WNΩ (g) = Ωp wN (ˆ g, ω ˆ 2).

(5.240)

For dimensional reasons, the optimal ΩN increases with g for large g like ΩN ≈ g 1/q cN , so that gˆ = c−q g = cqN remain finite in the strong-coupling limit, N and σ = 1/ˆ whereas ω ˆ 2 goes to zero like 1/[cN (g/ω q )1/q ]2 . Hence WNΩN (g) ≈ g p/q cpN wN (c−q N , 0).

(5.241)

Here cN plays the role of the variational parameter to be determined by the lowest extremum or turning point of cpN wN (c−q N , 0). The full strong-coupling expansion is obtained by expanding wN (ˆ g, ω ˆ 2 ) in powers of ω ˆ 2 = (g/ω q gˆ)−2/q at a fixed gˆ. The result is WN (g) = g with

p/q

"

g ωq

d dˆ ω2

!n

¯b0 (ˆ g ) + ¯b1 (ˆ g)

1 ¯bn (ˆ g) = n!

with respect to ω ˆ 2 . Explicitly:

−2/q

+ ¯b2 (ˆ g)

(n) wN (ˆ g, ω ˆ 2 )

l−n N X X 1 (n) wN (ˆ g , 0) = (−1)l+n aj n! j=0 l=0

g ωq

−4/q

#

+ ... ,

gˆ(2n−p)/q ,

(5.242)

(5.243)

ω ˆ 2 =0

(p − qj)/2 l−j

!

l−j n

!

(−ˆ g )j .

(5.244)

¯ g ) may be written as functions of the parameter c: Since gˆ = c−q N , the coefficients bn (ˆ ¯bn (c) =

N X l=0

al

N −l X j=0

(p − lq)/2 j

!

j (−1)j−n cp−lq−2n . n

(5.245)

The values of c which optimize WN (g) for fixed g yield the desired values of cN . The optimization may be performed stepwise using directly the expansion coefficients ¯bn (c). First we optimize the leading coefficient b0 (c) as a function of c and identifying the smallest of them as cN . Next we have to take into account that for large but finite α, the trial frequency Ω has corrections to the behavior gˆ1/q c. The coefficient c will depend on gˆ like c(ˆ g ) = c + γ1

gˆ ωq

!−2/q

+ γ2

gˆ ωq

!−4/q

+ ...,

(5.246)

426


requiring a re-expansion of c-dependent coefficients ¯bn in (5.242). The expansion coefficients c and γn for n = 1, 2, . . . are determined by extremizing ¯b2n (c). The final result can again be written in the form (5.242) with ¯b(c)n replaced by the final bn : " # −2/q −4/q g g p/q WN (g) = g b0 + b1 + b2 + ... . (5.247) ωq ωq The final bn are determined by the equations shown in Table 5.10. The two leading coefficients receive no correction and are omitted. The extremal values of gˆ will have a strong-coupling expansion corresponding to (5.246): " # −2/q −4/q g g −q + δ2 +··· . (5.248) gˆ = cN 1 + δ1 ωq ωq

Table 5.10 Equations determining coefficients bn in strong-coupling expansion (5.247) from the functions ¯bn (c) in (5.245) and their derivatives. For brevity, we have suppressed the argument c in the entries.

n bn 2 ¯b2 + γ1¯b′1 + 12 γ12¯b′′0 (3) 3 ¯b3 + γ2¯b′1 + γ1¯b′2 + γ1 γ2¯b′′0 + 12 γ12¯b′′1 + 16 γ13¯b0 4 ¯b4 + γ3¯b′1 + γ2¯b′2 + γ1¯b′3 + ( 12 γ22 + γ1 γ3 )¯b′′0 (3) (3) (4) +γ1 γ2¯b′′ + 1 γ 2¯b′′ + 1 γ 2 γ2¯b0 + 1 γ 3¯b1 + 1 γ 4¯b0 1

2 1 2

2 1

6 1

24 1

−γn−1 ¯b′ /¯b′′ 1 0 (3) ¯ (b′2 + γ1¯b′′1 + 12 γ12¯b0 )/¯b′′0 (3) (¯b′3 + γ2¯b′′1 + γ1¯b′′2 + γ1 γ2¯b0 (3) (4) + 1 γ 2¯b1 + 1 γ 3¯b0 )/¯b′′ 2 1

6 1

0

The convergence of the general strong-coupling expansion is similar to the one observed for the anharmonic oscillator. This will be seen in Subsection 17.10.5. The general strong-coupling expansion has important applications in the theory of critical phenomena. This theory renders expansions of the above type for the socalled critical exponents, which have to be evaluated at infinitely strong (bare) couplings of scalar field theories with gφ4 interactions. The results of these applications are better than those obtained previously with a much more involved theory based on a combination of renormalization group equations and Padé-Borel resummation techniques [28]. The critical exponents have power series expansions in powers of g/ω in the physically most interesting three-dimensional systems, where ω 2 /2 is the factor in front of the quadratic field term φ2 . The important phenomenon observed in such systems is the appearance of anomalous dimensions. These imply that the expansion terms (g/ω)n cannot simply be treated with the square-root trick (5.236). The anomalous dimension requires that (g/ω)n must be treated as if it were (g/ω q )n when applying the square-root trick. Thus we must use the anomalous square-root trick " 2/q #q/2 ω ω →Ω 1+ . (5.249) Ω H. Kleinert, PATH INTEGRALS

5.18 Variational Interpolation between Weak and Strong-Coupling Expansions

427

The power 2/q appearing in the strong-coupling expansion (5.234) is experimentally observable since it governs the approach of the system to the scaling limit. This exponent is usually denoted by the letter ω, and is referred to as the Wegner exponent [29]. This exponent ω is not to be confused with the frequency ω in the present discussion. In superfluid helium, for example, this critical exponent is very close to the value 4/5, implying q ≈ 5/2. The Wegner exponent of fluctuating quantum fields cannot be deduced, as in quantum mechanics, from simple scaling analyses of the action. It is, however, calculable by applying variational perturbation theory to the logarithmic derivative of the power series of the other critical exponents. These are called β-functions and have to vanish for the correct ω. This procedure is referred to as dynamical determination of ω and has led to values in excellent agreement with experiment [17].

5.18

Variational Interpolation between Weak and StrongCoupling Expansions

The possibility of calculating the strong-coupling coefficients from the perturbation coefficients can be used to find a variational interpolation of a function with known weak- and strong-coupling coefficients [18]. Such pairs of expansions are known for many other physical systems, for example most lattice models of statistical mechanics [19]. If applied to the ground state energy of the anharmonic oscillator, this method converges exponentially fast [15]. The weak-coupling expansion of the ground state energy of the anharmonic oscillator has the form (5.233). In natural units with h ¯ = M = ω = 1, the lowest coefficient a0 is trivially determined to be a0 = 1/2 by the ground state energy of the harmonic oscillator. If we identify α = g/4 with the coupling constant in (5.233), to save factors 1/4, the first coefficient is a1 = 3/4 [see (5.190)]. We have seen before in Section 5.12 that even the lowest order variational perturbation theory yields leading strong-coupling coefficient in excellent agreement with the exact one [with a maximal error of ≈ 2%, see Eq. (5.167)]. In Fig. 5.23 we have plotted the relative deviation of the variational approximation from the exact one in percent. The strong-coupling behavior is known from (5.226). It starts out like g 1/3 , followed by powers of g −1/3 , g −1 , g −5/3. Comparison with (5.234) shows that this corresponds to p = 1 and q = 3. The leading coefficient is given in Table 5.9 with extreme accuracy: b0 = 0.667 986 259 155 777 108 270 962 016 919 860 . . . . In a variational interpolation, this value is used to determine an approximate a1 (forgetting that we know the exact value aex 1 = 3/4). The energy (5.238) reads for N = 1 (with α = g/4 instead of g): Ω 1 a1 W1 (α, Ω) = + a0 + 2 α. 2 2Ω Ω

(5.250)

Equation (5.245) yields, for n = 0: c a1 b0 = a0 + 2 . 2 c

(5.251)

428


(0)

W1 /Eex

log g/4

Figure 5.23 Ratio of approximate and exact ground state energy of anharmonic oscillator from lowest-order variational interpolation. Dashed curve shows first-order FeynmanKleinert approximation W1 (g). The accuracy is everywhere better than 99.5 %. For comparison, we also display the much worse (although quite good) variational perturbation result using the exact aex 1 = 3/4.

Minimizing b0 with respect to c we find c = c1 ≡ 2(a1 /2a0 )1/3 with b0 = 3a0 c1 /4 = 3(a20 a1 /2)1/3 /2. Inserting this into (5.251) fixes a1 = 2(2/3b0 )3 /a20 = 0.773 970 . . . , quite close to the exact value 3/4. With our approximate a1 we calculate W1 (α, Ω) at its minimum, where   

Ω1 =  

√2 ω cosh 3 √2 ω cos 3

h

h

1 3

1 acosh(g/g (0)) 3

arccos(g/g (0) )

i

i

for

g > g (0) , g < g (0) ,

(5.252)

√ with g (0) ≡ 2ω 3 a0 /3 3a1 . The result is shown in Fig. (5.23). Since the difference with respect to the exact solution would be too small to be visible on a direct plot 0 of the energy, we display the ratio with respect to the exact energy W1 (g)/Eex . The accuracy is everywhere better than 99.5 %.

5.19

Systematic Improvement of Excited Energies

The variational method for the energies of excited states developed in Section 5.12 can also be improved systematically. Recall the n-dependent level shift formulas (3.515) and (3.516), according to which ∆E (n) = ∆1 E (n) + ∆2 E (n) + ∆3 E (n) = ∆Vnn − +

∆Vnk ∆Vkn k6=n Ek − En X

X ∆Vnk ∆Vkn ∆Vnk ∆Vkl ∆Vln − ∆Vnn . 2 (E − E )(E − E ) (E − E ) k n l n k n k6=n l6=n k6=n XX

(5.253)


429

5.20 Variational Treatment of Double-Well Potential

By applying the substitution rule (5.188) to the total energies E (n) = ω(n + 1/2) + ∆E (n) , and by expanding each term in powers of g up to g 3 , we find the contributions to the level shift g [3(2n2 + 2n + 1)a4 + (2n + 1)a2 r 2 ], 4 2 g = − [2(34n3 + 51n2 + 59n + 21)a8 4

∆1 E (n) = ∆2 E (n)

+4 · 3(2n2 + 2n + 1)a6 r 2 + (2n + 1)a4 r 4 ] ∆3 E (n) =

1 , h ¯Ω

3

g [4 · 3(125n4 + 250n3 + 472n2 + 347n + 111)a12 4 +4 · 5(34n3 + 51n2 + 59n + 21)a10 r 2 1 +16 · 3(2n2 + 2n + 1)a8 r 4 + 2 · (2n + 1)a6 r 6 ] 2 2 , h ¯ Ω

(5.254)

which for n = 0 reduce to the corresponding terms in (5.192). The extremization in Ω leads to energies which lie only very little above the exact values for all n. This is illustrated in Table 5.11 for n = 8 (compare with the energies in Table 5.5). A sum (n) over the Boltzmann factors e−βE3 produces an approximate partition function Z3 which deviates from the exact one by less than 50.1%. It will be interesting to use the improved variational approach for the calculation of density matrices, particle distributions, and magnetization curves.

5.20

Variational Treatment of Double-Well Potential

Let us also calculate the approximate effective classical potential of third order W3 (x0 ) for the double-well potential ω2 g M 2ω4 V (x) = M x2 + x4 + , ω 2 = −1. (5.255) 2 4 4g In the expression (5.197), the sign change of ω 2 affects only the coupling g2 (x0 ), which becomes g2 (x0 ) = M [−1 − Ω2 (x0 )] + 3gx20 = gr2 /2 + 3gx20

(5.256)

[recall (5.176)]. Note the constant energy M 2 ω 4 /4g in V (x) which shifts the minima of the potential to zero [compare (5.78)]. To see the improved accuracy of W3 with respect to the first approximation W1 (x0 ) discussed in Section 5.7 [corresponding to the first line of (5.197)], we study the limit of zero temperature where the accuracy is expected to be the worst. In this limit, W3 (x0 ) reduces to (5.201) and is easily minimized in x0 and Ω. At larger coupling constants g > gc ≈ 0.3, the energy has a minimum at x0 = 0. For g ≤ gc , there is an additional symmetric pair of minima at x0 = ±xm 6= 0 (recall Figs. 5.5 and 5.6). The resulting W3 (0) is plotted in Fig. 5.24 together with W1 (0). The figure also contains the first excited energy which is obtained by setting ω 2 = −1 in r2 = 2M (ω 2 − Ω2 )/g of Eqs. (5.253)–(5.255). For small couplings g, the energies W1 (0), W3 (0), . . . diverge and the minima at x = ±xm of Eq. (5.82) become relevant. Moreover, there is quantum tunneling across the central barrier from

430


Table 5.11 Higher approximations to excited energy with n = 8 of anharmonic oscillator (8) at various coupling constants g. The third-order approximation E3 (g) is compared with (8) (8) the exact values Eex (g), with the approximation E1 (g) of the last section, and with the (8) lower approximation of even order E2 (g) (all in units of h ¯ ω). g/4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 50 100 500 1000

(8)

Eex (g) 13.3790 15.8222 17.6224 19.0889 20.3452 21.4542 22.4530 23.3658 24.2091 24.9950 51.9865 88.3143 111.128 189.756 239.012

(8)

E1 (g) 13.3235257 15.7327929 17.5099190 18.9591071 20.2009502 21.2974258 22.2851972 23.1879959 24.0221820 24.7995745 51.5221384 87.5058600 110.105819 188.001018 236.799221

(8)

E2 (g) 13.3766211 15.8135994 17.6099785 19.0742800 20.3287326 21.4361207 22.4335694 23.3451009 24.1872711 24.9720376 51.9301030 88.2154879 111.002842 189.540577 238.740320

(8)

E3 (g) 13.3847643 15.8275802 17.6281810 19.0958388 20.3531080 21.4629384 22.4625543 23.3760415 24.2199988 25.0064145 51.9986710 88.3500454 111.173183 189.833415 239.109584

one minimum to the other which takes place for g ≤ gc ≈ 0.3 and is unaccounted for by W3 (0) and W3 (xm ). Tunneling leads to a level splitting to be calculated in Chapter 17. In this chapter, we test the accuracy of W1 (xm ) and W3 (xm ) by comparing them with the averages of the two lowest energies. Figure 5.24 shows that the accuracy of the approximation W3 (xm ) is quite good. Note that the approximation W1 (xm ) does not possess the correct slope in g, which is missed by 25%. In fact, a Taylor expansion of W1 (xm ) reads √ √ 2 3 9 2 2 27 3 W1 (xm ) = − g− g − g + ... , (5.257) 2 16 128 256 whereas the true expansion starts out with E (0) =

√ 2 1 − g + ... . 2 4

(5.258)

The optimal frequency associated with (5.257) has the expansion √ √ 3 27 2 2 27 3 Ω1 (xm ) = 2 − g − g − g + ... . 4 64 32 Let us also compare the x0 -behavior of W3 (x0 ) with that of the true effective classical potential calculated numerically by Monte Carlo simulations. The curves are plotted in Fig. 5.5, and the agreement is seen to be excellent. There are significant deviations only for low temperatures with > 20. β∼ At zero temperature, there exists a simple way of recovering the effective classical potential from the classical potential calculated up to two loops in Eq. (3.767). As we learned in Eq. (3.830), x0 coincides at zero temperature. with X in Eq. (3.767) Thus we merely have to employ the square-root trick (5.186) to the effective potential (3.767) and interchange X by x0 to obtain the variational approximation W1 (x0 ) to the effective classical potantial to be varied. Explicitly, H. Kleinert, PATH INTEGRALS

431

5.20 Variational Treatment of Double-Well Potential

E

E

Figure 5.24

Lowest two energies in double-well potential as function of coupling strength g. The approximations are W1 (0) (dashed line) and W3 (0) (solid line). The dots indicate numeric results of the Schrödinger equation. The lower part of the figure shows W1 (xm ) and W3 (0) in comparison with the average of the Schr¨ odinger energies (small dots). Note that W1 misses the slope by 25%. Tunneling causes a level splitting to be calculated in Chapter 17 (dotted curves).

we replace, as in (5.188), the one-loop contribtions, ground state energies ¯hωT,L in (3.767) by q 2 2 2 (X) − Ω2 ) ≈ Ω Ω2T,L + (ωT,L T,L + (ωT,L (X) − ΩT,L )/2ΩT,L , and exchange ωL,T (X) of the T,L remaining two-loop terms by ΩT,L . This yields for the D-dimensional rotated double-well potential (the Mexical hat potential in Fig. 3.13]: (

( 2 ΩL ωL (X) ΩT ωT2 (X) ¯h2 1 (4) W (x0 ) = v(X) + h ¯ + + (D − 1)¯ h + + v (X) T →0 4 4ΩL 4 4ΩT 8(2M )2 Ω2L ) D2 − 1 v ′′ (X) v ′ (X) 2(D − 1) v ′′′ (X) 2v ′′ (X) 2v ′ (X) + − + − + Ω2T X2 X3 ΩL ΩT X X2 X3 ( ) ′′ 2 ) ′ ¯h2 1 3(D − 1) 1 v (X) v (X) 3 [v ′′′ (X)]2 + − + O(¯ h ) . (5.259) − 6(2M )3 3Ω4L 2ΩT + ΩL ΩL Ω2T X X2

X→x0

This has to be extremized in ΩT,L at fixed x0 .

432


5.21

Higher-Order Effective Classical Potential for Nonpolynomial Interactions

The systematic improvement of the Feynman-Kleinert approximation in Section 5.13 was based on Feynman diagrams and therefore applicable only to polynomial potentials. If we want to calculate higher-order effective classical potentials for nonpolynomial interactions such as the Coulomb interaction, we need a generalization of the smearing rule (5.30) to the correlation functions of interaction potentials which occur in the expansion (5.180). The second-order term, for example, requires the calculation of

x0 2 x0 Aint Ω,c =

where

Z

h ¯β

dτ

0

Z

h ¯β

0

x

0 x0 x0 dτ ′ hVint (x(τ ))Vint (x(τ ′ ))iΩ,c ,

(5.260)

1 x0 Vint (x) = V (x) − M Ω2 (x0 )(x − x0 )2 . 2

(5.261)

Thus we need an efficient smearing formula for local expectations of the form 1 hF1 (x(τ1 )) . . . Fn (x(τn ))ixΩ0 = x0 ZΩ I 1 × Dx(τ )F1 (x(τ1 )) · · · Fn (x(τn ))δ(x − x0 ) exp − AxΩ0 [x(τ )] , ¯h

(5.262)

where AxΩ0 [x(τ )] and ZΩx0 are the local action and partition function of Eqs. (5.3) and (5.4). After rearranging the correlation functions to connected ones according to Eqs. (5.182) we find the cumulant expansion for the effective classical potential [see (5.180)]

V

eff,cl

(x0 ) = FΩx0

1 + ¯hβ

Zh¯ β x0 x0 dτ1 hVint (x(τ1 ))iΩ,c 0

1 − 2 2¯h β

Zh¯ β Zh¯ β x0 x0 x0 dτ1 dτ2 hVint (x(τ1 ))Vint (x(τ2 ))iΩ,c

1 + 3 6¯h β

Zh¯ β

0

0

(5.263)

0

dτ1

Zh¯ β 0

dτ2

Zh¯ β 0

x

x0 x0 x0 0 dτ3 hVint (x(τ1 ))Vint (x(τ2 ))Vint (x(τ3 ))iΩ,c + ... .

It differs from the previous expansion (5.180) for polynomial interactions by the potential V (x) not being expanded around x0 . The first term on the right-hand side is the local free energy (5.6).

5.21.1

Evaluation of Path Integrals

The local pair correlation function was given in Eq. (5.19): x

x

hδx(τ )δx(τ ′ )iΩ0 ≡ h[x(τ ) − x0 ][x(τ ′ ) − x0 ]iΩ0 =

¯h (2)x0 G (τ, τ ′ ) = a2τ τ ′ (x0 ), M Ω

(5.264)

with [recall (5.19)–(5.24)] a2τ τ ′ (x0 ) =

¯ h cosh[Ω(x0 )(τ −τ ′ − ¯hβ/2)] 1 − , 2M Ω(x0 ) sinh[Ω(x0 )¯ hβ/2] M βΩ2 (x0 )

τ ∈ (0, ¯hβ).

(5.265)


5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions

433

Higher correlation functions are expanded in products of these according to Wick’s rule (3.302). For an even number of δx(τ )’s one has X x hδx(τ1 )δx(τ2 ) · · · δx(τn )iΩ0 = a2τp(1) τp(2) (x0 ) · · · a2τp(n−1) τp(n) (x0 ) , (5.266) pairs

where the sum runs over all (n − 1)!! pair contractions. For an exponential, Wick’s rule implies * " Z #+x0 " # Z Z h¯ β h ¯β 1 h¯ β ′ 2 ′ dτ dτ j(τ )aτ τ ′ (x0 )j(τ ) . (5.267) exp i dτ j(τ ) δx(τ ) = exp − 2 0 0 0 Ω

Inserting j(τ ) =

Pn

ki δ(τ − τi ), this gives for the expectation value of a sum of exponentials   * " n #+x0 n X n X X 1 exp i ki δx(τi ) = exp − a2τi τj (x0 )ki kj  . (5.268) 2 i=1 i=1 j=1 i=1

Ω

R By Fourier-decomposing the functions F (x(τ )) = (dk/2π)F (k) exp ik [x0 + δx(τ )] in (5.262), we obtain from (5.268) the new smearing formula   +∞ n Z Y  x hF1 (x(τ1 )) · · · Fn (x(τn ))iΩ0 = d δxk Fk (x0 + δxk )   k=1 −∞ ) ( n n 1XX 1 −2 δxk aτk τk′ (x0 ) δxk′ , (5.269) × r h i exp − 2 ′ =1 n 2 k=1 k (2π) Det aτk τk′ (x0 ) 2 where a−2 τi τj (x0 ) is the inverse of the n × n -matrix aτi τj (x0 ). This smearing formula determines the harmonic expectation values in the variational perturbation expansion (5.263) as convolutions with Gaussian functions. For n = 1 and only the diagonal elements a2 (x0 ) = a2τ τ (x0 ) appear in the smearing formula (5.269), which reduces to the previous one in Eq. (5.30) [F (x(τ )) = V (x(τ ))]. For polynomials F (x(τ )), we set x(τ ) = x0 + δx(τ ) and expand in powers of δx(τ ), and see that the smearing formula (5.269) reproduces the Wick expansion (5.266). For two functions, the smearing formula (5.269) reads explicitly +∞ +∞ Z Z 1 dx1 dx2 F1 (x1 )F2 (x2 ) p (2π)2 [a4 (x0 ) − a4τ1 τ2 (x0 )] −∞ −∞ 2 2 2 a (x0 )(x1 −x0 ) −2aτ1τ2 (x0 )(x1 −x0 )(x2 − x0 )+a2 (x0 )(x2 −x0 )2 × exp − . 2[a4 (x0 ) − a4τ1 τ2 (x0 )] x

hF1 (x(τ1 ))F2 (x(τ2 ))iΩ0 =

Specializing F2 (x(τ2 )) to quadratic functions in x(τ ), we obtain from this D Ex0 a4τ1 τ2 (x0 ) 2 x0 2 F1 (x(τ1 )) [x(τ2 ) − x0 ] = hF1 (x(τ1 ))iΩ a (x0 ) 1 − 4 a (x0 ) Ω D Ex0 a4 (x ) τ1 τ2 0 + F1 (x(τ1 )) [x(τ1 ) − x0 ]2 , a4 (x0 ) Ω

(5.270)

(5.271)

and D Ex0 [x(τ1 ) − x0 ]2 [x(τ2 ) − x0 ]2 = a4 (x0 ) + 2a4τ1 τ2 (x0 ) . Ω

(5.272)

434


5.21.2

Higher-Order Smearing Formula in D Dimensions

The smearing formula can easily be generalized to D-dimensional systems, where the local pair correlation function (5.264) becomes a D × D -dimensional matrix: x

hδxi (τ )δxj (τ ′ )iΩ0 = a2ij;τ τ ′ (x0 ).

(5.273)

For rotationally-invariant systems, the matrix can be decomposed in the same way as the trial frequency Ω2ij (x0 ) in (5.95) into longitudinal and transversal components with respect to x0 : x0 ) + a2T ;τ τ ′ (r0 )PT ;ij (ˆ x0 ), a2ij;τ τ ′ (x0 ) = a2L;τ τ ′ (r0 )PL;ij (ˆ

(5.274)

where PL;ij (ˆ x0 ) and PT ;ij (ˆ x0 ) are longitudinal and transversal projection matrices introduced in (5.96). Denoting the matrix (5.274) by a2τ,τ ′ (x0 ), we can write the D-dimensional generalization of the smearing formula (5.275) as   +∞ n Z Y  0 hF1 (x(τ1 )) · · · Fn (x(τn ))ix = d δx F (x + δx ) k k 0 k Ω   k=1 −∞ ( ) n n 1 1XX −2 × r δxk aτk τk′ (x0 ) δxk′ . (5.275) h i exp − 2 ′ =1 n 2 k=1 k (2π) Det aτk τk′ (x0 )

The inverse D × D -matrix a−2 τk τk′ (x0 ) is formed by simply inverting the n × n -matrices −2 −2 aL;τk τk′ (r0 ), aT ;τk τk′ (r0 ) in the projection formula (5.274) with projection matrices PL (ˆ x0 ) and PT (ˆ x0 ): −2 a−2 x0 ) + a−2 x0 ). (5.276) τk τk′ (x0 ) = aL;τk τk′ (r0 )PL (ˆ T ;τk τk′ (r0 )PT (ˆ In D dimensions, the trial potential contains a D × D frequency matrix and reads M 2 Ω (x0 )(xi − x0i )(xj − x0j ) , 2 ij with the analogous decomposition Ω2ij (x0 ) = Ω2L (x0 )

x0i x0j δij − . r02

(5.277)

M 2 Ω (x0 )(xi − x0i )(xj − x0j ) . 2 ij

(5.278)

x0i x0j + Ω2T (x0 ) r02

The interaction potential (5.261) becomes x0 Vint (x) = V (x) −

To first order, the anisotropic smearing formula (5.275) reads r

hF1 (x(τ1 )iΩ0T ,ΩL =

+∞ Z 1 (x1L − r0 )2 x21T d3 x1 F1 (x1 ) p exp − − , 2a2L 2a2T (2π)3 a4T a2L

(5.279)

−∞

with the special cases D E r0 2 [δx(τ1 )]T

ΩT ,ΩL

=

D

2a2T ,

2

[δx(τ1 )]L

E r0

ΩT ,ΩL

= a2L .

(5.280)

Inserting this into formula (5.263) we obtain the first-order approximation for the effective classical potential x0

W1 (x0 ) = FΩ

1 + ¯hβ

Zh¯ β x0 x0 dτ1 hVint (x(τ1 ))iΩ,c ,

(5.281)

0



435

in agreement with the earlier result (5.97). To second-order, the smearing formula (5.275) yields [33] hF1 (x(τ1 )F2 (x(τ2 )irΩ0T ,ΩL

+∞ +∞ Z Z 3 = d x1 d3 x2 F1 (x1 )F2 (x2 ) −∞

−∞

2 2 aT x1T − 2a2T τ1 τ2 x1T x2T + a2T x22T q exp − × 2(a4T − a4T τ1 τ2 ) (2π)3 (a4T − a4T τ1 τ2 ) a4L − a4Lτ1 τ2 2 aL (x1L − r0 )2 − 2a2Lτ1 τ2 (x1L − r0 )(x2L − r0 ) + a2L (x2L − r0 )2 × exp − , 2(a4L − a4Lτ1 τ2 ) 1

(5.282)

so that rule (5.271) for expectation values generalizes to D

2

F1 (x(τ1 ) [δx(τ2 )]T

D

2

F1 (x(τ1 ) [δx(τ2 )]L

a4 r = 2a2T 1 − T τ41 τ2 hF1 (x(τ1 ))iΩ0T ,ΩL aT D E r0 a4 + T τ41 τ2 F1 (x(τ1 )) [δx(τ1 )]2T , aT ΩT ,ΩL a4Lτ1 τ2 r 2 = aL 1 − hF1 (x(τ1 ))iΩ0T ,ΩL a4L

r0 a4 + Lτ41 τ2 F1 (x(τ1 )[δx(τ1 )]2L ΩT ,ΩL . aL

E r0

ΩT ,ΩL

E r0

ΩT ,ΩL

(5.283)

(5.284)

Specializing F (x) to quadratic function, we obtain the generalizations of (5.272) D E r0 2 2 [δx(τ1 )]T [δx(τ2 )]T ΩT ,ΩL D E r0 2 2 [δx(τ1 )]T [δx(τ2 )]L ΩT ,ΩL D E r0 2 2 [δx(τ1 )]L [δx(τ2 )]L ΩT ,ΩL

5.21.3

=

4a4T + 4a4T τ1 τ2 ,

(5.285)

=

2a2T a2L ,

(5.286)

=

a4L + 2a4Lτ1 τ2 .

(5.287)

Isotropic Second-Order Approximation to Coulomb Problem

To demonstrate the use of the higher-order smearing formula (5.275), we calculate the effective classical potential of the three-dimensional Coulomb potential V (x) = −

e2 |x|

(5.288)

to second order in variational perturbation theory, thus going beyond the earlier results in Eq. (5.53) and Section 5.10. The interaction potential corresponding to (5.278) is x0 Vint (x) = −

M 2 e2 − Ω (x0 )(x − x0 )2 . |x| 2

(5.289)

For simplicity, we consider only the isotropic approximation with only a single trial frequency. Then all formulas derived in the beginning of this section have a trivial extension to three dimensions. Better results will, of course be obtained with two trial frequencies Ω2L (r0 ) and Ω2T (r0 ) of Section 5.9.

436


The Fourier transform 4πe2 /|k|2 of the Coulomb potential e2 /|x| is most conveniently written in a proper-time type of representation as 4πe2 V (k) = 2

Z

∞

dσe−σk

2

/2−ikx

,

(5.290)

0

where σ has the dimension length square. The lowest-order smeared potentials were calculated before in Section 5.10. For brevity, we consider here only the isotropic approximation in which longitudinal and transverse trial frequencies are identified [compare (5.112)]: D E x0 2 [x(τ1 ) − x0 ] = 3a2 (x0 ) ,

Ω

1 |x(τ1 )|

x0 Ω

# " 1 |x0 | = erf p . |x0 | 2a2 (x0 )

(5.291)

The first-order variational approximation to the effective classical potential (5.281) is then given by the earlier-calculated expression (5.115). To second order in variational perturbation theory we calculate expectation values D E x0 2 2 [x(τ1 ) − x0 ] [x(τ2 ) − x0 ] Ω x0 1 2 [x(τ1 ) − x0 ] |x(τ2 )| Ω

= =

9a4 (x0 ) + 6a4τ1 τ2 (x0 ) , 2[3a4 (x0 ) − a4τ1 τ2 (x0 )] p , πa6 (x0 )

(5.292) (5.293)

which follow from the obvious generalization of (5.271), (5.272) to three dimensions. More involved is the Coulomb-Coulomb correlation function

1 1 |x(τ1 )| |x(τ2 )|

x0 Ω

+∞ +∞ Z Z 1 dσ1 dσ2 p 3 2 2 [a (x0 ) + σ1 ][a (x0 ) + σ2 ] − a4τ1 τ2 (x0 ) 0 0 a2 (x0 ) + σ1 /2 + σ2 /2 − a2τ1 τ2 (x0 ) × exp −x20 2 . (5.294) [a (x0 ) + σ1 ][a2 (x0 ) + σ2 ] − a4τ1 τ2 (x0 )

=

1 2π

Using these smearing results we calculate the second connected correlation functions of the interaction potential (5.289) appearing in (5.263) and find the effective classical potential to second order in variational perturbation theory " # M e2 Ω(x0 ) 3M 2 Ω3 (x0 ) 4 − W2 (x0 ) = W1 (x0 ) + p l (x0 ) 4¯ h ¯h 2πa6 (x0 ) −

e4 2¯h

Zh¯ β 0

dτ

1 1 |x(τ )| |x(0)|

x0

,

(5.295)

Ω

with the abbreviation ¯h 4 + h ¯ 2 β 2 Ω2 (x0 ) − 4 cosh ¯hβΩ(x0 ) + h ¯ βΩ(x0 ) sinh h ¯ βΩ(x0 ) 4 l (x0 ) ≡ , 8βM 2 Ω3 (x0 ) sinh[¯ hβΩ(x0 )/2]

(5.296)

the symbol indicating that this is a quantity of dimension length to the forth power. After an extremization of (5.301) with respect to the trial frequency Ω(x0 ), which has to be done numerically, we obtain the second-order approximation for the effective classical potential of the Coulomb system plotted in Fig. 5.25 for various temperatures. The curves lie all below the first-order ones, and the difference between the two decreases with increasing temperature and increasing distance from the origin. H. Kleinert, PATH INTEGRALS

437

5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions 0

W1 (r0 ) -0.2

C

-0.4

W2 (r0 ) C

C CW

CW

-0.6

XXX y

-0.8

V (r0 )

-1 0.5

1

1.5

2

2.5

3

3.5

4

r0

Figure 5.25 Isotropic approximation to effective classical potential of Coulomb system in the first (lines) and second order (dots). The temperatures are 10−4 , 10−3 , 10−2 , 10−1 , and ∞ from top to bottom in atomic units. Compare also Fig. 5.9.

5.21.4

Anisotropic Second-Order Approximation to Coulomb Problem

The first-order effective classical potential W1 (x0 ) was derived in Eqs. (5.97) and (5.117)–(5.121). To obtain the second-order approximation W2 (x0 ), we insert the Coulomb potential in the representation (5.290) into the second-order smearing formula (5.282), and find

1 [δx(τ2 )]2T |x(τ1 )|

r0

1 2 [δx(τ2 )]L |x(τ1 )|

r0

=

r

=

r

ΩT ,ΩL

ΩT ,ΩL

2a2L π

Z1

2a2L π

Z1

dλ e

−

r2 0 2a2 L

λ2

−

r2 0 2a2 L

λ2

a6L + a4Lτ1 τ2 [r02 λ4 − a2L λ2 ] . a4L [(a2T − a2L )λ2 + a2L ]

0

dλ e

0

2a4T τ1 τ2 λ2 2a2T − 2 2 2 2 2 (aT − aL )λ + aL [(aT − a2L )λ2 + a2L ]2

,

(5.297)

These are special cases of the more general expectation value

a2L r02

exp −

+∞ r0 Z 2[a4L − a4Lτ1 τ2 + 2a2L σ] 1 1 q F (x(τ2 )) = 2 dσ |x(τ1 )| 2π 4 − a4 2 σ] a4 − a4 2 ΩT ,ΩL [a + 2a T T τ1 τ2 T L Lτ1 τ2 + 2aL σ 0 Z (a2L +2σ)(xL −r0 )2 +2a2Lτ1τ2 r0 (xL −r0 ) (a2T +2σ)x2T 3 × d xF (x) exp − 4 − , (5.298) 2[aT −a4T τ1 τ2 +2a2T σ] 2[a4L −a4Lτ1 τ2 +2a2Lσ]

which furthermore leads to

1 1 |x(τ1 )| |x(τ2 )|

r0

ΩT ,ΩL

2 = π

+∞ +∞ Z Z dσ1 dσ2 0

0

[a2T

1 ×q [a2L + 2σ1 ][a2L + 2σ2 ] − a4Lτ1 τ2

2σ1 ][a2T

1 + 2σ2 ] − a4T τ1 τ2

+ r02 [a2L + σ1 + σ2 − a2Lτ1 τ2 ] exp − 2 . [aL + 2σ1 ][a2L + 2σ2 ] − a4Lτ1 τ2

(5.299)

438


From these smearing results we calculate the second-order approximation to the effective classical potential x0

W2 (x0 ) = FΩ

1 + ¯hβ

Zh¯ β x0 x0 dτ1 hVint (x(τ1 ))iΩ,c − 0

1 2¯h2 β

Zh¯ β Zh¯ β x0 x0 x0 dτ1 dτ2 hVint (x(τ1 ))Vint (x(τ2 ))iΩ,c . (5.300) 0

0

The result is W2ΩT ,ΩL (r0 ) = W1ΩT ,ΩL (r0 ) r Z1 4 2 4 2 2 2 e2 M 2a2L [r0 λ − a2L λ2 ] 2ΩT lT4 λ2 ΩL l L + dλ − 4 2 e−r0 λ /2aL 2¯h π [(a2T − a2L )λ2 + a2L ]2 aL [(aT − a2L )λ2 + a2L ] 0

4 e4 M 2 [2Ω3T lT4 + Ω3L lL ] − 2 − 4¯h 2¯ h β

Zh¯ β 0

dτ1

Zh¯ β 0

dτ2

1 1 |x(τ1 )| |x(τ2 )|

r0

,

(5.301)

ΩT ,ΩL ,c

with the abbreviation 4 lT,L

¯h 4 + h ¯ 2 β 2 Ω2T,L − 4 cosh ¯hβΩT,L + h ¯ βΩT,L sinh h ¯ βΩT,L = , 8βM 2 Ω3T,L sinh[¯ hβΩT,L /2]

(5.302)

which is a quantity of dimension (length)4 . After an extremization of (5.115) and (5.301) with respect to the trial frequencies ΩT , ΩL which has to be done numerically, we obtain the second-order approximation for the effective classical potential of the Coulomb system plotted in Fig. 5.26 for various temperatures. The second order curves lie all below the first-order ones, and the difference between the two decreases with increasing temperature and increasing distance from the origin.

5.21.5

Zero-Temperature Limit

As a cross check of our result we take (5.301) to the limit T → 0. Just as in the lowest-order discussion in Sect. (5.4), the x0 -integral can be evaluated in the saddle-point approximation which becomes exact in this limit, so that the minimum of WN (x0 ) in x0 yields the nth approximation (0) to the free energy at T = 0 and thus the nth approximations EN the ground state energy E (0) of the Coulomb system. In this limit, the results should coincide with those derived from a direct variational treatment of the Rayleigh-Schrödinger perturbation expansion in Section 3.18. With the help of such a treatment, we shall also carry the approximation to the next order, thereby illustrating the convergence of the variational perturbation expansions. For symmetry reasons, the minimum of the effective classical potential occurs for all temperatures at the origin, such that we may restrict (5.115) and (5.301) to this point. Recalling the zero-temperature limit of the two-point correlations (5.19) from (3.246), lim a2τ τ ′ (x0 ) =

β→∞

¯h exp {−Ω(x0 ) |τ − τ ′ |} , 2M Ω(x0 )

(5.303)

we immediately deduce for the first order approximation (5.115) with Ω = Ω(0) the limit r 3 2 MΩ 2 (0) Ω E1 (Ω) = lim W1 (0) = ¯hΩ − √ e . (5.304) β→∞ 4 ¯h π In the second-order expression (5.301), the zero-temperature limit is more tedious to take. Performing the integrals over σ1 and σ2 , we obtain the connected correlation function s x0 a2τ1 τ2 (0) 1 1 1 2 2 = 4 − 4 arctan −1− 2 . (5.305) |x(τ1 )| |x(τ2 )| Ω,c aτ1 τ2 (0) πaτ1 τ2 (0) aτ1 τ2 (0) πaτ1 τ2 (0) H. Kleinert, PATH INTEGRALS


439

Inserting (5.303), setting τ1 = 0 and integrating over the imaginary times τ = τ2 ∈ [0, ¯hβ], we find Zh¯ β

dτ

0

1 1 |x(τ )| |x(0)|

x0

4M 2 β→∞ h ¯ βΩ Ω,c ≈

¯ 2 β 2 Ω2 h 2 − eh¯ βΩ − 1 − ¯hβΩ − π π



p 1 1 1  × eh¯ βΩ arcsin 1 − e−2¯hβΩ + ln α(β) − [ln α(β)]2 − 2 8 2

Z1

α(β)

with the abbreviation

  ln u  du  , 1+u  

√ 1 − 1 − e−2¯hβΩ √ α(β) = . 1 + 1 − e−2¯hβΩ

(5.306)

(5.307)

Inserting this into (5.301) and going to the limit β → ∞ we obtain r MΩ 2 4 9 3 π M 4 (0) Ω E2 (Ω) = lim W2 (0) = ¯hΩ − √ e − 1 + ln 2 − e . β→∞ 16 2 π ¯h π 2 ¯h2

(5.308)

Postponing for a moment the extremization of (5.304) and (5.308) with respect to the trial frequency Ω, let us first rederive this result from a variational treatment of the ordinary RayleighSchrödinger perturbation expansion for the ground state energy in Section 3.18. According to the replacement rule (5.186), we must first calculate the ground state energy for a Coulomb potential in the presence of a harmonic potential of frequency ω: V (x) =

e2 M 2 2 ω x − . 2 |x|

(5.309)

-0.2 W1ΩT =ΩL (r0 ) A W1ΩT 6=ΩL (r0 ) A A A A A A A AU A A A y AU X XX XX XXX y X XX W2ΩT =ΩL (r0 ) XXX X Ω 6=Ω W2 T L (r0 )

-0.4

-0.6

-0.8

-1

-1.2

X y XX XXX X

0

1

2

V (r0 )

3

4

5

r0 Figure 5.26 Isotropic and anisotropic approximations to effective classical potential of Coulomb system in first and second order at temperature 0.1 in atomic units. The lowest line represents the high temperature limit in which all isotropic and anisotropic approximations coincide.

440


√ After this, we make the trivial replacement ω → Ω2 + ω 2 − Ω2 and re-expand the energy in powers of ω 2 − Ω2 , considering this quantity as being of the order e2 and truncating the reexpansion accordingly. At the end we go to ω = 0, since the original Coulomb system contains no oscillator potential. The result of this treatment will be precisely the expansions (5.304) and (5.308). (0) The Rayleigh-Schrödinger perturbation expansion of the ground state energy EN (ω) for the potential (5.309) in Section 3.18 requires knowledge of the matrix elements of the Coulomb potential (5.288) with respect to the eigenfunctions of the harmonic oscillator with the frequency ω: Vn,l,m;n′ ,l′ ,m′ =

Z2π

dϕ

0

Zπ

dϑ sin ϑ

0

Z∞

∗ dr r2 ψn,l,m (r, ϑ, ϕ)

−e2 ψn′ ,l′ ,m′ (r, ϑ, ϕ), r

(5.310)

0

where [see (9.67), (9.68), and (9.53)] s ψn,l,m (r, ϑ, ϕ)

=

r (l+1)/2 2n! Mω Mω 2 4 r Γ(n + l + 3/2) ¯h ¯h Mω 2 Mω 2 l+1/2 ×Ln r exp − r Yl,m (ϑ, ϕ) . ¯h 2¯ h

(5.311)

Here n denotes the radial quantum number, Lα n (x) the Laguerre polynomials and Yl,m (ϑ, ϕ) the spherical harmonics obeying the orthonormality relation Z2π 0

dϕ

Zπ

∗ dϑ sin ϑ Yl,m (ϑ, ϕ)Yl′ ,m′ (ϑ, ϕ) = δl,l′ δm,m′ .

(5.312)

0

Inserting (5.311) into (5.310), and evaluating the integrals, we find s r M ω Γ(l + 1)Γ(n + 1/2) Γ(n′ + l + 3/2) 2 Vn,l,m;n′ ,l′ ,m′ = −e π¯h Γ(l + 3/2) n!n′ !Γ(n + l + 3/2) 1 3 1 ′ × 3 F2 −n , l + 1, ; l + , − n; 1 δl,l′ δm,m′ , 2 2 2

(5.313)

with the generalized hypergeometric series [compare (1.450)] 3 F2 (α1 , α2 , α3 ; β1 , β2 ; x)

=

∞ X (α1 )k (α2 )k (α3 )k xk (β1 )k (β2 )k k!

(5.314)

k=0

and the Pochhammer symbol (α)k = Γ(α + k)/Γ(α). These matrix elements are now inserted into the Rayleigh-Schrödinger perturbation expansion for the ground state energy X V0,0,0;n,l,m Vn,l,m;0,0,0 ′ E (0) (ω) = E0,0,0 + V0,0,0;0,0,0 + E0,0,0 − En,l,m n,l,m

− +

X

′

X

′

V0,0,0;0,0,0

n,l,m

V0,0,0;n,l,m Vn,l,m;0,0,0 2

[E0,0,0 − En,l,m ]

X V0,0,0;n,l,m Vn,l,m;n′ ,l′ ,m′ Vn′ ,l′ ,m′ ;0,0,0 ′ + ... , [E0,0,0 − En,l,m ] [E0,0,0 − En′ ,l′ ,m′ ] ′ ′ ′

(5.315)

n,l,m n ,l ,m

the denominators containing the energy eigenvalues of the harmonic oscillator 3 En,l,m = h ¯ ω 2n + l + . 2

(5.316)



441

Figure 5.27 Approach of the variational approximations of first, second, and third order to the correct ground state energy −0.5, in atomic units. The primed sums in (5.315) run over all values of the quantum numbers n, l = −∞, . . . , +∞ and m = −l, . . . , +l, excluding those for which the denominators vanish. For the first three orders we obtain from (5.313)–(5.316) r r 3 Mω 2 4 M3 6 2 π M 4 (0) E (ω) = ¯hω − √ e − 1 + ln 2 − e −c e + ... , (5.317) 2 2 π ¯h π 2 ¯h ¯h7 ω with the constant c

∞ X ∞ ∞ X X 1 1 · 3 · · · (2n − 1) 1 · 3 · · · · (2n − 1) − 2 · 4 · · · 2n n2 (n + 1/2) n=1 ′ 2 · 4 · · · 2n π 3/2 n=1 n =1 ) 1 · 3 · · · (2n′ − 1) 3 F2 −n′ , l + 1, 12 ; l + 23 , 12 − n; 1 × ≈ 0.031801 . 2 · 4 · · · 2n′ n n′ (n + 1/2)

=

1

(

(5.318)

Since we are interested only in the energies in the pure Coulomb system with ω = 0, the variational re-expansion √ procedure described after (5.309) becomes particularly simple: We simply have to replace ω by Ω2 − Ω2 which is appropriately re-expanded in the second Ω2 , thereby considering Ω2 as a quantity of order e2 . For the first term in the energy (5.317) which is proportional to Ω itself this amounts to a multiplication by a factor (1 − 1)1/2 which is re-expanded in the second 1 5 = 16 . The term 3ω/2 in (5.317) becomes therefore “1” up to the third order as 1 − 12 − 18 − 16 1/2 15/32ω. By the same rule, the factor ω in the second term of the energy (5.317) goes over into 3 21 Ω1/2 (1 − 1)1/4 , re-expanded to second order in the second ”1”, i.e., into Ω1/4 (1 − 41 − 32 ) = 32 . The next term in (5.317) happens to be independent of ω and needs no re-expansion, whereas the last term remains unchanged since it is already of highest order in e2 . In this way we obtain from (5.317) the third-order variational perturbation expansion r r 15 21 MΩ 2 4 π M 4 M3 6 (0) E3 (Ω) = ¯hΩ − √ e − 1 + ln 2 − e −c e . (5.319) 2 32 ¯h π 2 ¯h 16 π ¯h7 Ω Extremizing (5.304), (5.308), and (5.317) successively with respect to the trial frequency Ω we find to orders 1, 2, and 3 the optimal values M e4 , ¯h3 with c′ ≈ 0.52621. The corresponding approximations to the ground state energy are Ω 1 = Ω2 =

16 M e4 , 9π ¯h3

(0)

EN (ΩN ) = −γN

Ω3 = c′

M e4 , ¯h2

with the constants 4 5 + 4 ln 2 γ1 = ≈ 0.42441 , γ2 = − 2 ≈ 0.47409 , γ3 ≈ 0.49012 , 3π π approaching exponentially fast the exact value γ = 0.5, as shown in Fig. 5.27.

(5.320)

(5.321)

(5.322)

442

5.22


Polarons

An important role in the development of variational methods for the approximate solution of path integrals was played by the polaron problem [34]. Polarons arise when electrons travel through ionic crystals thereby producing an electrostatic deformation in their neighborhood. If Pi (x, t) denotes electric polarization density caused by the displacement of the positive against the negative ions, an electron sees a local ionic charge distribution ρi (x, t) = ∇ · Pi (x, t),

(5.323)

which gives rise to an electric potential satisfying ∇2 A0 (x, t) = 4π ∇ · Pi (x, t).

(5.324)

The Fourier transform of this, ≡

Z

∞

Pik (t) =

Z

∞

A0k (t)

−∞

and that of Pi (x, t),

are related by

−∞

d3 x A0 (x, t)e−ikp ,

(5.325)

d3 x Pi (x, t)e−ikx ,

(5.326)

4π ik · Pik (t). (5.327) k2 Only longitudinal phonons which have Pik (t) ∝ k and correspond to density fluctuations in the crystal contribute. For these, an electron at position x(t) experiences an electric potential A0k (t) = −

A0 (x, t) = −

X k

X 4π eikx eikx A0k (t) √ = i Pk (τ ) √ . V V k |k|

(5.328)

In the regime of optical phonons, each Fourier component oscillates with approximately the same frequency ω, the frequency of longitudinal optical phonons. The variables Pik (t) have therefore a Lagrangian L(t) =

i 1 Xh ˙ i ˙ i (t) − ω 2Pi (t)Pi (t) , P−k (t)P k −k k 2µ k

(5.329)

∗

with some material constant µ and Pi−k (t) = Pik (t), since the polarization is a real field. This can be expressed in terms of measurable properties of the crystal. For this we note that the interaction of the polarization field with a given total charge distribution ρ(x, t) is described by a Lagrangian Lint (t) = −

Z

d3 x ρ(x, t)V (x, t).


443

5.22 Polarons

Inserting (5.325) and performing a partial Z 1 (5.331) Lint (t) = 4π d3 x 2 ∇ρ(x, t) · Pi (x′ , t). ∇ Recalling the Gauss law ∇ · D(x, t) = 4πρ(x, t) we identify the factor of Pi (x, t) with the total electric displacement field and write Lint (t) = 4π

Z

d3 x D(x, t) · Pi (x, t).

In combination with (5.329) this leads to an equation of motion i 1 hï Pk (t) + ω 2 Pik (t) = Dk (t). µ

(5.332)

(5.333)

If we go over to the temporal Fourier components Piω′ ,k of the ionic polarization, we find the relation 1 2 ω − ω ′2 Piω′ ,k = Dω′ ,k . (5.334) µ For very slow deformations, this becomes ω2 i P ′ ≈ D0,k . µ ω ,k

(5.335)

Using the general relation Dω′ ,k = Eω′ ,k + 4πPω′ ,k ,

(5.336)

where 4πPω′ ,k contains both ionic and electronic polarizations, we obtain 1 4πPω′ ,k = 1 − Dω′ ,k , (5.337) ǫω′ with ǫω′ being the dielectric constant at frequency ω ′. For a slowly moving electron, the lattice deformations have small frequencies, and we can write the time-dependent equation 1 4πPk (t) ≈ 1 − Dk (t). (5.338) ǫ0 By comparison with Eq. (5.335) we determine the parameter µ. Before we can do so, however, we must subtract from (5.338) the contribution of the electrons. These fulfill the approximate time-dependent equation 1 el 4πPk (t) ≈ 1 − Dk (t), (5.339) ǫ∞ where ǫ∞ is the dielectric constant at high frequency where only electrons can follow the field oscillations. The purely ionic polarization field is therefore given by 1 1 i 4πPk (t) ≈ − Dk (t). (5.340) ǫ∞ ǫ0 By comparison with (5.334) we identify ω2 µ≈ 4π

1 1 − . ǫ∞ ǫ0

(5.341)

444


5.22.1

Partition Function

The partition function of the combined system of an electron and the oscillating polarization is therefore described by the imaginary-time path integral Z = +

Z

3

D x

X k

YZ k

1 − h ¯

3

D Pk exp

Z

hβ ¯ 0

dτ

(

M 2 x˙ (τ ) 2

i X 4π 1 h˙i eikx(τ ) ˙ i (t) − ω 2Pi (t)Pi (t) + ie √ P−k (t)P P (τ ) k k −k k 2µ V k |k|

)!

, (5.342)

where V is the volume of the system. The path integral is Gaussian in the Fourier components Pk (τ ). These can therefore be integrated out with the rules of Subsection 3.8.2. For the correlation function of the polarizations we shall use the representation of the Green function (3.248) as a sum of periodic repetitions of the zero-temperature Green function hPk (τ )P−k (τ )i =

∞ h ¯ X ′ e−ω|τ −τ +n¯hβ| . 2ω n=−∞

(5.343)

Abbreviating ∞ X

′

n=−∞

′

−τ | e−ω|τ −τ +n¯hβ| ≡ e−ω|τ = per

1 1 −ω|τ −τ ′ | ′ e − e−ω(¯hβ−|τ −τ |) , (5.344) 2ω 1 − e−¯hβω

the right-hand side being valid for h ¯ β > τ, τ ′ > 0, we find Z=

Z

3

(

1 ¯hβ M 2 dτ x˙ (τ ) h ¯ 0 2 ) Z ¯hβ Z ¯ 1 X 4π ¯hβ µ 2 h ′ ik[x(τ )−x(τ ′ )] −ω|τ −τ ′ | + 2 4πe dτ dτ e eper . (5.345) 2ω V k k2 0 0 2¯ h

D x exp −

Z

Performing the sum over all wave vectors k using the formula 1 X 4π ikx 1 e = , 2 V k k |x|

(5.346)

we obtain the path integral Z=

Z

(

1 D x exp − h ¯ 3

"Z

0

M a dτ x˙ 2 (τ ) − √ 2 2 2

hβ ¯

Z

0

hβ ¯

dτ

Z

0

hβ ¯

′

−τ | e−ω|τ per dτ |x(τ )−x(τ ′ )| ′

#)

, (5.347)

where

µ h ¯ √ 4πe2 2. (5.348) h ¯ 2ω √ The factor 2 is a matter of historic convention. Staying with this convention, we use the characteristic length scale (9.73) associated with the mass M and the frequency ω: s h ¯ λω ≡ . (5.349) Mω a≡


445

5.22 Polarons

This length scale will appear in the wave functions of the harmonic oscillator in Eq. (9.73). Using this we introduce a dimensionless coupling constant α defined by 1 1 1 α≡√ − 2 ǫω ǫ0

e2 . h ¯ ωλω

(5.350)

A typical value of α is 5 for sodium chloride. In different crystals it varies between 1 and 20, thus requiring a strong-coupling treatment. In terms of α, one has a=h ¯ ω 2 λω α.

(5.351)

The expression (5.348) is the famous path integral of the polaron problem written down in 1955 by Feynman [20] and solved approximately by a variational perturbation approach. In order to allow for later calculations of a particle density in an external potential, we decompose the paths in a Fourier series with fixed endpoints x(β) ≡ xb = xa : x(τ ) = xa +

∞ X

xn sin νn τ,

n=1

νn ≡ nπ/¯ hβ.

(5.352)

The path integral is then the limit N → ∞ of the product of integrals Z

D3x ≡

Z

2

d xa q

2π¯ h2 β/M

3

 Z ∞ Y   q

3

d xn

4π/Mνn2 β

n=1



3.



(5.353)

The correctness of this measure is verified by considering the free particle in which case the action is A0 =

∞ 1X A0 x2 , 2 n=1 n n

with

A0n ≡

M h ¯ β νn2 . 2

(5.354)

The Fourier components xn can be integrated leaving a final integral Z=

Z

d2 xa q

2π¯ h2 β/M

3,

(5.355)

which is the correct partition function of a free particle [compare with the onedimensional expression (3.808)]. The endpoints xb = xa do not appear in the integrand of (5.347) as a manifestation of translational invariance. The integral over the endpoints produces therefore a total volume factor V . We may imagine performing the path integral with fixed endpoints which produces the particle density.

446


5.22.2

Harmonic Trial System

The harmonic trial system used by Feynman as a starting point of his variational treatment has the generating functional Z

Ω,C

Z

[j] =

3

D x exp

(

"

#

1 ¯hβ M 2 − dτ x˙ (τ ) − j(τ )x(τ ) h ¯ 0 2 ) Z Z ¯hβ C M ¯hβ ′ ′ 2 −Ω|τ −τ ′ | − dτ dτ [x(τ ) − x(τ )] eper . (5.356) 2 h ¯ 0 0 Z

The external current is Fourier-decomposed in the same way as x(τ ) in (5.352). To preserve translational invariance, we assume the current to vanish at the endpoints: ja = 0. Then the first two terms in the action in (5.356) are ∞ 1X A0n x2n − β jn xn . A[ j ] = 2 n=1

(5.357)

The Fourier decomposition of the double integral in (5.356) reads Z

hβ ¯ 0

dτ

Z

hβ ¯

0

dτ

′

"

∞ X

′

n=1

(sin νn τ − sin νn τ ) xn

#2

′

−τ | e−Ω|τ . per

(5.358)

With the help of trigonometric identities and a change of variables to σ = (τ + τ ′ )/2 and ∆τ = (τ − τ ′ )/2, this becomes 2

Z

0

hβ ¯

dσ

Z

2¯ hβ

0

d∆τ

"

∞ X

cos νn σ sin(νn ∆τ /2) xn

n=1

#2

−Ω|∆τ | eper .

(5.359)

Integrating out σ leaves h ¯β

Z

0

2¯ hβ

d∆τ

∞ X

| sin2 (νn ∆τ /2) x2n e−Ω|∆τ , per

(5.360)

n=1

and performing the integral over δτ gives for (5.358) the result ∞ X h ¯β ′ −2¯ hβΩ 1−e 1+2 e−n ¯hβΩ 2 n′ =1

!

∞ X

x2n

n=1

νn2 . νn2 + Ω2

(5.361)

Hence we can write the interaction term in (5.356) as −

∞ Cβ Mβ X ν2 x2n 2 n 2 , 4Ω n=1 νn + Ω

Cβ ≡ C 1 − e−2¯hβΩ coth

h ¯ βΩ . 2

(5.362)

!

(5.363)

This changes A0n in (5.357) into M¯ hβνn2 Cβ 1 An ≡ 1+ 2 2 Ω νn + Ω2

!

=

A0n

Cβ 1 1+ . 2 Ω νn + Ω2


447

5.22 Polarons

The trial partition function without external source is then approximately equal to Z

Ω,C

≡

T ≈0

Z

3

D x

Z

d2 xa q

2π¯ h2 β/M

3

∞ Y

n=1

s

3

A0n . An

(5.364)

The product is calculated as follows: ∞ Y

n=1

s

3

∞ Y A0n = An n=1

−3

s

1 Cβ 1+ 2 Ω νn + Ω2

= e−βF

Ω,C

,

(5.365)

resulting in the approximate free energy F Ω,C =

T ≈0

∞ νn2 + Γ2β 1X log 2 , β n=1 νn + Ω2

(5.366)

where we have introduced the function of the trial frequency Ω: Γ2β (Ω) ≡ Ω2 + Cβ /Ω.

(5.367)

With the help of formula (2.166) we find therefore F Ω,C =

3 sinh h ¯ βΓβ log . 2β sinh h ¯ βΩ

(5.368)

For simplicity, we shall from now on consider only the low-temperature regime where Cβ − > C, Γβ → Ω2 + C/Ω, and the free energy (5.368) becomes approximately F Ω,C =

T ≈0

3¯ h (Γ − Ω) ≡ E0Ω,C . 2

(5.369)

The right-hand side is the ground state energy of the harmonic trial system (5.356). In Feynman’s variational approach, the ground state energy of the polaron is smaller than this given by the minimum of [compare (5.18), (5.32) and (5.45)] Ω,C Ω,C − ∆Eint,harm , E0 ≤ E0Ω,C + ∆Eint

(5.370)

where the two additional terms are the limits β → ∞ of the harmonic expectation values Ω,C ∆Eint

*

1 1 a =− hAint iΩ,C ≡ − √ h ¯β h ¯β 2 2

Z

0

hβ ¯

dτ

Z

0

hβ ¯

′

e−ω|τ −τ | dτ |x(τ ) − x(τ ′ )| ′

+Ω,C

, (5.371)

and Ω,C ∆Eint,harm

1 1 =− hAint,harm iΩ,C ≡ h ¯β h ¯β

*

+Ω,C C Z ¯hβ Z ¯hβ ′ ′ 2 −Ω|τ −τ ′ | dτ dτ [x(τ )−x(τ )] e . 2 0 0 (5.372)

448


The calculation of the first expectation value is most easily done using the Fourier decomposition (5.345), where we must find the expectation value D

eik[x(τ )−x(τ

′ )]

EΩ,C

.

(5.373)

In the trial path integral (5.356), the exponential corresponds to a source h

i

j(τ ′′ ) = h ¯ k δ (3) (τ − τ ′′ ) − δ (3) (τ ′ − τ ′′ ) ,

(5.374)

in terms of which (5.373) reads D R

ei


Introducing the correlation function hxi (τ )xj (τ ′ )i

Ω,C

EΩ,C

.

(5.375)

≡ δij GΩ,Γ (τ, τ ′ ),

(5.376)

and using Wick’s rule (3.306) for harmonically fluctuating paths, the expectation value (5.375) is equal to D R i

e


EΩ,C

(

1 = exp − 2 2¯ h

Z

hβ ¯

0

dτ

Z

hβ ¯

0

′

dτ j(τ ) G

Ω,Γ

′

′

)

(τ, τ ) j(τ ) . (5.377)

Inserting the special source (5.373), we obtain D

eik[x(τ )−x(τ

′ )]

EΩ,C

h

i

¯ Ω,Γ (τ, τ ′ ) , = I Ω,C (k, τ, τ ′ ) ≡ exp k2 G

where the exponent contains the subtracted Green function

¯ Ω,Γ(τ, τ ′ ) ≡ GΩ,Γ(τ, τ ′ )− 1 GΩ,Γ (τ, τ )− 1 GΩ,Γ (τ ′ , τ ′ ). G 2 2

(5.378)

(5.379)

The Green function GΩ,Γ(τ, τ ′ ) itself has the Fourier expansion GΩ,Γ (τ, τ ′ ) = h ¯

∞ X

∞ h ¯ X sin νn τ sin νn τ ′ νn2 + Ω2 = sin νn τ sin νn τ ′ . 2 (ν 2 + Γ2 ) 2A /¯ h β M ν n β n=1 n=1 n n

(5.380)

It solves the Euler-Lagrange equation which extremizes the action in (5.356) for a source j(τ ) = M (τ − τ ′ ): (

−∂τ2

+ 2C

Decomposing

Z

hβ ¯ 0

′

dτ [xi (τ ) − xi (τ

′

−τ ′ | )] e−Ω|τ per

)

GΩ,Γ (τ, τ ′ ) = δ(τ − τ ′ ).

Γ2β − Ω2 νn2 + Ω2 Ω2 1 1 = × + × 2 , 2 2 2 2 2 2 νn (νn + Γβ ) Γβ νn Γβ νn + Γ2β

(5.381)

(5.382)


449

5.22 Polarons

we obtain a combination of ordinary Green functions of the second-order operator differential equation (3.236), but with Dirichlet boundary conditions. For such Green functions, the spectral sum over n was calculated in Section 3.4 for real time [see (3.36) and (3.145)]. The imaginary-time result is Gω2 (τ, τ ′ ) =

∞ X

sin νn τ sin νn τ ′ sinh ω(¯ hβ − τ ) sinh ωτ ′ = , for τ > τ ′ > 0. (5.383) 2 + ω2 ν ω sinh ω¯ h β n n=1

In the low-temperature limit, this becomes Gω2 (τ, τ ′ ) = such that

1 −ω(τ −τ ′ ) ′ e − e−ω(τ +τ ) , for τ > τ ′ > 0, 2ω

(5.384)

¯ Γ2 (τ, τ ′ ) = 1 e−Γ(τ −τ ′ ) − 1 − e−Γ(τ +τ ′ ) + 1 e−2Γτ + 1 e−2Γτ ′ , for τ > τ ′ > 0. G 2Γ 2 2 (5.385) 1 ′ In the limit Γ → 0, this becomes − 2 |τ − τ |. We therefore obtain at zero temperature

h ¯ Ω2 ¯ Γ2 − Ω2 ¯ ′ G (τ, τ ) + G0 (τ, τ ′ ) (5.386) 0 T =0 2M Γ2 Γ2 ( ) Γ2 −Ω2 1 −2Γτ 1 −2Γτ ′ Ω2 h ¯ ′ −Γ|τ −τ ′ | −Γ(τ +τ ′ ) |τ −τ | + 1−e +e − e − e , =− 2M Γ2 Γ3 2 2 "

¯ Ω,Γ (τ, τ ′ ) = G

#

D

′

E

to be inserted into (5.378) to get the expectation value eik[x(τ )−x(τ )] . The last three terms can be avoided by shifting the time interval under consideration and thus the Fourier expansion (5.352) from (0, h ¯ β) to (−¯ hβ/2, h ¯ β/2), which changes Green function (5.387) to ∞ X

sin νn (τ + h ¯ β/2) sin νn (τ ′ + h ¯ β/2) G (τ, τ ) = νn2 + ω 2 n=1 sinh ω(¯ hβ/2 − τ ) sinh ω(τ ′ + h ¯ β/2) = , for τ > τ ′ > 0. ω sinh ω¯ hβ ω2

′

(5.387)

We have seen before at the end of Section 3.20 that such a shift is important when discussing the limit T → 0 which we want to do in the sequel. With the symmetric limits of integration, the Green function (5.384) looses its last term [compare with (3.147) for real times] and (5.386) simplifies to ¯ Ω,Γ (τ, τ ′ ) ≈ G

T =0

h ¯ − 2M

(

Ω2 Γ2 − Ω2 ′ −Γ|τ −τ ′ | |τ − τ | + 1 − e . (5.388) Γ2 Γ3 )

At any temperature, we have the complicated expression for τ > τ ′ : ¯ ¯ Ω,Γ (τ, τ ′ ) = − h G 2M

(

Ω2 1 2 τ − τ′ − (τ − τ ′ ) 2 Γβ h ¯β "

#

(5.389)

Γ2 − Ω2 − 2 β [sinh Γβ (¯ hβ/2−τ ) sinh Γβ (τ ′ + h ¯ β/2)−(τ ′ → τ )−(τ → τ ′ ) Γβ sinh h ¯ βΓβ

#)

.

450


With the help of the Fourier integral (5.346) we find from this the expectation value of the interaction in (5.347): *

+

1 = |x(τ ) − x(τ ′ )|

Z

Z d3 k 4π D ik[x(τ )−x(τ ′ )] E d3 k 4π Ω,C e = I (k, τ, τ ′ ). (5.390) (2π)3 k2 (2π)3 k2

For zero temperature, this leads directly to the expectation value of the interaction in (5.347): Z

0

hβ ¯

dτ

Z

hβ ¯

0

dτ

′

*

′

e−ω|τ −τ | |x(τ ) − x(τ ′ )|

+

d3 k 4π k2 G¯ Ω,Γ (∆τ,0) e (2π)3 k2 0 Z ∞ h ¯β e−ω∆τ ≈ 4√ d∆τ q . (5.391) ¯ Ω,Γ (∆τ, 0) 2πωλω 0 −2G ≈ 2¯ hβ

Z

hβ/2 ¯

d∆τ e−ω∆τ

Z

The expectation value of the harmonic trial interaction in (5.356), on the other hand, is simply found from the correlation function (5.376) [or equivalently from the second derivative of I Ω,C (k, τ, τ ′ ) with respect to the momenta]: E 2 Ω,C

D

[x(τ ) − x(τ ′ )]

Z

hβ ¯

¯ Ω,Γ (τ, τ ′ ). = −6G

(5.392)

At low temperatures, this leads to an integral CM 2¯ h

Z

0

hβ ¯

dτ

0

E 2 Ω,C

D

dτ ′ [x(τ ) − x(τ ′ )]

′

−τ | e−Ω|τ = h ¯β per T =0

3C . 4ΩΓ

(5.393)

This expectation value contributes to the ground state energy a term Ω,C ∆Eint, var =

3¯ hC . 4ΩΓ

(5.394)

Note that this term can be derived from the √ derivative of the ground state energy (5.368) as C∂C E0Ω,C . Together with −a/2 2 times the result of (5.391), the inequality (5.370) for the ground state energy becomes E0 ≤

3¯ h αω ¯ω √ (Γ − Ω)2 − h 4Γ πω

Z

∞ 0

e−ω∆τ d∆τ q . ¯ Ω,Γ (∆τ, 0) −2G

(5.395)

This has to be minimized in Ω and C, or equivalently, in Ω and Γ. Considering the low-temperature limit, we have taken the upper limit of integration to infinity (the frequency ω corresponds usually to temperatures of the order of 1000 K). For small α, the optimal parameters Ω and Γ differ by terms of order α. We can therefore expand the integral in (5.395) and find that the minimum lies, in natural units with h ¯ = ω = 1, at Ω = 3 and Γ = 3[1 + 2α(1 − P )/3Γ, where 1/2 P = 2[(1 − Γ) − 1]. From this we obtain the upper bound E0 ≤ −α −

α2 + . . . ≈ −α − 0.0123α2 + . . . . 81

(5.396)


451

5.22 Polarons

This agrees well with the perturbative result [21] Ewex = −α − 0.0159196220α2 − 0.000806070048α3 − O(α4).

(5.397)

The second term has the exact value "

√ 1 √ − log 1 + 3 2/4 α2 . 2 #

(5.398)

In the strong-coupling region, the best parameters are Ω = 1, Γ = 4α2 /9π−[4(log 2+ γ/2) − 1], where γ ≈ 0.5773156649 is the Euler-Mascheroni constant (2.461). At these values, we obtain the upper bound 1 α2 E0 ≤ − −3 + log 2 + O(α−2 ) ≈ −0.1061α − 2.8294 + O(α−2 ). 3π 4

(5.399)

This agrees reasonably well with the precise strong-coupling expansion [23]. Esex = −0.108513α2 − 2.836 − O(α−2).

(5.400)

The numerical results for variational parameters and energy are shown in Table 5.12.

Table 5.12 Numerical results for variational parameters and energy.

α Γ 1 3.110 3 3.421 5 4.034 7 5.810 9 9.850 11 15.41 15 30.08

5.22.3

(2)

Ω E0 2.871 -1.01 2.560 -3.13 2.140 -5.44 1.604 -8.11 1.282 -11.5 1.162 -15.7 1.076 -26.7

∆E0 Etot correction -0.0035 -1.02 0.35 % -0.031 -3.16 1.0 % -0.083 -5.52 1.5 % -0.13 -8.24 1.6 % -0.17 -11.7 1.4 % -0.22 -15.9 1.4 % -0.39 -27.1 1.5 %

Effective Mass

By performing a shift in the velocity of the path integral (5.347), Feynman calculated also an effective mass for the polaron. The result is M eff



α ω = M  1 + √ Γ2 3 πω

Z

∞ 0



(∆τ )2 e−ω∆τ  d∆τ q . ¯ Ω,Γ (∆τ, 0) −2G

(5.401)

The reduced effective mass m ≡ M eff /M has the weak-coupling expansion mw = 1 +

α + 2.469136 × 10−2 α2 + 3.566719 × 10−3 α3 + . . . 6

(5.402)

452


and behaves for strong couplings like 16 4 4 α − (1 + log 4) α2 + 11.85579 + . . . 81π 2 3π ≈ 0.020141α4 − 1.012775α2 + 11.85579 + . . . .

ms ≈

(5.403)

The exact expansions are [31]

5.22.4

α + 2.362763 × 10−2 α2 + O(α4 ), 6 = 0.0227019α4 + O(α2).

mex = 1+ w

(5.404)

mex s

(5.405)

Second-Order Correction

With some effort, also the second-order contribution to the variational energy has been calculated at zero temperature [32]. It gives a contribution to the ground state energy (2)

∆E0

= −

E 1 1 D 2 Ω,C (A − A ) . int int,harm c 2¯ hβ h ¯2

(5.406)

Recall the definitions of the interactions in Eqs. (5.371) and (5.372). There are three terms i 1 1 D 2 EΩ,C h (2,1) Ω,C 2 = − A − hA i , (5.407) ∆E0 int int 2¯ hβ h ¯2 (2,2)

∆E0

= 2

o 1 1 n Ω,C Ω,C Ω,C hA A i − hA i hA i , (5.408) int int,harm int int,harm 2¯ hβ h ¯2

and (2,3)

∆E0

= −

EΩ,C h i 1 1 D 2 Ω,C 2 A − hA i . int,harm int,harm 2¯ hβ h ¯2

(5.409)

The second term can be written as (2,2)

∆E0

=−

o 1 1 n Ω,C −2C∂ hA i , C int 2¯ hβ h ¯2

(5.410)

the third as

o 1 1 n Ω,C {1 − C∂ ] hA i . (5.411) C int,harm 2¯ hβ h ¯2 The final expression is rather involved and given in Appendix 5C. The second-order correction leads to the second term (5.398) found in perturbation theory. In the strong coupling limit, it changes the leading term −α2 /3π ≈ −0.1061 in (5.399) into √ √ √ √ 2− 3 3 ∞ −17+64 arcsin( 2 )−32 log(4 2− ) 1 2X (2n)! 2 − − = , (5.412) 4n 2 4π π n=1 2 (n!) n(2n + 1) 4π (2,3)

∆E0

=−

which is approximately equal to −0.1078. The corrections are shown numerically in the previous Table 5.12. H. Kleinert, PATH INTEGRALS

453

5.22 Polarons

5.22.5

Polaron in Magnetic Field, Bipolarons, Small Polarons, Polaronic Excitons, and More

Feynman’s solution of the polaron problem has instigated a great deal of research on this subject [34]. There are many publications dealing with a polaron in a magnetic field. In particular, there was considerable discussion on the validity of the JensenPeierls inequality (5.10) in the presence of a magnetic field until it was shown by Larsen in 1985 that the variational energy does indeed lie below the exact energy for sufficiently strong magnetic fields. On the basis of this result he criticized the entire approach. The problem was, however, solved by Devreese and collaborators who determined the range of variational parameters for which the inequality remained valid. In the light of the systematic higher-order variational perturbation theory developed in this chapter we do not consider problems with the inequality any more as an obstacle to variational procedures. The optimization procedure introduced in Section 5.13 for even and odd approximations does not require an inequality. We have seen that for higher orders, the exact result will be approached rapidly with exponential convergence. The inequality is useful only in Feynman’s original lowest-order variational approach where it is important to know the direction of the error. For higher orders, the importance of this information decreases rapidly since the convergence behavior allows us to estimate the limiting value quantitatively, whereas the inequality tells us merely the sign of the error which is often quite large in the lowest-order variational approach, for instance in the Coulomb system. There is also considerable interest in bound states of two polarons called bipolarons. Such investigations have become popular since the discovery of hightemperature superconductivity.3

5.22.6

Variational Interpolation for Polaron Energy and Mass

Let us apply the method of variational interpolation developed in Section 5.18 to the polaron. Starting from the presently known weak-coupling expansions (5.397) and (5.404) we fix a few more expansion coefficients such that the curves fit also the strong-coupling expansions (5.400) and (5.405). We find it convenient to make the series start out with α0 by removing an overall factor −α from E and deal with the quantity −Ewex /α. Then we see from (5.400) that the correct leading power in the strong-coupling expansion requires taking p = 1, q = 1. The knowledge of b0 and b1 allows us to extend the known weak coupling expansion (5.397) by two further expansion terms. Their coefficients a3 , a4 are solutions of the equations [recall (5.245)] 15 a2 2a3 a4 35 a0 c + a1 + + 2 + 3, 128 8 c c c 35 a0 5 a2 a3 = − 3 − 3. 32 c 4c c

b0 =

(5.413)

b1

(5.414)

454


The constant c governing the growth of ΩN for α → ∞ is obtained by extremizing b0 in c, which yields the equation 35 15 a2 4a3 4a4 − 3 − 5 = 0. a0 − 128 8 c2 c c

(5.415)

The simultaneous solution of (5.413)–(5.415) renders c4 = 0.09819868, a3 = 6.43047343 × 10−4 , a4 = −8.4505836 × 10−5 .

(5.416)

The re-expanded energy (5.238) reads explicitly as a function of α and Ω (for E including the earlier-removed factor −α) 35 35 35 7 5 − + − a1 α2 Ω− + 128 32Ω 64Ω3 32Ω5 128Ω7 15 5 3 2 1 3 4 5 1 + + a2 α − − + a α − + − a α . (5.417) 3 4 8Ω 4Ω3 8Ω5 Ω2 Ω4 Ω3

W4 (α, Ω) = a0 α −

Extremizing this we find Ω4 as a function of α [it turns out to be quite well approximated by the simple function Ω4 ≈ c4 α+1/(1+0.07α)]. This is to be compared with the optimal frequency obtained from minimizing the lower approximation W2 (α, Ω): 4a2 2 Ω22 = 1 + x + 3a0

s

1+

4a2 2 x 3a0

2

− 1,

(5.418)

q

which behaves like c2 α+1+. . . with c2 = 8a2 /3a0 ≈ 0.120154. The resulting energy is shown in Fig. 5.28, where it is compared with the Feynman variational energy. For completeness, we have also plotted the weak-coupling expansion, the strongcoupling expansion, the lower approximation W2 (α), and two Padé approximants given in Ref. [22] as upper and lower bounds to the energy. Consider now the effective mass of the polaron, where the strong-coupling behavior (5.405) fixes p = 4, q = 1. The coefficient b0 allows us to determine an approximate coefficient a3 and to calculate the variational perturbation expansion W3 (α). From (5.245) we find the equation b0 = −a1 c3 /8 + a3 c,

(5.419) q

q

whose minimum lies at c3 = 8a2 /3a0 where b0 = 32a33 /27a1 . Equating b0 of Eq. (5.419) with the leading coefficient in the strong-coupling expansion (5.405), we obtain a3 = [27a1 b20 /32]1/3 ≈ 0.0416929. The variational expression for the polaron mass is from (5.238) Ω3 3Ω 3 W3 (α, ω) = a0 + a1 α − + + + a2 α2 + a3 α3 Ω. 8 4 8Ω !

(5.420)


455

5.22 Polarons α

E

Figure 5.28 Variational interpolation of polaron energy (solid line) between the weakcoupling expansion (dashed) and the strong-coupling expansion (short-dashed) shown in comparison with Feynman’s variational approximation (fat dots), which is an upper bound to the energy. The dotted curves are upper and lower bounds coming from Padé approximants [22]. The dot-dashed curve shows the variational perturbation theory W2 (α) which does not make use of the strong-coupling information.

This is extremal at 4a3 2 x + Ω23 = 1 + 3a1

s

1+

4a3 2 x 3a1

2

− 1.

(5.421)

q

From this we may find once more c3 = 8a2 /3a0 . The approximation W3 (α) = W3 (α, Ω3 ) for the polaron mass is shown in Fig. 5.29, where it is compared with the weak and strong-coupling expansions and with Feynman’s variational result. To see better the differences between the curves which all grow fast with α, we have divided out the asymptotic behavior mas = 1 + b0 α4 before plotting the data. As for the energy, we have again displayed two Padé approximants given in Ref. [22] as upper and lower bounds to the energy. Note that our interpolation differs considerably from Feynman’s and higher order expansion coefficients in the weak- or the strongcoupling expansions will be necessary to find out which is the true behavior of the model. Our curve has, incidentally, the strong-coupling expansion ms = 0.0227019α4 + 0.125722α2 + 1.15304 + O(α−2 ),

(5.422)

the second term ∝ α2 -term being in sharp contrast with Feynman’s expression (5.403). On the weak-coupling side, a comparison of our expansion with Feynman’s in Eq. (5.402) shows that our coefficient a3 ≈ 0.0416929 is about 10 times larger than his.

456


m/(1 + b0 α4 )

α

Figure 5.29 Variational interpolation of polaron effective mass between the weak(dashed) and strong-coupling expansions (short-dashed). To see better the differences between the strongly rising functions, we have divided out the asymptotic behavior mas = 1 + b0 α4 before plotting the curves. The fat dots show Feynman’s variational approximation. The dotted curves are upper and lower bounds coming from Padé approximants [22].

Both differences are the reason for our curve forming a positive arch in Fig. 2, whereas Feynman’s has a valley. It will be interesting to find out how the polaron mass really behaves. This would be possible by calculating a few more terms in either the weak- or the strong-coupling expansion. Note that our interpolation algorithm is much more powerful than Padé’s. First, we can account for an arbitrary fractional leading power behavior αp as α → ∞. Second, the successive lower powers in the strong-coupling expansion can be spaced by an arbitrary 2/q. Third, our functions have in general a cut in the complex αplane approximating the cuts in the function to be interpolated (see the discussion in Subsection 17.10.4). Padé approximants, in contrast, have always an integer power behavior in the strong-coupling limit, a unit spacing in the strong-coupling expansion, and poles to approximate cuts.

5.23

Density Matrices

In path integrals with fixed end points, the separate treatment of the path average R x0 ≡ 0¯hβ dτ x(τ ) looses its special virtues. Recall that the success of this separation in the variational approach was based on the fact that for fixed x0 , the fluctuation square width a2 (x0 ) shrinks to zero for large temperatures like h ¯ 2 /12MkB T [recall (5.25)]. A similar shrinking occurs for paths whose endpoints held fixed, which is the case in path integrals for the density. Thus there is no need for a separate treatment of x0 , and one may develop a variational perturbation theory for fixed endpoints H. Kleinert, PATH INTEGRALS

457

5.23 Density Matrices

instead. These may, moreover, be taken to be different from one another xb 6= xa , thus allowing us to calculate directly density matrices.4 The density matrix is defined by the normalized expression 1 (5.423) ρ˜(xb , xa ) , Z where ρ˜(xb , xa ) is the unnormalized transition amplitude given by the path integral ρ(xb , xa ) =

Z

¯ β|xa 0) = ρ˜(xb , xa ) = (xb h

(xa ,0);(xb ,¯ h/kB T )

Dx exp {−A[x]/¯ h} ,

(5.424)

summing all paths with the fixed endpoints x(0) = xa and x(¯ h/kB T ) = xb . The diagonal matrix elements of the density matrix in the integrand yield, of course, the particle density (5.90). The diagonal elements coincide with the partition function density z(x) introduced in Eq. (2.326). The partition function divided out in (5.423) is found from the trace Z=

Z∞

dx ρ˜(x, x).

(5.425)

−∞

5.23.1

Harmonic Oscillator

As usual in the variational approach, we shall base the approximations to be developed on the exactly solvable density matrix of the harmonic oscillator. For the sake of generality, this will be assumed to be centered around xm , with an action AΩ,xm [x] =

Z

h/kB T ¯

0

dτ

1 1 M x˙ 2 (τ ) + MΩ2 [x(τ ) − xm ]2 . 2 2

(5.426)

Its unnormalized density matrix is [see (2.403)] s

MΩ 2π¯ h sinh h ¯ Ω/kB T ( ) h i MΩ 2 2 × exp − (˜ xb + xã ) cosh h ¯ Ω/kB T − 2˜ xb xã , (5.427) 2¯ h sinh h ¯ Ω/kB T

ρ˜0Ω,xm (xb , xa ) =

with the abbreviation x˜(τ ) ≡ x(τ ) − xm .

(5.428)

At fixed endpoints xb , xa and oscillation center xm , the quantum-mechanical correlation functions are given by the path integral m h O1 (x(τ1 )) O2(x(τ2 )) · · · iΩ,x xb ,xa =

×

Z

(xa ,0);(xb ,¯ h/kB T )

4

1 ρ˜0Ω,xm (xb , xa )

Dx O1 (x(τ1 )) O2 (x(τ2 )) · · · exp {−AΩ,xm [x]/¯ h} . (5.429)

H. Kleinert, M. Bachmann, and A. Pelster, Phys. Rev. A 60 , 3429 (1999) (quant-ph/9812063).

458


The path x(τ ) at a fixed imaginary time τ has a distribution p(x, τ ) ≡ h δ(x −

m x(τ )) iΩ,x xb ,xa

(˜ x − xcl (τ ))2 exp − =q , 2b2 (τ ) 2πb2 (τ ) 1

"

#

(5.430)

where xcl (τ ) is the classical path of a particle in the harmonic potential xcl (τ ) =

x˜b sinh Ωτ + xã sinh Ω(¯h/kB T − τ ) , sinh h ¯ Ω/kB T

(5.431)

and b2 (τ ) is the square width (

)

h ¯ h ¯Ω cosh[Ω(2τ − h ¯ /kB T )] b (τ ) = coth . − 2MΩ kB T sinh h ¯ Ω/kB T 2

(5.432)

In contrast to the square width a2 (x0 ) in Eq. (5.24) this depends on the Euclidean time τ , which makes calculations more cumbersome than before. Since the τ lies in the interval 0 ≤ τ ≤ h ¯ /kB T , the width (5.432) is bounded by b2 (τ ) ≤

h ¯ h ¯Ω tanh , 2MΩ 2kB T

(5.433)

thus sharing with a2 (x0 ) the property of remaining finite at all temperatures. The temporal average of (5.432) is b2

kB T = h ¯

Z

h/kB T ¯

0

h ¯ h ¯Ω kB T dτ b (τ ) = coth − 2MΩ kB T h ¯Ω 2

!

.

(5.434)

Just as a2 (x0 ), this goes to zero for T → ∞. Note however, that the asymptotic behavior is b2 − −−→ h ¯ Ω/6kB T, (5.435) 2

T →∞

which is twice as big as that of a (x0 ) in Eq. (5.25) (see Fig. 5.30).

5.23.2

Variational Perturbation Theory for Density Matrices

To obtain a variational approximation for the density matrix, we separate the full action into the harmonic trial action and a remainder A[x] = AΩ,xm [x] + Aint [x],

(5.436)

with an interaction Aint [x(τ )] =

Z

hβ ¯ 0

dτ Vint (x(τ )),

(5.437)

where the interaction potential is the difference between the original one V (x) and the inserted displaced harmonic oscillator: 1 Vint (x(τ )) = V (x(τ )) − MΩ2 [x(τ ) − xm ]2 . 2


459


1.0

a2tot

l2

fluctuation width a2tot,cl

0.5

b2

a2 (x0 ) 0.0

0.0

1.0

kB T /¯hΩ

2.0

Figure 5.30 Temperature dependence of fluctuation widths of any point x(τ ) on the path in a harmonic oscillator (l2 is the generic square length in units of h ¯ /M Ω). The quantity 2 2 a (dashed) is the quantum-mechanical width, whereas a (x0 ) (dash-dotted) shares the width after separating out the fluctuations around the path average x0 . The quantity a2cl (long-dashed) is the width of the classical distribution, and b2 (solid curve) is the fluctuation width at fixed ends which is relevant for the calculation of the density matrix by variational perturbation theory (compare Fig. 3.14).

The path integral (5.424) is then expanded perturbatively around the harmonic expression (5.427) as ρ˜(xb , xa ) =

ρ˜0Ω,xm (xb , xa )

EΩ,xm 1 1 D 2 m 1 − h Aint [x] iΩ,x + A [x] − . . . , (5.439) int xb ,xa xb ,xa h ¯ 2¯ h2

with the harmonic expectation values defined in (5.429). The sum can be evaluated as an exponential of its connected parts, going over to the cumulant expansion: EΩ,xm 1 1 D 2 m ρ˜(xb , xa ) = ρ˜0Ω,xm (xb , xa ) exp − h Aint [x] iΩ,x xb ,xa ,c + 2 Aint [x] x ,xa ,c − . . . ,(5.440) b h ¯ 2¯ h

where the cumulants are defined as usual [see (3.482), (3.483)]. The series (5.440) is truncated after the N-th term, resulting in the N-th order approximant for the quantum statistical density matrix ρÑΩ,xm (xb , xa )

=

ρ˜0Ω,xm (xb , xa ) exp

"

N X

n=1

(−1)n m h Anint [x] iΩ,x xb ,xa ,c , n!¯ hn #

which explicitly depends on the two variational parameters Ω and xm .

(5.441)

460


By analogy with classical statistics, where the Boltzmann distribution in configuration space is controlled by the classical potential V (x) according to [recall (2.346)] s

ρ˜cl (x) =

M exp [−βV (x)] , 2π¯ h2 β

(5.442)

we shall now work with the alternative type of effective classical potential V˜ eff,cl (xa , xb ) introduced in Subsection 3.25.4. It governs the unnormalized density matrix [see Eq. (3.834)] ρ˜(xb , xa ) =

s

h i M ˜ eff,cl (xb , xa ) . exp −β V 2π¯ h2 β

(5.443)

Variational approximations to V˜ eff,cl (xb , xa ) of Nth order are obtained from (5.427), (5.441), and (5.443) as a cumulant expansion n o 1 sinh h ¯ βΩ MΩ ln + (˜ x2b + x˜2a ) cosh h ¯ βΩ − 2˜ xb xã 2β h ¯ βΩ 2¯ hβ sinh h ¯ βΩ N n 1 X (−1) m − h Anint [x] iΩ,x (5.444) xb ,xa ,c . β n=1 n!¯ hn

˜ NΩ,xm (xb , xa ) = W

They have to be optimized in the variational parameters Ω and xm for a pair of ˜ N (xb , xa ). The optimal values Ω(xa , xb ) endpoints xb , xa . The result is denoted by W and xm (xa , xb ) are denoted by ΩN (xa , xb ), xN m (xa , xb ). The Nth-order approximation for the normalized density matrix is then given by Ω2 ,xN m

ρN (xb , xa ) = ZN−1 ρÑ N

(xb , xa ),

(5.445)

where the corresponding partition function reads ZN =

Z

∞

−∞

Ω2 ,xN m

dxa ρÑ N

(xa , xa ).

(5.446)

In principle, one could also optimize the entire ratio (5.445), but this would be harder to do in practice. Moreover, the optimization of the unnormalized density matrix is the only option, if the normalization diverges due to singularities of the potential. This will be seen in Subsection 5.23.6 when discussing the hydrogen atom.

5.23.3

Smearing Formula for Density Matrices

In order to calculate the connected correlation functions in the variational perturbation expansion (5.441), we must find efficient formulas for evaluating expectation values (5.429) of any power of the interaction (5.437) m h Anint [x] iΩ,x xb ,xa

x ˜Zb ,¯ hβ

1 = Ω,xm ρ˜0 (xb , xa ) x˜

a ,0

D˜ x

" n Z Y

l=1

1 × exp − AΩ,xm [˜ x + xm ] . h ¯

0

hβ ¯

dτl Vint (˜ x(τl ) + xm )

#


461


This can be done by an extension of the smearing formula (5.30). For this we rewrite the interaction potential as Vint (˜ x(τl ) + xm ) =

Z∞

dzl Vint (zl + xm )

−∞

Z∞

−∞

"

dλl iλl zl e exp − 2π

Z

hβ ¯

0

#

dτ iλl δ(τ − τl )˜ x(τ ) , (5.448)

and introduce a current n X

J(τ ) =

l=1

i¯ hλl δ(τ − τl ),

(5.449)

so that (5.447) becomes m h Anint [x] iΩ,x xb ,xa =

n Y 1 ρ˜0Ω,xm (xb , xa ) l=1

"Z

0

#

dλl iλl zl Ω,xm dτl dzl Vint (zl + xmin ) K [j]. e −∞ −∞ 2π (5.450)

hβ ¯

Z

∞

Z

∞

The kernel K Ω,xm [j] represents the generating functional for all correlation functions of the displaced harmonic oscillator K

Ω,xm

[j] =

x ˜Zb ,¯ hβ

x ã ,0

(

1 D˜ x exp − h ¯

Z

0

hβ ¯

1 m ˙2 dτ x˜ (τ ) + MΩ2 x˜2 (τ ) + j(τ )˜ x(τ ) 2 2

)

. (5.451)

For zero current j, this generating functional reduces to the Euclidean harmonic propagator (5.427): K Ω,xm [j = 0] = ρ˜0Ω,xm (xb , xa ),

(5.452)

and the solution of the functional integral (5.451) is given by (recall Section 3.1) K

Ω,xm

[j] =

ρ˜0Ω,xm (xb , xa ) exp

"

1 Z ¯hβ − dτ j(τ ) xcl (τ ) h ¯ 0 # Z ¯hβ Z ¯hβ 1 (2) ′ ′ ′ + 2 dτ dτ j(τ ) GΩ2 (τ, τ ) j(τ ) , (5.453) 0 2¯ h 0 (2)

where xcl (τ ) denotes the classical path (5.431) and GΩ2 (τ, τ ′ ) the harmonic Green function with Dirichlet boundary conditions (3.386), to be written here as (2)

GΩ2 (τ, τ ′ ) =

h ¯ cosh Ω(|τ − τ ′ | − h ¯ β) − cosh Ω(τ + τ ′ − h ¯ β) . 2MΩ sinh h ¯ βΩ

(5.454)

The expression (5.453) can be simplified by using the explicit expression (5.449) for the current j. This leads to a generating functional K

Ω,xm

[j] =

ρ˜0Ω,xm (xb , xa )

exp −i

T

1 xcl − 2

T

G

,

(5.455)

462


where we have introduced the n-dimensional vectors = (λ1 , . . . , λn ) and xcl = T (xcl (τ1 ), . . . , xcl (τn )) with the superscript T denoting transposition, and the sym(2) metric n × n-matrix G whose elements are Gkl = GΩ2 (τk , τl ). Inserting (5.455) into (5.450), and performing the integrals with respect to λ1 , . . . , λn , we obtain the n-th order smearing formula for the density matrix m h Anint [x] iΩ,x xb ,xa

=

"Z n Y

l=1

hβ ¯

0

dτl

Z

∞

−∞

dzl Vint (zl + xm )  

#  

n X

1 1 × q exp − [zk − xcl (τk )] G−1 kl [zl − xcl (τl )]. (5.456)  2 n (2π) det G k,l=1

The integrand contains an n-dimensional Gaussian distribution describing both thermal and quantum fluctuations around the harmonic classical path xcl (τ ) of Eq. (5.431) in a trial oscillator centered at xm , whose width is governed by the Green functions (5.454). For closed paths with coinciding endpoints (xb = xa ), formula (5.456) leads to the n-th order smearing formula for particle densities 1 1I ρ(xa ) = ρ˜(xa , xa ) = Dx δ(x(τ = 0) − xa ) exp{−A[x]/¯h}, Z Z

(5.457)

which can be written as m h Anint [x] iΩ,x xa ,xa

=

" n Z Y

1

ρ0Ω,xm (xa ) l=1

0

hβ ¯

dτl

Z

∞

−∞

dzl Vint (zl + xm )





n 1 X  × q exp − zk a−2 kl zl , n+1 2 2 (2π) det a k,l=0

1

#

(5.458)

with z0 = xã . Here a denotes a symmetric (n + 1) × (n + 1)-matrix whose elements a2kl = a2 (τk , τl ) are obtained from the harmonic Green function for periodic paths (2) GΩ2 (τ, τ ′ ) of Eq. (3.301): a2 (τ, τ ′ ) ≡

h ¯ (2) h ¯ cosh Ω(|τ − τ ′ | − h ¯ β/2) GΩ2 (τ, τ ′ ) = . M 2MΩ sinh h ¯ βΩ/2

(5.459)

The diagonal elements a2 = a(τ, τ ) are all equal to the fluctuation square width (5.24). Both smearing formulas (5.456) and (5.458) allow us to calculate all harmonic expectation values for the variational perturbation theory of density matrices and particle densities in terms of ordinary Gaussian integrals. Unfortunately, in many applications containing nonpolynomial potentials, it is impossible to solve neither the spatial nor the temporal integrals analytically. This circumstance drastically increases the numerical effort in higher-order calculations. H. Kleinert, PATH INTEGRALS

463


5.23.4

First-Order Variational Approximation

The first-order variational approximation gives usually a reasonable estimate for any desired quantity. Let us investigate the classical and the quantum-mechanical limit of this approximation. To facilitate the discussion, we first derive a new representation for the first-order smearing formula (5.458) which allows a direct evaluation of the imaginary time integral. The resulting expression will depend only on temperature, whose low- and high-temperature limits can easily be extracted. Alternative First-Order Smearing Formula For simplicity, we restrict ourselves to the case of particle densities and allow only symmetric potentials V (x) centered at the origin. If V (x) has only one minimum at the origin, then also xm will be zero. If V (x) has several symmetric minima, then xm goes to zero only at sufficiently high temperatures as in Section 5.7. To first order, the smearing formula (5.458) reads h Aint [x] iΩ xa ,xa =

1 Ω ρ0 (xa )

Z¯hβ

dτ

0

Z∞

−∞

dz 1 1 (z 2 +x2a )a00 −2zxa a01 Vint (z) q exp − . 2π 2 a200 −a201 a2 −a2 00

)

(

01

(5.460)

Expanding the exponential with the help of Mehler’s formula (2.290), we obtain the following expansion in terms of Hermite polynomials Hn (x): h Aint [x] iΩ xa ,xa=

∞ X

n=0





h ¯ β (n)  z  C Hn q 2n n! β 2a2 00

Z∞

dz q

2πa200

−∞

−z 2 /2a200

Vint (z) e



z



. Hn  q 2a200

(5.461)

Its temperature dependence stems from the diagonal elements of the harmonic Green (n) function (5.459). The dimensionless functions Cβ are defined by (n) Cβ

hβ ¯ 1 Z = dτ h ¯β 0

a201 a200

!n

.

(5.462)

Inserting (5.459) and performing the integral over τ , we obtain (n) Cβ

=

n X n

1 2n

n

cosh h ¯ βΩ/2 k=0

k

sinh h ¯ βΩ(n/2 − k) . h ¯ βΩ(n/2 − k)

(5.463)

At high temperatures, these functions of β go to unity: (n)

lim Cβ = 1.

β→0

(5.464)

464


(n)

Cβ

1.0

0 1

0.8

2 0.6

3 4 5 6 7 n=8

0.4

0.2

0.0

0.0

2.0

4.0

6.0

¯hβΩ

8.0

(n)

Figure 5.31 Temperature-dependence of first 9 functions Cβ , where β = 1/kB T .

Their zero-temperature limits are (n)

lim Cβ =

β→∞

  

1, 2  ,  h ¯ βΩn

n = 0, (5.465)

n > 0.

According to (5.444), the first-order approximation to the new effective potential is given by h ¯ βΩ ¯ βΩ MΩ 2 ˜ 1Ω (xa ) = 1 ln sinh h + xa tanh + VaΩ2 (xa ), W 2β h ¯ βΩ h ¯β 2

(5.466)

with the smeared interaction potential VaΩ2 (xa ) =

1 h Aint [x] iΩ xa ,xa . h ¯β

(5.467)

It is instructive to discuss separately the limits β → 0 and β → ∞ to see the effects of pure classical and pure quantum fluctuations. a) Classical Limit of Effective Classical Potential In the classical limit β → 0, the first-order effective classical potential (5.466) reduces to ˜ 1Ω,cl (xa ) = 1 MΩ2 x2 + lim V Ω2 (xa ). W (5.468) a a β→0 2 H. Kleinert, PATH INTEGRALS

465


The second term is determined by inserting the high-temperature limit of the total fluctuation width (5.26): kB T a2tot,cl = , (5.469) MΩ2 and of the polynomials (5.464) into the expansion (5.461), leading to ∞ X

s



1 MΩ2 β   VaΩ2 (xa ) ≈ H xa n n T →∞ 2 n=0 2 n!

Z∞

−∞

dz q

2π/MΩ2 β

−M Ω2 βz 2 /2

Vint (z)e

Hn

MΩ2 β z . 2 !

(5.470)

Then we make use of the completeness relation for Hermite polynomials ∞ 1 1 2 X √ e−x Hn (x) Hn (x′ ) = δ(x − x′ ), n π 2 n! n=0

(5.471)

which may be derived from Mehler’s formula (2.290) in the limit b → 1− , to reduce the smeared interaction potential VaΩ2 (xa ) to the pure interaction potential (5.438): lim VaΩ2 (xa ) = Vint (xa ).

β→0

(5.472)

Recalling (5.438) we see that the first-order effective classical potential (5.468) approaches the classical one: ˜ 1Ω,cl (xa ) = V (xa ). lim W

β→0

(5.473)

This is a consequence of the vanishing fluctuation width b2 of the paths around the classical orbits. This property is universal to all higher-order approximations to the effective classical potential (5.444). Thus all correction terms with n > 1 must disappear in the limit β → 0, lim

β→0

∞ −1 X (−1)n Ω n n h Aint [x] ixa ,xa ,c = 0. β n=2 n!¯ h

(5.474)

b) Zero-Temperature Limit At low temperatures, the first-order effective classical potential (5.466) becomes ¯Ω ˜ 1Ω,qm (xa ) = h W + lim VaΩ2 (xa ). β→∞ 2

(5.475)

The zero-temperature limit of the smeared potential in the second term defined in (5.467) follows from Eq. (5.461) by taking into account the limiting procedure for the (n) polynomials Cβ in (5.465) and the zero-temperature limit of the total fluctuation width (5.26), which is equal to the zero-temperature limit of a2 (x0 ): a2tot,0 = a2tot = T =0

466


h ¯q/2MΩ. Thus we obtain with H0 (x) = 1 and the inverse length κ ≡ 1/λΩ = MΩ/¯ h [recall (2.296)]: lim VaΩ2 (xa ) =

β→∞

Z∞

dz

−∞

s

κ2 H0 (κz)2 exp{−κ2 z 2 } Vint (z). π

(5.476)

Introducing the harmonic eigenvalues EnΩ

1 =h ¯Ω n + , 2

(5.477)

and the harmonic eigenfunctions [recall (2.294) and (2.295)] ψnΩ (x)

1 =√ n!2n

κ2 π

!1/4

1

e− 2 κ

2 x2

Hn (κx),

(5.478)

we can re-express the zero-temperature limit of the first-order effective classical potential (5.475) with (5.476) by ˜ 1Ω,qm (xa ) = E Ω + h ψ Ω | Vint | ψ Ω i. W 0 0 0

(5.479)

This is recognized as the first-order Rayleigh-Schrödinger perturbative result for the ground state energy. For the discussion of the quantum-mechanical limit of the first-order normalized density, n

o

exp − ¯h1 h Aint [x] iΩ ρ˜1Ω (xa ) xa ,xa Ω Ω n o, ρ1 (xa ) = = ρ0 (xa ) R ∞ Ω 1 Ω Z dx ρ (x ) exp − h A [x] i a a int 0 −∞ xa ,xa h ¯

(5.480)

we proceed as follows. First we expand (5.480) up to first order in the interaction, leading to 

Ω  ρΩ 1 (xa ) = ρ0 (xa ) 1 −



1 h Aint [x] iΩ xa ,xa − h ¯

Z∞

−∞



Ω  .(5.481) dxa ρΩ 0 (xa ) h Aint [x] ixa ,xa

Inserting (5.428) and (5.461) into the third term in (5.481), and assuming Ω not to depend explicitly on xa , the xa -integral reduces to the orthonormality relation for Hermite polynomials 1 √ 2n n! π

Z∞

2

dxa Hn (xa )H0 (xa )e−xa = δn0 ,

(5.482)

−∞

so that the third term in (5.481) eventually becomes −

Z∞

−∞

Ω dxa ρΩ 0 (xa ) h Aint [x] ixa ,xa = −β

Z∞

−∞

dz

s

κ2 Vint (z) exp{−κ2 z 2 } H0 (κz). (5.483) π H. Kleinert, PATH INTEGRALS

467


But this is just the n = 0 -term of (5.461) with an opposite sign, thus canceling the zeroth component of the second term in (5.481), which would have been divergent for β → ∞. The resulting expression for the first-order normalized density is 

Ω  ρΩ 1 (xa ) = ρ0 (xa ) 1 −

∞ X

n=1

β (n) C Hn (κxa ) n 2 n! β

Z∞

dz

−∞

(n)

The zero-temperature limit of Cβ

s



κ2 Vint (z) exp(−κ2 z 2 ) Hn (κz) . π

(5.484)

is from (5.465) and (5.477) (n)

lim βCβ =

β→∞

2 , EnΩ − E0Ω

(5.485)

so that we obtain from (5.484) the limit ρΩ 1 (xa )

=

ρΩ 0 (xa )

×

Z∞

−∞

dz

"

s

1−2 κ2 π

∞ X

n=1

1 2n n!

EnΩ

1 Hn (κxa ) − E0Ω 2 2

#

Vint (z) exp{−κ z }Hn (κz) H0 (κz) .

(5.486)

Taking into account the harmonic eigenfunctions (5.478), we can rewrite (5.486) as 2 Ω 2 Ω ρΩ 1 (xa ) = |ψ0 (xa )| = [ψ0 (xa )] − 2ψ0 (xa )

X

n>0

ψnΩ (xa )

h ψnΩ | Vint | ψ0Ω i , EnΩ − E0Ω

(5.487)

which is just equivalent to the harmonic first-order Rayleigh-Schrödinger result for particle densities. Summarizing the results of this section, we have shown that our method has properly reproduced the high- and low-temperature limits. Due to relation (5.487), the variational approach for particle densities can be used to determine approximately the ground state wave function ψ0 (xa ) for the system of interest.

5.23.5

Smearing Formula in Higher Spatial Dimensions

Most physical systems possess many degrees of freedom. This requires an extension of our method to higher spatial dimensions. In general, we must consider anisotropic harmonic trial systems, where the previous variational parameter Ω2 becomes a D × D -matrix Ω2µν with µ, ν = 1, 2, . . . , D. a) Isotropic Approximation An isotropic trial ansatz Ω2µν = Ω2 δµν

(5.488)

468


can give rough initial estimates for the properties of the system. In this case, the n-th order smearing formula (5.458) generalizes directly to n 1 Y = Ω ρ0 (ra ) l=1

h Anint [r] iΩ ra ,ra

"Z

hβ ¯

0

dτl

1

× q D (2π)n+1 det a2

Z

D

d zl Vint (zl )

# 



n 1 X  exp − zk a−2 kl zl , 2 k,l=0

(5.489)

with the D-dimensional vectors zl = (z1l , z2l , . . . , zDl )T . Note, that Greek labels µ, ν, . . . = 1, 2, . . . , D specify spatial indices and Latin labels k, l, . . . = 0, 1, 2, . . . , n refer to the different imaginary times. The vector z0 denotes ra , the matrix a2 is the same as in Subsection 5.23.3. The harmonic density reads ρΩ 0 (r) =

s

1 2πa200

D





D 1 X exp − 2 x2  . 2 a00 µ=1 µ

(5.490)

b) Anisotropic Approximation In the discussion of the anisotropic approximation, we shall only consider radiallysymmetric potentials V (r) = V (|r|) because of their simplicity and their major occurrence in physics. The trial frequencies decompose naturally into a radial frequency ΩL and a transverse one ΩT as in (5.95): Ω2µν

=

Ω2L

xaµ xaν xaµ xaν + Ω2T δµν − 2 ra ra2

!

,

(5.491)

¯ n = U xn with ra = |ra |. For practical reasons we rotate the coordinate system by x so that ¯ra points along the first coordinate axis, (¯ra )µ ≡ z¯µ0 =

(

ra , µ = 1, 0, 2 ≤ µ ≤ D,

(5.492)

and Ω2 -matrix is diagonal: 

Ω2 =

       

Ω2L 0 0 2 0 ΩT 0 0 0 Ω2T .. .. .. . . . 0 0 0

··· ··· ··· .. .

0 0 0 .. .

· · · Ω2T

        

= U Ω2 U −1 .

(5.493)

After this rotation, the anisotropic n-th order smearing formula in D dimensions reads L,T h Anint [r] iΩ ra ,ra

=

×

1

" n Z Y

Ω ρ0 L,T (¯ra ) l=1

(det a2L )−1/2

hβ ¯ 0

dτl

Z

#

D

d z¯l Vint (|¯ zl |) (2π)−D(n+1)/2 − 12

(det a2T )−(D−1)/2 e

n P

k,l=0

z¯1k aL −2 z¯ − 12 kl 1l

e

(5.494)

D P P n

µ=2

k,l=1

z¯µk aT −2 z¯ kl µl

.


469


The components of the longitudinal and transversal matrices a2L and a2T are a2Lkl = a2L (τk , τl ),

a2T kl = a2T (τk , τl ) ,

(5.495)

where the frequency Ω in (5.459) must be substituted by the new variational paramΩ eters ΩL , ΩT , respectively. For the harmonic density in the rotated system ρ0 L,T (¯r) which is used to normalize (5.494), we find Ω ρ0 L,T (¯r)

5.23.6

=

s

1 2πa2L00

s

1 2πa2T 00

D−1





D 1 1 X 2  exp − 2 x¯1 − x¯2µ  . 2 2 aL00 2 aT 00 µ=2

(5.496)

Applications

For applications, we employ natural units with h ¯ = kB = M = 1. In order to develop some feeling how the extension of the variational procedure to higher order works, we approximate at first the particle density in the double-well potential up to second order. After that we extend the first-order calculation of the electron density of the hydrogen atom in Section 5.10 to finite temperatures.

a) Double-Well A detailed analysis of the first-order approximation shows that the particle density in the doublewell potential is nearly exact for all temperatures if we use the two variational parameters Ω2 and xm , whereas one variational parameter Ω2 leads to larger deviations at low temperatures and coupling strengths. In this regime, the density has a maximum far away from origin xa = 0, and the displacement of the trial oscillator xm is essential for a good variational approximation. In higher orders, however, the dependence on xm becomes weaker and weaker For this reason, already the second-order calculation may be done by optimizing with respect to Ω while keeping xm fixed at the origin.

First-Order Approximation In the case of the double-well potential 1 1 1 V (x) = − ω 2 x2 + gx4 + , 2 4 4g

(5.497)

with coupling constant g, we obtain for the expectation of the interaction (5.461) to first order, setting also ω = 1, ! ! 1 1 xa −xm 1 xa −xm (1) (2) Ω,xm h Aint [x] ixa ,xa = βg0 + g1 Cβ H1 p 2 + g2 Cβ H2 p 2 2 2 4 2a00 2a00 ! ! 1 xa −xm 1 xa −xm (3) (4) + g3 Cβ H3 p 2 + g4 Cβ H4 p 2 , (5.498) 8 16 2a00 2a00

with

g0

=

g1

=

g2

=

g3

=

g4

=

3 1 1 1 −a200 (Ω2 + 1) + ga400 + 3ga200 x2m + gx4m + − x2m , 2 2 2g 2 q q 3 − 2a200 xm + g(2a200 )3/2 xm + g 2a200 x3m , 4 −a200 (Ω2 + 1) + 3ga400 + 3ga200 x2m , g(2a200 )3/2 xm , ga400 .

470


1 b) xa = √ g

a) xa = 0

0.5

0.25 0.5

˜1 W 0.6

˜1 W 1.0 0.8 0.6

Ω2

0.5

1.0 −1.0

1.5 Ω2 −1.0

0.0

0.0

2.5

xm

xm

2.0

˜ Ω,xm (xa ) to effective classical potential as a Figure 5.32 Plots of first-order approximation W 1 function of the two variational parameters Ω(xa ), xm (xa ) at β = 10 and g = 0.4 for two different values of xa . Inserting (5.498) in (5.467), we obtain the unnormalized double-well density ρ˜1Ω,xm (xa ) = √

1 exp[−βW1Ω,xm (xa )], 2πβ

(5.499)

with the first-order approximation to the effective classical potential of the alternative type of Subsection 3.25.4): ˜ Ω,xm (xa ) = 1 ln sinh βΩ + Ω (xa − xm )2 tanh βΩ + 1 h Aint [x] iΩ,xm . W 1 xa ,xa 2 βΩ β 2 β

(5.500)

˜ Ω,xm (xa ), the normalized first-order particle density ρ1 (xa ) is found by dividing After optimizing W 1 ρ˜1 (xa ) by the first-order partition function 1 Z1 = √ 2πβ

Z∞

˜ 1 (xa )]. dxa exp[−β W

(5.501)

−∞

˜ Ω,xm (xa ) we usually obtain a unique minimum at some Ω1 (xa ) and x1 (xa ). When optimizing W m 1 Only rarely must a turning point or a vanishing higher derivative be used. The dependence of the ˜ Ω,xm (xa ) on the variational parameters Ω(xa ) and xm (xa ) first-order effective classical potential W 1 is shown in the three-dimensional plots of Fig. 5.32 for β = 10 at two typical values of xa . Darker ˜ Ω,xm . After having determined roughly the area around the regions indicate smaller values of W 1 expected minimum, we determine the optima numerically. Note that for symmetry reasons, xm (xa ) = −xm (−xa ),

(5.502)

Ω(xa ) = Ω(−xa ).

(5.503)

and

˜ 1 (xa ) are shown in Fig. 5.33 Some first-order approximations to the effective classical potential W which are obtained by optimizing with respect to Ω(xa ) and xm (xa ). H. Kleinert, PATH INTEGRALS

471

5.23 Density Matrices ˜ 1 (xa ) W

1.5 g = 0.1

1.0

0.2

0.3 0.4 0.6

0.5 √ −2/ g

√ −1/ g

0 xa

√ +1/ g

√ +2/ g

˜ 1 (xa ) for different First-order approximation to effective classical potential W coupling strengths g as a function of the position xa at β = 10 by optimizing in both variational parameters Ω and xm (solid curves) in comparison with the approximations obtained by variation in Ω only (dashed curves).

Figure 5.33

The sharp maximum occurring for weak-coupling is a consequence of the reflection property (5.502) enforcing a vanishing xm (xa = 0). In the strong-coupling regime, on the other hand, where xm (xa = 0) ≈ 0, the sharp top is absent. This behavior is illustrated in the right-hand parts of Figs. 5.34 and 5.35 at different temperatures. The influence of the center parameter xm decreases for increasing values of g and decreasing height 1/4g of the central barrier (see Fig. 5.33). The same thing is true at high temperatures and large values of xa , where the precise knowledge of the optimal value of xm is irrelevant. In these limits, the particle density can be determined without optimizing in xm , setting simply xm = 0, where the expectation value Eq. (5.498) reduces to q q 1 (2) 1 (4) Ω h Aint [x] ixa ,xa = Cβ H2 xa / 2a200 (g1 + 3g2 ) + g2 Cβ H4 xa / 2a200 4 16 1 3 + β g1 + g2 + g3 , (5.504) 2 4 with the abbreviations g1 = −a200 (Ω2 + 1),

g2 = ga400 ,

g3 =

1 . 4g

Inserting (5.504) in (5.467) we obtain the unnormalized double-well density 1 ˜ Ω (xa )], ρ˜1Ω (xa ) = √ exp[−β W 1 2πβ

(5.505)

with the first-order effective classical potential ˜ 1Ω (xa ) = 1 ln sinh βΩ + Ω x2a tanh βΩ + 1 h Aint [x] iΩ W xa ,xa . 2 βΩ β 2 β

(5.506)

472


b)

a) xm (xa )

Ω2 (xa ) 3.0

2.0

β=5 8 10

β=5 3.00 20 ?

8

100

100

2.95

1.0

10 20 √ −1/ g

0 xa

√ +1/ g

√ 1/ g xa

0

√ 2/ g

Figure 5.34 a) Trial frequency Ω(xa ) at different temperatures and coupling strength g = 0.1. b) Minimum of trial oscillator xm (xa ) at different temperatures and coupling g = 0.1.

a)

b)

Ω2 (xa )

xm (xa ) 8

5.5

0.00

5.4

100

8

10 6

10 100

√ −2/ g

0 xa

20

−0.01

20 5.3

β=5

β=5

√ 2/ g

0

√ 2/ g

√ 4/ g xa

Figure 5.35 a) Trial frequency Ω(xa ) at different temperatures and coupling strength g = 10. b) Minimum of trial oscillator xm (xa ) at different temperatures and coupling g = 10.


473


ρ(xa ) ρ1 (xa )

0.2

exact

0.1

0.0 0.0

1.0

xa

2.0

3.0

Figure 5.36 First-order approximation to particle density for β = 10 and g = 0.4 compared with the exact particle density in a double-well from numerical solution of the Schrödinger equation. All values are in natural units. The optimization in xm = 0 gives reasonable results for moderate temperatures at couplings such as g = 0.4, as shown in Fig. 5.36 by a comparison with the exact density which is obtained from numerical solutions of the Schrödinger equation. An additional optimization in xm cannot be distinguished on the plot. An example where the second variational parameter xm becomes important is shown in Fig. 5.37, where we compare the first-order approximation with one (Ω) and two variational parameters (Ω, xm ) with the exact density for different temperatures at the smaller coupling strength g = 0.1. In Fig. 5.34 we see that for xa > 0, the optimal xm -values lie close to the right hand minimum of the double-well potential, which we only want to consider here. √ The minimum is located at 1/ g ≈ 3.16. We observe that, with two variational parameters, the first-order approximation is nearly exact for all temperatures, in contrast to the results with only one variational parameter at low temperatures (see the curve for β = 20).

Second-Order Approximation In second-order variational perturbation theory, the differences between the optimization procedures using one or two variational parameters become less significant. Thus, we restrict ourselves to the optimization in Ω(xa ) and set xm = 0. The second-order density 1 ˜ 2Ω (xa )] ρ˜2Ω (xa ) = √ exp[−β W 2πβ

(5.507)

with the second-order approximation of the effective classical potential

˜ 2Ω (xa ) = 1 ln sinh βΩ + Ω x2a tanh βΩ + 1 h Aint [x] iΩ − 1 A2int [x] Ω W xa ,xa xa ,xa ,c 2 βΩ β 2 β 2β

(5.508)

474


ρ1 (xa ) 0.4 β = 20 0.3

3 @ R @

6

0.2

β=1 6

0.1

β = 0.25 0.0

0.0

2.0

xa

4.0

6.0

Figure 5.37

First-order approximation to particle densities of the double-well for g = 0.1 obtained by optimizing with respect to two variational parameters Ω and xm (dashed curves) and with only Ω2 (dash-dotted) vs. exact distributions (solid) for different temperatures. The parameter xm is very important for low temperatures. requires evaluating the smearing formula (5.456) for n = 1, which is given in (5.504), and for n = 2 which will now be calculated. Going immediately to the cumulant we have

Ω A2int [x] xa ,xa ,c =

Zh¯ β

dτ1

0

Zh¯ β

dτ2

0

(

1 2 (Ω + 1)2 [I22 (τ1 , τ2 ) − I2 (τ1 )I2 (τ2 )] 4

(5.509)

) 1 1 2 2 − g(Ω + 1) [I24 (τ1 , τ2 ) − I2 (τ1 )I4 (τ2 )] + g [I44 (τ1 , τ2 ) − I4 (τ1 )I4 (τ2 )] , 4 16

with Im (τk ) =

(a400

−

a40k )m

2 ∂m j + 2xa a20k j exp , ∂j m 2a200 (a400 − a40k ) j=0

k = 1, 2,

(5.510)

and Imn (τ1 , τ2 ) =

(−det A)m+n

∂m ∂n F (j1 , j2 ) exp , ∂j1m ∂j2n 2a200 (det A)2 j1 =j2 =0

(5.511)

where det A = a600 + 2a201 a202 a212 − a200 (a401 + a402 + a412 ). The generating function is H. Kleinert, PATH INTEGRALS

475


ρ2 (xa ) β = 20 5 0.2

1

0.25

0.1

0.0

0.0

1.0

xa

2.0

3.0

Figure 5.38 Second-order approximation to particle density (dashed) compared to exact results from numerical solutions of the Schrödinger equation (solid) in a double-well at different inverse temperatures. The coupling strength is g = 0.4. F (j1 , j2 )

= a400 (j12 + j22 ) − 2a600 (a201 j1 + a202 j2 )xa +2a200 a212 j1 j2 + (a401 + a402 + a412 )(a201 j1 + a202 j2 )xa −(a201 j1 + a202 j2 )(a201 j1 + a202 j2 + 4a201 a202 a212 xa ).

(5.512)

All necessary derivatives and the imaginary time integrations in (5.509) have been calculated analytically. After optimizing the unnormalized second-order density (5.507) in Ω we obtain the results depicted in Fig. 5.38. Comparing the second-order results with the exact densities obtained from numerical solutions of the Schrödinger equation, we see that the deviations are strongest in the region of intermediate β, as expected. Quantum-mechanical limits are reproduced very well, classical limits exactly.

b) Hydrogen Atom With the insights gained in the last section by discussing the double-well potential, we are prepared to apply our method to the electron in the hydrogen atom which is exposed to the attractive Coulomb interaction V (r) = −

e2 . r

(5.513)

Apart from its physical significance, the theoretical interest in this problem originates from the non-polynomial nature of the attractive Coulomb interaction. The usual Wick rules or Feynman diagrams do not allow to evaluate harmonic expectation values in this case. Only by the aid of the above-mentioned smearing formula we are able to compute the variational expansion. Since we learned from the double-well potential that the importance of the second variational parameter rm diminishes for a decreasing height of the central barrier, it is sufficient for the Coulomb potential

476


with an absent central barrier to set rm = 0 and to take into account only one variational parameter Ω2 . By doing so we will see in the first order that the anisotropic variational approximation becomes significant at low temperatures, where radial and transversal quantum fluctuations have quite different weights. The effect of anisotropy disappears completely in the classical limit.

Isotropic First-Order Approximation In the first-order approximation for the unnormalized density, we must calculate the harmonic expectation value of the action Aint [r] =

Zh¯ β

dτ1 Vint (r(τ1 )),

(5.514)

e2 1 + rT Ω2 r , r 2

(5.515)

0

with the interaction potential Vint (r) = −

where the matrix Ω2µν has the form (5.491). Applying the isotropic smearing formula (5.489) for N = 1 to the harmonic term in (5.514) we easily find

r2 (τ1 )

Ω

ra ,ra

=3

a401 2 a400 − a401 + r . a200 a400 a

(5.516)

For the Coulomb potential we obtain the local average

e2 r(τ1 )

Ω

e2 a200 = erf ra a201

ra ,ra

a201

p ra 2a200 (a400 − a401 )

!

.

(5.517)

The time integration in (5.514) cannot be done in an analytical manner and must be performed numerically. Alternatively we can use the expansion method introduced in Subsection 5.23.4 for evaluating the smearing formula in three dimensions which yields p 2 ∞ −ra /2a200 X H2n+1 (ra / 2a200 ) (2n) Ω Ω −1 e h Aint [r] ira ,ra = [ρ0 (ra )] Cβ π 2 a200 ra n=0 22n+1 (2n + 1)! Z∞

×

0

q 2 dy y Vint ( 2a200 y)e−y H2n+1 (y).

(5.518)

This can be rewritten in terms of Laguerre polynomials Lµn (r) as r ∞ q 2a200 1 X (−1)n n! (2n) Ω h Aint [r] ira ,ra = Cβ H2n+1 (ra / 2a200 ) π ra n=0 (2n + 1)! ×

Z∞ 0

q 1/2 dy y 1/2 Vint ( 2a200 y 1/2 )e−y L1/2 n (y)L0 (y).

(5.519)

Using the integral formula (9.54) and inserting the interaction potential (5.515) we find ∞ q e2 X (−1)n (2n − 1)!! (2n) Ω h Aint [r] ira ,ra = − √ C H (r / 2a200 ) 2n+1 a β πra n=0 2n (2n + 1)! q q q 3 1 1 (2) (0) 6 2 2 4 − 2a00 Ω Cβ H1 (ra / 2a00 ) + Cβ H3 (ra / 2a00 ) . 4 ra 6

(5.520)


477


g1 (ra ) 105

×104 K

1 2

104 103

4

102 8 101 100

0.0

2.0

4.0

6.0

8.0

10.0

ra

14.0

Figure 5.39

Radial distribution function for an electron-proton pair. The first-order results obtained with isotropic (dashed curves) and anisotropic (solid) variational perturbation theory are compared with Storer’s numerical results (dotted, see Fig. 5.10) and the earlier approximation in Fig. 5.9 (dash–dotted). The first term comes from the Coulomb potential, the second from the harmonic potential. Inserting (5.520) in (5.441), we compute the first-order isotropic form of the radial distribution function p 3 (5.521) g(r) = 2πβ ρ˜(r) .

This can be written as

˜ 1Ω (ra )], g1Ω (ra ) = exp[−β W

(5.522)

with the isotropic first-order approximation of the effective classical potential ˜ 1Ω (ra ) = 3 ln sinh βΩ + Ω ra2 tanh βΩ + 1 h Aint [r] iΩ W ra ,ra , 2β βΩ β 2 β

(5.523)

which is shown in Fig. 5.39 for various temperatures. The results compare well with Storer’s precise numerical results (see Fig. 5.10). Near the origin, our results are better than those obtained from ˜ 1 (r0 ) in Fig. 5.9. the lowest-order effective classical potential W

Anisotropic First-Order Approximation The above results can be improved by taking care of the anisotropy of the problem. For the harmonic part of the action (5.514), Aint [r] = AΩ [r] + AC [r],

(5.524)

the smearing formula (5.494) yields the expectation value q 1 1 (2) (0) (0) (2) ΩL,T 2 2 2 2 2 h AΩ [r] ira ,ra = − ΩL aL 00 Cβ + Cβ,L H2 (ra / 2aL00 ) +2ΩT aT 00 (Cβ −Cβ,T ) , (5.525) 2 2

478

5 Variational Perturbation Theory (n)

where the Cβ,L(T ) are the polynomials (5.463) with Ω replaced by the longitudinal or transverse frequency. For the Coulomb part of action, the smearing formula (5.494) leads to a double integral L,T h AC [r] iΩ ra ,ra

= −e

2

Zh¯ β 0

dτ1

s

2 2 πaL00 (1 − a4L )

Z1 0

2

4

2

2

4

e−ra aL λ /2aL00 (1−aL ) h a2 (1−a4 ) i, dλ 1 + λ2 aT2 00 (1−aT4 ) − 1 L 00

(5.526)

L

with the abbreviations a2L ≡ a2L01 /a2L00 , a2T ≡ a2T 01 /a2T 00 . The integrals must be done numerically. Once this is done, we obtain the first-order approximation of the radial distribution function as Ω ˜ ΩL,T (ra )], g1 L,T (ra ) = exp[−β W 1

(5.527)

with ˜ ΩL,T (ra ) = 1 ln sinh βΩL + 1 ln sinh βΩT + ΩL r2 tanh βΩL + 1 h Aint [r] iΩL,T . W 1 ra ,ra β βΩL 2β βΩT β a 2 β

(5.528)

This is optimized in ΩL (ra ), ΩT (ra ) with the results shown in Fig. 5.39. The anisotropic approach improves the isotropic result for temperatures below 104 K.

Appendix 5A

Feynman Integrals for T = / 0 without Zero Frequency

The Feynman integrals needed in variational perturbation theory of the anharmonic oscillator at nonzero temperature can be calculated in close analogy to those of ordinary perturbation theory in Section 3.20. The calculation proceeds as explained in Appendix 3D, except that the lines represent now the thermal correlation function (5.19) with the zero-frequency subtracted from the spectral decomposition: " # ˜ − τ ′ | − ¯hβ/2) 1 1 cosh Ω(|τ (2)x0 ′ − . GΩ˜ (τ, τ ) = ˜ ˜ hβ/2) ˜2 2Ω sinh(Ω¯ ¯hβ Ω With the dimensionless variable x ≡ ¯hβ ω ˜ , the results for the quantities a2L V defined of each Feynman diagram with L lines and V vertices as in (3.547), but now without the zero Matsubara frequency, are [compare with the results (3D.3)–(3D.11)] ¯ 1 x h x coth − 1 , ω ˜x 2 2 2 ¯h 1 1 = 4 + x2 − 4 cosh x + x sinh x , x 2 ω ˜ 8x sinh 2 3 ¯h 1 1 x x 3x = −3 x cosh + 2 x3 cosh + 3 x cosh ω ˜ 64x sinh3 x 2 2 2 2 x x 3x +48 sinh + 6 x2 sinh − 16 sinh , 2 2 2 4 ¯h 1 1 = −864 + 18 x4 + 1152 cosh x + 32 x2 cosh x ω 768x3 sinh4 x 2 −288 cosh 2x − 32 x2 cosh 2x − 288 x sinh x + 24 x3 sinh x +144 x sinh 2x + 3 x3 sinh 2x ,

a2 =

(5A.1)

a42

(5A.2)

a63

a82

(5A.3)

(5A.4)


Appendix 5A

a10 3

a12 3

a62 a83

a10 3′

Feynman Integrals for T 6= 0 without Zero Frequency

5 ¯h 1 1 x x x = 672 x cosh − 8 x3 cosh + 24 x5 cosh x ω ˜ 4096x3 sinh5 2 2 2 2 3x 3x 5x 5x −1008 x cosh + 3 x3 cosh + 336 x cosh + 5 x3 cosh 2 2 2 2 x x x 3x 2 4 −7680 sinh − 352 x sinh + 72 x sinh + 3840 sinh 2 2 2 2 3x 3x 5x 5x 2 4 2 +224 x sinh +12 x sinh −768 sinh − 64 x sinh , 2 2 2 2 6 ¯h 1 1 = −107520 − 7360 x2 + 624 x4 + 96 x6 4 ω ˜ 49152x sinh6 x 2 +161280 cosh x + 1200 x2 cosh x − 777 x4 cosh x + 24 x6 cosh x −64512 cosh 2x − 5952 x2 cosh 2x + 144 x4 cosh 2x + 10752 cosh 3x

+28800 x sinh x + 1312 x2 cosh 3x + 9 x4 cosh 3x − 1120 x3 sinh x +324 x5 sinh x + 23040 x sinh 2x − 320 x3 sinh 2x − 5760 x sinh 3x −160 x3 sinh 3x , 3 ¯h 1 1 −24−4 x2 +24 cosh x+ x2 cosh x− 9 x sinh x , = x 2 2 ω ˜ 24x sinh 2 4 1 1 x x 3x ¯h = 45 x cosh − 6 x3 cosh − 45 x cosh ω ˜ 288x2 sinh3 x 2 2 2 2 x 3x 3x x −432 sinh − 54 x2 sinh + 144 sinh + 4 x2 sinh , 2 2 2 2 5 ¯h 1 1 = −3456 − 414 x2 − 6 x4 + 4608 cosh x+ ω ˜ 2304x3 sinh4 x 2 496 x2 cosh x − 1152 cosh 2x − 82 x2 cosh 2x − 1008 x sinh x− 16 x3 sinh x + 504 x sinh 2x + 5 x3 sinh 2x .

479

(5A.5)

(5A.6)

(5A.7)

(5A.8)

Six of these integrals are the analogs of those in Eqs. (3.547). In addition there are the three integrals a62 , a83 , and a10 3′ , corresponding to the three diagrams ,

,

,

(5A.9)

respectively, which are needed in Subsection 5.14.2. They have been calculated with zero Matsubara frequency in Eqs. (3D.8)–(3D.11). In the low-temperature limit where x = Ω¯hβ → ∞, the x-dependent factors in Eqs. (5A.1)– (5A.8) converge towards the same constants (3D.13) as those with zero Matsubara frequency, and the same limiting relations hold as in Eqs. (3.550) and (3D.14). The high-temperature limits x → 0, however, are quite different from those in Eq. (3D.16). The present Feynman integrals all vanish rapidly for increasing temperatures. For L lines and V V L vertices, ¯hβ(1/˜ ω)V −1 a2L V goes to zero like β (β/12) . The first V factors are due to the V -integrals over τ , the second are the consequence of the product of n/2 factors a2 . Thus a2L V behaves like L ¯h a2L xV −1+L . (5A.10) V ∝ ω ˜ Indeed, the x-dependent factors in (5A.1)–(5A.8) vanish now like x/12, x3 /720, x5 /30240,

480

5 Variational Perturbation Theory x5 /241920, x7 /11404800, 193x8 /47551795200, x4 /30240, x6 /1814400, x7 /59875200,

(5A.11)

respectively. When expanding (5A.1)–(5A.8) into a power series, the lowest powers cancel each other. For the temperature behavior of these Feynman integrals see Fig. 5.40. We have plotted the reduced Feynman integrals a ˆ2L V (x) in which the low-temperature behaviors (3.550) and (3D.14) have been divided out of a2L . V 1 0.8

a ˆ2L V

0.6 0.4 0.2

a2 0.5

1

2

1.5

L/x

Figure 5.40 Plot of the reduced Feynman integrals a ˆ2L V (x) as functions of L/x = LkB T /¯ hω. The integrals (3D.4)–(3D.11) are indicated by decreasing dash-lenghts. Compare Fig. 3.16. The integrals (5A.2) and (5A.3) for a42 and a63 can be obtained from the integral (5A.1) for a2 via the operation n n ¯hn ∂ ¯hn 1 ∂ − 2 = − , (5A.12) n! ∂ω ˜ n! 2˜ ω ∂ω ˜ with n = 1 and n = 2, respectively. This is derived following the same steps as in Eqs. (3D.18)– (3D.20). The absence of the zero Matsubara frequency does not change the argument. Also, as in Eqs. (5.195)–(3D.21), the same type of expansion allows us to derive the three integrals from the one-loop diagram (3.546).

Appendix 5B

Proof of Scaling Relation for Extrema of WN

Here we prove the scaling relation (5.215), according to which the derivative of the N th approximation WN to the ground state energy can be written as [14] g N d WNΩ = pN (σ), (5B.1) dΩ Ω3 where pN (σ) is a polynomial of order N in the scaling variable σ = Ω(Ω2 − 1)/g. For the sake of generality, we consider an anharmonic oscillator with a potential gxP whose power P is arbitrary. The ubiquitous factor 1/4 accompanying g is omitted, for convenience. The energy eigenvalue of the ground state (or any excited state) has an N th order perturbation expansion N l X g EN (g) = ω El , (5B.2) ω (P +2)/2 l=0 where El are rational numbers. After the replacement (5.188), the series is re-expanded at fixed r in powers of g up to order N , and we obtain WNΩ = Ω

N X l=0

εl (σ)

g Ω(P +2)/2

l

,

(5B.3)


Proof of Scaling Relation for the Extrema of WN

Appendix 5B

481

with the re-expansion coefficients [compare (5.207)] εl (σ) =

l X

Ej

j=0

(1 −

P +2 2 j)/2

l−j

(−σ)l−j .

(5B.4)

Here σ is a scaling variable for the potential gxP generalizing (5.208) (note that it is four times as big as the previous σ, due to the different normalization of g): σ≡

Ω(P −2)/2 (Ω2 − ω 2 ) . g

(5B.5)

We now show that the derivative dWN (g, Ω)/dΩ has the following scaling form generalizing (5B.1): N dWNΩ g = pN (σ), (5B.6) dΩ Ω(P +2)/2 where pN (σ) is the following polynomial of order N in the scaling variable σ: pN (σ)

dεN +1 (σ) dσ N X (1 − P +2 j)/2 2 = 2 Ej (N + 1 − j) (−σ)N −j . N +1−j = −2

(5B.7)

j=0

The proof starts by differentiating (5B.3) with respect to Ω, yielding N l X dWNΩ P +2 dεl g εl (σ) − . = lεl (σ) + Ω dΩ 2 dΩ Ω(P +2)/2

(5B.8)

l=0

Using the chain rule of differentiation we see from (5B.4) that (P +2)/2 dεl Ω P −2 dεl Ω = 2 + σ , dΩ g 2 dσ

(5B.9)

and (5B.8) can be rewritten as dWNΩ dΩ

=

(P +2)/2 N l X P +2 Ω P −2 dεl g 1− l εl (σ) + 2 + σ . 2 g 2 dσ Ω(P +2)/2 l=0

(5B.10)

After rearranging the sum, this becomes dWNΩ dΩ

= + +

−1 dε0 g (P +2)/2 dσ Ω N −1 l X P − 2 dεl dεl+1 g P +2 1− l εl + σ +2 2 2 dσ dσ Ω(P +2)/2 l=0 N P +2 P − 2 dεN g 1− N εN + σ . 2 2 dσ Ω(P +2)/2 2

(5B.11)

The first term vanishes trivially since ε0 happens to be independent of σ. The sum in the second line vanishes term by term: P +2 P − 2 dεl dεl+1 1− l εl + σ +2 = 0, l = 1 . . . N − 1. (5B.12) 2 2 dσ dσ

482


To see this we form the derivative 2

l X dεl+1 (1 − P 2+2 j)/2 (j − l − 1) (−σ)l−j , =2 Ej l+1−j dσ j=0

2

(5B.13)

and use the identity (1 − P +2 2 j)/2 l+1−j

=

P −2 2 j

+ 2l − 1 j−l−1

(1 −

P +2 2 j)/2

l−j

(5B.14)

to rewrite (5B.13) as l

X dεl+1 2 = Ej dσ j=0

(1 −

P +2 2 j)/2

l−j

P −2 j + 2l − 1 (−σ)l−j , 2

(5B.15)

implying l

X P − 2 dεl σ = Ej 2 dσ j=0

(1 −

P +2 2 j)/2

l−j

P −2 (l − j)(−σ)l−j . 2

(5B.16)

By combining this with (5B.4), (5B.13), we obtain Eq. (5B.12) which proves that the second line in (5B.11) vanishes. Thus we are left with the last term on the right-hand side of Eq. (5B.11). Using (5B.12) for l = N leads to N dε dWNΩ g N +1 (σ) = −2 . (5B.17) dΩ dσ Ω(P +2)/2 When expressing dεN +1 (σ)/dσ with the help of (5B.4), we arrive at dWNΩ dΩ

=

2

g Ω(P +2)/2

N N X

Ej

j=0

(1 − P +2 2 j)/2 N +1−j

(N + 1 − j) (−σ)N −j . (5B.18)

This proves the scaling relation (5B.6) with the polynomial (5B.7). The proof can easily be extended to physical quantities QN (g) with a different physical dimension α, which have an expansion QN (g) = ω α

N X l=0

El

g ω (P +2)/2

l

,

α 6= 1,

(5B.19)

rather than (5B.2). In this case the quantity [QN (g)]1/α has again an expansion like (5B.2). By rewriting QN (g) as {[QN (g)]1/α }α and forming the derivative using the chain rule we see that the derivative vanishes whenever the polynomial pN (σ) vanishes, which is formed from [QN (g)]1/α as in Eq. (5B.7).

Appendix 5C

Second-Order Shift of Polaron Energy

For brevity, we introduce the dimensionless variable ρ ≡ ω∆τ and ¯ Ω,Γ (ρ, 0). F [ρ] ≡ −2Γ2 G

(5C.1)

Going to natural units with h ¯ = M = ω = 1, Feynman’s variational energy (5.395) takes the form Z ∞ 3 1 2 E0 = (Γ − Ω) − √ Γ dρ e−ρ F −1/2 (ρ). (5C.2) 4Γ π 0 H. Kleinert, PATH INTEGRALS

483

Notes and References The second-order correction (5.406) reads Z ∞ 412 2 1 3 Γ I + √ (Γ2 − Ω2 ) dρ e−ρ F −1/2 (ρ) − (Γ2 − Ω2 )2 3 π 2Γ π 16Γ 0 Z ∞ Γ2 −Ω2 −Γρ 1 Ω2 2 2 −ρ −3/2 − √ (Γ −Ω ) dρ e F (ρ) 1+ 1−e−Γρ + ρe , Γ Γ 4 π 0 (2)

∆E0

=−

where I denotes the integral Z Z Z 1 ∞ ∞ ∞ arcsin Q I= dρ1 dρ2 dρ3 e−ρ1 F −1/2 (ρ1 )e−ρ2 F −1/2 (ρ2 ) −1 , 4 0 Q 0 0

(5C.3)

(5C.4)

with Q = Q1 for ρ3 − ρ2 + ρ1 ≥ 0 and ρ3 − ρ2 ≥ 0, Q = Q2 for ρ3 − ρ2 + ρ1 ≥ 0 and ρ3 − ρ2 < 0, Q = Q3 for ρ3 − ρ2 + ρ1 < 0 and ρ3 − ρ2 < 0,

(5C.5)

and Q1 = Q2 = × Q3 = ×

1 −1/2 2 −1/2 2 Γ2 − Ω2 −Γρ3 F (ρ1 )F (ρ2 ) e (1 − e−Γρ1 )(eΓρ − 1), 2 Γ 1 −1/2 F (ρ1 )F −1/2 (ρ2 ) 2 i Γ2 − Ω2 h −Γ(ρ2 −ρ3 ) −Γρ3 −Γ(ρ1 −ρ2 ) −Γρ1 2 e −e (1 + e −e ) − 2Ω (ρ2 − ρ3 ) , Γ 1 −1/2 F (ρ1 )F −1/2 (ρ2 ) 2 i Γ2 − Ω2 h −Γρ3 −Γρ1 −Γ(ρ2 −ρ3 ) Γρ1 2 −e (1 − e )−e (e − 1) − 2Ω ρ1 . Γ

(5C.6)

(5C.7)

(5C.8)

Notes and References The first-order variational approximation to the effective classical partition function V eff cl (x0 ) presented in this chapter was developed in 1983 by R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin. de/~kleinert/159). For further development see: H. Kleinert, Phys. Lett. B 181, 324 (1986) (ibid.http/151); A 118, 195 (1986) (ibid.http/151); W. Janke and B.K. Chang, Phys. Lett. B 129, 140 (1988); W. Janke, in Path Integrals from meV to MeV , ed. by V. Sa-yakanit et al., World Scientific, Singapore, 1990. A detailed discussion of the accuracy of the approach in comparison with several other approximation schemes is given by S. Srivastava and Vishwamittar, Phys. Rev. A 44, 8006 (1991). For a similar, independent development containing applications to simple quantum field theories, see R. Giachetti and V. Tognetti, Phys. Rev. Lett. 55, 912 (1985); Int. J. Magn. Mater. 54-57, 861 (1986); R. Giachetti, V. Tognetti, and R. Vaia, Phys. Rev. B 33, 7647 (1986); Phys. Rev. A 37, 2165 (1988); Phys. Rev. A 38, 1521, 1638 (1988); Physica Scripta 40, 451 (1989). R. Giachetti, V. Tognetti, A. Cuccoli, and R. Vaia, lecture presented at the XXVI Karpacz School of Theoretical Physics, Karpacz, Poland, 1990.

484


See also R. Vaia and V. Tognetti, Int. J. Mod. Phys. B 4, 2005 (1990); A. Cuccoli, V. Tognetti, and R. Vaia, Phys. Rev. B 41, 9588 (1990); A 44, 2743 (1991); A. Cuccoli, A. Maradudin, A.R. McGurn, V. Tognetti, and R. Vaia, Phys. Rev. D 46, 8839 (1992). The variational approach has solved some old problems in quantum crystals by extending in a simple way the classical methods into the quantum regime. See V.I. Yukalov, Mosc. Univ. Phys. Bull. 31, 10-15 (1976); S. Liu, G.K. Horton, and E.R. Cowley, Phys. Lett. A 152, 79 (1991); A. Cuccoli, A. Macchi, M. Neumann, V. Tognetti, and R. Vaia, Phys. Rev. B 45, 2088 (1992). The systematic extension of the variational approach was developed by H. Kleinert, Phys. Lett. A 173, 332 (1992) (quant-ph/9511020). See also J. Jaenicke and H. Kleinert, Phys. Lett. A 176, 409 (1992) (ibid.http/217); H. Kleinert and H. Meyer, Phys. Lett. A 184, 319 (1994) (hep-th/9504048). A similar convergence mechanism was first observed within an order-dependent mapping technique in the seminal paper by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). For an introduction into various resummation procedures see C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. The proof of the convergence of the variational perturbation expansion to be given in Subsection 17.10.5 went through the following stages: First a weak estimate was found for the anharmonic integral: I.R.C. Buckley, A. Duncan, H.F. Jones, Phys. Rev. D 47, 2554 (1993); C.M. Bender, A. Duncan, H.F. Jones, Phys. Rev. D 49, 4219 (1994). This was followed by a similar extension to the quantum-mechanical case: A. Duncan and H.F. Jones, Phys. Rev. D 47, 2560 (1993); C. Arvanitis, H.F. Jones, and C.S. Parker, Phys.Rev. D 52, 3704 (1995) (hep-ph/9502386); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. 241, 152 (1995) (hep-th/9407027). The exponentially fast convergence observed in the calculation of the strong-coupling coefficients of Table 5.9 was, however, not explained. The accuracy in the table was reached by working up to the order 251 with 200 digits in Ref. [13]. The analytic properties of the strong-coupling expansion were studied by C.M. Bender and T.T. Wu, Phys. Rev. 184, 1231 (1969); Phys. Rev. Lett. 27, 461 (1971); Phys. Rev. D 7, 1620 (1973); ibid. D 7, 1620 (1973); C.M. Bender, J. Math. Phys. 11, 796 (1970); T. Banks and C.M. Bender, J. Math. Phys. 13, 1320 (1972); J.J. Loeffel and A. Martin, Cargèse Lectures on Physics (1970); D. Bessis ed., Gordon and Breach, New York 1972, Vol. 5, p.415; B. Simon, Ann. Phys. (N.Y.) 58, 76 (1970); Cargèse Lectures on Physics (1970), D. Bessis ed., Gordon and Breach, New York 1972, Vol. 5, p. 383. The problem of tunneling at low barriers (sliding) was solved by H. Kleinert, Phys. Lett. B 300, 261 (1993) (ibid.http/214). See also Chapter 17. Some of the present results are contained in H. Kleinert, Pfadintegrale in Quantenmechanik, Statistik und Polymerphysik , B.-I. Wissenschaftsverlag, Mannheim, 1993. A variational approach to tunneling is also used in chemical physics: M.J. Gillan, J. Phys. C 20, 362 (1987);



485

G.A. Voth, D. Chandler, and W.H. Miller, J. Chem. Phys. 91, 7749 (1990); G.A. Voth and E.V. O’Gorman, J. Chem. Phys. 94, 7342 (1991); G.A. Voth, Phys. Rev. A 44, 5302 (1991). Variational approaches without the separate treatment of x0 have been around in the literature for some time: T. Barnes and G.I. Ghandour, Phys. Rev. D 22, 924 (1980); B.S. Shaverdyan and A.G. Usherveridze, Phys. Lett. B 123, 316 (1983); K. Yamazaki, J. Phys. A 17, 345 (1984); H. Mitter and K. Yamazaki, J. Phys. A 17, 1215 (1984); P.M. Stevenson, Phys. Rev. D 30, 1712 (1985); D 32, 1389 (1985); P.M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); D 36, 2415 (1987); W. Namgung, P.M. Stevenson, and J.F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M.H. Thoma, Z. Phys. C 44, 343 (1991); I. Stancu and P.M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A.N. Sissakian, I.L. Solovtsov, and O.Y. Shevchenko, Phys. Lett. B 313, 367 (1993). Different applications of variational methods to density matrices are given in V.B. Magalinsky, M. Hayashi, and H.V. Mendoza, J. Phys. Soc. Jap. 63, 2930 (1994); V.B. Magalinsky, M. Hayashi, G.M. Martinez Pe˜ na, and R. Reyes Sánchez, Nuovo Cimento B 109, 1049 (1994). The particular citations in this chapter refer to the publications [1] H. Kleinert, Phys. Lett. A 173, 332 (1993) (quant-ph/9511020). [2] The energy eigenvalues of the anharmonic oscillator are taken from F.T. Hioe, D. MacMillan, and E.W. Montroll, Phys. Rep. 43, 305 (1978); W. Caswell, Ann. Phys. (N.Y.) 123, 153 (1979); R.L. Somorjai, and D.F. Hornig, J. Chem. Phys. 36, 1980 (1962). See also K. Banerjee, Proc. Roy. Soc. A 364, 265 (1978); R. Balsa, M. Plo, J.G. Esteve, A.F. Pacheco, Phys. Rev. D 28, 1945 (1983); and most accurately ˇ ıˇzek, J. Math. Phys. 32, 3392 (1991); F. Vinette and J. C´ ˇ ıˇzek, J. Math. Phys. 34, 571 (1993). E.J. Weniger, J. C´ [3] R.P. Feynman, Statistical Mechanics, Benjamin, Reading, 1972, Section 3.5. [4] M. Hillary, R.F. O’Connell, M.O. Scully, and E.P. Wigner, Phys. Rep. 106, 122 (1984). [5] H. Kleinert, Phys. Lett. A 118, 267 (1986) (ibid.http/145). [6] For a detailed discussion of the effective classical potential of the Coulomb system see W. Janke and H. Kleinert, Phys. Lett. A 118, 371 (1986) (ibid.http/153). [7] C. Kouveliotou et al., Nature 393, 235 (1998); Astroph. J. 510, L115 (1999); K. Hurley et al., Astroph. J. 510, L111 (1999); V.M. Kaspi, D. Chakrabarty, and J. Steinberger, Astroph. J. 525, L33 (1999); B. Zhang and A.K. Harding, (astro-ph/0004067). [8] The perturbation expansion of the ground state energy in powers of the magnetic field B was driven to high orders in ˇ ıˇzek, M. Clay, M.L. Glasser, P. Otto, J. Paldus, and E. Vrscay, J.E. Avron, B.G. Adams, J. C´

486

5 Variational Perturbation Theory Phys. Rev. Lett. 43, 691 (1979); ˇ ıˇzek, P. Otto, J. Paldus, R.K. Moats, and H.J. Silverstone, B.G. Adams, J.E. Avron, J. C´ Phys. Rev. A 21, 1914 (1980). This was possible on the basis of the dynamical group O(4,1) and the tilting operator (13.181) found by the author in his Ph.D. thesis. See H. Kleinert, Group Dynamics of Elementary Particles, Fortschr. Physik 6, 1 (1968) (ibid.http/1); H. Kleinert, Group Dynamics of the Hydrogen Atom, Lectures in Theoretical Physics, edited by W.E. Brittin and A.O. Barut, Gordon and Breach, N.Y. 1968, pp. 427-482 (ibid.http/4).

[9] Precise numeric calculations of the ground state energy of the hydrogen atom in a magnetic field were made by H. Ruder, G. Wunner, H. Herold, and F. Geyer, Atoms in Strong Magnetic Fields (SpringerVerlag, Berlin, 1994). [10] M. Bachmann, H. Kleinert, and A. Pelster, Phys. Rev. A 62, 52509 (2000) (quantph/0005074), Phys. Lett. A 279, 23 (2001) (quant-ph/000510). [11] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965. [12] J.C. LeGuillou and J. Zinn-Justin, Ann. Phys. (N.Y.) 147, 57 (1983). [13] W. Janke and H. Kleinert, Phys. Rev. Lett. 75, 2787 (1995) (quant-ph/9502019). [14] The high accuracy became possible due to a scaling relation found in W. Janke and H. Kleinert, Phys. Lett. A 199, 287 (1995) (quant-ph/9502018). [15] For the proof of the exponentially fast convergence see H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) (quant-ph/9502019); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. 249, 109 (1996) (hep-th/9505084). The proof will be given in Subsection 17.10.5. [16] The oscillatory behavior around the exponential convergence shown in Fig. 5.22 was explained in H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) (quant-ph/9502019) in terms of the convergence radius of the strong-coupling expansion (see Section 5.15). [17] H. Kleinert, Phys. Rev. D 60 , 085001 (1999) (hep-th/9812197); Phys. Lett. B 463, 69 (1999) (cond-mat/9906359). See also Chapters 19–20 in the textbook H. Kleinert and V. Schulte-Frohlinde, Critical Properties of Φ4 -Theories, World Scientific, Singapore 2001 (ibid.http/b8) [18] H. Kleinert, Phys. Lett. A 207, 133 (1995). [19] See for example the textbooks H. Kleinert, Gauge Fields in Condensed Matter , Vol. I Superflow and Vortex Lines, Vol. II Stresses and Defects, World Scientific, Singapore, 1989 (ibid.http/b1). [20] R.P. Feynman, Phys. Rev. 97, 660 (1955). [21] S. Höhler and A. M¨ ullensiefen, Z. Phys. 157 , 159 (1959); M.A. Smondyrev, Theor. Math. Fiz. 68 , 29 (1986); O.V. Selyugin and M.A. Smondyrev, Phys. Stat. Sol. (b) 155 , 155 (1989). [22] N.N. Bogoliubov (jun) and V.N. Plechko, Teor. Mat. Fiz. [Sov. Phys.-Theor. Math. Phys.], 65, 423 (1985); Riv. Nuovo Cimento 11, 1 (1988). [23] S.J. Miyake, J. Phys. Soc. Japan, 38, 81 (1975). H. Kleinert, PATH INTEGRALS


487

[24] J.E. Avron, I.W. Herbst, B. Simon, Phys. Rev. A 20, 2287 (1979). [25] I.D. Feranshuk and L.I. Komarov, J. Phys. A: Math. Gen. 17, 3111 (1984). [26] H. Kleinert, W. K¨ urzinger, and A. Pelster, J. Phys. A: Math. Gen. 31, 8307 (1998) (quantph/9806016). [27] H. Kleinert, Phys. Lett. A 173, 332 (1993) (quant-ph/9511020). [28] H. Kleinert, Phys. Rev. D 57, 2264 (1998) and Addendum: Phys. Rev. D 58, 107702 (1998). [29] F.J. Wegner, Phys. Rev. B 5, 4529 (1972); B 6, 1891 (1972). [30] H. Kleinert, Phys. Lett. A 207, 133 (1995) (quant-ph/9507005). [31] J. Rössler, J. Phys. Stat. Sol. 25, 311 (1968). [32] J.T. Marshall and L.R. Mills, Phys. Rev. B 2 , 3143 (1970). [33] Higher-order smearing formulas for nonpolynomial interactions were derived in H. Kleinert, W. K¨ urzinger and A. Pelster, J. Phys. A 31, 8307 (1998) (quant-ph/9806016). [34] The polaron problem is solved in detail in the textbook R.P. Feynman, Statistical Mechanics, Benjamin, New York, 1972, Chapter 8. Extensive numerical evaluations are found in T.D. Schultz, Phys. Rev. 116, 526 (1959); and in M. Dineykhan, G.V. Efimov, G. Ganbold, and S.N. Nedelko, Oscillator Representation in Quantum Physics, Springer, Berlin, 1995. An excellent review article is J.T. Devreese, Polarons, Review article in Encyclopedia of Applied Physics, 14, 383 (1996) (cond-mat/0004497). This article contains ample references on work concerning polarons in magnetic fields, for instance F.M. Peeters, J.T. Devreese, Phys. Stat. Sol. B 110, 631 (1982); Phys. Rev. B 25, 7281, 7302 (1982); Xiaoguang Wu, F.M. Peeters, J.T. Devreese, Phys. Rev. B 32, 7964 (1985); F. Brosens and J.T. Devreese, Phys. Stat. Sol. B 145, 517 (1988). For discussion of the validity of the Jensen-Peierls inequality (5.10) in the presence of a magnetic field, see J.T. Devreese and F. Brosens, Solid State Communs. 79, 819 (1991); Phys. Rev. B 45, 6459 (1992); Solid State Communs. 87, 593 (1993); D. Larsen in Landau Level Spectroscopy, Vol. 1, G. Landwehr and E. Rashba (eds.), North Holland, Amsterdam, 1991, p. 109. The paper D. Larsen, Phys. Rev. B 32, 2657 (1985) shows that the variational energy can lie lower than the exact energy. The review article by Devreese contains numerous references on bipolarons, small polarons, and polaronic excitations. For instance: J.T. Devreese, J. De Sitter, M.J. Goovaerts, Phys. Rev. B 5, 2367 (1972); L.F. Lemmens, J. De Sitter, J.T. Devreese, Phys. Rev. B 8, 2717 (1973); J.T. Devreese, L.F. Lemmens, J. Van Royen, Phys. Rev. B 15, 1212 (1977); J. Thomchick, L.F. Lemmens, J.T. Devreese, Phys. Rev. B 14, 1777 (1976); F.M. Peeters, Xiaoguang Wu, J.T. Devreese, Phys. Rev. B 34, 1160 (1986); F.M. Peeters, J.T. Devreese, Phys. Rev. B 34, 7246 (1986); B 35, 3745 (1987); J.T. Devreese, S.N. Klimin, V.M. Fomin, F. Brosens, Solid State Communs. 114, 305 (2000). There exists also a broad collection of articles in

488

5 Variational Perturbation Theory E.K.H. Salje, A.S. Alexandrov, W.Y. Liang (eds.), Polarons and Bipolarons in High-Tc Superconductors and Related Materials, Cambridge University Press, Cambridge, 1995. A generalization of the harmonic trial path integral (5.356), in which the exponential func′ tion e−Ω|τ −τ | at zero temperature is replaced by f (|τ − τ ′ |), has been proposed by M. Saitoh, J. Phys. Soc. Japan. 49, 878 (1980), and further studied by R. Rosenfelder and A.W. Schreiber, Phys. Lett. A 284, 63 (2001) (cond-mat/ 0011332). In spite of a much higher numerical effort, this generalization improves the ground state energy only by at most 0.1 % (the weak-coupling expansion coefficient −0.012346 in (5.396) is changed to −0.012598, while the strong-coupling coefficients in (5.399) are not changed at all. For the effective mass, the lowest nontrivial weak-coupling coefficient of 12 in (5.402) is changed by 0.0252 % while the strong-coupling coefficients in (5.403) are not changed at all.



Aevo rarissima nostro, simplicitas. Simplicity, a very rare thing in our age. Ovid, Ars Amatoria, Book 1, 241

6 Path Integrals with Topological Constraints The path integral representations of the time evolution amplitudes considered so far were derived for orbits x(t) fluctuating in Euclidean space with Cartesian coordinates. Each coordinate runs from minus infinity to plus infinity. In many physical systems, however, orbits are confined to a topologically restricted part of a Cartesian coordinate system. This changes the quantum-mechanical completeness relation and with it the derivation of the path integral from the time-sliced time evolution operator in Section 2.1. We shall consider here only a point particle moving on a circle, in a half-space, or in a box. The path integral treatment of these systems is the prototype for any extension to more general topologies.

6.1

Point Particle on Circle

For a point particle on a circle, the orbits are specified in terms of an angular variable ϕ(t) ∈ [0, 2π] subject to the topological constraint that ϕ = 0 and ϕ = 2π be identical points. The initial step in the derivation of the path integral for such a system is the same as before: The time evolution operator is decomposed into a product NY +1 i i ˆ ˆ |ϕa i ≡ hϕb | exp − ǫH |ϕa i. hϕb tb |ϕa ta i = hϕb |exp − (tb − ta )H h ¯ h ¯ n=1

(6.1)

The restricted geometry shows up in the completeness relations to be inserted between the factors on the right-hand side for n = 1, . . . , N: Z

0

2π

dϕn |ϕn ihϕn | = 1.

(6.2)

If the integrand is singular at ϕ = 0, the integrations must end at an infinitesimal piece below 2π. Otherwise there is the danger of double-counting the contributions from the identical points ϕ = 0 and ϕ = 2π. The orthogonality relations on these intervals are hϕn |ϕn−1i = δ(ϕn − ϕn−1 ), 489

ϕn ∈ [0, 2π).

(6.3)

490

6 Path Integrals with Topological Constraints

The δ-function can be expanded into a complete set of periodic functions on the circle: δ (ϕn − ϕn−1 ) =

∞ X

1 exp[imn (ϕn − ϕn−1 )]. mn =−∞ 2π

(6.4)

For a trivial system with no Hamiltonian, the scalar products (6.4) lead to the following representation of the transition amplitude: (ϕb tb |ϕa ta )0 =

N Z Y

n=1

2π

0

dϕn

" NY +1 X mn

n=1

N +1 X 1 exp i mn (ϕn − ϕn−1 ) . 2π n=1

#

"

#

(6.5)

We now introduce a Hamiltonian H(p, ϕ). At each small time step, we calculate i ˆ (ϕn tn |ϕn−1 tn−1 ) = hϕn | exp − ǫH(p, ϕ) |ϕn−1 i h ¯ i ˆ h∂ϕn , ϕn ) hϕn |ϕn−1i. = exp − ǫH(−i¯ h ¯

Replacing the scalar products by their spectral representation (6.4), this becomes i ˆ ϕ) |ϕn−1 i (ϕn tn |ϕn−1 tn−1 ) = hϕn | exp − ǫH(p, h ¯ X ∞ i ˆ 1 h∂ϕn , ϕn ) exp[imn (ϕn − ϕn−1 )]. (6.6) = exp − ǫH(−i¯ h ¯ mn =−∞ 2π

By applying the operator in front of the sum to each term, we obtain (ϕn tn |ϕn−1

∞ X

iǫ 1 tn−1 ) = exp imn (ϕn − ϕn−1 ) − H(¯ hmn , ϕn ) . (6.7) h ¯ mn =−∞ 2π

The total amplitude can therefore be written as (ϕb tb |ϕa ta ) ≈

N Z Y

n=1

2π

0

(

× exp i

dϕn

" ∞ NY +1 X n=1

N +1 X n=1

1 mn =−∞ 2π

#

(6.8)

1 mn (ϕn − ϕn−1 ) − ǫH(¯ hmn , ϕn ) h ¯

)

.

This is the desired generalization of the original path integral from Cartesian to cyclic coordinates. As a consequence of the indistinguishability of ϕ(t) and ϕ(t) + 2πn, the momentum integrations have turned into sums over integer numbers. The sums √ reflect the fact that the quantum-mechanical wave functions (1/ 2π) exp(ipϕ ϕ/¯ h) are single-valued. The discrete momenta enter into (6.8) via a “momentum step sum” rather than a proper path integral. At first sight, such an expression looks somewhat hard to deal with in practical calculations. Fortunately, it can be turned into a more comfortable H. Kleinert, PATH INTEGRALS

491

6.1 Point Particle on Circle

equivalent form, involving a proper continuous path integral. This is possible at the expense of a single additional infinite sum which guarantees the cyclic invariance in the variable ϕ. To find the equivalent form, we recall Poisson’s formula (1.197), ∞ X

∞ X

e2πikl =

m=−∞

l=−∞

δ(k − m),

(6.9)

to make the right-hand side of (6.4) a periodic sum of δ-functions, so that (6.3) becomes ∞ X

hϕn |ϕn−1 i =

l=−∞

δ(ϕn − ϕn−1 + 2πl).

(6.10)

A Fourier decomposition of the δ-functions yields ∞ Z X

hϕn |ϕn−1i =

dkn exp[ikn (ϕn − ϕn−1 ) + 2πikn l]. 2π

∞

l=−∞ −∞

(6.11)

Note that the right-hand side reduces to (6.4) when applying Poisson’s summation formula (6.9) to the l-sum, which produces a sum of δ-functions for the integer values of kn = mn = 0, ±1, ±2, . . . . Using this expansion rather than (6.4), the amplitude (6.5) with no Hamiltonian takes the form (ϕb tb |ϕa ta )0 =

N Z Y

n=1

2π

0

 NY +1 Z  dϕn

∞

−∞

n=1



∞ PN+1 dkn X  ei n=1 [kn (ϕn −ϕn−1 )+2πkn ln ] . (6.12) 2π ln =−∞

In this expression, we observe that the sums over ln can be absorbed into the variables ϕn by extending their range of integration from [0, 2π) to (−∞, ∞). Only in P the last sum lN+1 , this is impossible, and we arrive at (ϕb tb |ϕa ta )0 =

∞ Y N Z X

l=−∞ n=1

∞

−∞

" NY +1 Z

dϕn

n=1

∞

−∞

#

dkn i PN+1 kn (ϕn −ϕn−1 +2πlδn,N+1 ) e n=1 .(6.13) 2π

ˆ ≡ 0 -amplitude of an ordinary particle The right-hand side looks just like an H which would read (ϕb tb |ϕa ta )0,noncyclic =

N Z Y

n=1

∞

−∞

dϕn

" NY +1 Z n=1

∞

−∞

#

dkn i PN+1 kn (ϕn −ϕn−1 ) e n=1 . (6.14) 2π

The amplitude (6.13) differs from this by the sum over paths running over all periodic repetitions of the final point ϕb + 2πn, tb . The amplitude (6.13) may therefore be written as a sum over all periodically repeated final points of the amplitude (6.14): (ϕb tb |ϕa ta )0 =

∞ X

l=−∞

(ϕb + 2πl, tb |ϕa ta )0,noncyclic .

(6.15)

492


In each term on the right-hand side, the Hamiltonian can be inserted as usual, and we arrive at the time-sliced formula (ϕb tb |ϕa ta ) ≈

∞ Y N Z X

∞

−∞

l=−∞ n=1

dϕn

" NY +1 Z n=1

∞ −∞

dpn 2π¯ h

#

+1 i NX × exp [pn (ϕn − ϕn−1 + 2πlδn,N +1 ) − ǫH(pn , ϕn )] . h ¯ n=1

(

)

(6.16)

In the continuum limit, this tends to the path integral ǫ→0

(ϕb tb |ϕa ta ) − −−→

∞ Z X

l=−∞ ϕa ;ϕb +2πl

Dϕ(t)

Z

Dp(t) i exp 2π¯ h h ¯

Z

tb

ta

dt[pϕ˙ −H(p, ϕ)] . (6.17)

The way in which this path integral has replaced the sum over all paths on the circle ϕ ∈ [0, 2π) by the sum over all paths with the same action on the entire ϕ-axis is illustrated in Fig. 6.1. As an example, consider a free particle moving on a circle with a Hamiltonian H(p, ϕ) =

p2 . 2M

(6.18)

The ordinary noncyclic path integral is (ϕb tb |ϕa ta )noncyclic

i M (ϕb − ϕa )2 exp =q . h ¯ 2 tb − ta 2π¯ hi(tb − ta )/M "

1

#

(6.19)

Using Eq. (6.15), the cyclic amplitude is given by the periodic Gaussian i M (ϕb − ϕa + 2πl)2 (ϕb tb |ϕa ta ) = q exp . h ¯ 2 tb − ta 2π¯ hi(tb − ta )/M l=−∞ ∞ X

1

#

"

(6.20)

The same amplitude could, of course, have been obtained by a direct quantummechanical calculation based on the wave functions 1 ψm (ϕ) = √ eimϕ (6.21) 2π and the energy eigenvalues

h ¯2 2 H= m. 2M Within operator quantum mechanics, we find

(6.22)

i ˆ |ϕa i (ϕb tb |ϕa ta ) = hϕb |exp − (tb − ta )H h ¯ " =

∞ X

m=−∞ ∞ X

∗ ψm (ϕb )ψm (ϕa ) exp

ih ¯ 2 m2 − (tb − ta ) h ¯ 2M

#

1 h ¯ m2 = exp im(ϕb − ϕa ) − i (tb − ta ) . 2M m=−∞ 2π "

#

(6.23)


493

6.2 Infinite Wall

Figure 6.1 Path with 3 jumps from 2π to 0 at tj1 , tj2 , tj3 , and with one jump from 0 to 2π at t¯1 . It can be drawn as a smooth path in the extended zone scheme, arriving at ϕ(n,¯n) = ϕb + (n − n ¯ )2π, where n and n ¯ count the number of jumps of the first and the second type, respectively.

If the sum over m is converted into an integral over p and a dual l-sum via Poisson’s formula (6.9), this coincides with the previous result: (ϕb tb |ϕa ta ) =

∞ Z X

∞

l=−∞ −∞

p2 dp i exp p(ϕb − ϕa + 2πl) − (tb − ta ) 2π¯ h h ¯ 2M (

"

∞ X

i M (ϕb − ϕa + 2πl)2 q = exp . h ¯ 2 tb − ta 2π¯ hi(tb − ta )/M l=−∞

6.2

1

"

#

#)

(6.24)

Infinite Wall

In the case of an infinite wall, only a half-space, say x = r > 0, is accessible to the particle, and the completeness relation reads Z

0

∞

dr|rihr| = 1.

(6.25)

For singular integrands, the origin has to be omitted from the integration. The orthogonality relation is hr|r ′i = δ(r − r ′ );

r, r ′ > 0.

(6.26)

Given a free particle moving in such a geometry, we want to calculate N +1 Y

i ˆ (rb tb |ra ta ) = hrb | exp − Hǫ |ra i. h ¯ n=1

(6.27)

494


As usual, we insert N completeness relations between the N + 1 factors. In the case of a vanishing Hamiltonian, the amplitude (6.27) becomes N Z Y

(rb tb |ra ta )0 =

n=1

0

∞

drn

NY +1 n=1

hrn |rn−1 i = hrb |ra i.

(6.28)

For each scalar product hrn |rn−1 i = δ(rn − rn−1 ), we substitute its spectral representation appropriate to the infinite-wall boundary at r = 0. It consists of a superposition of the free-particle wave functions vanishing at r = 0: ′

hr|r i = 2 =

Z

Z

dk (6.29) sin kr sin kr ′ π dk [exp ik(r − r ′) − exp ik(r + r ′ )] = δ(r − r ′ ) − δ(r + r ′ ). 2π

∞

0 ∞

−∞

This Fourier representation does a bit more than what we need. In addition to the δ-function at r = r ′ , there is also a δ-function at the unphysical reflected point r = −r ′ . The reflected point plays a similar role as the periodically repeated points in the representation (6.11). For the same reason as before, we retain the reflected points in the formula as though r ′ were permitted to become zero or negative. Thus we rewrite the Fourier representation (6.29) as hr|r ′i =

X Z

∞

x=±r −∞

dp i exp p(x − x′ ) + iπ(σ(x) − σ(x′ )) 2π¯ h h ¯

x′ =r ′

,

(6.30)

where σ(x) ≡ Θ(−x)

(6.31)

with the Heaviside function Θ(x) of Eq. (1.309). For symmetry reasons, it is convenient to liberate both the initial and final positions r and r ′ from their physical half-space and to introduce the localized states |xi whose scalar product exists on the entire x-axis: hx|x′ i =

X Z

∞

x′′ =±x −∞ ′

dp i exp p(x′′ − x′ ) + iπ(σ(x′′ ) − σ(x′ )) 2π¯ h h ¯

= δ(x − x ) − δ(x + x′ ).

(6.32)

With these states, we write hr|r ′i = hx|x′ i|x=r,x′=r′ .

(6.33)

We now take the trivial transition amplitude with zero Hamiltonian (rb tb |ra ta )0 = δ(rb − ra ),

(6.34)

extend it with no harm by the reflected δ-function (rb tb |ra ta )0 = δ(rb − ra ) − δ(rb + ra ),


495

6.2 Infinite Wall

and factorize it into many time slices: N Z Y

(rb tb |ra ta )0 =

∞

0

n=1

drn

" NY +1 X

(xn ǫ|xn−1 0)0

xn =±rn

n=1

#

(6.36)

(rb = rN +1 , ra = r0 ), where the trivial amplitude of a single slice is (xn ǫ|xn−1 0)0 = hxn |xn−1 i,

x ∈ (−∞, ∞).

(6.37)

With the help of (6.32), this can be written as (rb tb |ra ta )0 =

N Z Y

n=1

∞

0

× exp

" NY +1 X

drn

n=1

(N +1 X i

h ¯

n=1

xn =±rn

Z

∞ −∞

dpn 2π¯ h

#

p(xn − xn−1 ) + iπ(σ(xn ) − σ(xn−1 ))

)

.

(6.38)

The sumR over the reflected points xn = ±rn is now combined, at each n, with the integral 0∞ drn to form an integral over the entire x-axis, including the unphysical half-space x < 0. Only the last sum cannot be accommodated in this way, so that we obtain the path integral representation for the trivial amplitude (rb tb |ra ta )0 =

N Z Y

X

xb =±rb n=1

× exp

∞

−∞

(N +1 X i n=1

h ¯

dxn

" NY +1 Z n=1

∞

−∞

dpn 2π¯ h

#

p(xn − xn−1 ) + iπ(σ(xn ) − σ(xn−1 ))

)

. (6.39)

The measure of this path integral is now of the conventional type, integrating over all paths which fluctuate through the entire space. The only special feature is the final symmetrization in xb = ±rb . It is instructive to see in which way the final symmetrization together with the phase factor exp[iπσ(x)] = ±1 eliminates all the wrong paths in the extended space, i.e., those which cross the origin into the unphysical subspace. This is illustrated in Fig. 6.2. Note that having assumed xa = ra > 0, the initial phase σ(xa ) can be omitted. We have kept it merely for symmetry reasons. In the continuum limit, the exponent corresponds to an action Aσ0 [p, x]

=

Z

tb

ta

dt[px˙ + h ¯ π∂t σ(x)] ≡ A0 [p, x] + Aσtopol .

(6.40)

The first term is the usual canonical expression in the absence of a Hamiltonian. The second term is new. It is a pure boundary term: Aσtopol [x] = h ¯ π(σ(xb ) − σ(xa )),

(6.41)

which keeps track of the topology of the half space x > 0 embedded in the full space x ∈ (−∞, ∞). This is why the action carries the subscript “topol”.

496


Figure 6.2 Illustration of path counting near reflecting wall. Each path touching the wall once is canceled by a corresponding path of equal action crossing the wall once into the unphysical regime (the path is mirror-reflected after the crossing). The phase factor exp[iπσ(xb )] provides for the opposite sign in the path integral. Only paths not touching the wall at all cannot be canceled in the path integral.

The topological action (6.41) can be written formally as a local coupling of the velocity at the origin: Aσtopol [x] = −π¯ h

Z

tb

ta

dtx(t)δ(x(t)). ˙

(6.42)

This follows directly from σ(xb ) − σ(xa ) =

Z

xb

xa

dxσ ′ (x) =

Z

xb

xa

dxΘ′ (−x) = −

Z

xb

xa

dxδ(x).

(6.43)

Consider now a free point particle in the right half-space with the usual Hamiltonian p2 . (6.44) H= 2M The action reads A[p, x] =

Z

tb

ta

dt[px˙ − p2 /2M − h ¯ π x(t)δ(x(t))], ˙

(6.45)

and the time-sliced path integral looks like (6.39), except for additional energy terms −p2n /2M in the action. Since the new topological term is a pure boundary term, all the extended integrals in (6.39) can be evaluated right away in the same way as for a free particle in the absence of an infinite wall. The result is (rb tb |ra ta ) =

X

xb =±rb

1 q

2π¯ hi(tb − ta )/M

i M (xb − xa )2 × exp + iπ(σ(xb ) − σ(xa )) h ¯ 2 tb − ta "

1

=q 2π¯ hi(tb − ta )/M

(

#

i M (rb − ra )2 exp − (rb → −rb ) h ¯ 2 tb − ta "

#

(6.46) )


497

6.3 Point Particle in Box

with xa = ra . This is indeed the correct result: Inserting the Fourier transform of the Gaussian (Fresnel) distribution we see that dp i 2 (rb tb |ra ta ) = exp p(rb − ra ) − (rb → −rb ) e−ip (tb −ta )/2M ¯h h h ¯ −∞ 2π¯ " # Z ∞ dp i p2 sin(prb /¯ (tb − ta ) , = 2 h) sin(pra /¯ h) exp − (6.47) 2π¯ h h ¯ 2M 0 Z

∞

which is the usual spectral representation of the time evolution amplitude. Note that the first part of (6.46) may be written more symmetrically as 1 X i M (xb − xa )2 exp + iπ(σ(xb ) − σ(xa )) . (rb tb |ra ta ) = q h ¯ 2 tb − ta 2π¯ hi(tb − ta )/M 2 xa =±ra "

1

#

xb =±rb

(6.48)

In this form, the phase factors eiπσ(x) are related to what may be considered as even and odd “spherical harmonics” in one dimension [more after (9.60)] 1 x) = √ (Θ(x) ± Θ(−x)), Ye,ø (ˆ 2 namely, 1 x) = √ , Ye (ˆ 2

1 Yø (ˆ x) = √ eiπσ(x) . 2

(6.49)

The amplitude (6.48) is therefore simply the odd “partial wave” of the free-particle amplitude X (rb tb |ra ta ) = Yo∗ (ˆ xb )hxb tb |xa ta iYø (ˆ xa ), (6.50) x ˆb ,ˆ xa |xb |=rb ,|xa |=ra

which is what we would also have obtained from Schrödinger quantum mechanics.

6.3

Point Particle in Box

If a point particle is confined between two infinitely high walls in the interval x ∈ (0, d), we speak of a particle in a box .1 The box is a geometric constraint. Since the wave functions vanish at the walls, the scalar product between localized states is given by the quantum-mechanical orthogonality relation for r ∈ (0, d): hr|r ′i = 1

2 X sin kν r sin kν r ′ , d kν >0

(6.51)

See W. Janke and H. Kleinert, Lett. Nuovo Cimento 25 , 297 (1979) (http://www.physik.fu-berlin.de/~kleinert/64).

498


where kν runs over the discrete positive momenta π ν, d

kν =

ν = 1, 2, 3, . . . .

(6.52)

We can write the restricted sum in (6.51) also as a sum over all momenta kν with ν = 0, ±1, ±2, . . .: i 1 X h ikν (r−r′ ) ′ hr|r ′i = e − eikν (r+r ) . (6.53) 2d kν With the help of the Poisson summation formula (6.9), the right-hand side is converted into an integral and an auxiliary sum: ∞ Z X

′

hr|r i =

∞

l=−∞ −∞

i dk h ik(r−r′ +2dl) ′ e − eik(r+r +2dl) . 2π

(6.54)

Using the potential σ(x) of (6.31), this can be re-expressed as ′

hr|r i =

∞ Z X

X

dk ik(x−x′ +2dl)+iπ(σ(x)−σ(x′ )) e . 2π

∞

x=±r l=−∞ −∞

(6.55)

The trivial path integral for the time evolution amplitude with a zero Hamiltonian is again obtained by combining a sequence of scalar products (6.51): "Z N Y

(rb tb |ra ta )0 =

d

0 n=1 " Z d N Y

=

n=1

0

#

drn hrn |rn−1 i drn

# N +1 Y n=1

(6.56)

2X sin kνn rn sin kνn−1 rn−1 . d kν

The alternative spectral representation (6.55) allows us to extend the restricted integrals over xn and sums over kν to complete phase space integrals, and we may write (rb tb |ra ta )0 =

X

∞ Y N Z X

xb =±rb l=−∞ n=1

∞

−∞

dxn

" NY +1 Z n=1

∞

−∞

#

dpn i N exp A , 2π¯ h h ¯ 0

(6.57)

with the time-sliced H ≡ 0 -action: AN 0 =

N +1 X n=1

[pn (xn − xn−1 ) + h ¯ π(σ(xn ) − σ(xn−1 ))] .

(6.58)

The final xb is summed over all periodically repeated endpoints rb + 2dl and their reflections −rb + 2dl. We now add dynamics to the above path integral by introducing some Hamiltonian H(p, x), so that the action reads A=

Z

tb

ta

dt[px˙ − H(p, x) − h ¯ π xδ(x)]. ˙


499

6.3 Point Particle in Box

Figure 6.3 Illustration of path counting in a box. A path reflected once on the upper and once on the lower wall of the box is eliminated by a path with the same action running to (1) (0) (1) xb and to x ¯b , x ¯b . The latter receive a negative sign in the path integral from the phase factor exp[iπσ(xb )]. Only paths remaining completely within the walls have no partner for cancellation.

Figure 6.4 A particle in a box is topologically equivalent to a particle on a circle with an infinite wall at one point.

The amplitude is written formally as the path integral (rb tb |ra ta ) =

∞ X

X

l=−∞ xb =±rb +2dl

Z

Dx

Z

Dp i exp A . 2π¯ h h ¯

(6.60)

In the time-sliced version, the action is AN = AN 0 −ǫ

N +1 X

H(pn , xn ).

(6.61)

n=1

The way in which the sum over the final positions xb = ±rb + 2dl together with the phase factor exp[iπσ(xb )] eliminates the unphysical paths is illustrated in Fig. 6.3. The mechanism is obviously a combination of the previous two. A particle in a box of length d behaves like a particle on a circle of circumference 2d with a periodic boundary condition, containing an infinite wall at one point. This is illustrated in Fig. 6.4. The periodicity in 2d selects the momenta kν = (π/d)ν,

ν = 1, 2, 3, . . . ,

500


as it should. For a free particle with H = p2 /2M, the integrations over xn , pn can be done as usual and we obtain the amplitude (xa = ra ) (rb tb |ra ta ) =

∞ X

X

l=−∞ xb =±rb +2dl

1 q

2π¯ hi(tb − ta )/M

"

e

2 i M (xb −xa +2dl) h ¯ 2 tb −ta

#

− (xb → −xb ) . (6.62)

A Fourier transform and an application of Poisson’s formula (6.9) shows that this is, of course, equal to the quantum-mechanical expression i ˆ |ra i (rb tb |ra ta ) = hrb |exp − (tb − ta )H h ¯ " # ∞ 2X kν2 = sin kν rb sin kν ra exp −i¯ h (tb − ta ) . d ν=1 2M

(6.63)

In analogy with the discussion in Section 2.6, we identify in the exponentials the eigenvalues of the energy levels labeled by ν − 1 = 0, 1, 2, . . . : E (ν−1) =

h ¯ 2 kν2 , 2M

ν = 1, 2, 3 . . . .

(6.64)

The factors in front determine the wave functions associated with these energies: ψ (ν−1) (x) =

6.4

2 sin kν x. d

(6.65)

Strong-Coupling Theory for Particle in Box

The strong-coupling theory developed in Chapter 5 open up the possibility of treating quantum-mechanical systems with hard-wall potentials via perturbation theory. After converting divergent weak-coupling expansions into convergent strongcoupling expansions, the strong-coupling limit of a function can be evaluated from its weak-coupling expansion with any desired accuracy. Due to the combination with the variational procedure, new classes of physical systems become accessible to perturbation theory. For instance, the important problem of the pressure exerted by a stack of membranes upon enclosing walls has been solved by this method.2 Here we illustrate the working of that theory for the system treated in the previous section, the point particle in a one-dimensional box. This is just a quantum-mechanical exercise for the treatment of physically more interesting problems. The ground state energy of this system has, according to Eq. (6.64), the value E (0) = π 2 /2d2 . For simplicity, we shall now use natural units in which we can omit Planck and Boltzmann constants everywhere, setting them equal to unity: h ¯ = 1, kB = 1. We shall now demonstrate how this result is found via strong-coupling theory from a perturbation expansion. 2

See Notes and References. H. Kleinert, PATH INTEGRALS

501

6.4 Strong-Coupling Theory for Particle in Box

6.4.1

Partition Function

The discussion becomes simplest by considering the quantum statistical partition function of the particle. It is given by the Euclidean path integral (always in natural units) Z R Z=

1

Du(τ )e 2

h ¯β

0

dτ (∂u)2

,

(6.66)

where the shifted particle coordinate u(τ ) ≡ x(τ )−d/2 is restricted to the symmetric interval −d/2 ≤ u(τ ) ≤ d/2. Since such a hard-wall restriction is hard to treat analytically in (6.66), we make the hard-walls soft by adding to the Euclidean action E in the exponent of (6.66) a potential term diverging near the walls. Thus we consider the auxiliary Euclidean action 1 Ae = 2 where V (u) is given by ω2 V (u) = 2

Z

0

hβ ¯

n

o

dτ [∂u(τ )]2 + V (u(τ )) ,

d πu tan π d

!2

(6.67)

ω2 2 2 4 = u + gu + . . . . 2 3

(6.68)

On the right-hand side we have introduced a parameter g ≡ π 2 /d2 .

6.4.2


The expansion of the potential in powers of g can now be treated perturbatively, leading to an expansion of Z around the harmonic part of the partition function. In this, the integrations over u(τ ) run over the entire u-axis, and can be integrated out as described in Section 2.15. The result is [see Eq. (2.481)] Zω = e−(1/2)Tr log(∂

2 +ω 2 )

.

(6.69)

For β → ∞, the exponent gives a free energy density F = −β −1 log Z equal to the ground state energy of the harmonic oscillator ω Fω = . (6.70) 2 The treatment of the interaction terms can be organized in powers of g, and give rise to an expansion of the free energy with the generic form F = Fω + ω

∞ X

k=1

ak

g ω

k

.

(6.71)

The calculation of the coefficients ak in this expansion proceeds as follows. First we expand the potential in (6.67) to identify the power series for the interaction energy ω2 dτ gv4 u4 + g 2 v6 u6 + g 3 v8 u8 + . . . 2 ∞ Z ω2 X = dτ g k v2k+2 [u2 (τ )]k+1 , 2 k=1

Aint = e

Z

(6.72)

502


with coefficients v4 = v16 = v24 = v30 = v34 =

2 17 62 1382 21844 929569 , v6 = , v8 = , v10 = , v12 = , v14 = , 3 45 315 14175 467775 42567525 6404582 443861162 18888466084 113927491862 , v18 = , v20 = , v22 = , 638512875 97692469875 9280784638125 126109485376875 58870668456604 8374643517010684 689005380505609448 , v26 = , v28 = , 147926426347074375 48076088562799171875 9086380738369043484375 129848163681107301953 1736640792209901647222 , v32 = , 3952575621190533915703125 122529844256906551386796875 418781231495293038913922 , ... . (6.73) 68739242628124575327993046875 R 2 k+1

The interaction terms dτ [u (τ )] and their products are expanded according to Wick’s rule in Section 3.10 into sums of products of harmonic two-point correlation functions Z dk eik(τ1 −τ2 ) e−ω|τ1 −τ2 | hu(τ1 )u(τ2 )i = = . (6.74) 2π k 2 + ω 2 2ω Associated local expectation values are hu2 i = 1/2ω, and dk k =0 2 2π k + ω 2 Z dk k 2 ω h∂u∂ui = =− , 2 2 2π k + ω 2 hu∂ui =

Z

(6.75)

where the last integral is calculated using dimensional regularization in which dk k α = 0 for all α. The Wick contractions are organized with the help of the Feynman diagrams as explained in Section 3.20. Only the connected diagrams contribute to the free energy density. The graphical expansion of free energy up to four loops is

R

ω ω2 F = + 2 2

!

1 − 2!

ω2 2

!2

1 + 3!

ω2 2

!3

n

+ g 2 v6 15

gv4 3

n

g 2v42 [72

g 3v43

+ 24

2592

o

+ g 3 v8 105 h

] +g 3 2v4 v6 540

+ 1728

+3456

io

+ 360 + 1728

o

.

(6.76)

Note different numbers of loops contribute to the terms of order g n . The calculation of the diagrams in Eq. (6.76) is simplified by the factorization property: If a diagram consists of two subdiagrams touching each other at a single vertex, the associated Feynman integral factorizes into those of the subdiagrams. In each diagram, the last t-integral yields an overall factor β, due to translational invariance along the t-axis, the others produce a factor 1/ω. Using the explicit expression (6.75) for the lines in the diagrams, we find the following values for the Feynman integrals: = β

1 , 16ω 5

=β

1 , 64ω 8 H. Kleinert, PATH INTEGRALS

503


1 , 32ω 5 1 = β , 32ω 6 1 = β , 32ω 6

3 , 128ω 8 5 =β , 8 · 64ω 8 3 =β . 8 · 64ω 8

= β

=β

(6.77)

Adding all contributions in (6.76), we obtain up to the order g 3 : (

21 1 3 g 15 + F3 = ω + v4 v6 − v42 2 8 ω 16 32

g ω

2

45 333 3 105 + v8 − v4 v6 + v 32 8 128 4

)

g 3 , ω (6.78)

which has the generic form (6.71). We can go to higher orders by extending the Bender-Wu recursion relation (3C.20) for the ground state energy of the quartic anharmonic oscillator as follows: ′

′

2p′ Cnp = (p′ + 1)(2p′ + 1)Cnp + ′

C00 = 1, Cnp = 0

n n−1 X 1X p′ −k−1 p′ v2k+2 Cn−k − Ck1 Cn−k , 1 ≤ p′ ≤ 2n, 2 k=1 k=1

(n ≥ 1, p′ < 1).

(6.79)

After solving these recursion relations, the coefficients ak in (6.71) are given by ak = (−1)k+1 Ck,1 . For brevity, we list here the first sixteen expansion coefficients for F , calculated with the help of MATHEMATICA of REDUCE programs:3 1 1 1 1 a0 = , a1 = , a2 = , a3 = 0, a4 = − , a5 = 0, 2 4 16 256 1 5 a6 = , a7 = 0, a8 = − , a9 = 0, 2048 65536 7 21 a10 = , a11 = 0, a12 = − , a13 = 0, 524288 8388608 33 429 a14 = , a15 = 0, a16 = − ,... . 67108864 4294967296

6.4.3

(6.80)

Variational Strong-Coupling Approximations

We are now ready to calculate successive strong-coupling approximations to the function F (g). It will be convenient to remove the expected correct d dependence π 2 /d2 from F (g), and study the function F˜ (¯ g ) ≡ F (g)/g which depends only on the dimensionless reduced coupling constant g¯ = g/ω. The limit ω → 0 corresponds to a strong-coupling limit in the reduced coupling constant g¯. According to the general theory of variational perturbation theory and its strong-coupling limit in Sections 5.14 and 5.17, the Nth order approximation to the strong-coupling limit of F˜ (¯ g ), to be denoted by F˜ ∗ , is found by replacing, in the series truncated after the 3

The programs can be downloaded from www.physik.fu-berlin.de/˜kleinert/b5/programs

504


Nth term, FÑ (g/ω), the frequency ω by the identical expression where r 2 ≡ 2M(Ω2 − ω 2 )/g.

q

Ω2 − gr 2/2M, (6.81)

For a moment, this is treated as an independent variable, whereas Ω is a dummy parameter. Then the square root is expanded binomially in powers of g, and q 2 ˜ FN (g/ Ω − gr 2 /2M) is re-expanded up to order g N . After that, r is replaced by its proper value. In this way we obtain a function FÑ (g, Ω) which depends on Ω, which thus becomes a variational parameter. The best approximation is obtained by extremizing FÑ (g, Ω) with respect to ω. Setting ω = 0, we go to the strong-coupling limit g → ∞. There the optimal Ω grows proportionally to g, so that g/Ω = c−1 is finite, and the variational expression FÑ (g, Ω) becomes a function of fN (c). In this limit, the above re-expansion amounts simply to replacing each power ω n in each expansion terms of FÑ (¯ g ) by the binomial expansion of (1 − 1)−n/2 truncated after the (N − n)th term, and replacing g¯ by c−1 . The first nine variational functions fN (c) are listed in Table 6.1. The functions fN (c) are minimized starting from f2 (c) and searching the minimum of each successive f3 (c), f3 (c), . . . nearest to the previous one. The functions fN (c) together with their minima are plotted in Fig. 6.5. The minima lie at Table 6.1 First eight variational functions fN (c).

f2 (c) = f3 (c) = f4 (c) = f5 (c) = f6 (c) = f7 (c) = f8 (c) = f9 (c) =

1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4

+ + − − + + − −

1 + 316c 16 c 3 + 532c 32 c 1 15 c + 128 + 35 256 c3 c 256 5 35 c + 256 + 63 512 c3 c 512 1 35 315 c − 2048 + 2048 + 231 2048 c5 c3 c 2048 7 105 693 c − 4096 + 4096 + 429 4096 c5 c3 c 4096 5 63 1155 3003 c + 16384 − 32768 + 16384 + 6435 65536 c7 c5 c3 c 65536 45 231 3003 6435 c + 32768 − 65536 + 32768 + 12155 131072 c7 c5 c3 c 131072

c 0.8

0.85

0.9

0.95 0.498

fN (c)

0.496 0.494 0.492 0.49

Figure 6.5 Variational functions fN (c) for particle between walls up to N = 16 are shown together with their minima whose y-coordinates approach rapidly the correct limiting value 1/2. H. Kleinert, PATH INTEGRALS

505


(N, fNmin) = (2, 0.466506), (3, 0.492061), (4, 0.497701), (5, 0.499253), (6, 0.499738), (7, 0.499903), (8, 0.499963), (9, 0.499985), (10, 0.499994), (11, 0.499998), (12, 0.499999), (13, 0.5000), (14, 0.50000), (15, 0.50000), (16, 0.5000).

(6.82)

They converge exponentially fast against the known result 1/2, as shown in Fig. 6.6.

6.4.4

Special Properties of Expansion

The alert reader will have noted that the expansion coefficients (6.80) possess two special properties: First, they lack the factorial growth at large orders which would be found for a single power [u2 (τ )]k+1 of the interaction potential, as mentioned in Eq.(3C.27) and will be proved in Eq. (17.323). The factorial growth is canceled by the specific combination of the different powers in the interaction (6.72), making the series (6.71) convergent inside a certain circle. Still, since this circle has a finite radius (the ratio test shows that it is unity), this convergent series cannot be evaluated in the limit of large g which we want to do, so that variational strong-coupling theory is not superfluous. However, there is a second remarkable property of the coefficients (6.80): They contain an infinite number of zeros in the sequence of coefficients for each odd number, except for the first one. We may take advantage of this property by separating off the irregular term a1 g = g/4 = π 2 /4d2 , setting α = g 2 /4ω 2 , and rewriting F˜ (¯ g ) as 1 1 ˜ F (α) = 1 + √ h(α) , 4 α

h(α) ≡

N X

22n+1 a2n αn .

(6.83)

n=0

Inserting the numbers (6.80), the expansion of h(α) reads h(α) = 1 +

α3 5 4 7 5 21 6 33 7 429 8 α α2 − + − α + α − α + α − α + ... . 2 8 16 128 256 1024 2048 32768 (6.84) √

We now realize that this is the binomial power series expansion of 1 + α. Substituting this into (6.83), we find the exact ground state energy for the Euclidean action (6.67) ! ! r r π2 1 π2 d4 (0) 2 E = 2 1+ 1+ = 2 1 + 1 + 4ω 4 . (6.85) 4d α 4d π

e−N

0.5

0.00005

0.499998 0.499996

min fN

0.499994 0.499992

Figure 6.6 exact value.

Exponentially fast convergence of strong-coupling approximations towards

506


Here we can go directly to the strong-coupling limit α → ∞ to recover the exact ground state energy E (0) = π 2 /2d2 . The energy (6.85) can of course be obtained directly by solving the Schrödinger equation associated with the potential (6.72), 1 d2 ∂2 λ(1 − λ) − 2+ − 1 ψ(x) = 2 Eψ(x), (6.86) 2 2 ∂x π cos x where we have replaced u → dx/π and set ω 2 d4 /π 4 ≡ λ(λ − 1), so that ! r 4 1 d 1 + 1 + 4ω 2 4 . λ= 2 π

(6.87)

Equation (6.86) is of the Pöschl-Teller type [see Subsection 14.4.5], and has the ground state wave function, to be derived in Eq. (14.162), ψ0 (x) = const × cosλ x ,

(6.88)

with the eigenvalue π 2 E (0) /d2 = (λ2 − 1)/2, which agrees of course with √ Eq. (6.85). α in F of Eq. (6.85), by If we were to apply the variational procedure to the series h(α)/ √ replacing the factor 1/ω 2n contained in each power αn by Ω = Ω2 − rα and re-expanding now in powers of α rather than g, we would find that all approximation hN (c) would possess a minimum with unit value, such that the corresponding extremal functions fN (c) yield the correct final energy in each order N .

6.4.5

Exponentially Fast Convergence

With the exact result being known, let us calculate the exponential approach of the variational approximations observed in Fig. (6.6). Let us write the exact energy (6.85) as E (0) = After the replacement ω →

p 1 (g + g 2 + 4ω 2 ). 4

p Ω2 − ρg, this becomes E (0) =

p Ω g+4 , gˆ + gˆ2 − 4ρˆ 4

(6.89)

(6.90)

where gˆ ≡ g/Ω2 . The N th-order approximant fN (g) of E (0) is obtained by expanding (6.91) in powers of gˆ up to order N , N X fN (g) = Ω hk (ρ)ˆ gk , (6.91) 0

2

2

and substituting ρ by 2M r = (1 − ω ˆ )/ˆ g [compare (6.81)], with ω ˆ 2 ≡ ω 2 /Ω2 . The resulting function of gˆ is then optimized. It is straightforward to find an integral representation for FN (g). Setting rˆ g ≡ z, we have I 1 dz 1 − z N +1 FN = f (z), (6.92) 2πi C0 z N +1 1 − z

where the contour C0 refers to small circle around the origin and ! r Ω z z2 F (z) = + − 4z + 4 4 r r2 p 1 = z + (z − z1 )(z − z2 ) , 4r

(6.93)


507

Notes and References p with branch points at z1,2 = 2r2 1 ± 1 − 1/r2 . For z < 1, we rewrite

1 − z N +1 = (1 − z)(1 + z + . . . + z N ) = (1 − z)(N + 1) −(1 − z)2 N + (N − 1)z + . . . + z N −1

(6.94)

1 − z N +1 = (1 − z)(N + 1) + O(|1 − z|2 N 2 ).

(6.95)

and estimate this for z ≈ 1 as

Dividing the approximant (6.92) by Ω, and indicating this by a hat, we use (6.94) to write FˆN as a sum over the discontinuities across the two branch cuts: I (N + 1) dz Fˆ (z) ˆ (N + 1) ˆ (N ) FˆN = F (z) = F (0) N +1 2πi N! C0 z 2 Z ∞ X dz ˆ F (z). (6.96) = (N + 1) N z +1 i=1 zi The integrals yield a constant plus a product ∆FˆN ≈

(N + 1)(N − 23 )! 1 1 , 2 N N! (r ) (1 + r2 )N

(6.97)

which for large N can be approximated using Stirling’s formula (5.204) by ∆FˆN ≈

2 A √ e−r N . N

(r2 )N

(6.98)

In the strong-coupling limit of interest here, ω ˆ 2 = 0, and r = 1/ˆ g = Ω/g = c. In Fig. 6.5 we see that the optimal c-values tend to unity for N → ∞, so that ∆fˆN goes to zero like e−N , as observed in Fig. 6.6.

Notes and References There exists a large body of literature on this subject, for example L.S. Schulman, J. Math. Phys. 12, 304 (1971); M.G.G. Laidlaw and C. DeWitt-Morette, Phys. Rev. D 3, 1375 (1971); J.S. Dowker, J. Phys. A 5, 936 (1972); P.A. Horvathy, Phys. Lett. A 76, 11 (1980) and in Differential Geometric Methods in Math. Phys., Lecture Notes in Mathematics 905, Springer, Berlin, 1982; J.J. Leinaas and J. Myrheim, Nuovo Cimento 37, 1, (1977). The latter paper is reprinted in the textbook F. Wilczek, Fractional Statistics and Anyon Superconductivity, World Scientific, 1990. See further P.A. Horvathy, G. Morandi, and E.C.G. Sudarshan, Nuovo Cimento D 11, 201 (1989), and the textbook L.S. Schulman, Techniques and Applications of Path Integration, Wiley, New York, 1981. It is possible to account for the presence of hard walls using infinitely high δ-functions: C. Grosche, Phys. Rev. Lett. 71, 1 (1993); Ann. Phys. 2, 557 (1993); (hep-th/9308081); (hepth/9308082); (hep-th/9402110); M.J. Goovaerts, A. Babcenco, and J.T. Devreese, J. Math. Phys. 14, 554 (1973); C. Grosche, J. Phys. A Math. Gen. 17, 375 (1984).

508


The physically important problem of membranes between walls has been discussed in W. Helfrich, Z. Naturforsch. A 33, 305 (1978); W. Helfrich and R.M. Servuss, Nuovo Cimento D 3, 137 (1984); W. Janke and H. Kleinert, Phys. Lett. 58, 144 (1987) (http://www.physik.fu-berlin.de/ ~kleinert/143); W. Janke, H. Kleinert, and H. Meinhardt, Phys. Lett. B 217, 525 (1989) (ibid.http/184); G. Gompper and D.M. Kroll, Europhys. Lett. 9, 58 (1989); R.R. Netz and R. Lipowski, Europhys. Lett. 29. 345 (1995); F. David, J. de Phys. 51, C7-115 (1990); H. Kleinert, Phys. Lett. A 257 , 269 (1999) (cond-mat/9811308); M. Bachmann, H. Kleinert, A. Pelster, Phys. Lett. A 261 , 127 (1999) (cond-mat/9905397). The problem has been solved with the help of the strong-coupling variational perturbation theory developed in Chapter 5 by H. Kleinert, Phys. Lett. A 257, 269 (1999) (cond-mat/9811308); M. Bachmann, H. Kleinert, and A. Pelster, Phys. Lett. A 261, 127 (1999) (cond-mat/9905397). The quantum-mechanical calculation presented in Section 6.4 is taken from H. Kleinert, A. Chervyakov, and B. Hamprecht, Phys. Lett. A 260, 182 (1999) (condmat/9906241).



Mirum, quod divina natura dedit agros. It’s wonderful that divine nature has given us fields. Varro, 82 B.C.

7 Many Particle Orbits – Statistics and Second Quantization Realistic physical systems usually contain groups of identical particles such as specific atoms or electrons. Focusing on a single group, we shall label their orbits by x(ν) (t) with ν = 1, 2, 3, . . . , N. Their Hamiltonian is invariant under the group of all N! permutations of the orbital indices ν. Their Schrödinger wave functions can then be classified according to the irreducible representations of the permutation group. Not all possible representations occur in nature. In more than two space dimensions, there exists a superselection rule, whose origin is yet to be explained, which eliminates all complicated representations and allows only for the two simplest ones to be realized: those with complete symmetry and those with complete antisymmetry. Particles which appear always with symmetric wave functions are called bosons. They all carry an integer-valued spin. Particles with antisymmetric wave functions are called fermions 1 and carry a spin whose value is half-integer. The symmetric and antisymmetric wave functions give rise to the characteristic statistical behavior of fermions and bosons. Electrons, for example, being spin-1/2 particles, appear only in antisymmetric wave functions. The antisymmetry is the origin of the famous Pauli exclusion principle, allowing only a single particle of a definite spin orientation in a quantum state, which is the principal reason for the existence of the periodic system of elements, and thus of matter in general. The atoms in a gas of helium, on the other hand, have zero spin and are described by symmetric wave functions. These can accommodate an infinite number of particles in a single quantum state giving rise to the famous phenomenon of Bose-Einstein condensation. This phenomenon is observable in its purest form in the absence of interactions, where at zero temperature all particles condense in the ground state. In interacting systems, Bose-Einstein statistics can lead to the stunning quantum state of superfluidity. The particular association of symmetry and spin can be explained within relativistic quantum field theories in spaces with more than two dimensions where it is shown to be intimately linked with the locality and causality of the theory. 1

Had M. Born as editor of Zeitschrift f¨ ur Physik not kept a paper by P. Jordan in his suitcase for half a year in 1925, they would be called jordanons. See the bibliographical notes by B. Schroer (hep-th/0303241).

509

510

7 Many Particle Orbits — Statistics and Second Quantization

In two dimensions there can be particles with an exceptional statistical behavior. Their properties will be discussed in Section 7.5. In Chapter 16, such particles will serve to explain the fractional quantum Hall effect. The problem to be solved in this chapter is how to incorporate the statistical properties into a path integral description of the orbits of a many-particle system. Afterwards we describe the formalism of second quantization or field quantization in which the path integral of many identical particle orbits is abandoned in favor of a path integral over a single fluctuating field which is able to account for the statistical properties in a most natural way.

7.1

Ensembles of Bose and Fermi Particle Orbits

For bosons, the incorporation of the statistical properties into the orbital path integrals is quite easy. Consider, for the moment, distinguishable particles. Their many-particle time evolution amplitude is given by the path integral (1) (N ) (N ) (xb , . . . , xb ; tb |x(1) a , . . . , xa ; ta )

N Z Y

=

ν=1

D (ν)

D x

(N) /¯ h

eiA

,

(7.1)

with an action of the typical form A(N ) =

Z

tb

ta

 " N X M (ν)

dt 

ν=1



N  1 X ′ x˙ (ν)2 − V (x(ν) ) − Vint (x(ν) − x(ν ) ) , 2 2 ν6=ν ′ =1

#

(7.2)

where V (x(ν) ) is some common background potential for all particles interacting via ′ the pair potential Vint (x(ν) − x(ν ) ). We shall ignore interactions involving more than two particles at the same time, for simplicity. If we want to apply the path integral (7.1) to indistinguishable particles of spin zero, we merely have to add to the sum over all paths x(ν) (t) running to the final (ν) positions xb the sum of all paths running to the indistinguishable permuted final (p(ν)) positions xb . The amplitude for n bosons reads therefore (1)

(N )

(N ) (xb , . . . , xb ; tb |x(1) a , . . . , xa ; ta ) =

X

p(ν)

(p(1))

(xb

(p(N ))

, . . . , xb

(N ) ; tb |x(1) a , . . . , xa ; ta ), (7.3)

where p(ν) denotes the N! permutations of the indices ν. For bosons of higher spin, the same procedure applies to each subset of particles with equal spin orientation. A similar discussion holds for fermions. Their Schrödinger wave function requires complete antisymmetrization in the final positions. Correspondingly, the amplitude (p(ν)) (7.1) has to be summed over all permuted final positions xb , with an extra minus sign for each odd permutation p(ν). Thus, the path integral involves both sums and differences of paths. So far, the measure of path integration has always been a true sum over paths. For this reason it will be preferable to attribute the alternating sign to an interaction between the orbits, to be called a statistics interaction. This interaction will be derived in Section 7.4. H. Kleinert, PATH INTEGRALS

511

7.1 Ensembles of Bose and Fermi Particle Orbits

For the statistical mechanics of Bose- and Fermi systems consider the imaginarytime version of the amplitude (7.3): (1)

(N )

(N ) (xb , . . . , xb ; h ¯ β|x(1) a , . . . , xa ; 0) =

X

(p(1))

ǫp(ν) (xb

(p(N ))

, . . . , xb

(N ) ;h ¯ β|x(1) a , . . . , xa ; 0),

p(ν)

(7.4)

where ǫp(ν) = ±1 is the parity of even and odd permutations p(ν), to be used for Bosons and Fermions, respectively. Its spatial trace integral yields the partition function of N-particle orbits: Z (N ) =

1 N!

Z

dD x(1) · · · dD x(N ) (x(1) , . . . , x(N ) ; h ¯ β|x(1) , . . . , x(N ) ; 0).

(7.5)

A factor 1/N! accounts for the indistinguishability of the permuted final configurations. For free particles, each term in the sum (7.4) factorizes: (p(1))

(xb

(p(N ))

, . . . , xb

(p(1))

(N ) ;h ¯ β|x(1) a , . . . , xa ; 0)0 = (xb

(p(N ))

h ¯ β|x(1) a 0)0 · · · (xb

) h ¯ β|x(N a 0)0 ,(7.6)

where each factor has a path integral representation (p(ν)) (xb h ¯ β|x(ν) a

0)0 =

Z

(p(ν))

x(ν) (¯ hβ)=xb (ν)

x(ν) (0)=xa

D (ν)

D x

"

1 exp − h ¯

Z

hβ ¯

0

#

M dτ x˙ (ν)2 (τ ) , 2

(7.7)

which is solved by the imaginary-time version of (2.125):   

h

(p(ν))

1 1 M xb (p(ν)) (xb h ¯ β|x(ν) 0) = exp − q 0 a D  ¯ 2  h 2π¯ h2 β/M

2   −x(ν) a

h ¯β

The partition function can therefore be rewritten in the form (N )

Z0

1 1 =q ND N! 2π¯ h2 β/M

Z

i 

.

(7.8)

 

2 (p(ν)) (ν)   x −x 1 M dD x(1) · · · dD x(N ) ǫp(ν) exp − .   ¯ 2 h ¯β  h  ν=1 p(ν)

X

N Y

  

h

i

(7.9) This is a product of Gaussian convolution integrals which can easily be performed as before when deriving the time evolution amplitude (2.70) for free particles with the help of Formula (2.69). Each convolution integral simply extends the temporal length in the fluctuation factor by h ¯ β. Due to the indistinguishability of the particles, only a few paths will have their end points connected to their own initial points, i.e., they satisfy periodic boundary conditions in the interval (0, h ¯ β). The sum over permutations connects the final point of some paths to the initial point of a different path, as illustrated in Fig. 7.1. Such paths satisfy periodic boundary conditions on an interval (0, w¯ hβ), where w is some integer number. This is seen most clearly by drawing the paths in Fig. 7.1 in an extended zone scheme shown in Fig. 7.2, which

512


Figure 7.1 Paths summed in partition function (7.9). Due to indistinguishability of particles, final points of one path may connect to initial points of another.

Figure 7.2 Periodic representation of paths summed in partition function (7.9), once in extended zone scheme, and once on D-dimensional hypercylinder embedded in D + 1 dimensions. The paths are shown in Fig. 7.1. There is now only one closed path on the cylinder. In general there are various disconnected parts of closed paths.

is reminiscent of Fig. 6.1. The extended zone scheme can, moreover, be placed on a hypercylinder, illustrated in the right-hand part of Fig. 7.2. In this way, all paths decompose into mutually disconnected groups of closed paths winding around the cylinder, each with a different winding number w [1]. An example for a connected path which winds three times 3 around the D-dimensional cylinder contributes to the partition function a factor [using Formula (2.69)]: (N ) 3

∆Z0

1 =q 3D 2π¯ h2 β/M   

h

(2)

1M x  ¯ 2  h

× exp −

(1)

−x h ¯β

2 (3) (2)   1 M x −x D (1) D (2) D (3) d x d x d x exp −   ¯ 2 h ¯β  h 

  

Z

i2    

  

h

(1)

(N ) w

(3)

1 M x −x  ¯ 2 h ¯β  h

exp −

For cycles of length w the contribution is ∆Z0

h

= Z0 (wβ) ,

i

i2  

VD =q D .(7.10)   2 2π¯ h 3β/M (7.11) H. Kleinert, PATH INTEGRALS

513

7.1 Ensembles of Bose and Fermi Particle Orbits

where Z0 (wβ) is the partition function of a free particle in a D-dimensional volume VD for an imaginary-time interval w¯ hβ: VD Z0 (wβ) = q D. 2 2π¯ h wβ/M

(7.12)

q

hβ) ≡ 2π¯ h2 β/M associated with the In terms of the thermal de Broglie length le (¯ temperature T = 1/kB β [recall (2.345)], this can be written as Z0 (wβ) =

VD . D le (w¯ hβ)

(7.13)

There is an additional factor 1/w in Eq. (7.11), since the number of connected windings of the total w! closed paths is (w − 1)!. In group theoretic language, it is the number of cycles of length w, usually denoted by (1, 2, 3, . . . , w), plus the (w − 1)! permutations of the numbers 2, 3, . . . , w. They are illustrated in Fig. 7.3 for w = 2, 3, 4. In a decomposition of all N! permutations as products of cycles, the number of elements consisting of C1 , C2 , C3 , . . . cycles of length 1, 2, 3, . . . contains N! Cw w=1 Cw !w

M(C1 , C2 , . . . , CN ) = QN

(7.14)

elements [2]. With the knowledge of these combinatorial factors we can immediately write down the canonical partition function (7.9) of N bosons or fermions as the sum of all orbits around the cylinder, decomposed into cycles: (N ) Z0 (β)

1 X = ǫp(ν) M(C1 , . . . , CN ) N! p(ν)

N Y

w=1 N = Σw wCw

h

Z0 (wβ)

iCw

.

(7.15)

The sum can be reordered as follows: (N ) Z0

1 = N!

X

M(C1 , . . . , CN ) ǫw,C1 ,...,Cn

N h Y

Z0 (wβ)

w=1

C1 ,...,CN N = Σw wCw

iCw

.

(7.16)

The parity ǫw,C1 ,...,Cn of permutations is equal to (±1)Σw (w+1)Cw . Inserting (7.14), the sum (7.16) can further be regrouped to (N )

Z0 (β) =

X


=

X


N h iCw Y 1 Σw (w+1)Cw (±1) Z (wβ) 0 Cw w=1 Cw ! w w=1

QN

N Y

1 Z0 (wβ) (±1)w−1 w w=1 Cw !

Cw

.

(7.17)

(0)

For N = 0, this formula yields the trivial partition function Z0 (β) = 1 of the no(1) particle state, the vacuum. For N = 1, i.e., a single particle, we find Z0 (β) = Z0 (β).

514


Figure 7.3 Among the w! permutations of the different windings around the cylinder, (w − 1)! are connected. They are marked by dotted frames. In the cycle notation for permutation group elements, these are (12) for two elements, (123), (132) for three elements, (1234), (1243), (1324), (1342), (1423), (1432) for four elements. The cycles are shown on top of each graph, with trivial cycles of unit length omitted. The graphs are ordered according to a decreasing number of cycles. (N )

The higher Z0 temperature

can be written down most efficiently if we introduce a characteristic

Tc(0)

2π¯ h2 N ≡ kB M VD ζ(D/2) "

#2/D

,

(7.18)

and measure the temperature T in units of Tc(0) , defining a reduced temperature (1) t ≡ T /Tc(0) . Then we can rewrite Z0 (β) as tD/2 VD . Introducing further the Ndependent variable " #2/D N τN ≡ t, (7.19) ζ(D/2) (1)

D/2

we find Z0 = τ1 (2)

Z0

(3)

Z0

(4)

Z0

. A few low-N examples are for bosons and fermions: D/2

+ τ2D ,

D/2

+ 2−1−D/2 τ3D ± 3−1 2−1 τ3

= ±2−1−D/2 τ2 = ±3−1−D/2 τ3

D/2

= ±2−2−D τ4

3D/2

+ (2−3−D + 3−1−D/2 ) τ4D ±2

,

−2− D 2

(7.20) 3D/2

τ4

+ 3−1 2−3 τ42D . H. Kleinert, PATH INTEGRALS

515

7.1 Ensembles of Bose and Fermi Particle Orbits (N )

From Z0 (β) we calculate the specific heat [recall (2.594)] of the free canonical ensemble: d2 d2 (N ) (N ) (N ) C0 = T 2 [T log Z0 ] = τN 2 [τN log Z0 ], (7.21) dT dτN and plot it [3] in Fig. 7.4 against t for increasing particle number N. In the limit N → ∞, the curves approach a limiting form with a phase transition at T = Tc(0) , which will be derived from a grand-canonical ensemble in Eqs. (7.67) and (7.70). The partition functions can most easily be calculated with the help of a recursion (0) relation [4, 5], starting from Z0 ≡ 1: (N ) Z0 (β)

2 1.75 1.5 1.25 1 0.75 0.5 0.25

N 1 X (N −n) = (±1)n−1 Z0 (nβ)Z0 (β). N n=1

(N )

C0

(7.22)

/N kB

N =∞

N = 10

0.5

1

1.5

2

2.5

3

0 T /T(c)

Figure 7.4 Plot of the specific heat of free Bose gas with N = 10, 20, 50, 100, 500, ∞ particles. The curve approaches for large T the Dulong-Petit limit 3kB N/2 corresponding to the three harmonic kinetic degrees of freedom in the classical Hamiltonian p2 /2M . There are no harmonic potential degrees of freedom.

This relation is proved with the help of the grand-canonical partition function (N ) which is obtained by forming the sum over all canonical partition functions Z0 (β) with a weight factor z N : ZG 0 (β) ≡

∞ X

(N )

Z0 (β) z N .

(7.23)

N =0

The parameter z is the Boltzmann factor of one particle with the chemical potential µ: z = z(β) ≡ eβµ . (7.24) It is called the fugacity of the ensemble. Inserting the cycle decompositions (7.17), the sum becomes ZG 0 (β) =

X

N Y

C1 ,...,CN w=1 N = Σw wCw

1 Z0 (wβ)ewβµ (±1)w−1 Cw ! w

Cw

.

(7.25)

516


The right-hand side may be rearranged to ∞ X ∞ Y

1 Z0 (wβ)ewβµ (±1)w−1 ZG 0 (β) = w w=1 Cw =0 Cw ! = exp

"

∞ X

(±1)w−1

w=1

Cw

#

Z0 (wβ) wβµ . e w

(7.26)

From this we read off the grand-canonical free energy [recall (1.542)] of noninteracting identical particles ∞ 1 1 X Z0 (wβ) wβµ FG (β) ≡ − log ZG 0 (β) = − (±1)w−1 e . β β w=1 w

(7.27)

This is simply the sum of the contributions (7.11) of connected paths to the canonical partition function which wind w = 1, 2, 3, . . . times around the cylinder [1, 6]. Thus we encounter the same situation as observed before in Section 3.20: the free energy of any quantum-mechanical system can be obtained from the perturbation expansion of the partition function by keeping only the connected diagrams. The canonical partition function is obviously obtained from (7.27) by forming the derivative: 1 ∂N (N ) Z0 (β) = Z (β) . (7.28) G 0 N N! ∂z z=0

It is now easy to derive the recursion relation (7.22). From the explicit form (7.27), we see that ∂ ∂ ZG 0 = − βFG ∂z ∂z

!

ZG 0 .

(7.29)

Applying to this N − 1 more derivatives yields N −1 X ∂ N −1 ∂ (N − 1)! ZG 0 = − N −1 ∂z ∂z l=0 l!(N − l − 1)!

#

"

∂ l+1 ∂ N −l−1 βF ZG0 . G ∂z l+1 ∂z N −l−1 !

(N )

To obtain from this Z0 we must divide this equation by N! and evaluate the derivatives at z = 0. From (7.27) we see that the l + 1st derivative of the grandcanonical free energy is

Thus we obtain

∂ l+1 βF = −(±1)l l! Z0 ((l + 1)β). G ∂z l+1 z=0

(7.30)

−1 1 ∂N 1 NX 1 ∂ N −l−1 l Z = (±1) Z ((l + 1)β) Z . G 0 0 G0 N N −l−1 N! ∂z N l=0 (N − l − 1)! ∂z z=0 z=0


517

7.2 Bose-Einstein Condensation

Inserting here (7.28) and replacing l → n−1 we obtain directly the recursion relation (7.22). The grand-canonical free energy (7.27) may be simplified by using the property Z0 (wβ) = Z0 (β)

1 w D/2

(7.31)

√ D of the free-particle partition function (7.12), to remove a factor 1/ w from Z0 (wβ). This brings (7.27) to the form ∞ X 1 ewβµ FG = − Z0 (β) (±1)w−1 D/2+1 . β w w=1

(7.32)

The average number of particles is found from the derivative with respect to the chemical potential2 N =−

∞ X ∂ ewβµ FG = Z0 (β) (±1)w−1 D/2 . ∂µ w w=1

(7.33)

The sums over w converge certainly for negative or vanishing chemical potential µ, i.e., for fugacities smaller than unity. In Section 7.3 we shall see that for fermions, the convergence extends also to positive µ. If the particles have a nonzero spin S, the above expressions carry a multiplicity factor gS = 2S + 1, which has the value 2 for electrons. The grand-canonical free energy (7.32) will now be studied in detail thereby revealing the interesting properties of many-boson and many-fermion orbits, the ability of the former to undergo Bose-Einstein condensation, and of the latter to form a Fermi sphere in momentum space.

7.2

Bose-Einstein Condensation

We shall now discuss the most interesting phenomenon observable in systems containing a large number of bosons, the Bose-Einstein condensation process.

7.2.1

Free Bose Gas

For bosons, the above thermodynamic functions (7.32) and (7.33) contain the functions ζν (z) ≡

∞ X

zw . ν w=1 w

(7.34)

These start out for small z like z, and increase for z → 1 to ζ(ν), where ζ(z) is Riemann’s zeta function (2.513). The functions ζν (z) are called Polylogarithmic functions in the mathematical literature [7], where they are denoted by Liν (z). 2

In grand-canonical ensembles, one always deals with the average particle number hN i for which one writes N in all thermodynamic equations [recall (1.548)]. This should be no lead to confusion.

518


w ν They are related to the Hurwitz zeta function ζ(ν, a, z) ≡ ∞ w=0 z /(w + a) as ζν (z) = zζ(ν, 1, z). The functions φ(z, ν, a) = ζ(ν, a, z) are also known as Lerch functions. In terms of the functions ζν (z), and the explicit form (7.12) of Z0 (β), we may write FG and N of Eqs. (7.32) and (7.33) simply as 1 1 VD FG = − Z0 (β)ζD/2+1 (z) = − D ζD/2+1 (z), (7.35) β β le (¯ hβ) VD ζD/2 (z). N = Z0 (β)ζD/2(z) = (7.36) leD (¯ hβ) The most interesting range where we want to know the functions ζν (z) is for negative small chemical potential µ. There the convergence is very slow and it is useful to find a faster-convergent representation. As in Subsection 2.15.6 we rewrite the sum over w for z = eβµ as an integral plus a difference between sum and integral

P

βµ

ζν (e ) ≡

Z

0

∞

Z ∞ ∞ X ewβµ ewβµ dw ν + − dw . w wν 0 w=1 !

(7.37)

The integral yields Γ(1 − ν)(−βµ)ν−1 , and the remainder may be expanded sloppily in powers of µ to yield the Robinson expansion (2.573): βµ

ζν (e ) = Γ(1 − ν)(−βµ)

ν−1

+

∞ X

1 (βµ)k ζ(ν − k). k=0 k!

(7.38)

There exists a useful integral representation for the functions ζν (z): 1 iν (βµ), Γ(ν)

ζν (z) ≡

(7.39)

where iν (α) denotes the integral iν (α) ≡

Z

εν−1 dε ε−α , e −1

∞

0

(7.40)

containing the Bose distribution function (3.93): nbε =

1 eε−α

−1

.

(7.41)

Indeed, by expanding the denominator in the integrand in a power series 1 eε−α

−1

=

∞ X

e−wε ewα ,

(7.42)

w=1

and performing the integrals over ε, we obtain directly the series (7.34). It is instructive to express the grand-canonical free energy FG in terms of the functions iν (α). Combining Eqs. (7.35) with (7.39) and (7.40), we obtain Z ∞ iD/2+1 (βµ) 1 1 VD εD/2 FG = − Z0 (β) =− dε . q D β Γ(D/2+1) βΓ(D/2+1) eε−βµ − 1 0 2 2π¯ h β/M

(7.43)


519


The integral can be brought to another form by partial integration, using the fact that 1 eε−βµ

−1

=

∂ log(1 − e−ε+βµ ). ∂ε

(7.44)

The boundary terms vanish, and we find immediately: FG =

Z ∞ VD 1 dε εD/2−1 log(1 − e−ε+βµ ). q D βΓ(D/2) 0 2π¯ h2 β/M

(7.45)

This expression is obviously equal to the sum over momentum states of oscillators with energy h ¯ ωp ≡ p2 /2M, evaluated in the thermodynamic limit N → ∞ with fixed particle density N/V , where the momentum states become continuous: FG =

1X log(1 − e−β¯hωp +βµ ). β p

(7.46)

This is easily verified if we rewrite the sum with the help of formula (1.555) for the surface of a unit sphere in D dimensions and a change of variables to the reduced particle energy ε = βp2 /2M as an integral X p

Z dD p 1 (2M/β)D/2 1 Z D−1 → VD = V D SD dp p = V D SD dε εD/2−1 (2π¯ h)D (2π¯ h)D (2π¯ h)D 2 Z ∞ 1 VD = dε εD/2−1. (7.47) q D Γ(D/2) 0 2 2π¯ h β/M Z

Another way of expressing this limit is X p

→

Z

∞

0

dε Nε ,

(7.48)

where Nε is the reduced density of states per unit energy interval: Nε ≡

1 VD D/2−1 . q D ε Γ(D/2) 2π¯ h2 β/M

(7.49)

The free energy of each oscillator (7.46) differs from the usual harmonic oscillator expression (2.477) by a missing ground-state energy h ¯ ωp /2. The origin of this difference will be explained in Sections 7.7 and 7.14. The particle number corresponding to the integral representations (7.45) and (7.46) is N =−

∂ 1 VD FG = q D ∂µ Γ(D/2) 2π¯ h2 β/M

Z

0

∞

dε

X 1 εD/2−1 ≈ . (7.50) β¯ hωp −βµ − 1 eε−βµ − 1 p e

520


1 0.8 0.6

ζ5/2 (z)/ζ(5/2)

0.4

ζ3/2 (z)/ζ(3/2)

0.2 0.2

0.4

0.6

0.8

1

z

Figure 7.5 Plot of functions ζν (z) for ν = 3/2 and 5/2 appearing in Bose-Einstein thermodynamics.

For a given particle number N, Eq. (7.36) allows us to calculate the fugacity as a function of the inverse temperature, z(β), and from this the chemical potential µ(β) = β −1 log z(β). This is most simply done by solving Eq. (7.36) for β as a function of z, and inverting the resulting function β(z). The required functions ζν (z) are shown in Fig. 7.5. There exists a solution z(β) only if the total particle number N is smaller than the characteristic function defined by the right-hand side of (7.36) at unit fugacity z(β) = 1, or zero chemical potential µ = 0: N
Tc , the chemical potential at fixed N satisfies the equation β∂β (βµ) =

D ζD/2 (z) . 2 ζD/2−1 (z)

(7.68)

This follows directly from the vanishing derivative β∂β N = 0 implied by the fixed particle number N. Applying the derivative to Eq. (7.36) and using the relation z∂z ζν (z) = ζν−1 (z), as well as β∂β f (z) = z∂z f (z) β∂β (βµ), we obtain β∂β N = [β∂β Z0 (β)]ζD/2 (z)+Z0 (β)β∂β ζD/2 (z) D = − Z0 (β)ζD/2 (z)+Z0 (β) ζD/2−1(z)β∂β (βµ) = 0, 2

(7.69)

thus proving (7.68). H. Kleinert, PATH INTEGRALS

523


The specific heat C at a constant volume in units of kB is found from the derivative C = T ∂T S|N = −β 2 ∂β E|N , using once more (7.68): (D + 2)D ζD/2+1 (z) D 2 ζD/2 (z) C = kB N , − 4 ζD/2 (z) 4 ζD/2−1 (z) "

#

T > Tc .

(7.70)

At high temperatures, C tends to the Dulong-Petit limit DkB N/2 since for small z all ζν (z) behave like z. Consider now the physical case D = 3, where the second denominator in (7.70) contains ζ1/2 (z). As the temperature approaches the critical point from above, z tends to unity from below and ζ1/2 (z) diverges. Thus 1/ζ1/2 (1) = 0 and the second term in (7.70) disappears, yielding a maximal value in three dimensions Cmax = kB N

15 ζ5/2 (1) ≈ kB N 1.92567. 4 ζ3/2 (1)

(7.71)

This value is the same as the critical value of Eq. (7.67) below Tc . The specific heat is therefore continuous at Tc . It shows, however, a marked kink. To calculate the jump in the slope we calculate the behavior of the thermodynamic quantities for > T . As T passes T from below, the chemical potential starts becoming smaller T ∼ c c than zero, and we can expand Eq. (7.54) T 1= Tc

3/2 "

#

∆ζ3/2 (z) 1+ , ζ3/2 (1)

(7.72)

where the symbol ∆ in front of a quantity indicates that the same quantity at zero chemical potential is subtracted. Near Tc , we can approximate

T Tc

3/2

−1≈−

∆ζ3/2 (z) . ζ3/2 (1)

(7.73)

We now use the Robinson expansion (7.38) to approximate for small negative µ: ζ3/2 (eβµ ) = Γ(−1/2)(−βµ)1/2 +ζ(3/2)+βµζ(1/2)+ . . . , (7.74) √ with Γ(−1/2) = −2 √π. The right-hand side of (7.73) becomes therefore −∆ζ3/2 (z)/ζ3/2 (1) = −2 π/ζ(3/2)(−βµ)1/2. Inserting this into Eq. (7.73), we obtain the temperature dependence of −µ for T > ∼ Tc : "

1 −µ ≈ kB Tc ζ 2(3/2) 4π

T Tc

3/2

#2

−1 .

(7.75)

The leading square-root term on the right-hand side of (7.74) can also be derived from the integral representation (7.40) which receives for small α its main contribution from α ≈ 0, where ∆i3/2 (α) can be approximated by ∆i3/2 (α) =

Z

0

∞

dz z 1/2

1 ez−α −1

−

1 ≈α ez −1

Z

0

∞

dz

√ 1 = −π −α. (7.76) z 1/2 (z − α)

524


Using the relation (7.61) for D = 3, we calculate the derivative of the energy with respect to the chemical potential from ∂E 3 ∂FG =− . ∂µ T,V 2 ∂µ T,V

(7.77)

This allows us to find the internal energy slightly above the critical temperature Tc , where −µ is small, as "

3 3 E ≈ E< + Nµ = E< − NkB Tc ζ 2(3/2) 2 8π

T Tc

3/2

#2

−1 .

(7.78)

Forming the derivative of this with respect to the temperature we find that the slope of the specific heat below and above Tc jumps at Tc by ∂C ∆ ∂T

!

≈

27 2 kB kB ζ (3/2) N ≡ 3.6658 N , 16π Tc Tc

(7.79)

the individual slopes being from (7.66) with D = 3 (using ∂C/∂T = ∂ 2 E/∂T 2 ): ∂C< ∂T

!

=

15 3 ζ(5/2) NkB NkB ≈ 2.8885 , 4 2 ζ(3/2) Tc Tc

T < Tc ,

(7.80)

and from (7.78): ∂C> ∂T

!

=

∂C< ∂T

!

∂C −∆ ∂T

!

≈ −0.7715

NkB , Tc

T > Tc .

(7.81)

The specific heat of the three-dimensional Bose gas is plotted in Fig. 7.6, where it is compared with the specific heat of superfluid helium for the appropriate atomic parameters n = 22.22 nm−3 , M = 6.695 × 10−27 kg, where the critical temperature is Tc ≈ 3.145K, which is somewhat larger than Tc ≈ 2.17 K of helium. There are two major disagreements due to the strong interactions in helium. First, the small-T behavior is (T /Tc )3/2 rather than the physical (T /Tc )3 due to phonons. Second, the Dulong-Petit limit of an interacting system is closer to that of harmonic oscillators which is twice as big as the free-particle case. Recall that according to the Dulong-Petit law (2.595), C receives a contribution NkB T /2 per harmonic degree of freedom, potential as well as kinetic. Free particle energies are only harmonic in the momentum. In 1995, Bose-Einstein condensation was observed in a dilute gas in a way that fits the above simple theoretical description [9]. When 87 Rb atoms were cooled down in a magnetic trap to temperatures less than 170 nK, about 2000 atoms were observed to form a condensate, a kind of “superatom”. Recently, such condensates have been set into rotation and shown to become perforated by vortex lines [10] just like rotating superfluid helium II. H. Kleinert, PATH INTEGRALS

525


Figure 7.6 Specific heat of ideal Bose gas with phase transition at Tc . For comparison, we have also plotted the specific heat of superfluid helium for the same atomic parameters. The experimental curve is reduced by a factor 2 to have the same Dulong-Petit limit as free bosons.

7.2.2

Bose Gas in Finite Box

The condensation process can be understood better by studying a system in which there is a large but finite number of bosons enclosed in a large cubic box of size L. Then the sum on the right-hand side of Eq. (7.50) gives the contribution of the discrete momentum states to the total particle number. This implies that the function box ζD/2 (z) in Eq. (7.36) has to be replaced by a function ζD/2 (z) defined by the sum over the discrete momentum vectors pn = h ¯ π(n1 , n2 , . . . , nD )/L with ni = 1, 2, 3, . . . : N=

X VD 1 box (z) = ζ . 2 /2M −βµ D/2 D βp n le (¯ hβ) −1 pn e

(7.82)

This can be expressed in terms of the one-dimensional auxiliary partition function of a particle in a one-dimensional “box”: Z1 (b) ≡

∞ X

n=1

e−bn

2 /2

,

b ≡ β¯ h2 π 2 /ML2 = πle2 (¯ hβ)/2L2 ,

(7.83)

as N=

X VD box ζ (z) = Z1D (wb)z w . D/2 D le (¯ hβ) w

(7.84)

The function Z1 (b) is related to the elliptic theta function ϑ3 (u, z) ≡ 1 + 2

∞ X

n=1

2

z n cos 2nu

(7.85)

526


by Z1 (b) = [ϑ3 (0, e−b/2 ) − 1]/2. The small-b behavior of this function is easily calculated following the technique of Subsection 2.15.6. We rewrite the sum with the help of Poisson’s summation formula (1.205) as a sum over integrals

−b/2

ϑ3 (0, e

∞ X

) =

−k 2 b/2

e

k=−∞

=

s

=

Z ∞ X

∞

m=−∞ −∞

dk e−k

2 b/2+2πikm

∞ X 2π 2 2 1+2 e−2π m /b . b m=1

!

(7.86)

Thus, up to exponentially small corrections, we may replace ϑ3 (0, e−b/2 ) by √ so that for small b (i.e., large L/ β): Z1 (b) =

r

1 π 2 − + O(e−2π /b ). 2b 2

q

2π/b,

(7.87)

For large b, Z1 (b) falls exponentially fast to zero. In the sum (7.82), the lowest energy level with p1,...,1 = h ¯ π(1, . . . , 1)/L plays a special role. Its contribution to the total particle number is the number of particles in the condensate: Ncond (T ) =

1 eDb/2−βµ

−1

=

zD , 1 − zD

zD ≡ eβµ−Db/2 .

(7.88)

box This number diverges for zD → 1, where the box function ζD/2 (z) has a pole 1/(Db/2 − βµ). This pole prevents βµ from becoming exactly equal to Db/2 when solving the equation (7.82) for the particle number in the box. For a large but finite system near T = 0, almost all particles will go into the condensate, so that Db/2−βµ will be very small, of the order 1/N, but not zero. The thermodynamic limit can be performed smoothly by defining a regularized function box ζ¯D/2 (z) in which the lowest, singular term in the sum (7.82) is omitted. Let us define the number of normal particles which have not condensed into the state of zero momentum as Nn (T ) = N − Ncond (T ). Then we can rewrite Eq. (7.82) as an equation for the number of normal particles

Nn (T ) =

VD ¯box ζ (z(β)), D le (¯ hβ) D/2

(7.89)

which reads more explicitly Nn (T ) = SD (zD ) ≡

∞ X

w [Z1D (wb)ewDb/2 − 1]zD .

(7.90)

w=1

A would-be critical point may now be determined by setting here zD = 1 and equating the resulting Nn with the total particle number N. If N is sufficiently large, we need only the small-b limit of SD (1) which is calculated in Appendix 7A H. Kleinert, PATH INTEGRALS

527


[see Eq. (7A.12)], so that the associated temperature Tc(1) is determined from the equation s 3 π 3π N= ζ(3/2) + (1) log C3 bc + . . . , (7.91) 2bc 4bc where C3 ≈ 0.0186. In the thermodynamic limit, the critical temperature Tc(0) is obtained by ignoring the second term, yielding N=

s

π (0)

2bc

3

ζ(3/2),

(7.92)

in agreement with Eq. (7.52), if we recall b from (7.83). Using this we rewrite (7.91) as !3/2 Tc(1) 3 π 1≡ + log C3 b(0) (7.93) c . (0) 2N 2b(0) Tc c Expressing b(0) c in terms of N from (7.92), this implies δTc(1) (0)

Tc

≈

1 2 N 2/3 log . ζ 2/3 (3/2)N 1/3 πC3 ζ 2/3 (3/2)

(7.94)

Experimentally, the temperature Tc(1) is not immediatly accessible. What is easy to find is the place where the condensate density has the largest curvature, i.e., where d3 Ncond /dT 3 = 0. The associated temperature Tcexp is larger than Tc(1) by a factor 1 + O(1/N), so that it does not modify the leading finite-size correction to order in 1/N 1/3 The proof is given in Appendix 7B.

7.2.3

Effect of Interactions

Superfluid helium has a lower transition temperature than the ideal Bose gas. This should be expected since the atomic repulsion impedes the condensation process of the atoms in the zero-momentum state. Indeed, a simple perturbative calculation based on a potential whose Fourier transform behaves for small momenta like h

i

2 V˜ (k) = g 1 − reff k2 /6 + . . .

(7.95)

gives a negative shift ∆Tc ≡ Tc − Tc(0) proportional to the particle density [11]: ∆Tc (0)

Tc

=−

1 N 2 , 2 Mreff g V 3¯ h

(7.96)

where V is the three-dimensional volume. When discussing the interacting system we shall refer to the previously calculated critical temperature of the free system as Tc(0) . This result follows from the fact that for small g and reff , the free-particle energies ǫ0 (k) = h ¯ 2 k2 /2M are changed to ǫ(k) = ǫ(0)+¯ h2 k2 /2M ∗ with a renormalized inverse 2 effective mass 1/M ∗ = [1 − Mreff gN/3¯ h2 V ]/M. Inserting this into Eq. (7.18), from

528


which we may extract the equation for the temperature shift ∆Tc /Tc(0) = M/M ∗ −1, we obtain indeed the result (7.96). The parameter reff is called the effective range of the potential. The parameter g in (7.95) can be determined by measuring the s-wave scattering length a in a two-body scattering experiment. The relation is obtained from a solution of the Lippmann-Schwinger equation (1.522) for the T -matrix. In a dilute gas, this yields in three dimensions g=

2π¯ h2 a. M/2

(7.97)

The denominator M/2 is the reduced mass of the two identical bosons. In D = 2 dimensions where g has the dimension energy×lengthD , Eq. (7.97) is replaced by [12, 13, 14, 15, 17] π D/2−1 2π¯ h2 D−2 h2 D−2 D 2π¯ ≡ γ(D, na ) . a a 22−D Γ(1 − D/2)(naD )D−2 + Γ(D/2 − 1) M/2 M/2 (7.98) In the limit D → 2, this becomes g=

g=−

2π¯ h2 /M , ln(eγ na2 /2)

(7.99)

where γ is the Euler-Mascheroni constant. The logarithm in the denominator implies that the effective repulsion decreases only very slowly with decreasing density [16]. For a low particle density N/V , the effective range reff becomes irrelevant and the shift ∆Tc depends on the density with a lower power (N/V )κ/3 , κ < 3. The lowdensity limit can be treated by keeping in (7.95) only the first term corresponding to a pure δ-function repulsion V (x − x′ ) = g δ (3) (x − x′ ).

(7.100)

For this interaction, the lowest-order correction to the energy is in D dimensions ∆E = g

Z

dD x n2 (x),

(7.101)

where n(x) is the local particle density. For a homogeneous gas, this changes the grand-canonical free energy from (7.35) to "

ζD/2 (z) 1 VD FG = − D ζD/2+1 (z) + g VD D β le (¯ hβ) le (¯ hβ)

#2

+ O(g 2 ),

(7.102)

where we have substituted n(x) by the constant density N/VD of Eq. (7.36). We now introduce a length parameter α proportional to the coupling constant g: g≡

2 α D l (¯ hβ). β le (¯ hβ) e


529


In three dimensions, α coincides in the dilute limit (small naD ) with the s-wave scattering length a of Eq. (7.97). In in D dimensions, the relation is "

a α = γ(D, naD ) le (¯ hβ)

#D−3

a.

(7.104)

In two dimensions, this has the limit α=−

le (¯ hβ) . γ ln(e na2 /2)

(7.105)

We further introduce the reduced dimensionless coupling parameter α ˆ ≡ α/le (¯ hβ) = γ(D, naD ) [a/le (¯ hβ)]D−2 , for brevity [recall (7.104)]. In terms of α, ˆ the grandcanonical free energy (7.102) takes the form FG = −

o 1 VD n 2 ζ (z) − 2 α[ζ ˆ (z)] + O(α2 ). D/2+1 D/2 β leD (¯ hβ)

(7.106)

A second-order perturbation calculation extends this by a term [18, 19] ∆FG = − (1)

o 1 VD n 2 8 α ˆ h (z) , D β leD (¯ hβ)

(7.107)

(2)

where hD (z) = hD (z) + hD (z) is the sum of two terms. The first is simply (1)

hD (z) ≡ [ζD/2 (z)]2 ζD/2−1 (z),

(7.108)

the second has been calculated only for D = 3: (2) h3 (z)

≡

∞ X

n1 ,n2 ,n3 =1

√

z n1 +n2 +n3 . n1 n2 n3 (n1 +n2 )(n1 +n3 )

(7.109)

The associated particle number is (

dhD (z) VD N= D ζD/2 (z) − 4αζ ˆ D/2−1 (z)ζD/2 (z) + 8α ˆ2z le (¯ hβ) dz

)

+ O(α3 ).

(7.110)

For small α, we may combine the equations for FG and N and derive the following simple relation between the shift in the critical temperature and the change in the particle density caused by the interaction ∆Tc 2 ∆n ≈− , (7.111) (0) D n Tc where ∆n is the change in the density at the critical point caused by the interaction. The equation is correct to lowest order in the interaction strength. The calculation of ∆Tc /Tc(0) in three dimensions has turned out to be a difficult problem. The reason is that the perturbation series for the right-hand side of (7.111) is found to be an expansion in powers (T − Tc(0) )−n which needs a strong-coupling

530


evaluation for T → Tc(0) . Many theoretical papers have given completely different result, even in sign, and Monte Carlo data to indicate sign and order of magnitude. The ideal tool for such a calculation, field theoretic variational perturbation theory, has only recently been developed [20], and led to a result in rough agreement with Monte Carlo data [21]. Let us briefly review the history. All theoretical results obtained in the literature have the generic form ∆Tc (0)

Tc

= c[ζ(3/2)]κ/3

a ℓc

κ

= c aκ

N V

κ/3

,

(7.112)

where the right-hand part of the equation follows from the middle part via Eq. (7.53). In an early calculation [22] based on the δ-function potential (7.100), κ was found to be 1/2, with a downward shift of Tc . More recent studies, however, have lead to the opposite sign [23]–[38]. The exponents κ found by different authors range from κ = 1/2 [23, 18, 31] to κ = 3/2 [19]. The most recent calculations yield κ = 1 [24]–[28], i.e., a direct proportionality of ∆Tc /Tc(0) to the s-wave scattering length a, a result also found by Monte Carlo simulations [29, 30, 38], and by an extrapolation of experimental data measured in the strongly interacting superfluid 4 He after diluting it with the help of Vycor glass [40]. As far as the proportionality constant c is concerned, the literature offers various values which range for κ = 1 from c = 0.34 ± 0.03 [29] to c = 5.1 [40]. A recent negative value c ≈ −0.93 [41] has been shown to arise from a false assumption on the relation between canonical and grand-canonical partition function [42]. An older Monte Carlo result found c ≈ 2.3 [30] lies close to the theoretical results of Refs. [23, 27, 28], who calculated c1 ≈ −8ζ(1/2)/3ζ 1/3(3/2) ≈ 2.83, 8π/3ζ 4/3 (3/2) ≈ 2.33, 1.9, respectively, while the extrapolation of the experimental data on 4 He in Vycor glass favored c = 5.1, near the theoretical estimate c = 4.66 of Stoof [24]. The latest Monte Carlo data, however, point towards a smaller value c ≈ 1.32 ± 0.02, close to theoretical numbers 1.48 and 1.14 in Refs. [35, 39] and 1.14 ± 0.11 in Ref. [21]. There is no space here to discuss in detail how the evaluation is done. We only want to point out an initially surprising result that in contrast to a potential with a finite effective range in (7.96), the δ-function repulsion (7.100) with a pure phase shift causes no change of Tc linear an1/D if only the first perturbative correction to the grand-canonical free energy in Eq. (7.106) is taken into account. A simple nonperturbative approach which shows this goes as follows: We observe that the one-loop expression for the free energy may be considered as the extremum with respect to σ of the variational expression FGσ

1 VD σ2 = − D ζD/2+1 (zeσ ) − , β le (¯ hβ) 8α ˆ "

#

(7.113)

which is certainly correct to first order in a. We have introduced the reduced dimensionless coupling parameter α ˆ ≡ α/le (¯ hβ) = γ(D, naD ) [a/le (¯ hβ)]D−2 , for brevity [recall (7.104)]. The extremum lies at σ = Σ which solves the implicit equation Σ ≡ −4αζ ˆ D/2 (zeΣ ).


531


The extremal free energy is FGΣ = −

o 1 VD n 2 ζ (Z) − 2 α[ζ ˆ (Z)] , D/2+1 D/2 β leD (¯ hβ)

Z ≡ zeΣ .

(7.115)

The equation for the particle number, obtained from the derivative of the free energy (7.113) with respect to −µ, reads N =

VD ζD/2 (Z), D le (¯ hβ)

(7.116)

and fixes Z. This equation has the same form as in the free case (7.36), so that the phase transition takes place at Z = 1, implying the same transition temperature as in the free system to this order. As discussed above, a nonzero shift (7.112) can only be found after summing up many higher-order Feynman diagrams which all diverge at the critical temperature. Let us also point out that the path integral approach is not an adequate tool for discussing the condensed phase below the critical temperature, for which infinitely many particle orbits are needed. In the condensed phase, a quantum field theoretic formulation is more appropriate (see Section 7.6). The result of such a discussion is that due to the positive value of c in (7.112), the phase diagram has an unusual shape shown in Fig. 7.7. The nose in the phase transition curve implies that the system can undergo Bose-Einstein condensation slightly above Tc(0) if the interaction is increased (see Ref. [43]). aeff /aeff (T = 0) 1 normal

0.8 0.6 0.4

superfluid

0.2 (0)

0.2 0.4 0.6 0.8

1

1.2 1.4

T /Tc

Figure 7.7 Reentrant transition in phase diagram of Bose-Einstein condensation for different interaction strengths. Curves were obtained in Ref. [43] from a variationally improved one-loop approximation to field-theoretic description with properly imposed posi(0) tive slope at Tc (dash length increasing with order of variational perturbation theory). (0) Short solid curve and dashed straight line starting at Tc are due to Ref. [38] and [21]. Diamonds correspond to the Monte-Carlo data of Ref. [29] and dots stem from Ref. [44], both scaled to their critical value aeff (T = 0) ≈ 0.63.

532


An important consequence of a repulsive short-range interaction is a change in the particle excitation energies below Tc . It was shown by Bogoliubov [46] that this changes the energy from the quadratic form ǫ(p) = p2 /2M to ǫ(p) =

q

ǫ2 (p) + 2gnǫ(p),

q

(7.117)

q

¯ 4πan/M|p| for small p, the slope which starts linearly like ǫ(p) = gn/M|p| = h defining the second sound velocity. In the strongly interacting superfluid helium, the momentum dependence has the form shown in Fig. 7.8. It was shown by Bogoliubov

ǫ(p)/kB o

K

p/¯h (rA−1 )

Figure 7.8 Energies of elementary excitations of superfluid 4 He measured by neutron scattering showing excitation energy of NG bosons [after R.A. Cowley and A.D. Woods, Can. J. Phys 49, 177 (1971)].

[46] that a system in which ǫ(p) > vc |p| for some finite critical velocity vc will display superfluidity as long as it moves with velocity v = ∂ǫ(p)/∂p smaller than vc . This follows by forming the free energy in a frame moving with velocity v. It is given by the Legendre transform: f ≡ ǫ(p) − v · p, (7.118) which has a minimum at p = 0 if as long as |v| < vc , implying that the particles do not move. Seen from the moving frame, they keep moving with a constant velocity −v, without slowing down. This is in contrast to particles with the free spectrum ǫ(p) = p2 /2M for which f has a minimum at p = Mv, implying that these particles always move with the same velocity as the moving frame. Note that the second sound waves of long wavelength have an energy ǫ(p = c|p| just like light waves in the vacuum, with light velocity exchanged by the sound velocity. Even though the superfluid is nonrelativistic, the sound waves behave like relativistic particles. This phenomenon is known from the sound waves in crystals which behave similar to relativistic massless particles. In fact, the Debye theory of specific heat is very similar to Planck’s black-body theory, except that the lattice size appears explicitly as a short-wavelength cutoff. It has recently been speculated that the relativistically invariant world we observe is merely the long-wavelength limit of a H. Kleinert, PATH INTEGRALS

533


world crystal whose lattice constant is very small [47], of the order of the Planck length ℓP ≈ 8.08 × 10−33 cm.

7.2.4

Bose-Einstein Condensation in Harmonic Trap

In a harmonic magnetic trap, the path integral (7.7) of the individual orbits becomes (p(ν)) (xb h ¯ β|x(ν) a

Z

0)ω =

(ν)

x(p(ν)) (τ b )=xb (ν)

x(ν) (τa )=xa

(

1 D x exp − h ¯ D (ν)

Z

0

)

i M h (ν)2 dτ x˙ + ω 2 x(ν)2 , (7.119) 2

hβ ¯

and is solved by [recall (2.403)] (p(ν)) h ¯ β|x(ν) (xb a

1

0)ω = q D 2π¯ h/M

s

ω sinh β¯ hω

D

(7.120)

(

)

1 Mω (p(ν)) 2 (p(ν)) (ν) 2 × exp − [(x + x(ν) ) cosh β¯ hω−2xb xa ] . a 2¯ h sinh β¯ hω b

The partition function (7.5) can therefore be rewritten in the form Zω(N )

7.2.5

ND

s

Mω 1 = dD x(1) · · · dD x(N ) 2π¯ h sinh β¯ hω N! ( ) X 1 Mω (p(ν)) 2 (ν) 2 (p(ν)) (ν) [(x × exp − + x ) cosh β¯ hω − 2x x ] . (7.121) 2¯ h sinh β¯ hω p(ν) Z

Thermodynamic Functions

With the same counting arguments as before we now obtain from the connected paths, which wind some number w = 1, 2, 3, . . . times around the cylinder, a contribution to the partition function ∆Zω(N ) w

1 = Zω (wβ) = w

Z

dD x zω (wβ; x)

1 , w

(7.122)

where

1 (7.123) [2 sinh(β¯ hω/2)]D is the D-dimensional harmonic partition function (2.399), and zω (wβ; x) the associated density [compare (2.326) and (2.404)]: Zω (β) =

zω (β; x) =

s

D

"

#

ωM Mω β¯ hω 2 exp − tanh x . 2π¯ h sinh β¯ hω h ¯ 2

(7.124)

Its spatial integral is the partition function of a free particle at an imaginary-time interval β [compare (2.404)]. The sum over all connected contributions (7.122) yields the grand-canonical free energy ∞ ∞ Z 1 X ewβµ 1 X ewβµ FG = − Zω (wβ) =− dD x zω (wβ; x) . β w=1 w β w=1 w

(7.125)

534


Note the important difference between this and the free-boson expression (7.27). Whereas in (7.27), the winding number appeared as a factor w −D/2 which was removed from Z0 (wβ) by writing Z0 (wβ) = Z0 (β)/w D/2 which lead to (7.32), this is no longer possible here. The average number of particles is thus given by N =−

∞ X ∂ FG = Zω (wβ)ewβµ = ∂µ w=1

Z

dD x

∞ X

zω (wβ; x)ewβµ.

(7.126)

w=1

Since Zω (wβ) ≈ e−wD¯hωβ/2 for large w, the sum over w converges only for µ < D¯ hω/2. Introducing the fugacity associated with the ground-state energy zD (β) = e−(D¯hω/2−µ)β = e−Dβ¯hω/2 z,

(7.127)

by analogy with the fugacity (7.24) of the zero-momentum state, we may rewrite (7.126) as ∂ N = − FG = Zω (β)ζD (β¯ hω; zD ). (7.128) ∂µ hω; zD ) are generalizations of the functions (7.34): Here ζD (β¯ ζD (β¯ hω; zD ) ≡

∞ X

w=1

"

sinh(ω¯ hβ/2) sinh(wω¯ hβ/2)

#D

ewβµ = Zω−1 (β)

∞ X

w=1 (1

1

−

zw , D/2 D −2wβ¯ h ω e )

(7.129)

which reduce to ζD (z) in the trapless limit ω → 0, where zD → z. Expression (7.128) is the closest we can get to the free-boson formula (7.36). We may define local versions of the functions ζD (β¯ hω; zD ) as in Eq.(7.124): ∞ X ζD (β¯ hω; zD ; x) ≡ Zω−1 (β) w=1 (1

1 e−2wβ¯hω )D/2

ωM π¯ h

D/2

e−M ω tanh(wβ¯hω/2)x

2 /¯ h

− in terms of which the particle number (7.128) reads Z Z ∂ N = − FG = dD x nω (x) ≡ Zω (β) dD x ζD (β¯ hω; zD ; x), ∂µ

w zD ,(7.130)

(7.131)

and the free energy (7.125) becomes FG =

Z

1 d x f (x) ≡ − Zω (β) dD x β Z

D

Z

0

zD

dz ζD (β¯ hω; z; x). z

(7.132)

For small trap frequency ω, the function (7.130) has a simple limiting form: ω≈0

ζD (β¯ hω; zD ; x) ≈

β¯ hω 2π

!D/2 q

∞ 1 X 1 −wβ(M ω2 x2 /2−µ) e , D/2 λD ω w=1 w

(7.133)

where λω is the oscillator length scale h ¯ /Mω of Eq. (2.296). Together with the prefactor Zω (¯ hβ) in (7.131), which for small ω becomes Zω (¯ hβ) ≈ 1/(β¯ hω)D , this yields the particle density n0 (β; z; x) =

∞ X 1 1 −wβ[V (x)−µ] 1 e = D ζD/2 (e−β[V (x)−µ] ), D D/2 le (¯ hβ) w=1 w le (¯ hβ)

(7.134)


535


where V (x) = Mω 2 x2 /2 is the oscillator potential and le (¯ hβ) the thermal length scale (2.345). There is only one change with respect to the corresponding expression for the density in the homogeneous gas [compare (7.36)]: the fugacity z = eβµ in the argument by the local fugacity z(x) ≡ e−β[V (x)−µ] .

(7.135)

The function ζD (β¯ hω; zD ) starts out like zD , and diverges for zD → 1 like D [2 sinh(β¯ hω/2)] /(zD −1). This divergence is the analog of the divergence of the box function (7.82) which reflects the formation of a condensate in the discrete ground state with particle number [compare (7.88)] Ncond =

1 eDβ¯hω/2−βµ − 1

.

(7.136)

This number diverges for µ → D¯ hω/2 or zD → 1. In a box with a finite the number of particles, Eq. (7.36) for the particle number was replaced by the equation for the normal particles (7.89) containing the regbox (z). In the thermodynamic limit, this turned into the ularized box functions ζ¯D/2 function ζD/2 (z) in which the momentum sum was evaluated as an integral. The present functions ζD (β¯ hω; zD ) governing the particle number in the harmonic trap play precisely the role of the previous box functions, with a corresponding singular term which has to be subtracted, thus defining a regularized function zD ζ¯D (β¯ hω; zD ) ≡ ζD (β¯ hω; zD )−Zω−1 (β) = Zω−1 (β)SD (β¯ hω; zD ), (7.137) 1−zD where S(β¯ hω, zD ) is the sum S(β¯ hω, zD ) ≡

∞ X

w=1

"

1 (1−e−wβ¯hω )D

#

w − 1 zD

(7.138)

The local function (7.130) has a corresponding divergence and can be regularized by a similar subtraction. The sum (7.138) governs directly the number of normal particles Nn (T ) = Zω (β)ζ¯D (β¯ hω; zD ) ≡ SD (β¯ hω, zD ).

(7.139)

The replacement of the singular equation (7.128) by the regular (7.139) of completely analogous to the replacement of the singular (7.82) by the regular (7.89) in the box.

7.2.6

Critical Temperature

Bose-Einstein condensation can be observed as a proper phase transition in the thermodynamic limit, in which N goes to infinity, at a constant average particle D density in the trap defined by N/λD ω ∝ Nω . In this limit, ω goes to zero and the D sum (7.139) becomes ζD (zD )/β¯ hω . The associated particle equation Nn =

1 ζD (zD ) (β¯ hω)D

(7.140)

536


can be solved only as long as zD < 1. Above Tc , this equation determines zD as a function of T from the condition that all particles are normal, Nn = N. Below Tc , it determines the temperature dependence of the number of normal particles by inserting zD = 1, where Nn =

1 ζ(D). (β¯ hω)D

(7.141)

The particles in the ground state from a condensate, whose fraction is given by Ncond (T )/N ≡ 1 − Nn (T )/N. This is plotted in Fig. 7.9 as a function of the temperature for a total particle number N = 40 000. The critical point with T = Tc , µ = µc = Dω/2 is reached if Nn is equal to the total particle number N where kB Tc(0)

"

N =h ¯ω ζ(D)

#1/D

.

(7.142)

This formula has a solution only for D > 1. Inserting (7.142) back into (7.162), we may re-express the normal fraction as a function of the temperature as follows: Nn(0) ≈ N

T (0)

Tc

!D

,

(7.143)

and the condensate fraction as (0)

Ncond T ≈1− (0) N Tc

!D

.

(7.144)

Including the next term in (7.162), the condensate fraction becomes (0)

Ncond T ≈1− (0) N Tc + δTc

!D

.

(7.145)

It is interesting to re-express the critical temperature (7.142) in terms of the particle density at the origin which is at the critical point, according to Eq. (7.134), n0 (0) =

1 ζ(D/2), leD (¯ hβc )

(7.146)

where we have shortened the notation on an obvious way. From this we obtain kB Tc(0)

2π¯ h2 n0 (0) = M ζ(D/2) "

#2/D

.

(7.147)

Comparing this with the critical temperature without a trap in Eq. (7.18) we see that both expressions agree if we replace N/V by the uniform density n0 (0). As an H. Kleinert, PATH INTEGRALS

537


Figure 7.9 Condensate fraction Ncond /N ≡ 1 − Nn /N as function of temperature for total number of N particles. The long- and short-dashed curves on the left-hand show the zeroth and first-order approximations (7.144) and (7.163). The dotted curve displays the free-boson behavior (7.58). The right-hand figure shows experimental data for 40 000 87 Rb atoms near T by Ensher et. al., Phys. Rev. Lett. 77 , 4984 (1996). The solid c curve describes noninteracting Bose gas in a harmonic trap [cf. (7.144)]. The dotted curve corrects for finite-N effects [cf. (7.163)]. The dashed curve is the best fit to the data. The transition lies at Tc ≈ 280 nK, about 3% below the dotted curve.

obvious generalization, we conclude that Bose condensation in any trap will set in when the density at the lowest point reaches the critical value determined by (7.147) [and (7.18)]. Note the different power D and argument in ζ(D) in Eq. (7.142) in comparison with the free-boson argument D/2 in Eq. (7.18). This has the consequence that in contrast to the free Bose gas, the gas in a trap can form a condensate in two dimensions. There is now, however, a problem in D = 1 dimension where (7.142) gives a vanishing transition temperature. The leading-order expression (7.142) for critical temperature can also be calculated from a simple statistical consideration. For small ω, the density of states available to the bosons is given by the classical expression (4.210): M ρcl (E) = 2π¯ h2

D/2

1 Γ(D/2)

Z

dD x [E − V (x)]D/2−1 .

(7.148)

The number of normal particles is given by the equation Nn =

Z

∞

Emin

dE

ρcl (E) , eE/kB T − 1

(7.149)

where Emin is the classical ground state energy. For a harmonic trap, the spatial integral in (7.148) can be done and is proportional to E D , after which the integral over E in (7.149) yields [kB T /¯ hω]D ζ(D) = (T /Tc(0) )D N, in agreement with (7.143).

538


Alternatively we may use the phase space formula ∞ X dD p 1 = (2π¯ h)D eβ[p2 /2M +V (x)] − 1 n=1 Z ∞ X 1 = dD x e−nβV (x) , q D 2 n=1 2π¯ h nβ/M

Nn =

Z

dD x

where the spatial integration produces a factor side becomes again (T /Tc(0) )D N.

7.2.7

Z

dD x

dD p −nβ[p2 /2M +V (x)] e (2π¯ h )D (7.150)

q

2π/Mω 2 nβ so that the right-hand

More General Anisotropic Trap

The equation (7.150) for the particle number can be easily calculated for a more general trap where the potential has the anisotropic power behavior D M 2 2X |xi | V (x) = ˜ ω ˜ a 2 ai i=1

!p i

,

(7.151) h

where ω ˜ is some frequency parameter and a ˜ is the geometric average a˜ ≡ ΠD i=1 ai Inserting (7.151) into (7.150) we encounter a product of integrals D Z Y

∞

i=1 −∞

dx e−nβM ω˜

2a ˜2 (|x

p i |/ai ) i /2

=

D Y

ai Γ(1 + 1/pi), ˜ 2a ˜2 /2)1/pi i=1 (βM ω

i1/D

.

(7.152)

˜

so that the right-hand side of (7.150) becomes (T /Tc(0) )D N, with the critical temperature kB Tc(0)

M a˜2 ω ˜2 = 2

h ¯ω ˜ M a˜2 ω ˜2

!D/D˜ "

˜ is the dimensionless parameter where D

Nπ D/2 Q ˜ D Γ(1 + 1/pi ) ζ(D) i=1

#1/D˜

D X 1 ˜ ≡D+ D . 2 i=1 pi

,

(7.153)

(7.154)

which takes over the role of D in the harmonic formula (7.142). A harmonic trap with different oscillator frequencies ω1 , . . . , ωD along the D Cartesian axes, ˜ = D, and formula is a special case of (7.151) with pi ≡ 2, ωi2 = ω ˜ 2a˜2 /a2i and D (7.153) reduces to (7.142) with ω replaced by the geometric average of the frequencies ω ˜ ≡ (ω1 · · · ωD )1/D . The parameter a˜ disappears from the formula. A free Bose gas Q D D in a box of size VD = D ˜ is described by (7.151) in the limit pi → ∞ i=1 (2ai ) = 2 a ˜ = D/2. Then Eq. (7.153) reduces to where D kB Tc(0)

π¯ h2 N = 2 2M a˜ ζ(D/2) "

#2/D

2π¯ h2 N = M VD ζ(D/2) "

#2/D

,

(7.155)


539


in agreement with (7.53) and (7.18). Another interesting limiting case is that of a box of length L = 2a1 in the xdirection with p1 = ∞, and two different oscillators of frequency ω2 and ω3 in the 2 other two directions. To find Tc(0) for such a Bose gas we identify ω ˜ 2a ˜2 /a22,3 = ω2,3 in the potential (7.151), so that ω ˜ 4 /˜ a2 = ω22 ω32/a21 , and obtain kB Tc(0)

7.2.8

π¯ h =h ¯ω ˜ 2M ω ˜

!1/5 "

N a1 ζ(5/2)

#2/5

2πλω1 λω2 =h ¯ω ˜ L2

!1/5 "

N ζ(5/2)

#2/5

. (7.156)

Rotating Bose-Einstein Gas

Another interesting potential can be prepared in the laboratory by rotating a Bose condensate [48] with an angular velocity Ω around the z-axis. The vertical trapping frequencies is ωz ≈ 2π × 11.0 Hz ≈ 0.58× nK, the horizontal one is ω⊥ ≈ 6 × ωz . The centrifugal forces create an additional repulsive harmonic potential, bringing the rotating potential to the form 4 κ r⊥ Mωz2 2 2 z + 36ηr⊥ + V (x) = 2 2 λ2ωz

!

h ¯ ωz = 2

2 4 r⊥ κ r⊥ z2 + 36η + λ2ωz λ2ωz 2 λ4ωz

!

,

(7.157)

2 2 where r⊥ = x2 + y 2 , η ≡ 1 − Ω2 /ω⊥ , κ ≈ 0.4, and λωz ≡ 3.245 µm ≈ 1.42 × 10−3 K. For Ω > ω⊥ , η turns negative and the potential takes the form of a Mexican hat as 2 shown in Fig. 3.13, with a circular minimum at rm = −36ηλ2ωz /κ. For large rotation speed, the potential may be approximated by a circular harmonic well, so that we may apply formula (7.156) with a1 = 2πrm , to obtain the η-independent critical temperature

kB Tc(0)

≈h ¯ ωz

1/5 "

κ π

N ζ(5/2)

#2/5

.

(7.158)

For κ = 0.4 and N = 300 000, this yields Tc ≈ 53nK. q At the critical rotation speed Ω = ω⊥ , the potential is purely quartic r⊥ = (x2 +y 2). To estimate Tc(0) we approximate it for a moment by the slightly different potential (7.151) with the powers p1 = 2, p2 = 4, p3 = 4, a1 = λωz , a2 = a3 = λωz (κ/2)1/4 , so that formula (7.153) becomes kB Tc(0)

=h ¯ ωz

"

π2κ 16Γ4 (5/4)

#1/5 "

N ζ(5/2)

#2/5

.

(7.159)

It is easy to change this result so that it holds for the potential ∝ r 4 = (x + y)4 rather than x4 + y 4: we multiply the right-hand side of equation (7.149) for N by a factor R 4 2π rdrdxdy e−r π 3/2 R = . (7.160) dxdy e−x4 −y4 Γ[5/4]2

540


This factor arrives inversely in front of N in Eq. (7.161), so that we obtain the critical temperature in the critically rotating Bose gas kB Tc(0)

=h ¯ ωz

4κ π

1/5 "

N ζ(5/2)

#2/5

.

(7.161)

The critical temperature at Ω = ω⊥ is therefore by a factor 41/5 ≈ 1.32 larger than at infinite Ω. Actually, this limit is somewhat academic in a semiclassical approximation since the quantum nature of the oscillator should be accounted for.

7.2.9

Finite-Size Corrections

Experiments never take place in the thermodynamic limit. The particle number is finite and for comparison with the data we must calculate finite-size corrections coming from finite N where ll ω ≈ 1/N 1/D is small but nonzero. The transition is no longer sharp and the definition of the critical temperature is not precise. As in the thermodynamic limit, we shall identify it by the place where zD = 1 in Eq. (7.139) for Nn = N. For D > 3, the corrections are obtained by expanding the first term in the sum (7.138) in powers of ω and performing the sums over w and subtracting P ζ(0) for the sum ∞ w=1 1:

β¯ hω (ω¯ hβ)2 1 Dζ(D−1) + D(3D−1)ζ(D−2) + . . . ζ(D) + Nn (Tc ) = (β¯ hω)D 2 24 − ζ(0). (7.162) The higher expansion terms contain logarithmically divergent expressions, for instance in one dimension the first term ζ(1), and in three dimensions ζ(D −2) = ζ(1). These indicate that the expansion powers of β¯ hω is has been done improperly at a singular point. Only the terms whose ζ-function have a positive argument can be trusted. A careful discussion along the lines of Subsection 2.15.6 reveals that ζ(1) must be replaced by ζreg (1) = − log(β¯ hω) + const., similar to the replacement (2.582). The expansion is derived in Appendix 7A. For D > 1, the expansion (7.162) can be used up to the ζ(D − 1) terms and yields the finite-size correction to the number of normal particles "

#

Nn = N

T (0) Tc

!D

+

D ζ(D − 1) 1 . 2 ζ 1−1/D (D) N 1/D

(7.163)

Setting Nn = N, we obtain a shifted critical temperature by a relative amount δTc (0) Tc

=−

1 ζ(D − 1) 1 + ... . (D−1)/D 2ζ (D) N 1/D

(7.164)

In three dimensions, the first correction shifts the critical temperature Tc(0) downwards by 2% for 40 000 atoms. Note that correction (7.164) has no direct ω-dependence whose size enters only implicitly via Tc(0) of Eq. (7.142). In an anisotropic harmonic trap, the temperature H. Kleinert, PATH INTEGRALS

541


shift ˜ /¯ ω where ω ¯ is the arithmetic mean ω ¯ ≡ P would carry a dimensionless factor ω D i=1 ωi /D. The higher finite-size corrections for smaller particle numbers are all calculated in Appendix 7A. The result can quite simply be deduced by recalling that according to the Robinson expansion (2.575), the first term in the naive, wrong power P P −wb k /w ν = ∞ series expansion of ζν (e−b ) = ∞ w=1 e k=0 (−b) ζ(ν − k)/k! is corrected ν−1 by changing the leading term ζ(ν) to Γ(1 − ν)(−b) + ζ(ν) which remains finite for all positive integer ν. Hence we may expect that the correct equation for the critical temperature is obtained by performing this change in Eq. (7.162) on each ζ(ν). This expectation is confirmed in Appendix 7A. It yields for D = 3, 2, 1 the equations for the number of particles in the excited states (β¯ hω)2 1 β¯ hω Nn = − (log β¯ hω − γ) − ζ(0) + ζ(−1) + . . . , (7.165) β¯ hω 2 12 " # 1 1 7(β¯ hω)2 Nn = ζ(2) − β¯ hω log β¯ hω − γ + − ζ(0) + . . . , (7.166) (β¯ hω)2 2 12 " # 1 3β¯ hω 19 2 ζ(2) − (β¯ hω) log β¯ Nn = ζ(3) + hω − γ + + . . . , (7.167) (β¯ hω)3 2 24 "

#

where γ = 0.5772 . . . is the Euler-Mascheroni number (2.461). Note that all nonlogarithmic expansion terms coincide with those of the naive expansion (7.162). These equations may be solved for β at Nn = N to obtain the critical temperature of the would-be phase transition. Once we study the position of would-be transitions at finite size, it makes sense to include also the case D = 1 where the thermodynamic limit has no transition at all. There is a strong increase of number of particles in the ground state at a “critical temperature” determined by equating Eq. (7.165) with the total particle number N, which yields kB Tc(0) = h ¯ ωN

1 1 ≈h ¯ ωN , (− log β¯ hω + γ) log N

D = 1.

(7.168)

Note that this result can also be found also from the divergent naive expansion (7.162) by inserting for the divergent quantity ζ(1) the dimensionally regularized expression ζreg (1) = − log(β¯ hω) of Eq. (2.582).

7.2.10

Entropy and Specific Heat

By comparing (7.126) with (7.125) we see that the grand-canonical free energy can be obtained from Nn of Eq. (7.162) by a simple multiplication with −1/β and an increase of the arguments of the zeta-functions ζ(ν) by one unit. Hence we have, up to first order corrections in ω FG (β, µc) = −

1 β(β¯ hω)D

ζ(D + 1) + β¯ hω

D ζ(D) + . . . . 2

(7.169)

542


From this we calculate immediately the entropy S = −∂T FG = kB β 2 ∂β FG as 1 S = −kB (D + 1)βFG = kB (D + 1) (β¯ hω)D

D ζ(D + 1) + β¯ hω ζ(D) + . . . . (7.170) 2

In terms of the lowest-order critical temperature (7.142), this becomes  

T

S = kB N(D + 1) 

(0)

Tc

!D

"

ζ(D + 1) D ζ(D) + ζ(D) 2 N

#1/D

T (0)

Tc

!D−1

From this we obtain the specific heat C = T ∂T S below Tc :  

C = kB N(D+1) D 

T (0)

Tc

!D

"

ζ(D+1) D(D−1) ζ(D) + ζ(D) 2 N

#1/D

T (0)

Tc

 

+ . . . . (7.171)

!D−1

 

+ . . . .(7.172)

At the critical temperature, this has the maximal value



ζ(D+1) D−1 ζ 1+1/D (D) D ζ(D−1) 1 1+ − . (7.173) Cmax≈ kB N(D+1)D ζ(D) 2 ζ(D+1) 2 ζ 1−1/D(D) N 1/D (

"

#

)

In three dimensions, the lowest two approximations have their maximum at (0) Cmax ≈ kB N 10.805,

(1) Cmax ≈ kB N 9.556.

(7.174)

Above Tc , we expand the total particle number (7.126) in powers of ω as in (7.162). The fugacity of the ground state is now different from unity: D 1 N(β, µ) = ζD (zD ) + β¯ hω ζD−1 (zD ) + . . . . D (β¯ hω) 2

(7.175)

The grand-canonical free energy is 1 FG (β, µ) = − β(β¯ hω)D

D ζD+1(zD ) + β¯ hω ζD (zD ) + . . . , 2

(7.176)

and the entropy S(β, µ) = kB

1 1 2 (D + 1)ζ (z ) + β¯ h ωD − 2βµ ζD (zD ) + . . . . (7.177) D+1 D (β¯ hω)D 2

The specific heat C is found from the derivative −β∂β S|N as C(β, µ) = kB

1 (β¯ hω)D

(D+1)D ζD+1 (zD )

i 1h + D D 2 + 1 β¯ hω− Dβµ − β∂β (βµ) ζD (zD )+. . . . 2

(7.178)


543


The derivative β∂β (βµ) is found as before from the condition: ∂N(β, µ(β))/∂β = 0, implying 1 D 0 = − DζD (zD ) + β¯ hω (D − 1)ζD−1(zD ) D (β¯ hω) 2 D D + ζD−1 (zD ) + β¯ hω ζD−2 (zD ) β∂β (βµ)−β¯ hω + ... , 2 2

so that we obtain

(7.179)

D ζD−2 (zD ) + . . . 2 β∂β (βµ) = D . (7.180) D ζD−1(zD ) + β¯ h ωζD−2(zD ) + . . . 2 Let us first consider the lowest approximation, where 1 ζD (z), (7.181) N(β, µ) ≈ (β¯ hω)D 1 1 1 ζD+1 (z) FG (β, µ) ≈ − ζD+1 (z) = −N , (7.182) D β (β¯ hω) β ζD (z) " # " # 1 D+1 D + 1 ζD+1 (z) 1 S(β, µ) ≈ ζD+1 (z) − µN = NkB − βµ , T β (β¯ hω)D (β¯ hω)D ζD (z) (7.183) so that from (7.60) ζD (zD ) + β¯ hω

E(β, µ) ≈

D 1 D ζD+1 (z) . ζD+1(z) = N D β (β¯ hω) β ζD (z)

(7.184)

The chemical potential at fixed N satisfies the equation β∂β (βµ) = D

ζD (z) . ζD−1(z)

(7.185)

The specific heat at a constant volume is "

#

ζD+1(z) ζD (z) C = kB N (D + 1)D − D2 . ζD (z) ζD−1(z)

(7.186)

At high temperatures, C tends to the Dulong-Petit limit DNkB since for small z all ζν (z) behave like z. This is twice a big as the Dulong-Petit limit of the free Bose gas since there are twice as many harmonic modes. As the temperature approaches the critical point from above, z tends to unity from below and we obtain a maximal value in three dimensions (0) Cmax

"

#

ζ4 (1) ζ3 (1) = kB N 12 −9 ≈ kB N 4.22785. ζ3 (1) ζ2 (1)

(7.187)

The specific heat for a fixed large number N of particles in a trap has a much sharper peak than for the free Bose gas. The two curves are compared in Fig. 7.10, where we also show how the peak is rounded for different finite numbers N.3 3

P.W. Courteille, V.S. Bagnato, and V.I. Yukalov, Laser Physics 2 , 659 (2001).

544

7 Many Particle Orbits — Statistics and Second Quantization 10

CN /kB N

8

CN /kB N N =∞

harmonic trap

6

N = 10 000 N = 1000

4

N = 100

2

free Bose gas 0.2

0.4

0.6

0.8

1

1.2 (0)

1.4 (0)

T /Tc

T /Tc

Figure 7.10 Peak of specific heat for infinite (left-hand plot) and various finite numbers 100, 1000, 10 000 of particles N (right-hand plots) in harmonic trap. The large-N curve is compared with that of a free Bose gas.

7.2.11

Interactions in Harmonic Trap

Let us now study the effect of interactions on a Bose gas in an isotropic harmonic trap. This is most easily done by adding to the free part (7.132) the interaction (7.101) with n(x) taken from (7.131), to express the grand-canonical free energy by analogy with (7.102) as (

1 d x − Zω (β) β

)

dz FG = ζD (β¯ hω; z; x) + g [Zω (β)ζD (β¯ hω; zD ; x)]2 . z 0 (7.188) Using the relation (7.103), this takes a form more similar to (7.106): Z

D

1 FG = − Zω (β) dD x β

(Z

Z

zD

)

dz ζD (β¯ hω; z; x)−2ˆ a leD (¯ hβ)Zω (β) [ζD (β¯ hω; zD ; x)]2 . z 0 (7.189) As in Eq. (7.113), we now construct the variational free energy to be extremized with respect to Rthe local parameter σ(x). Moreover, we shall find it convenient to express ζ(z) as 0z (dz ′ /z ′ )ζ(z ′ ). This leads to the variational expression Z

FGσ

1 = − Zω (β) β

Z

zD

3

dx

"Z

zD eσ(x)

0

dz σ 2 (x) ζD (β¯ hω; z; x) − , z 8˜ a #

(7.190)

where zD = e−(D¯hω/2−µ)β = e−Dβ¯hω/2 z and a˜ ≡ a ˆ leD (¯ hβ)Zω (β) =

a lD (¯ hβ)Zω (β). le (¯ hβ) e

(7.191)

The extremum lies at σ(x) = Σ(x) where by analogy with (7.114): Σ(x) ≡ −4˜ aζD (β¯ hω; zD eΣ(x) ; x).

(7.192)

For a small trap frequency ω, we use the function ζD (β¯ hω; zD ; x) in the approximate form (7.133), written as H. Kleinert, PATH INTEGRALS

545


ζD (β¯ hω; zD ; x) ≡

∞ X

w=1

s

D

ω 2Mβ w −M wβω2 x2 /2 zD e . 2πw

(7.193)

In this approximation, Eq. (7.192) becomes ω≈0

Σ(x) ≈ −4˜ a

∞ X

w=1

s

D

ω 2Mβ w wΣ(x) −M wβω2 x2 /2 z e e . 2πw D

(7.194)

For small a ˜, Σ(x) is also small, so that the factor ewΣ(x) on the right-hand side is close to unity and can be omitted. This will be inserted into the equation for the particle number above Tc : N=

Z

dD x n(x) = Zω (β)

Z

dD x ζ¯D (β¯ hω; zD eΣ(x) ; x).

(7.195)

Recall that in the thermodynamic limit for D > 1 where the phase transition prophω; zD ; x) and ζ¯D (β¯ hω; zD ; x) coincide, due to (7.139) and (7.141). erly exists, ζD (β¯ From (7.195) we may derive the following equation for the critical temperature as a function of zD : Zω (β)

1=

(0)

Zω (βc )

Z

dD x

ζ¯D (β¯ hω; zD eΣ(x) ; x) , (0) ¯ ω; 1) ζ¯D (βc h

(7.196)

where Tc(0) is the critical temperature in the trap without the repulsive interaction. The critical temperature Tc of the interacting system is reached if the second argument of ζ¯D (β¯ hω; zD eΣ(x) ; x) hits the boundary of the unit convergence radius of the expansion (7.130) for x = 0, i.e., if zD eΣ(0) = 1. Thus we find the equation for Tc : 1=

Zω (βc ) (0)

Zω (βc )

Z

dD x

ζ¯D (βc h ¯ ω; eΣc (x)−Σc (0) ; x) , (0) ζ¯D (βc h ¯ ω; 1)

(7.197)

where the subscript of Σc (x) indicates that β in (7.194) has been set equal to βc . In particular, a ˜ contained in Σc (x) is equal to a ˜c ≡ a/ℓc . Since this is small by assumption, we expand the numerator of (7.197) as 1≈

Zω (βc ) (0)

Zω (βc )

(

1+

1 (0) ζ¯D (βc h ¯ ω; 1)

Z

d

D

x ∆ζ¯D (βc(0) h ¯ ω; 1; x)

)

,

(7.198)

where the integral has the explicit small-ω form Z

dD x

∞ X

w=1

v D u u ω 2 Mβ (0) c t

2πw

(0)

e−M wβc

ω 2 x2 /2

ew[Σc (x)−Σc (0)] − 1 .

(7.199)

In the subtracted term we have used the fact that for small ω, ζ¯D (βc(0)h ¯ ω; 1) ≈ ζ¯D (βch ¯ ω; 1) ≈ ζ(D) is independent of β [see (7.139) and (7.141)].

546


Next we approximate near Tc(0) : Zω (βc ) (0) Zω (βc )

≡1+D

∆Tc

βc(0) h ¯ ω/2

(0) Tc

(0) tanh(βc h ¯ ω/2)

ω≈0

≈ 1+D

∆Tc (0)

Tc

,

(7.200)

such that Eq. (7.198) can be solved for ∆Tc /Tc(0) : ∆Tc (0) Tc

≈−

1 1 D ζ(D)

Z

dD x ∆ζ¯D (βc(0) h ¯ ω; 1; x).

(7.201)

On the right-hand side, we now insert (7.199) with the small quantity Σc (x) approximated by Eq. (7.194) at βc(0) , in which the factor ewΣc (x) on the right-hand side is replaced by 1, and find for the integral (7.199): −4˜ ac

Z

D

d x

v D u (0) ∞ u 2 X (0) t ω Mβc −M wβc ω 2 x2 /2

e

2πw

w=1

w

v D u (0) ∞ u 2 X t ω Mβc

w ′ =1

2πw ′

(0)

M w ′ βc ω 2 x2 /2

e

−1 .

The integral leads to

−4˜ ac

v D u u ω 2 Mβ (0) X ∞ c t

2π

1

w,w ′=1 w

D/2−1 w ′D/2

"

#

1 1 − ≡ −4ˆ ac S(D), (7.202) (w + w ′)D/2 w D/2

where S(D) abbreviates the double sum, whose prefactor has been simplifies to −4ˆ ac (0) (0) −D using (7.191) and the fact that for small ω, Zω (βc ) ≈ (ω¯ hβc ) . For D = 3, the double sum has the value S(3) ≈ 1.2076−ζ(2)ζ(3/2) ≈ −3.089. Inserting everything into (7.201), we obtain for small a and small ω the shift in the critical temperature ∆Tc (0) Tc

≈

4ˆ ac S(D) a ≈ − 3.427 . D ζ(D) D=3 ℓc

(7.203)

In contrast to the free Bose gas, where a small δ-function repulsion does not produce any shift using the same approximation as here [recall (7.116)], and only a high-loop calculation leads to an upwards shift proportional to a, the critical temperature of the trappedqBose gas is shifted downwards. We can express ℓc in terms of the length scale λω ≡ h ¯ /Mω associated with the harmonic oscillator [recall Eq. (2.296)] and √ rewrite ℓc = λω 2πβch ¯ ω. Together with the relation (7.142), we find ∆Tc (0) Tc

1 a 1/6 a 1/6 ≈ −3.427 √ N ≈ −1.326 N . λω 2π[ζ(3)]1/6 λω

(7.204)

Note that since ω is small, the temperature shift formula (7.201) can also be expressed in terms of the zero-ω density (7.134) as ∆Tc

ω, a ˜≈0

≈

2g

R

d3 x [∂µ n0 (x)] [n0 (x) − n0 (0)] R , d3 x ∂T n0 (x)

(7.205)


547


where we have omitted the other arguments β and z of n0 (β; z; x), for brevity. To derive this formula we rewrite the grand-canonical free energy (7.190) as " # 1 Z 3 Z zeβν(x) dz ′ ν 2 (x) ′ FG = − dx n0 (β; z ; x) − β , β z′ 4g 0 so that the particle number equation Eq. (7.195) takes the form

N=

Z

(7.206)

d3 x n0 (β; zeβν(x) ; x).

(7.207)

Extremizing FG in ν(x) yields the self-consistent equation ν(x) = −2gn0 (β; zeβν(x) ; x).

(7.208)

As before, the critical temperature is reached for zeβν(0) = 1, implying that N=

Z

d3 x n0 (βc ; ze−2βg[n0 (x)−n0 (0)] ; x).

(7.209)

In the exponent we have omitted again the arguments β and z of n0 (β; z; x). If we now impose the condition of constant N, ∆N = (∆Tc ∂T + ∆µ∂µ )N = 0, and insert ∆µ = −2g[n0 (x) − n0 (0)], we find (7.205). Inserting into (7.205) the density (7.134) for the general trap (7.151), we find the generalization of (7.203): ∆Tc (0)

Tc

∞ X 4ˆ ac 1 1 1 1 ≈ − , ˜ ˜ D/2−1 ′D/2 ˜ ζ(D) ˜ w,w′=1 w w (w + w ′ )D−D/2 w D−D/2 D

"

#

(7.210)

which vanishes for the homogeneous gas, as concluded before on the basis of Eq. (7.116). Let us compare the result (7.204) with the experimental temperature shift for 87 Rb in a trap with a critical temperature Tc ≈ 280 nK which lies about 3% below the noninteracting Bose gas temperature (see Fig. 7.9). Its thermal de Broglie length is calculated best in atomic units. Then the fundamental length scale is the Bohr radius aH = h ¯ /Mp cα, where Mp is the proton mass and α ≈ 1/137.035 the fine-structure constant. The fundamental energy scale is EH = Me c2 α2 . Writing now the thermal de Broglie length at the critical temperature as √ a 2π¯ h2 EH = = 2π ℓc MkB Tc kB Tc s

we estimate with q

q

EH /kB Tc ≈

q

q

s

Me aH , 87Mp

(7.211)

27.21 eV/(280 × 10−9 /11 604.447 eV) ≈ 1.06 × 106

and Me /87Mp ≈ 0.511eV/(87 × 938.27eV) ≈ 0.002502 such that ℓc ≈ 6646aH . The triplet s-wave scattering length of 87 Rb is a ≈ (106 ± 4) aH such that we find from (7.203) ∆Tc ≈ −5.4%, (7.212) (0) Tc which is compatible with the experimentally data of the trap in Fig. 7.9. Let us finally mention recent studies of more realistic systems in which bosons in a trap interact with longer-range interactions [45].

548

7.3


Gas of Free Fermions

For fermions, the thermodynamic functions (7.32) and (7.33) contain the functions ζνf (z) ≡

∞ X

(−1)w−1

w=1

zw , wν

(7.213)

which starts out for small z like z. For z = 1, this becomes ζνf (1) =

∞ ∞ X X 1 1 1 1 1 1 1−ν 1−ν − 2 = 1 − 2 ζ(ν). (7.214) − ν + ν − ν + ... = ν ν 1 2 3 4 k=0 k k=0 k

In contrast to ζν (z) in Eq. (7.34) this function is perfectly well-defined for all chemical potentials by analytic continuation. The reason is the alternating sign in the series (7.213). The analytic continuation is achieved by expressing ζνf (z) as an integral by analogy with (7.39), (7.40): ζnf (z) ≡

1 f i (α), Γ(n) n

ifn (α) ≡

Z

(7.215)

where ifn (α) are the integrals ∞

0

dε

εn−1 . eε−α + 1

(7.216)

In the integrand we recognize the Fermi distribution function of Eq. (3.111), in which ω¯ hβ is replaced by ε − α: 1

nfε =

. (7.217) eε−α + 1 The quantity ε plays the role of a reduced energy ε = E/kB T , and α is a reduced chemical potential α = µ/kB T . Let us also here express the grand-canonical free energy FG in terms of the functions ifn (α). Combining Eqs. (7.32), (7.213), and (7.216) we obtain for fermions with gS = 2S + 1 spin orientations: 1 gS 1 g S VD FG = − Z0 (β) ifD/2+1 (βµ) = − q D β Γ(D/2+1) βΓ(D/2+1) 2π¯ h2 β/M

Z

∞

0

dε

εD/2 , eε−βµ + 1 (7.218)

and the integral can be brought by partial integration to the form FG = −

1 g S VD q D βΓ(D/2) 2π¯ h2 β/M

Z

0

∞

dε εD/2−1 log(1 + e−ε+βµ ).

(7.219)

Recalling Eq. (7.47), this can be rewritten as a sum over momenta of oscillators with energy h ¯ ωp ≡ p2 /2M: gS X FG = − log(1 + e−β¯hωp +βµ ). (7.220) β p H. Kleinert, PATH INTEGRALS

549

7.3 Gas of Free Fermions

This free energy will be studied in detail in Section 7.14. The particle number corresponding to the integral representations (7.219) and (7.220) is g S VD ∂ 1 N = − FG = q D ∂µ Γ(D/2) 2π¯ h2 β/M

Z

∞

0

dε

X εD/2−1 1 = g . (7.221) S ε+βµ β¯ h ω +βµ p e −1 −1 p e

Recalling the reduced density of states, this may be written with the help of the Fermi distribution function nfω of Eq. (7.217) as N = g S VD

Z

∞

0

dε Nε nfε .

(7.222)

The Bose function contains a pole at α = 0 which prevents the existence of a solution for positive α. In the analytically continued fermionic function (7.215), on the other hand, the point α = 0 is completely regular. Consider now a Fermi gas close to zero temperature which is called the degenerate limit. Then the reduced variables ε = E/kB T and α = µ/kB T become very large and the distribution function (7.217) reduces to nfε

=

(

1 εα

)

= Θ(ε − α).

(7.223)

All states with energy E lower than the chemical potential µ are filled, all higher states are empty. The chemical potential µ at zero temperature is called Fermi energy EF : µ ≡ EF . (7.224) T =0

The Fermi energy for a given particle number N in a volume VD is found by performing the integral (7.222) at T = 0: N = g S VD where

Z

0

EF

D/2

1 gS VD EF 1 g S VD dεNε = q √ D D = Γ(D/2+1) 2π¯ Γ(D/2+1) 4π h2 /M pF ≡

q

2MEF

pF h ¯

D

, (7.225)

(7.226)

is the Fermi momentum associated with the Fermi energy. Equation (7.225) is solved for EF by 2π¯ h2 Γ(D/2 + 1) EF = M gS "

#2/D

N VD

2/D

,

(7.227)

1/D

(7.228)

and for the Fermi momentum by √

"

Γ(D/2 + 1) pF = 2 π¯ h gS

#1/D

N VD

h ¯.

550


Note that in terms of the particle number N, the density of states per unit energy interval and volume can be written as dN Nε ≡ 2 VD

s

kB T EF

D/2

εD/2−1 .

(7.229)

As the gas is heated slightly, the degeneracy in the particle distribution function in Eq. (7.223) softens. The degree to which this happens is governed by the size of the ratio kT /EF . It is useful to define a characteristic temperature, the so-called Fermi temperature EF 1 pF 2 = TF ≡ . (7.230) kB kB 2M For electrons in a metal, pF is of the order of h ¯ /1rA. Inserting M = me = 9.109558× −28 −16 10 g, further kB = 1.380622 × 10 erg/K and h ¯ = 6.0545919 × 10−27 erg sec, we see that TF has the order of magnitude TF ≈ 44 000K.

(7.231)

Hence, even far above room temperatures the relation T /TF ≪ 1 is quite well fulfilled and can be used as an expansion parameter in evaluating the thermodynamic properties of the electron gas at nonzero temperature. Let us calculate the finite-T effects in D = 3 dimensions. From Eq. (7.225) we obtain gS V 2 f µ √ i N = N(T, µ) ≡ 3 . (7.232) le (¯ hβ) π 3/2 kB T Expressing the particle number in terms of the Fermi energy with the help of Eq. (7.225), we obtain for the temperature dependence of the chemical potential an equation analogous to (7.54): 1=

kB T EF

!3/2

3 f µ i3/2 2 kB T

.

(7.233)

To evaluate this equation, we write the integral representation (7.216) as (α + x)n−1 ex + 1 −α Z α Z ∞ (α − x)n−1 (α + x)n−1 = dx −x + dx . e +1 ex + 1 0 0 Z

ifn (α) =

∞

dx

(7.234)

where ε = x + α. In the first integral we substitute 1/(e−x + 1) = 1 − 1/(ex + 1), and obtain ifn (α) =

Z

α 0

n−1

dx x

+

Z

0

∞

(α+x)n−1 − (α−x)n−1 Z ∞ (α−x)n−1 dx + dx x . ex + 1 e +1 α

(7.235)


551

7.3 Gas of Free Fermions

In the limit α→∞, only the first term survives, whereas the last term is exponentially small, so that it can be ignored in a series expansion in powers of 1/α. The second term is expanded in such a series: ∞ X n−1

X xk (n−1)! n−1−k (1 − 2−k )ζ(k + 1). = 2 α x+1 e (n−1−k)! k 0 k=odd k=odd (7.236) In the last equation we have used the integral formula for Riemann’s ζ-function4

2

αn−1−k

Z

∞

Z

0

∞

dx

dx

xν−1 = µ−ν (1 − 21−ν )ζ(ν). eµx + 1

(7.237)

At even positive and odd negative integer arguments, the zeta function is related to the Bernoulli numbers by Eq. (2.561). The lowest values of ζ(x) occurring in the expansion (7.236) are ζ(2), ζ(4), . . . , whose values were given in (2.563), so that the expansion of ifn (α) starts out like ifn (α) =

1 n 1 7 α + 2(n−1) ζ(2)αn−2 + 2(n−1)(n−2)(n−3) ζ(4)αn−4 + . . . . (7.238) n 2 8

Inserting this into Eq. (7.233) where n = 3/2, we find the low-temperature expansion 1=

kB T EF

!3/2

"

3 2 µ 2 3 kB T

3/2

µ π2 + 12 kB T

−1/2

µ 7π 4 + 3 · 320 kB T

−5/2

#

... , (7.239)

implying for µ the expansion 

π2 µ = EF 1 − 12

kB T EF

!2

7π 4 + 720

kB T EF

!4



+ . . . .

(7.240)

These expansions are asymptotic. They have a zero radius of convergence, diverging for any T . They can, however, be used for calculations if T is sufficiently small or at all T after a variational resummation à la Section 5.18. We now turn to the grand-canonical free energy FG . In terms of the function (7.216), this reads 1 gS V 1 FG = − 3 if (α). (7.241) β le (¯ hβ) Γ(5/2) 5/2 Using again (7.238), this has the expansion 

5π 2 FG (T, µ, V ) = FG (0, µ, V ) 1 + 8

where

FG (0, µ, V ) 4

kB T µ

!2

7π 4 − 384

kB T µ

!4



+ . . . ,

√ 2 2M 3/2 2 µ 3/2 3/2 ≡ − gS V (µ) µ = − N µ. 5 5 EF 3π 2 h ¯3


(7.242)

552


By differentiating FG with respect to the temperature at fixed µ, we obtain the low-temperature behavior of the entropy S = kB

π 2 kB T N + ... . 2 EF

(7.243)

From this we find a specific heat at constant volume C = T

π 2 kB T ∂S T ≈0 ≈ S = k N + ... . B ∂T V,N 2 EF

(7.244)

This grows linearly with increasing temperature and saturates at the constant value 3kB N/2 which obeys the Dulong-Petit law of Section 2.12 corresponding to three kinetic and no potential harmonic degrees of freedom in the classical Hamiltonian p2 /2M. See Fig. 7.11 for the full temperature behavior. The linear behavior is

Figure 7.11 Temperature behavior of specific heat of free Fermi gas. As in the free Bose gas, the Dulong-Petit rule gives a high-temperature limit 3kB N/2 for the three harmonic kinetic degrees of freedom in the classical Hamiltonian p2 /2M . There are no harmonic potential degrees of freedom.

due to the progressive softening near the surface of the Fermi distribution which makes more and more electrons thermally excitable. It is detected experimentally in metals at low temperature where the contribution of lattice vibrations freezes out as (T /TD )3 . Here TD is the Debye temperature which characterizes the elastic stiffness of the crystal and ranges from TD ≈ 90K in soft metals like lead over TD ≈ 389K for aluminum to TD ≈ 1890K for diamond. The experimental size of the slope is usually larger than the free electron gas value in (7.244). This can be explained mainly by the interactions with the lattice which result in a large effective mass Meff > M. Note that the quantity FG (0, µ, V ) is temperature dependent via the chemical potential µ. Inserting (7.240) into (7.242) we find the complete T -dependence 

5π 2 FG (T, µ, V ) = FG (0, EF , V ) 1 + 12

kB T EF

!2

π4 − 16

kB T EF

!4



+ . . . ,

(7.245)


553

7.4 Statistics Interaction

where

2 FG (0, EF , V ) = − NEF . (7.246) 5 As in the boson gas, we have a relation (7.61) between energy and grandcanonical free energy: 3 (7.247) E = − FG , 2 such that equation (7.245) supplies us with the low-temperature behavior of the internal energy: 

2

3 5π E = NEF 1 + 5 12

kB T EF

!2

4

π 16

−

kB T EF

!4



+ . . . .

(7.248)

The first term is the energy of the zero-temperature Fermi sphere. Using the relation cV = ∂E/V ∂T , the second term yields once more the leading T → 0 -behavior (7.244) of specific heat. This behavior of the specific heat can be observed in metals where the conduction electrons behave like a free electron gas. Due to Bloch’s theorem, a single electron in a perfect lattice behaves just like a free particle. For many electrons, this is still approximately true, if the mass of the electrons is replaced by an effective mass. Another important macroscopic system where (7.244) can be observed is a liquid consisting of the fermionic isotope 3 He. There are two electron spins and an odd number of nucleon spins which make this atom a fermion. Also there the strong interactions in the liquid produce a screening effect which raises to an effective value of the mass to 8 times that of the atom.

7.4

Statistics Interaction

First, we consider only two identical particles; the generalization to n particles will be obvious. For simplicity, we ignore the one-body potentials V (x(ν) ) in (7.2) since they cause only inessential complications. The total orbital action is then A=

Z

tb

ta

M (1) (1)2 M (2) (2)2 dt x˙ + x˙ − Vint (x(1) − x(2) ) . 2 2 "

#

(7.249)

The standard change of variables to center-of-mass and relative coordinates X = (M (1) x(1) + M (2) x(2) )/(M (1) + M (2) ),

x = (x(1) − x(2) ),

(7.250)

respectively, separates the action into a free center-of-mass and a relative action A = ACM + Arel =

Z

tb

ta

M ˙2 dt X + 2

Z

tb

ta

µ dt x˙ 2 − Vint (x) , 2

(7.251)

with a total mass M = M (1) +M (2) and a reduced mass µ = M (1) M (2) /(M (1) +M (2) ). Correspondingly, the time evolution amplitude of the two-body system factorizes

554


into that of an ordinary free particle of mass M, (Xb tb |Xa ta ), and a relative amplitude (xb tb |xa ta ). The path integral for the center-of-mass motion is solved as in Chapter 2. Only the relative amplitude is influenced by the particle statistics and needs a separate treatment for bosons and fermions. First we work in one dimension only. Many of the formulas arising in this case are the same as those of Section 6.2, where we derived the path integral for a particle moving in a half-space x = r > 0; only the interpretation is different. We take care of the indistinguishability of the particles by restricting x to the positive semiaxis x = r ≥ 0; the opposite vector −x describes an identical configuration. The completeness relation of local states reads therefore ∞

Z

0

dr|rihr| = 1.

(7.252)

To write down the orthogonality relation, we must specify the bosonic or fermionic nature of the wave functions. Since these are symmetric or antisymmetric, respectively, we express hrb |ra i in terms of the complete set of functions with these symmetry properties: hrb |ra i = 2

Z

∞

0

dp π¯ h

(

cos prb /¯ h cos pra /¯ h sin prb /¯ h sin pra /¯ h

)

.

(7.253)

This may be rewritten as dp ip(rb −ra )/¯h e ± eip(rb +ra )/¯h = δ(rb − ra ) ± δ(rb + ra ). (7.254) h −∞ 2π¯ The infinitesimal time evolution amplitude of relative motion is then, in the canonical formulation,

hrb |ra i =

Z

∞

ˆ

(rn ǫ|rn−1 0) = hrn |e−iǫHrel /¯h |rn−1 i Z ∞ dp ip(rn −rn−1 )/¯h = e ± eip(rn +rn−1 )/¯h e−iǫHrel (p,rn )/¯h , h −∞ 2π¯

(7.255)

where Hrel (p, x) is the Hamiltonian of relative motion associated with the action Arel in Eq. (7.251). By combining N + 1 factors, we find the time-sliced amplitude (rb tb |ra ta ) = (

"

i × exp h ¯

N Z Y

n=1 N +1 X

∞

0

drn

" NY +1 Z

∞

−∞

n=1

dpn 2π¯ h

#

(7.256)

+1 i NX pn (rn − rn−1 ) ± exp pn (rn + rn−1 ) h ¯ n=1 n=1

#

"

#)

i

e− h¯ ǫ

PN+1 n=1

Hrel (pn ,rn )

,

valid for bosons and fermions, respectively. By extending the radial integral over the entire space it is possible to remove the term after the ±sign by writing (rb tb |ra ta ) =

X

N Z Y

xb =±rb n=1

∞

−∞

dxn

" NY +1 Z n=1

∞

−∞

dpn 2π¯ h

#

(7.257)

+1 i NX × exp [pn (xn − xn−1 ) − ǫHrel (pn , xn ) + h ¯ π(σ(xn ) − σ(xn−1 ))] , h ¯ a=1

(

)


555

7.4 Statistics Interaction

where the function σ(x) vanishes identically for bosons while being equal to σ(x) = Θ(−x)

(7.258)

for fermions, where Θ(x) is the Heaviside function (1.309). As usual, we have identified xb ≡ xN +1 and xa ≡ x0 which is equal to ra . The final sum over xb = ±rb accounts for the indistinguishability of the two orbits. The phase factors eiπσ(xn ) give the necessary minus signs when exchanging two fermion positions. Let us use this formula to calculate explicitly the path integral for a free twoparticle relative amplitude. In the bosonic case with a vanishing σ-term, we simply obtain the free-particle amplitude summed over the final positions ±rb : i µ (rb − ra )2 exp + (rb → −rb ) . (rb tb |ra ta ) = q h ¯ 2 tb − ta 2π¯ hi(tb − ta )/µ (

1

"

#

)

(7.259)

For fermions, the phases σ(xn ) in (7.257) cancel each other successively, except for the boundary term eiπ(σ(xb )−σ(xa )) . (7.260) When summing over xb = ±rb in (7.257), this causes a sign change of the term with xb = −rb and leads to the antisymmetric amplitude i µ (rb − ra )2 (rb tb |ra ta ) = q exp − (rb → −rb ) . h ¯ 2 (tb − ta ) 2π¯ hi(tb − ta )/µ (

1

"

#

)

(7.261)

Let us also write down the continuum limit of the time-sliced action (7.257). It reads Z

A = Arel + Af =

tb

ta

dt [px˙ − Hrel (p, x) + h ¯ π x(t)∂ ˙ x σ(x(t))] .

(7.262)

The last term is the desired Fermi statistics interaction. It can also be written as Af = −¯ hπ

Z

tb

ta

dtx(t)δ(x(t)) ˙ =h ¯π

Z

tb

ta

dt∂t Θ(−x(t)).

(7.263)

The right-hand expression shows clearly the pure boundary character of Af , which does not change the equations of motion. Such an interaction is called a topological interaction. Since the integrals in (7.257) over x and p now cover the entire phase space and σ(x) enters only at the boundaries of the time axis, it is possible to add to the action any potential Vint (r). As long as the ordinary path integral can be performed, also the path integral with the additional σ-terms in (7.257) can be done immediately. It is easy to generalize this result to any number of fermion orbits x(ν) (t), ν = P ′ 1, . . . , n. The statistics interaction is then ν 0.

(7.314)

This is obviously a special case of the matrix formula Z=

"Z N Y

n=0

1 da∗n dan − Pn,m a∗n Anm am e = , π det A #

(7.315)

in which the matrix A = Ad has only diagonal elements with a positive real part. Now we observe that the measure of integration is certainly invariant under any unitary transformation of the components an : an →

X

Un,n′ an′ .

(7.316)

n′

So is the determinant of A: det A → det (U Ad U † ) = det Ad .


565

7.7 Fluctuating Bose Fields

But then formula (7.315) holds for any matrix A which can be diagonalized by a unitary transformation and has only eigenvalues with a positive real part. In the present case, the possibility of diagonalizing A is guaranteed by the fact that A satisfies AA† = A† A, i.e., it is a normal matrix. This property makes the Hermitian and anti-Hermitian parts of A commute with each other, allowing them to be diagonalized simultaneously. In the partition function (7.309), the (N + 1) × (N + 1) matrix A has the form 

1 0 0  1 0  −1 + ǫω   0 −1 + ǫω 1  A = ǫ(1 − ǫω)∇ + ǫω =  0 0 −1 + ǫω   ..  .  0

0

0

... ... ... ...

0 0 0 0

. . . −1 + ǫω

−1 + ǫω  0    0  . 0   ..  .  

1 (7.318)

This matrix acts on a complex vector space. Its determinant can immediately be calculated by a repeated expansion along the first row, giving detN +1 A = 1 − (1 − ǫω)N +1 .

(7.319)

Hence we obtain the time-sliced partition function ZωN =

1 1 = . 1 − (1 − ǫω)N +1 detN +1 [ǫ(1 − ǫω)∇ + ǫω]

(7.320)

It is useful to introduce the auxiliary frequency 1 ω ¯ e ≡ − log(1 − ǫω). ǫ

(7.321)

The subscript e records the Euclidean nature of the time [in analogy with the frequencies ω ˜ e of Eq. (2.391)]. In terms of ω ¯ e , ZωN takes the form ZωN =

1 . 1 − e−β¯hω¯ e

(7.322)

This is the well-known partition function of Bose particles for a single state of energy ω ¯ e . It has the expansion Zω = 1 + e−β¯hω¯ e + e−2β¯hω¯e + . . . ,

(7.323)

in which the nth term exhibits the Boltzmann factor for an occupation of a particle state by n particles, in accordance with the Hamiltonian operator ˆω = h ˆ =h H ¯ω ¯eN ¯ω ¯ e a† a.

(7.324)

566


In the continuum limit ǫ → 0, the auxiliary frequency tends to ω, ǫ→0

−−→ ω, ω ¯e −

(7.325)

and ZωN reduces to

1 . (7.326) 1 − e−β¯hω The generalization of the partition function to a system with a time-dependent frequency Ω(τ ) reads Zω =

ZωN

"Z N Y

=

n=0

da†n dan 1 exp − AN , π h ¯ #

(7.327)

with the sliced action ¯ AN ω =h

N X

n=1

[a∗n (an − an−1 ) + ǫΩn a∗n an−1 ] ,

(7.328)

or, expressed in terms of the difference operator ∇, AN ¯ǫ ω = h

N X

n=1

h

i

a∗n (1 − ǫΩn )∇ + Ωn an .

(7.329)

The result is ZωN =

1 1 = . QN detN +1 [ǫ(1 − ǫΩ)∇ + ǫΩ] 1 − n=0 (1 − ǫΩn )

(7.330)

Here we introduce the auxiliary frequency ¯e ≡ − Ω

N X 1 log(1 − ǫΩn ), (N + 1)ǫ n=0

(7.331)

which brings ZωN to the form ZωN =

1 . 1 − e−β¯hΩ¯ e

(7.332)

For comparison, let us also evaluate the path integral directly in the continuum limit. Then the difference operator (7.318) becomes the differential operator (1 − ǫω)∇ + ω → ∂τ + ω,

(7.333)

acting on periodic complex functions e−iωm τ with the Matsubara frequencies ωm . Hence the continuum partition function of a harmonic oscillator could be written as Zω

Da∗ Da = exp − π 1 = Nω . det (∂τ + ω) I

"

Z

0

hβ ¯

∗

∗

dτ (a ∂τ a + ωa a)

#


567

7.7 Fluctuating Bose Fields

The normalization constant is fixed by comparison with the time-sliced result. The operator ∂τ + ω has the eigenvalues −iωm + ω. The product of these is calculated by considering the ratios with respect to the ω = 0 -values ∞ Y

−iωm + ω sinh(¯ hωβ/2) . = −iωm h ¯ ωβ/2 m=−∞,6=0

(7.335)

This product is the ratio of functional determinants det (∂τ + ω) sinh(¯ hωβ/2) =ω , ′ det (∂τ ) h ¯ ωβ/2

(7.336)

where the prime on the determinant with ω = 0 denotes the omission of the zero frequency ω0 = 0 in the product of eigenvalues; the prefactor ω accounts for this. Note that this ratio formula of continuum fluctuation determinants gives naturally only the harmonic oscillator partition function (7.306), not the secondquantized one (7.322). Indeed, after fixing the normalization factor Nω in (7.334), the path integral in the continuum formulation can be written as Zω

Z ¯hβ Da∗ Da = exp − dτ (a∗ ∂τ a + ωa∗ a) π 0 ′ kB T det (∂τ ) 1 = = . h ¯ det (∂τ + ω) 2 sinh(¯ hωβ/2) "

I

#

(7.337)

In the continuum, the relation with the oscillator fluctuation factor can be established most directly by observing that in the determinant, the operator ∂τ +ω can be replaced by the conjugate operator −∂τ + ω, since all eigenvalues come in complexconjugate pairs, except for the m = 0 -value, which is real. Hence the determinant of ∂τ + ω can be substituted everywhere by det (∂τ + ω) = det (−∂τ + ω) = rewriting the partition function (7.337) as Zω

q

det (−∂τ2 + ω 2),

(7.338)

kB T det ′ (∂τ ) = h ¯ det (∂τ + ω) kB T det ′ (−∂τ2 ) = h ¯ det (−∂τ2 + ω 2 ) "

#1/2

=

1 , 2 sinh(¯ hωβ/2)

(7.339)

where the second line contains precisely the oscillator expressions (2.388). A similar situation holds for an arbitrary time-dependent frequency where the partition function is Zω =

I

Da∗ (τ )Da(τ ) exp − π (

Z

kB T det ′ (−∂τ2 ) = h ¯ det (−∂τ2 + Ω2 (τ )) "

hβ ¯

h

†

†

i

dτ a ∂τ a + Ω(τ )a a

0

#1/2

)

1 det (−∂τ2 + ω 2 ) = 2sinh(¯ hωβ/2) det (−∂τ2 + Ω2 (τ )) "

#1/2

. (7.340)

568


While the oscillator partition function can be calculated right-away in the continuum limit after forming ratios of eigenvalues, the second-quantized path integral depends sensitively on the choice a†n an−1 in the action (7.310). It is easy to verify that the alternative slicings a†n+1 an and a†n an would have led to the partition functions [eβ¯hω − 1]−1 and [2 sinh(¯ hωβ/2)]−1, respectively. The different time slicings produce obviously the same physics as the corresponding time-ordered Hamiltonian operators ˆ = Tˆ a H ˆ† (t)ˆ a(t′ ) in which t′ approaches t once from the right, once from the left, and once symmetrically from both sides. It is easy to decide which of these mathematically possible approaches is the physically correct one. Classical mechanics is invariant under canonical transformations. Thus we require that path integrals have the same invariance. Since the classical actions (7.300) and (7.301) arise from oscillator actions by the canonical transformation (7.296), the associated partition functions must be the same. This fixes the time-slicing to the symmetric one. Another argument in favor of this symmetric ordering was given in Subsection 2.15.4. We shall see that in order to ensure invariance of path integrals under coordinate transformations, which is guaranteed in Schrödinger theory, path integrals should be defined by dimensional regularization. In this framework, the symmetric fixing emerges automatically. It must, however, be pointed out that the symmetric fixing gives rise to an important and poorly understood physical problem in many-body theory. Since each harmonic oscillator in the world has a ground-state energy ω, each momentum state of each particle field in the world possesses a nonzero vacuum energy h ¯ ω (thus for each element in the periodic system). This would lead to a divergence in the cosmological constant, and thus to a catastrophic universe. So far, the only idea to escape this is to imagine that the universe contains for each Bose field a Fermi field which, as we shall see in Eq. (7.427), contributes a negative vacuum energy to the ground state. Some people have therefore proposed that the world is described by a theory with a broken supersymmetry, where an underlying supersymmetric action contains fermions and bosons completely symmetrically. Unfortunately, all theories proposed so far possess completely unphysical particle spectra.

7.8

Coherent States

As long as we calculate the partition function of the harmonic oscillator in the variables a∗ (τ ) and a(τ ), the path integrals do not differ from those of the harmonicoscillator (except for the possibly absent ground-state energy). The situation changes if we want to calculate the path integral (7.334) for specific initial and final values aa = a(τa ) and ab = a(τb ), implying also a∗a = a∗ (τa ) and a∗b = a∗ (τb ) by complex conjugation. In the definition of the canonical path integral in Section 2.1 we had to choose between measures (2.44) and (2.45), depending on which of the two completeness relations Z

dx |xihx| = 1,

Z

dp |pihp| = 1 2π


569

7.8 Coherent States

we wanted to insert into the factorized operator version of the Boltzmann factor ˆ ˆ e−β H into products of e−ǫH . The time-sliced path integral (7.309), on the other hand, runs over a∗ (τ ) and a(τ ) corresponding to an apparent completeness relation Z

dp |x pihx p| = 1. 2π

dx

(7.342)

This resolution of the identity is at first sight surprising, since in a quantummechanical system either x or p can be specified, but not both. Thus we expect (7.342) to be structurally different from the completeness relations in (7.341). In fact, (7.342) may be called an overcompleteness relation. In order to understand this, we form coherent states [49] similar to those used earlier in Eq. (3A.5) [49]: † −z ∗ a ˆ

|zi ≡ ezâ

† +z ∗ a ˆ

hz| ≡ h0|e−zâ

|0i,

.

(7.343)

The Baker-Campbell-Hausdorff formula (2.9) allows us to rewrite † −z ∗ a ˆ

ezâ

= ez

∗ z[ˆ a† ,ˆ a]/2

†

ezâ e−z

∗a ˆ

= e−z

∗ z/2

†

∗

ezâ e−z aˆ .

(7.344)

Since a ˆ annihilates the vacuum state, we may expand |zi = e−z

∗ z/2

†

ezâ |0i = e−z

∗ z/2

∞ X

zn √ |ni. n! n=0

(7.345)

The states |ni and hn| can be recovered from the coherent states |zi and hz| by the operations:

z ∗ z/2

|ni = |zie

← n ∂z

1 √ , z=0 n!

1 ∗ hn| = √ ∂zn∗ ez z/2 hz| n!

.

(7.346)

z=0

ˆ the trace can be calculated from the integral over the diagonal For an operator O, elements ˆ= tr O

dz ∗ dz ˆ = hz|O|zi π

Z

Z

∞ dz ∗ dz −z ∗ z X z∗ m zn ˆ √ √ hm|O|ni. e π m! n! m,n=0

(7.347)

Setting z = reiφ , this becomes ˆ= tr O

Z

dr

2

"

∞ (m+n)/2 dφ −i(m−n)φ −r2 X 1 1 ˆ √ √ hm|O|ni. e r2 e 2π m! n! m,n=0

#

(7.348)

The integral over φ gives a Kronecker symbol δm,n and the integral over r 2 cancels the factorials, so that we remain with the diagonal sum ˆ= tr O

Z

dr 2 e−r

2

∞ X

n=0

r2

n

∞ X 1 ˆ ˆ hn|O|ni = hn|O|ni. n! n=0

(7.349)

570


The sum on the right-hand side of (7.345) allows us to calculate immediately the scalar product of two such states: ∗

∗

∗

hz1 |z2 i = e−z1 z1 /2−z2 z2 /2+z1 z2 .

(7.350)

We identify the states in formula (7.342) with these coherent states: √ |x pi ≡ |zi, where z ≡ (x + ip)/ 2.

(7.351)

Then (7.345) can be written as −(x2 +p2 )/4

|x pi = e

∞ X

(x + ip)n √ |0i, 2n n! n=0

(7.352)

and Z ∞ X dp dp (x − ip)m (x + ip)n 2 2 √ √ dx |x pihx p| = dx |x pihx p|e−(x +p )/2 |mihn|. 2π 2π 2m m! 2n n! m,n=0 (7.353) √ Setting (x − ip)/ 2 ≡ reiφ , this can be rewritten as Z

Z

∞ (m+n)/2 1 dp dφ −i(m−n)φ −r2 X 1 √ √ |mihn|. dx e r2 |x pihx p| = dr 2 e 2π 2π m! n! m,n=0

Z

"

#

(7.354)

The angular integration enforces m = n, and the integrals over r 2 cancel the factorials, as in (7.348), thus proving the resolution of the identity (7.342), which can also be written as Z dz ∗ dz |zihz| = 1. (7.355) π This resolution of the identity can now be inserted into a product decomposition of a Boltzmann operator ˆ

ˆ

ˆ

ˆ

hzb |e−β Hω |za i = hzb |e−β Hω /(N +1) e−β Hω /(N +1) · · · e−β Hω /(N +1) |za i,

(7.356)

to arrive at a sliced path integral [compare (2.2)–(2.4)] ˆω −β H

hzb |e

|za i =

"Z N Y

n=1

dzn∗ dzn π

# N +1 Y n=1

ˆ

hzn |e−ǫHω |zn−1 i, z0 = za , zN +1 = zb , ǫ ≡ β/(N +1). (7.357) ˆω −ǫH

We now calculate the matrix elements hzn |e

|zn−1 i and find

ˆ

ˆ ω |zn−1 i = hzn |zn−1 i − ǫhzn |H ˆ ω |zn−1 i. hzn |e−ǫHω |zn−1 i ≈ hzn |1 − ǫH

(7.358)

Using (7.350) we find hzn |zn−1 i = e−zn zn /2−zn−1 zn−1 /2+zn zn−1 = e−(1/2)[zn (zn −zn−1 )−(zn −zn−1 )zn−1 ] . ∗

∗

∗

∗

∗

∗

(7.359)


571

7.8 Coherent States

The matrix elements of the operator Hamiltonian (7.298) is easily found. The coherent states (7.345) are eigenstates of the annihilation operator aˆ with eigenvalue z: ∞ ∞ X X zn zn −z ∗ z/2 −z ∗ z/2 q √ a ˆ|zi = e aˆ|ni = e |n − 1i = z|zi. (7.360) n! (n − 1)! n=0 n=1 Thus we find immediately

1 ˆ ω |zn−1 i = h ¯ ωhzn |(ˆ a† a ˆ+a ˆ aˆ† )|zn−1 i = h ¯ ω zn† zn−1 + . (7.361) hzn |H 2 Inserting this together with (7.359) into (7.358), we obtain for small ǫ the path integral "Z # N Y dzn∗ dzn −ANω [z ∗ ,z]/¯h ˆω −β H hzb |e |za i = e , (7.362) π n=1 with the time-sliced action N +1 h i X 1 ∗ 1 N ∗ ∗ ∗ ¯ǫ zn ∇zn − (∇zn )zn−1 + ω zn zn−1 + . (7.363) Aω [z , z] = h 2 n=1 2

The gradient terms can be regrouped using formula (2.35), and rewriting its rightP +1 hand side as pN +1 xN +1 − p0 x0 + N n=1 (pn − pn−1 )xn−1 . This leads to ∗ AN ω [z , z]

N +1 X h ¯ 1 = (−zb∗ zb + za∗ za ) + h ¯ǫ zn∗ ∇zn + ω zn∗ zn−1 + 2 2 n=1

.

(7.364)

Except for the surface terms which disappear for periodic paths, this action agrees with the time-sliced Euclidean action (7.310), except for a trivial change of variables a → z. As a brief check of formula (7.362) we set N = 0 and find h ¯ 1 0 ∗ ∗ ∗ ∗ ∗ Aω [z , z] = (−zb zb + za za ) + h ¯ zb (zb − za ) + ω zb za + , (7.365) 2 2 and the short-time amplitude (7.364) becomes 1 1 ˆ hzb |e−ǫHω |za i = exp − (zb∗ zb + za∗ za ) + zb∗ za − ǫ¯ hω zb∗ za + . (7.366) 2 2 Applying the recovery operations (7.346) we find ˆ

h

∗

∗

ˆ

h0|e−ǫHω |0i = e(zb zb +za za )/2 hzb |e−ǫHω |za i ˆ

h1|e−ǫHω |1i = ˆ

i

= e−ǫ¯hω/2 , z ∗ =0,z=0 h i← ˆω (zb∗ zb +za∗ za )/2 −ǫH ∂z e hzb |e |za i ∂ ∗z = z ∗ =0,z=0 ˆω −ǫH

h0|e−ǫHω |1i = h1|e

|0i = 0.

(7.367) e−ǫ3¯hω/2 ,

(7.368) (7.369)

Thus we have shown that for fixed ends, the path integral gives the amplitude for an initial coherent state |za i to go over to a final coherent state |zb i. The partition function (7.337) is obtained from this amplitude by forming the diagonal integral Z dz ∗ dz ˆ Zω = hz|e−β Hω |zi. (7.370) π

572

7.9


Second-Quantized Fermi Fields

The existence of the periodic system of elements is based on the fact that electrons can occupy each orbital state only once (counting spin-up and -down states separately). Particles with this statistics are called fermions. In the above Hilbert space in which n-particle states at a point x are represented by oscillator states |n, xi, this implies that the particle occupation number n can take only the values n = 0 (no electron), n = 1 (one electron). It is possible to construct such a restricted many-particle Hilbert space explicitly ˆ by subjecting the quantized fields ψˆ† (x), ψ(x) or their Fourier components aˆ†p , a ˆp to anticommutation relations, instead of the commutation relations (7.282), i.e., by postulating ˆ t), ψˆ† (x′ , t)]+ = δxx′ , [ψ(x, [ψˆ† (x, t), ψˆ† (x′ , t)]+ = 0, ˆ t), ψ(x ˆ ′ , t)]+ = 0, [ψ(x,

(7.371)

or for the Fourier components [ˆ ap (t), a ˆ†p′ (t)]+ = δpp′ , [ˆ a†p (t), a ˆ†p′ (t)]+ = 0, [ˆ ap (t), a ˆp′ (t)]+ = 0.

(7.372)

ˆ ˆ B] ˆ + denotes the anticommutator of the operators Aˆ and B Here [A, ˆ B] ˆ + ≡ AˆB ˆ +B ˆ A. ˆ [A,

(7.373)

Apart from the anticommutation relations, the second-quantized description of Fermi fields is completely analogous to that of Bose fields in Section 7.6.

7.10

Fluctuating Fermi Fields

The question arises as to whether it is possible to find a path integral formulation which replaces the anticommuting operator structure. The answer is affirmative, but at the expense of a somewhat unconventional algebraic structure. The fluctuating paths can no longer be taken as c-numbers. Instead, they must be described by anticommuting variables.

7.10.1

Grassmann Variables

Mathematically, such objects are known under the name of Grassmann variables. They are defined by the algebraic property θ1 θ2 = −θ2 θ1 ,


573

7.10 Fluctuating Fermi Fields

which makes them nilpotent: θ2 = 0.

(7.375)

These variables have the curious consequence that an arbitrary function of them possesses only two Taylor coefficients, F0 and F1 , F (θ) = F0 + F1 θ.

(7.376)

They are obtained from F (θ) as follows: F0 = F (0),

(7.377)

F1 = F ′ ≡

∂ F. ∂θ

The existence of only two parameters in F (θ) is the reason why such functions naturally collect amplitudes of two local fermion states, F0 for zero occupation, F1 for a single occupation. It is now possible to define integrals over functions of these variables in such a way that the previous path integral formalism remains applicable without a change in the notation, leading to the same results as the second-quantized theory with anticommutators. Recall that for ordinary real functions, integrals are linear functionals. We postulate this property also for integrals with Grassmann variables. Since an arbitrary function of a Grassmann variable F (θ) is at most linear in θ, its integral isR completely determined by specifying only the two fundamental integrals R dθ and dθ θ. The values which render the correct physics with a conventional path integral notation are dθ √ = 0, 2π Z dθ √ θ = 1. 2π Z

(7.378) (7.379)

Using the linearity property, an arbitrary function F (θ) is found to have the integral dθ √ F (θ) = F1 = F ′ . (7.380) 2π Thus, integration of F (θ) coincides with differentiation. This must be remembered whenever Grassmann integration variables are to be changed: The integral is transformed with the inverse of the usual Jacobian. The obvious equation Z

Z

dθ √ F (c · θ) = c · F ′ = c · 2π

Z

dθ′ √ F (θ′ ) 2π

(7.381)

#−1

(7.382)

for any complex number c implies the relation Z

dθ √ F (θ′ (θ)) = 2π

Z

dθ′ dθ √ 2π dθ′ "

F (θ′ ).

574


For ordinary integration variables, the Jacobian dθ/dθ′ would appear without the power −1. When integrating over a product of two functions F (θ) and G(θ), the rule of integration by parts holds with the opposite sign with respect to that for ordinary integrals: " # Z Z dθ ∂ dθ ∂ √ G(θ) F (θ) = √ G(θ) F (θ). (7.383) ∂θ 2π 2π ∂θ There exists a simple generalization of the Dirac δ-function to Grassmann variables. We shall define this function by the integral identity Z

dθ′ √ δ(θ − θ′ )F (θ′ ) ≡ F (θ). 2π

(7.384)

Inserting the general form (7.376) for F (θ), we see that the function δ(θ − θ′ ) = θ′ − θ

(7.385)

satisfies (7.384). Note that the δ-function is a Grassmann variable and, in contrast to Dirac’s δ-function, antisymmetric. Its derivative has the property δ ′ (θ − θ′ ) ≡ ∂θ δ(θ − θ′ ) = −1.

(7.386)

It is interesting to see that δ ′ shares with Dirac’s δ ′ the following property: Z

dθ′ √ δ ′ (θ − θ′ )F (θ′ ) = −F ′ (θ), 2π

(7.387)

with the opposite sign of the Dirac case. This follows from the above rule of partial integration, or simpler, by inserting (7.386) and the explicit decomposition (7.376) for F (θ). The integration may be extended to complex Grassmann variables which are combinations of two real Grassmann variables θ1 , θ2 : 1 a∗ = √ (θ1 − iθ2 ), 2 The measure of integration is defined by Z

da∗ da ≡ π

Z

1 a = √ (θ1 + iθ2 ). 2

dθ2 dθ1 ≡− 2πi

Z

(7.388)

da da∗ . π

(7.389)

Using (7.378) and (7.379) we see that the integration rules for complex Grassmann variables are da∗ da da∗ da = 0, a = 0, π π Z Z da∗ da ∗ dθ2 dθ1 aa = iθ1 θ2 = 1. π 2πi Z

Z

Z

da∗ da ∗ a = 0, π

(7.390) (7.391)


575


Every function of a∗ a has at most two terms: F (a∗ a) = F0 + F1 a∗ a.

(7.392)

In particular, the exponential exp{−a∗ Aa} with a complex number A has the Taylor series expansion ∗ e−a Aa = 1 − a∗ Aa. (7.393)

Thus we find the following formula for the Gaussian integral: Z

da∗ da −a∗ Aa e = A. π

(7.394)

The integration rule (7.390) can be used directly to calculate the Grassmann version of the product of integrals (7.315). For a matrix A which can be diagonalized by a unitary transformation, we obtain directly f

Z =

" Y Z n

da∗n dan iΣn,n′ a∗n An,n′ an′ e = det A. π #

(7.395)

Remarkably, the fermion integration yields precisely the inverse of the boson result (7.315).

7.10.2

Fermionic Functional Determinant

Consider now the time-sliced path integral of the partition function written like (7.309) but with fermionic anticommuting variables. In order to find the same results as in operator quantum mechanics it is necessary to require the anticommuting Grassmann fields a(τ ), a∗ (τ ) to be antiperiodic on the interval τ ∈ (0, h ¯ β), i.e., a(¯ hβ) = −a(0),

(7.396)

aN +1 = −a0 .

(7.397)

or in the sliced form Then the exponent of (7.395) has the same form as in (7.315), except that the matrix A of Eq. (7.398) is replaced by 

1 0 0  1 0  −1 + ǫω   0 −1 + ǫω 1  f A = ǫ(1 − ǫω)∇τ + ǫω =  0 0 −1 + ǫω   .  ..  0

0

0

... ... ... ...

0 0 0 0

. . . −1 + ǫω

1 − ǫω 0    0   , 0    ..  .  

1 (7.398)

where the rows and columns are counted from 1 to N + 1. The element in the upper right corner is positive and thus has the opposite sign of the bosonic matrix

576


in (7.318). This makes an important difference: While for ω = 0 the bosonic matrix gave det (−ǫ∇)ω=0 = 0, (7.399) due to translational invariance in τ , we now have det (−ǫ∇)ω=0 = 2.

(7.400)

The determinant of the fermionic matrix (7.398) can be calculated by a repeated expansion along the first row and is found to be detN +1 A = 1 + (1 − ǫω)N +1 .

(7.401)

Hence we obtain the time-sliced fermion partition function Zωf,N = detN +1 [ǫ(1 − ǫω)∇ + ǫω] = 1 + (1 − ǫω)N +1 .

(7.402)

As in the boson case, we introduce the auxiliary frequency 1 ω ¯ e ≡ − log(1 − ǫω) ǫ

(7.403)

ZωN = 1 + e−β¯hω¯e .

(7.404)

and write Zωf,N in the form

This partition function displays the typical property of Fermi particles. There are only two terms, one for the zero-particle and one for the one-particle state at a point. Their energies are 0 and h ¯ω ¯ e , corresponding to the Hamiltonian operator ˆω = h ˆ =h H ¯ω ¯eN ¯ω ¯ e a† a.

(7.405)

In the continuum limit ǫ → 0, where ω ¯ e → ω, the partition function Zω N goes over into Zω = 1 + e−β¯hω . (7.406) Let us generalize also the fermion partition function to a system with a timedependent frequency Ω(τ ), where it reads Zωf,N

=

"Z N Y

n=0

da∗n dan 1 exp − AN , π h ¯ ω #

(7.407)

with the sliced action AN ¯ ω =h

N X

n=1

[a∗n (an − an−1 ) + ǫΩn a∗n an−1 ] ,

(7.408)

or, expressed in terms of the difference operator ∇, AN ¯ǫ ω = h

N X

n=1

h

i

a∗n (1 − ǫΩn )∇ + Ωn an .

(7.409)


577


The result is Zωf,N

= detN +1 [ǫ(1 − ǫΩ)∇ + ǫω] = 1 −

N Y

(1 − ǫΩn ).

(7.410)

n=0

As in the bosonic case, it is useful to introduce the auxiliary frequency ¯e ≡ − Ω

N X 1 log(1 − ǫΩn ), (N + 1)ǫ n=0

and write Zωf,N in the form

(7.411)

¯

Zωf,N = 1 + e−β¯hΩe .

(7.412)

If we attempt to write down a path integral formula for fermions directly in the continuum limit, we meet the same phenomenon as in the bosonic case. The difference operator (7.398) turns into the corresponding differential operator (1 − ǫω)∇ + ω → ∂τ + ω,

(7.413) f

which now acts upon periodic complex functions e−iωm τ with the odd Matsubara frequencies f = π(2m + 1)kB T /¯ h, m = 0, ±1, ±2, . . . . (7.414) ωm The continuum partition function can be written as a path integral Zωf

Da∗ Da = exp − π = Nω det (∂τ + ω), "

I

Z

0

hβ ¯

∗

∗

dτ (a ∂τ a + ωa a)

#

(7.415)

with some normalization constant Nω determined by comparison with the timesliced result. To calculate Zωf , we take the eigenvalues of the operator ∂τ + ω, which f are now −iωm + ω, and evaluate the product of ratios ∞ Y

f −iωm +ω = cosh(¯ hωβ/2). f −iωm m=−∞

(7.416)

This corresponds to the ratio of functional determinants det (∂τ + ω) = cosh(¯ hωβ/2). det (∂τ )

(7.417)

In contrast to the boson case (7.336), no prime is necessary on the determinant of ∂τ since there is no zero frequency in the product of eigenvalues (7.416). Setting Nω = 1/2det (∂τ ), the ratio formula produces the correct partition function Zωf = 2cosh(¯ hωβ/2).

(7.418)

578


Thus we may write the free-fermion path integral in the continuum form explicitly as follows: Da∗ Da = exp − π = 2 cosh(¯ hωβ/2). "

I

Zωf

Z

hβ ¯

0

∗

#

∗

dτ (a ∂τ a + ωa a) = 2

det (∂τ + ω) det (∂τ ) (7.419)

The determinant of the operator ∂τ + ω can again be replaced by det (∂τ + ω) = det (−∂τ + ω) =

q

det (−∂τ2 + ω 2 ).

(7.420)

As in the bosonic case, this Fermi analog of the harmonic oscillator partition function agrees with the results of dimensional regularization in Subsection 2.15.4 which will ensure invariance of path integrals under a change of variables, as will be seen in Section 10.6. The proper fermionic time-sliced partition function corresponding to the dimensional regularization in Subsection 2.15.4 is obtained from a fermionic version of the time-sliced oscillator partition function by evaluating h

¯ + ǫ2 ω 2) Zωf,N = detN +1 (−ǫ2 ∇∇ ≡

N Y

m=0

h

2(1 −

f cos ωm ǫ)

2

i1/2

+ǫ ω

=

N h Y

¯ f + ǫ2 ω 2 ǫ2 Ωfm Ω m

m=0

i 2 1/2

=

N Y

m=0

i1/2

f ǫωm + ǫ2 ω 2 2 sin 2

"

2

#1/2

,

(7.421)

f . The result is with a product over the odd Matsubara frequencies ωm

Zωf,N = 2cosh(¯ hω ˜ e β),

(7.422)

sinh(˜ ωe /2) = ǫω/2.

(7.423)

with ω ˜ e given by This follows from the Fermi analogs of the product formulas (2.392), (2.394):7 N/2−1

Y

m=0 (N −1)/2

Y

m=0



sin2 x

1 −

sin2 (2m+1)π 2(N +1)

1 −

sin2



sin2 x (2m+1)π 2(N +1)



=



= cos(N + 1)x,

 

cos(N + 1)x , cos x

N = even,

(7.424)

N = odd.

(7.425)

For odd N, where all frequencies occur twice, we find from (7.425) that N Y

m=0 7



1 −

sin2 x sin2 (2m+1)π 2(N +1)

1/2 

= cos(N + 1)x,

(7.426)

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 1.391.2, 1.391.4. H. Kleinert, PATH INTEGRALS

579


and thus, with (7.423), directly (7.422). For even N, where the frequency with m = N/2 occurs only once, formula (7.424) gives once more the same answer, thus proving (7.426) for even and odd N. There exists no real fermionic oscillator action since x2 and x˙ 2 would vanish identically for fermions, due to the nilpotency (7.375) of Grassmann variables. The product of eigenvalues in Eq. (7.421) emerges naturally from a path integral in which the action (7.408) is replaced by a symmetrically sliced action. An important property of the partition function (7.418) of (7.422) is that the ground-state energy is negative: E (0) = −

h ¯ω . 2

(7.427)

As discussed at the end of Section 7.7, such a fermionic vacuum energy is required for each bosonic vacuum energy to avoid an infinite vacuum energy of the world, which would produce an infinite cosmological constant, whose experimentally observed value is extremely small.

7.10.3

Coherent States for Fermions

For the bosonic path integral (7.304) we have studied in Section 7.8, the case that the endpoint values aa = a(τa ) and ab = a(τb ) of the paths a(τ ) are held fixed. ˆ The result was found to be the matrix element of the Boltzmann operator e−β Hω ∗ † between coherent states |ai = e−a a/2 eaâ |0i [recall (7.345)]. There exists a similar interpretation for the fermion path integral (7.415) if we hold the endpoint values aa = a(τa ) and ab = a(τb ) of the Grassmann paths fixed. By analogy with Eq. (7.345) we introduce coherent states [50] |ζi ≡ e−ζ

∗ ζ/2

†

ea ζ |0i = e−ζ

The corresponding adjoint states read hζ| ≡ e−ζ

∗ ζ/2

h0|eζ

∗a

= e−ζ

|0i − ζ|1i .

(7.428)

(7.429)

∗ ζ/2

∗ ζ/2

h0| + ζ ∗ h1| .

Note that for consistency of the formalism, the Grassmann elements ζ anticommute with the fermionic operators. The states |0i and h1| and their conjugates h0| and h1| can be recovered from the coherent states |ζi and hζ| by the operations:

ζ ∗ ζ/2

|ni = |ζie

← n ∂ζ

1 √ , ζ=0 n!

1 ∗ hn| = √ ∂ζn∗ eζ ζ/2 hζ| n!

.

(7.430)

ζ=0

These formula simplify here to

ζ ∗ ζ/2

|0i = |ζie |1i =

,

ζ=0 ← ∗ |ζieζ ζ/2 ∂ ζ , ζ=0

ζ ∗ ζ/2

h0| = e

h1| = ∂ζ ∗ eζ

hζ|

∗ ζ/2

,

(7.431)

ζ=0

hζ|

ζ=0

.

(7.432)

580


ˆ the trace can be calculated from the integral over the antidiFor an operator O, agonal elements ˆ= tr O

Z

dζ ∗dζ ˆ h−ζ|O|ζi = π

Z

dζ ∗dζ −ζ ∗ ζ ˆ |0i − ζ|1i . (7.433) e h0| − ζ ∗ h1| O π

Using the integration rules (7.390) and (7.391), this becomes ˆ = h0|O|0i ˆ + h1|O|1i. ˆ tr O

(7.434)

The states |ζi form an overcomplete set in the one-fermion Hilbert space. The scalar products are [compare (7.350)]: ∗

∗

∗

hζ1 |ζ2 i = e−ζ1 ζ1 /2−ζ2 ζ2 /2+ζ1 ζ2 ∗

∗

(7.435)

∗

= e−ζ1 (ζ1 −ζ2 )/2+(ζ1 −ζ2 )ζ2 /2 .

The resolution of the identity (7.355) is now found as follows [recall (7.390)]: Z i dζ ∗dζ dζ ∗ dζ −ζ ∗ ζ h |ζihζ| = e |0ih0| − ζ|1ih0| + ζ ∗|0ih1| π π Z i dζ ∗dζ h = |0ih0| + |1ih0|ζ + ζ ∗|0ih1| + ζζ ∗ |0ih0| + |1ih0| = 1. (7.436) π We now insert this resolution of the identity into the product of Boltzmann factors Z

ˆ

ˆ

ˆ

ˆ

hζb |e−β Hω |ζai = hζb |e−ǫHω e−ǫHω · · · e−ǫHω |ζa i

(7.437)

where ǫ ≡ β/(N + 1), and obtain by analogy with (7.362) the time-sliced path integral "Z # N Y dzn∗ dzn −Ae [z ∗ ,z]/¯h ˆω −β H hzb |e |za i = e , (7.438) π n=1 with the a time-sliced action similar to the bosonic one in (7.364): ∗ AN ω [ζ , ζ]

N +1 X h ¯ 1 = (−ζb∗ ζb + ζa∗ζa ) + h ¯ǫ ζn∗∇ζn + ω ζn∗ ζn−1 + 2 2 n=1

.

(7.439)

Except for the surface term which disappears for antiperiodic paths, this agrees with the time-sliced Euclidean action (7.310), except for a trivial change of variables a → ζ. We have shown that as in the Bose case the path integral with fixed ends gives the amplitude for an initial coherent state |ζa i to go over to a final coherent state |ζb i. The fermion partition function (7.419) is obtained from this amplitude by ˆ forming the trace of the operator e−β Hω , which by formula (7.434) is given by the integral over the antidiagonal matrix elements dζ ∗ dζ ˆ h−ζ|e−β Hω |ζi. (7.440) π The antidiagonal matrix elements lead to antiperiodic boundary conditions of the fermionic path integral. Zωf =

Z


581

7.11 Hilbert Space of Quantized Grassmann Variable

7.11

Hilbert Space of Quantized Grassmann Variable

To understand the Hilbert space associated with a path integral over a Grassmann variable we recall that a path integral with zero Hamiltonian serves to define the Hilbert space via all its scalar products as shown in Eq. (2.18): (xb tb |xa ta ) =

Z

Dx

Z

Z Dp ˙ = hxb |xa i = δ(xb − xa ). exp i dt p(t)x(t) 2π¯ h

(7.441)

A momentum variable inside the integral corresponds to a derivative operator pˆ ≡ −i¯ h∂x outside the amplitude, and this operator satisfies with xˆ = x the canonical commutation relation [ˆ p, xˆ] = −i¯ h [see (2.19)]. In complete analogy it is possible to create the Hilbert space of spinor indices with the help of a path integral over anticommuting Grassmann variables. In order to understand the Hilbert space, we shall consider three different cases.

7.11.1

Single Real Grassmann Variable

First we consider the path integral of a real Grassmann field with zero Hamiltonian Z

"

Dθ i exp 2π h ¯

Z

#

i¯ h ˙ dt θ(t)θ(t) . 2

(7.442)

From the Lagrangian

i ˙ ¯ θ(t)θ(t) L(t) = h 2 we obtain a canonical momentum pθ =

∂L i¯ h = − θ. 2 ∂ θ˙

(7.443)

(7.444)

Note the minus sign in (7.444) arising from the fact that the derivative with resect to θ˙ anticommutes with the variable θ on its left. The canonical momentum is proportional to the dynamical variable. The system is therefore subject to a constraint χ = pθ +

i¯ h θ = 0. 2

(7.445)

In the Dirac classification this is a second-class constraint, in which case the quantization proceeds by forming the classical Dirac brackets rather than the Poisson brackets (1.21), and replacing them by ±i/¯ h times commutation or anticommutation relations, respectively. For n dynamical variables qi and m constraints χp the Dirac brackets are defined by {A, B}D = {A, B} − {A, χp }C pq {χq , B},

(7.446)

where C pq is the inverse of the matrix C pq = {χp , χq }.

(7.447)

582


For Grassmann variables pi , qi , the Poisson bracket (1.21) carries by definition an overall minus sign if A contains an odd product of Grassmann variables. Applying this rule to the present system we insert A = pθ and B = θ into the Poisson bracket (1.21) we see that it vanishes. The constraint (7.445), on the other hand, satisfies (

)

i¯ h i¯ h {χ, χ} = pθ + θ, pθ + θ = −i¯ h{pθ , θ} = −i¯ h. 2 2

(7.448)

Hence C = −i¯ h with an inverse i/¯ h. The Dirac bracket is therefore i −i {pθ , θ}D = {pθ , θ} − {pθ , χ}{χ, θ} = 0 − h ¯ h ¯

!

i¯ h h ¯ (¯ h) = − . 2 2

(7.449)

ˆ B] ˆ + , we therefore obtain the canonWith the substitution rule {A, B}D → (−i/¯ h)[A, ical equal-time anticommutation relation for this constrained system: h ˆ + = − i¯ [ˆ p(t), θ(t)] , 2

(7.450)

ˆ θ(t)] ˆ + = 1. [θ(t),

(7.451)

or, because of (7.444),

The proportionality of pθ and θ has led to a factor 1/2 on the right-hand side with respect to the usual canonical anticommutation relation. Let ψ(θ) be an arbitrary wave function of the general form (7.376): ψ(θ) = ψ0 + ψ1 θ.

(7.452)

The scalar product in the space of all wave functions is defined by the integral ′

hψ |ψi ≡

Z

dθ ′∗ ψ (θ)ψ(θ) = ψ0′ ∗ ψ1 + ψ1′ ∗ ψ0 . 2π

(7.453)

In the so-defined Hilbert space, the operator θˆ is diagonal, while the operator pˆ is given by the differential operator pˆ = i¯ h∂θ ,

(7.454)

to satisfy (7.450). The matrix elements of the operator pˆ are hψ ′ |ˆ p|ψi ≡

Z

dθ ′∗ ∂ h ψ(θ) = i¯ ψ (θ)i¯ hψ1′ ∗ ψ1 . 2π ∂θ

(7.455)

By calculating ′

hˆ pψ |ψi ≡

Z

"

dθ ∂ i¯ h ψ ′ (θ) 2π ∂θ

#∗

ψ(θ) = −i¯ hψ1′ ∗ ψ1 ,

(7.456)



583

we see that the operator pˆ is anti-Hermitian, this being in accordance with the opposite sign in the rule (7.383) of integration by parts. Let |θi be the local eigenstates of which the operator θˆ is diagonal: ˆ = θ|θi. θ|θi

(7.457)

The operator θˆ is Hermitian, such that hθ|θˆ = hθ|θ = θhθ|.

(7.458)

The scalar products satisfy therefore the usual relation (θ′ − θ)hθ′ |θi = 0.

(7.459)

On the other hand, the general expansion rule (7.376) tells us that the scalar product S = hθ′ |θi must be a linear combination of S0 + S1 θ + S1′ θ′ + S2 θθ′ . Inserting this into (7.459), we find (7.460) hθ′ |θi = −θ′ + θ + S2 θθ′ , where the proportionality constants S0 and S1 are fixed by the property hθ′ |θi =

Z

dθ′′ ′ ′′ ′′ hθ |θ ihθ |θi. 2π

(7.461)

The constant S2 is an arbitrary real number. Recalling (7.385) we see that Eq. (7.460) implies that the scalar product hθ′ |θi is equal to a δ-function: hθ′ |θi = δ(θ − θ′ ),

(7.462)

just as in ordinary quantum mechanics. Note the property hθ′ |θi∗ = −hθ|θ′ i = h−θ| − θ′ i,

(7.463)

and the fact that since the scalar product hθ′ |θi is a Grassmann object, a Grassmann variable anticommutes with the scalar product. Having assumed in (7.458) that the Grassmann variable θ can be taken to the left of the bra-vector hθ|, the ket-vector |θi must be treated like a Grassmann variable, i.e., ˆ = θ|θi = −|θiθ. θ|θi

(7.464)

The momentum operator has the following matrix elements hθ′ |ˆ p|θi = −i¯ h∂θ′ hθ′ |θi = i¯ h.

(7.465)

Let |pi be an eigenstate of pˆ with eigenvalue ip, then its scalar product with |θi satisfies hθ|ˆ p|pi =

Z

dθ′ hθ|ˆ p|θ′ ihθ′ |pi = i¯ h 2π

Z

dθ′ ′ hθ |pi = i¯ h∂θ′ hθ′ |pi, 2π

(7.466)

584


the last step following from the rule (7.380). Solving (7.466) we find hθ|pi = eiθp/¯h ,

(7.467)

the right-hand side being of course equal to 1 + iθp/¯ h. It is easy to find an orthonormal set of basis vectors in the space of wave functions (7.452): 1 1 (7.468) ψ+ (θ) ≡ √ (1 + θ) , ψ− (θ) ≡ √ (1 − θ) . 2 2 We can easily check that these are orthogonal to each other and that they have the scalar products Z dθ ∗ ψ (θ)ψ± (θ) = ±1. (7.469) 2π ± The Hilbert space contains states of negative norm which are referred to as ghosts. Because of the constraint, only half of the Hilbert space is physical. For more details on these problems see the literature on supersymmetric quantum mechanics.

7.11.2

Quantizing Harmonic Oscillator with Grassmann Variables

Let us now turn to the more important physical system containing two Grassmann variables θ1 and θ2 , combined to complex Grassmann variables (7.388). The Lagrangian is assumed to have the same form as that of an ordinary harmonic oscillator: h i L(t) = h ¯ a∗ (t)i∂t a(t) − ωa∗ (t)a(t) . (7.470) We may treat a(t) and a∗ (t) as independent variables, such that there is no constraint in the system. The classical equation of motion ia(t) ˙ = ωa(t)

(7.471)

is solved by a(t) = e−iωt a(0),

a† (t) = e−iωt a† (0).

(7.472)

The canonical momentum reads pa (t) =

∂L(t) = −i¯ ha(t), ∂ a(t) ˙

(7.473)

and the system is quantized by the equal-time anticommutation relation [ˆ pa (t), a ˆ(t)]+ = −i¯ h,

(7.474)

[ˆ a† (t), a ˆ(t)]+ = 1.

(7.475)

or In addition we have [ˆ a(t), a ˆ(t)]+ = 0,

[ˆ a† (t), a ˆ† (t)]+ = 0.


585


Due to these anticommutation relations, the time-independent number operator ˆ ≡ a† (t)a(t) N

(7.477)

satisfies the commutation relations ˆ a† (t)] = a† (t), [N,

ˆ a(t)] = −a(t). [N,

(7.478)

We can solve the algebra defined by (7.475), (7.476), and (7.478) for any time, say t = 0, in the usual way, defining a ground state |0i by the condition and an excited state |1i as

a|0i = 0,

(7.479)

|1i ≡ a† |0i.

(7.480)

ˆ = ωN ˆ possesses the These are the only states, and the Hamiltonian operator H eigenvalues 0 and ω on them. Let ψ(a) be wave functions in the representation where the operator a is diagonal. The canonically conjugate operator pâ = i¯ ha ˆ† has then the form pâ ≡ −i¯ h∂a . (7.481)

7.11.3

Spin System with Grassmann Variables

For the purpose of constructing path integrals of relativistic electrons later in Chapter 19 we discuss here another system with Grassmann variables. Pauli Algebra First we introduce three real Grassmann fields θi , i = 1, 2, 3, and consider the path integral " Z # 3 Z Y i¯ h i ˙i i i Dθ exp dt θ (t)θ (t) . (7.482) h ¯ 4 i=1 The equation of motion is

θ˙i (t) = 0,

(7.483)

so that θi (t) are time independent variables. The three momentum operators lead now to the three-dimensional version of the equal-time anticommutation relation (7.451) [θî (t), θˆj (t)]+ = 2δ ij , (7.484) where the time arguments can be omitted due to (7.483). The algebra is solved with the help of the Pauli spin matrices (1.445). The solution of (7.484) is obviously i hB|θî |Ai = σBA ,

A, B = 1, 2.

(7.485)

Let us now add in the exponent of the trivial path integral a Hamiltonian HB = −S · B(t),

(7.486)

586


where

i S i ≡ − ǫijk θj θk (7.487) 4 plays the role of a spin vector. This can be verified by calculating the canonical commutation relations between the operators [Sî , Sˆj ] = iǫijk Sˆk ,

(7.488)

[Sî , θˆj ] = iǫijk θˆk .

(7.489)

and

ˆ describes the coupling of a spin vector to a magnetic field B(t). Thus, the operator H Using the commutation relation (7.488), we find the Heisenberg equation (1.283) for the Grassmann variables: ˙ =B× , (7.490)

which goes over into a similar equation for the spin vector: S˙ = B × S.

(7.491)

The important observation is now that the path integral (7.492) with the magnetic Hamiltonian (7.486) with fixed ends θbi = θi (τb ) and θai = θi (τa ) written as Z

θbi =θ i (τb )

θai =θ i (τa )

"

i D θ exp h ¯ 3

Z

dt

i¯ h i ˙i i θ θ + B i ǫijk θj θk 4 4

!#

,

(7.492)

represents the matrix (recall Appendix 1A) Tˆ exp

i h ¯

Z

dt B(t) ·

2

,

(7.493)

where Tˆ is the time-ordering operator defined in Eq. (1.241). This operator is necessary for a time-dependent B(t) field since the matrices B(t) /2 do not in general commute with each other. Whenever we encounter time ordered exponentials of integrals over matrices, these can be transformed into a fluctuating path integral over Grassmann variables. In the applications to come, we need only the trace of the matrix (7.493). According to Eq. (7.440), this is found from the integral over (7.492) with θbi = −θai , which means performing the integral over all antiperiodic Grassmann paths θi (τ ). If we want to find individual matrix elements, we have to make use of suitably extended recovery formulas (7.431) and (7.432). The result may be expressed in a slightly different notation using the spin tensors S ij ≡ ǫijk S k =

1 i j θθ, 2i

(7.494)

whose matrix elements satisfy the rotation algebra

[Sîj , Sˆkl ] = i δ ik Sˆjl − δ il Sˆjk + δ jl Sîk − δ jk Sîl ,

(7.495)


587


and which have the matrix representation 1 ij 1 σk hB|Sîj |Ai = σBA ≡ [σ i , σ j ]BA = ǫijk . 2 4i 2

(7.496)

Note the normalization σ 12 = σ 3 .

(7.497)

Introducing the analogous magnetic field tensor F ij ≡ ǫijk B k ,

(7.498)

we can write the final result also in the tensorial form "

Z θi =θi (τb ) i Z iZ i¯ h i ˙i i jk j k b Tˆ exp dt F ij (t)σ ij = D 3 θ exp dt θθ + F θθ 4¯ h h ¯ 4 4 θai =θ i (τa )

!#

.(7.499)

The trace of the left-hand side is, of course, given by the path integral over all antiperiodic Grassmann paths on the right-hand side. Dirac Algebra There exists a similar path integral suitable for describing relativistic spin systems. If we introduce four Grassmann variables θµ , µ = 0, 1, 2, 3, the path integral 3 Z Y

µ=0

Dθµ exp

leads to an equation of motion

i h ¯

Z

i dt − θµ (t)θ˙µ (t) 4

,

(7.500)

θ˙µ (t) = 0,

(7.501)

and an operator algebra at equal times {θˆµ (t), θˆν (t)} = 2g µν ,

(7.502)

where gµν is the metric in Minkowski space    

g µν = 

1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1



  . 

(7.503)

β, α = 1, 2, 3, 4,

(7.504)

The time argument in (7.502) can again be dropped due to (7.501). The algebra (7.502) is solved by the matrix elements [recall (7.485)] hβ|θˆµ(t)|αi = (γ µ )βα ,

where γ µ , γ5 are composed of 2 × 2-matrices 0, 1, and σ i as follows: 0

γ ≡

0 1 −1 0

!

i

, γ ≡

0 − σi

σi 0

!

0 1 2 3

, γ5 ≡ iγ γ γ γ =

−1 0 0 1

!

≡ γ 5. (7.505)

588


One may also introduce an additional Grassmann variable θ5 such that the path integral Z 3 Z Y i i ˙ Dθ5 exp dt θ5 (t)θ5 (t) , (7.506) h ¯ 4 µ=0 produces the matrix elements of γ5 (t): hβ|θˆ5 (t)|αi = (γ5 )βα ,

β, α = 1, 2, 3, 4.

(7.507)

The Grassmann variables θµ and θ5 anticommute with each other, and so do the matrices γ5 γ µ and γ5 . In terms of the Grassmann variables θµ it is possible to rewrite a four-dimensional version of the time ordered 2 × 2 matrix integral (7.493) as a path integral without time ordering: Tˆ exp

(

"

Z i Z iZ i¯ h i dt F µν Σµν = D 4 θ exp dt − θµ θ˙µ + Fµν θµ θν 2¯ h h ¯ 4 4

#)

,

(7.508)

where Σµν is the Minkowski space generalization of the spin tensor matrix σ ij /2 in (7.496): i (7.509) Σµν ≡ [γ µ , γ ν ] = −Σνµ . 4 As a check we use (7.505) and find 12

Σ

1 = 2

σ 12 0 0 σ 12

!

σ3 0 0 σ3

1 = 2

!

,

(7.510)

in agreement with (7.497). As remarked before, we shall need in the applications to come only the trace of the matrix (7.493), which is found, according to Eq. (7.440), from the integral over (7.492) with θbi = −θai , i.e., by performing the integral over all antiperiodic Grassmann paths θi (τ ). If we want to find individual matrix elements, we have to make use of suitably extended recovery formulas (7.431) and (7.432). The path integral over all antiperiodic paths for Fµν = 0 fixes the normalization. The left-hand side is equal to 4, so that we have Z

(

i D θ exp h ¯ 4

Z

"

i¯ h dt − θµ θ˙µ 4

#)

= 4.

(7.511)

This normalization factor agrees with Eqs. (7.415) and (7.418), where we found the path integral of single complex fermion field to carry a normalization factor 2. For four real fields this corresponds to a factor 4. In the presence of a nonzero field tensor, the result of the path integral (7.508) is therefore Z

(i/4¯ h)

Dx e

R

dt{[−i¯ hθµ θ˙µ +iFµν θ µ θ ν ]}

= 4 Det

1/2

1 δµν ∂t − Fµν (x(τ )) . h ¯

(7.512)


589


Relation between Harmonic Oscillator, Pauli and Dirac Algebra There exists a simple relation between the path integrals of the previous three paragraphs. We simply observe that the combinations 1 1 a ˆ = σ 1 + iσ 2 = σ + = 2 2

0 0 1 0

†

!

1 1 , a ˆ = σ 1 − iσ 2 = σ − = 2 2

0 1 0 0

!

, (7.513)

satisfy the anticommutation rules [ˆ a, a ˆ† ]+ = 1,

[ˆ a† , a ˆ† ]+ = 1,

[ˆ a, a ˆ]+ = 0.

(7.514)

The vacuum state annihilated by a is the spin-down spinor, and the one-particle state is the spin-up spinor: 0 1

|0i =

!

|1i =

,

1 0

!

.

(7.515)

A similar construction can be found for the Dirac algebra. There are now two types of creation and annihilation operators 1 1 a ˆ = γ 1 + iγ 2 = 2 2 †

0 σ+ −σ + 0

!

0 −σ − σ− 0

1 1 , a ˆ = −γ 1 + iγ 2 = 2 2

!

, (7.516)

and 1 1 b = γ0 + γ3 = 2 2

ˆ†

0 iσ 2 σ + −iσ 2 σ + 0

!

!

0 iσ 2 σ − . −iσ 2 σ − 0 (7.517)

1 1 , ˆb = γ 1 − γ 3 = 2 2

The states |na , ab i with na quanta a and nb quanta b are the following: 

1   |0, 0i = √  2

0 1 0 1



  , 





1   |0, 1i = b† |0, 0i = √  2

0 −1 0 1



  , 

1   |1, 0i = a† |0, 0i = √  2

1 0 −1 0



(7.518)

1 0 1 0

  , 



  . 

(7.519)



1   |1, 1i = a† b† |0, 0i = √  2

From these relations we can easily deduce the proper recovery formulas generalizing (7.431) and (7.432).

590


7.12

External Sources in a∗ , a -Path Integral

In Chapter 3, the path integral of the harmonic oscillator was solved in the presence of an arbitrary external current j(τ ). This yielded the generating functional Z[j] for the calculation of all correlation functions of x(τ ). In the present context we are interested in the generating functional of correlations of a(τ ) and a∗ (τ ). Thus we also need the path integrals quadratic in a(τ ) and a∗ (τ ) coupled to external currents. Consider the Euclidean action A[a∗ , a] + Asource = h ¯

Z

hβ ¯

0

dτ [a∗ (τ )∂τ a(τ ) + ωa∗ a(τ ) − (η ∗ (τ )a(τ ) + cc)], (7.520)

with periodic boundary conditions for a(τ ) and a∗ (τ ). For simplicity, we shall use the continuum formulation of the partition function, so that our results will correspond to the harmonic oscillator time slicing, with the energies En = (n + 1/2)¯ hω. There is no problem in going over to the second-quantized formulation with the energies En = n¯ hω. The partition function is given by the path integral Zω [η ∗ , η] =

I

1 Da∗ Da exp − (A[a∗ , a] + Asource ) . π h ¯

(7.521)

As in Chapter 3, we define the functional matrix between a∗ (τ ), a(τ ′ ) as Dω,e (τ, τ ′ ) ≡ (∂τ + ω)δ(τ − τ ′ ),

τ, τ ′ ∈ (0, h ¯ β).

(7.522)

The inverse is the Euclidean Green function −1 Gpω,e (τ, τ ′ ) = Dω,e (τ, τ ′ ),

(7.523)

satisfying the periodic boundary condition. We now complete the square and rewrite the action (7.520), using the shifted fields ′

a (τ ) = a(τ ) −

Z

0

hβ ¯

dτ ′ Gpω,e (τ, τ ′ )η(τ ′ )

(7.524)

as Ae = h ¯

Z

0

hβ ¯

dτ

Z

0

hβ ¯

dτ ′ [a′∗ (τ )Dω,e (τ, τ ′ )a′ (τ ′ ) − η ∗ (τ )Gpω,e (τ, τ ′ )η(τ ′ )].

(7.525)

On an infinite β-interval the Green function can easily be written down in terms of the Heaviside function (1.309): ′

Gω,e (τ, τ ′ ) = Gω,e (τ − τ ′ ) = e−ω(τ −τ ) Θ(τ − τ ′ ).

(7.526)

As we have learned in Section 3.3, the periodic Green function is obtained from this by forming a periodic sum Gpω,e (τ, τ ′ ) = Gpω,e (τ − τ ′ ) =

∞ X

n=−∞

e−ω(τ −τ

′ −n¯ hβ)

Θ(τ − τ ′ − n¯ hβ),

(7.527)


7.12 External Sources in a∗ , a -Path Integral

591

which is equal to Gpω,e (τ ) =

e−ω(τ −¯hβ/2) = (1 + nbω )e−ωτ , 2sinh(ω¯ hβ/2)

(7.528)

where nω is the Bose-Einstein distribution function 1 nω = β¯hω (7.529) e −1 [compare (3.92) and (3.93)]. The same considerations hold for anticommuting variables with antiperiodic boundary conditions, in which case we find once more the action (7.525) with the antiperiodic Green function [compare (3.112)] Gaω,e (τ ) =

e−ω(τ −¯hβ/2) = (1 − nfω )e−ωτ , 2cosh(ω¯ hβ/2)

(7.530)

where nω is the Fermi-Dirac distribution function [see (3.111)]: 1

nfω =

. (7.531) eβ¯hω + 1 In either case, we may decompose the currents in (7.525) into real and imaginary parts j and k via q

ω/2M¯ h(j − iωMk),

η =

(7.532)

q

η∗ =

ω/2M¯ h(j + iωMk),

and write the source part of the original action (7.520) in the form A

source

=h ¯

Z

hβ ¯

0

∗

∗

dτ (a η + η a) =

Z

0

hβ ¯

dτ (jx + kp).

(7.533)

q

Hence, the real current η = η ∗ = ω/2M¯ h j corresponds to the earlier source term (3.2). Inserting this current into the action (7.525), it yields the quadratic source term Z ¯hβ Z τ −ω(τ −τ ′ ) 1 s ′ e dτ dτ j(τ )j(τ ′ ). (7.534) Ae = − 2Mω 0 1 − e−β¯hω 0 This can also be rewritten as Z τ ∞ Z ¯ hω 1 X ′ Ase = − dτ dτ ′ e−ω(τ −τ +n¯hβ) j(τ )j(τ ′ ), (7.535) 2Mω n=0 0 0 which becomes for a current periodic in h ¯ β:

hω ¯ τ 1 ′ =− dτ dτ ′ e−ω(τ −τ ) j(τ )j(τ ′ ). 2Mω 0 −∞ ′ Interchanging τ and τ , this is also equal to

Ase

1 =− 4Mω in agreement with (3.276). Ase

Z

Z

Z

0

hω ¯

dτ

Z

∞ −∞

′

dτ ′ e−ω|τ −τ | j(τ )j(τ ′ ),

(7.536)

(7.537)

592


7.13

Generalization to Pair Terms

There exists an important generalization of these considerations to the case of the R 2 ∗ quadratic frequency term h ¯ dτ Ω (τ )a (τ )a(τ ) being extended to the more general quadratic form Z 1 ∗ 1 2 ∗ 2 ∗2 h ¯ dτ Ω (τ )a (τ )a(τ ) + ∆ (τ )a (τ ) + ∆(τ )a (τ ) . (7.538) 2 2 The additional off-diagonal terms in a∗ , a are called pair terms. They play an important role in the theory of superconductivity. The basic physical mechanism for this phenomenon will be explained in Section 17.10. Here we just mention that the lattice vibrations give rise to the formation of bound states between pairs of electrons, called Cooper pairs. By certain manipulations of the path integral of the electron field in the second-quantized interpretation, it is possible to introduce a complex pair field ∆x (t) at each space point which is coupled to the electron field in an action of the type (7.538). The partition function to be studied is then of the generic form Z=

I

Z ¯hβ Da∗ Da exp − dτ (a∗ ∂τ a + Ω2 a∗ a + 12 ∆∗ a2 + 12 ∆ a∗2 ) . π 0 "

#

(7.539)

It is easy to calculate this partition function on the basis of the previous formulas. To this end we rewrite the action in the matrix form, using the field doublets f (τ ) ≡

a(τ ) a∗ (τ )

!

,

(7.540)

as

!

h ¯ ¯hβ ∂τ + Ω(τ ) ∆(τ ) Ae = dτ f ∗T (τ ) f (τ ), (7.541) ∆∗ (τ ) ∓(∂τ ± Ω(τ )) 2 0 where the derivative terms require a partial integration to obtain this form. The partition function can be written as Z

Z= with the matrix M=

Z

Df ∗ Df − 12 f ∗ M f e , π

∂τ + Ω(τ ) ∆(τ ) ∗ ∓(∂τ ± Ω(τ )) ∆ (τ )

(7.542) !

.

(7.543)

The fields f ∗ and f are not independent of each other, since ∗

f =

0 1 1 0

!

f.

(7.544)

Thus, there are only half as many independent integrations as for a usual complex field. This has the consequence that the functional integration in (7.542) gives only the square root of the determinant of M, Z=N

b,f

det

∂τ + Ω(τ ) ∆(τ ) ∗ ∆ (τ ) ∓(∂τ + Ω(τ ))

!∓1/2

,

(7.545)


593

7.13 Generalization to Pair Terms

with the normalization factors N b,f being fixed by comparison with (7.337) and (7.419). A fluctuation determinant of this type occurs in the theory of superconductivity [32], with constant parameters ω, ∆. When applied to functions oscillating like f e−iωm τ , the matrix M becomes M=

f −iωm +ω ∆ f ∆∗ ∓(−iωm ± ω)

!

.

(7.546)

It is brought to diagonal form by what is called, in this context, a Bogoliubov transformation [33] d

M →M =

f + ω∆ 0 −iωm f 0 ∓(−iωm ± ω∆ )

where ω∆ is the frequency ω∆ =

√

!

,

ω 2 + ∆2 .

(7.547)

(7.548)

In a superconductor, these frequencies correspond to the energies of the quasiparticles associated with the electrons. They generalize the quasi-particles introduced in Landau’s theory of Fermi liquids. The partition function (7.539) is then given by Zω,∆ = N b,f det = N

b,f

∂τ + ω ∆ ∗ ∆ ∓∂τ ± ω

h

det (−∂τ2

+

i∓1/2 2 ω∆ )

=

!∓1/2 (

[2 sinh(¯ hβω∆ /2)]−1 , 2 cosh(¯ hβω∆ /2),

(7.549)

for bosons and fermions, respectively. This is equal to the partition function of a symmetrized Hamilton operator 1 ˆ = ω∆ ∆ a H ˆ† a ˆ + aâ ˆ† = ω∆ aˆ† a ˆ± , 2 2

with the eigenvalue spectrum ω∆ n ± n = 0, 1 for fermions: Zω,∆ =

∞,1 X

1 2

(7.550)

, where n = 0, 1, 2, 3 . . . for bosons and

e−¯hω∆ (n±1/2)β .

(7.551)

n

In the second-quantized interpretation where zero-point energies are omitted, this becomes Zω,∆ =

(

(1 − e−¯hβω∆ )−1 , (1 + e−¯hβω∆ ).

(7.552)

594


7.14

Spatial Degrees of Freedom

In the path integral treatment of the last sections, the particles have been restricted to a particular momentum state p. In a three-dimensional volume, the fields a∗ (τ ), ∆(τ ) and their frequency ω depend on p. With this trivial extension one obtains a free quantum field theory.

7.14.1

Grand-Canonical Ensemble of Particle Orbits from Free Fluctuating Field

The free particle action becomes a sum over momentum states ∗

Ae [a , a] = h ¯

hβ ¯

Z

0

Xh

i

a∗p ∂τ ap + ω(p)a∗p ap .

dτ

p

(7.553)

The time-sliced partition function is given by the product Z =

Yn p

=

  

The free energy

o∓1

det [ǫ(1 − ǫω)∇ + ǫω(p)] h

exp − exp

h

P

p

P

log 1 − e−¯hβ ω¯e (p)

p

i

log 1 + e−¯hβ ω¯e (p)

i

(7.554) for bosons, for fermions.

F = −kB T log Z

is for bosons

F = kB T

X p

and for fermions

F = −kB T

(7.555)

log 1 − e−¯hβ ω¯e (p) ,

X

(7.556)

log 1 + e−¯hβ ω¯e (p) .

p

(7.557)

In a large volume, the momentum sum may be replaced by the integral X p

→

Z

dD p V . (2π¯ h)D

(7.558)

For infinitely thin time slices, ǫ → 0 and ω ¯ e (p) reduces to ω(p) and the expressions (7.556), (7.557) turn into the usual free energies of bosons and fermions. They agree completely with the expressions (7.46) and (7.220) derived from the sum over orbits. Next we may introduce a fluctuating field in space and imaginary time 1 X ipx ψ(x, τ ) = √ e ap (τ ), V p

(7.559)

and rewrite the action (7.553) in the local form ∗

Ae [ψ , ψ] =

Z

0

hβ ¯

dτ

Z

h ¯2 d x ψ (x, τ )¯ h∂τ ψ(x, τ ) + ∇ψ ∗ (x, τ )∇ψ(x, τ ) . (7.560) 2M D

"

∗

#


595

7.14 Spatial Degrees of Freedom

The partition function is given by the functional integral Z=

I

DψDψ ∗ e−Ae [ψ

∗ ,ψ]/¯ h

.

(7.561)

Thus we see that the functional integral over a fluctuating field yields precisely the same partition function as the sum over a grand-canonical ensemble of fluctuating orbits. Bose or Fermi statistics are naturally accounted for by using complex or Grassmann field variables with periodic or antiperiodic boundary conditions, respectively. The theory based on the action (7.560) is completely equivalent to the second-quantized theory of field operators. In order to distinguish the second-quantized or quantum field description of many particle systems from the former path integral description of many particle orbits, the former is referred to as the first-quantized approach, or also the world-line approach. The action (7.560) can be generalized further to include an external potential ˆ ) = p ˆ 2 + V (x, τ ) V (x, τ ), i.e., it may contain a general Schrödinger operator H(τ instead of the gradient term: ∗

Ae [ψ , ψ] =

Z

0

hβ ¯

Z

n

h

i

o

ˆ ) − µ ψ(x, τ ) . (7.562) dτ dD x ψ ∗ (x, τ )¯ h∂τ ψ(x, τ )+ψ ∗ (x, τ ) H(τ

For the sake of generality we have also added a chemical potential to enable the study of grand-canonical ensembles. This action can be used for a second-quantized description of the free Bose gas in an external magnetic trap potential V (x), which would, of course, lead to the same results as the first-quantized approach in Section 7.2.4. The free energy associated with this action 1 ˆ ) − µ] h∂τ + H(τ (7.563) F = Tr log[¯ β was calculated in Eq. (3.139) as an expansion "Z # ∞ R h¯ β ′′ hβ ¯ X 1 1 ˆ ′′ ˆ ) −1 F = Tr dτ H(τ Tr Tˆ e−n 0 dτ [H(τ )−µ]/¯h , 2¯ hβ β n=1 n 0

(7.564)

The sum can be evaluated in the semiclassical expansion developed in Section 4.9. For simplicity, we consider here only time-independent external potentials, where ˆ we must calculate Tr [e−nβ (H−µ) ]. Its semiclassical limit was given in Eq. (4.251) continued to imaginary time. From this we obtain ∞ Z n X 1 1 1 D [z(x)] ˆ F = Tr H − q d x D/2+1 , D 2 β n 2π¯ h2 β/M n=1

(7.565)

where z(x) ≡ e−β[V (x)−µ] is the local fugacity (7.24). The sum agrees with the previous first-quantized result in (7.132) and (7.133). The first term is due to the symmetric treatment of the fields in the action (7.562) [recall the discussion after Eq. (7.340)]. One may calculate quantum corrections to this expansion by including the higher gradient terms of the semiclassical expansion (4.251). If the potential is timedependent, the expansion (4.251) must be generalized accordingly.

596


7.14.2

First versus Second Quantization

There exists a simple set of formulas which illustrates nicely the difference between first and second quantization, i.e., between path and field quantization. Both may be though of as being based on two different representations of the Dirac δ-function. The first-quantized representation is δ (D) (xb − xa ) =

Z

x(tb )=xb

x(ta )=xa

DD x

I

D D p (i/¯h) Rttb dt px˙ a e , (2π¯ h)D

(7.566)

the second-quantized representation i¯ hδ

(D)

(xb − xa )δ(tb − ta ) =

I

(i/¯ h)

DψDψ ∗ ψ(xb , tb )ψ ∗ (xa , ta )e

R

dD x

R∞

−∞

dtψ∗ (x,t)ψ(x,t)

.

(7.567) The first representation is turned into a transition amplitude by acting upon it with ˆ the time-evolution operator e−iH(tb −ta ) , which yields ˆ

(xb tb |xa ta ) = eiH(tb −ta ) δ (D) (xb − xa ) =

Z

I

DD x

D D p (i/¯h) Rttb dt(px−H) ˙ a e . (7.568) D (2π¯ h)

By multiplying this with the Heaviside function Θ(tb − ta ), we obtain the solution of the inhomogeneous Schrödinger equation ˆ (i¯ h∂t − H)Θ(t hδ (D) (xb − xa )δ(tb − ta ). b − ta )(xb tb |xa ta ) = i¯

(7.569)

This may be expressed as a path integral representation for the resolvent *

+ i¯ h x t = Θ(tb xb tb i¯ ˆ a a h∂ − H t

− ta )

Z

x(tb )=xb

x(ta )=xa

D

D x

I

D D p (i/¯h) Rttb dt(px−H) ˙ a e . (7.570) D (2π¯ h)

The same quantity is obtained from the second representation (7.567) by changing ˆ the integrand in the exponent from ψ ∗ (x, t)ψ(x, t) to ψ ∗ (x, t)(i¯ h∂t − H)ψ(x, t): *

+ i¯ h xb tb x t ˆ a a i¯ h∂t − H I

=

(i/¯ h)

DψDψ ∗ ψ(xb , tb )ψ ∗ (xa , ta )e

R

dD x

R∞

−∞

ˆ dtψ∗ (x,t)(i¯ h∂t −H)ψ(x,t)

.

(7.571)

This is the second-quantized functional integral representation of the resolvent.

7.14.3

Interacting Fields

The interaction between particle orbits in a grand-canonical ensemble can be accounted for by anharmonic terms in the particle fields. A pair interaction between orbits, for example, corresponds to a fourth-order self interaction. An example is the interaction in the Bose-Einstein condensate corresponding to the energy in Eq. (7.101). Expressed in terms of the fields it reads H. Kleinert, PATH INTEGRALS

597

7.14 Spatial Degrees of Freedom

Z

∗ Aint e [ψ , ψ] = −

0

g dτ ∆E = − 2

hβ ¯

Z

0

hβ ¯

Z

dτ d3 x ψ ∗ (x, τ + η)ψ ∗ (x, τ + η)ψ(x, τ )ψ(x, τ ),

(7.572) where η > 0 is an infinitesimal time shift. It is then possible to develop a perturbation theory in terms of Feynman diagrams by complete analogy with the treatment in Section 3.20 of the anharmonic oscillator with a fourth-order self interaction. The free correlation function is the momentum sum of oscillator correlation functions: hψ(x, τ )ψ ∗ (x′ , τ ′ )i =

X

p,p′

′ ′

hap (τ )a∗p′ (τ ′ )iei(px−p x ) =

′

hap (τ )a∗p (τ ′ )ieip(x−x ) . (7.573)

X p

The small η > 0 in (7.572) is necessary to specify the side of the jump of the correlation functions (recall Fig. 3.2). The expectation value of ∆E is given by (7.101), with a prefactor g rather than g/2 due to the two possible Wick contractions. Inserting the periodic correlation function (7.528), we obtain the Fourier integral hψ(x, τ )ψ ∗ (x′ , τ ′ )i =

X

(1 + nωp )e−ωp (τ −τ

′ )+ip(x−x′ )

.

(7.574)

p

Recalling the representation (3.284) of the periodic Green function in terms of a sum over Matsubara frequencies, this can also be written as hψ(x, τ )ψ ∗ (x′ , τ ′ )i =

−1 1 X ′ ′ e−iωm (τ −τ )+ip(x−x ) . h ¯ β ωm ,p iωm − ωp

(7.575)

The terms in the free energies (7.102) and (7.107) with the two parts (7.108) and (7.109) can then be shown to arise from the Feynman diagrams in the first line of Fig. 3.7. In a grand-canonical ensemble, the energy h ¯ ωp in (7.575) is replaced by h ¯ ωp − µ. The same replacement appears in ωp of Eq. (7.574) which brings the distribution function nωp to [recall (7.529)] nωp −µ/¯h =

1 z −1 eβ¯hωp

−1

,

(7.576)

where z is the fugacity z = eβµ . The expansion of the Feynman integrals in powers of z yields directly the expressions (7.102) and (7.107).

7.14.4

Effective Classical Field Theory

For the purpose of studying phase transitions, a functional integral over fields ψ(x, τ ) with an interaction (7.572) must usually be performed at a finite temperature. Then is often advisable to introduce a direct three-dimensional extension of the effective classical potential V eff cl (x0 ) introduced in Section 3.25 and used efficiently in Chapter 5. In a field theory we can set up, by analogy, an effective classical action which is a functional of the three-dimensional field with zero Matsubara frequency φ(x) ≡ ψ0 (x). The advantages come from the reasons discussed in Section 3.25, that

598


the zero-frequency fluctuations have a linearly diverging fluctuation width at high temperature, following the Dulong-Petit law. Thus only the nonzero-modes can be treated efficiently by the perturbative methods explained in Subsection 3.25.6. By analogy with the splitting of the measure of path integration in Eq. (3.805), we may factorize the functional integral (7.561) into zero- and nonzero-Matsubara frequency parts as follows: Z=

I

DψDψ ∗e−Ae [ψ

∗ ,ψ]

=

I

Dψ0 Dψ0∗

I

D ′ ψD ′ψ ∗ e−Ae [ψ

∗ ,ψ]

,

(7.577)

and introduce the Boltzmann factor [compare (3.810)] contain the effective classical action I eff cl ∗ ∗ B[ψ0∗ , ψ0 ] ≡ e−A [ψ0 ,ψ0 ] ≡ D ′ ψD ′ ψ ∗ e−Ae [ψ ,ψ] , (7.578) to express the partition function as a functional integral over time-independent fields in three dimensions as: Z=

I

eff cl [ψ ∗ ,ψ ] 0 0

Dψ0 Dψ0∗ e−A

.

(7.579)

In Subsection 3.25.1 we have seen that the full effective classical potential V eff cl (x0 ) in Eq. (3.809) reduces in the high-temperature limit to the initial potential V (x0 ). For the same reason, the full effective classical action in the functional integral (7.577) can be approximated at high temperature by the bare effective classical action, which is simply the zero-frequency part of the initial action: cl ∗ [ψ0 , ψ0 ] = β Aeff b

Z

3

dx

ψ0∗ (x)

1 2 2πa ∗ − ∇ − µ ψ0 (x) + [ψ0 (x)ψ0 (x)]2 . (7.580) 2m m

This follows directly from the fact that, at high temperature, the fluctuations in the functional integral (7.578) are strongly suppressed by the large Matsubara frequencies in the kinetic terms. Remarkably, the absence of a shift in the critical temperature in the first-order energy (7.113) deduced from Eq. (7.116) implies that the chemical potential in the effective classical action does not change at this order [34]. For this reason, the lowest-order shift in the critical temperature of a weakly interacting Bose-Einstein condensate can be calculated entirely from the three-dimensional effective classical field theory (7.577) with the bare effective classical action (7.580) in the Boltzmann factor. The action (7.580) may be brought to a more conventional form by introducing the differently normalized two-component fields φ = (φ1 , φ2 ) related to the original √ complex field ψ by ψ(x) = MT [φ1 (x) + iφ2 (x)]. If we also define a square mass m2 ≡ −2Mµ and a quartic coupling u = 48πaMT , the bare effective classical action reads cl ∗ Aeff [ψ0 , ψ0 ] b

= A[φ] =

Z

1 1 u d x |∇φ|2 + m2 φ2 + (φ2 )2 . 2 2 4! 3

(7.581)


599

7.15 Bosonization

In the field theory governed by the action (7.581), the relation (7.111) for the shift of the critical temperature to lowest order in the coupling constant becomes ∆Tc (0)

Tc

2 mTc(0) D 2 E 4π (mTc(0) )2 ∆φ2 ≈ − ∆φ = − 4! 3 n 3 n u * + 2 2 4π (2π) ∆φ = − 4! an1/3 , 3 [ζ(3/2)]4/3 u *

+

a (7.582)

where h∆φ2 i is the shift in the expectation value of φ2 caused by the interaction. Since a repulsive interaction pushes particles apart, h∆φ2 i is negative, thus explaining the positive shift in the critical temperature. The evaluation of the expectation value h∆φ2 i from the path integral (7.577) with the bare three-dimensional effective classical action (7.581) in the exponent can now proceed within one of the best-studied field theories in the literature [51]. The theoretical tools for calculating strong-coupling results in this theory are well developed, and this has made it possible to drive the calculations of the shift to the five-loop order [21]. Some low-order corrections to the effective classical action (7.580) have been calculated from the path integral (7.578) in Ref. [34].

7.15

Bosonization

The path integral formulation of quantum-mechanical systems is very flexible. Just as integrals can be performed in different variables of integration, so can path integrals in different path variables. This has important applications in many-body systems, which show a rich variety of so-called collective phenomena. Practically all fermion systems show collective excitations such as sound, second sound, and spin waves. These are described phenomenologically by bosonic fields. In addition, there are phase transitions whose description requires a bosonic order parameter. Superconductivity of electron systems is a famous example where a bosonic order parameter appears in a fermion system—the energy gap. In all these cases it is useful to transform the initial path variables to so-called collective path variables, in higher dimensions these become collective fields [32]. Let us illustrate the technique with simple a model defined by the Lagrangian L(t) = a∗ (t)i∂t a(t) −

ε ∗ [a (t)a(t)]2 , 2

(7.583)

where a∗ , a are commuting or anticommuting variables. All Green functions can be calculated from the generating functional Z[η ∗ , η] = N

Z

Z

Da∗ Da exp i

dt (L + η ∗ a + a∗ η) ,

(7.584)

where we have omitted an irrelevant overall factor. In operator language, the model is defined by the Hamiltonian operator ˆ = ε(ˆ H a† a ˆ)2 /2,

(7.585)

600


where a ˆ† , a ˆ are creation and annihilation operator of either a boson or a fermion at a point. In the boson case, the eigenstates are 1 a† )n |0i, |ni = √ (ˆ n!

n = 0, 1, 2, . . . ,

(7.586)

with an energy spectrum n2 En = ε . 2

(7.587)

In the fermion case, there are only two solutions |0i

with E0 = 0, ε with E1 = . 2

|1i = a† |0i

(7.588) (7.589)

Here the Green functions are obtained from the generating functional Z

Z[η † , η] = h0|Tˆ exp i

dt(η ∗ a ˆ+a ˆ† η) |0i,

(7.590)

where Tˆ is the time ordering operator (1.241). The functional derivatives with respect to the sources η ∗ , η generate all Green functions of the type (3.296) at T = 0.

7.15.1

Collective Field

At this point we introduce an additional collective field via the HubbardStratonovich transformation [52] Z Z εZ ρ2 (t) ∗ 2 − ρ(t)a∗ (t)a(t) exp −i dt[a (t)a(t)] = N Dρ(t) exp i dt 2 2ε (

"

#)

, (7.591)

where N is some irrelevant factor. Equivalently we multiply the partition function (7.584) with the trivial Gaussian path integral 1=N

Z

Dρ(t) exp

Z

i2 1 h dt ρ(t) − εa† (t)a(t) , 2ε

(7.592)

to obtain the generating functional Z[η † , η] = N × exp

(Z

Z

Da∗ DaDρ

ρ2 (t) dt a (t)i∂t a(t) − ερ(t)a (t)a(t) + + η ∗ (t)a(t) + a∗ (t)η(t) 2ε "

∗

∗

#)

.(7.593)

From (7.591) we see that the collective field ρ(t) fluctuates harmonically around ε times particle density. By extremizing the action in (7.593) with respect to ρ(t) we obtain the classical equality: ρ(t) = εa∗ (t)a(t).


601

7.15 Bosonization

The virtue of the Hubbard-Stratonovich transformation is that the fundamental variables a∗ , a appear now quadratically in the action and can be integrated out to yield a path integral involving only the collective variable ρ(t): ∗

Z[η , η] = N

Z

Dρ exp iA[ρ] −

Z

′ ∗

′

′

dtdt η (t)Gρ (t, t )η(t ) ,

(7.595)

with the collective action A[ρ] = ±iTr log

iG−1 ρ

+

Z

dt

ρ2 (t) , 2

(7.596)

where Gρ denotes the Green function of the fundamental path variables in an external ρ(t) background potential, which satisfies [compare (3.75)] [i∂t − ρ(t)] Gρ (t, t′ ) = iδ(t − t′ ).

(7.597)

The Green function was found in Eq. (3.124). Here we find the solution once more in a different way which will be useful in the sequel. We introduce an auxiliary field ϕ(t) =

Z

t

ρ(t′ )dt′ ,

(7.598)

in terms of which Gρ (t, t′ ) is simply ′

Gρ (t, t′ ) = e−iϕ(t) eiϕ(t ) G0 (t − t′ ),

(7.599)

with G0 being the free-field propagator of the fundamental particles. At this point one has to specify the boundary condition on G0 (t − t′ ). They have to be adapted to the physical situation of the system. Suppose the time interval is infinite. Then G0 is given by Eq. (3.100), so that we obtain, as in (3.124), ¯ − t′ ). Gρ (t, t′ ) = e−iϕ(t) eiϕ(t ) Θ(t ′

(7.600)

¯ defined in Eq. (1.309) corresponds to ρ(t) coupling to the The Heaviside function Θ symmetric operator combination (ˆ a† aˆ +ˆ aa ˆ† )/2. The products a ˆ† a ˆ in the Hamiltonian (7.585), however, have the creation operators left of the annihilation operators. In this case we must use the Heaviside function Θ(t − t′ ) of Eq. (1.300). If this function is inserted into (7.600), the right-hand side of (7.600) vanishes at t = t′ . Using this property we see that the functional derivative of the tracelog in (7.596) vanishes: o δ n δ ±iTr log(iG−1 ) = {±iTr log [i∂t −ρ(t)]} = ∓Gρ (t, t′ )|t′ =t = 0. (7.601) ρ δρ(t) δρ(t)

This makes the tracelog an irrelevant constant. The generating functional is then simply †

Z[η , η] = N

Z

Z i Z ′ 2 Dϕ(t) exp dt ϕ(t) ˙ − dtdt′ η † (t)η(t′ )e−iϕ(t) e−iϕ(t ) Θ(t − t′ ) , 2ε (7.602)

602


where we have used the Jacobian Dρ = Dϕ Det(∂t ) = Dϕ,

(7.603)

since Det(∂t ) = 1 according to (2.529). Observe that the transformation (7.598) provides ϕ(t) with a standard kinetic term ϕ˙ 2 (t). The original theory has been transformed into a new theory involving the collective field ϕ. In a three-dimensional generalization of this theory to electron gases, the field ϕ describes density waves [32].

7.15.2

Bosonized versus Original Theory

The first term in the bosonized action in (7.602) is due to a Lagrangian L0 (t) =

1 ϕ(t) ˙ 2 2ε

(7.604)

describing free particles on the infinite ϕ-axis. Its correlation function is divergent and needs a small frequency, say κ, to exist. Its small-κ expansion reads ′

dω i ′ e−iω(t−t ) 2 2 2π ω − κ + iǫ ε −κ|t−t′ | ε i = e = − ε |t − t′ | + O(κ). 2κ 2κ 2

hϕ(t)ϕ(t )i = ε

Z

(7.605)

The first term plays an important role in calculating the bosonized correlation functions generated by Z[η ∗ , η] in Eq. (7.602). Differentiating this n times with respect to η and η ∗ and setting them equal to zero we obtain ′

′

G(n) (tn , . . . , t1 ; t′n , . . . , t′1 ) = (−1)n he−iϕ(tn ) · · · e−iϕ(t1 ) eiϕ(tn ) · · · eiϕ(t1 ) i X × ǫp Θ(tp1 − t1 ) · · · Θ(tpn − t1 ),

(7.606)

p

where the sum runs over all permutations p of the time labels of t1 . . . tn making sure that these times are all later than the times t′1 . . . t′n . The factor ǫp reflects the commutation properties of the sources η, η ∗, being equal to 1 for all permutations if η and η ∗ commute. For Grassmann variables η, η ∗ , the factor ǫp is equal to −1 if the permutation p is odd. The expectation value is defined by the path integral R 1 Z ˙ 2 h ... i ≡ Dϕ(t) . . . e(i/2ε) dt ϕ(t) . Z[0, 0]

(7.607)

Let us calculate the expectation value of the exponentials in (7.606), writing it as

"

exp i

2n X i=1

qi ϕ(ti )

#

=

exp

"

XZ i

dt δ(t − ti )qi ϕ(t)

#+

,

(7.608)


603

7.15 Bosonization

where we have numbered the times as t1 , t2 , . . . t2n rather than t1 , t′1 , t2 , t′2 , . . . tn , t′n . Half of the 2n “charges” qi are 1, the other half are −1. Using Wick’s theorem in the form (3.307) we find P 2n

ei

q ϕ(ti ) i=1 i





2n X X 1Z = exp − dtdt′ qi δ(t − ti )hϕ(t)ϕ(t′ )i(t′ ) qj δ(t − tj ) 2 i=1 j





2n 1 X = exp − qi qj hϕ(ti )ϕ(tj )i . 2 i,j=1

(7.609)

Inserting here the small-κ expansion (7.605), we see that the 1/κ-term gives a prefactor !2 ε X exp − qi , (7.610) 4κ i which vanishes since the sum of all “charges” qi is zero. Hence we can go to the limit κ → 0 and obtain

"

exp i

2n X

qi ϕ(ti )

i=1

#



= exp ε

i 2

X i>j



qi qj |ti − tj | .

(7.611)

Note that the limit κ → 0 makes all correlation functions vanish which do not contain an equal number of a∗ (t) and a(t) variables. It ensures “charge conservation”. For one positive and one negative charge we obtain, after a multiplication by a Heaviside function Θ(t1 − t′1 ), the two-point function ′

G(2) (t, t′ ) = e−iε(t−t )/2 Θ(t − t′ ).

(7.612)

This agrees with the operator result derived from the generating functional (7.590), where ′ G(2) (t, t′ ) = h0|Tˆ a ˆ(t)ˆ a† (t′ )|0i = Θ(t − t′ )h0|ˆ a(t)ˆ a† (t′ )|0ie−iε(t−t )/2 Θ(t − t′ ). (7.613)

Inserting the Heisenberg equation [recall (1.277)] ˆ

ˆ

′

a(t) = eitH ae−itH ,

ˆ

′

ˆ

a† (t′ ) = eit H a† e−it H .

(7.614)

we find †a ˆ)2 t/2

h0|Tˆ a ˆ(t)ˆ a† (t′ )|0i = Θ(t − t′ )h0|eiε(â ′

i

†a ˆ)2 (t−t′ )

ae− 2 (â

− 2i (ˆ a† a ˆ)2 (t−t′ )

= Θ(t − t )h1|e

†a ˆ)2 t′ /2

aˆ† e−i(â ′

|0i

−iε(t−t′ )/2

|1i = Θ(t − t )e

. (7.615)

Let us compare the above calculations with an operator evaluation of the bosonized theory. The Hamiltonian operator associated with the Lagrangian (7.604) ˆ = εˆ is H p2 /2. The states in the Hilbert space are eigenstates of the momentum operator pˆ = −i∂/∂ϕ: 1 {ϕ|p} = √ eipϕ . 2π

(7.616)

604


Here, curly brackets are a modified Dirac notation for states of the bosonized theory, which distinguishes them from the states of the original theory created by products of operators aˆ† (t) acting on |0i. In operator language, the generating functional of the theory (7.602) reads 1 Z[η , η] = {0|Tˆ exp − {0|0}

†

Z

′ †

′

−iϕ(t) ˆ iϕ(t ˆ ′)

dtdt η (t)η(t )e

e

′

Θ(t − t ) |0}, (7.617)

where ϕ(t) ˆ are free-particle operators. We obtain all Green functions of the initial operators aˆ(t), a ˆ† (t) by forming functional derivatives of the generating functional Z[η † , η] with respect to η † , η. Take, for instance, the two-point function δ (2) Z 1 ˆ ˆ ′) h0|Tˆ a ˆ(t)ˆ a (t )|0i = − † = eiϕ(t |0}Θ(t − t′ ). (7.618) {0|e−iϕ(t) ′ δη (t)δη(t ) η† ,η=0 {0|0} †

′

Inserting here the Heisenberg equation (1.277), p ˆ2

p ˆ2

ϕ(t) ˆ = eiε 2 t ϕ(0) ˆ e−iε 2 t ,

(7.619)

we can evaluate the matrix element in (7.618) as follows: p ˆ2

p ˆ2

′

p ˆ2 ′

p ˆ2

′

ˆ ˆ ˆ ˆ {0|eiε 2 t 2e−iϕ(0) e−iε 2 (t−t ) eiϕ(0) e−iε 2 t |0} = {0|e−iϕ(0) e−iε 2 (t−t ) eiϕ(0) |0}. (7.620) ˆ The state eiϕ(0) |0} is an eigenstate of pˆ with unit momentum |1} and the same norm as |0}, so that (7.620) equals

1 ′ ′ {1|1} e−iε(t−t )/2 = e−iε(t−t )/2 {0|0}

(7.621)

and the Green function (7.618) becomes, as in (7.613) and (7.615): h0|Tâˆ(t)ˆ a† (t′ )|0i = e−iε(t−t )/2 Θ(t − t′ ). ′

(7.622)

Observe that the Fermi and Bose statistics of the original operators a ˆ, a ˆ† enters to result only via the commutation properties of the sources. There exists also the opposite phenomenon that a bosonic theory possesses solutions which behave like fermions. This will be discussed in Section 8.12.

Appendix 7A

Treatment of Singularities in Zeta-Function

Here we show how to evaluate the sums which determine the would-be critical temperatures of a Bose gas in a box and in a harmonic trap. H. Kleinert, PATH INTEGRALS

Appendix 7A

7A.1


605

Finite Box

According to Eqs. (7.83), (7.88), and (7.90), the relation between temperature T = h ¯ 2 π 2 /bM L2 kB and the fugacity zD at a fixed particle number N in a finite D-dimensional box is determined by the equation zD N = Nn (T ) + Ncond(T ) = SD (zD ) + , (7A.1) 1 − zD where SD (zD ) is the subtracted infinite sum SD (zD ) ≡

∞ X

w [Z1D (wb)ewDb/2 − 1]zD ,

(7A.2)

w=1

P∞ 2 containing the Dth power of one-particle partition function in the box Z1 (b) = k=1 e−bk /2 . The would-be critical temperature is found by equating this sum at zD = 1 with the total particle number N . We shall rewrite Z1 (b) as h i Z1 (b) = e−b/2 1 + e−3b/2 σ1 (b) , (7A.3) where σ1 (b) is related to the elliptic theta function (7.85) by σ1 (b) ≡

∞ X

e−(k

2

−4)b/2

k=2

=

i e2b h ϑ3 (0, e−b/2 ) − 1 − 2e−b/2 . 2

According to Eq. (7.87), this has the small-b behavior r π 2b 1 σ1 (b) = e − e3b/2 − e2b + . . . . 2b 2

(7A.4)

(7A.5)

The omitted terms are exponentially small as long as b < 1 [see the sum over m in Eq. (7.86)]. For large b, these terms become important to ensure an exponentially fast falloff like e−3b/2 . Inserting (7A.3) into (7A.2), we find SD (1) ≡ D

∞ X

w=1

σ1 (wb)e−3wb/2 +

D−1 2 (D−1)(D−2) 3 σ1 (wb)e−6wb/2 + σ1 (wb)e−9wb/2 . 2 6

(7A.6)

Inserting here the small-b expression (7A.5), we obtain S2 (1) ≡ S3 (1) ≡

r ∞ X π wb π wb 1 wb e − e + e − 1 + ... , (7A.7) 2wb 2wb 4 w=1 ! r r 3 ∞ X π 3 π 3wb/2 3 π 3wb/2 1 3wb/2 3wb/2 e − e + e − e − 1 + . . . , (7A.8) 2wb 2 2wb 4 2wb 8 w=1

the dots indicating again exponentially small terms. The sums are convergent only for negative b, this being a consequence of the approximate nature of these expressions. If we evaluate in Pthem ∞ this regime, the sums produce Polylogarithmic functions (7.34), and we find, using also w=1 1 = ζ(0) = −1/2 from (2.514), r π 1 S2 (1) = ζ1 (eb ) − ζ1/2 (eb ) + ζ0 (eb ) − ζ(0) + . . . . (7A.9) 2b 4 r 3 r π 3 π 3 π 1 3b/2 3b/2 S3 (1) = ζ3/2 (e )− ζ1 (e )+ ζ1 (e3b/2 ) − ζ0 (e3b/2 ) − ζ(0) + . . . . (7A.10) 2b 2 2b 4 2b 8

606


These expressions can now be expanded in powers of b with the help of the Robinson expansion (2.573). Afterwards, b is continued analytically to positive values and we obtain r π π 1 S2 (1) = − log(C2 b) − ζ(1/2) + (3 − 2π) + O(b1/2 ), 2b 2b 8 r r 3 π 3π 3 π 9 ζ(3/2) + log(C3 b) + ζ(1/2)(1 + π) + (1 + π) + O(b1/2 ). S3 (1) = 2b 4b 4 2b 16

(7A.11) (7A.12)

′ The constants C2,3 , C2,3 inside the logartithms turn out to be complex, implying that the limiting expressions (7A.7) and (7A.8) cannot be used reliably. A proper way to proceed gors as follows. We subtract from SD (1) terms which remove the small-b singularties by means of modifications of (7A.7) and (7A.8) which have the same small-b expansion up to b0 :

r ∞ X π π 4π − 3 −wb − + e + ... , 2wb 2wb 4 w=1 "r # r 3 ∞ X π π 3 π 3 9 S˜3 (1) ≡ − + (1 + 2π) − (1 + 2π) e−3wb/2 + . . . . 2wb 2 2wb 4 2wb 8 w=1

S˜2 (1) ≡

(7A.13) (7A.14)

In these expressions, the sums over w can be performed for positive b yielding r π 4π − 3 π −b ˜ S2 (1) ≡ ζ1 (e ) − ζ1/2 (e−b ) + ζ0 (e−b ) + . . . , (7A.15) 2b 2b 4 r r 3 3 π 3 9 π π S˜3 (1) ≡ ζ3/2 (e−3b/2 ) − ζ1 (e−3b/2 ) + (1+2π) ζ1/2 (e−3b/2 )− (1+2π)ζ0 (e−3b/2 ) + . . . . 2b 2 2b 4 2b 8 (7A.16) Inserting again the Robinson expansion (2.573), we obtain once more the above expansions (7A.11) and (7A.12), but now with the well-determined real constants C˜2 = e3/2π−2+

√ 2

≈ 0.8973,

√ 3 C˜3 = e−2+1/ 3−1/π ≈ 0.2630. 2

(7A.17)

The subtracted expressions SD (1) − S˜D (1) are smooth near the origin, so that the leading small-b behavior of the sums over these can simply be obtained from a numeric integral over w: Z ∞ Z ∞ 1.1050938 ˜ dw[S2 (1) − S2 (1)] = − , dw[S3 (1) − S˜3 (1)] = 3.0441. (7A.18) b 0 0 These modify the constants C˜2,3 to C2 = 1.8134,

C3 = 0.9574.

(7A.19)

The corrections to the sums over SD (1) − S˜D (1) are of order b0 an higher and already included in the expansions (7A.11) and (7A.12), which were only unreliable as far as C2,3 os concerned. Let us calculate from (7A.12) the finite-size correction to the critical temperature by equat(0) ing S3 (1) with N . Expressing this in terms of bc via (7.92), and introducing the ratio (0) ˆbc ≡ bc /bc which is close to unity, we obtain the expansion in powers of the small quantity (0) 2/3 2bc /π = [ζ(3/2)/N ] : ˆb3/2 = 1 + c

s

(0)

2bc π

p (0) 3 ˆbc 2bc 3 ˆ log(C3 b(0) ζ(1/2)(1 + π)ˆbc + . . . . c bc ) + 2ζ(3/2) π 4ζ(3/2)

(7A.20)


Appendix 7A

607


To lowest order, the solution is simply s (0) 1 ˆbc = 1 + 2bc log(C3 b(0) c ) + ... , π ζ(3/2)

(7A.21)

yielding the would-be critical temperature to first in 1/N 1/3 as stated in (7.94). To next order, we insert into the last term the zero-order solution ˆbc ≈ 1, and in the second term the first-order solution (7A.21) to find ˆb3/2 c

s

(0)

=

3 1+ 2

2bc π

+

2bc 3 π 4ζ(3/2)

(0)

1 log(C3 b(0) c ) ζ(3/2) ζ(1/2)(1 + π) +

(0)

i 1 h 2 (0) (0) 2 log(C3 bc ) + log (C3 bc ) + . . . . ζ(3/2)

(7A.22)

(0)

2/3

Replacing bc by (2/π) [ζ(3/2)/N ] , this gives us the ratio (Tc /Tc )3/2 between finite- and (0) infinite size critical temperatures Tc and Tc . The first and second-order corrections are plotted in Fig. 7.12, together with precise results from numeric solution of the equation N = S3 (1). 1.4 1.3 (0)

Tc /Tc 1.2 1.1

0.05

0.1

0.15

1/N 1/3

Figure 7.12 Finite-size corrections to the critical temperature for N = 300 to infinity calculated once from the formula N = S3 (1) (solid curve) and once from the expansion (0) (0) (7A.22) (short-dashed up to order [bc ]1/2 ∝ 1/N 1/3 , long-dashed up to the order bc ∝ 1/N 2/3 ). The fat dots show the peaks in the second derivative d2 Ncond (T )/dT 2 . The small dots show the corresponding values for canonical ensembles, for comparison.

7A.2

Harmonic Trap

The sum relevant for the would-be phase transition in a harmonic trap is (7.138), SD (b, zD ) =

∞ X

w=1

"

1 D (1−e−wb)

#

w − 1 zD ,

(7A.23)

which determines the number of normal particles in the harmonic trap via Eq. (7.139). We consider only the point zD = 1 which determines the critical temperature by the condition Nn = N . Restricting ourselves to the physical cases D = 1, 2, 3, we rewrite the sum as ∞ X

(D − 1) −2wb (D − 1)(D − 2) −3wb 1 −wb SD (b, 1) = D e − e + e . 2 6 (1 − e−wb )D w=1

(7A.24)

608


According to the method developed in the evaluation of Eq. (2.573) we obtain such a sum in two steps. First we go to small b where the sum reduces to an integral over w. After this we calculate the difference between sum and integral by a naive power series expansion. As it stands, the sum (7A.24) cannot be converted into an integral due to singularities at w = 0. These have to be first removed by subtractions. Thus we decompose SD (b, 1) into a subtracted sum plus a remainder as D D(3D − 1) ∆D−2 SD (b, 1),(7A.25) SD (b, 1) = S¯D (b, 1) + ∆D SD (b, 1) + b ∆D−1 SD (b, 1) + b2 2 24 where S¯D (b, 1) = ×

∞ X

D − 1 −2wb (D − 1)(D − 2) −3wb D e−wb − e + e 2 6 w=1 1 1 D D(3D − 1) − D D − − (1 − e−wb )D w b 2wD−1 bD−1 24wD−2 bD−2

(7A.26)

is the subtracted sum and ∆D′ SD (b, 1) ≡

D D−1 (D − 1)(D − 2) −b −2b −3b ′ ′ ′ ζ (e ) − ζ (e ) + ζ (e ) D D D bD 2 6

(7A.27)

collects the remainders. The subtracted sum can now be done in the limit of small b as an integral over w, using the well-known integral formula for the Beta function: Z ∞ Γ(a)Γ(1 − b) e−ax dx = B(a, 1 − b) = . (7A.28) −x )b (1 − e Γ(1 + a − b) 0 This yields the small-b contributions to the subtracted sums 1 7 S¯1 (b, 1) → γ− ≡ s1 , b→0 b 12 1 9 S¯2 (b, 1) → γ +log 2− ≡ s2 , b→0 b 8 1 19 S¯3 (b, 1) → γ +log 3− ≡ s3 , b→0 b 24

(7A.29)

where γ = 0.5772 . . . is the Euler-Mascheroni number (2.461). The remaining sum-minus-integral is obtained by a series expansion of 1/(1 − e−wb )D in powers of b and performing the sums over w using formula (2.576). However, due to the subtractions, the corrections are all small of order (1/bD )O(b3 ), and will be ignored here. Thus we obtain SD (b, 1) =

sD 1 + ∆SD (b, 1) + D O(b3 ). D b b

(7A.30)

We now expand ∆D′ SD (b, 1) using Robinson’s formula (2.575) up to b2 /bD and find ∆D′ S1 (b, 1) = ∆D′ S2 (b, 1) = ∆D′ S3 (b, 1) = where ζ1 (e−b ) =

1 ζD′ (e−b ), b 1 2ζD′ (e−b ) − ζD′ (e−2b ) , b2 1 3ζD′ (e−b ) − 3ζD′ (e−2b ) + ζD′ (e−3b ) , b3

(7A.31) (7A.32)

b b2 − log 1 − e−b = − log b + − + ... , 2 24 H. Kleinert, PATH INTEGRALS

Experimental versus Theoretical Would-be Critical Temperature609

Appendix 7B

ζ2 (e−b ) = ζ3 (e−b ) =

b2 + ... , ζ(2) + b(log b − 1) − 4 b b2 3 ζ(3) − ζ(2) − log b − + ... , 6 2 2

(7A.33) (7A.34)

The results are S1 (b, 1) = S2 (b, 1) = S3 (b, 1) =

1 b 1 b2 1 b3

b b2 (− log b + γ) + − + ... , 4 144 1 7b2 ζ(2) − b log b − γ + + + ... , 2 24 3b 19 ζ(3) + ζ(2) − b2 log b − γ + + ... , 2 24

(7A.35)

as stated in Eqs. (7.165)–(7.167). Note that the calculation cannot be shortened by simply expanding the factor 1/(1 − e−wb )D in the unsubtracted sum (7A.24) in powers of w, which would yield the result (7A.25) without the first term S¯1 (b, 1), and thus without the integrals (7A.29).

Appendix 7B

Experimental versus Theoretical Would-be Critical Temperature

In Fig. 7.12 we have seen that there is only a small difference between the theoretical wouldbe critical temperature Tc of a finite system calculated from the equation N = S2 (1) and and the experimental Tcexp determined from the maximum of the second derivative of the condensate fraction ρ ≡ Ncond/N . Let us extimate the difference. The temperature Tc is found by solving the equation [recall (7.90)] ∞ X N = SD (1) ≡ [Z1D (wbc )ewDb/2 − 1]. (7B.1) w=1

To find the latter we must solve the full equation (7A.1) and search for the maximum of d2 Ncond (T )/dT 2 . The last term in (7A.1) can be expressed in terms of the condensate fraction ρ(T ) as Ncond (T ) = ρ(T )N . Near the critical point, we set δzD ≡ 1 − zD and see that it is equal to δzD = 1/(1 + N ρ) √ In the critical regime, ρ and 1/N ρ are both of the same order 1/ N . Thus we expand SD (zD ) to lowest order in δzD = 1/N ρ + . . . , and replace (7A.1) by N ≈ [SD (1) + ∂b SD (1)δb − ∂zD SD (1) (1/N ρ − 1/N + . . .)] + ρN,

(7B.2)

b∂b SD (1) δb ∂z SD (1) 1 ≈ D − ρ+ ... , SD (1) b SD (1) N ρ

(7B.3)

or

In this leading-order discussion we may ignore the finite-size correction to SD (1), i.e., the difference (0) (1) (0) between Tc and Tc , so that SD (1) = (π/2bc )D/2 , and we may approximate the left-hand side (0) of (7B.3) by (D/2)δT /Tc . Abbreviating ∂zD SD (1)/SD (1) by A which is of order unity, we must find ρ(T ) from the the equation D A A t≈ −ρ− , 2 Nρ N

(7B.4)

(0)

where t ≡ δT /Tc . This yields r r r 3 A D D2 N 2 D4 N 4 ρ(t) = − t+ t − t + ... . N 4 32 A 2048 A

(7B.5)

610


The maximum of d2 ρ(t)/dt2 lies at t = 0, so that the experimental would-be critical coincides with (1) the theoretical Tc to this order. √ If we carry the expansion of (7B.2) to the second order in ρ ≈ 1/N ρ ≈ 1/ N , we find a shift of the order 1/N . As an example, take D = 3, assume A = 1, and use the full T -dependence rather than the lowest-order expansion in Eq. (7B.4):

T (0) Tc

3/2

= (1 + t)3/2 =

1 − ρ. Nρ

(7B.6)

For simplicity, we ignore the 1/N terms in (7B.3). The resulting ρ(t) and d2 ρ/dt2 are plotted in Fig. 7.13. The maximum of d2 ρ/dt2 lies at t ≈ 8/9N , which can be ignored if we know Tc only up to order O(N −1/3 ) or O(N −2/3 ), as in Eqs. (7.91) and (7A.22). 1 0.8 0.6 0.4 0.2

15 10

(0)

ρ(T /Tc )

(0)

ρ(T /Tc )

5 0.5 0.2 0.4 0.6 0.8 1 1.2

(0) T /Tc

1

1.5

2

2.5

(0)

-5

T /Tc

Figure 7.13 Plots of condensate fraction and its second derivative for Bose Gas in a finite box following Eq. (7B.6) for N = 100 and 1000. Note that a convenient way to determine the would-be critical temperature from numerical data is via the maximal curvature of the chemical potential µ(T ), i.e., from the maximum of −d2 µ(T )/dT 2 . Since zD = eµ−Db/2 = 1 − δzD ≈ 1 + µ − Db/2, the second derivative of µ is related to that of δzD by d2 µ d2 δzD − 2 ≈ − Db(0) (7B.7) c . dt dt2 The second term is of the order 1/N 2/D and can be ignored as we shall see immediately. The equation shows that the second derivative of −µ coincides with that of δzD , which is related to d2 ρ/dt2 by " # r 2 d2 1 1 2 dρ 1 d2 ρ D2 N 3D2 N 2 d2 δzD 4 ≈ = − = 1 − t + O(t ) . (7B.8) dt2 dt2 N ρ N ρ3 dt ρ2 dt2 16 A3 32 A (1)

This is again maximal at t = 0, implying that the determination of Tc by this procedure coincides with the previous one. The neglected term of order 1/N 2/D has a relative suppression of 1/N 2/D+1/2 , which can be ignored for larger N .

Notes and References Path integrals of many identical particles are discussed in the texbook R.P. Feynman, Statistical Mechanics, Benjamin, Reading, 1972. See also the papers by L.S. Schulman, Phys. Rev. 176, 1558 (1968); M.G.G. Laidlaw and C. DeWitt-Morette, Phys. Rev. D 3, 5 (1971); F.J. Bloore, in Differential Geometric Methods in Math. Phys., Lecture Notes in Mathematics 905, Springer, Berlin, 1982; H. Kleinert, PATH INTEGRALS


611

P.A. Horvathy, G. Morandi, and E.C.G. Sudarshan, Nuovo Cimento D 11, 201 (1989). Path Integrals over Grassmann variables are discussed in detail in F.A. Berezin, The Method of Second Quantization, Nauka, Moscow, 1965; J. Rzewuski, Field Theory, Hafner, New York, 1969; L.P. Singh and F. Steiner, Phys. Lett. B 166, 155 (1986); M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press, 1992. For fractional statistics see the papers by F. Wilczek and A. Zee, Phys. Rev. Lett. 51, 2250 (1983); Y.S. Wu, ibid., 53, 111 (1984); and the reprint collection by A. Shapere and F. Wilczek and Geometric Phases in Physics, World Scientific, Singapore, 1989, p. 284. The simple derivation of the statistics interaction given in this text is taken from H. Kleinert, Phys. Lett. B 235, 381 (1989) (wwwK/104, where (wwwK is short for http://www.physik.fu-berlin.de/~kleinert). See also the notes and references in Chapter 16. I thank Prof. Carl Wieman for his permission to publish Fig. 7.9 of the JILA BEC group. The particular citations in this chapter refer to the publications [1] R.P. Feynman, Statistical Mechanics, Benjamin, Reading, 1972. [2] J. Flachsmeyer, Kombinatorik , Dtsch. Verl. der Wiss., Berlin, 1972. p. 128, Theorem 11.3. The number nnz of permutations of n elements consisting of z cycles satisfies the recursion n−1 relation nnz = nn−1 , with the initial conditions nnn = 1 and nn1 = (n − 1)!. z−1 + (n − 1) nz [3] The Mathematica program for this can be downloaded from the internet (wwwK/b5/pgm7). [4] F. Brosens, J.T. Devreese, L.F. Lemmens, Phys. Rev. E 55 , 227 (1997). See also K. Glaum, Berlin Ph.D. thesis (in preparation). [5] P. Bormann and G. Franke, J. Chem. Phys. 98, 2484 (1984). [6] S. Bund and A. Schakel, Mod. Phys. Lett. B 13, 349 (1999). [7] See E.W. Weisstein’s internet page mathworld.wolfram.com. [8] See, for example, Chapter 11 in H. Kleinert, Gauge Fields in Condensed Matter , op. cit., Vol. I, World Scientific, Singapore, 1989 (wwwK/b1). [9] The first observation was made at JILA with 87 Ru: M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science 269, 198 (1995). It was followed by a condensate of 7 Li at Rice University: C.C. Bradley, C.A. Sackett, J.J. Tollet, and R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995), and in 30 Na at MIT: K.B. Davis, M.-O. Mewes, M.R. Andrews, and N.J. van+Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [10] J.R. Abo-Shaeer, C. Raman, J.M. Vogels, and W. Ketterle, Science 292, 476 (2001). [11] A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems McGraw–Hill, New York, 1971, Sec. 28. [12] V.N. Popov, Theor. Math. Phys. 11, 565 (1972); Functional Integrals in Quantum Field Theory and Statistical Physics (Reidel, Dordrecht, 1983). [13] M. Schick, Phys. Rev. A 3, 1067 (1971).

612


[14] J.O. Andersen, U. Al Khawaja, and H.T.C. Stoof, Phys. Rev. Lett. 88, 070407 (2002). [15] H.T.C. Stoof and M. Bijlsma, Phys. Rev. E 47, 939 (1993); U. Al Khawaja, J.O. Andersen, N.P. Proukakis, and H.T.C. Stoof, Phys. Rev. A A, 013615 (2002). [16] D.S. Fisher and P.C. Hohenberg, Phys. Rev. B 37, 4936 (1988). [17] F. Nogueira and H. Kleinert, Phys. Rev. B 73, 104515 (2006) (cond-mat/0503523). [18] K. Huang, Phys. Rev. Lett. 83, 3770 (1999). [19] K. Huang, C.N. Yang, Phys. Rev. 105, 767 (1957); K. Huang, C.N. Yang, J.M. Luttinger, Phys. Rev. 105, 776 (1957); T.D. Lee, K. Huang, C.N. Yang, Phys. Rev. 106, 1135 (1957); K. Huang, in Studies in Statistical Mechanics, North-Holland, Amsterdam, 1964, Vol. II. [20] H. Kleinert, Strong-Coupling Behavior of Phi4 -Theories and Critical Exponents, Phys. Rev. D 57 , 2264 (1998); Addendum: Phys. Rev. D 58, 107702 (1998) (cond-mat/9803268); Seven Loop Critical Exponents from Strong-Coupling φ4 -Theory in Three Dimensions, Phys. Rev. D 60 , 085001 (1999) (hep-th/9812197); Theory and Satellite Experiment on Critical Exponent alpha of Specific Heat in Superfluid Helium, Phys. Lett. A 277, 205 (2000) (cond-mat/9906107); Strong-Coupling φ4 -Theory in 4 − ǫ Dimensions, and Critical Exponent , Phys. Lett. B 434 , 74 (1998) (cond-mat/9801167); Critical Exponents without betaFunction, Phys. Lett. B 463, 69 (1999) (cond-mat/9906359). See also the textbook: H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories, World Scientific, Singapore, 2001 (wwwK/b8). [21] H. Kleinert, Five-Loop Critical Temperature Shift in Weakly Interacting Homogeneous BoseEinstein Condensate (cond-mat/0210162). [22] T. Toyoda, Ann. Phys. (N.Y.) 141, 154 (1982). [23] A. Schakel, Int. J. Mod. Phys. B 8, 2021 (1994); Boulevard of Broken Symmetries, (condmat/9805152). The upward shift ∆Tc /Tc by a weak δ-function repulsion reported in this paper is, however, due to neglecting a two-loop contribution to the free energy. If this is added, the shift disappears. See the remarks below in [34]. [24] H.T.C. Stoof, Phys. Rev. A 45, 8398 (1992). [25] M. Holzmann, P. Grueter, and F. Laloe, Eur. Phys. J. B 10, 239 (1999). [26] G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloe, and D. Vautherin, Phys. Rev. Lett. 83, 1703 (1999). [27] G. Baym, J.-P. Blaizot, and J. Zinn-Justin, Europhys. Lett. 49, 150 (2000) (cond– mat/9907241) [28] M. Holzmann, G. Baym, J.-P. Blaizot, and F. Laloe, Phys. Rev. Lett. 87, 120403 (2001) (cond-mat/0107129). [29] P. Grueter, D. Ceperley, and F. Laloe, Phys. Rev. Lett. 79, 3549 (1997). [30] M. Holzmann and W. Krauth, Phys. Rev. Lett. 83, 2687 (1999). [31] M. Bijlsma and H.T.C. Stoof, Phys. Rev. A 54, 5085 (1996). [32] H. Kleinert, Collective Quantum Fields, Fortschr. Phys. 26, 565 (1978) http://www.physik.fu-berlin.de/~kleinert/55). [33] N.N. Bogoliubov, Zh. Eksp. Teor. Fiz. 34, 58 (1958). H. Kleinert, PATH INTEGRALS


613

[34] P. Arnold and B. Tomáˇsik, Phys. Rev. A 62, 063604 (2000). This paper starts out from the 3+1-dimensional initial theory and derives from it the threedimensional effective classical field theory as defined in Subsection 7.14.4. This program was initiated much earlier by A. Schakel, Int. J. Mod. Phys. B 8, 2021 (1994); and explained in detail in Ref. [23]. Unfortunately, the author did not go beyond the oneloop level so that he found a positive shift ∆Tc /Tc , and did not observe its cancellation at the two-loop level. See the recent paper A. Schakel, Zeta Function Regularization of Infrared Divergences in Bose-Einstein Condensation, (cond-mat/0301050). [35] F.F. de Souza Cruz, M.B. Pinto, R.O. Ramos, Phys. Rev. B 64, 014515 (2001); F.F. de Souza Cruz, M.B. Pinto, R.O. Ramos, P. Sena, Phys. Rev. A 65, 053613 (2002) (cond-mat/0112306). [36] P. Arnold, G. Moore, and B. Tomáˇsik, Phys. Rev. A 65, 013606 (2002) (cond-mat/0107124). [37] G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloe, and D. Vautherin (cond-mat/0107129). [38] P. Arnold and G. Moore, Phys. Rev. Lett. 87, 120401 (2001); V.A. Kashurnikov, N.V. Prokofév, and B.V. Svistunov, Phys. Rev. Lett. 87, 120402 (2001). [39] J.L. Kneur, M.B. Pinto, R.O. Ramos, Phys. Rev. A 68, 43615 (2003) (cond-mat/0207295). [40] J.D. Reppy, B.C. Crooker, B. Hebral, A.D. Corwin, J. He, and G.M. Zassenhaus, cond– mat/9910013 (1999). [41] M. Wilkens, F. Illuminati, and M. Krämer, J. Phys. B 33, L779 (2000) (cond-mat/0001422). [42] E.J. Mueller, G. Baym, and M. Holzmann, J. Phys. B 34, 4561 (2001) (cond-mat/0105359) [43] H. Kleinert, S. Schmidt, and A. Pelster, Phys. Rev. Lett. 93, 160402 (2004) (condmat/0307412). [44] E.L. Pollock and K.J. Runge, Phys. Rev. B 46, 3535 (1992). [45] J. Tempere, F. Brosens, L.F. Lemmens, and J.T. Devreese, Phys. Rev. A 61, 043605 (2000) (cond-mat/9911210). [46] N.N. Bogoliubov, On the Theory of Superfluidity , Izv. Akad. Nauk SSSR (Ser. Fiz.) 11, 77 (1947); Sov.Phys.-JETP 7, 41(1958). See also J.G. Valatin, Nuovo Cimento 7, 843 (1958); K. Huang, C.N. Yang, J.M. Luttinger, Phys. Rev. 105, 767, 776 (1957). [47] See pp. 1356–1369 in H. Kleinert, Gauge Fields in Condensed Matter , op. cit., Vol. II, World Scientific, Singapore, 1989 (wwwK/b2); H. Kleinert, Gravity as Theory of Defects in a Crystal with Only SecondGradient Elasticity, Ann. d. Physik, 44, 117 (1987) (wwwK/172); H. Kleinert and J. Zaanen, (gr-qc/0307033). [48] V. Bretin, V.S. Stock, Y. Seurin, F. Chevy, and J. Dalibard, Fast Rotation of a BoseEinstein Condensate, Phys. Rev. Lett. 92, 050403 (2004). [49] For the beginnings see J.R. Klauder, Ann. of Physics 11, 123 (1960). The present status is reviewed in J.R. Klauder, The Current State of Coherent States, (quant-ph/0110108). [50] G. Junker and J.R. Klauder, Eur. Phys. J. C 4, 173 (1998) (quant-phys/9708027). [51] See the textbook cited at the end of Ref. [20].

614


[52] R. Stratonovich Sov. Phys. Dokl. 2, 416 (1958): J. Hubbard, Phys. Rev. Lett. 3, 77 (1959); B. M¨ uhlschlegel, J. Math. Phys. , 3, 522 (1962); J. Langer, Phys. Rev. A 134, 553 (1964); T.M. Rice, Phys. Rev. A 140 1889 (1965); J. Math. Phys. 8, 1581 (1967); A.V. Svidzinskij, Teor. Mat. Fiz. 9, 273 (1971); D. Sherrington, J. Phys. C 4, 401 (1971).


H. Kleinert, PATH INTEGRALS June 16, 2007 ( /home/kleinert/kleinert/books/pathis/pthic8.tex) Every path hath a puddle. George Herbert, Outlandish Proverbs, 1640

8 Path Integrals in Polar and Spherical Coordinates Many physical systems possess rotational symmetry. In operator quantum mechanics, this property is of great help in finding wave functions and energies of a system. If a rotationally symmetric Schrödinger equation is transformed to spherical coordinates, it separates into a radial and several angular differential equations. The latter are universal and have well-known solutions. Only the radial equation contains specific information on the dynamics of the system. Being an ordinary one-dimensional Schrödinger equation, it can be solved with the usual techniques. In the path integral approach, a similar coordinate transformation is possible, although it makes things initially more complicated rather than simpler. First, the use of non-Cartesian coordinates causes nontrivial problems of the kind observed in Chapter 6, where the configuration space was topologically constrained. Such problems can be solved as in Chapter 6 using the knowledge of the correct procedure in Cartesian coordinates. A second complication is more severe: When studying a system at a given angular momentum, the presence of a centrifugal barrier destroys the possibility of setting up a time-sliced path integral of the Feynman type as in Chapter 2. The recent solution of the latter problem has paved the way for two major advances in path integration which will be presented in Chapters 10, 11, and 12.

8.1

Angular Decomposition in Two Dimensions

Consider a two-dimensional quantum-mechanical system with rotational invariance. In Schrödinger quantum mechanics, it is convenient to introduce polar coordinates x = r(cosϕ, sinϕ),

(8.1)

and to split the differential equation into a radial and an azimuthal one which are solved separately. Let us try to follow the same approach in path integrals. To avoid the complications associated with path integrals in the canonical formulation [1],all calculations will be done in the Lagrange formulation. It will, moreover, be advantageous to work with the imaginary-time amplitude (the thermal density 615

616

8 Path Integrals in Polar and Spherical Coordinates

matrix) to avoid carrying around factors of i. Thus we start out with the path integral (xb τb |xa τa ) =

Z

D 2 x(τ )exp −

1 Z τb M 2 dτ x˙ + V (r) h ¯ τa 2

.

(8.2)

It is time-sliced in the standard way into a product of integrals (xb τb |xa τa ) ≈

"Z N Y

n=1

+1 d2 xn ǫ NX M exp − (xn − xn−1 )2 + V (rn ) 2π¯ hǫ/M h ¯ n=1 2ǫ2

#

(

)

. (8.3)

When going over to polar coordinates, the measure of integration changes to N Z Y

∞

n=1 0

drn rn

Z

2π

0

dϕn 2π¯ hǫ/M

and the kinetic term becomes i 1M 1M h 2 2 2 exp − (xn −xn−1 ) = exp − r + rn−1 −2rn rn−1 cos(ϕn −ϕn−1 ) . (8.4) h ¯ 2ǫ h ¯ 2ǫ n

To do the ϕn -integrals, it is useful to expand (8.4) into a factorized series using the formula acosϕ

e

=

∞ X

Im (a)eimϕ ,

(8.5)

m=−∞

where Im (z) are the modified Bessel functions. Then (8.4) becomes 1M exp − (xn − xn−1 )2 h ¯ 2ǫ X ∞ 1M 2 M 2 = exp − (rn + rn−1 ) Im rn rn−1 eim(ϕn −ϕn−1 ) . h ¯ 2ǫ h ¯ǫ m=−∞

(8.6)

In the discretized path integral (8.3), there are N + 1 factors of this type. The Nintegrations over the ϕn -variables can now be performed and produce N Kronecker δ’s: N Y

2πδmn ,mn−1 .

(8.7)

n=1

These can be used to eliminate all but one of the sums over m, so that we arrive at the amplitude "Z # ∞ N ∞ dr r 2π Y X 1 2π n n (xb τb |xa τa ) ≈ eim(ϕb −ϕa ) 2π¯ hǫ/M n=1 0 2π¯ hǫ/M m=−∞ 2π

×

NY +1 n=1

1M 2 M 2 exp − (rn + rn−1 ) Im rn rn−1 h ¯ 2ǫ h ¯ǫ

+1 ǫ NX exp − V (rn ) . h ¯ n=1

"

#

(8.8)


617

8.1 Angular Decomposition in Two Dimensions

We now define the radial time evolution amplitudes by the following expansion with respect to the azimuthal quantum numbers m: ∞ X

(xb τb |xa τa ) =

m=−∞

√

1 1 (rb τb |ra τa )m eim(ϕb −ϕa ) . rb ra 2π

(8.9)

The amplitudes (rb τb |ra τa )m are obviously given by the radial path integral 1

(rb τb |ra τa )m ≈ q 2π¯ hǫ/M ×

NY +1 n=1

N Y

n=1

 Z 

0

drn

∞

q

2π¯ hǫ/M

 

M Mrn rn−1 exp − (rn − rn−1 )2 I˜m 2ǫ¯ h h ¯ǫ

+1 ǫ NX exp − V (rn ) . (8.10) h ¯ n=1

(

Here we have introduced slightly different modified Bessel functions √ I˜m (z) ≡ 2πze−z Im (z).

)

(8.11)

They will also be called “Bessel functions”, for short. They have the asymptotic behavior z→∞ m2 −1/4 m2 − 1/4 I˜m (z) − −−→ 1 − + . . . = e− 2z + . . . , 2z √ z→0 I˜m (z) − −−→ 2 π(z/2)m+1/2 + . . . .

(8.12) (8.13)

In the case of a free particle with V (r) = 0, it is easy to perform all the intermediate integrals over rn in (8.10). Two neighboring figures in the product require the integral ! !√ ! √ ′′ ′ Z ∞ r ′′ 2 r2 1 1 r r r ′′ r ′ r′r r′r ′ ′2 dr exp − − exp −r + Im Im 2ǫ2 2ǫ1 2ǫ2 2ǫ1 ǫ2 ǫ2 ǫ1 ǫ1 0 √ " # ! r ′′ 2 + r 2 r ′′ r r ′′ r = exp − Im . (8.14) 2 (ǫ1 + ǫ2 ) ǫ1 + ǫ2 ǫ1 + ǫ2 For simplicity, the units in this formula are M = 1, h ¯ = 1. The right-hand side of (8.14) follows directly from the formula Z

0

∞

drre−r

2 /ǫ

ǫ 2 2 Iν (βr) Iν (αr) = e(α +β )ǫ/4 Iν (ǫαβ/2) , 2

(8.15)

after identifying ǫ, α, β as ǫ = 2ǫ1 ǫ2 / (ǫ1 + ǫ2 ) , α = r/ǫ1 , β = r ′′ /ǫ2 .

(8.16)

618


Thus, the integrals in (8.10) with V (r) = 0 can successively be performed yielding the thermal amplitude for τb > τa : " # √ M rb ra M rb2 + ra2 M rb ra (rb τb |ra τa )m = exp − Im . (8.17) h ¯ τb − τa 2¯ h (τb − τa ) h ¯ τb − τa Note that the same result could have been obtained more directly from the imaginary-time amplitude of a free particle in two dimensions, M M (xb − xa )2 , (xb τb |xa τa ) = exp − 2π¯ h(τb − τa ) 2¯ h τb − τa "

#

(8.18)

by rewriting the right-hand side as M M rb2 + ra2 M rb ra exp − exp cos (ϕb − ϕa ) , 2π¯ h (τb − τa ) 2¯ h τb − τa 2¯ h τb − τa !

and expanding the second exponential according to (8.5) into the series ∞ X

m=−∞

Im

M rb ra eim(ϕb −ϕa ) . h ¯ τb − τa

(8.19)

A comparison of the coefficients with those in (8.9) gives the radial amplitudes (8.17). Due to (8.14), the radial amplitude satisfies a fundamental composition law corresponding to (2.4), which reads for τb > τ > τa Z

0

8.2

∞

drr (rb τb |r τ )m (r τ |ra τa )m = (rb τb |ra τa )m .

(8.20)

Trouble with Feynman’s Path Integral Formula in Radial Coordinates

In the above calculation we have shown that the expression (8.10) is certainly the correct radial path integral. It is, however, not of the Feynman type. In operator quantum mechanics we learn that the action of a particle moving in a potential V (r) at a fixed angular momentum L3 = m¯ h contains a centrifugal barrier h ¯ 2 (m2 − 2 1/4)/2Mr and reads Am =

Z

τb

τa

dτ

M 2 h ¯ 2 m2 − 1/4 r˙ + + V (r) . 2 2M r2 !

(8.21)

This is shown by separating the Hamiltonian operator into radial and azimuthal coordinates, over fixing the azimuthal angular momentum L3 , and choosing for it the quantum-mechanical value h ¯ m. According to Feynman’s rules, the radial amplitude therefore should simply be given by the path integral (rb τb |ra τa )m =

Z

0

∞

1 Dr exp − Am . h ¯


619

8.2 Trouble with Feynman’s Path Integral Formula in Radial Coordinates

The reader may object to using the word “classical” in the presence of a term proportional to h ¯ 2 in the action. In this section, however, h ¯ m is merely meant to be a parameter specifying the azimuthal momentum pϕ ≡ h ¯ m in the classical centrifugal barrier p2ϕ /2Mr 2 . It is parametrized in terms of a dimensionless number m which does not necessarily have the integer values required by the quantization of the azimuthal motion. By naively time-slicing (8.22) according to Feynman’s rules of Section 2.1 we would have defined it by the finite-N expression 1

(rb τb |ra τa )m ≈ q 2π¯ hǫ/M

N Y

n=1

 Z 

0

drn

∞

q

2π¯ hǫ/M

 

+1 M h ¯ 2 m2 − 1/4 1 NX (rn − rn−1 )2 + ǫ × exp − + ǫV (rn ) h ¯ n=1 2ǫ 2M rn rn−1

(

"

#)

. (8.23)

Actually, the denominators in the centrifugal barrier could have been chosen to be rn2 . This would make a negligible difference for small ǫ. Note that in contrast to a standard Feynman path integral in one dimension, the integrations over r cover only the semi-axis r ≥ 0 rather than the complete r-axis. This represents no problem since we have learned in Chapter 6 how to treat such half-spaces. The expression (8.23) is now a place for an unpleasant surprise: For m = 0, a time-sliced Feynman path integral formula cannot possibly exist since the potential has an abyss at small r. This leads to a phenomenon which will be referred to as the path collapse, to be understood physically and resolved later in Chapter 12. At this place we merely point out the mathematical origin of the problem, by comparing the naively time-sliced expression (8.23) with the certainly correct one (8.10). The singularity would be of no consequence if the two expressions were to converge towards each other in the continuum limit ǫ → 0. At first sight, this seems to be the case. After all, ǫ is assumed to be infinitesimally small so that we may replace the “Bessel function” I˜m (Mrn rn−1 /¯ hǫ) by its asymptotic form (8.12), I˜m

Mrn rn−1 h ¯ǫ

h ¯ m2 − 1/4 − −−→ exp −ǫ . 2M rn rn−1 ǫ→0

!

(8.24)

For a fixed set of rn , i.e., for a given path, the continuum limit ǫ → 0 makes the integrands (8.10) and (8.23) coincide, the difference being of the order ǫ2 . Unfortunately, the path integral requires the limit to be taken after the integrations over the drn . The integrals, however, do not exist at m = 0. For paths moving very close to the singularity at r = 0, the approximation (8.24) breaks down. In fact, the large-z expansion m2 − 1/4 (m2 − 1/4)(m2 − 1/9) I˜m (z) = 1 − + − ... 2z 2!(2z)2

(8.25)

with z = Mrn rn−1 /¯ hǫ is never convergent even for a very small ǫ. The series shows only an asymptotic convergence (more on this subject in Section 17.9). If we want

620


√ to evaluate I˜m (z) = 2πze−z Im (z) for all z we have to use the convergent power series expansion of Im (z) around z = 0: m "

z Im (z) = 2

2

1 1 z + 0!m! 1!(m + 1)! 2

4

1 z + 2!(m + 2)! 2

#

+ ... .

It is known from the Schrödinger theory that the leading power z m determines the threshold behavior of the quantum-mechanical particle distribution near the origin. This is qualitatively different from the exponentially small distribution exp [−ǫ¯ h(m2 − 1/4)/2Mr 2 ] contained in each time slice of the Feynman formula (8.23) for |m| > 1/2. The root of these troubles is an anomalous behavior in the high-temperature limit of the partition function. In this limit, the imaginary time difference τb −τa = h ¯ /kB T is very small and it is usually sufficient to keep only a single slice in a time-sliced path integral (see Sections 2.9, 2.13). If this were true also here, the formula (8.23) would lead, in the absence of a potential V (r), to the classical particle distribution [compare (2.341)] h ¯ m2 − 1/4 exp −ǫ . (ra τa + ǫ|ra τa ) = q 2M ra2 2π¯ hǫ/M !

1

(8.26)

If we subtract the barrier-free distribution, this amounts to the classical partition function1 Zcl =

Z

∞

0

s

dr m2 − 1/4 1 1 √ exp −a −1 = − m2 − , 2 2r 2 4 2πa "

!

#

(8.27)

where we have abbreviated the factor ǫ¯ h/M by a. The integral is temperatureindependent and converges only for |m| > 1/2. Compare this result with the proper high-temperature limit of the exact partition function calculated with the use of (8.17) and with the same subtraction as before. It reads for all T Z 1 ∞ Z= dze−z [Im (z) − I1/2 (z)]. (8.28) 2 0 As in the classical expression, there is no temperature dependence. The integral Z

0

∞

−1/2

dze−αz Iµ (z) = (α2 − 1)

(α +

√

−µ

α2 − 1)

(8.29)

[see formula (2.467)] converges for arbitrary real ν and α > 1, and gives in the limit α→1 1 Z = − (m − 1/2). (8.30) 2 √ p √ R∞ 2 2 This follows by expanding the formula 0 dxe−a/x −bx = π/4be−2 ab in powers of b, subtracting the a = 0 -term, and taking the limit b → 0. 1


8.2 Trouble with Feynman’s Path Integral Formula in Radial Coordinates

621

This is different from the classical result (8.27) and agrees with it only in the limit of large m.2 Thus we conclude that a time-sliced path integral containing a centrifugal barrier can only give the correct amplitude when using the Euclidean action AÑ m =

N +1 X n=1

M M (rn − rn−1 )2 − h ¯ logI˜m rn rn−1 2ǫ h ¯ǫ

,

(8.31)

in which the neighborhood of the singularity is treated quantum-mechanically. The naively time-sliced classical action in (8.23) is of no use. The centrifugal barrier renders therefore a counterexample to Feynman rules of path integration according to which quantum-mechanical amplitudes should be obtainable from a sum over all histories of exponentials which involve only the classical expression for the short-time actions. It is easy to see where the derivation of the time-sliced path integral in Section 2.1 breaks down. There, the basic ingredient was the Trotter product formula (2.26), which for imaginary time reads ˆ

ˆ

e−β(T +V ) = lim

N →∞

ˆ

ˆ

ˆ N +1

ˆ

e−ǫT e−ǫV ˆ

ˆ

An exact identity is e−β(T +V ) ≡ e−ǫT e−ǫV eiǫ

2X

ǫ ≡ β/(N + 1).

,

(8.32)

N +1

with X given in Eq. (2.10) consisting of a sum of higher and higher commutators between Vˆ and Tˆ . The Trotter formula neglects these commutators. In the presence of a centrifugal barrier, however, this is not permitted. Although the neglected commutators carry increasing powers of the small quantity ǫ, they are more and more divergent at r = 0, like ǫn /r2π . The same terms occur in the asymptotic expansion (8.25) of the “Bessel function”. In the proper action (8.31), these terms are present. It should be noted that for a Hamiltonian possessing a centrifugal barrier Vcb in addition to an arbitrary smooth potential V , i.e., for a Hamiltonian operator of the ˆ = Tˆ + Vˆcf + Vˆ , the Trotter formula is applicable in the form form H e−βH = lim

N →∞

ˆ

ˆ N +1

ˆ

e−ǫ(T +Vcb ) e−ǫV

,

ǫ ≡ β/(N + 1).

(8.33)

It leads to a valid time-sliced path integral formula (rb τb |ra τa )m ≈ 2

N Z Y

n=1

0

∞

drn

" NY +1 Z n=1

#

dpn 1 exp − AÑ [p, r] , 2π¯ h h ¯ m

(8.34)

If m were not merely a dimensionless number parametrizing an arbitrary centrifugal barrier with fixed p2ϕ /2M r2 , as it is in this section, the classical limit at a fixed pϕ would eliminate the problem since for h ¯ → 0, the number m would become infinitely large, leading to the correct high-temperature limit of the partition function. For a fixed finite m, however, the discrepancy is unavoidable.

622


with the sliced action AÑ m [p, r]

=

N +1 X n=1

"

−ipn (rn − rn−1 ) + ǫ

p2n 2M

M −h ¯ logI˜m rn rn−1 + ǫV (rn ) . h ¯ǫ

(8.35)

After integrating out the momenta, this becomes 1

N Y

 Z 

(8.36)

M Mrn rn−1 (rn − rn−1 )2 − h ¯ logI˜m + ǫV (rn ) . 2ǫ h ¯ǫ

(8.37)

n=1

with the action

N +1 X n=1



1 − AÑ [r] , h ¯ m

(rb τb |ra τa )m ≈ q 2π¯ hǫ/M

AÑ ˙ = m [r, r]

drn

0

∞

q

2π¯ hǫ/M

 exp

The path integral formula (8.36) can in principle be used to find the amplitude for a fixed angular momentum of some solvable systems. An example is the radial harmonic oscillator at an angular momentum m, although it should be noted that this particularly simple example does not really require calculating the integrals in (8.36). The result can be found much more simply from a direct angular momentum decomposition of the amplitude (2.170). After a continuation to imaginary times t = −iτ and an expansion of part of the exponent with the help of (8.5), it reads for D = 2 (xb τb |xa τa ) =

Mω 1 Mω 2 2 e− 2¯h coth[ω(τb −τa )](rb +ra ) 2π h ¯ sinh[ω(τb − τa )] ! ∞ X Mωrb ra × Im eim(ϕb −ϕa ) . h ¯ sinh[ω(τ − τ )] b a m=−∞

By comparison with (8.9), we extract the radial amplitude ! √ ω rb ra M Mωrb ra (rb τb |ra τa )m = Im . h ¯ coth[ω(τb − τa )] h ¯ sinh[ω(τb − τa )] The limit ω → 0 gives the free-particle result √ M rb ra M rb ra (rb τb |ra τa )m = Im . h ¯ τb − τa h ¯ τb − τa

8.3

(8.38)

(8.39)

(8.40)

Cautionary Remarks

It is important to emphasize that we obtained the correct amplitudes by performing the time slicing in Cartesian coordinates followed by the transformation to the polar coordinates in the time-sliced expression. Otherwise we would have easily missed H. Kleinert, PATH INTEGRALS

623

8.3 Cautionary Remarks

the factor −1/4 in the centrifugal barrier. To see what can go wrong let us proceed illegally and do the change of variables in the initial continuous action. Thus we try to calculate the path integral (xb τb |xa τa ) =

Z

X

∞

l=−∞,∞ 0 Z τ 1 b

× exp

h ¯

τa

dτ

Dr r

Z

∞

−∞

Dϕ

M 2 (r˙ + r 2 ϕ˙ 2 ) + V (r) 2

.

(8.41)

ϕ(τb )=ϕb +2πl

The summation over all periodic repetitions of the final azimuthal angle ϕ accounts for its multivaluedness according to the rules of Section 6.1. If the expression (8.41) is time-sliced straightforwardly it reads 



Z ∞ N Y 1 drn rn Z ∞ dϕn   q q hǫ/M n=1 0 2π¯ hǫ/M −∞ 2π¯ hǫ/M ϕN+1 =ϕb +2πl,l=−∞,∞ 2π¯ X

+1 1 NX M ×exp − ǫ [(rn − rn−1 )2 + rn2 (ϕn − ϕn−1 )2 ] + V (rn ) 2 h ¯ n=1 2ǫ

(

)

, (8.42)

where rb = rN +1 , ϕa = ϕ0 , ra = r0 . The integrals can be treated as follows. We introduce the momentum integrals over (pϕ )n conjugate to ϕn , writing for each n [with the short notation (pϕ )n ≡ pn ] rn q

2 2 (ϕ −ϕ −(M/2¯ hǫ)rn n n−1 )

2π¯ hǫ/M

e

=

Z

∞

−∞

dpn 1 ǫ p2n i exp − + pn (ϕn − ϕn−1 ) . 2 2π¯ h h ¯ 2M rn h ¯ (

)

(8.43)

After this, the integrals over ϕn (n = 1 . . . N) in (8.42) enforce all pn to be equal to each other, i.e., pn ≡ p for n = 1, . . . , N + 1, and we remain with "Z # ∞ Z ∞ N ∞ 1 1 X dp (i/¯h)p(ϕb −ϕa +2πl) Y drn (xb τb |xa τa ) ≈ e 2π¯ hǫ/M rb l−∞ −∞ 2π¯ h 2π¯ hǫ/M 0 n=1 +1 ǫ NX M (rn − rn−1 )2 p2 × exp − + + V (rn ) h ¯ n=1 2 ǫ2 2Mrn2

(

"

#)

.

(8.44)

Performing the sum over l with the help of Poisson’s formula (6.9) changes the integral over dp/2π¯ h into a sum over integers m and yields the angular momentum decomposition (8.9) with the partial-wave amplitudes (rb τb |ra τa )m ≈

√

"Z # N ∞ 1 1 Y drn rb ra 2π¯ hǫ/M rb n=1 0 2π¯ hǫ/M

+1 ǫ NX M (rn − rn−1 )2 h ¯ 2 m2 × exp − + + V (rn ) h ¯ n=1 2 ǫ2 2M rn2

(

"

#)

.

(8.45)

624


This wrong result differs from the correct one in Eq. (8.10) in three respect. First, it does not possess the proper h log I˜m (Mrb ra /¯ hǫ). Second, there is q centrifugal term −¯ a spurious overall factor ra /rb . Third, in comparison with the limiting expression (8.24), the centrifugal barrier lacks the term 1/4. It is possibleq to restore the term 1/4 by observing that in the time-sliced expression, the factor rb /ra is equal to NY +1 n=1

s

NY +1 rb rn −rn−1 = 1+ ra rn−1 n=1

1 = exp − 2

N +1 X n=1

!−1/2

=

NY +1

1 rn −rn−1 1 (rn −rn−1 )2 1− √ + −... 2 rn rn−1 8 rn rn−1

!

n=1

!

rn − rn−1 + ... . √ rn rn−1

(8.46)

This, in turn, can be incorporated into the kinetic term of (8.45) via a quadratic completion leading to  



+1 ǫ NX M rn − rn−1 h ¯  +i exp − √ h ¯ n=1 2 ǫ 2M rn rn−1

!2



h ¯ 2 m2 − 1/4  + .  2M rn2

(8.47)

The centrifugal barrier is now correct, but the kinetic term is wrong. In fact, it does not even correspond to a Hermitian Hamiltonian operator, as can be seen by introducing momentum integrations and completing the square to ǫ p2n pn − i¯ h exp −ipn (rn − rn−1 ) + h ¯ 2M 2Mrn (

"

#)

.

(8.48)

The last term is an imaginary energy. Only by dropping it artificially would the time-sliced action acquire the Feynman form (8.23), while still being beset with the problem of nonexistence for m = 0 (path collapse) and the nonuniform convergence of the path integrations to be solved in Chapter 12. The lesson of this is the following: A naive time slicing cannot be performed in curvilinear coordinates. It can safely be done in the Cartesian formulation. Fortunately, a systematic modification of the naive slicing rules has recently been found which makes them applicable to non-Cartesian systems. This will be shown in Chapters 10 and 11. In the sequel it is useful to maintain, as far as possible, the naive notation for the radial path integral (8.23) and the continuum limit of the action (8.21). The places where care has to be taken in the time-slicing process will be emphasized by setting the centrifugal barrier in quotation marks and defining “ h Mrn rn−1 ¯ 2 m2 − 1/4 ” ≡ −¯ h log I˜m . ǫ 2M rn rn−1 h ¯ǫ

(8.49)

Thus we shall write the properly sliced action (8.37) as AÑ ˙ = m [r, r]

N +1 X n=1

"

“ h M ¯ 2 m2 − 1/4 ” 2 (rn − rn−1 ) + ǫ + ǫV (rn ) , 2ǫ 2µ rn rn−1 #


625

8.4 Time Slicing Corrections

and emphasize the need for the non-naive time slicing of the continuum action correspondingly: Am =

8.4

Z

τb

τa

¯ 2 m2 − 1/4 ” M 2 “h dτ + V (r) . r˙ + 2 2M r2 "

#

(8.51)

Time Slicing Corrections

It is interesting to find the origin of the above difficulties. For this purpose, we take the Cartesian kinetic terms expressed in terms of polar coordinates (xn − xn−1 )2 = (rn − rn−1 )2 + 2rn rn−1 [1 − cos(ϕn − ϕn−1 )],

(8.52)

and treat it perturbatively in the coordinate differences [2]. Expanding the cosine into a power series we obtain the time-sliced action AN =

+1 +1 M NX M NX (xn − xn−1 )2 = (rn − rn−1 )2 2ǫ n=1 2ǫ n=1 1 1 + 2rn rn−1 (ϕn − ϕn−1 )2 − (ϕn − ϕn−1 )4 + . . . .(8.53) 2! 4!

In contrast to the naively time-sliced expression (8.42), we now keep the quartic term (ϕn − ϕn−1 )4 . To see how it contributes, consider a single intermediate integral Z

dϕn−1 q

2πǫ/a

e−(a/2ǫ)[(ϕn −ϕn−1 )

].

4 2 +a (ϕ −ϕ n 4 n−1 ) +...

(8.54)

The first term in the exponent restricts the width of the fluctuations of the difference ϕn − ϕn−1 to ǫ (8.55) h(ϕn − ϕn−1 )2 i0 = . a √ If we rescale the arguments, ϕn → ǫun , the integral takes the form Z

dun−1 q

2π/a

2

e−(a/2)[(un −un−1 )

+ǫa4 (un −un−1 )4 +...]

.

(8.56)

This shows that each by an √ higher power in the difference un − un−1 is suppressed √ additional factor ǫ. We now expand the integrand in powers of ǫ and use the integrals  2    u  a−1            4    Z  3a−2  du −(a/2)u2  u   q e = (8.57) . ..  ,    ..    .  2π/a   

u

2n

  

  

−n

(2n − 1)!!a

  

with odd powers of u giving trivially 0, to find an expansion of the integral (8.56). It begins as follows: a 1 − ǫ a4 3a−2 + O(ǫ2 ). (8.58) 2

626


This can be thought of as coming from the equivalent integral Z

dϕn−1 q

2πǫ/a

2

e−(a/2ǫ)(ϕn −ϕn−1 )

−3a4 ǫ/2a+...

.

(8.59)

The quartic term in (8.54), ∆A =

a a4 (ϕn − ϕn−1 )4 , 2ǫ

(8.60)

has generated an effective action-like term in the exponent: Aeff = ǫ

3a4 . 2a

(8.61)

This is obviously due to the expectation value h∆Ai0 of the quartic term and we can record, for later use, the perturbative formula Aeff = h∆Ai0 .

(8.62)

If u is a vector in D dimensions to be denoted by u, with the quadratic term being (a/2ǫ)(un − un−1 )2 , the integrals (8.57) are replaced by Z

dD u q

−(a/2)u De

2π/a

      2     

ui uj ui uj uk ul .. . ui1 · . . . · ui2n

a−1 δij a−2 (δij δkl + δik δjl + δil δjk ) = ..       .        

       

a−n δi1 ...i2n

     

, (8.63)

    

where δi1 ...i2n will be referred to as contraction tensors, defined iteratively by the recursion relation δi1 ...i2n = δi1 i2 δi3 i4 ...i2n + δi1 i3 δi2 i4 ...i2n + . . . + δi1 i2n δi2 i3 ...i2n−1 .

(8.64)

A comparison with the Wick expansion (3.303) shows that this recursion relation amounts to δi1 ...i2n possessing a Wick-like expansion into the sum of products of Kronecker δ’s, each representing a pair contraction. Indeed, the integral formulas (8.63) can be derived by adding a source term j · u to the exponent in the integrand, 2 completing the square, and differentiating the resulting e(1/2a)j with respect to the “current” components ji . For vectors ϕi , a possible quartic term in the exponent of (8.54) may have the form ∆A =

a (a4 )ijkl (ϕn − ϕn−1 )i (ϕn − ϕn−1 )j (ϕn − ϕn−1 )k (ϕn − ϕn−1 )l . 2ǫ

(8.65)

Then the factor 3a4 in Aeff of Eq. (8.61) is replaced by the three contractions (a4 )ijkl (δij δkl + 2 more pair terms). H. Kleinert, PATH INTEGRALS

627

8.4 Time Slicing Corrections

Applying the simple result (8.61) to the action (8.53), where a = (M/¯ h)2rn rn−1 and a4 = −1/4!, we find that the naively time-sliced kinetic term of the ϕ field M rn rn−1 (ϕn − ϕn−1 )2 2ǫ is extended to M 1/4 rn rn−1 (ϕn − ϕn−1 )2 − ǫ¯ h2 + ... . 2ǫ 2Mrn rn−1 Thus, the lowest perturbative correction due to the fourth-order expansion term of cos(ϕn − ϕn−1 ) supplies precisely the 1/4-term in the centrifugal barrier which was missing in (8.45). Proceeding in this fashion, the higher powers in the expansion of cos(ϕn − ϕn−1 ) give higher and higher contributions (ǫ/rn rn−1 )n . Eventually, they would of course produce the entire asymptotic expansion of the “Bessel function” in the correct time-sliced action (8.37). Note that the failure of this series to converge destroys the justification for truncating the perturbation series after any finite number of terms. In particular, the knowledge of the large-order behavior (the “tail end” of the series) [3] is needed to recover the correct threshold behavior ∝ r m in the amplitudes observed in (8.26). The reader may rightfully object that the integral (8.56) should really contain an exponential factor exp [−(a/2)(un−1 − un−2 )2 ] from the adjacent time slice which also contains the variable un−1 and which has been ignored in the integral (8.56) over un−1 . In fact, with the abbreviations u¯n−1 ≡ (un + un−2)/2, δ ≡ un−1 − u¯n−1 , ∆ ≡ un − un−2 , the complete integrand containing the variable un−1 can be written as Z

dδ q

2π/a

2

e−aδ e−a∆

2 /4

a 1 − ǫa4 [(−δ + ∆/2)4 + (δ + ∆/2)4 ] + . . . . 2

(8.66)

When doing the integral over δ, each even power δ 2n gives a factor (1/2a)(2n − 1)!! and we observe that the mean value of the fluctuating un−1 is different from what it was above, when we singled out the expression (8.56) and ignored the un−1 dependence of the adjacent integral. Instead of un in (8.56), the mean value of un−1 is now the average position of the neighbors, u¯n−1 = (un + un−2)/2. Moreover, instead of the width of the un−1 fluctuations being h(un − u¯n−1)2 i0 ∼ 1/a, as in (8.55), it is now given by half this value: h(un−1 − u¯n−1 )2 i0 =

1 . 2a

(8.67)

At first, these observations seem to invalidate the above perturbative evaluation of (8.56). Fortunately, this objection ignores an important fact which cancels the apparent mistake, and the result (8.62) of the sloppy derivation is correct after all. The argument goes as follows: The integrand of a single time slice is a sharply peaked function of the coordinate difference whose width is of the order ǫ and goes to zero in the continuum limit. If such a function is integrated together with some

628


smooth amplitude, it is sensitive only to a small neighborhood of a point in the amplitude. The sharply peaked function is a would-be δ-function that can be effectively replaced by a δ-function plus correction terms which contain increasing derivatives √ of δ-functions multiplied by corresponding powers of ǫ. Indeed, let us take the integrand of the model integral (8.54), 1

e−(a/2ǫ)[(ϕn −ϕn−1 )

],

4 2 +a (ϕ −ϕ n 4 n−1 ) +...

q

2πǫ/a

(8.68)

and integrate it over ϕn−1 together with a smooth amplitude ψ(ϕn−1 ) which plays the same role of a test function in mathematics [recall Eq. (1.162)]: Z

dϕn−1

∞

−∞

q

e−(a/2ǫ)[(ϕn −ϕn−1 )

] ψ(ϕ

4 2 +a (ϕ −ϕ n 4 n−1 ) +...

2πǫ/a

n−1 ).

(8.69)

For small ǫ, we expand ψ(ϕn−1 ) around ϕn , 1 ψ(ϕn−1 ) = ψ(ϕn ) − (ϕn − ϕn−1 )ψ ′ (ϕn ) + (ϕn − ϕn−1 )2 ψ ′′ (ϕn ) + . . . , 2 (8.70) and (8.69) becomes (1 − Aeff )ψ(ϕn ) +

ǫ ′′ ψ (ϕn ) + . . . . 2a

(8.71)

This shows that the amplitude for a single time slice, when integrated together with a smooth amplitude, can be expanded into a series consisting of a δ-function and its derivatives: 1 q

e−(a/2ǫ)[(ϕn −ϕn−1 )

]

4 2 +a (ϕ −ϕ n 4 n−1 ) +...

2πǫ/a

= (1 − Aeff )δ(ϕn − ϕn−1 ) +

ǫ ′′ δ (ϕn − ϕn−1 ) + . . . . 2a

(8.72)

The right-hand side may be viewed as the result of a simpler would-be δ-function 1 q

2πǫ/a

2

e−(a/2ǫ)(ϕn −ϕn−1 ) e−Aeff ,

(8.73)

correct up to terms of order ǫ. This is precisely what we found in (8.59). The problems observed above arise only if the would-be δ-function in (8.72) is integrated together with another sharply peaked neighbor function which is itself a would-be δ-function. Indeed, in the theory of distributions, it is strictly forbidden to form integrals over products of two proper distributions. For the would-be distributions at hand the rule is not quite as strict and integrals over products can be formed. The crucial expansion (8.72), however, is no longer applicable if the H. Kleinert, PATH INTEGRALS

629

8.5 Angular Decomposition in Three and More Dimensions

accompanying function is a would-be δ-function, and a more careful treatment is required. The correctness of formula (8.62) derives from the fact that each time slice has, for sufficiently small ǫ, a large number of neighbors at earlier and later times. If the integrals are done for all these neighbors, they render a smooth amplitude before and a smooth amplitude after the slice under consideration. Thus, each intermediate integral in the time-sliced product contains a would-be δ-function multiplied on the right- and left-hand side with a smooth amplitude. In each such integral, the replacement (8.59) and thus formula (8.62) is correct. The only exceptions are time slices near the endpoints. Their integrals possess the above subtleties. The relative number of these, however, goes to zero in the continuum limit ǫ → 0. Hence they do not change the final result (8.62). For completeness, let us state that the presence of a cubic term in the single-sliced action (8.68) has the following δ-function expansion 1 q

e−(a/2ǫ)[(ϕn −ϕn−1 )

]

3 4 2 +a (ϕ −ϕ n 3 n−1 ) +a4 (ϕn −ϕn−1 ) +...

2πǫ/a

= (1 − Aeff )δ(ϕn − ϕn−1 ) + 3a3

(8.74)

ǫ ǫ ′ δ (ϕn − ϕn−1 ) + δ ′′ (ϕn − ϕn−1 ) + . . . 2a 2a

corresponding to an “effective action” Aeff =

ǫ 15 3a4 − a23 . 2a 4

(8.75)

Using (8.75), the left-hand side of (8.74) can also be replaced by the would-be δfunction 1 q

2

2πǫ/a

e−(a/2ǫ)(ϕn −ϕn−1 ) e−Aeff [1 −

3a3 (ϕn − ϕn−1 ) + . . .], 2

(8.76)

which has the same leading terms in the δ-function expansion. In D dimensions, the term 3a3 (ϕn − ϕn−1 ) has the general form [(a3 )ijj + (a3 )jij + (a3 )jji](ϕn − ϕn−1 )i , and the term 15a23 in Aeff becomes (a3 )ijk (a3 )i′ j ′ k′ (δii′ δjj ′ δkk′ + 14 more pair terms).

8.5

Angular Decomposition in Three and More Dimensions

Let us now extend the two-dimensional development of Section 8.2 and study the radial path integrals of particles moving in three and more dimensions. Consider the amplitude for a rotationally invariant action in D dimensions (xb τb |xa τa ) =

Z

1 D x(τ ) exp − h ¯ D

Z

τb

τa

M 2 x˙ + V (r) 2

.

(8.77)

630


By time-slicing this in Cartesian coordinates, the kinetic term gives an integrand +1 1 M NX 2 (r 2 + rn−1 − 2rn rn−1 cos ∆ϑn ) , exp − h ¯ 2ǫ n=1 n

"

#

(8.78)

where ∆ϑn is the relative angle between the vectors xn and xn−1 .

8.5.1

Three Dimensions

In three dimensions, we go over to the spherical coordinates x = r(cos θ cos ϕ, cos θ sin ϕ, sin θ)

(8.79)

cos ∆ϑn = cos θn cos θn−1 + sin θn sin θn−1 cos(ϕn − ϕn−1 ).

(8.80)

and write

The integration measure in the time-sliced version of (8.77), 1 q

2π¯ hǫ/M

becomes

1 q

2π¯ hǫ/M

3

3

N Z Y

n=1

N Z Y

n=1

d3 xn q

2π¯ hǫ/M

3,

drn rn2 d cos θn dϕn q

2π¯ hǫ/M

3

(8.81)

.

(8.82)

To perform the integrals, we use the spherical analog of the expansion (8.5) h cos ∆ϑn

e

r

=

∞ π X Il+1/2 (h)(2l + 1)Pl (cos ∆ϑn ), 2h l=0

(8.83)

where Pl (z) are the Legendre polynomials. These, in turn, can be decomposed into spherical harmonics m

Ylm (θ, ϕ) = (−1)

"

2l + 1 (l − m)! 4π (l + m)!

#1/2

Plm (cos θ)eimϕ ,

(8.84)

with the help of the addition theorem l X 2l + 1 ∗ Pl (cos ∆ϑn ) = Ylm (θn , ϕn )Ylm (θn−1 , ϕn−1 ), 4π m=−l

(8.85)

the sum running over all azimuthal (magnetic) quantum numbers m. The right-hand side of Plm (z) contains the associated Legendre polynomials Plm (z) =

(1 − z 2 )m/2 (l − m)! dl+m 2 (z − 1)l , 2l l! (l + m)! dz l+m

(8.86)


631


which are solutions of the differential equation3 "

1 d d − sin θ sin θ dθ dθ

m2 + P m (cos θ) = l(l + 1)Plm(cos θ). sin2 θ l

!

#

(8.87)

Thus, the expansion (8.83) becomes h cos ∆ϑn

e

r

=

∞ l X X π ∗ Il+1/2 (h) Ylm (θn , ϕn )Ylm (θn−1 , ϕn−1 ). 4π 2h l=0 m=−l

(8.88)

Inserted into (8.78), it leads to the time-sliced path integral  Z N Y  

4π (xb τb |xa τa ) ≈ q 3 2π¯ hǫ/M ×

" NY +1 n=1

h ¯ ǫπ 2Mrn rn−1

!1/2

n=1

∞

0

2π¯ hǫ/M

ln X

∞ X



drn rn2 d cos θn dϕn 4π   q 3 Iln +1/2

ln =0 mn =−ln

M rn rn−1 h ¯ǫ

+1 2 ǫ NX M rn2 + rn−1 × Yln mn (θn , ϕn )Yln mn (θn−1 , ϕn−1 ) exp − +V (rn ) . h ¯ n=1 2 ǫ2

#

∗

(

"

#)

(8.89)

The intermediate ϕn - and cos θn -integrals can now all be done using the orthogonality relation Z

1

−1

d cos θ

Z

π

−π

∗ dϕ Ylm (θ, ϕ)Yl′ m′ (θ, ϕ) = δll′ δmm′ .

(8.90)

Each ϕn -integral yields a product of Kronecker symbols δln ln−1 δmn mn−1 . Only the initial and the final spherical harmonics survive, YlN+1 mN+1 and Yl0∗m0 , since they are not subject to integration. Thus we arrive at the angular momentum decomposition

(xb τb |xa τa ) =

∞ X l X

1 ∗ (rb τb |ra τa )l Ylm (θb , ϕb )Ylm (θa , ϕa ), l=0 m=−l rb ra

(8.91)

with the radial amplitude 4πrb ra

(rb τb |ra τa )l ≈ q 3 2π¯ hǫ/M × 3

NY +1 n=1

N Y

n=1

  " Z +1 2 drn rn 4π  NY   q 3

2π¯ hǫ/M

n=1

h ¯ǫ 2Mrn rn−1

#

(8.92)

+1 M M ǫ NX ˜ Il+1/2 ( rn rn−1 ) exp − (rn − rn−1 )2 + V (rn ) 2 h ¯ǫ h ¯ n=1 2ǫ

Note that ylm (cos θ) =

(

h √ sin θPlm (cos θ) satisfies − dθd2 −

differential equation will be used later in Eq. (8.196).

1 4

+

m2 −1/4 sin2 θ

i

)

.

ylm = l(l + 1)ylm . This

632


N +1 2 The factors N rn rn−1 pile up to 1/rb ra and cancel the prefactor rb ra . a rn / a Together with the remaining product, the integration measure takes the usual onedimensional form   Z ∞ N Y dr 1 n  . q q (8.93) 0 2π¯ hǫ/M n=1 2π¯ hǫ/M

Q

Q

If we were to use here the large-argument limit (8.24) of the Bessel function, the integrand would become exp(−AN h), with the time-sliced radial action l /¯ AN l

=ǫ

N +1 X n=1

"

M h ¯ 2 l(l + 1) 2 (r − r ) + + V (rn ) . n n−1 2ǫ2 2M rn rn−1 #

(8.94)

1 − AN h ¯ l

(8.95)

The associated radial path integral (rb τb |ra τa )l ≈

1 q

2πǫ¯ h/M

N Y

n=1

 

Z

drn

∞

q

2π¯ hǫ/M

0



 exp

agrees precisely with what would have been obtained by naively time-slicing the continuum path integral (rb τb |ra τa )l =

Z

∞

0

1

Dr(τ )e− h¯ Al [r] ,

(8.96)

with the radial action Al [r] =

Z

τb

τa

h ¯ 2 l(l + 1) M 2 dτ r˙ + + V (r) . 2 2M r 2 "

#

(8.97)

In particular, this would contain the correct centrifugal barrier Vcf =

h ¯ 2 l(l + 1) . 2M r 2

(8.98)

However, as we know from the discussion in Section 8.2, Eq. (8.95) is incorrect and must be replaced by (8.92), due to the non uniformity of the continuum limit ǫ → 0 in the integrand of (8.92).

8.5.2

D Dimensions

The generalization to D dimensions is straightforward. The main place where the dimension enters is the expansion of M

eh cos ∆ϑn = e− h¯ ǫ rn rn−1 cos ∆ϑn ,

(8.99)

in which ∆ϑn is the relative angle between D-dimensional vectors xn and xn−1 . The expansion reads [compare with (8.5) and (8.83)] eh cos ∆ϑn =

∞ X l=0

al (h)

l + D/2 − 1 1 (D/2−1) C (cos ∆ϑn ), D/2 − 1 SD l

(8.100)



633

where SD is the surface of a unit sphere in D dimensions (1.555), and al (h) ≡ (2π)D/2 h1−D/2 Il+D/2−1 (h) h

h

≡ e a ˜l (h) = e

2π h

(D−1)/2

I˜l+D/2−1 (h).

(8.101)

(α)

The functions Cl (cos ϑ) are the ultra-spherical Gegenbauer polynomials. The ex(ν) pansion (8.100) follows from the completeness of the polynomials Cl (cos ϑ) at fixed ν, using the integration formulas4 Z

π 0

Z

(ν)

dϑ sinν ϑeh cos ϑ Cl (cos ϑ) = π π

0

(ν)

21−ν Γ(2ν + l) −ν h Iν+l (h), l!Γ(ν)

(ν)

dϑ sinν ϑCl (cos ϑ)Cl′ (cos ϑ) = π

21−2ν Γ(2ν + l) δll′ . l!(l + ν)Γ(ν)2

(8.102) (8.103)

The Gegenbauer polynomials are related to Jacobi polynomials, which are defined in terms of hypergeometric functions (1.450) by5 (α,β)

Pl

(z) ≡

1 Γ(l + 1 + β) F (−l, l + 1 + α + β; 1 + β; (1 + z)/2). l! Γ(1 + β)

(8.104)

The relation is (ν)

Cl (z) =

Γ(2ν + l)Γ(ν + 1/2) (ν−1/2,ν−1/2) (z). P Γ(2ν)Γ(ν + l + 1/2) l

(8.105)

This follows from the defining equation6 (ν)

Cl (z) =

1 Γ(l + 2ν) F (−l, l + 2ν; 1/2 + ν; (1 + z)/2). l! Γ(2ν)

(8.106)

For D = 2 and 3, one has7 1 (ν) 1 lim Cl (cos ϑ) = cos lϑ, ν 2l

ν→0 (1/2)

Cl

(0,0)

(cos ϑ) = Pl

(cos ϑ) = Pl (cos ϑ),

(8.107) (8.108)

and the expansion (8.100) reduces to (8.5) and (8.7), respectively. For D = 4 (1)

Cl (cos ϑ) = 4

sin(l + 1)β . sin β

I. S. Gradshteyn and I. M. Ryzhik, op. cit., Formulas 7.321 and 7.313. M. Abramowitz and I. Stegun, op. cit., Formula 15.4.6. 6 ibid., Formula 15.4.5. 7 I.S. Gradshteyn and I.M. Ryzhik, op. cit.,ibid., Formula 8.934.4. 5

(8.109)

634


According to an addition theorem, the Gegenbauer polynomials can be decomposed into a sum of pairs of D-dimensional ultra-spherical harmonics Ylm (ˆ x).8 The label m stands collectively for the set of magnetic quantum numbers ˆ of a m1 , m2 , m3 , ..., mD−1 with 1 ≤ m1 ≤ m2 ≤ . . . ≤ |mD−2 |. The direction x vector x is specified by D − 1 polar angles xˆ1 = sin ϕD−1 · · · sin ϕ1 , xˆ2 = sin ϕD−1 · · · cos ϕ1 , .. . xˆD = cos ϕD−1 ,

(8.110)

with the ranges 0 ≤ ϕ1 < 2π, 0 ≤ ϕi < π, i 6= 1.

(8.111) (8.112)

The ultra-spherical harmonics Ylm (ˆ x) form an orthonormal and complete set of functions on the D-dimensional unit sphere. For a fixed quantum number l of total angular momentum, the label m can take dl =

(2l + D − 2)(l + D − 3)! l!(D − 2)!

(8.113)

different values. The functions are orthonormal, Z

with Z

R

∗ ˆ Ylm dx (ˆ x)Yl′ m′ (ˆ x) = δll′ δmm′ ,

(8.114)

ˆ denoting the integral over the surface of the unit sphere: dx

ˆ = dx

Z

dϕD−1 sinD−2 ϕD−1

Z

dϕD−2 sinD−3 ϕD−2 · · ·

Z

dϕ2 sin ϕ2

Z

dϕ1 . (8.115)

By doing this integral over a unit integrand we find the value SD of (1.555). Since √ ˆ , the integral (8.114) implies that Y00 (ˆ Y00 (ˆ x) is independent of x x) = 1/ SD . The integral over the sphere in D-dimensions can be decomposed recursively into ˆ , in the space an angular integration with respect to any selected direction, say u followed by an integral over a sphere of radius sin ϕD−1 in the remaining D − 1ˆ . If x ˆ ⊥ denotes the unit vector covering the directions in this dimensional space to u ˆ = (cos ϕD u ˆ + sin ϕD x ˆ , and can factorize the remaining space, one decomposes x integral measure as Z

ˆ = dD−1 x

Z

dϕD−1 sinD−2 ϕD−1

Z

ˆ⊥ . dD−2 x

(8.116)

8

See H. Bateman, Higher Transcendental Functions, McGraw-Hill, New York, 1953, Vol. II, Ch. XI; N.H. Vilenkin, Special Functions and the Theory of Group Representations, Am. Math. Soc., Providence, RI, 1968. H. Kleinert, PATH INTEGRALS


635

For clarity, the dimensionalities of initial and remaining surfaces have been noted ˆ and d x ˆ⊥. as superscripts on the measure symbols d x For the surface of the sphere, this corresponds to the recursion relation [compare (1.555)] √ πΓ((D − 1)/2) SD = × SD−1 , (8.117) Γ(D/2) which is solved by (1.555). ˆ have a parametrization in terms of polar angles In four dimensions, unit vectors x ˆ = (cos θ, sin θ cos ψ, sin θ sin ψ cos ϕ, sin θ sin ψ sin ϕ), x

(8.118)

with the integration measure dˆ x = dθ sin2 θdψ sin ψ dϕ.

(8.119)

It is, however, more convenient to go over to another parametrization in terms of the three Euler angles which are normally used in the kinematic description of the spinning top. In terms of these, the unit vectors have the components xˆ1 xˆ2 xˆ3 xˆ4

= = = =

cos(θ/2) cos[(ϕ + γ)/2], − cos(θ/2) sin[(ϕ + γ)/2], sin(θ/2) cos[(ϕ − γ)/2], sin(θ/2) sin[(ϕ − γ)/2],

(8.120)

with the angles covering the intervals θ ∈ [0, π), ϕ ∈ [0, 2π), γ ∈ [−2π, 2π).

(8.121)

We have renamed the usual Euler angles α, β, γ introduced in Section 1.15 calling them ϕ, θ, γ, since the formulas to be derived for them will be used in a later application in Chapter 13 [see Eq. (13.97)]. There the first two Euler angles coincide with the polar angles ϕ, θ of a position vector in a three-dimensional space. It is important to note that for a description of the entire surface of the sphere, the range of the angle γ must be twice as large as for the classical spinning top. The associated group space belongs to the covering group, of the rotation group which is equivalent to the group of unimodular matrices in two dimensions called SU(2). It is defined by the matrices g(ϕ, θ, γ) = exp(iϕσ3 /2) exp(iθσ2 /2) exp(iγσ3 /2),

(8.122)

where σi are the Pauli spin matrices (1.445). In this parametrization, the integration measure reads 1 dˆ x = dθ sin θ dϕ dγ. (8.123) 8 When integrated over the surface, the two measures give the same result S4 = 2π 2 . The Euler parametrization has the advantage of allowing the spherical harmonics in

636


four dimensions to be expressed in terms of the well-known representation functions of the rotation group introduced in (1.442), (1.443): Yl,m1 ,m2 (ˆ x) =

s

l + 1 l/2 D (ϕ, θ, γ) = 2π 2 m1 m2

s

l + 1 l/2 d (θ)ei(m1 ϕ+m2 γ) . 2π 2 m1 m2

(8.124)

For even and odd l, the numbers m1 , m2 are both integer or half-integer, respectively. In arbitrary dimensions D > 2, the ultra-spherical Gegenbauer polynomials satisfy the following addition theorem X 2l + D − 2 1 (D/2−1) ∗ Cl (cos ∆ϑn ) = Ylm (ˆ xn )Ylm (ˆ xn−1 ). D − 2 SD m

(8.125)

For D = 3, this reduces properly to the well-known addition theorem for the spherical harmonics l X 1 ∗ Ylm (ˆ xn )Ylm (ˆ xn−1 ). (2l + 1)Pl (cos ∆ϑn ) = 4π m=−l

(8.126)

For D = 4, it becomes9 l/2 X l + 1 (1) l+1 l/2 ∗ C (cos ∆ϑ ) = D l/2 (ϕn , θn , γn )Dm (ϕn−1 , θn−1 , γn−1 ), n 1 m2 2π 2 l 2π 2 m1 ,m2 =−l/2 m1 m2

(8.127) where the angle ∆ϑn is related to the Euler angles of the vectors xn , xn−1 by cos ∆ϑn = cos(θn /2) cos(θn−1 /2) cos[(ϕn − ϕn−1 + γn − γn−1 )/2] + sin(θn /2) sin(θn−1 /2) cos[(ϕn − ϕn−1 − γn + γn−1 )/2].

(8.128)

Using (8.125), we can rewrite the expansion (8.100) in the form h(cos ∆ϑn −1)

e

=

∞ X l=0

a ˜l (h)

X

∗ Ylm (ˆ xn )Ylm (ˆ xn−1 ).

(8.129)

m

This is now valid for any dimension D, including the case D = 2 where the left-hand side of (8.125) involves the limiting procedure (8.107). We shall see in Chapter 9 in connection with Eq. (9.61) that it also makes sense to apply this expansion to the case D = 1 where the “partial-wave expansion” degenerates into a separation of even and odd wave functions. In four dimensions, we shall mostly prefer the expansion eh(cos ∆ϑn −1) =

∞ X l=0

l/2

a ˜l (h)

X l+1 l/2 ∗ D l/2 (ϕn , θn , γn )Dm (ϕn−1 , θn−1 , γn−1), 1 m2 2π 2 m1 ,m2 =−l/2 m1 m2

(8.130) 9

(1)

Note that Cl (cos ∆ϑn ) coincides with the trace over the representation functions (1.443) of Pl/2 l/2 the rotation group, i.e., it is equal to m=−l/2 dm,m (∆ϑn ). H. Kleinert, PATH INTEGRALS

637


where the sum over m1 , m2 runs for even and odd l over integer and half-integer numbers, respectively. The reduction of the time evolution amplitude in D dimensions to a radial path integral proceeds from here on in the same way as in two and three dimensions. The generalization of (8.89) reads  Z N Y  

1 (xb τb |xa τa ) ≈ q D 2π¯ hǫ/M ×

NY +1

×

X

n=1



n=1

2π¯ hǫ Mrn rn−1



∞ 0



drn rnD−1 dˆ xn  q D 2π¯ hǫ/M

!(D−1)/2

∞ X

M I˜D/2−1+ln rn rn−1 h ¯ǫ ln =0

Yln mn (ˆ xn )Yl∗n mn (ˆ xn−1 )

mn

#

(8.131)

+1 M ǫ NX exp − (rn − rn−1 )2 + V (rn ) 2 h ¯ n=1 2ǫ

(

)

.

By performing the angular integrals and using the orthogonality relations (8.114), the product of sums over ln , mn reduces to a single sum over l, m, just as in the three-dimensional amplitude (8.91). The result is the spherical decomposition (xb τb |xa τa ) =

1 (D−1)/2

(rb ra )

∞ X l=0

(rb τb |ra τa )l

X

∗ Ylm (ˆ xb )Ylm (ˆ xa ),

(8.132)

m

where (rb τb |ra τa )l is the purely radial amplitude 1

(rb τb |ra τa )l ≈ q 2π¯ hǫ/M

N Y

n=1

 

Z

0

drn

∞

q

2π¯ hǫ/M



 exp

1 − AN [r] , h ¯ l

(8.133)

with the time-sliced action AN l [r]

= ǫ

N +1 X n=1

#

M h ¯ M (rn − rn−1 )2 − logI˜l+D/2−1 rn rn−1 + V (rn ) . (8.134) 2 2ǫ ǫ h ¯ǫ

+1 (D−1)/2 D−1 As before, the product N has removed the product N n=1 1/(rn rn−1 ) n=1 rn in the measure as well as the factor (rb ra )(D−1)/2 in front of it, leaving only the standard one-dimensional measure of integration. In the continuum limit ǫ → 0, the asymptotic expression (8.24) for the Bessel function brings the action to the form

Q

AN ¯] l [r, r

≈ǫ

N +1 X n=1

"

Q

M h ¯ 2 (l + D/2 − 1)2 − 1/4 2 (r − r ) + + V (rn ) . (8.135) n n−1 2ǫ2 2M rn rn−1 #

This looks again like the time-sliced version of the radial path integral in D dimensions Z 1 (rb τb |ra τa )l = Dr(τ ) exp − Al [r] , (8.136) h ¯

638


with the continuum action Al [r] =

Z

τb

τa

M 2 “h ¯ 2 (l + D/2 − 1)2 − 1/4 ” dτ + V (r) . r˙ + 2 2M r2 "

#

(8.137)

As in Eq. (8.50), we have written the centrifugal barrier as “ h ” ¯2 [(l + D/2 − 1)2 − 1/4] , 2 2Mr

(8.138)

to emphasize the subtleties of the time-sliced radial path integral, with the understanding that the time-sliced barrier reads [as in (8.51)] “

8.6

” ǫ¯ h2 M [(l + D/2 − 1)2 − 1/4)] ≡ −¯ hlogI˜l+D/2−1 ( rn rn−1 ). 2Mrn rn−1 h ¯ǫ

(8.139)

Radial Path Integral for Harmonic Oscillator and Free Particle in D Dimensions

For the harmonic oscillator and the free particle, there is no need to perform the radial path integral (8.133) with the action (8.134). As in (8.38), we simply take the known amplitude in D dimensions, (2.170), continue it to imaginary times t = −iτ , and expand it with the help of (8.129): 1

(xb τb |xa τa ) = q D 2π¯ h/M (

s

ω sinh[ω(τb − τa )]

D

(8.140) )

1 Mω (rb2 + ra2 ) cosh[ω(τb − τa )] × exp − h ¯ sinh[ω(τb − τa )] ! ∞ X X Mωrb ra ∗ × al Ylm (ˆ xb )Ylm (ˆ xa ). h ¯ sinh[ω(τ − τ )] b a m l=0

Comparing this with Eq. (8.132) and remembering (8.101), we identify the radial amplitude as √ M ω rb ra (rb τb |ra τa )l = h ¯ sinh[ω(τb − τa )] ! Mωrb ra −(M ω/2¯ h) coth[ω(τb −τa )](rb2 +ra2 ) × e Il+D/2−1 , (8.141) h ¯ sinh[ω(τb − τa )] generalizing (8.39). The limit ω → 0 yields the amplitude for a free particle ! √ rb ra −M (r2 +ra2 )/2¯h(τb −τa ) M Mrb ra b (rb τb |ra τa )l = e Il+D/2−1 . (8.142) h ¯ (τb − τa ) h ¯ (τb − τa ) Comparing this with (8.40) on the one hand and Eqs. (8.139), (8.137) with (8.49), (8.51) on the other hand, we conclude: An analytical continuation in D yields the H. Kleinert, PATH INTEGRALS

8.7 Particle near the Surface of a Sphere in D Dimensions

639

path integral for a linear oscillator in the presence of an arbitrary 1/r2 -potential as follows: 1 τb M 2 “h ¯ 2 µ2 − 1/4 ” M 2 2 + ω r (rb τb |ra τa )l = Dr(τ ) exp − dτ r˙ + h ¯ τa 2 2M r2 2 0 ! √ ω rb ra Mωrb ra M −(M ω/2¯ h) coth[ω(τb−τa )](rb2 +ra2 ) = e Iµ . h ¯ sinh[ω(τb − τa )] h ¯ sinh[ω(τb − τa )] (8.143) Z

∞

"

!#

Z

Here µ is some strength parameter which initially takes the values µ = l + D/2 − 1 with integer l and D. By analytic continuation, the range of validity is extended to all real µ > 0. The justification for the continuation procedure follows from the fact that the integral formula (8.14) holds for arbitrary m = µ ≥ 0. The amplitude (8.143) satisfies therefore the fundamental composition law (8.20) for all real m = µ ≥ 0. The harmonic oscillator with an arbitrary extra centrifugal barrier potential l2 ¯ 2 extra2 (8.144) Vextra (r) = h 2Mr has therefore the radial amplitude (8.143) with µ=

q

2 (l + D/2 − 1)2 + lextra .

(8.145)

For a finite number N + 1 of time slices, the radial amplitude is known from the angular momentum expansion of the finite-N oscillator amplitude (2.192) in its obvious extension to D dimensions. It can also be calculated directly as in Appendix 2B by a successive integration of (8.131), using formula (8.14). The iteration formulas are the Euclidean analogs of those derived in Appendix 2B, with the prefactor of the √ amplitude being 2πN12 NN2 +1 rb ra , with the exponent −aN +1 (rb2 + ra2 )/¯ h, and with the argument of the Bessel function 2bN +1 rb ra /¯ h. In this way we obtain precisely the expression (8.143), except that sinh[ω(τb −τa )] is replaced by sinh[˜ ω (N +1)ǫ]ǫ/ sinh ω ˜ǫ and cosh[ω(τb − τa )] by cosh[˜ ω (N + 1)ǫ].

8.7

Particle near the Surface of a Sphere in D Dimensions

With the insight gained in the previous sections, it is straightforward to calculate exactly a certain class of auxiliary path integrals. They involve only angular variables and will be called path integrals of a point particle moving near the surface of a sphere in D dimensions. The resulting amplitudes lead eventually to the physically more relevant amplitudes describing the behavior of a particle on the surface of a sphere. On the surface of a sphere of radius r, the position of the particle as a function of time is specified by a unit vector u(t). The Euclidean action is M 2 A= r 2

Z

τb

τa

dτ u˙ 2 (τ ).

(8.146)

640


The precise way of time-slicing this action is not known from previous discussions. It cannot be deduced from the time-sliced action in Cartesian coordinates, nor from its angular momentum decomposition. A new geometric feature makes the previous procedures inapplicable: The surface of a sphere is a Riemannian space with nonzero intrinsic curvature. Sections 1.13 to 1.15 have shown that the motion in a curved space does not follow the canonical quantization rules of operator quantum mechanics. The same problem is encountered here in another form: Right in the beginning, we are not allowed to time-slice the action (8.146) in a straightforward way. The correct slicing is found in two steps. First we use the experience gained with the angular momentum decomposition of time-sliced amplitudes in a Euclidean space to introduce and solve the earlier mentioned auxiliary time-sliced path integral near the surface of the sphere. In a second step we shall implement certain corrections to properly describe the action on the sphere. At the end, we have to construct the correct measure of path integration which will not be what one naively expects. To set up the auxiliary path integral near the surface of a sphere we observe that the kinetic term of a time slice in D dimensions +1 M NX 2 (rn2 + rn−1 − 2rn rn−1 cos ∆ϑn ) 2ǫ n=1

(8.147)

decomposes into radial and angular parts as −

+1 +1 M NX M NX 2 (rn2 + rn−1 − 2rn rn−1 ) + 2rn rn−1 (1 − cos ∆ϑn ). 2ǫ n=1 2ǫ n=1

(8.148)

The angular factor can be written as −

+1 M NX ˆ n−1 )2 , rn rn−1 (ˆ xn − x 2ǫ n=1

(8.149)

ˆn, x ˆ n−1 are the unit vectors pointing in the directions of xn , xn−1 [recall where x (8.110)]. Restricting all radial variables rn to the surface of a sphere of a fixed radius ˆ with u leads us directly to the time-sliced path integral near the r and identifying x surface of the sphere in D dimensions: 1

(ub τb |ua τa ) ≈ q D−1 2π¯ hǫ/Mr 2 with the sliced action AN =

 Z N Y   q

n=1

dun

2π¯ hǫ/Mr 2



 D−1  exp

+1 M 2 NX r (un − un−1 )2 . 2ǫ n=1

1 − AN , h ¯

(8.150)

(8.151)

The measure dun denotes infinitesimal surface elements on the sphere in D dimensions [recall (8.115)]. Note that although the endpoints un lie all on the sphere, the paths remain only near the sphere since the path sections between the points leave H. Kleinert, PATH INTEGRALS

8.7 Particle near the Surface of a Sphere in D Dimensions

641

the surface and traverse the embedding space along a straight line. This will be studied further in Section 8.8. As mentioned above, this amplitude can be solved exactly. In fact, for each time interval ǫ, the exponential Mr 2 Mr 2 exp − (un − un−1 )2 = exp − (1 − cos ∆ϑn ) 2¯ hǫ h ¯ǫ "

#

"

#

(8.152)

can be expanded into spherical harmonics according to formulas (8.100)–(8.101), ∞ X Mr 2 l + D/2 − 1 1 (D/2−1) exp − (un − un−1 )2 = a ˜l (h) Cl (cos ∆ϑn ) 2¯ hǫ D/2 − 1 S D l=0

"

#

=

∞ X

a ˜l (h)

l=0

X

∗ Ylm (un )Ylm (un−1 ),

(8.153)

m

where

2π (D−1)/2 ˜ Mr 2 Il+D/2−1 (h), h = . (8.154) h h ¯ǫ For each adjacent pair (n + 1, n), (n, n − 1) of such factors in the sliced path integral, the integration over the intermediate un variable can be done using the orthogonality relation (8.114). In this way, (8.150) produces the time-sliced amplitude a ˜l (h) =

(ub τb |ua τa ) =

h 2π

!(N +1)(D−1)/2

∞ X

a ˜l (h)N +1

X

∗ Ylm (ub )Ylm (ua ).

(8.155)

m

l=0

We now go to the continuum limit N → ∞, ǫ = (τb − τa )/(N + 1) → 0, where [recall (8.11)] h 2π

!(N +1)(D−1)/2

!#N +1

Mr 2 a˜l (h) = I˜l+D/2−1 h ¯ǫ ( ) ǫ→0 (l + D/2 − 1)2 − 1/4 − −−→ exp −(τb − τa )¯ h . 2Mr 2 N +1

"

(8.156)

Thus, the final time evolution amplitude for the motion near the surface of the sphere is ∞ X

"

#

X h ¯ L2 ∗ (ub τb |ua τa ) = exp − (τ − τ ) Ylm (ub )Ylm (ua ), b a 2 2Mr m l=0

(8.157)

with L2 ≡ (l + D/2 − 1)2 − 1/4.

(8.158)

For D = 3, this amounts to an expansion in terms of associated Legendre polynomials ( ) ∞ X 2l + 1 h ¯ L2 (ub τb |ua τa ) = exp − (τb − τa ) 4π 2Mr 2 l=0 ×

l X

(l − m)! m Pl (cos θb )Plm (cos θa )eim(ϕb −ϕa ) . (l + m)! m=−l

(8.159)

642


If the initial point lies at the north pole of the sphere, this simplifies to ∞ X

"

#

2l + 1 h ¯ L2 exp − (ub τb |ˆ za τa ) = (τb − τa ) Pl (cos θb )Pl (1), 4π 2Mr 2 l=0

(8.160)

where Pl (1) = 1. By rotational invariance the same result holds for arbitrary directions of ua , if θb is replaced by the difference angle ϑ between ub and ua . In four dimensions, the most convenient expansion uses again the representation functions of the rotation group, so that (8.157) reads ∞ X

"

h ¯ L2 (ub τb |ua τa ) = exp − (τb − τa ) 2Mr 2 l=0

#

(8.161)

l/2

X l+1 l/2 ∗ × 2 D l/2 (ϕn , θn , γn )Dm (ϕn−1 , θn−1 , γn−1). 1 m2 2π m1 ,m2 =−l/2 m1 m2

These results will be needed in Sections 8.9 and 10.4 to calculate the amplitudes on the surface of a sphere. First, however, we extract some more information from the amplitudes near the surface of the sphere.

8.8

Angular Barriers near the Surface of a Sphere

In Section 8.5 we have projected the path integral of a free particle in three dimensions into a state of fixed angular momentum l finding a radial path integral containing a singular potential, the centrifugal barrier. This could not be treated via the standard time-slicing formalism. The projection of the path integral, however, supplied us with a valid time-sliced action and yielded the correct amplitude. A similar situation occurs if we project the path integral near the surface of a sphere into a fixed azimuthal quantum number m. The physics very near the poles of a sphere is almost the same as that on the tangential surfaces at the poles. Thus, at a fixed two-dimensional angular momentum, the tangential surfaces contain centrifugal barriers. We expect analogous centrifugal barriers at a fixed azimuthal quantum number m near the poles of a sphere at a fixed azimuthal quantum number m. These will be called angular barriers.

8.8.1

Angular Barriers in Three Dimensions

Consider first the case D = 3 where the azimuthal decomposition is (ub τb |ua τa ) =

X m

(sin θb τb | sin θa τa )m

1 im(ϕb −ϕa ) e . 2π

(8.162)

It is convenient to introduce also the differently normalized amplitude (θb τb |θa τa )m ≡

q

sin θb sin θa (sin θb τb | sin θa τa )m ,

(8.163)


643

8.8 Angular Barriers near the Surface of a Sphere

in terms of which the expansion reads (ub τb |ua τa ) =

X m

√

1 1 (θb τb |θa τa )m eim(ϕb −ϕa ) . 2π sin θb sin θa

(8.164)

While the amplitude (sin θb τb | sin θa τa )m has the equal-time limit (sin θb τ | sin θa τ )m =

1 δ(θb − θa ) sin θa

(8.165)

corresponding to the invariant measure of the θ-integration on the surface of the R sphere dθ sin θ, the new amplitude (θb τb |θa τ )m has the limit (θb τ |θa τ )m = δ(θb − θa )

(8.166)

with a simple δ-function, just as for a particle moving on the coordinate interval R θ ∈ (0, 2π) with an integration measure dθ. The renormalization is analogous to that of the radial amplitudes in (8.9). The projected amplitude can immediately be read off from Eq. (8.157): q

(θb τb |θa τa )m =

sin θb sin θa

∞ X

"

#

h ¯ l(l + 1) ∗ × exp − (τb − τa ) 2πYlm (θb , 0)Ylm (θa , 0). 2 2Mr l=m

(8.167)

In terms of associated Legendre polynomials [recall (8.84)], this reads (θb τb |θa τa )m =

∞ X

(

)

h ¯ l(l + 1) sin θb sin θa exp − (τb − τa ) 2Mr 2 l=m (2l + 1) (l − m)! m × P (cos θb )Plm (cos θa ). 2 (l + m)! l

q

(8.168)

Let us look at the time-sliced path integral associated with this amplitude. We start from Eq. (8.150) for D = 3, "Z # N Y 1 d cos θn dϕn 1 N (ub τb |ua τa ) ≈ exp − A , 2π¯ hǫ/Mr 2 n=1 2π¯ hǫ/Mr 2 h ¯

(8.169)

and use the addition theorem cos ∆ϑn = cos θn cos θn−1 + sin θn sin θn−1 cos(ϕn − ϕn−1 )

(8.170)

to expand the exponent as Mr 2 Mr 2 exp − (un − un−1 )2 = exp − (1 − cos ∆ϑn ) 2¯ hǫ h ¯ǫ " # Mr 2 = exp − (1 − cos θn cos θn−1 − sin θn sin θn−1 ) h ¯ǫ ∞ X 1 ×√ I˜m (hn )eim(ϕn −ϕn−1 ) , 2πhn m=−∞ "

#

"

#

(8.171)

644


where hn is defined as

Mr 2 sin θn sin θn−1 . (8.172) h ¯ǫ By doing successively the ϕn -integrations, we wind up with the path integral for the projected amplitude hn ≡

1

(θb τb |θa τa )m ≈ q 2πǫ¯ h/Mr2

where AN m is the sliced action AN m

=

N +1 X

(

n=1

N Y

n=1

 Z 

0

dθn

π

q

2πǫ¯ h/Mr2



 exp

1 − AN , h ¯ m

Mr 2 [1 − cos(θn − θn−1 )] − h ¯ log I˜m (hn ) . ǫ )

For small ǫ, this can be approximated (setting ∆θn ≡ θn − θn−1 ) by AN m

≈ǫ

N +1 X n=1

(

(8.173)

(8.174)

Mr 2 1 h ¯2 m2 − 1/4 2 4 (∆θn ) − (∆θn ) + . . . + , (8.175) 2ǫ2 12 2Mr 2 sin θn sin θn−1

)

with the continuum limit Am =

Z

τb

τa

dτ

h ¯2 h ¯ 2 m2 − 1/4 Mr 2 ˙2 θ − + . 2 8Mr 2 2Mr 2 sin2 θ !

(8.176)

This action has a 1/ sin2 θ -singularity at θ = 0 and θ = π, i.e., at the north and south poles of the sphere, whose similarity with the 1/r2 -singularity of the centrifugal barrier justifies the name “angular barriers”. By analogy with the problems discussed in Section 8.2, the amplitude (8.173) with the naively time-sliced action (8.175) does not exist for m = 0, this being the path collapse problem to be solved in Chapter 12. With the full time-sliced action (8.174), however, the path integral is stable for all m. In this stable expression, the successive integration of the intermediate variables using formula (8.14) gives certainly the correct result (8.168). To do such a calculation, we start out from the product of integrals (8.173) and expand in each factor Im (hn ) with the help of the addition theorem s

2ζ ζ cos θn cos θn−1 e Im (ζ sin θn sin θn−1 ) π ∞ X (l − m)! m = Il+1/2 (ζ)(2l + 1) Pl (cos θn )Plm (cos θn−1 ), (l + m)! l=m

(8.177)

where ζ ≡ Mr 2 /¯ hǫ. This theorem follows immediately from a comparison of two expansions e−ζ(1−cos ∆θn ) = e−ζ[1−cos θn cos θn−1 −sin θn sin θn−1 cos(ϕn −ϕn−1 )] (8.178) ∞ X 1 × √ I˜m (ζ sin θn sin θn−1 )eim(ϕn −ϕn−1 ) , 2πζ sin θn sin θn−1 m=−∞ −ζ(1−cos ∆θn )

e

−ζ

= e

s

∞ π X (2l + 1)Il+1/2 (ζ)Pl (cos ∆θn ). 2ζ l=0

(8.179)


645


The former is obtained with the help (8.5), the second is taken from (8.83). After the comparison, the Legendre polynomialis expanded via the addition theorem (8.85), which we rewrite with (8.84) as l X

(l − m)! m Pl (θn )Plm (θn−1 )eim(ϕn −ϕn−1 ) . m=−l (l + m)!

Pl (cos ∆θn ) =

(8.180)

We now recall the orthogonality relation (8.50), rewritten as Z

d cos θ m (l + m)! 2 m δll′ . 2 Pl (cos θ)Pl′ (cos θ) = (l − m)! 2l + 1 sin θ

1

−1

(8.181)

This allows us to do all angular integrations in (8.174). The result (θb τb |θa τa )m =

q

∞ X

sin θb sin θa

[I˜m+l+1/2 (ζ)]N +1

l=m

(2l + 1) (l − m)! m P (cos θb )Plm (cos θa ) × 2 (l + m)! l

(8.182)

is the solution of the time-sliced path integral (8.173). In the continuum limit, [I˜m+l+1/2 (ζ)]N +1 is dominated by the leading asymptotic term of (8.12) so that [I˜m+l+1/2 (ζ)]

N +1

"

#

h ¯ ≈ exp − L2 (τb − τa ) , 2Mr 2

(8.183)

leading to the previously found expression (8.168). We have gone through this calculation in detail for the following purpose. Later applications will require an analytic continuation of the path integral from integer values of m to arbitrary real values µ ≥ 0. With the present calculation, such a continuation is immediately possible by rewriting (8.182) with the help of the relation (l + m)! Plm (z) = (−)m Pl−m (8.184) (l − m)! as (θb τb |θa τa )µ = ×

q

sin θb sin θa

∞ X

[Iñ+µ+1/2 (ζ)]N +1

n=0

(2n + 2µ + 1) (n + 2µ)! −µ −µ Pn+µ (cos θb )Pn+µ (cos θa ). 2 n!

(8.185)

Here, µ can be an arbitrary real number if the factorials (n + 2µ)! and n! are defined as Γ(n + 2µ + 1) and Γ(n + 1). In the continuum limit, (8.185) becomes (θb τb |θa τa )µ

∞ X

"

h ¯ (n + µ)(n + µ + 1) = sin θb sin θa exp − (τb − τa ) 2Mr 2 n=0 (2n + 2µ + 1) (n + 2µ)! −µ −µ × Pn+µ (cos θb )Pn+µ (cos θa ). 2 n! q

#

(8.186)

646


We prove this to solve the time-sliced path integral (8.173) for arbitrary real values of m = µ [4] by using the addition theorem10 (sin α sin β) ×

−µ

iz cos α cos β

Jµ (z sin α sin β)e

22µ+1 Γ2 (µ + 1/2) √ = 2πz

∞ X

in n!(n + µ + 1/2) Jn+µ+1/2 (z)Cn(µ+1/2) (cos α)Cn(µ+1/2) (cos β). Γ(n + 2µ + 1) n=0

(8.187)

After substituting z by ζe−iπ/2 this turns into (sin α sin β)−µ Iµ (ζ sin α sin β) exp(ζ cos α cos β) = ×

22µ+1 Γ2 (µ + 1/2) √ 2πζ

∞ X

n!(n + µ + 1/2) In+µ+1/2 (ζ)Cn(µ+1/2) (cos α)Cn(µ+1/2) (cos β). Γ(n + 2µ + 1) n=0

(8.188)

The Gegenbauer polynomials Cn(µ+1/2) (z) can be expressed, for arbitrary µ, by means of Eq. (8.105) in terms of Jacobi polynomials Pn(µ,µ) , and these further in terms of −µ Legendre functions Pn+µ , using the formula Pn(µ,µ) (z) = (−2)µ

(n + µ)! −µ (1 − z 2 )−µ/2 Pn+µ (z). n!

(8.189)

Thus11 Cn(µ+1/2) (z)

Γ(n + 2µ + 1)Γ(µ + 1) 1 − z 2 = Γ(2µ + 1)n! 4 "

#−µ/2

−µ Pn+µ (z).

(8.190)

We can now perform the integrations in the time-sliced path integral by means of the known continuation of the orthogonality relation (8.181) to arbitrary real values of µ: Z

1

−1

n! d cos θ −µ 2 −µ δnn′ . 2 Pn+µ (cos θ)Pn′ +µ (cos θ) = (n + 2µ)! 2n + 1µ + 1 sin θ

(8.191)

−m Note that for noninteger µ, the Legendre functions Pn+m (cos θ) are no longer polynomials as in (1.417). Instead, they are defined in terms of the hypergeometric function as follows:

Pνµ (z) =

1 1+z Γ(1 − µ) 1 − z

µ/2

F (−ν, ν + 1; 1 − µ; (1 − z)/2).

(8.192)

The integral formula (8.191) is a consequence of the orthogonality of the Gegenbauer polynomials (8.103), which is applied here in the form Z

1

−1

(µ+1/2)

dz (1 − z 2 )µ Cn(µ+1/2) (z)Cn′

(z) = δnn′

π2−2µ Γ(2µ + 2 + n) . n!(n + µ)[Γ(µ + 1/2)]2

(8.193)

10

G.N. Watson, Theory of Bessel Functions, Cambridge University Press, 1952, Ch. 11.6, Eq. (11.9). 11 I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.936. H. Kleinert, PATH INTEGRALS

647


Using (8.188), (8.190) and (8.191), the integrals in the product (8.206) can all be performed as before, resulting in the amplitude (8.185) with the continuum limit (8.186), both valid for arbitrary real values of m = µ ≥ 0. The continuation to arbitrary real values of µ has an important application: The action (8.176) of the projected motion of a particle near the surface of the sphere coincides with the action of a particle moving in the so-called Pöschl-Teller potential [5]: h ¯ 2 s(s + 1) V (θ) = (8.194) 2Mr 2 sin2 θ with the strength parameter s = m − 1/2. After the continuation of arbitrary real m = µ ≥ 0, the amplitude (8.186) describes this system for any potential strength. This fact will be discussed further in Chapter 14 where we develop a general method for solving a variety of nontrivial path integrals. Note that the amplitude (sin θb τb | sin θa τa )m satisfies the Schrödinger equation "

1 1 d h ¯2 d m2 − sin θ + +h ¯ ∂τ (sin θ τ | sin θa τa )m Mr 2 2 sin θ dθ dθ 2 sin2 θ =h ¯ δ(τ − τa )δ(cos θ − cos θa ). !

#

(8.195)

This follows from the differential equation obeyed by the Legendre polynomials Plm (cos θ) in (8.87). The new amplitude √ (θ τm|θa τa )m , on the other hand, satisfies the equation [corresponding to that of sin θPl (cos θ) in the footnote to Eq. (8.87)] "

1 d 1 m2 − 1/4 h ¯2 − − + +h ¯ ∂τ (θ τ |θa τa )m = h ¯ δ(τ − τa )δ(θ − θa ). (8.196) Mr 2 2 dθ2 8 2 sin2 θ !

8.8.2

#

Angular Barriers in Four Dimensions

In four dimensions, the angular momentum decomposition reads in terms of Euler angles [see (8.161)] ∞ X

(

h ¯ L2 (ub τb |ua τa ) = exp − (τb − τa ) 2Mr 2 l=0

)

l/2

X l+1 im1 (ϕb −ϕa )+im2 (γb −γa ) × dl/2 (θb )dl/2 , m1 m2 (θa )e 2π 2 m1 m2 =−l/2 m1 m2

(8.197)

with L2 ≡ (l + 1)2 − 1/4 = 4(l/2)(l/2 + 1) + 3/4

(8.198)

and m1 , m2 running over integers or half-integers depending on l/2. We now define the projected amplitudes by the expansion (ub τb |ua τa ) = 8

X

m1 m2

(sin θb τb | sin θa τa )m1 m2

1 im1 (ϕb −ϕa ) 1 im2 (γb −γa ) e e . 2π 4π

(8.199)

648


As in (8.163), it is again convenient to introduce the differently normalized amplitude (θb τb |θa τa )m defined by (θb τb |θa τa )m1 m2 ≡

q

sin θb sin θa (sin θb τb | sin θa τa )m1 m2 ,

(8.200)

in terms of which the expansion becomes [compare (8.162)] (ub τb |ua τa ) =

X m

√

8 1 1 (θb τb |θa τa )m1 m2 eim(ϕb −ϕa ) eim(γb −γa ) . 2π 4π sin θb sin θa

(8.201)

A comparison with (8.197) gives immediately the projected amplitude (θb τb |θa τa )m1 m2 = ×

∞ X l

(

exp −

q

sin θb sin θa

(8.202)

h ¯ [(l + 1)2 − 1/4] l + 1 l/2 (τb − τa ) dm1 m2 (θb )dl/2 m1 m2 (θa ), 2 2Mr 2 )

in which l is summed in even steps from the larger value of |2m1 |, |2m2 | to infinity. Let us write down the time-sliced path integral leading to this amplitude. According to (8.150)–(8.152), it is given by 1

(ub τb |ua τa ) ≈ q 3 2π¯ hǫ/Mr 2

 Z N Y  

n=1

π

0

Z

2π

Z

0

4π

0



1 N .  exp − A h ¯

dθn sin θn dϕn dγn  q

8 2π¯ hǫ/Mr 2

3

(8.203)

In each time slice we make use of the addition theorem (8.128) and expand the exponent with (8.6) as Mr 2 Mr 2 exp − (un − un−1 )2 = exp − (1 − cos ∆ϑn ) 2¯ hǫ h ¯ǫ ( ) Mr 2 = exp − [1 − cos(θn /2) cos(θn−1 /2) − sin(θn /2) sin(θn−1 /2)] 2¯ hǫ ∞ X 1 1 q ×q I˜|m1 +m2 | (hcn )I˜|m1 −m2 | (hsn ) (8.204) c s 2πhn 4πhn m1 ,m2 =−∞ "

#

"

#

× exp{im1 (ϕn − ϕn−1 ) + im2 (γn − γn−1 )},

where hcn and hsn are given by hcn =

Mr 2 cos(θn /2) cos(θn−1 /2), h ¯ǫ

hsn =

Mr 2 sin(θn /2) sin(θn−1 /2). h ¯ǫ

(8.205)

By doing successively the ϕn - and γn -integrations, we wind up with the path integral for the projected amplitude 1

(θb τb |θa τa )m1 m2 ≈ q 2πǫ¯ h/4Mr 2

N Y

n=1

 

Z

0

dθn

π

q

2πǫ¯ h/4Mr 2



 exp

1 − AN , (8.206) h ¯ m1 m2


649


where AN m1 m2 is the sliced action AN m1 m2 =

N +1 X n=1

(

Mr 2 [1 − cos[(θn − θn−1 )/2] ǫ −

h ¯ log I˜|m1 +m2 | (hcn )

−

h ¯ log I˜|m1 −m2 | (hsn )

)

.

(8.207)

For small ǫ, this can be approximated (setting ∆θn ≡ θn − θn−1 ) by AN m1 m2

→ǫ

N +1 X n=1

(

Mr 2 1 2 [(∆θ /2) − (∆θn /2)4 + . . .] n 2ǫ2 12

h ¯2 h ¯2 (m1 + m2 )2 − 1/4 (m1 − m2 )2 − 1/4 + + , (8.208) 2Mr 2 cos(θn /2) cos(θn−1 /2) 2Mr 2 sin(θn /2) sin(θn−1 /2) )

with the continuum limit Am1 m2 =

Z

τb

τa

Mr 2 ˙2 h ¯2 h ¯ 2 |m1 + m2 |2 − 1/4 θ − + 8 8Mr 2 2Mr 2 cos2 (θ/2)

dτ

h ¯ 2 |m1 − m2 |2 − 1/4 + 2Mr 2 sin2 (θ/2)

!

.

(8.209)

After introducing the auxiliary mass µ = M/4

(8.210)

and rearranging the potential terms, we can write the action equivalently as Am1 m2 =

Z

τb τa

dτ

h ¯2 h ¯ 2 m21 + m22 − 1/4 − 2m1 m2 cos θ µr 2 ˙2 θ − + . (8.211) 2 32µr 2 2µr 2 sin2 θ !

Just as in the previous system, this action contains an angular barrier 1/ sin2 θ at θ = 0, and θ = π, so that the amplitude (8.206) with the naively time-sliced action (8.175) does not exist for m1 = m2 or m1 = −m2 , due to path collapse. Only with the properly time-sliced action (8.207) is the path integral stable and solvable by successive integrations with the result (8.202). As before, the path integral (8.206) is initially only defined and solved by (8.202) if both m1 and m2 have integer or half-integer values. The path integral and its solution can, however, be continued to arbitrary real values of m1 = µ1 ≥ 0 and its m2 = µ2 ≥ 0. For this we rewrite (8.202) in the form [4] (θb τb |θa τa )µ1 µ2 = ×

∞ X

q

sin θb sin θa

(8.212)

h ¯ [(n + µ1 + 1)2 − 1/4] n + µ1 + 1 n+µ1 1 exp − (τb − τa ) dµ1 µ2 (θb )dn+µ µ1 µ2 (θa ), 2 2Mr 2 n=0 (

)

650


assuming that µ1 ≥ µ2 . The products of the rotation functions dλµ1 µ2 (θ) have a well-defined analytic continuation to arbitrary real values of the indices µ1 , µ2 , λ, as can be seen by expressing them in terms of Jacobi polynomials via formula (1.443). To perform the path integral in the analytically continued case, we use the expansion valid for all µ+ , µ− ,12 z Jµ (z cos α cos β)Jµ− (z sin α sin β) = cosµ+ α cosµ+ β sinµ− α sinµ− β 2 + ×

∞ X

(−1)n (µ+ + µ− + 2n + 1)Jµ+ +µ− +2n+1 (z)

n=0

Γ(n + µ+ + µ− + 1)Γ(n + µ− + 1) n!Γ(n + µ+ + 1)[Γ(µ− + 1)]2 × F (−n, n + µ+ + µ− + 1; µ− + 1; sin2 α) × F (−n, n + µ+ + µ− + 1; µ− + 1; sin2 β) ×

(8.213)

with ζ ≡ Mr 2 /¯ hǫ. The hypergeometric functions appearing on the right-hand side have a first argument with a negative integer value. They are therefore proportional to the Jacobi polynomials Pn(µ− ,µ+ ) : Pn(µ− ,µ+ ) (x) =

1 Γ(n + µ− + 1) F (−n, n + µ+ + µ− + 1; 1 + µ− ; (1 − x)/2) (8.214) n! Γ(µ− + 1)

[recall (1.443) and the identity Pn(µ− ,µ+ ) (x) = (−)n Pn(µ− ,µ+ ) (−x)]. Inserting z = iζ, α = θn /2, β = θn−1 /2, and expressing the Jacobi polynomials in terms of rotation functions continued to real-valued µ1 , µ2 , we obtain from (8.213) for µ1 ≥ µ2 !

!

θn θn−1 θn θn−1 Iµ+ ζ cos cos Iµ− ζ sin sin 2 2 2 2 ∞ 4X (n + µ1 + µ2 )!(n + µ1 − µ2 )! n+µ1 1 = I2n+µ+ +µ− +1 (ζ) dµ1 µ2 (θn )dn+µ µ1 µ2 (θn−1 ). (8.215) ζ n=0 (n + 2µ1 )!n! Now we make use of the orthogonality relation [compare (1.452)] Z

1

−1

′

n +µ1 1 d cos θ dn+µ µ1 µ2 (θ)dµ1 µ2 (θ) = δnn′

2 , 2n + 1

(8.216)

which for real µ1 , µ2 follows from the corresponding relation for Jacobi polynomials13 Z

1

−1

(µ ,µ+ )

dx (1 − x)µ− (1 + x)µ+ Pn(µ− ,µ+ ) (x)Pn′ − = δnn′

12 13

(x)

2µ+ +µ− +1 Γ(µ+ + n + 1)Γ(µ− + n + 1) , n!(µ+ + µ− + 1 + 2n)Γ(µ+ + µ− + n + 1)

(8.217)

G.N. Watson, op. cit., Chapter 11.6, Gl. (11.6), (1). I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 7.391. H. Kleinert, PATH INTEGRALS

651


valid for Re µ+ > −1 , Re µ− > −1. Performing all θn -integrations in (8.206) yields the time-sliced amplitude (θb τb |θa τa )µ1 µ2 =

q

sin θb sin θa

∞ h X

N +1 n+µ1 1 I˜2n+µ+ +µ− +1 (ζ) dn+µ µ1 µ2 (θb )dµ1 µ2 (θa ), (8.218)

n=0

i

valid for all real µ1 ≥ µ2 ≥ 0. In the continuum limit, this becomes (θb τb |θa τa )µ1 µ2 =

q

sin θb sin θa

∞ X

n+µ1 1 e−En (τb −τa )/¯h dn+µ µ1 µ2 (θb )dµ1 µ2 (θa ),

(8.219)

n=0

with En =

h ¯ [(2n + µ+ + µ− + 1)2 − 1/4], 2Mr 2

(8.220)

which proves (8.212). Apart from the projected motion of a particle near the surface of the sphere, the amplitude (8.212) describes also a particle moving in the general Pöschl-Teller potential14 " # h ¯2 s1 (s1 + 1) s2 (s2 + 1) VPT ′ (β) = . (8.221) + 2Mr 2 sin2 (β/2) cos2 (β/2) Due to the analytic continuation to arbitrary real m1 , m2 the parameters s1 and s2 are arbitrary with the potential strength parameters s1 = m1 + m2 − 1/2 and s2 = m1 − m2 − 1/2. This will be discussed further in Chapter 14. Recalling the differential equation (1.451) satisfied by the rotation functions l/2 dm1 m2 (θ), we see that the original projected amplitude (8.206) obeys the Schrödinger equation "

h ¯2 1 d d 3 m21 + m22 − 2m1 m2 cos θ sin θ + + − +h ¯ ∂τ 2µr 2 sin θ dθ dθ 16 sin2 θ × (sin θ τ | sin θa τa )m1 m2 = h ¯ δ(τ − τa )δ(cos θ − cos θa ). !

#

(8.222)

The extra term 3/16 is necessary to account for the energy difference between the motion near the surface of a sphere in four dimensions, whose energy is (¯ h2 /2µr 2 )[(l/2)(l/2 + 1) + 3/16] [see (8.157)], and that of a symmetric spinning top with angular momentum L = l/2 in three dimensions, whose energy is (¯ h2 /2µr 2 )(l/2)(l/2 + 1), as shown in the next section in detail. The amplitude (θb τb |θa τa )m1 m2 defined in (8.200) satisfies the differential equation "

h ¯2 d2 1 m21 + m22 − 1/4 − 2m1 m2 cos θ − − + 2µr 2 dθ2 16 sin2 θ × (θ τ |θa τa )m1 m2 = h ¯ δ(τ − τa )δ(θ − θa ).

!

+h ¯ ∂τ

#

(8.223)

This is, of course, precisely the Schrödinger equation associated with the action (8.211). 14

See Footnote 15.

652

8.9


Motion on a Sphere in D Dimensions

The wave functions in the time evolution amplitude near the surface of a sphere are also correct for the motion on a sphere. This is not true for the energies, for which the amplitude (8.157) gives h ¯2 El = (L2 )l , (8.224) 2Mr 2 2 with (L22 )l = (l + D/2 − 1)2 − 1/4, l = 0, 1, 2, . . . . (8.225) As we know from Section 1.14, the energies should be equal to El =

h ¯2 ˆ2 (L )l , 2Mr 2

(8.226)

ˆ 2 )l denotes the eigenvalues of the square of the angular momentum operator. where (L In D dimensions, the eigenvalues are known from the Schrödinger theory to be ˆ 2 )l = l(l + D − 2), (L

l = 0, 1, 2, . . . .

(8.227)

Apart from the trivial case D = 1, the two energies are equal only for D = 3, where ˆ 2 )l = l(l + 1). For all other dimensions, we shall have to remove the (L22 )l ≡ (L difference ˆ 2 − L22 = ∆(L22 )l ≡ L

1 D − −1 4 2

2

=−

(D − 1)(D − 3) . 4

(8.228)

The simplest nontrivial case where the difference appears is for D = 2 where the role of l is played by the magnetic quantum number m and (L22 )m = m2 − 1/4, whereas ˆ 2 )m = m2 . the correct energies should be proportional to (L Two changes are necessary in the time-sliced path integral to find the correct energies. First, the time-sliced action (8.151) must be modified to measure the proper distance on the surface rather than the Euclidean distance in the embedding space. Second, we will have to correct the measure of path integration. The modification of the action is simply +1 M 2 NX (∆ϑn )2 AN = r , (8.229) on sphere ǫ 2 n=1 in addition to AN =

+1 M 2 NX r (1 − cos ∆ϑn ). ǫ n=1

(8.230)

Since the time-sliced path integral was solved exactly with the latter action, it is convenient to expand the true action around the solvable one as follows: AN on sphere

+1 M 2 NX 1 = r (1 − cos ∆ϑn ) + (∆ϑn )4 − . . . . ǫ 24 n=1

(8.231)


8.9 Motion on a Sphere in D Dimensions

653

There is no need to go to higher than the fourth order in ∆ϑn , since these do not contribute to the relevant order ǫ. For D = 2, the correction of the action is sufficient to transform the path integral near the surface of the sphere into one on the sphere, which in this reduced dimension is merely a circle. On a circle, ∆ϑn = ϕn − ϕn−1 and the measure of path integration becomes 1 q

2π¯ hǫ/Mr 2

NY +1 Z π/2 n=1

−π/2

dϕn q

2π¯ hǫ/Mr 2

.

(8.232)

The quartic term (∆ϑn )4 = (ϕn − ϕn−1 )4 can be replaced according to the rules of perturbation theory by its expectation [see (8.62)] h(∆ϑn )4 i0 = 3

ǫ¯ h . Mr 2

(8.233)

The correction term in the action +1 M 2 NX 1 ∆qu A = r (∆ϑn )4 ǫ n=1 24 N

(8.234)

has, therefore, the expectation h ¯ 2 /4 h∆qu A i0 = (N + 1)ǫ . 2Mr 2 N

(8.235)

This supplies precisely the missing term which raises the energy from the near -thesurface value Em = h ¯ 2 (m2 − 1/4)/2Mr 2 to the proper on-the-sphere value Em = 2 2 2 h ¯ m /2Mr . In higher dimensions, the path integral near the surface of a sphere requires a ˆ 2 and L2 is negative. Since the second correction. The difference (8.228) between L 2 expectation of the quartic correction term alone is always positive, it can certainly not explain the difference.15 Let us calculate first its contribution at arbitrary D. For very small ǫ, the fluctuations near the surface of the sphere lie close to the D − 1 -dimensional tangent space. Let ∆xn be the coordinates in this space. Then we can write the quartic correction term as ∆qu AN ≈

+1 M NX 1 (∆xn )4 , ǫ n=1 24r 2

(8.236)

where the components (∆xn )i have the correlations h(∆xn )i (∆xn )j i0 = 15

h ¯ǫ δij . M

(8.237)

This was claimed by G. Junker and A. Inomata, in Path Integrals from meV to MeV, edited by M.C. Gutzwiller, A. Inomata, J.R. Klauder, and L. Streit (World Scientific, Singapore, 1986), p.333.

654


Thus, according to the rule (8.62), ∆qu AN has the expectation h ¯2 h∆A i0 = (N + 1)ǫ ∆qu L22 , 2Mr 2 N

(8.238)

where ∆qu L22 is the contribution of the quartic term to the value L22 : ∆qu L22 =

D2 − 1 . 12

(8.239)

This result is obtained using the contraction rules for the tensor ǫ¯ h M

h∆xi ∆xj ∆xk ∆xl i0 =

!2

(δij δkl + δik δjl + δil δjk ),

(8.240)

which follow from the integrals (8.63). Incidentally, the same result can also be derived in a more pedestrian way: The term (∆xn )4 can be decomposed into D − 1 quartic terms of the individual components ∆xni , and (D − 1)(D − 2) mixed quadratic terms (∆xni )2 (∆xnj )2 with i 6= j. The former have an expectation (D − 1) · 3(ǫ¯ h/Mr)2 , the latter (D − 1)(D − 2) · (ǫ¯ h/Mr)2 . When inserted into (8.236), they lead to (8.238). Thus we remain with a final difference in D dimensions: 1 ∆f L22 = ∆L22 − ∆qu L22 = − (D − 1)(D − 2). 3

(8.241)

This difference can be removed only by the measure of the path integral. Near the sphere we have used the measure  Z N Y   q

d

D−1

un

2π¯ hǫ/Mr 2

n=1



D−1  .

(8.242)



In Chapter 10 we shall argue that this measure is incorrect. We shall find that the measure (8.242) receives a correction factor N Y

n=1

D−2 1+ (∆ϑn )2 6

(8.243)

[see the factor (1 + i∆AǫJ ) of Eq. (10.150)]. Setting (∆ϑn )2 = (∆xn /r)2 , the expectation of this factor becomes N Y

n=1

"

(D − 2)(D − 1) ǫ¯ h 1+ 2 6r M

#

(8.244)

corresponding to a correction term in the action h∆AN f i0 = (N + 1)ǫ

h ¯2 ∆f L22 , 2Mr 2


8.9 Motion on a Sphere in D Dimensions

655

with ∆f L22 given by (8.241). This explains the remaining difference between the ˆ 2. eigenvalues (L2 )l and (L) l In summary, the time evolution amplitude on the D-dimensional sphere reads [6] " # ∞ ˆ2 X X h ¯L ∗ (ub τb |ua τa ) = exp − (τ − τ ) Ylm (ub )Ylm (ua ), (8.246) b a 2 2Mr m l=0 with

ˆ 2 = l(l + D − 2), L

(8.247)

which are precisely the eigenvalues of the squared angular momentum operator of Schrödinger quantum mechanics. For D = 3 and D = 4, the amplitude (8.246) coincides with the more specific representations (8.160) and (8.161), if L22 is replaced ˆ2. by L Finally, let us emphasize that in contrast to the amplitude (8.157) near the surface of the sphere, the normalization of the amplitude (8.246) on the sphere is Z

dD−1 ub (ub τb |ua τa ) = 1.

(8.248)

This follows from the integral Z

dD−1ub

X

∗ Ylm (ub )Ylm (ua ) = δl0

Z

∗ dD−1 ub Y00 (ub )Y00 (ua )

= δl0

Z

dD−1 ub 1/SD = δl0 .

m

(8.249)

This is in contrast to the amplitude near the surface which satisfies Z

d

D−1

(D/2 − 1)2 − 1/4 ub (ub τb |ua τa ) = exp − (τb − τa ) . 2µr 2 "

#

(8.250)

We end this section with the following observation. In the continuum, the Euclidean path integral on the surface of a sphere can be rewritten as a path integral in flat space with an auxiliary path integral over a Lagrange multiplier λ(τ ) in the form16 (xb τb |xa τa ) =

Z

i∞

−i∞

Z

Dλ(τ ) 1 D x(τ ) exp − 2πi¯ h h ¯ 2

Z

τb

τa

dτ

(

)!

M 2 λ(τ ) 2 x˙ + [x (τ ) − r 2 ] . 2 2r (8.251)

A naive time slicing of this expression would not yield the correct energy spectrum on the sphere. The slicing would lead to the product of integrals (ub tb |ua ta ) ≈ 16

N Z Y

n=1

D

d xn

"Z Y N n=1

#

dλn i N exp A , 2πi¯ h/ǫ h ¯

(8.252)

The field-theoretic generalization of this path integral, in which τ is replaced by a d-dimensional spatial vector x, is known as the O(D)-symmetric nonlinear σ-model in d dimensions. In statistical mechanics it corresponds to the well-studied classical O(D) Heisenberg model in d dimensions.

656


with u ≡ x/|x| and the time-sliced action N

A =

N +1 X n=1

"

#

M λn (xn − xn−1 )2 + ǫ (x2n − r 2 ) . 2ǫ 2r

(8.253)

Integrating out the λn ’s would produce precisely the expression (8.150) with the action (8.151) near the surface of the sphere. The δ-functions arising from the λn integrations would force only the intermediate positions xn to lie on the sphere; the sliced kinetic terms, however, would not correspond to the geodesic distance. Also, the measure of path integration would be wrong.

8.10

Path Integrals on Group Spaces

In Section 8.3, we have observed that the surface of a sphere in four dimensions is equivalent to the covering group of rotations in three dimensions, i.e., with the group SU(2). Since we have learned how to write down an exactly solvable timesliced path integral near and on the surface of the sphere, the equivalence opens up the possibility of performing path integrals for the motion of a mechanical system near and on the group space of SU(2). The most important system to which the path integral on the group space of SU(2) can be applied is the spinning top, whose Schrödinger quantum mechanics was discussed in Section 1.15. Exploiting the above equivalence we are able to describe the same quantum mechanics in terms of path integrals. The theory to be developed for this particular system will, after a suitable generalization, be applicable to systems whose dynamics evolves on any group space. First, we discuss the path integral near the group space using the exact result of the path integral near the surface of the sphere in four dimensions. The crucial observation is the following: The time-sliced action near the surface AN =

+1 +1 M 2 NX Mr 2 NX r (un − un−1 )2 = (1 − cos ∆ϑn ) 2ǫ n=1 ǫ n=1

(8.254)

can be rewritten in terms of the group elements g(ϕ, θ, γ) defined in Eq. (8.122) as +1 M 2 NX 1 −1 A = r 1 − tr(gn gn−1 ) , ǫ 2 n=1 N

(8.255)

with the obvious notation gn = g(ϕn , θn , γn ).

(8.256)

This follows after using the explicit matrix form for g, which reads g(ϕ, θ, γ) = exp(iϕσ3 /2) exp(iθσ2 /2) exp(iγσ3 /2) ! ! eiϕ/2 0 cos(θ/2) sin(θ/2) = − sin(θ/2) cos(θ/2) 0 e−iϕ/2

(8.257) iγ/2

e

0

0 −iγ/2

e

!

.


657

8.10 Path Integrals on Group Spaces

After a little algebra we find 1 −1 tr(gn gn−1 ) = cos(θn /2) cos(θn−1 /2) cos[(ϕn − ϕn−1 + γn − γn−1 )/2] 2 + sin(θn /2) sin(θn−1 /2) cos[(ϕn − ϕn−1 − γn + γn−1 )/2],

(8.258)

just as in (8.128). The invariant group integration measure is usually defined to be normalized to unity, i.e., Z

1 Z π Z 2π Z 4π 1 Z 3 dg ≡ dθ sin θdϕdγ = 2 d u = 1. 16π 2 0 0 0 2π

(8.259)

We shall renormalize the time evolution amplitude (ub τb |ua τa ) near the surface of the four-dimensional sphere accordingly, making it a properly normalized amplitude for the corresponding group elements (gb τb |ga τa ). Thus we define (ub τb |ua τa ) ≡

1 (gb τb |ga τa ). 2π 2

(8.260)

The path integral (8.150) then turns into the following path integral for the motion near the group space [compare also (8.203)]: 2π

2

(gb τb |ga τa ) ≈ q 3 2π¯ hǫ/Mr 2

N Y

n=1

 Z   q

2

2π dgn

2π¯ hǫ/Mr 2



 3  exp

1 − AN . h ¯

(8.261)

Let us integrate this expression within the group space language. For this we expand the exponential as in (8.130): Mr 2 1 −1 exp − 1 − tr(gn gn−1 ) h ¯ǫ 2 (

=

∞ X

a ˜l (h)

l=0

)

=

∞ X l=0

a ˜l (h)

l + 1 (1) C (cos ∆ϑn ) 2π 2 l

(8.262)

l/2 X l+1 l/2 ∗ D l/2 (ϕn , θn , γn )Dm (ϕn−1 , θn−1 , γn−1 ). 1 m2 2π 2 m1 ,m2 =−l/2 m1 m2

In general terms, the right-hand side corresponds to the well-known character expansion for the group SU(2): ∞ h 1X −1 −1 exp tr(gn gn−1 ) = (l + 1)Il+1(h)χ(l/2) (gn gn−1 ). 2 h l=0

"

#

(8.263)

Here χl/2 (g) are the so-called characters, the traces of the representation matrices of the group element g, i.e., l/2 χ(l/2) (g) = Dmm (g).

(8.264)

658


The relation between the two expansions is obvious if we use the representation l/2 properties of the Dm functions and their unitarity to write 1 m2 l/2

l/2∗

−1 χ(l/2) (gn gn−1 ) = Dmm′ (gn )Dmm′ (gn−1 ).

(8.265)

This leads directly to (8.262) [see also the footnote to (8.127)]. Having done the character expansion in each time slice, the intermediate group integrations can all be performed using the orthogonality relations of group characters Z

′

dgχ(L) (g1 g −1)χ(L ) (gg2−1) = δLL′

1 (L) χ (g1 g2−1 ). dL

(8.266)

The result of the integrations is, of course, the same amplitude as before in (8.161): ∞ X

"

h ¯ L2 (gb τb |ga τa ) = exp − (τb − τa ) 2Mr 2 l=0 × (l + 1)

l/2 X

m1 ,m2 =−l/2

#

(8.267)

l/2 l/2 ∗ Dm (ϕn , θn , γn )Dm (ϕn−1 , θn−1 , γn−1). 1 m2 1 m2

Given this amplitude near the group space we can find the amplitude for the motion on the group space, by adding to the energy near the sphere E = h ¯ 2 [(l/2 + 1)2 −1/4]/2Mr 2 the correction ∆E = h ¯ 2 ∆L22 /2Mr 2 associated with Eq. (8.228). For ˆ 2 = L2 +∆L2 = (l/2)(l/2+1), D = 4, L22 = (l/2)(l/2+1)+3/4 has to be replaced by L 2 2 and the energy changes by 3¯ h2 ∆E = − . (8.268) 8M Otherwise the amplitude is the same as in (8.267) [6]. Character expansions of the exponential of the type (8.263) and the orthogonality relation (8.266) are general properties of group representations. The above timesliced path integral can therefore serve as a prototype for the quantum mechanics of other systems moving near or on more general group spaces than SU(2). Note that there is no problem in proceeding similarly with noncompact groups [7]. In this case we would start out with a treatment of the path integral near and on the surface of a hyperboloid rather than a sphere in four dimensions. The solution would correspond to the path integral near and on the group space of the covering group SU(1,1) of the Lorentz group O(2,1). The main difference with respect to the above treatment would be the appearance of hyperbolic functions of the second Euler angle θ rather than trigonometric functions. An important family of noncompact groups whose path integral can be obtained in this way are the Euclidean groups [8] consisting of rotations and translations. ˆ , whose representation on Their Lie algebra comprises the momentum operators p ˆ = −i¯ the spatial wave functions has the Schrödinger form p h∇. Thus, the canonical commutation rules in a Euclidean space form part of the representation algebra of these groups. Within a Euclidean group, the separation of the path integral into a radial and an azimuthal part is an important tool in obtaining all group representations. H. Kleinert, PATH INTEGRALS

659

8.11 Path Integral of Spinning Top

8.11

Path Integral of Spinning Top

We are now also in a position to solve the time-sliced path integral of a spinning top by reducing it to the previous case of a particle moving on the group space SU(2). Only in one respect is the spinning top different: the equivalent “particle” does not move on the covering space SU(2) of the rotation group, but on the rotation group O(3) itself. The angular configurations with Euler angles γ and γ +2π are physically indistinguishable. The physical states form a representation space of O(3) and the time evolution amplitude must reflect this. The simplest possibility to incorporate the O(3) topology is to add the two amplitudes leading from the initial configuration ϕa , θa , γa to the two identical final ones ϕb , θb , γb and ϕb , θb , γb + 2π. This yields the amplitude of the spinning top: (ϕb , θb , γb τb |ϕb , θb , γb τa )top = (ϕb , θb , γb τb |ϕb , θb , γb τa ) + (ϕb , θb , γb + 2π τb |ϕb , θb , γb τa ).

(8.269)

l/2

The sum eliminates all half-integer representation functions Dmm′ (θ) in the expansion (8.267) of the amplitude. Instead of the sum we could have also formed another representation of the operation γ → γ + 2π, the antisymmetric combination (ϕb , θb , γb τb |ϕb , θb , γb τa )fermionic = (ϕb , θb , γb τb |ϕb , θb , γb τa ) − (ϕb , θb , γb + 2π τb |ϕb , θb , γb τa ).

(8.270)

Here the expansion (8.267) retains only the half-integer angular momenta l/2. As discussed in Chapter 7, half-integer angular momenta are associated with fermions such as electrons, protons, muons, and neutrinos. This is indicated by the subscript “fermionic”. In spite of this, the above amplitude cannot be used to describe a single fermion since this has only one fixed spin l/2, while (8.270) contains all possible fermionic spins at the same time. In principle, there is no problem in also treating the non-spherical top. In the formulation near the group space, the gradient term in the action, 1 1 −1 1 − tr(gn gn−1 ) , 2 ǫ 2

(8.271)

has to be separated into time-sliced versions of the different angular velocities. In the continuum these are defined by ωa = −itr (σ˙ a g −1 ),

a = ξ, η, ζ.

(8.272)

The gradient term (8.271) has the symmetric continuum limit ω˙ a2. With the different moments of inertia Iξ , Iη , Iζ , the asymmetric sliced gradient term reads 1 1 1 −1 −1 Iξ 1 − tr(gn σξ gn−1 ) + Iη 1 − tr(gn ση gn−1 ) 2 ǫ 2 2 1 −1 + Iζ 1 − tr(gn σζ gn−1 ) , 2

(8.273)

660


rather than (8.271). The amplitude near the top is an appropriate generalization of (8.267). The calculation of the correction term ∆E, however, is more complicated than before and remains to be done, following the rules explained above.

8.12

Path Integral of Spinning Particle

The path integral of a particle on the surface of a sphere contains states of all integer angular momenta l = 1, 2, 3, . . . . The path integral on the group space SU(2) contains also all half-integer spins s = 12 , 32 , 52 , . . . . The question arises whether it is possible to set up a path integral which contains only a single spinning particle, for instance of spin s = 1/2. Thus we need a path integral which for each time slice spans precisely one irreducible representation space of the rotation group, consisting of the 2s + 1 states |s s3 i for s3 = −s, . . . , s. In order to sum over paths, we must parametrize this space in terms of a continuous variable. This is possible by selecting a particular spin state, for example the state |ssi pointing in the z-direction, and rotating it into an arbitrary direction u = (sin θ cos ϕ, sin θ sin ϕ, cos θ) with the help of some rotation, for instance |θ ϕi ≡ Rs (θ, ϕ)|ssi ≡ e−iS3 ϕ e−iS2 θ |ssi,

(8.274)

where Si are matrix generators of the rotation group of spin s, which satisfy the ˆ i in (1.410). The states (8.274) are nonabelian commutation rules of the generators L versions of the coherent states (7.343). They can be expanded into the 2s + 1 spin states |s s3i as follows: |θ ϕi =

s X

s3 =−s

ˆ ϕ)|ssi = |s s3ihs s3 |R(θ,

s X

s3 =−s

|s s3 ie−is3 ϕ dss3 s (θ),

(8.275)

where djmm′ (θ) are the representation matrices of e−iS2 θ with angular momentum j given in Eq. (1.443). For s = 1/2, where the matrix djmm′ has the form (1.444), the states (8.275) are |θϕi = eiϕ/2 cos θ/2| 21 21 i − e−iϕ/2 sin θ/2| 21 − 12 i.

(8.276)

At this point it is useful to introduce the so-called monopole spherical harmonics defined by s 2j + 1 imϕ j Ymj q (θ, ϕ) ≡ e dm q (θ). (8.277) 4π They satisfy the orthogonality relation Z

1

−1

d cos θ

Z

2π

0

′

dϕ Ymj ∗q (θ, ϕ)Ymj ′ q (θ, ϕ) = δjj ′ δmm′ ,

(8.278)

and the completeness relation X j,m

Ymj q (θ, ϕ)Ymj ∗q (θ′ , ϕ′ ) = δ(cos θ − cos θ′ )δ(ϕ − ϕ′ ).

(8.279)


661

8.12 Path Integral of Spinning Particle

We define now the covariant looking states s

|ui ≡

s X 2j + 1 |θ ϕi = |s s3i Yss3∗s (θ, ϕ), 4π s3 =−s

(8.280)

and write the angular integral as an integral over the surface of the unit sphere: Z

1

−1

d cos θ

Z

2π

0

dϕ =

Z

3

2

d u δ(u − 1) ≡

Z

du.

(8.281)

From (8.278) we deduce that the states |ui are complete in the space of spin-s states: Z

du |uihu| = =

s X

s X

s3 =−s s′3 =−s s X

s3 =−s

|s s3 i

Z

1

−1

d cos θ

Z

2π

0

dϕYss3∗s (θ, ϕ)Yss′3 s (θ, ϕ)hss′3|

|s s3 i hs s3| = 1s .

(8.282)

The states are not orthogonal, however. Writing Yss3 s (θ, ϕ) as Yss3 s (u), we see that hu|u′ i =

s X

s X

s3 =−s s′3 =−s

Yss3 s (u)hs s3 |ss′3 iYss′3s (u′ ) =

s X

Yss3 s (u)Yss3∗s (u′ ).

(8.283)

s3 =−s

The right-hand side can be calculated as follows: 2j + 1 2j + 1 ′ ′ hss|eiθS2 eiϕS3 e−iϕ S3 e−iθ S2 |ssi = hss|e−isAS3 e−iβS2 |ssi 4π 4π !s 2j + 1 −isAS3 s 2j + 1 −isAS3 1 + u · u′ e e = dss (β) = , 4π 4π 2

(8.284)

ˆ) is the area of the spherical where β is the angle between u and u′ , and A(u, u′ , z triangle on the unit sphere formed by the three points in the argument. For a radius R, the area is equal to R2 time the angular excess E of the triangle, defined as the amount by which the sum of the angles in the triangle is larger that π. An explicit formula for E is the spherical generalization of Heron’s formula for the area A of a triangle [9]: A=

q

s(s − a)(s − b)(s − c),

s = (a + b + c)/2 = semiperimeter.

(8.285)

The angular excess on a sphere is E tan = 4

s

tan

φs φs − φa φs − φb φs − φc tan tan tan , 2 2 2 2

(8.286)

where φa , φb , φc are the angular lengths of the sides of the triangle and φs = (φa + φb + φc )/2

(8.287)

662


is the angular semiperimeter on the sphere. We can now set up a path integral for the scalar product (8.284): hub |ua i =

"

N Z Y

n=1

#

dun hub |uN ihuN |uN −1 ihuN −1 | · · · |u1 ihu1 |ua i.

(8.288)

For large N, the intermediate un -vectors will all lie close to their neighbors and we can write approximately 2s + 1 i∆An 2s + 1 i∆An + 2s un (un −un−1 ) e [1 + 12 un (un − un−1 )]s ≈ e . (8.289) 4π 4π Let us take this expression to the continuum limit. We introduce a time parameter labeling the chain of un -vectors by tn = nǫ, and find the small-ǫ approximation hun |un−1 i ≈

2s + 1 s exp iǫ s cos θ ϕ˙ − ǫ2 θ˙2 + sin2 θ ϕ˙ 2 + . . . 4π 4

.

(8.290)

The first term is obtained from the scalar product

hθ ϕ|i∂t |θ ϕi = hss|eiS2 θ eiS3 ϕ i∂t e−iS3 ϕ e−iS2 θ |ssi

˙ 2 e−iS2 θ |ssi = hss|eiS2 θ eiS3 ϕ ϕS ˙ 3 e−iS3 ϕ e−iS2 θ + e−iS3 ϕ θS ˙ 2 |ssi = s cos θ ϕ. = hss| (cos θS3 − sin θS1 ) ϕ˙ + θS ˙

(8.291)

This result is actually not completely correct. The reason is that the angular variables in the states (8.274) are cyclic variables. For integer spins, θ and ϕ are cyclic in 2π, for half-integer spins in 4π. Thus there can be jumps by 2π or 4π in these angles which do not change the states (8.274). In writing down the approximation (8.290) we must assume that we are at a safe distances from such singularities. If we get close to them, we must change the direction of the quantization axis. Keeping this in mind we can express the scalar product in the limit ǫ → 0 by the path integral Z R i

hub |ua i =

where

R

Du e h¯

tb ta

dt ¯ hs cos θ ϕ˙

,

(8.292)

Du is defined by the limit N → ∞ of the product of integrals Z

Du ≡ lim

N →∞

"

N 2s + 1 Y 4π n=1

Z

#

dun .

(8.293)

The path integral fixes the Hilbert space of the spin theory. It is the analog of the zero-Hamiltonian path integral in Eqs. (2.17) and (2.18). Comparing (8.292) with (2.18), we see that s¯ h cos θ plays the role of a canonically conjugate momentum of the variable ϕ. For a specific spin dynamics we must add, as in (2.15), a Hamiltonian H(cos θ, ϕ), and arrive at the general path integral representation for the time evolution amplitude of a spinning particle [10] (ub tb |ua ta ) =

Z

i

Du e

R tb ta

dt [¯ hs cos θ ϕ−H(θ,ϕ)]/¯ ˙ h

.

(8.294)


663

8.12 Path Integral of Spinning Particle

The above path integral has a remarkable property which is worth emphasizing. For half-integer spins s = 12 , 32 , 52 , . . . it is able to describe the physics of a fermion in terms of a field theory involving a unit vector field u which describes the direction of the spin state: ˆ ˆ −1 (θ, ϕ)S ˆ R(θ, ˆ ϕ)|ssi = Ri j (θ, ϕ)hss|Sj |ssi = su. hu|S|ui = hss|R

(8.295)

ˆ The matrices Ri j (θ, ϕ) This follows from the vector property of the spin matrices S. are the defining 3 × 3 matrices of the rotation group (the so-called adjoint representation). Thus we describe a fermion in terms of a Bose field. The above path integral is only the simplest illustration for a more general phenomenon. In 1961, Skyrme pointed out that a certain field configuration of pions is capable of behaving in many respects like a nucleon [11], in particular its fermionic properties. In two dimensions, Bose field theories are even more powerful and can describe particles with any commutation rule, called anyons in Section 7.5. This will be shown in Chapter 16. In Chapter 10 we shall see that the action in (8.292) can be interpreted as the action of a particle of charge e on the surface of the unit sphere whose center contains a fictitious magnetic monopole of charge g = −4π¯ hcs/e. The associated (g) vector potential A (u) will be given in Eq. (10A.59). Coupling this minimally to a particle of charge e as in Eq. (2.627) on the surface of the sphere yields the action A0 =

e c

Z

tb

ta

A(g) (u) · u˙ = h ¯s

Z

tb

ta

dt cos θ ϕ˙ ,

(8.296)

where the magnetic flux is supplied to the monopole by two infinitesimally thin flux tubes, the famous Dirac strings, one from below and one from above. The field A(−s) in (8.296) is the average of the two expressions in (10A.61) for g = −4π¯ hcs/e. We can easily change the supply line to a single string from above, by choosing the states ˆ ϕ)|ssi = e−iS3 ϕ e−iS2 θ eiS3 ϕ |ssi = |θ ϕieisϕ , |θ ϕi′ ≡ R(θ,

(8.297)

rather than |θ ϕi of Eq. (8.274) for the construction of the path integral. The physics is the same since the string is an artifact of the choice of the quantization axis. In terms of Cartesian coordinates, the action (8.296) with a flux supplied from the north pole can also be expressed in terms of the vector u(t) as [compare (10A.59)] A0 = h ¯s

Z

dt

ˆz × u(t) ˙ · u(t). 1 − uz (t)

(8.298)

This expression is singular on the north pole of the unit sphere. The singularity can be rotated into an arbitrary direction n, leading to A0 = h ¯s

Z

dt

n × u(t) ˙ · u(t). 1 − n · u(t)

(8.299)

664


If u gets close to n we must change the direction of n. The action (8.299) is referred to as Wess-Zumino action. In Chapter 10 we shall also calculate the curl of the vector potential A(g) of a monopole of magnetic charge g and find the radial magnetic field accompanied by a singular string contribution along the direction n of flux supply [compare (10A.54)]: B

(g)

= ∇×A

(g)

u =g − 4πg |u|

Z

∞ 0

ˆ δ (3) (u − s n ˆ ), ds n

(8.300)

The singular contribution is an artifact of the description of the magnetic field. The line from zero to infinity is called a Dirac string. Since the magnetic field has no divergence, the magnetic flux emerging at the origin of the sphere must be imported from somewhere at infinity. In the field (8.301) the field is imported along the straight line in the direction u. Indeed, we can easily check that the divergence of (8.301) is zero. For a closed orbit, the interaction (8.296) can be rewritten by Stokes’ theorem as A0 =

e c

Z

dt A(g) (t) · u(t) =

e c

Z

h

i

dS · ∇ × A(g) =

e c

Z

dS · B(g) ,

(8.301)

where dS runs over the surface enclosed by the orbit. This surface may or may not contain the Dirac string of the monopole, in which case A0 differ by 4πge/c. A path integral over closed orbits of the spinning particle R

ZQM =

I

du ei(A0 +A)/¯h ,

(8.302)

is therefore invariant under changes of the position of the Dirac string if the monopole charge g satisfies the Dirac charge quantization condition ge = s, h ¯c

(8.303)

with s = half-integer or integer. Dirac was the first to realize that as a consequence of quantum mechanics, an electrically charged particle whose charge satisfies the quantization condition (8.303) sees only the radial monopole field in (8.301), not the field in the string. The string can run along any line L without being detectable. This led him to conjecture that there could exist magnetic monopoles of a specific g, which would explain that all charges in nature are integer multiples of the electron charge [15]. More on this subject will be discussed in Section 16.2. In Chapter 10 we shall learn how to define a monopole field A(g) which is free of the artificial string singularity [see Eq. (10A.58)]. With the new definition, the divergence of B is a δ-function at the origin: ∇ × B(u) = 4πg δ (3) (u).


665

8.13 Berry Phase

8.13

Berry Phase

This phenomenon has a simple physical basis which can be explained most clearly by means of the following gedanken experiment. Consider a thin rod whose dynamics is described by a unit vector field u(t) with an action A=

M 2

Z

h

i

dt u˙ 2 (t) − V (u2 (t)) ,

u2 (t) ≡ 1,

(8.305)

where u(t) is a unit vector along the rod. This is the same Lagrangian as for a particle on a sphere as in (8.146) (recall p. 655). Let us suppose that the thin rod is a solenoid carrying a strong magnetic field, and containing at its center a particle of spin s = 1/2. Then (8.305) is extended by the action [13] Z A0 =

dt ψ ∗ (t) [i¯ h∂t + γu(t) · ] ψ(t),

(8.306)

where σ i are the Pauli spin matrices (1.445). For large coupling strength γ and sufficiently slow rotations of the solenoid, the direction of the fermion spin will always be in the ground state of the magnetic field, i.e., its direction will follow the direction of the solenoid adiabatically, pointing always along u(t). If we parametrize u(t) in terms of spherical angles θ(t), φ(t) as n = (sin θ cos φ, sin θ sin φ, cos θ), the associated wave function satisfies u(t) · ψ(u(t)) = (1/2)ψ(u(t)), and reads ψ(u(t)) =

e−iφ(t)/2 cos θ(t)/2 eiφ(t)/2 sin θ(t)/2

!

.

(8.307)

Inserting this into (8.306) we obtain [compare (8.291)] A0 =

Z

1 ˙ ≡h dt ψ(u(t)) i¯ h∂t ψ(u(t)) = h ¯ cos θ(t) φ(t) ¯ β(t). 2

(8.308)

The action coincides with the previous expression (8.296). The angle β(t) is called Berry phase [14]. In this simple model it is obvious why the bosonic theory of the solenoid behaves like a spin-1/2 particle: It simply inherits the physical properties of the enslaved spinor. The reason why it is a monopole field that causes the spin-1/2 behavior will become clear in another way in Section 14.6. There we shall solve the path integral of a charged particle in a monopole field and show that it behaves like a fermion if its charge e and the monopole charge g have half-integer products q ≡ eg/¯ hc.

8.14

Spin Precession

The Wess-Zumino action A0 adds an interesting kinetic term to the equation of motion of a solenoid. Extremizing A0 + A yields

¨ + u˙ 2 u = −∂u V (u) − M u

δ A0 . δu(t)

(8.309)

666


The functional derivative of A0 is most easily calculated starting from the general expression in (8.296) δ h ¯ δui(t)

Z

tb

ta

(g)

(g)

dt A(g) · u˙ = h ¯ ∂ui Aj u˙j − ∂t Ai =h ¯

h

i

∇ × A(g) × u .

=h ¯

h

(g)

(g)

∂ui Aj − ∂uj Ai

u˙ j

i

(8.310)

i

Inserting here the curl of Eq. (8.301), while staying safely away from the singularity, ˙ and the equation of motion (8.309) turns the last term in (8.309) becomes h ¯ gu × u, into ¨ + u˙ 2 u = −∂u V (u) − h ˙ M u ¯ g u × u. (8.311) Multiplying this vectorially by u(t) we find [15, 16] h

i

˙ − u˙ u2 . ¨ = −u × ∂u V (u) − h Mu × u ¯ g u (u · u) Since u2 = 1, this reduces to

¨ −h Mu × u ¯ g u˙ = −u × ∂u V (u(t)).

(8.312)

(8.313)

If M is small we obtain for a particle of spin s, where g = −s, the so-called LandauLifshitz equation [17, 18] h ¯ s u˙ = −u × ∂u V (u). (8.314) This is a useful equation for studying magnetization fields in ferromagnetic materials. The interaction energy of a spinning particle with an external magnetic field has the general form Hint = − · B, = −γS, (8.315) where is the magnetic moment, S the vector of spin matrices, and γ the gyromagnetic ratio. Since u is the direction vector of the spin, we may identify the magnetic moment of the spin s as = γs¯ hu, (8.316) so that the interaction energy becomes Hint = −γs¯ h u · B.

(8.317)

Inserting this for V (u) in (8.314) yields the equation of motion u˙ = −γ B × u,

(8.318)

showing a rate of precession = −γB. For comparison we recall the derivation of this result in the conventional way from the Heisenberg equation of motion (1.272): ˆ S]. h ¯ S˙ = i[H,



667

ˆ the interaction energy (8.315) and using the commutation relations Inserting for H (1.410) of the rotation group for the spin matrices S, this yields the Heisenberg equation for spin precession S˙ = −γ B × S, (8.320) in agreement with the Landau-Lifshitz equation (8.314). This shows that the WessZumino term in the action A0 + A has the ability to render quantum equations of motion from a classical action. This allows us, in particular, to mimic systems of halfinteger spins, which are fermions, with a theory containing only a bosonic directional vector field u(t). This has important applications in statistical mechanics where models of interacting quantum spins for ferro- and antiferromagnets à la Heisenberg can be studied by applying field theoretic methods to vector field theories.

Notes and References The path integral in radial coordinates was advanced by S.F. Edwards and Y.V. Gulyaev, Proc. Roy. Soc. London, A 279, 229 (1964); D. Peak and A. Inomata, J. Math. Phys. 19, 1422 (1968); A. Inomata and V.A. Singh, J. Math. Phys. 19, 2318 (1978); C.C. Gerry and V.A. Singh, Phys. Rev. D 20, 2550 (1979); Nuovo Cimento 73B, 161 (1983). The path integral with a Bessel function in the measure has a predecessor in the theory of stochastic differential equations: J. Pitman and M. Yor, Bessel Processes and Infinitely Divisible Laws, in Stochastic Integrals, Springer Lecture Notes in Mathematics 851, ed. by D. Williams, 1981, p. 285. Difficulties with radial path integrals have been noted before by W. Langguth and A. Inomata, J. Math. Phys. 20, 499 (1979); F. Steiner, in Path Integrals from meV to MeV, edited by M. C. Gutzwiller, A. Inomata, J.R. Klauder, and L. Streit (World Scientific, Singapore, 1986), p. 335. The solution to the problem was given by H. Kleinert, Phys. Lett. B 224, 313 (1989) (http://www.physik.fu-berlin.de/~kleinert/195). It will be presented in Chapters 12 and 14. The path integral on a sphere and on group spaces was found by H. Kleinert, Phys. Lett. B 236, 315 (1990) (ibid.http/202). Compare also with earlier attempts by L.S. Schulman, Phys. Rev. 174, 1558 (1968), who specified the correct short-time amplitude but did not solve the path integral. The individual citations refer to [1] C.C. Gerry, J. Math. Phys. 24, 874 (1983); C. Garrod, Rev. Mod. Phys. 38, 483 (1966). [2] These observations are due to S.F. Edwards and Y.V. Gulyaev, Proc. Roy. Soc. London, A 279, 229 (1964). See also D.W. McLaughlin and L.S. Schulman, J. Math. Phys. 12, 2520 (1971). [3] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of Φ4 -Theories, World Scientific, Singapore 2001, pp. 1–487 (http://www.physik.fu-berlin.de/~kleinert/b8). [4] H. Kleinert and I. Mustapic, J. Math. Phys. 33, 643 (1992) (http://www.physik.fu-ber lin.de/~kleinert/207).

668


[5] G. Pöschl and E. Teller, Z. Phys. 83, 143 (1933). See also S. Fl¨ ugge, Practical Quantum Mechanics, Springer, Berlin, 1974, p. 89. [6] H. Kleinert, Phys. Lett. B 236, 315 (1990) (ibid.http/202). [7] M. Böhm and G. Junker, J. Math. Phys. 28, 1978 (1987). Note, however, that these authors do not really solve the path integral on the group space as they claim but only near the group space. Also, many expressions are meaningless due to path collapse. [8] M. Böhm and G. Junker, J. Math. Phys. 30, 1195 (1989). See also remarks in [7]. [9] D.D. Ballow and F.H. Steen, Plane and Spherical Trigonometry with Tables, Ginn, New York , 1943. [10] J.R. Klauder, Phys. Rev. D 19, 2349 (1979); A. Jevicki and N. Papanicolaou, Nucl. Phys. B 171, 382 (1980). [11] T.H.R. Skyrme, Proc. R. Soc. London A 260, 127 (1961). [12] P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931); Phys. Rev. 74, 817 (1948), Phys. Rev. 74, 817 (1948). [13] M. Stone, Phys. Rev. D 33, 1191 (1986). [14] For the Berry phase and quantum spin systems see the reprint collection A. Shapere and F. Wilczek, Geometric Phases in Physics, World Scientific, Singapore, 1989, and the textbook E. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, MA, 1991. [15] H. Poincaré, Remarques sur une expérience de Birkeland, Rendues, 123, 530 (1896). [16] E. Witten, Nucl. Phys. B 223, 422 (1983). [17] L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon, London, 1975, Chapter 7. [18] The ability of the monopole action to generate the Landau-Lifshitz equation in a classical equation of motion was apparently first recognized by C.F. Valenti and M. Lax, Phys. Rev. B 16, 4936-4944 (1977).


H. Kleinert, PATH INTEGRALS June 16, 2007 ( /home/kleinert/kleinert/books/pathis/pthic9.tex) Tollimur in caelum curvato gurgite, et idem Subducta ad manes imos descendimus unda. We are carried up to the heaven by the circling wave, and immediately the wave subsiding, we descend to the lowest depths. Virgil, Aeneid, 3, 564

9 Wave Functions The fundamental quantity obtained by solving a path integral is the time evolution amplitude or propagator of a system (xb tb |xa ta ). In Schrödinger quantum mechanics, on the other hand, one has direct access on the energy spectrum and the wave functions of a system [see (1.94)]. This chapter will explain how to extract this information from the time evolution amplitude (xb tb |xa ta ). The crucial quantity for this purpose the Fourier transform of (xb tb |xa ta ), the fixed-energy amplitude introduced in Eq. (1.313): (xb |xa )E =

Z

∞ ta

dtb exp {iE(tb − ta )/¯ h} (xb tb |xb ta ),

(9.1)

which contains as much information on the system as (xb tb |xa ta ), and gives, in particular, a direct access to the energy spectrum and the wave functions of the system. This is done via the the spectral decomposition (1.321). Alternatively, we can work with the causal propagator at imaginary time, (xb τb |xa τa ) =

Z

dD p i p2 exp − x ) − p(x (τb − τa ) , b a (2π¯ h)D h ¯ 2M¯ h "

#

(9.2)

and calculate the fixed-energy amplitude by the Laplace transformation (xb |xa )E = −i

9.1

Z

∞ τa

dτb exp {E(τb − τa )/¯ h} (xb τb |xb τa ).

(9.3)

Free Particle in D Dimensions

For a free particle in D dimensions, the fixed-energy amplitude was calculated in Eqs. (1.344) and (1.351). It will be instructive to rederive the same result once more using the development in Section 8.5.1. Here we start directly from the spectral representation (1.321), which for a free particle takes the explicit form Eq. (1.340): (xb |xa )E =

Z

dD k ik(xb −xa ) i¯ h e . 2 2 D (2π) E−h ¯ k /2M + iη 669

(9.4)

670

9 Wave Functions

The momentum integral can now be done as follows. The exponential function exp (ikR) is written as exp (ikR cos ϑ), where R is the distance vector xb − xa and ϑ the angle between k and R. Then we use formula (8.100) with the coefficients (8.101) and the hyperspherical harmonics Ylm (ˆ x) of Eq. (8.125) and expand eikR =

X

X

al (ikR)

∗ ˆ ˆ lm Ylm (k)Y (R).

(9.5)

m

l=0

The integral over k follows now directly from the decomposition into size and diˆ and the orthogonality property (8.114) of the hyperspherical rection dD k = dkdk, harmonics, according to which Z

q

ˆ Ylm (k) ˆ = δl0 δm0 SD , dk

(9.6)

√ with SD of Eq. (1.555). Since Y00 (ˆ x) = 1/ SD , we obtain 2Mi (xb |xa )E = − (2π)D

Z

κ≡

q

where

∞

dkk D−1

0

k2

1 a0 (ikR), + κ2

(9.7)

−2ME/¯ h2 ,

(9.8)

as in (1.342). Inserting a0 (ikR) from (8.101), a0 (ikR) = (2π)D/2 JD/2−1 (kR)/(kR)D/2−1 , we find

2Mi (xb |xa )E = − R1−D/2 (2π)D/2

Z

0

∞

dkk D/2

k2

1 JD/2−1 (kR). + κ2

(9.9)

(9.10)

The integral Z

0

∞

dk

k ν+1 aν−µ bµ J (kb) = Kν−µ (ab) ν (k 2 + a2 )µ+1 2µ Γ(µ + 1)

(9.11)

yields once more the fixed-energy amplitude (1.343). In two dimensions, the amplitude (1.343) becomes [recall (1.346)] (xb |xa )E = −

iM K0 (κ|xb − xa |). π¯ h

(9.12)

It can be decomposed into partial waves by inserting |xb − xa | =

q

rb2 + ra2 − 2ra rb cos(ϕb − ϕa ).

(9.13)

Then a well-known addition theorem for Bessel functions yields the expansion K0 (κ|xb − xa |) =

∞ X

Im (κr< )Km (κr> )eim(ϕb −ϕa ) ,

(9.14)

m=−∞


9.1 Free Particle in D Dimensions

671

where r< and r> are the smaller and larger values of ra and rb , respectively. Hence the fixed-energy amplitude turns into (xb |xa )E = −

2iM X 1 Im (κr< )Km (κr> ) eim(ϕb −ϕa ) . h ¯ m 2π

(9.15)

This is an analytic function in the complex E-plane. The parameter κ is real for E < 0. For E > 0, √the square root (9.8) allows for two imaginary solutions, κ± ≡ e∓iπ/2 k ≡ e∓iπ/2 2ME/¯ h, so that the amplitude has a right-hand cut. Its discontinuity specifies the continuum of free-particle states. On top of the cut, we use the analytic continuation formulas (valid for −π/2 < arg z ≤ π)1 Iµ (e−iπ/2 z) = e−iπµ/2 Jµ (z), π Kµ (e−iπ/2 z) = i eiπµ/2 Hµ(1) (z) , 2

(9.16)

to find the fixed-energy amplitude above the cut (xb |xa )E+iη =

πM X 1 (1) Jm (kr< )Hm (kr> ) eim(ϕb −ϕa ) . h ¯ m 2π

(9.17)

The reflection properties2 (2)

Hµ(1) (eiπ z) = −H−µ (z) ≡ −e−iπµ Hµ(2) (z), Jµ (eiπ z) = eiπµ Jµ (z)

(9.18)

yield the amplitude below the cut (xb |xa )E−iη = −

πM X 1 (2) Jm (kr< )Hm (kr> ) eim(ϕb −ϕa ) . h ¯ m 2π

(9.19)

The discontinuity across the cut follows from the relation Jµ (z) =

i 1 h (1) Hµ (z) + Hµ(2) (z) 2

(9.20)

and reads disc (xb |xa )E =

∞ 2πM X 1 Jm (krb )Jm (kra ) eim(ϕb −ϕb ) . h ¯ m=−∞ 2π

(9.21)

According to (1.326), the integral over the discontinuity yields the completeness relation dE disc (xb |xa )E h −∞ 2π¯ Z ∞ ∞ dE 2πM X 1 = Jm (krb )Jm (kra ) eim(ϕb −ϕb ) = δ(xb − xa ). h h ¯ m=−∞ 2π −∞ 2π¯

Z

1 2

∞

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 8.406.1 and 8.407.1. ibid., Formulas 8.476.1 and 8.476.8.

(9.22)

672

9 Wave Functions

After replacing the energy integral by a k-integral, Z

∞

−∞

dE 2πM = 2π¯ h h ¯

Z

0

∞

dkk,

(9.23)

we can also write Z

∞

−∞

Z dE d2 k ik(xb −xa ) 1 Z∞ disc (xb |xa )E = e = dkkJ0 (k|xb − xa |) 2π¯ h (2π)2 2π 0 Z ∞ ∞ X 1 dkkJm (krb )Jm (kra ) eim(ϕb −ϕa ) = 2π m=−∞ 0 1 = √ δ(rb − ra )δ(ϕb − ϕa ). (9.24) rb ra

The last two lines exhibit the well-known completeness relation of the radial wave functions of a free particle.3

Harmonic Oscillator in D Dimensions

9.2

The wave functions of the one-dimensional harmonic oscillator have already been derived in Section 2.6 from a spectral decomposition of the time evolution amplitude. This was possible with the help of Mehler’s formula. In D dimensions, the fixed-energy amplitude is the best starting point for determining the wave functions. We take the radial propagator (8.141) obtained from the angular momentum decomposition (8.140) or, for the sake of greater generality, the radial amplitude (8.143) with an additional centrifugal barrier, continue it to imaginary time τ = it, and go over to its Laplace transform q

q

(rb |ra )E,l = −i Mω/¯ h Mωrb ra /¯ h 1 Mω 2

×e− h¯

Z

∞

τa

coth[ω(τb −τa )](rb2 +ra2 )

1 sin[ω(τb − τa )] ! Mωrb ra . h ¯ sinh[ω(τb − τa )]

dτb eE(τb −τa )/¯h

Iµ

(9.25)

To evaluate the τ -integral we make use of a standard integral formula for Bessel functions4 Z

0

∞

dx [coth(x/2)]2ν e−β cosh x Jµ (α sinh x) q q Γ((1 + µ)/2 − ν) 2 2 2 2 = Wν,µ/2 α + β + β M−ν,µ/2 α + β − β ,(9.26) αΓ(µ + 1)

where Wν,µ/2 (z), M−ν,µ/2 (z) are the Whittaker functions. The formula is valid for Re β > |Re α|,

1 Re (µ/2 − ν) > − . 2

3

Compare with Eqs. (3.112) and (3.139) in J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1975. 4 I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 6.669.1. H. Kleinert, PATH INTEGRALS

9.2 Harmonic Oscillator in D Dimensions

673

By a change of variables q

α2 + β 2 ± β = tαb,a ,

and

sinh x = (sinh y)−1 , cosh x = coth y, coth(x/2) = ey , coth x = cosh y, with dx = dy/ sinh y, Eq. (9.26) goes over into Z

∞

0

! √ α α dy 2νy b a e exp [− 12 t(αb − αa ) coth y] Jµ t sinh y sinh y Γ ((1 + µ)/2 − ν) Wν,µ/2 (tαb )M−ν,µ/2 (−tαa ). = √ t αb αa Γ(µ + 1)

(9.27)

Using the identity M−ν,µ/2 (z) ≡ e−i(µ+1)π/2 Mν,µ/2 (−z)

(9.28)

and changing the sign of αa in (9.27), this can be turned into ! √ Z ∞ t α α dy 2νy b a e exp [− 12 t(αb + αa ) coth y] Iµ sinh y sinh y 0 Γ ((1 + µ)/2 − ν) = √ Wν,µ/2 (tαb )Mν,µ/2 (tαa ), t αb αa Γ(µ + 1)

(9.29)

with the range of validity αb > αa > 0, Re [(1 + µ)/2 − ν] > 0, Re t > 0, |arg t| < π.

(9.30)

Setting

M 2 Mω 2 ωrb , αa = r , ν = E/2ω¯ h h ¯ h ¯ a in (9.29) brings the radial amplitude (9.25) to the form (valid for rb > ra ) y = ω(tb − ta ), αb =

(9.31)

1 1 Γ((1 + µ)/2 − ν) Mω 2 Mω 2 (rb |ra )E,l = −i √ Wν,µ/2 rb Mν,µ/2 r . (9.32) ω rb ra Γ(µ + 1) h ¯ h ¯ a

The Gamma function has poles at ν = νr ≡ (1 + µ)/2 + nr

(9.33)

for integer values of the so-called radial quantum number of the system nr = 0, 1, 2, . . . . The poles have the form ν∼ν

Γ ((1 + µ)/2 − ν) ∼ r −

(−1)nr 1 . nr ! ν − νr

(9.34)

674

9 Wave Functions

Inserting here the particular value of the parameter µ for the D-dimensional oscillator which is µ = D/2 + l − 1, and remembering that ν = E/2ω¯ h, we find the energy spectrum E=h ¯ ω (2nr + l + D/2) . (9.35) The principal quantum number is defined by n ≡ 2nr + l

(9.36)

and the energy depends on it as follows: En = h ¯ ω(n + D/2).

(9.37)

For a fixed principal quantum number n = 2nr + l, the angular momentum runs through l = 0, 2, . . . , n for even, and l = 1, 3, . . . , n for odd n. There are (n + 1)(n + 2)/2 degenerate levels. From the residues 1/(ν − νr ) ∼ 2¯ hω/(E − En ), we extract the product of radial wave functions at given nr , l: 1 2(−1)nr rb ra Γ(µ + 1)nr ! h)M(1+µ)/2+nr , µ2 (Mωra 2 /¯ h). × W(1+µ)/2+nr , µ2 (Mωrb 2 /¯

Rnr l (rb ) Rnr l (ra ) = √

(9.38)

It is now convenient to express the Whittaker functions in terms of the confluent hypergeometric or Kummer functions:5 W(1+µ)/2+nr , µ2 (z) = e−z/2 z (1+µ)/2 U(−nr , 1 + µ, z),

(9.39)

M(1+µ)/2+nr , µ2 (z) = e−z/2 z (1+µ)/2 M(−nr , 1 + µ, z).

(9.40)

The latter equation follows from the relation M−(1+µ)/2−nr , µ2 (z) = e−z/2 z (1+µ)/2 M(1 + µ + nr , 1 + µ, z),

(9.41)

after replacing nr → −nr − µ − 1. For completeness, we also mention the identity M(a, b, z) = ez M(b − a, b, −z),

(9.42)

M(1 + µ + nr , 1 + µ, z) = ez M(−nr , 1 + µ, −z).

(9.43)

so that This permits us to rewrite (9.41) as M−(1+µ)/2−nr , µ2 (z) = ez/2 z (1+µ)/2 M(−nr , 1 + µ, −z),

(9.44)

which turns into (9.40) by using (9.28) and appropriately changing the indices. 5



675

The Kummer function M(a, b, z) has the power series a(a + 1) z a M(a, b, z) ≡ 1 F1 (a; b; z) = 1 + z + + ..., b b(b + 1) z!

(9.45)

2

showing that M(1+µ)/2+nr , µ2 (Mωra 2 /2¯ h) is an exponential e−M ωra /¯h times a polynomial in ra of order 2nr . A similar expression is obtained for the other factor W(1+µ)/2+nr , µ2 (Mωrb 2 /¯ h) of Eq. (9.39). Indeed, the Kummer function U(a, b, z) is related to M(a, b, z) by6 "

#

π M(a, b, z) M(1 + a − b, 2 − b, z) U(a, b, z) = . − z 1−b sin πb Γ(1 + a − b)Γ(b) Γ(a)Γ(2 − b)

(9.46)

Since a = −nr with integer nr and 1/Γ(a) = 0, we see that only the first term in the brackets is present. Then the identity Γ(−µ)Γ(1 + µ) = π/ sin[π(1 + µ)] leads to the relation U(−nr , 1 + µ, z) =

Γ(−µ) M(−nr , 1 + µ, z), Γ(−nr − µ)

(9.47)

which is a polynomial in z of order nr . Thus we have the useful formula W(1+µ)/2+nr , µ2 (zb )M(1+µ)/2+nr , µ2 (za ) =

Γ(−µ) e−(zb +za )/2 Γ(−nr − µ)

×(zb za )(1+µ)/2 M(−nr , 1 + µ, zb )M(−nr , 1 + µ, za ).

(9.48)

We can therefore re-express Eq. (9.38) as Rnr l (rb ) Rnr l (ra ) = 2

s 2

Mω 2(−)nr Γ(−µ) h ¯ Γ(−nr − µ)Γ(1 + µ) nr !

×e−M ω(rb +ra )/2¯h (Mωrb ra /¯ h)1/2+µ ×M(−nr , 1 + µ, Mωrb2 /¯ h)M(−nr , 1 + µ, Mωra2/¯ h). We now insert

(−)nr Γ(−µ) Γ(nr + 1 + µ) = , Γ(−nr − µ) Γ(1 + µ)

(9.49)

(9.50)

setting µ = D/2 + l − 1, and identify the wave functions as Rnr l (r) = Cnr l (Mω/¯ h)1/4 (Mωr 2 /¯ h)l/2+(D−1)/4 ×e−M ωr 6

2 /2¯ h

M(−nr , l + D/2, Mωr 2/¯ h),

(9.51)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965, Formula 13.1.3.

676

9 Wave Functions

with the normalization factor Cnr l

√

2 = Γ(1 + µ)

By introducing the Laguerre polynomials Lµn (z) ≡

s

(nr + µ)! . nr !

(9.52)

7

(n + µ)! M(−n, µ + 1, z), n!µ!

(9.53)

and using the integral formula8 Z

0

∞

dze−z z µ Lµn (z)Lµn′ (z) = δnn′

(n + µ)! , n!

(9.54)

we find that the radial wave functions satisfy the orthonormality relation Z

∞ 0

drRnr l (r)Rn′r l (r) = δnr n′r .

(9.55)

The radial imaginary-time evolution amplitude has now the spectral representation ∞ (rb τb |ra τa )l =

X

Rnr l (rb )Rnr l (ra )e−En (τb −τa )/¯h ,

(9.56)

nr =0

with the energies En = h ¯ ω(n + D/2) = h ¯ ω (2nr + l + D/2) .

(9.57)

The full causal propagator is given, as in (8.91), by (xb τb |xa τa ) =

∞ X X 1 ∗ (rb τb |ra τa )l Ylm (ˆ xb )Ylm (ˆ xa ). (D−1)/2 (rb ra ) m l=0

(9.58)

From this, we extract the wave functions ψnr lm (x) =

1 r (D−1)/2

Rnr l (r)Ylm(ˆ x).

(9.59)

They have the threshold behavior r l near the origin. The one-dimensional oscillator may be viewed as a special case of these formulas. For D = 1, the partial wave expansion amounts to a separation into even and odd wave functions. There are two “spherical harmonics”, 1 Ye,ø (ˆ x) = √ (Θ(x) ± Θ(−x)), 2

(9.60)

7

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.970 (our definition differs from that in L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965, Eq. (d.13). The µ relation is Lµn = (−)µ /(n + µ)!LL.L. n+µ ). 8 ibid., Formula 7.414.3. H. Kleinert, PATH INTEGRALS


677

and the amplitude has the decomposition (xb τb |xa τa ) = (rb τb |ra τa )e Ye (ˆ xb )Ye (ˆ xa ) + (rb τb |ra τa )ø Yø (ˆ xb )Yø (ˆ xa ),

(9.61)

with the “radial” amplitudes (rb τb |ra τa )e,ø = (xb τb |xa τa ) ± (−xb τb |xa τa ).

(9.62)

These are known from Eq. (2.170) to be 1 (rb τb |ra τa )e,ø = √ 2π

v u u t

Mω/¯ h sinh[ω(τb − τa )]

(9.63)

Mω 2 × exp − (rb + ra2 ) cot ω(τb − τa ) 2 2¯ h

(

cosh sinh

)

(Mωrb ra /¯ h) .

The two cases coincide with the integrand of (9.25) for l = 0 and 1, respectively, since µ = l + D/2 − 1 takes the values ±1/2 and √

2 zI∓ 1 (z) = 2 2π

(

cosh z , sinh z .

(9.64)

h ¯ ω(2nr + 12 ) even, h ¯ ω(2nr + 32 ) odd,

(9.65)

The associated energy spectrum (9.35) is E=

(

with the radial quantum number nr = 0, 1, 2, . . . . The two cases follow the single formula E=h ¯ ω(n + 12 ), (9.66) where the principal quantum number n = 0, 1, 2, . . . is related to nr by n = 2nr and n = 2nr + 1, respectively. The radial wave functions (9.51) become Rnr ,e (r) = (Mω/¯ h) Rnr ,ø (r) =

v u u 1/4 t 2Γ(nr

+ 12 ) M(−nr , 12 , Mωr 2 /¯ h), πnr !

v u u 1/4 t 2Γ(nr (Mω/¯ h)

+ 32 ) (π/4)nr !

s

Mωr 2 M(−nr , 32 , Mωr 2 /¯ h). h ¯

(9.67) (9.68)

The special Kummer functions appearing here are Hermite polynomials n! (−)n H2n (x), (2n)! √ n! M(−n, 32 , x2 ) = (−)n H2n+1 (x)/2 x. (2n + 1)! M(−n, 12 , x2 ) =

(9.69) (9.70)

678

9 Wave Functions

Using the identity Γ(z)Γ(z + 12 ) = (2π)1/2 2−2z+1/2 Γ(2z),

(9.71)

we obtain in either case the radial wave functions [to be compared with the onedimensional wave functions (2.295)] √ 2 2 (9.72) Rn (r) = Nn 2λω−1/2 e−r /2λω Hn (r/λω ), n = 0, 1, 2, . . . with λω ≡

s

h ¯ , Mω

1 Nn = q √ . 2n n! π

(9.73)

This formula holds for both even and odd wave functions with nr = 2n and nr = 2n + 1, respectively. It is easy to check that they possess the correct normalization √ R∞ 2 2 0 drRn (r) = 1. Note that the “spherical harmonics” (9.60) remove a factor in (9.72), but compensate for this by extending the x > 0 integration to the entire x-axis by the “one-dimensional angular integration”.

Free Particle from ω → 0 -Limit of Oscillator

9.3

The results obtained for the D-dimensional harmonic oscillator in the last section can be used to find the amplitude and wave functions of a free particle in D dimensions in radial coordinates. This is done by taking the limit ω → 0 at fixed energy E. In the amplitude (9.32) with Wν,µ/2 (z), Mν,µ/2 (z) substituted according to (9.39), we rewrite nr as (E/ω¯ h − l − 1) /2 and go to the limit ω → 0 at a fixed energy E. Replacing Mωr 2 /¯ h by k 2 r 2 /2nr ≡ z/nr (where z = k 2 r 2 /2, and using E = p2 /2M=¯ h2 k 2 /2M), we apply the limiting formulas9 lim {Γ(1 − nr − b)U (−a, b, ∓z/nr )} ( √ 2Kb−1 (2 z) 1 − 2 (b−1) √ =z (1) −iπeiπb Hb−1 (2 z) (Im z > 0),

nr →∞

lim M (−a, b, ∓z/nr ) /Γ(b) = z

− 12 (b−1)

nr →∞

(

√ Ib−1 (2 √z) Jb−1 (2 z),

(9.74)

(9.75)

and obtain the radial wave functions directly from (9.51) and (9.75): nr− − − →∞

Rnr l (r) − −−→ Cnr l (Mωr 2 /¯ h)(µ/2+1/2) k 2 r 2 /2 where

9

Γ(1 + µ)Jµ (kr),

(9.76)

√ E µ/2 1 − −−→ 2 . 2¯ hω Γ(1 + µ)

(9.77)

q √ nr− − − →∞ Rnr l (r) − −−→ r 1/2 Mω/¯ h 2Jµ (kr).

(9.78)

Cnr l Hence

−µ/2

nr− − − →∞

M. Abramowitz and I. Stegun, op. cit., Formulas 13.3.1–13.3.4.


9.3 Free Particle from ω → 0 -Limit of Oscillator

679

Inserting these wave functions into the radial time evolution amplitude X

(rb τb |ra τa )l =

Rnr l (rb )Rnr l (rb )e−En (τb −τa )/¯h ,

(9.79)

nr

and replacing the sum over nr by the integral 0∞ dk h ¯ k/Mω [in accordance with 2 2 the nr → ∞ limit of Enr = ω¯ h(2nr + l + D/2) → h ¯ k /2M], we obtain the spectral representation of the free-particle propagator R

√

(rb τb |ra τa )µ =

rb ra

Z

∞

0

h ¯ k2

dk kJµ (krb )Jµ (kra )e− 2M (τb −τa ) .

(9.80)

For comparison, we derive the same results directly from the initial spectral representation (9.2) in one dimension (xb τb |xa τa ) =

Z

∞

−∞

dk h ¯ k2 exp ik(xb − xa ) − (τb − τa ) . 2π 2M "

#

(9.81)

Its “angular decomposition” is a decomposition with respect to even and odd wave functions h ¯ k2 dk [cos k(rb − ra ) ± cos k(rb + ra )]e− 2M (τb −τa ) π ( 0 ) Z ∞ h ¯ k2 dk cos krb cos kra = 2 e− 2M (τb −τa ) . sin krb sin kra π 0

(rb τb |ra τa )e,ø =

Z

∞

(9.82)

In D dimensions we use the expansion (8.100) for eikx to calculate the amplitude in the radial form dD k ik(xb −xa ) − h¯ k2 (τb −τa ) (xb τb |xa τa ) = e e 2M (2π)D Z ∞ h ¯ k2 1 1 = dkk 2ν Jν (kR)e− 2M (τb −τa ) , D/2 ν (2π) (kR) 0 Z

(9.83)

with ν ≡ D/2 − 1. With the help of the addition theorem for Bessel functions10 (8.187) we rewrite ∞ 1 2ν Γ(ν) X (ν) J (kR) = (ν + l)Jν+l (krb )Jν+l (kra )Cl (∆ϑ) ν ν 2 ν (kR) (k rb ra ) l=0

(9.84)

and expand further according to ∞ X 1 (2π)D/2 X ∗ J (kR) = J (kr )J (kr ) Ylm (ˆ xb )Ylm (ˆ xa ), ν ν+l b ν+l a (kR)ν (k 2 rb ra )ν l=0 m

(9.85)

to obtain the radial amplitude (rb τb |ra τa )l = 10

√

rb ra

Z

0

∞

h ¯ k2

dkkJν+l (krb )Jν+l (kra )e− 2M (τb −τa ) ,

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.532.

(9.86)

680

9 Wave Functions

just as in (9.80). For D = 1, this reduces to (9.82) using the particular Bessel functions √

9.4

2 zJ∓1/2 (z) = √ 2π

(

cos z sin z

)

.

(9.87)

Charged Particle in Uniform Magnetic Field

Let us also find the wave functions of a charged particle in a magnetic field. The amplitude was calculated in Section 2.18. Again we work with the imaginary-time version. Factorizing out the free motion along the direction of the magnetic field, we write ⊥ (xb τb |xa τa ) = (zb τb |za τa )(x⊥ (9.88) b τb |xa τa ), with

M (zb − za )2 exp − , (zb τb |za τa ) = q 2¯ h τb − τa 2π¯ h(τb − τa )/M 1

(

)

(9.89)

and have for the amplitude in the transverse direction ⊥ (x⊥ b τb |xa τa ) =

h i M ω(τb − τa )/2 exp −A⊥ /¯ h , l 2π¯ h(τb − τa ) sinh [ω(τb − τa )/2]

(9.90)

with the classical transverse action A⊥ l =

o Mω n 1 ⊥ ⊥ 2 ⊥ ⊥ coth [ω(τ − τ )/2] (x − x ) + x × x . b a 2 b a a b 2

(9.91)

This result is valid if the vector potential is chosen as 1 A = B × x. 2

(9.92)

A = (0, Bx, 0),

(9.93)

In the other gauge with there is an extra surface term, and A⊥ cl is replaced by Mω ⊥ A˜⊥ (xb yb − xa ya ). cl = Acl + 2

(9.94)

The calculation of the wave functions is quite different in these two gauges. In the gauge (9.93) we merely recall the expressions (2.655) and (2.657) and write down the integral representation ⊥ (x⊥ b τb |xa τa )

=

Z

dpy ipy (yb −ya )/¯h e (xb τb |xa τa )x0 =py /M ω , 2π¯ h

(9.95)

with the oscillator amplitude in the x-direction (xb τb |xa τa )x0 =

s

Mω 1 exp − Aos , 2π¯ h sinh[ω(τb − τa )] h ¯ cl

(9.96)


681

9.4 Charged Particle in Uniform Magnetic Field

and the classical oscillator action centered around x0 Aos cl =

Mω {[(xb − x0 )2 + (xa − x0 )2 ] cosh[ω(τb − τb )] 2 sinh[ω(τb − τa )] − 2(xb − x0 )(xa − x0 )}. (9.97)

The spectral representation of the amplitude (9.96) is then (xa τb |xa τa )x0 =

∞ X

n=0

ψn (xb − x0 )ψn (xa − x0 )e−(n+ 2 )ω(τb −τa ) , 1

(9.98)

where ψn (x) are the oscillator wave functions (2.295). This leads to the spectral representation of the full amplitude (9.88) (xb τb |xa τa ) =

Z

×

dpz Z dpy ipz (zb −za )/¯h e 2π¯ h 2π¯ h

∞ X

n=0

(9.99) 1

ψn (xb − py /Mω)ψn (xa − py /Mω)e−(n+ 2 )ω(τb −τa ) .

The combination of a sum and two integrals exhibits the complete set of wave functions of a particle in a uniform magnetic field. Note that the energy En = (n + 12 )¯ hω

(9.100)

is highly degenerate; it does not depend on py . In the gauge A = 12 B × x, the spectral decomposition looks quite different. To derive it, the transverse Euclidean action is written down in radial coordinates [compare Eq. (2.661)] as A⊥ cl

h i M ω = coth [ω(τb − τa )/2] rb 2 + ra 2 − 2rb ra cos(ϕb − ϕa ) 2 2

− iωrb ra sin(ϕb − ϕa ) .

(9.101)

This can be rearranged to Mω coth [ω(τb − τa )/2] (rb 2 + ra 2 ) 2 2 M ω cos [ϕb − ϕa − iω(τb − τa )/2] . − 2 sinh [ω(τb − τa )/2]

A⊥ cl =

(9.102)

⊥

We now expand e−Acl /¯h into a series of Bessel functions using (8.5) −A⊥ h cl /¯

e

Mω = exp − coth [ω(τb − τa )/2] (rb 2 + ra 2 ) (9.103) 2¯ h2 ! ∞ X Mω rb ra × Im emω(τb −τa )/2 eim(ϕb −ϕa ) . 2¯ h sinh [ω(τ − τ )/2] b a m=−∞

682

9 Wave Functions

The fluctuation factor is the same as before. Hence we obtain the angular decomposition of the transverse amplitude ⊥ (x⊥ b τb |xa τa ) = √

1 1 X (rb τb |ra τa )m eim(ϕb −ϕa ) , rb ra m 2π

(9.104)

where √

!

Mω η Mωrb ra Mω (rb τb |ra τa )m = rb ra exp − coth η(rb2 + ra2 ) Im emη , 2¯ hη sinh η 2¯ h2 2¯ h sinh η (9.105)

with η ≡ ω(τb − τa )/2.

(9.106)

To find the spectral representation we go to the fixed-energy amplitude (rb |ra )m,E = −i

Z

∞

τa

dτb eE(τb −τa )/¯h (rb τb |ra τa )m

(9.107)

! Z √ M ∞ 1 −(M ω/4¯h) coth η(r2 +ra2 ) Mωrb ra 2νη b = −i rb ra dη e e Im . h ¯ 0 sinhη 2¯ hsinhη

The integral is done with the help of formula (9.29) and yields Γ

1 2

−ν+

|m| 2

√ M (rb |ra )m,E = −i rb ra h ¯ (Mω/2¯ h)rb ra Γ(|m| + 1) Mω 2 Mω 2 ×Wν,|m|/2 r Mν,|m|/2 r , 2¯ h b 2¯ h a with ν≡

E m + . ω¯ h 2

(9.108)

(9.109)

The Gamma function Γ(1/2 − ν − |m|/2) has poles at ν = νr ≡ nr +

1 |m| + 2 2

(9.110)

of the form Γ(1/2 − ν − |m|/2) ≈ −

1 (−1)nr (−1)nr ω¯h ∼− . nr ! ν − νr nr ! E − Enr m

(9.111)

The poles lie at the energies Enr m

!

1 |m| m =h ¯ ω nr + + − . 2 2 2

(9.112)


683


These are the well-known Landau levels of a particle in a uniform magnetic field. The Whittaker functions at the poles are (for m > 0) M−ν,m/2 (z) = ez/2 z

1+m 2

M(−nr , 1 + m, −z), 1+m (nr + m)! Wν,m/2 (z) = e−z/2 z 2 (−)nr M(−nr , 1 + m, z). m!

(9.113) (9.114)

The fixed-energy amplitude near the poles is therefore (rb |ra )m,E ∼

i¯ h Rn m (rb )Rnr m (ra ), E − Enr m r

(9.115)

with the radial wave functions11 s √ Mω 1/2 (nr + |m|)! 1 Rnr m (r) = r h ¯ nr ! |m|!

Mω 2 × exp − r 4¯ h

Mω 2 r 2¯ h

|m|/2

(9.116) Mω 2 M −nr , 1 + |m|, r . 2¯ h

Using Eq. (9.53), they can be expressed in terms of Laguerre polynomials Lαn (z): Rnr m

s √ Mω 1/2 nr ! Mω 2 = r exp − r h ¯ (nr + |m|)! 4¯ h

Mω 2 × r 2¯ h

|m|/2

L|m| nr

Mω 2 r . 2¯ h

(9.117)

The integral (9.54) ensures the orthonormality of the radial wave functions Z

∞

0

drRnr m (r)Rn′r m (r) = δnr n′r .

(9.118)

A Laplace transformation of the fixed-energy amplitude (9.108) gives, via the residue theorem, the spectral representation of the radial time evolution amplitude (rb τb |ra τa ) =

X

Rnr m (rb )Rnr m (ra )e−Enr m (τb −τa )/¯h ,

(9.119)

nr m

with the energies (9.112). The full wave functions in the transverse subspace are, of course, 1 eimϕ ψnr m (x) = √ Rnr m (r) √ . (9.120) r 2π Comparing the energies (9.112) with (9.100), we identify the principal quantum number n as |m| m n ≡ nr + − . (9.121) 2 2 11

Compare with L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965, p. 427.

684

9 Wave Functions

Note that the infinite degeneracy of the energy levels observed in (9.100) with respect to py is now present with respect to m. This energy does not depend on m for m ≥ 0. The somewhat awkward m-dependence of the energy can be avoided by introducing, instead of m, another quantum number n′ related to n, m by m = n′ − n.

(9.122)

The states are then labeled by n, n′ with both n and n′ taking the values 0, 1, 2, 3, . . . . For n′ < n, one has n′ = nr and m = n′ − n < 0, whereas for n′ ≥ n one has n = nr and m = n′ − n ≥ 0. There exists a natural way of generating the wave functions ψnr m (x) such that they appear immediately with the quantum numbers n, n′ . For this we introduce the Landau radius a=

s

2¯ h = Mω

s

2¯ hc eB

(9.123)

as a length parameter and define the dimensionless transverse coordinates √ √ z = (x + iy)/ 2a, z ∗ = (x − iy)/ 2a. (9.124) It is then possible to prove that the ψnr m ’s coincide with the wave functions ∗

z∗ z

ψn,n′ (z, z ) = Nn,n′ e

1 − √ ∂z∗ 2

!n

1 − √ ∂z 2

!n′

∗

e−2z z .

(9.125)

The normalization constants are obtained by observing that the differential operators z∗ z

e

ez

∗z

!

1 1 ∗ − √ ∂z ∗ e−z z = √ (−∂z ∗ + z), 2 2 ! 1 1 ∗ − √ ∂z e−z z = √ (−∂z + z ∗ ) 2 2

(9.126)

behave algebraically like two independent creation operators 1 a ˆ† = √ (−∂z ∗ + z), 2 ˆb† = √1 (−∂z + z ∗ ), 2

(9.127)

whose conjugate annihilation operators are 1 aˆ = √ (∂z + z ∗ ), 2 1 ˆb = √ (∂z ∗ + z). 2


685


The ground state wave function annihilated by these is ∗

ψ0,0 (z, z ∗ ) = hz, z ∗ |0i ∝ e−z z .

(9.129)

We can therefore write the complete set of wave functions as ψn,n′ (z, z ∗ ) = Nnn′ a ˆ†nˆb†n ψ0,0 (z, z ∗ ). ′

(9.130)

Using the fact that a ˆ†∗ = ˆb† , ˆb†∗ = a ˆ† , and that partial integrations turn ˆb† , a ˆ† into a ˆ, ˆb, respectively, the normalization integral can be rewritten as Z

dx dy ψn1 ,n′ 1 (z, z ∗ )ψn2 ,n′2 (z, z ∗ ) = Nn1 n′ 1 Nn2 n′ 2 = Nn1 n′ 1 Nn2 n′ 2

Z Z

h

′

dxdy (a† )n1 (b† )n 1 e−z dxdye−2z

∗z

∗z

′

ih

′

(a† )n2 (b† )n 2 e−z ′

an1 bn 1 a†n2 b†n 2 .

∗z

i

(9.131)

Here the commutation relations between a ˆ† , ˆb† , a ˆ, ˆb serve to reduce the parentheses in the last line to n1 !n2 ! δn1 n′ 1 δn2 n′ 2 . (9.132) The trivial integral Z

−2z ∗ z

dxdy e

=π

Z

dr 2 e−r

2 /a2

= πa2

(9.133)

shows that the normalization constants are Nn,n′ = √

1 . πa2 n!n′ !

(9.134)

Let us prove the equality of ψnr m and ψn,n′ up to a possible overall phase. For this we first observe that z, ∂z ∗ and z ∗ , ∂z carry phase factors eiϕ and e−iϕ , respectively, so that the two wave functions have obviously the azimuthal quantum number m = n− n′ . Second, we make sure that the energies coincide by considering the Schrödinger equation corresponding to the action (2.635) 1 e −i¯ h∇ − A 2M c

2

ψ = Eψ.

(9.135)

In the gauge where A = (0, Bx, 0), it reads

h ¯2 eB ∂x 2 + (∂y − i x)2 + ∂z 2 ψ = Eψ, 2M c and the wave functions can be taken from Eq. (9.99). In the gauge where −

A = (−By/2, Bx/2, 0) ,

(9.136)

(9.137)

686

9 Wave Functions

on the other hand, the Schrödinger equation becomes, in cylindrical coordinates, "

h ¯2 1 1 ie¯ hB e2 B 2 2 − ∂r2 + ∂r + 2 ∂ϕ2 + ∂z2 − r ψ(r, z, ϕ) ∂ϕ + 2M r r 2Mc 8Mc2 = Eψ(r, z, ϕ).

#

(9.138)

Employing a reduced radial coordinate ρ = r/a and factorizing out a plane wave in the z-direction, eipz z/¯h , this takes the form "

∂ρ2

1 a2 1 + ∂ρ + 2 (2ME − p2z ) − ρ2 − 2i∂ϕ − 2 ∂ϕ2 ψ(r, ϕ) = 0. ρ ρ h ¯ #

(9.139)

The solutions are 2 /2

ψnr m (r, ϕ) ∝ eimϕ e−ρ

1 ρ|m|/2 M −nr , |m| + , ρ , 2

(9.140)

where the confluent hypergeometric functions M −nr , |m| + 12 , ρ are polynomials for integer values of the radial quantum number 1 1 1 nr = n + m − |m| − , 2 2 2

(9.141)

as in (9.116). The energy is related to the principal quantum number by n+ Since

1 a2 ≡ 2 2ME − p2z . 2 h ¯

(9.142)

2Ma2 1 = , 2 h ¯ω h ¯

(9.143)

the energy is

E = n+

1 p2 h ¯ω + z . 2 2M

(9.144)

We now observe that the Schrödinger equation (9.139) can be expressed in terms of the creation and annihilation operators (9.127), (9.128) as 1 p2 4 −(a a + 1/2) + E− z h ¯ω 2M "

†

!#

ψ(z, z ∗ ) = 0.

(9.145)

This proves that the algebraically constructed wave functions ψn,n′ in (9.130) coincide with the wave functions ψnr m of (9.116) and (9.140), up to an irrelevant phase. Note that the energy depends only on the number of a-quanta; it is independent of the number of b-quanta. H. Kleinert, PATH INTEGRALS

9.5 Dirac δ -Function Potential

9.5

687

Dirac δ -Function Potential

For a particle in a Dirac δ-function potential, the fixed-energy amplitudes (xb |xa )E can be calculated by performing a perturbation expansion around a free-particle amplitude and summing it up exactly. For any time-independent potential V (x), in addition to a harmonic potential Mω 2 x2 /2, the perturbation expansion in Eq. (3.475) can be Laplace-transformed in the imaginary time via (9.3) to find (xb |xa )E = (xb |xa )ω,E

iZ D − d x1 (xb |x1 )ω,E V (x1 )(x1 |xa )ω,E h ¯ Z

1 Z D + − 2 d x2 h ¯ + ... .

dD x1 (xb |x2 )ω,E V (x2 )(x2 |x1 )ω,E V (x1 )(x1 |xa )ω,E (9.146)

If the potential is a Dirac δ-function centered around X, V (x) = g δ (D) (x − X),

g≡

h ¯2 , Ml2−D

(9.147)

this series simplifies to (xb |xa )E =(xb |xa )ω,E −

ig g2 g(xb |X)ω,E (X|xa )ω,E − 2 (xb |X)ω,E (X|X)ω,E (X|xa )ω,E +. . . , h ¯ h ¯ (9.148)

and can be summed up to g (xb |X)ω,E (X|xa )ω,E . (9.149) h ¯ 1 + i g (X|X)ω,E h ¯ This is, incidentally, true if a δ-function potential is added to an arbitrary solvable fixed-energy amplitude, not just the harmonic one. If the δ-function is the only potential, we use formula (9.149) with ω = 0, so that (xb |xa )0,E reduces to the fixed-energy amplitude (9.12) of a free particle, and obtain directly (xb |xa )E = (xb |xa )ω,E − i

M κD−2 KD/2−1 (κR) h ¯ (2π)D/2 (κR)D/2−1 2M κD−2 KD/2−1 (κRa ) 2M κD−2 KD/2−1 (κRb ) i × i ig h ¯ (2π)D/2 (κRb )D/2−1 h ¯ (2π)D/2 (κRa )D/2−1 − , h ¯ g 2M κD−2 KD/2−1 (κδ) 1− h ¯ h ¯ π D/2 (κδ)D/2−1

(xb |xa )E = −2i

(9.150)

where R ≡ |xb − xa | and Ra,b ≡ |xa,b − X|, and δ is an infinitesimal distance regularizing a possible singularity at zero-distance. In D = 1 dimension, this reduces to M 1 −κR M 1 (xb |xa )E = −i e + i eκ(Rb +Ra ) , (9.151) h ¯ κ h ¯κ lκ + 1

688

9 Wave Functions

For an attractive potential with l < 0, the second term can be written as −i

1 h ¯ eκ(Rb +Ra ) , 2 2 κl E + h ¯ /2Ml2

(9.152)

h/2Ml2 . In its neighborhood, exhibiting a pole at the bound-state energy EB = −¯ the pole contribution reads 2 −(Rb +Ra )/l i¯ h e . 2 l E +h ¯ /2Ml2

(9.153)

This has precisely the spectral form (1.321) with the normalized bound-state wave function s 2 −|x−X|/l ψB (x) = e . (9.154) l In D = 3 dimensions, the amplitude (9.150) becomes (xb |xa )E = −i

M 1 −κR M eκRb eκRa 1 e . +i h ¯ 2πR h ¯ 2πRb 2πRa 1/l + e−κδ /2πδ

(9.155)

In the limit δ → 0, the denominator requires renormalization. We introduce a renormalized coupling length scale 1 1 ≡1+ , lr 2πδ

(9.156)

and rewrite the last factor in (9.155) as 1 . 1/lr − κ/2π

(9.157)

2 For lr < 0, this has a pole at the bound-state energy EB = −4π 2h ¯ 2 /2MlR of the form 1 −lr EB . (9.158) E − EB

The total pole term in (9.155) can therefore be written as ψB (xb )ψB∗ (xa ) with κB =

√

i¯ h , E − EB

(9.159)

2MEB /¯ h = 2π/lr and the normalized bound-state wave functions ψB (x) =

κ2B 4π 2

!1/4

e−κB |x−X| . r

(9.160)


689


In D = 2 dimensions, the situation is more subtle. It is useful to consider the amplitude (9.150) in D = 2 + ǫ dimensions where one has M 1 Kǫ/2 (κR) h ¯ π (2πκR)ǫ/2 Kǫ/2 (κRb )Kǫ/2 (κRa ) M2 1 1 + i 2 2 . (9.161) M 1 ¯ h ¯ π (2πκRb )ǫ/2 (2πκRa )ǫ/2 h + Kǫ/2 (κǫ) g h ¯ π (2πκδ)ǫ/2

(xb |xa )E = −i

Inserting here Kǫ/2 (κδ) ≈ (1/2)Γ(ǫ/2)(κδ/2)−ǫ/2, the denominator becomes h ¯ M Γ(ǫ/2) M 2 h ¯ ǫ + ≈ + 1 − log(πκδ) . ǫ/2 g 2¯ hπ (πκδ) g 2¯ hπ ǫ 2

(9.162)

Here we introduce a renormalized coupling constant 1 M1 1 = + 2 , gr g h ¯ ǫ

(9.163)

and rewrite the right-hand side as h ¯ M log πκδ. − gr 2¯ hπ

(9.164)

This has a pole at

1 2¯h2 π/M gr e , (9.165) πδ indicating a bound-state pole of energy EB = −¯ h2 κ2B /2M. We can now go to the limit of D = 2 dimensions and find that the pole term in (9.161) has the form i¯ h ψB (xb )ψB∗ (xa ) , (9.166) E − EB with the normalized bound-state wave function κB =

κB ψB (x) = √ K0 (κB |x − X|). π

(9.167)

Notes and References The wave functions derived in this chapter from the time evolution amplitude should be compared with those given in standard textbooks on quantum mechanics, such as L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965. The charged particle in a magnetic field is treated in §111. The δ-function potential was studied via path integrals by C. Grosche, Phys. Rev. Letters, 71, 1 (1993).

H. Kleinert, PATH INTEGRALS June 16, 2007 ( /home/kleinert/kleinert/books/pathis/pthic10.tex) Make not my paths offensive to the Gods. Aeschylos, Agamemnon, 891 b.c.

10 Spaces with Curvature and Torsion The path integral of a free particle in spherical coordinates has taught us an important lesson: In a Euclidean space, we were able to obtain the correct time-sliced amplitude in curvilinear coordinates by setting up the sliced action in Cartesian coordinates xi and transforming them to the spherical coordinates q µ = (r, θ, φ). It was crucial to do the transformation at the level of the finite coordinate differences, ∆xi → ∆q µ . This produced higher-order terms in the differences ∆q µ which had to be included up to the order (∆q)4 /ǫ. They all contributed to the relevant order ǫ. It is obvious that as long as the space is Euclidean, the same procedure can be used to find the path integral in an arbitrary curvilinear coordinate system q µ , if we ignore subtleties arising near coordinate singularities which are present in centrifugal barriers, angular barriers, or Coulomb potentials. For these, a special treatment will be developed in Chapters 12–14. We are now going to develop an entirely nontrivial but quite natural extension of this procedure and define a path integral in an arbitrary metric-affine space with curvature and torsion. It must be emphasized that the quantum theory in such spaces is not uniquely defined by the formalism developed so far. The reason is that also the original Schrödinger theory which was used in Chapter 2 to justify the introduction of path integrals is not uniquely defined in such spaces. In classical physics, the equivalence principle postulated by Einstein is a powerful tool for deducing equations of motion in curved space from those in flat space. At the quantum level, this principle becomes insufficient since it does not forbid the appearance of arbitrary coordinate-independent terms proportional to Planck’s quantum h ¯ 2 and the scalar curvature R to appear in the Schödinger equation. We shall set up a simple extension of Einstein’s equivalence principle which will allow us to carry quantum theories from flat to curved spaces which are, moreover, permitted to carry certain classes of torsion. In such spaces, not only the time-sliced action but also the measure of path integration requires a special treatment. To be valid in general it will be necessary to find construction rules for the time evolution amplitude which do not involve the crutch of Cartesian coordinates. The final formula will be purely intrinsic to the general metric-affine space [1]. 690

10.1 Einstein’s Equivalence Principle

691

A crucial test of the validity of the resulting path integral formula will come from applications to systems whose correct operator quantum mechanics is known on the basis of symmetries and group commutation rules rather than canonical commutation rules. In contrast to earlier approaches, our path integral formula will always yield the same quantum mechanics as operator quantum mechanics quantized via group commutation rules. Our formula can, of course, also be used for an alternative approach to the path integrals solved before in Chapter 8, where a Euclidean space was parametrized in terms of curvilinear coordinates. There it gives rise to a more satisfactory treatment than before, since it involves only the intrinsic variables of the coordinate systems.

10.1

Einstein’s Equivalence Principle

To motivate the present study we invoke Einstein’s equivalence principle, according to which gravitational forces upon a spinless mass point are indistinguishable from those felt in an accelerating local reference.1 They are independent of the atomic composition of the particle and strictly proportional to the value of the mass, the same mass that appears in the relation between force and acceleration, in Newton’s first law. The strict equality between the two masses, gravitational and inertial, is fundamental to Einstein’s equivalence principle. Experimentally, the equality holds to an extremely high degree of accuracy. Any possible small deviation can presently be attributed to extra non-gravitational forces. Einstein realized that as a consequence of this equality, all spinless point particles move in a gravitational field along the same orbits which are independent of their composition and mass. This universality of orbital motion permits the gravitational field to be attributed to geometric properties of spacetime. In Newton’s theory of gravity, the gravitational forces between mass points are inversely proportional to their distances in a Euclidean space. In Einstein’s geometric theory the forces are explained entirely by a curvature of spacetime. In general the spacetime of general relativity may also carry another geometric property, called torsion. Torsion is supposed to be generated by the spin densities of material bodies. Quantitatively, this may have only extremely small effects, too small to be detected by present-day experiments. But this is only due to the small intrinsic spin of ordinary gravitational matter. In exceptional states of matter such as polarized neutron stars or black holes, torsion can become relevant. It is now generally accepted that spacetime should carry a nonvanishing torsion at least locally at those points which are occupied by spinning elementary particles [55]. This follows from rather general symmetry considerations. The precise equations of motion for the torsion field, on the other hand, are still a matter of speculation. Thus it is an open question whether ¨ Quotation from his original paper Uber das Relativit¨ atsprinzip und die aus demselben gezogenen Folgerungen, Jahrbuch der Relativitd’t und Elektonik 4 , 411 (1907): “Wir . . . wollen daher im folgenden die völlige physikalische Gleichwertigkeit von Gravitationsfeld und entsprechender Beschleunigung des Bezugssystems annehmen”. 1

692

10 Spaces with Curvature and Torsion

or not the torsion field is able to propagate into the empty space away from spinning matter. Even though the effects of torsion are small we shall keep the discussion as general as possible and study the motion of a particle in a metric-affine space with both curvature and torsion. To prepare the grounds let us first recapitulate a few basic facts about classical orbits of particles in a gravitational field. For simplicity, we assume here only the three-dimensional space to have a nontrivial geometry.2 Then there is a natural choice of a time variable t which is conveniently used to parametrize the particle orbits. Starting from the free-particle action we shall then introduce a path integral for the time evolution amplitude in any metric-affine space which determines the quantum mechanics via the quantum fluctuations of the particle orbits.

10.2

Classical Motion of Mass Point in General Metric-Affine Space

On the basis of the equivalence principle, Einstein formulated the rules for finding the classical laws of motion in a gravitational field as a consequence of the geometry of spacetime. Let us recapitulate his reasoning adapted to the present problem of a nonrelativistic point particle in a non-Euclidean geometry.

10.2.1

Equations of Motion

Consider first the action of the particle along the orbit x(t) in a flat space parametrized with rectilinear, Cartesian coordinates: A=

Z

tb

ta

dt

M i 2 (x˙ ) , 2

i = 1, 2, 3.

(10.1)

It is transformed to curvilinear coordinates q µ , µ = 1, 2, 3, via some functions xi = xi (q),

(10.2)

leading to M gµν (q)q˙µ q˙ν , 2

(10.3)

gµν (q) = ∂µ xi (q)∂ν xi (q)

(10.4)

A=

Z

tb

ta

dt

where

is the induced metric for the curvilinear coordinates. Repeated indices are understood to be summed over, as usual. 2

The generalization to non-Euclidean spacetime will be obvious after the development in Chapter 19. H. Kleinert, PATH INTEGRALS

10.2 Classical Motion of Mass Point in General Metric-Affine Space

693

The length of the orbit in the flat space is given by l=

Z

tb

ta

q

dt gµν (q)q˙µ q˙ν .

(10.5)

Both the action (10.3) and the length (10.5) are invariant under arbitrary reparametrizations of space q µ → q ′µ . Einstein’s equivalence principle amounts to the postulate that the transformed action (10.3) describes directly the motion of the particle in the presence of a gravitational field caused by other masses. The forces caused by the field are all a result of the geometric properties of the metric tensor. The equations of motion are obtained by extremizing the action in Eq. (10.3) with the result 1 ¯ λνµ q˙λ q˙ν = 0. ∂t (gµν q˙ν ) − ∂µ gλν q˙λ q˙ν = gµν q¨ν + Γ 2

(10.6)

¯ λνµ ≡ 1 (∂λ gνµ + ∂ν gλµ − ∂µ gλν ) Γ 2

(10.7)

Here

is the Riemann connection or Christoffel symbol of the first kind [recall (1.70)]. With the help of the Christoffel symbol of the second kind [recall (1.71)] ¯ µ ≡ g µσ Γ ¯ λνσ , Γ λν

(10.8)

¯ µ q˙λ q˙ν = 0. q¨µ + Γ λν

(10.9)

we can write

The solutions of these equations are the classical orbits. They coincide with the extrema of the length of a line l in (10.5). Thus, in a curved space, classical orbits are the shortest lines, the geodesics [recall (1.72)]. The same equations can also be obtained directly by transforming the equation of motion from xï = 0

(10.10)

to curvilinear coordinates q µ , which gives xï =

∂xi µ ∂ 2 xi λ ν q ¨ + q˙ q˙ = 0. ∂q µ ∂q λ ∂q ν

(10.11)

At this place it is again useful to employ the quantities defined in Eq. (1.357), the basis triads and their reciprocals ∂xi e µ (q) ≡ µ , ∂q i

∂q µ ei (q) ≡ , ∂xi µ

(10.12)

694


which satisfy the orthogonality and completeness relations (1.358): ei µ ei ν = δ µ ν ,

ei µ ej µ = δi j .

(10.13)

The induced metric can then be written as gµν (q) = ei µ (q)ei ν (q).

(10.14)

Labeling Cartesian coordinates, upper and lower indices i are the same. The indices µ, ν of the curvilinear coordinates, on the other hand, can be lowered only by contraction with the metric gµν or raised with the inverse metric g µν ≡ (gµν )−1 . Using the basis triads, Eq. (10.11) can be rewritten as d i µ (e µ q˙ ) = ei µ q¨µ + ∂ν ei µ q˙µ q˙ν = 0, dt or as q¨µ + ei µ ∂λ ei κ q˙κ q˙λ = 0.

(10.15)

The quantity in front of q˙κ q˙λ is called the affine connection: Γλκ µ = ei µ ∂λ ei κ .

(10.16)

Due to (10.13), it can also be written as [compare (1.366)] Γλκ µ = −ei κ ∂λ ei µ .

(10.17)

Thus we arrive at the transformed flat-space equation of motion q¨µ + Γκλ µ q˙κ q˙λ = 0.

(10.18)

The solutions of this equation are called the straightest lines or autoparallels. If the coordinate transformation functions xi (q) are smooth and single-valued, their derivatives commute as required by Schwarz’s integrability condition (∂λ ∂κ − ∂κ ∂λ )xi (q) = 0.

(10.19)

Then the triads satisfy the identity ∂λ eiκ − ∂κ eiλ = 0,

(10.20)

implying that the connection Γλκ µ is symmetric in the lower indices. In fact, it ¯ µ . This follows coincides with the Riemann connection, the Christoffel symbol Γ λκ immediately after inserting gµν (q) = ei µ (q)ei ν (q) into (10.7) and working out all derivatives using (10.20). Thus, for a space with curvilinear coordinates q µ which can be reached by an integrable coordinate transformation from a flat space, the autoparallels coincide with the geodesics. H. Kleinert, PATH INTEGRALS


10.2.2

695

Nonholonomic Mapping to Spaces with Torsion

It is possible to map the x-space locally into a q-space with torsion via an infinitesimal transformation dxi = ei µ (q)dq µ .

(10.21)

We merely have to assume that the coefficient functions ei µ (q) do not satisfy the property (10.20) which follows from the Schwarz integrability condition (10.19): ∂λ ei κ (q) − ∂κ ei λ (q) 6= 0,

(10.22)

(∂λ ∂κ − ∂κ ∂λ )xi (q) 6= 0.

(10.23)

implying that second derivatives in front of xi (q) do not commute as in Eq. (10.19):

In this case we shall call the differential mapping (10.21) nonholonomic, in analogy with the nomenclature for nonintegrable constraints in classical mechanics. The property (10.23) implies that xi (q) is a multivalued function xi (q), of which we shall give typical examples below in Eqs. (10.44) and (10.55). Educated readers in mathematics have been wondering whether such nonholonomic coordinate transformations make any sense. They will understand this concept better if they compare the situation with the quite similar but much simpler creation of magnetic field in a field-free space by nonholonomic gauge transformations. More details are explained in Appendix 10A. From Eq. (10.22) we see that the image space of a nonholonomic mapping carries torsion. The connection Γλκ µ = ei µ ei κ,λ has a nonzero antisymmetric part, called the torsion tensor :3 1 1 Sλκ µ = (Γλκ µ − Γκλ µ ) = ei µ ∂λ ei κ − ∂κ ei λ . (10.24) 2 2 In contrast to Γλκ µ , the antisymmetric part Sλκ µ is a proper tensor under general coordinate transformations. The contracted tensor Sµ ≡ Sµλ λ

(10.25)

transforms like a vector, whereas the contracted connection Γµ ≡ Γµν ν does not. Even though Γµν λ is not a tensor, we shall freely lower and raise its indices using contractions with the metric or the inverse metric, respectively: Γµ ν λ ≡ g µκ Γκν λ , ¯ µν λ . Γµ ν λ ≡ g νκ Γµκ λ , Γµνλ ≡ gλκ Γµν κ . The same thing will be done with Γ In the presence of torsion, the affine connection (10.16) is no longer equal to the Christoffel symbol. In fact, by rewriting Γµνλ = eiλ ∂µ ei ν trivially as i h i h io 1 nh eiλ ∂µ ei ν + ∂µ eiλ ei ν + eiµ ∂ν ei λ + ∂ν eiµ ei λ − eiµ ∂λ ei ν + ∂λ eiµ ei ν 2 i h i h io 1 nh + eiλ ∂µ ei ν − eiλ ∂ν ei µ − eiµ ∂ν ei λ − eiµ ∂λ ei ν + eiν ∂λ ei µ − eiν ∂µ ei λ 2 (10.26)

Γµνλ =

3

Our notation for the geometric quantities in spaces with curvature and torsion is the same as in J.A. Schouten, Ricci Calculus, Springer, Berlin, 1954.

696


and using ei µ (q)ei ν (q) = gµν (q), we find the decomposition ¯ λ + Kµν λ , Γµν λ = Γ µν

(10.27)

where the combination of torsion tensors Kµνλ ≡ Sµνλ − Sνλµ + Sλµν

(10.28)

is called the contortion tensor . It is antisymmetric in the last two indices so that ¯ µν ν . Γµν ν = Γ

(10.29)

In the presence of torsion, the shortest and straightest lines are no longer equal. Since the two types of lines play geometrically an equally favored role, the question arises as to which of them describes the correct classical particle orbits. Intuitively, we expect the straightest lines to be the correct trajectories since massive particles possess inertia which tend to minimize their deviations from a straight line in spacetime. It is hard to conceive how a particle should know which path to take at each instant in time in order to minimize the path length to a distant point. This would contradict the principle of locality which pervades all laws of physics. Only in a spacetime without torsion is this possible, since there the shortest lines happen to coincide with straightest ones for purely mathematical reasons. In Subsection 10.2.3, the straightest lines will be derived from an action principle. In Einstein’s theory of gravitation, matter produces curvature in fourdimensional Minkowski spacetime, thereby explaining the universal nature of gravitational forces. The flat spacetime metric is    

ηab = 

1 −1

−1



−1

   

,

a, b = 0, 1, 2, 3.

(10.30)

ab

The Riemann-Cartan curvature tensor is defined as the covariant curl of the affine connection: Rµνλ κ = ∂µ Γνλ κ − ∂ν Γµλ κ − [Γµ , Γν ]λ κ ,

µ, ν, . . . = 0, 1, 2, 3.

(10.31)

The last term is written in a matrix notation for the connection, in which the tensor components Γµλ κ are viewed as matrix elements (Γµ )λ κ . The matrix commutator in (10.31) is then equal to [Γµ , Γν ]λ κ ≡ (Γµ Γν − Γν Γµ )λ κ = Γµλ σ Γνσ κ − Γνλ σ Γµσ κ .

(10.32)

Expressing the affine connection (10.16) in (10.31) with the help of Eqs. (10.16) in terms of the four-dimensional generalization of the triads (10.12) and their reciprocals (10.12), the tetrads ea µ and their reciprocals ea µ , we obtain the compact formula Rµνλ κ = ea κ (∂µ ∂ν − ∂ν ∂µ )ea λ .



697

For the mapping (10.21), this implies that not only the coordinate transformation xa (q), but also its first derivatives fail to satisfy Schwarz’s integrability condition: (∂µ ∂ν − ∂ν ∂µ )∂λ xa (q) 6= 0.

(10.34)

Such general transformation matrices ea µ (q) will be referred to as multivalued basis tetrads. A transformation for which xa (q) have commuting derivatives, while the first derivatives ∂µ xa (q) = ei µ (q) do not, carries a flat-space region into a purely curved one. Einstein’s original theory of gravity assumes the absence of torsion. The space properties are completely specified by the Riemann curvature tensor formed from the Riemann connection (the Christoffel symbol) ¯ νλκ − ∂ν Γ ¯ µλκ − [Γ ¯ µ, Γ ¯ ν ]λ κ . ¯ µνλκ = ∂µ Γ R

(10.35)

The relation between the two curvature tensors is ¯ κ+D ¯µ Kνλ κ − D ¯ ν Kµλ κ − [Kµ , Kν ]λ κ . Rµνλ κ = R µνλ

(10.36)

¯µ denote In the last term, the Kµλ κ ’s are viewed as matrices (Kµ )λ κ . The symbols D the covariant derivatives formed with the Christoffel symbol. Covariant derivatives act like ordinary derivatives if they are applied to a scalar field. When applied to a vector field, they act as follows: ¯µ vν ≡ ∂µ vν − Γ ¯ λ vλ , D µν ν ν ¯ ¯ Dµ v ≡ ∂µ v + Γµλν v λ .

(10.37)

The effect upon a tensor field is the generalization of this; every index receives a ¯ contribution. corresponding additive Γ Note that the Laplace-Beltrami operator (1.367) applied to a scalar field σ(q) can be written as ¯ µD ¯ ν σ. ∆ σ = g µν D

(10.38)

In the presence of torsion, there exists another covariant derivative formed with the affine connection Γµν λ rather than the Christoffel symbol which acts upon a vector field as Dµ vν ≡ ∂µ vν − Γµν λ vλ , Dµ v ν ≡ ∂µ v ν + Γµλ ν v λ .

(10.39)

Note by definition of Γλκ µ in (10.16) and (10.17), the covariant derivatives of ei µ and ei µ vanish: Dµ ei ν ≡ ∂µ ei ν − Γµν λ ei λ = 0,

Dµ ei ν ≡ ∂µ ei ν + Γµλ ν ei λ = 0.

(10.40)

698


This will be of use later. ¯ µνλκ , one can form the From either of the two curvature tensors, Rµνλ κ and R once-contracted tensors of rank two, the Ricci tensor Rνλ = Rµνλ µ ,

(10.41)

R = g νλRνλ .

(10.42)

and the curvature scalar

The celebrated Einstein equation for the gravitational field postulates that the tensor 1 Gµν ≡ Rµν − gµν R, 2

(10.43)

the so-called Einstein tensor , is proportional to the symmetric energy-momentum tensor of all matter fields. This postulate was made only for spaces with no torsion, ¯ µν and Rµν , Gµν are both symmetric. As mentioned before, it in which case Rµν = R is not yet clear how Einstein’s field equations should be generalized in the presence of torsion since the experimental consequences are as yet too small to be observed. In this text, we are not concerned with the generation of curvature and torsion but only with their consequences upon the motion of point particles. It is useful to set up two simple examples for nonholonomic mappings which illustrate the way in which these are capable of generating curvature and torsion from a Euclidean space. The reader not familiar with this subject is advised to consult a textbook on the physics of defects [2]. where such mappings are standard and of great practical importance; every plastic deformation of a material can only be described in terms of such mappings. As a first example consider the transformation in two dimensions i

dx =

(

dq 1 dq 2 + ǫ∂µ φ(q)dq µ

for i = 1, for i = 2,

(10.44)

with an infinitesimal parameter ǫ and the multi-valued function φ(q) ≡ arctan(q 2 /q 1 ).

(10.45)

The triads reduce to dyads, with the components e1 µ = δ 1 µ , e2 µ = δ 2 µ + ǫ∂µ φ(q) ,

(10.46)

and the torsion tensor has the components e1 λ Sµν λ = 0,

ǫ e2 λ Sµν λ = (∂µ ∂ν − ∂ν ∂µ )φ. 2



699

Figure 10.1 Edge dislocation in crystal associated with missing semi-infinite plane of atoms. The nonholonomic mapping from the ideal crystal to the crystal with the dislocation introduces a δ-function type torsion in the image space.

If we differentiate (10.45) formally, we find (∂µ ∂ν − ∂ν ∂µ )φ ≡ 0. This, however, is incorrect at the origin. Using Stokes’ theorem we see that Z

2

d q(∂1 ∂2 − ∂2 ∂1 )φ =

I

µ

dq ∂µ φ =

I

dφ = 2π

(10.48)

for any closed circuit around the origin, implying that there is a δ-function singularity at the origin with ǫ e2 λ S12 λ = 2πδ (2) (q). (10.49) 2 By a linear superposition of such mappings we can generate an arbitrary torsion in the q-space. The mapping introduces no curvature. In defect physics, the mapping (10.46) is associated with a dislocation caused by a missing or additional layer of atoms (see Fig. 10.1). When encircling a dislocation along a closed path C, its counter image C ′ in the ideal crystal does not form a closed path. The closure failure is called the Burgers vector bi ≡

I

C′

dxi =

I

C

dq µ ei µ .

(10.50)

It specifies the direction and thickness of the layer of additional atoms. With the help of Stokes’ theorem, it is seen to measure the torsion contained in any surface S spanned by C: i

b

=

I

S

2 µν

i

d s ∂µ e ν =

I

S

d2 sµν ei λ Sµν λ ,

(10.51)

where d2 sµν = −d2 sνµ is the projection of an oriented infinitesimal area element onto the plane µν. The above example has the Burgers vector bi = (0, ǫ).

(10.52)

A corresponding closure failure appears when mapping a closed contour C in the ideal crystal into a crystal containing a dislocation. This defines a Burgers vector: bµ ≡

I

C′

dq µ =

I

C

dxi ei µ .

(10.53)

700


Figure 10.2 Edge disclination in crystal associated with missing semi-infinite section of atoms of angle Ω. The nonholonomic mapping from the ideal crystal to the crystal with the disclination introduces a δ-function type curvature in the image space.

By Stokes’ theorem, this becomes a surface integral µ

b

=

I

2 ij

d s ∂i ej =

SI

= −

µ

S

I

S

d2 sij ei ν ∂ν ej µ

d2 sij ei ν ej λ Sνλ µ ,

(10.54)

the last step following from (10.17). As a second example for a nonholonomic mapping, we generate curvature by the transformation xi = δ i µ [q µ + Ωǫµ ν q ν φ(q)],

(10.55)

with the multi-valued function (10.45). The symbol ǫµν denotes the antisymmetric Levi-Civita tensor. The transformed metric 2Ω gµν = δµν − σ ǫµλ ǫνκ q λ q κ (10.56) q qσ is single-valued and has commuting derivatives. The torsion tensor vanishes since (∂1 ∂2 − ∂2 ∂1 )x1,2 are both proportional to q 2,1 δ (2) (q), a distribution identical to zero. The local rotation field ω(q) ≡ 12 (∂1 x2 − ∂2 x1 ), on the other hand, is equal to the multi-valued function −Ωφ(q), thus having the noncommuting derivatives: (∂1 ∂2 − ∂2 ∂1 )ω(q) = −2πΩδ (2) (q).

(10.57)

To lowest order in Ω, this determines the curvature tensor, which in two dimensions possesses only one independent component, for instance R1212 . Using the fact that gµν has commuting derivatives, R1212 can be written as R1212 = (∂1 ∂2 − ∂2 ∂1 )ω(q).

(10.58)

In defect physics, the mapping (10.55) is associated with a disclination which corresponds to an entire section of angle Ω missing in an ideal atomic array (see Fig. 10.2). It is important to emphasize that our multivalued basis tetrads ea µ (q) are not related to the standard tetrads or vierbein fields h αµ (q) used in the theory of gravitation with spinning particles. The difference is explained in Appendix 10B. H. Kleinert, PATH INTEGRALS


10.2.3

701

New Equivalence Principle

In classical mechanics, many dynamical problems are solved with the help of nonholonomic transformations. Equations of motion are differential equations which remain valid if transformed differentially to new coordinates, even if the transformation is not integrable in the Schwarz sense. Thus we postulate that the correct equations of motion of point particles in a space with curvature and torsion are the images of the equation of motion in a flat space. The equations (10.18) for the autoparallels yield therefore the correct trajectories of spinless point particles in a space with curvature and torsion. This postulate is based on our knowledge of the motion of many physical systems. Important examples are the Coulomb system which will be discussed in detail in Chapter 13, and the spinning top in the body-fixed reference system [3]. Thus the postulate has a good chance of being true, and will henceforth be referred to as a new equivalence principle.

10.2.4

Classical Action Principle for Spaces with Curvature and Torsion

Before setting up a path integral for the time evolution amplitude we must find an action principle for the classical motion of a spinless point particle in a space with curvature and torsion, i.e., the movement along autoparallel trajectories. This is a nontrivial task since autoparallels must emerge as the extremals of an action (10.3) involving only the metric tensor gµν . The action is independent of the torsion and carries only information on the Riemann part of the space geometry. Torsion can therefore enter the equations of motion only via some novel feature of the variation procedure. Since we know how to perform variations of an action in the Euclidean x-space, we deduce the correct procedure in the general metric-affine space by transferring the variations δxi (t) under the nonholonomic mapping q˙µ = ei µ (q)x˙ i

(10.59)

into the q µ -space. Their images are quite different from ordinary variations as illustrated in Fig. 10.3(a). The variations of the Cartesian coordinates δxi (t) are done at fixed endpoints of the paths. Thus they form closed paths in the x-space. Their images, however, lie in a space with defects and thus possess a closure failure indicating the amount of torsion introduced by the mapping. This property will be emphasized by writing the images δ S q µ (t) and calling them nonholonomic variations. The superscript indicates the special feature caused by torsion. Let us calculate them explicitly. The paths in the two spaces are related by the integral equation µ

µ

q (t) = q (ta ) +

Z

t

ta

dt′ ei µ (q(t′ ))x˙ i (t′ ).

(10.60)

702


For two neighboring paths in x-space differing from each other by a variation δxi (t), equation (10.60) determines the nonholonomic variation δ S q µ (t): δ S q µ (t) =

Z

t

ta

dt′ δ S [ei µ (q(t′ ))x˙ i (t′ )].

(10.61)

A comparison with (10.59) shows that the variation δ S and the time derivatives d/dt of q µ (t) commute with each other: δ S q˙µ (t) =

d S µ δ q (t), dt

(10.62)

d i δx (t). dt

(10.63)

just as for ordinary variations δxi : δ x˙ i (t) =

Let us also introduce auxiliary nonholonomic variations in q-space: δ¯q µ ≡ ei µ (q)δxi .

(10.64)

In contrast to δ S q µ (t), these vanish at the endpoints, ¯δ q(ta ) = δ¯q(tb ) = 0,

(10.65)

just as the usual variations δxi (t), i.e., they form closed paths with the unvaried orbits. Using (10.62), (10.63), and the fact that δ S xi (t) ≡ δxi (t), by definition, we derive from (10.61) the relation d S µ d δ q (t) = δ S ei µ (q(t))x˙ i (t) + ei µ (q(t)) δxi (t) dt dt d = δ S ei µ (q(t))x˙ i (t) + ei µ (q(t)) [ei ν (t)¯ δ q ν (t)]. dt

(10.66)

After inserting δ S ei µ (q) = −Γλν µ δ S q λ ei ν ,

d i e ν (q) = Γλν µ q˙λ ei µ , dt

(10.67)

this becomes d S µ d δ q (t) = −Γλν µ δ S q λ q˙ν + Γλν µ q˙λ ¯δq ν + ¯δ q µ . dt dt

(10.68)

It is useful to introduce the difference between the nonholonomic variation δ S q µ and an auxiliary closed nonholonomic variation δq µ : δ S bµ ≡ δ S q µ − ¯δ q µ .


703


Then we can rewrite (10.68) as a first-order differential equation for δ S bµ : d S µ δ b = −Γλν µ δ S bλ q˙ν + 2Sλν µ q˙λ δ¯q ν . dt

(10.70)

After introducing the matrices Gµ λ (t) ≡ Γλν µ (q(t))q˙ν (t)

(10.71)

Σµ ν (t) ≡ 2Sλν µ (q(t))q˙λ (t),

(10.72)

and

equation (10.70) can be written as a vector differential equation: d S δ b = −Gδ S b + Σ(t) δ¯q ν (t). dt

(10.73)

Although not necessary for the further development, we solve this equation by δ S b(t) =

Z

t

ta

dt′ U(t, t′ ) Σ(t′ ) δ¯q(t′ ),

(10.74)

with the matrix

U(t, t′ ) = T exp −

Z

t

t′

dt′′ G(t′′ ) .

(10.75)

In the absence of torsion, Σ(t) vanishes identically and δ S b(t) ≡ 0, and the variations δ S q µ (t) coincide with the auxiliary closed nonholonomic variations δq µ (t) [see Fig. 10.3(b)]. In a space with torsion, the variations δ S q µ (t) and δ¯q µ (t) are different from each other [see Fig. 10.3(c)]. Under an arbitrary nonholonomic variation δ S q µ (t) = δq µ + δ S bµ , the action (10.3) changes by δS A = M

Z

tb

ta

1 dt gµν q˙ν δ S q˙µ + ∂µ gλκ δ S q µ q˙λ q˙κ . 2

(10.76)

After a partial integration of the δ q-term ˙ we use (10.65), (10.62), and the identity ∂µ gνλ ≡ Γµνλ + Γµλν , which follows directly form the definitions gµν ≡ ei µ ei ν and Γµν λ ≡ ei λ ∂µ ei ν , and obtain S

Z

δ A=M

tb ta

"

dt −gµν

¯ λκ ν q˙λ q˙κ δ¯q µ + gµν q˙ν d δ S bµ + Γµλκ δ S bµ q˙λ q˙κ q¨ + Γ dt ν

!#

.

(10.77)

To derive the equation of motion we first vary the action in a space without torsion. Then δ S bµ (t) ≡ 0, and (10.77) becomes S

δ A = −M

Z

tb

ta

¯ λκ ν q˙λ q˙κ )¯ dtgµν (¨ qν + Γ δqν .

(10.78)

704


Figure 10.3 Images under holonomic and nonholonomic mapping of fundamental δfunction path variation. In the holonomic case, the paths x(t) and x(t) + δx(t) in (a) turn λ 6= 0, they into the paths q(t) and q(t) + δ¯q(t) in (b). In the nonholonomic case with Sµν go over into q(t) and q(t) + δS q(t) shown in (c) with a closure failure bµ at tb analogous to the Burgers vector bµ in a solid with dislocations.

Thus, the action principle δ S A = 0 produces the equation for the geodesics (10.9), which are the correct particle trajectories in the absence of torsion. In the presence of torsion, δ S bµ is nonzero, and the equation of motion receives a contribution from the second parentheses in (10.77). After inserting (10.70), the nonlocal terms proportional to δ S bµ cancel and the total nonholonomic variation of the action becomes δ S A = −M = −M

Z

tb

ta tb

Z

ta

¯ λκ ν + 2S ν λκ q˙λ q˙κ ¯δ q µ dtgµν q¨ν + Γ h

dtgµν q¨ν + Γλκ ν q˙λ q˙κ δ¯q µ .

i

(10.79)

¯ {λκ} ν +2S ν {λκ} . The second line follows from the first after using the identity Γλκ ν = Γ The curly brackets indicate the symmetrization of the enclosed indices. Setting δ S A = 0 and inserting for δ¯q(t) the image under (10.64) of an arbitrary δ-function variation δxi (t) ∝ ǫi δ(t − t0 ) gives the autoparallel equations of motions (10.18), which is what we wanted to show. The above variational treatment of the action is still somewhat complicated and calls for a simpler procedure [4]. The extra term arising from the second parenthesis in the variation (10.77) can be traced to a simple property of the auxiliary closed H. Kleinert, PATH INTEGRALS

705


nonholonomic variations (10.64). To find this we form the time derivative dt ≡ d/dt of the defining equation (10.64) and find dt δ¯q µ (t) = ∂ν ei µ (q(t)) q˙ν (t)δxi (t) + ei µ (q(t))dt δxi (t).

(10.80)

Let us now perform variation δ¯ and t-derivative in the opposite order and calculate dt ¯δq µ (t). From (10.59) and (10.13) we have the relation dt q λ (t) = ei λ (q(t)) dtxi (t) .

(10.81)

δ¯dt q µ (t) = ∂ν ei µ (q(t)) δ¯q ν dt xi (t) + ei µ (q(t))¯ δ dt xi .

(10.82)

Varying this gives Since the variation in xi -space commute with the t-derivatives [recall (10.63)], we obtain δ¯dt q µ (t) − dt δ¯q µ (t) = ∂ν ei µ (q(t)) δ¯q ν dt xi (t) − ∂ν ei µ (q(t)) q˙ν (t)δxi (t).

(10.83)

δ¯dt q µ (t) − dt δ¯q µ (t) = 2Sνλ µ q˙ν (t)¯ δ q λ (t).

(10.84)

After re-expressing δxi (t) and dt xi (t) back in terms of δ¯q µ (t) and dt q µ (t) = q˙µ (t), and using (10.17), (10.24), this becomes

Thus, due to the closure failure in spaces with torsion, the operations dt and δ¯ do not commute in front of the path q µ (t). In other words, in contrast to the open variations δ S (and of course the usual ones δ), the auxiliary closed nonholonomic variations δ¯ of velocities q˙µ (t) no longer coincide with the velocities of variations. This property is responsible for shifting the trajectory from geodesics to autoparallels. Indeed, let us vary an action A=

Zt2

dtL (q µ (t), q˙µ (t))

(10.85)

t1

directly by δ¯q µ (t) and impose (10.84), we find ¯δ A =

Zt2

t1

(

)

∂L µ ∂L d µ ∂L ν λ µ dt δ ¯ q + δ ¯ q +2 S q˙ δ¯q . νλ ∂q µ ∂ q˙µ dt ∂ q˙µ

(10.86)

After a partial integration of the second term using the vanishing δ¯q µ (t) at the endpoints, we obtain the Euler-Lagrange equation ∂L d ∂L ∂L − = −2Sµν λ q˙ν λ . µ µ ∂q dt ∂ q˙ ∂ q˙

(10.87)

This differs from the standard Euler-Lagrange equation by an additional contribution due to the torsion tensor. For the action (10.3), we thus obtain the equation of motion h i 1 M q¨µ + g µκ ∂ν gλκ − ∂κ gνλ − 2S µ νλ q˙ ν q˙λ = 0, (10.88) 2 which is once more Eq. (10.18) for autoparallels.

706

10.3


Path Integral in Spaces with Curvature and Torsion

We now turn to the quantum mechanics of a point particle in a general metric-affine space. Proceeding in analogy with the earlier treatment in spherical coordinates, we first consider the path integral in a flat space with Cartesian coordinates 1

′ ′

(x t|x t ) = q D 2πiǫ¯ h/M

N Z Y

n=1

∞

−∞

dxn

NY +1

K0ǫ (∆xn ),

(10.89)

n=1

where K0ǫ (∆xn ) is an abbreviation for the short-time amplitude K0ǫ (∆xn )

i ˆ 1 i M (∆xn )2 ≡ hxn | exp − ǫH |xn−1 i = q exp , (10.90) D h ¯ h ¯ 2 ǫ 2πiǫ¯ h/M

"

#

with ∆xn ≡ xn − xn−1 , x ≡ xN +1 , x′ ≡ x0 . A possible external potential has been omitted since this would contribute in an additive way, uninfluenced by the space geometry. Our basic postulate is that the path integral in a general metric-affine space should be obtained by an appropriate nonholonomic transformation of the amplitude (10.89) to a space with curvature and torsion.

10.3.1

Nonholonomic Transformation of Action

The short-time action contains the square distance (∆xn )2 which we have to transform to q-space. For an infinitesimal coordinate difference ∆xn ≈ dxn , the square distance is obviously given by (dx)2 = gµν dq µ dq ν . For a finite ∆xn , however, we know from Chapter 8 that we must expand (∆xn )2 up to the fourth order in ∆qn µ = qn µ − qn−1 µ to find all terms contributing to the relevant order ǫ. It is important to realize that with the mapping from dxi to dq µ not being holonomic, the finite quantity ∆q µ is not uniquely determined by ∆xi . A unique relation can only be obtained by integrating the functional relation (10.60) along a specific path. The preferred path is the classical orbit, i.e., the autoparallel in the q-space. It is characterized by being the image of a straight line in x-space. There the velocity x˙ i (t) is constant, and the orbit has the linear time dependence ∆xi (t) = x˙ i (t0 )∆t,

(10.91)

where the time t0 can lie anywhere on the t-axis. Let us choose for t0 the final time in each interval (tn , tn−1 ). At that time, x˙ in ≡ x˙ i (tn ) is related to q˙nµ ≡ q˙µ (tn ) by x˙ in = ei µ (qn )q˙nµ .

(10.92)

µ It is easy to express q˙nµ in terms of ∆qnµ = qnµ − qn−1 along the classical orbit. First we expand q µ (tn−1 ) into a Taylor series around tn . Dropping the time arguments, for brevity, we have

∆q ≡ q λ − q ′λ = ǫq˙λ −

ǫ2 λ ǫ3 ˙ λ q¨ + q¨ + . . . , 2! 3!


707

10.3 Path Integral in Metric-Affine Space

where ǫ = tn − tn−1 and q˙λ , q¨λ , . . . are the time derivatives at the final time tn . An expansion of this type is referred to as a postpoint expansion. Due to the arbitrariness of the choice of the time t0 in Eq. (10.92), the expansion can be performed around any other point just as well, such as tn−1 and t¯n = (tn + tn−1 )/2, giving rise to the so-called prepoint or midpoint expansions of ∆q. Now, the term q¨λ in (10.93) is given by the equation of motion (10.18) for the autoparallel q¨λ = −Γµν λ q˙µ q˙ν .

(10.94)

A further time derivative determines λ q¨˙ = −(∂σ Γµν λ − 2Γµν τ Γ{στ } λ )q˙µ q˙ν q˙σ .

(10.95)

Inserting these expressions into (10.93) and inverting the expansion, we obtain q˙λ at the final time tn expanded in powers of ∆q. Using (10.91) and (10.92) we arrive at the mapping of the finite coordinate differences: ∆xi = ei λ q˙λ ∆t

= ei λ ∆q λ −

(10.96) 1 1 Γµν λ ∆q µ ∆q ν + ∂σ Γµν λ +Γµν τ Γ{στ } λ ∆q µ ∆q ν ∆q σ +. . . , 2! 3!

where ei λ and Γµν λ are evaluated at the postpoint. It is useful to introduce ∆ξ µ ≡ ei µ ∆xi

(10.97)

as autoparallel coordinates or normal coordinates to parametrize the neighborhood of a point q. If the space has no torsion, they are also called Riemann normal coordinates or geodesic coordinates. The normal coordinates (10.97) are expanded in (10.149) in powers of ∆q µ around the postpoint. There exists also a prepoint version of ∆ξ in which all signs of ∆q are simply the opposite. The prepoint version, for instance, has the expansion: ∆ξ λ = ∆q λ +

1 1 Γµν λ ∆q µ ∆q ν + ∂σ Γµν λ +Γµν τ Γ{στ } λ ∆q µ ∆q ν ∆q σ +. . . . (10.98) 2! 3!

In contrast to the finite differences ∆q µ , the normal coordinates ∆ξ µ in the neighborhood of a point are vectors and thus allow for a covariant Taylor expansion of a function f (q µ + ∆q µ ). Its form is found by performing an ordinary Taylor expansion of a function F (x) in Cartesian coordinates F (x + ∆x) = F (x) + ∂i F (x)∆xi +

1 ∂i ∂j F (x)∆xi ∆xj + . . . , 2!

(10.99)

and transforming this to coordinates q µ . The function F (x) becomes f (q) = F (x(q)), and the derivatives ∂i1 ∂i2 · · · ∂in f (x) go over into covariant derivatives:

708


ei1 µ ei2 µ · · · ein µ Dµ1 Dµ2 · · · Dµn f (q). For instance, ∂i F (x) = ei µ ∂µ f (q) = ei µ Dµ f (q), and ∂i ∂j f (q) = ei µ ∂µ ej ν ∂ν f (q) = [ei µ ej ν ∂µ ∂ν + ei µ (∂µ ej ν )∂ν ] f (q) h

i

= ei µ ej ν ∂µ ∂ν − Γµν λ ∂λ f (q) = eiµ ej ν Dµ ∂ν f (q) = ei µ ej ν Dµ Dν f (q), (10.100) where we have used (10.17) to express ∂µ ej ν = −Γµσ ν ej σ , and changed dummy indices. The differences ∆xi in (10.99) are replaced by ei µ ∆ξ µ with the prepoint expansion (10.98). In this way we arrive at the covariant Taylor expansion f (q + ∆q) = F (x) + Dµ f (q)∆ξ i +

1 Dµ Dν f (q)∆ξ µ ∆ξ ν + . . . . 2!

(10.101)

Indeed, re-expanding the right-hand side in powers of ∆q µ via (10.98) we may verify that the affine connections cancel against those in the covariant derivatives of f (q), so that (10.101) reduces to the ordinary Taylor expansion of f (q + ∆q) in powers of ∆q. Note that the expansion (10.96) differs only slightly from a naive Taylor expansion of the difference around the postpoint: 1 1 ∆xi = xi (q) − xi (q − ∆q) = ei λ ∆q λ − ei ν,µ ∆q µ ∆q ν + ei ν,µσ ∆q µ ∆q ν ∆q σ + . . . , 2 3! (10.102) where a subscript λ separated by a comma denotes the partial derivative ∂λ = ∂/∂q λ , i.e., f,λ ≡ ∂λ f . The right-hand side can be rewritten with the help of the completeness relation (10.13) as 1 1 ∆xi = ei λ ∆q λ − ej λ ej ν,µ ∆q µ ∆q ν + ej λ ej ν,µσ ∆q µ ∆q ν ∆q σ + . . . 2 3!

. (10.103)

The expansion coefficients can be expressed in terms of the affine connection (10.16), using the derived relation ei λ ei ν,µσ = ∂σ (ei λ ei ν,µ ) − eiτ ei ν,µ ej τ ejλ ,σ = ∂σ Γµν λ + Γµν τ Γστ λ .

(10.104)

Thus we obtain i

i

∆x = e λ

1 1 ∆q − Γµν λ ∆q µ ∆q ν+ ∂σ Γµν λ +Γµν τ Γστ λ ∆q µ ∆q ν ∆q σ +. . . .(10.105) 2! 3!

λ

This differs from the true expansion (10.149) only by the absence of the symmetrization of the indices in the last affine connection. Inserting (10.96) into the short-time amplitude (10.90), we obtain i ˆ 1 iAǫ> (q,q−∆q)/¯ h K0ǫ (∆x) = hx| − ǫH |x − ∆xi = q , De h ¯ 2πiǫ¯ h/M

(10.106)


709


with the short-time postpoint action Aǫ> (q, q − ∆q) = (∆xi )2 = ǫ

M gµν q˙µ q˙ν 2

M = gµν ∆q µ ∆q ν − Γµνλ ∆q µ ∆q ν ∆q λ (10.107) 2ǫ 1 1 τ µ ν λ κ τ δ σ + gµτ ∂κ Γλν + Γλν Γ{κδ} + Γλκ Γµνσ ∆q ∆q ∆q ∆q + . . . . 3 4

Separating the affine connection into Christoffel symbol and torsion, this can also be written as M ¯ µνλ ∆q µ ∆q ν ∆q λ − ∆q) = gµν ∆q µ ∆q ν − Γ (10.108) 2ǫ 1 1 1 σ τ δ¯ τ σ¯ µ ν λ κ ¯ ¯ ¯ + gµτ ∂κ Γλν + Γλν Γδκ + Γλκ Γµνσ + S λκ Sσµν ∆q ∆q ∆q ∆q + . . . . 3 4 3

Aǫ> (q, q

Note that in contrast to the formulas for the short-time action derived in Chapter 8, the right-hand side contains only intrinsic quantities of q-space. For the systems treated there (which all lived in a Euclidean space parametrized with curvilinear coordinates), the present intrinsic result reduces to the previous one. Take, for example, a two-dimensional Euclidean space parametrized by radial coordinates treated in Section 8.1. The postpoint expansion (10.96) reads for the components r, φ of q˙λ ∆r r(∆φ)2 ∆r(∆φ)2 + − + ... , ǫ 2ǫ ǫ ∆φ ∆r∆φ (∆φ)3 ˙ − − + ... . φ = ǫ ǫr 6ǫ r˙ =

(10.109) (10.110)

Inserting these into the short-time action which is here simply Aǫ =

M 2 ǫ(r˙ + r 2 φ˙ 2 ), 2

(10.111)

we find the time-sliced action Aǫ =

M 1 ∆r 2 + r 2 (∆φ)2 − r∆r(∆φ)2 − r 2 (∆φ)4 + . . . . 2ǫ 12

(10.112)

A symmetrization of the postpoint expressions using the fact that r 2 stands for rn2 = rn (rn−1 + ∆rn ),

(10.113)

leads to the short-time action displaying the subscripts n Aǫ =

M 1 ∆rn2 + rn rn−1 (∆φn )2 − rn rn−1 (∆φn )4 + . . . . 2ǫ 12

(10.114)

This agrees with the previous expansion of the time-sliced action in Eq. (8.53). While the previous result was obtained from a transformation of the time-sliced

710


Euclidean action to radial coordinates, the short-time action here is found from a purely intrinsic formulation. The intrinsic method has the obvious advantage of not being restricted to a Euclidean initial space and therefore has the chance of being true in an arbitrary metric-affine space. At this point we observe that the final short-time action (10.107) could also have been introduced without any reference to the flat reference coordinates xi . Indeed, the same action is obtained by evaluating the continuous action (10.3) for the small time interval ∆t = ǫ along the classical orbit between the points qn−1 and qn . Due to the equations of motion (10.18), the Lagrangian L(q, q) ˙ =

M gµν (q(t)) q˙µ (t)q˙ν (t) 2

(10.115)

is independent of time (this is true for autoparallels as well as geodesics). The short-time action Aǫ (q, q ′ ) =

MZt dt gµν (q(t))q˙µ (t)q˙ν (t) 2 t−ǫ

(10.116)

can therefore be written in either of the three forms Aǫ =

M M M µ ν q )q¯˙ q¯˙ , ǫgµν (q)q˙µ q˙ν = ǫgµν (q ′ )q˙′µ q˙′ν = ǫgµν (¯ 2 2 2

(10.117)

where q µ , q ′µ , q¯µ are the coordinates at the final time tn , the initial time tn−1 , and the average time (tn + tn−1 )/2, respectively. The first expression obviously coincides with (10.107). The others can be used as a starting point for deriving equivalent prepoint or midpoint actions. The prepoint action Aǫ< arises from the postpoint one Aǫ> by exchanging ∆q by −∆q and the postpoint coefficients by the prepoint ones. The midpoint action has the most simple-looking appearance: ∆q ∆q A¯ǫ (¯ q+ , q¯ − )= (10.118) 2 2 M 1 gµν (¯ q )∆q µ ∆q ν + gκτ ∂λ Γµν τ +Γµν δ Γ{λδ} τ ∆q µ ∆q ν ∆q λ ∆q κ + . . . , 2ǫ 12 where the affine connection can be evaluated at any point in the interval (tn−1 , tn ). The precise position is irrelevant to the amplitude, producing changes only in higher than the relevant orders of ǫ. We have found the postpoint action most useful since it gives ready access to the time evolution of amplitudes, as will be seen below. The prepoint action is completely equivalent to it and useful if one wants to describe the time evolution backwards. Some authors favor the midpoint action because of its symmetry and the absence of cubic terms in ∆q µ in the expression (10.118). The different completely equivalent “anypoint” formulations of the same shorttime action, which is universally defined by the nonholonomic mapping procedure, must be distinguished from various so-called time-slicing “prescriptions” found in H. Kleinert, PATH INTEGRALS

711


the literature when setting up a lattice approximation to the Lagrangian (10.115). There, a midpoint prescription is often favored, in which one approximates L by M gµν (¯ q ) ∆q µ (t)∆q ν (t), 2ǫ2 and uses the associated short-time action L(q, q) ˙ → Lǫ (q, ∆q/ǫ) =

A¯ǫmpp = ǫLǫ (q, ∆q/ǫ)

(10.119)

(10.120)

in the exponent of the path integrand. The motivation for this prescription lies in the popularity of H. Weyl’s ordering prescription for products of position and momenta in operator quantum mechanics. From the discussion in Section 1.13 we know, however, that the Weyl prescription for the operator order in the kinetic q )ˆ pµ pˆν /2M does not lead to the correct Laplace-Beltrami operator in energy g µν (ˆ general coordinates. The discussion in this section, on the other hand, will show that the Weyl-ordered action (10.120) differs from the correct midpoint form (10.118) of the action by an additional forth-order term in ∆q µ , implying that the short-time action A¯ǫmpp does not lead to the correct physics. Worse shortcomings are found when slicing the short-time action following a pre- or postpoint prescription. There is, in fact, no freedom of choice of different slicing prescriptions, in contrast to ubiquitous statements in the literature. The short-time action is completely fixed as being the unique nonholonomic image of the Euclidean time-sliced action. This also solves uniquely the operator-ordering problem which has plagued theorists for many decades. In the following, the action Aǫ without subscript will always denote the preferred postpoint expression (10.107): Aǫ ≡ Aǫ> (q, q − ∆q).

10.3.2

(10.121)

Measure of Path Integration

We now turn to the integration measure in the Cartesian path integral (10.89) 1 q

2πiǫ¯ h/M

D

N Y

dD xn .

n=1

This has to be transformed to the general metric-affine space. We imagine evaluating the path integral starting out from the latest time and performing successively the integrations over xN , xN −1 , . . . , i.e., in each short-time amplitude we integrate over the earlier position coordinate, the prepoint coordinate. For the purpose of this Q QN +1 D i i discussion, we relabel the product N n=1 d xn by n=2 dxn−1 , so that the integration in each time slice (tn , tn−1 ) with n = N + 1, N, . . . runs over dxin−1 . In a flat space parametrized with curvilinear coordinates, the transformation of µ is obvious: the integrals over dD xin−1 into those over dD qn−1 NY +1 Z n=2

dD xin−1

=

NY +1 Z n=2

µ dD qn−1

det

h

eiµ (qn−1 )

i

.

(10.122)

712


The determinant of ei µ is the square root of the determinant of the metric gµν : det (ei µ ) =

q

det gµν (q) ≡

and the measure may be rewritten as NY +1 Z

dD xin−1

n=2

=

NY +1 Z

µ dD qn−1

n=2

q

g(q),

(10.123)

q

g(qn−1) .

(10.124)

µ This expression is not directly applicable. When trying to do the dD qn−1 -integrations µ successively, starting from the final integration over dq , the integration variable N i h i qn−1 appears for each n in the argument of det eµ (qn−1 ) or gµν (qn−1 ). To make this qn−1 -dependence explicit, we expand in the measure (10.122) eiµ (qn−1 ) = ei µ (qn − ∆qn ) around the postpoint qn into powers of ∆qn . This gives

1 dxi = eiµ (q − ∆q)dq µ = eiµ dq µ − ei µ,ν dq µ ∆q ν + ei µ,νλ dq µ ∆q ν ∆q λ + . . . ,(10.125) 2 omitting, as before, the subscripts of qn and ∆qn . Thus the Jacobian of the coordinate transformation from dxi to dq µ is 1 J0 = det (ei κ ) det δ κ µ − ei κ ei µ,ν ∆q ν + ei κ ei µ,νλ ∆q ν ∆q λ , 2

(10.126)

giving the relation between the infinitesimal integration volumes dD xi and dD q µ : NY +1 Z

dD xin−1 =

n=2

NY +1 Z

µ dD qn−1 J0n .

n=2

(10.127)

The well-known expansion formula det (1 + B) = exp tr log(1 + B) = exp tr(B − B 2 /2 + B 3 /3 − . . .)

(10.128)

allows us now to rewrite J0 as J0 = det (ei κ ) exp

i ǫ A , h ¯ J0

(10.129)

q

with the determinant det (eiµ ) = g(q) evaluated at the postpoint. This equation defines an effective action associated with the Jacobian, for which we obtain the expansion i i ǫ 1h AJ0 = −ei κ ei κ,µ ∆q µ + ei µ eiµ,νλ − ei µ ei κ,ν ej κ ej µ,λ ∆q ν ∆q λ + . . . . (10.130) h ¯ 2

The expansion coefficients are expressed in terms of the affine connection (10.16) using the relations: eiν,µ ei κ,λ = ei σ ei ν,µ ejσ ej κ,λ = Γµν σ Γλκσ eiµ ei ν,λκ = gµτ [∂κ (ei τ ei ν,λ ) − eiσ ei ν,λ ej σ ejτ ,κ ] = gµτ (∂κ Γλν τ + Γλν σ Γκσ τ ).

(10.131) (10.132)


713


The Jacobian action becomes therefore: i ǫ 1 AJ0 = −Γµν ν ∆q µ + ∂µ Γνκ κ ∆q ν ∆q µ + . . . . h ¯ 2

(10.133)

The same result would, incidentally, be obtained by writing the Jacobian in accordance with (10.124) as J0 =

q

g(q − ∆q),

(10.134)

which leads to the alternative formula for the Jacobian action i ǫ exp A = h ¯ J0

An expansion in powers of ∆q gives exp

q

g(q − ∆q) q

g(q)

.

(10.135)

i ǫ 1 q 1 q AJ¯0 = 1− q g(q),µ ∆q µ + q g(q),µν ∆q µ ∆q ν +. . . . (10.136) h ¯ g(q) 2 g(q)

Using the formula

1 √ 1 ¯ ν, √ ∂µ g = g στ ∂µ gστ = Γ µν g 2

(10.137)

this becomes exp

i ǫ ¯ νλ λ +Γ ¯ νλ λ )∆q µ ∆q ν + . . . , ¯ µν ν ∆q µ + 1 (∂µ Γ ¯ µσ σ Γ AJ¯0 = 1 − Γ h ¯ 2

(10.138)

i ǫ ¯ µν ν ∆q µ + 1 ∂µ Γ ¯ νλ λ ∆q µ ∆q ν + . . . . AJ¯0 = −Γ h ¯ 2

(10.139)

so that

¯ λµν ≡ Γµν λ , the Jacobian actions (10.133) and In a space without torsion where Γ (10.139) are trivially equal to each other. But the equality holds also in the presence ¯ λ + Kµν λ , of torsion. Indeed, when inserting the decomposition (10.27), Γµν λ = Γ µν into (10.133), the contortion tensor drops out since it is antisymmetric in the last two indices and these are contracted in both expressions. In terms of AǫJ0n , we can rewrite the transformed measure (10.122) in the more useful form NY +1 Z n=2

dD xin−1

=

NY +1 Z n=2

µ dD qn−1

det

h

eiµ (qn )

i

i ǫ exp A h ¯ J0n

.

(10.140)

In a flat space parametrized in terms of curvilinear coordinates, the right-hand sides of (10.122) and (10.140) are related by an ordinary coordinate transformation, and both give the correct measure for a time-sliced path integral. In a general

714


metric-affine space, however, this is no longer true. Since the mapping dxi → dq µ is nonholonomic, there are in principle infinitely many ways of transforming the path integral measure from Cartesian coordinates to a non-Euclidean space. Among these, there exists a preferred mapping which leads to the correct quantummechanical amplitude in all known physical systems. This will serve to solve the path integral of the Coulomb system in Chapter 13. The clue for finding the correct mapping is offered by an unaesthetic feature of Eq. (10.125): The expansion contains both differentials dq µ and differences ∆q µ . This is somehow inconsistent. When time-slicing the path integral, the differentials dq µ in the action are increased to finite differences ∆q µ . Consequently, the differentials in the measure should also become differences. A relation such as (10.125) containing simultaneously differences and differentials should not occur. It is easy to achieve this goal by changing the starting point of the nonholonomic mapping and rewriting the initial flat space path integral (10.89) as NY N Z ∞ +1 Y 1 d∆x K0ǫ (∆xn ). (x t|x′ t′ ) = q n D −∞ n=1 2πiǫ¯ h/M n=1

(10.141)

Since xn are Cartesian coordinates, the measures of integration in the time-sliced expressions (10.89) and (10.141) are certainly identical: N Z Y

n=1

D

d xn ≡

NY +1 Z

dD ∆xn .

(10.142)

n=2

Their images under a nonholonomic mapping, however, are different so that the initial form of the time-sliced path integral is a matter of choice. The initial form (10.141) has the obvious advantage that the integration variables are precisely the quantities ∆xin which occur in the short-time amplitude K0ǫ (∆xn ). Under a nonholonomic transformation, the right-hand side of Eq. (10.142) leads to the integral measure in a general metric-affine space NY +1 Z n=2

D

d ∆xn →

NY +1 Z

D

d ∆qn Jn ,

n=2

(10.143)

with the Jacobian following from (10.96) (omitting n) ∂(∆x) 1 J= =det (ei κ ) det δµ λ −Γ{µν}λ ∆q ν + ∂{σ Γµν}λ +Γ{µν τ Γ{τ |σ}}λ ∆q ν ∆q σ +. . . . ∂(∆q) 2 (10.144)

In a space with curvature and torsion, the measure on the right-hand side of (10.143) replaces the flat-space measure on the right-hand side of (10.124). The curly double brackets around the indices ν, κ, σ, µ indicate a symmetrization in τ and σ followed H. Kleinert, PATH INTEGRALS

715


by a symmetrization in µ, ν, and σ. With the help of formula (10.128) we now calculate the Jacobian action i i ǫ 1h AJ = −Γ{µν} µ ∆q ν + ∂{µ Γνκ} κ + Γ{νκ σ Γ{σ|µ}} κ − Γ{νκ} σ Γ{σµ} κ ∆q ν ∆q µ + . . . . h ¯ 2 (10.145)

This expression differs from the earlier Jacobian action (10.133) by the symmetrization symbols. Dropping them, the two expressions coincide. This is allowed if q µ are curvilinear coordinates in a flat space. Since then the transformation functions xi (q) and their first derivatives ∂µ xi (q) are integrable and possess commuting derivatives, the two Jacobian actions (10.133) and (10.145) are identical. There is a further good reason for choosing (10.142) as a starting point for the nonholonomic transformation of the measure. According to Huygens’ principle of wave optics, each point of a wave front is a center of a new spherical wave propagating from that point. Therefore, in a time-sliced path integral, the differences ∆xin play a more fundamental role than the coordinates themselves. Intimately related to this is the observation that in the canonical form, a short-time piece of the action reads Z

i ip2n dpn exp pn (xn − xn−1 ) − t . 2π¯ h h ¯ 2M¯ h "

#

Each momentum is associated with a coordinate difference ∆xn ≡ xn − xn−1 . Thus, we should expect the spatial integrations conjugate to pn to run over the coordinate differences ∆xn = xn − xn−1 rather than the coordinates xn themselves, which makes the important difference in the subsequent nonholonomic coordinate transformation. We are thus led to postulate the following time-sliced path integral in q-space: ′

1

ˆ

hq|e−i(t−t )H/¯h |q ′ i = q D 2πi¯ hǫ/M



NY +1 n=2

 

Z

q

g(qn )



i dD ∆qn q D e 2πiǫ¯ h/M



PN+1 n=1

(Aǫ +AǫJ )/¯ h

,

(10.146) where the integrals over ∆qn may be performed successively from n = N down to n = 1. Let us emphasize that this expression has not been derived from the flat space path integral. It is the result of a specific new quantum equivalence principle which rules how a flat space path integral behaves under nonholonomic coordinate transformations. It is useful to re-express our result in a different form which clarifies best the relation with the naively expected measure of path integration (10.124), the product of integrals N Z Y

n=1

D

d xn =

N Z Y

n=1

D

d qn

q

g(qn ) .

(10.147)

716


The measure in (10.146) can be expressed in terms of (10.147) as NY +1 Z n=2

D

q

d ∆qn g(qn ) =

N Z Y

D

d qn

n=1

q

−iAǫJ /¯ h

g(qn )e

0

.

The corresponding expression for the entire time-sliced path integral (10.146) in the metric-affine space reads ˆ h −i(t−t′ )H/¯

hq|e

1

|q ′ i = q D 2πi¯ hǫ/M

 Z N Y  

n=1



q

g(qn )

i dD qn q De 2πi¯ hǫ/M



PN+1 n=1

(Aǫ +∆AǫJ )/¯ h

,

(10.148) where ∆AǫJ is the difference between the correct and the wrong Jacobian actions in Eqs. (10.133) and (10.145): ∆AǫJ ≡ AǫJ − AǫJ0 .

(10.149)

¯ µν λ , this simplifies to In the absence of torsion where Γ{µν} λ = Γ i 1¯ µ ν ∆AǫJ = R µν ∆q ∆q , h ¯ 6

(10.150)

¯ µν is the Ricci tensor associated with the Riemann curvature tensor, i.e., the where R contraction (10.41) of the Riemann curvature tensor associated with the Christoffel ¯ µν λ . symbol Γ Being quadratic in ∆q, the effect of the additional action can easily be evaluated perturbatively using the methods explained in Chapter 8, according to which ∆q µ ∆q ν may be replaced by its lowest order expectation h∆q µ ∆q ν i0 = iǫ¯ hg µν (q)/M. Then ∆AǫJ yields the additional effective potential h ¯2 ¯ Veff (q) = − R(q), 6M

(10.151)

¯ is the Riemann curvature scalar. By including this potential in the action, where R the path integral in a curved space can be written down in the naive form (10.147) as follows: N Y





q

PN+1 ǫ g(qn ) i¯ hR(qn )/6M  i A [qn ]/¯ h n=1 dD qn q e e .  D n=1 2πiǫ¯ h/M (10.152) This time-sliced expression will from now on be the definition of a path integral in curved space written in the continuum notation as

1 ′ ˆ hq|e−i(t−t )H/¯h |q ′i = q D 2πi¯ hǫ/M

′

ˆ

 

Z

hq|e−i(t−t )H/¯h |q ′ i =

Z

q

i

D D q g(q) e

R tb ta

dt A[q]/¯ h

.

(10.153)


717

10.4 Completing Solution of Path Integral on Surface of Sphere

The integrals over qn in (10.152) are conveniently performed successively downq q wards over ∆qn+1 = qn+1 −qn at fixed qn+1 . The weights g(qn ) = g(qn+1 − ∆qn+1 ) require a postpoint expansion leading to the naive Jacobian J0 of (10.126) and the Jacobian action AǫJ0 of Eq. (10.133). It is important to observe that the above time-sliced definition is automatically invariant under coordinate transformations. This is an immediate consequence of the definition via the nonholonomic mapping from a flat-space path integral. It goes without saying that the path integral (10.152) also has a phase space version. It is obtained by omitting all (M/2ǫ)(∆qn )2 terms in the short-time actions Aǫ and extending the multiple integral by the product of momentum integrals NY +1 n=1

 

dpn q

2π¯ h g(qn )



 e(i/¯h)

PN+1 n=1

1 g µν (qn )pnµ pnν ] [pnµ ∆qµ −ǫ 2M .

(10.154)

When using this expression, all problems which were encountered in the literature with canonical transformations of path integrals disappear. An important property of the definition of the path integral in spaces with curvature and torsion as a nonholonomic image of a Euclidean path integral is that this image is automatically invariant under ordinary holonomic coordinate transformations.

10.4

Completing Solution of Path Integral on Surface of Sphere in D Dimensions

The measure of path integration in Eq. (10.146) allows us to finally complete the calculation, initiated in Sections 8.7–8.9, of the path integrals of a point particle on the surface of a sphere on group spaces in any number of dimensions. Indeed, using the result (10.151) we are now able to solve the problems discussed in Section 8.7 in conjunction with the energy formula (8.224). Thus we are finally in a position to find the correct energies and amplitudes of these systems. A sphere of radius r embedded in D dimensions has an intrinsic dimension D ′ ≡ D − 1 and a curvature scalar

′ ′ ¯ = (D − 1)D . R (10.155) r2 This is most easily derived as follows. Consider a line element in D dimensions

(dx)2 = (dx1 )2 + (dx2 )2 + . . . + (dxD )2

(10.156)

and restrict the motion to a spherical surface (x1 )2 + (x2 )2 + . . . + (xD )2 = r 2 ,

(10.157)

by eliminating xD . This brings (10.156) to the form ′

′

(x1 dx1 + dx2 + . . . + xD dxD )2 (dx) = (dx ) + (dx ) + . . . + (dx ) + , (10.158) r 2 − r ′2 2

1 2

2 2

D′ 2

718

10 Spaces with Curvature and Torsion ′

where r ′2 ≡ (x1 )2 + (x2 )2 + . . . + (xD )2 . The metric on the D ′ -dimensional surface is therefore gµν (x) = δµν +

xµ xν . r 2 − r ′2

(10.159)

¯ will be constant on the spherical surface, we may evaluate it for small xµ Since R (µ = 1, . . . , D ′) where gµν (x) ≈ δµν + xµ xν /r2 and the Christoffel symbols (10.7) are Γµν λ ≈ Γµνλ ≈ δµν xλ /r2 . Inserting this into (10.35) we obtain the curvature tensor for small xµ : ¯ µνλκ ≈ 1 (δµκ δνλ − δµλ δνκ ) . R r2

(10.160)

This can be extended covariantly to the full surface of the sphere by replacing δµλ by the metric gµλ (x): ¯ µνλκ (x) = 1 [gµκ (x)gνλ (x) − gµλ (x)gνκ (x)] , R r2

(10.161)

so that Ricci tensor is [recall (10.41)] ′ ¯ νκ (x) = R ¯ µνκ µ (x) = D − 1 gνκ (x). R r2

(10.162)

Contracting this with g νκ [recall (10.42)] yields indeed the curvature scalar (10.155). The effective potential (10.151) is therefore Veff

h ¯2 =− (D − 2)(D − 1). 6Mr 2

(10.163)

It supplies precisely the missing energy which changes the energy (8.224) near the sphere, corrected by the expectation of the quartic term ϑ4n in the action, to the proper value El =

h ¯2 l(l + D − 2). 2Mr 2

(10.164)

Astonishingly, this elementary result of Schrödinger quantum mechanics was found only a decade ago by path integration [5]. Other time-slicing procedures yield extra ¯ which are absent in our theory. Here terms proportional to the scalar curvature R, the scalar curvature is a trivial constant, so it would remain undetectable in atomic experiments which measure only energy differences. The same result will be derived in general in Eqs. (11.25) and in Section 11.3. Experimental arguments for the absence will be given in Subsection 13.10.3. An important property of this spectrum is that the ground state energy vanished for all dimensions D. This property would not have been found in the naive measure of path integration on the right-hand side of Eq. (10.147) which is used in most H. Kleinert, PATH INTEGRALS

719

10.5 External Potentials and Vector Potentials

works on this subject. The correction term (10.150) coming from the nonholonomic mapping of the flat-space measure is essential for the correct result. More evidence for the correctness of the measure in (10.146) will be supplied in Chapter 13 where we solve the path integrals of the most important atomic system, the hydrogen atom. We remark that for t → t′ , the amplitude (10.152) shows the states |qi to obey the covariant orthonormality relation hq|q ′i = The completeness relation reads Z

10.5

q

g(q)

−1

δ (D) (q − q ′ ).

(10.165)

q

dD q g(q)|qihq| = 1.

(10.166)

External Potentials and Vector Potentials

An important generalization of the above path integral formulas (10.146), (10.148), (10.152) of a point particle in a space with curvature and torsion includes the presence of an external potential and a vector potential. These allow us to describe, for instance, a particle in external electric and magnetic fields. The classical action is then Aem =

Z

tb

ta

dt

e Aµ (q(t))q˙µ − V (q(t)) . c

(10.167)

To find the time-sliced action we proceed as follows. First we set up the correct time-sliced expression in Euclidean space and Cartesian coordinates. For a single slice it reads, in the postpoint form, Aǫ =

M e e (∆xi )2 + Ai (x)∆xi − Ai,j (x)∆xi ∆xj − ǫV (x) + . . . . 2ǫ c 2c

(10.168)

As usual, we have neglected terms which do not contribute in the continuum limit. The derivation of this time-sliced expression proceeds by calculating, as in (10.116), the action Aǫ =

Z

t

t−ǫ

dtL(t)

(10.169)

along the classical trajectory in Euclidean space, where L(t) =

M 2 e ˙ − V (x(t)) x˙ (t) + A(x(t))x(t) 2 c

(10.170)

is the classical Lagrangian. In contrast to (10.116), however, the Lagrangian has now a nonzero time derivative (omitting the time arguments): d e e ˙ x + A(x)¨ L = M x¨ x + Ai,j (x)x˙ i x˙ j − Vi (x)x˙ i . dt c c

(10.171)

720


For this reason we cannot simply write down an expression such as (10.117) but we have to expand the Lagrangian around the postpoint leading to the series ǫ

A =

1 d dtL(t) = ǫL(t) − ǫ2 L(t) + . . . . 2 dt t−ǫ

Z

t

(10.172)

The evaluation makes use of the equation of motion e M xï = − (Ai,j (x) − Aj,i (x))x˙ j − Vi (x), c

(10.173)

from which we derive the analog of Eq. (10.96): First we have the postpoint expansion 1 ∆xi = −ǫx˙ i + ǫ2 xï + . . . 2 i e 2h ǫ (Ai,j − Aj,i)x˙ j + Vi (x) + . . . . = −ǫx˙ i − 2Mc

(10.174)

Inverting this gives ∆xi e − (Ai,j − Aj,i)∆xj + . . . . x˙ = − ǫ 2Mc i

(10.175)

When inserted into (10.172), this yields indeed the time-sliced short-time action (10.168). The quadratic term ∆xi ∆xj in the action (10.168) can be replaced by the perturbative expectation value ∆xi ∆xj → h∆xi ∆xj i = δij i

h ¯ǫ , M

(10.176)

so that Aǫ becomes Aǫ =

M e h ¯e (∆xi )2 + Ai (x)∆xi − iǫ Ai,i (x) − ǫV (x) + . . . . 2ǫ c 2Mc

(10.177)

Incidentally, the action (10.168) could also have been written as Aǫ =

M e (∆xi )2 + Ai (¯ x)∆xi − ǫV (x) + . . . , 2ǫ c

(10.178)

¯ is the midpoint value of the slice coordinates where x 1 ¯ = x − ∆x, x 2

(10.179)

1 ¯ (tn ) ≡ [x(tn ) + x(tn−1 )]. x 2

(10.180)

i.e., more explicitly,


10.6 Perturbative Calculation of Path Integrals in Curved Space

721

Thus, with an external vector potential in Cartesian coordinates, a midpoint “prescription” for Aǫ happens to yield the correct expression (10.178). Having found the time-sliced action in Cartesian coordinates, it is easy to go over to spaces with curvature and torsion. We simply insert the nonholonomic transformation (10.96) for the differentials ∆xi . This gives again the short-time action (10.107), extended by the interaction due to the potentials e e Aǫem = Aµ ∆q µ − ∂ν Aµ ∆q µ ∆q ν − ǫV (q) + . . . . c 2c

(10.181)

The second term can be evaluated perturbatively leading to e h ¯e ∂µ Aµ − ǫV (q) + . . . . Aǫem = Aµ ∆q µ − iǫ c 2Mc

(10.182)

The sum over all slices, AN em =

N +1 X n=1

Aǫem ,

(10.183)

has to be added to the action in each time-sliced expression (10.146), (10.148), and (10.152).

10.6

Perturbative Calculation of Path Integrals in Curved Space

In Sections 2.15 and 3.21 we have given a perturbative definition of path integrals which does not require the rather cumbersome time slicing but deals directly with a continuous time. We shall now extend this definition to curved space in such a way that it leads to the same result as the time-sliced definition given in Section 10.3. In particular, we want to ensure that this definition preserves the fundamental property of coordinate independence achieved in the time-sliced definition via the nonholonomic mapping principle, as observed at the end of Subsection 10.3.2. In a perturbative calculation, this property will turn out to be highly nontrivial. In addition, we want to be sure that the ground state energy of a particle on a sphere is zero in any number of dimensions, just as in the time-sliced calculation leading to Eq. (10.164). This implies that also in the perturbative definition of path integral, the operator-ordering problem will be completely solved.

10.6.1

Free and Interacting Parts of Action

The partition function of a point particle in a curved space with an intrinsic dimension D is given by the path integral over all periodic paths on the imaginary-time axis τ : Z=

Z

√ D D q g e−A[q] ,

(10.184)

722


where A[q] is the Euclidean action Z

A[q] =

β

0

1 dτ gµν (q(τ ))q˙µ (τ )q˙ν (τ ) + V (q(τ )) . 2

(10.185)

We have set h ¯ and the particle mass M equal to unity. For a space with constant curvature, this is a generalization of the action for a particle on a sphere (8.146), also called a nonlinear σ-model (see p. 655). The perturbative definition of Sections 2.15 and 3.21 amounts to the following calculation rules. Expand the metric gµν (q) and the potential V (q) around some point qaµ in powers of δq µ ≡ q µ − qaµ . After this, separate the action A[q] into a harmonically fluctuating part A(0) [qa ; δq] ≡

1 2

Z

0

β

h

i

dτ gµν (qa ) δ q˙µ (τ )δ q˙ν (τ ) + ω 2 δq µ (τ )δq ν (τ ) ,

(10.186)

and an interacting part Aint [qa ; δq] ≡ A[q] − A(0) [qa ; δq].

(10.187)

The second term in (10.186) is called frequency term or mass term. It is not invariant under coordinate transformations. The implications of this will be seen later. When studying the partition function in the limit of large β, the frequency ω cannot be set equal to zero since this would lead to infinities in the perturbation expansion, as we shall see below. A delicate problem is posed by the square root of the determinant of the metric in the functional integration measure in (10.184). In a purely formal continuous definition of the measure, we would write it as " Z Y

#

"

#

1X g(q(τ )) DD q g ≡ dD q(τ ) g(τ ) = dD q(τ ) g(qa ) exp log . 2 τ g(qa ) τ τ (10.188) The formal sum over all continuous times τ in the exponent corresponds to an inR tegral dτ divided by the spacing of the points, which on a sliced time axis would be the slicing parameter ǫ. Here it is dτ . The ratio 1/dτ may formally be identified R with δ(0), in accordance with the defining integral dτ δ(τ ) = 1. The infinity of δ(0) may be regularized in some way, for instance by a cutoff in the Fourier repreR sentation δ(0) ≡ dω/(2π) at large frequencies ω, a so-called UV-cutoff . Leaving the regularization unspecified, we rewrite the measure (10.188) formally as √

Z

Z

YZ

D

√

D q g≡

q

"

YZ τ

q

#

"

1 d q(τ ) g(qa ) exp δ(0) 2 q

D

Z

0

β

#

g(q(τ )) dτ log , g(qa )

(10.189)

and further as Z

DD q

q

g [q]

g(qa ) e−A

,


723


where we have introduced an effective action associated with the measure: Z β 1 g(q(τ )) Ag [q] = − δ(0) dτ log . 2 g(qa ) 0

(10.191)

For a perturbative treatment, this action is expanded in powers of δq(τ ) and is a functional of this variable: Z β 1 Ag [qa , δq] = − δ(0) dτ [log g(qa + δq(τ )) − log g(qa )]. 2 0

(10.192)

This is added to (10.187) to yield the total interaction int Aint tot [qa , δq] = A [qa , δq] + Ag [qa , δq].

(10.193)

The path integral for the partition function is now written as Z

Z =

q

DD q

(0) [q]

int

g(qa ) e−A

e−Atot [q] .

(10.194) int

According to the rules of perturbation theory, we expand the factor e−Atot in powers of the total interaction, and obtain the perturbation series Z =

Z

D

D q

q

g(qa ) 1 −

D

E

= Zω 1 − Aint tot +

Aint tot

1 2 −A(0) [q] + hAint tot i − . . . e 2

1 D int 2 E Atot − . . . , 2!

(10.195)

where Zω ≡ e−βFω =

Z

q

(0) [q]

D D q g(qa ) e−A

(10.196)

is the path integral over the free part, and the symbol h . . . i denotes the expectation values in this path integral h . . . i = Zω−1

Z

Dq

q

(0) [q]

g(qa ) ( . . . ) e−A D

With the usual definition of the cumulants Aint tot

D

Aint tot

E2

E

c

D

.

Z≡e

= exp −βFω −

D

E

Aint tot c

E

= Aint tot ,

, . . . [recall (3.482), (3.483)], this can be written as −βF

(10.197) D

2 Aint tot

1 D int 2 E + Atot − ... , c 2!

E

c

D

E

2 = Aint − tot

(10.198)

where Fω ≡ −β −1 log Z the free energy associated with Zω . The cumulants are now calculated according to Wick’s rule order by order in h ¯, treating the δ-function at the origin δ(0) as if it were finite. The perturbation series will contain factors of δ(0) and its higher powers. Fortunately, these unpleasant terms will turn out to cancel each other at each order in a suitably defined expansion

724


parameter. On account of these cancellations, we may ultimately discard all terms containing δ(0), or set δ(0) equal to zero, in accordance with Veltman’s rule (2.500). The harmonic path integral (10.196) is performed using formulas (2.481) and (2.497). Assuming for a moment what we shall prove below that we may choose coordinates in which gµν (qa ) = δµν , we obtain directly in D dimensions Zω =

Z

Dq e−Aω [q] = exp −

D Tr log(−∂ 2 + ω 2 ) ≡ e−βFω . 2

(10.199)

The expression in brackets specifies the free energy Fω of the harmonic oscillator at the inverse temperature β.

10.6.2

Zero Temperature

For simplicity, we fist consider the limit of zero temperature or β → ∞. Then Fω becomes equal to the sum of D ground state energies ω/2 of the oscillator, one for each dimension: 1D D Tr log(−∂ 2 + ω 2 ) → Fω = β→∞ 2 β 2

Z

∞

−∞

D

dk D log(k 2 + ω 2 ) = ω. 2π 2

(10.200)

E

2 The Wick contractions in the cumulants Aint tot c of the expansion (10.197) contain only connected diagrams. They contain temporal integrals which, after suitable partial integrations, become products of the following basic correlation functions ′ ′ G(2) µν (τ, τ ) ≡ hqµ (τ )qν (τ )i =

,

(10.201)

′ ′ ∂τ G(2) µν (τ, τ ) ≡ hq˙µ (τ )qν (τ )i =

,

(10.202)

′ ′ ∂τ ′ G(2) µν (τ, τ ) ≡ hqµ (τ )q˙ν (τ )i =

,

(10.203)

′ ′ ∂τ ∂τ ′ G(2) µν (τ, τ ) ≡ hq˙µ (τ )q˙ν (τ )i =

.

(10.204)

The right-hand sides define line symbols to be used for drawing Feynman diagrams for the interaction terms. ′ Under the assumption gµν (qa ) = δµν , the correlation function G(2) µν (τ, τ ) factorizes as ′ ′ G(2) µν (τ, τ ) = δµν ∆(τ − τ ),

(10.205)

with ∆(τ − τ ′ ) abbreviating the correlation the zero-temperature Green function Gpω2 ,e (τ ) of Eq. (3.246) (remember the present units with M = h ¯ = 1): ∆(τ − τ ′ ) =

Z

∞

−∞

′

dk eik(τ −τ ) 1 −ω|τ −τ ′ | = e . 2π k 2 + ω 2 2ω

(10.206)

As a consequence, the second correlation function (10.202) has a discontinuity ′ ∂τ G(2) µν (τ, τ ) = δµν i

Z

∞

−∞

′

dk eik(τ −τ ) k ˙ − τ ′ ) ≡ − 1 δµν ǫ(τ − τ ′ )e−ω|τ −τ ′ | , = δ ∆(τ µν 2π k 2 + ω 2 2 (10.207) H. Kleinert, PATH INTEGRALS

725


where ǫ(τ − τ ′ ) is the distribution defined in Eq. (1.311) which has a jump at τ = τ ′ from −1 to 1. It can be written as an integral over a δ-function: ′

ǫ(τ − τ ) ≡ −1 + 2

Z

τ

−∞

dτ ′′ δ(τ ′′ − τ ′ ).

(10.208)

The third correlation function (10.203) is simply the negative of (10.202): ′ (2) ′ ′ ˙ ∂τ ′ G(2) µν (τ, τ ) = −∂τ Gµν (τ, τ ) = −δµν ∆(τ − τ ).

(10.209)

At the point τ = τ ′ , the momentum integral (10.207) vanishes by antisymmetry: ′ (2) ′ ∂τ G(2) µν (τ, τ )|τ =τ ′ = −∂τ ′ Gµν (τ, τ )|τ =τ ′ = δµν

i

Z

∞

−∞

dk k ˙ = δµν ∆(0) = 0. (10.210) 2 2π k + ω 2

The fourth correlation function (10.204) contains a δ-function: ′ ∂τ ∂τ ′ G(2) µν (τ, τ )

=

= δµν

Z

∞

−∞

′

dk eik(τ −τ ) k 2 ¨ − τ ′ ) (10.211) = δµν = −δµν ∆(τ −∞ 2π k 2 + ω 2 ! dk ik(τ −τ ′ ) ω2 ′ e 1− 2 = δµν δ(τ − τ ′ ) − ω 2 G(2) µν (τ, τ ). 2π k + ω2 Z

′ −∂τ2 G(2) µν (τ, τ )

∞

The Green functions for µ = ν are plotted in Fig. 10.4.

0.4

∆(τ )

0.4

0.2 -2

-1

˙ ) −∆(τ

0.4 0.2

0.2 1

2

τ

-2

-1

¨ ) −∆(τ

1

2

τ

-2

-1

1

-0.2

-0.2

-0.2

-0.4

-0.4

-0.4

2

τ

Figure 10.4 Green functions for perturbation expansions in curvilinear coordinates in natural units with ω = 1. The third contains a δ-function at the origin.

The last equation is actually the defining equation for the Green function, which is always the solution of the inhomogeneous equation of motion associated with the harmonic action (10.186), which under the assumption gµν (qa ) = δµν reads for each component: −¨ q (τ ) + ω 2 q(τ ) = δ(τ − τ ′ ).

(10.212)

The Green function ∆(τ − τ ′ ) solves this equation, satisfying ¨ ) = ω 2∆(τ ) − δ(τ ). ∆(τ

(10.213)

When trying to evaluate the different terms produced by the Wick contractions, we run into a serious problem. The different terms containing products of time derivatives of Green functions contain effectively products of δ-functions and

726


Heaviside functions. In the mathematical theory of distributions, such integrals are undefined. We shall offer two ways to resolve this problem. One is based on extending the integrals over the time axis to integrals over a d-dimensional time space, and continuing the results at the end back to d = 1. The extension makes the path integral a functional integral of the type used in quantum field theories. It will turn out that this procedure leads to well-defined finite results, also for the initially divergent terms coming from the effective action of the measure (10.192). In addition, and very importantly, it guarantees that the perturbatively defined path integral is invariant under coordinate transformations. For the time-sliced definition in Section 10.3, coordinate independence was an automatic consequence of the nonholonomic mapping from a flat-space path integral. In the perturbative definition, the coordinate independence has been an outstanding problem for many years, and was only solved recently in Refs. [23]–[25]. In d-dimensional quantum field theory, path integrals between two and four spacetime dimensions have been defined by perturbation expansions for a long time. Initial difficulties in guaranteeing coordinate independence were solved by ’t Hooft and Veltman [29] using dimensional regularization with minimal subtractions. For a detailed description of this method see the textbook [30]. Coordinate independence emerges after calculating all Feynman integrals in an arbitrary number of dimensions d, and continuing the results to the desired physical integer value. Infinities occuring in the limit are absorbed into parameters of the action. In contrast, and surprisingly, numerous attempts [31]–[36] to define the simpler quantum-mechanical path integrals in curved space by perturbation expansions encountered problems. Although all final results are finite and unique, the Feynman integrals in the expansions are highly singular and mathematically undefined. When evaluated in momentum space, they yield different results depending on the order of integration. Various definitions chosen by the earlier authors were not coordinate-independent, and this could only be cured by adding coordinate-dependent “correction terms” to the classical action — a highly unsatisfactory procedure violating the basic Feynman postulate that physical amplitudes should consist of a sum over all paths with phase factors eiA/¯h containing only the classical actions along the paths. The calculations in d spacetime dimensions and the continuation to d = 1 will turn out to be somewhat tedious. We shall therefore find in Subsection 10.11.4 a method of doing the calculations directly for d = 1.

10.7

Model Study of Coordinate Invariance

Let us consider first a simple model which exhibits typical singular Feynman integrals encountered in curvilinear coordinates and see how these can be turned into a finite perturbation expansion which is invariant under coordinate transformations. For simplicity, we consider an ordinary harmonic oscillator in one dimension, with the action Z i 1 β h 2 dτ x˙ (τ ) + ω 2 x2 (τ ) . (10.214) Aω = 2 0 H. Kleinert, PATH INTEGRALS

727

10.7 Model Study of Coordinate Invariance

The partition function of this system is exactly given by (10.200): Zω =

Z

−Aω [x]

Dx e

D = exp − Tr log(−∂ 2 + ω 2 ) ≡ e−βFω . 2

(10.215)

A nonlinear transformation of x(τ ) to some other coordinate q(τ ) turns (10.215) into a path integral of the type (10.184) which has a singular perturbation expansion. For simplicity we assume a specific simple coordinate transformation preserving the reflection symmetry x ←→ −x of the initial oscillator, whose power series expansion starts out like 1 η η2 x(τ ) = fη (ηq(τ )) ≡ f (ηq(τ )) = q − q 3 + a q 5 − · · · , η 3 5

(10.216)

where η is an expansion parameter which will play the role of a coupling constant counting the orders of the perturbation expansion. An extra parameter a is introduced for the sake of generality. We shall see that it does not influence the conclusions. The transformation changes the partition function (10.215) into Z=

Z

Dq(τ ) e−AJ [q] e−A[q] ,

(10.217)

where is A[q] is the transformed action, whereas AJ [q] = −δ(0)

Z

dτ log

∂f (q(τ )) ∂q(τ )

(10.218)

is an effective action coming from the Jacobian of the coordinate transformation v u Y u ∂f (q(τ )) t .

J=

τ

∂q(τ )

(10.219)

The Jacobian plays the role of the square root of the determinant of the metric in (10.184), and AJ [q] corresponds to the effective action Ag [δq] in Eq. (10.192). The transformed action is decomposed into a free part 1 Aω [q] = 2

Z

0

β

dτ [q˙2 (τ ) + ω 2q 2 (τ )],

(10.220)

and an interacting part corresponding to (10.187), which reads to second order in η: int

A [q] =

Z

0

β

ω2 dτ −η q (τ )q˙ (τ ) + q 4 (τ ) 3 2a 6 1 1 2 4 2 2 +η + a q (τ )q˙ (τ ) + ω + q (τ ) . 2 18 5 (

"

2

2

#

(10.221)

This is found from (10.187) by inserting the one-dimensional metric g00 (q) = g(q) = [f ′ (ηq)]2 = 1 − 2ηq 2 + (1 + 2a)η 2 q 4 + . . . .

(10.222)

728


To the same order in η, the Jacobian action (10.218) is AJ [q] = −δ(0)

Z

0

β

dτ −ηq 2 (τ ) + η 2 a −

1 4 q (τ ) , 2

(10.223)

and the perturbation expansion (10.198) is to be performed with the total interaction int Aint tot [q] = A [q] + AJ [q].

(10.224)

For η = 0, the transformed partition function (10.217) coincides trivially with (10.215). When expanding Z of Eq. (10.217) in powers of η, we obtain sums of Feynman diagrams contributing to each order η n . This sum must vanish to ensure coordinate independence of the path integral. From the connected diagrams in the cumulants in (10.198) we obtain the free energy βF = βFω + β

X

n=1

D

η n Fn = βFω + Aint tot

E

c

−

1 D int 2 E Atot + ... . c 2!

(10.225)

The perturbative treatment is coordinate-independent if F does not depend on the parameters η and a of the coordinate transformation (10.216). Hence all expansion terms Fn must vanish. This will indeed happen, albeit in a quite nontrivial way.

10.7.1

Diagrammatic Expansion

The graphical expansion for the ground state energy will be carried here only up to three loops. At any order η n , there exist different types of Feynman diagrams with L = n + 1, n, and n − 1 number of loops coming from the interaction terms (10.221) and (10.223), respectively. The diagrams are composed of the three types of lines in (10.201)–(10.204), and new interaction vertices for each power of η. The diagrams coming from the Jacobian action (10.223) are easily recognized by an accompanying power of δ(0). D E First we calculate the contribution to the free energy of the first cumulant Aint tot c in the expansion (10.225). The associated diagrams contain only lines whose end points have equal times. Such diagrams will be called local . To lowest order in η, the cumulant contains the terms βF1 = η

Z

0

β

ω2 dτ − q (τ )q˙ (τ ) + q 4 (τ ) + δ(0)q 2 (τ ) 3 *

"

2

2

#

+

. c

There are two diagrams originating from the interaction, one from the Jacobian action: βF1 = − η

− η ω2

+ η δ(0)

.

(10.226)


729

10.7 Model Study of Coordinate Invariance

The first cumulant contains also terms of order η 2 : η2

Z

β

0

dτ

2a 6 1 1 1 4 + a q 4 (τ )q˙2 (τ ) + ω 2 + q (τ ) − δ(0) a − q (τ ) 2 18 5 2

. c

The interaction gives rise to two three-loop diagrams, the Jacobian action to a single two-loop diagram: (1) βF2

=η

2

"

1 3 +a 2

+15 ω

2

1 a + 18 5

#

1 −3 a− δ(0) 2

D

. (10.227)

E

2 We now come to the contribution of the second cumulant Aint tot c in the expansion (10.225). Keeping only terms contributing to order η 2 we have to calculate the expectation value

1 Zβ Zβ − η 2 dτ dτ ′ 2! 0 0

ω2 −q (τ )q˙ (τ ) − q 4 (τ ) + δ(0)q 2 (τ ) 3 " #+ ω2 4 ′ 2 ′ 2 ′ 2 ′ × −q (τ )q˙ (τ ) − q (τ ) + δ(0)q (τ ) . 3 c

*"

2

#

2

(10.228)

Only the connected diagrams contribute to the cumulant, and these are necessarily nonlocal. The simplest diagrams are those containing factors of δ(0): (2)

βF2

=−

η2 n 2 2δ (0) 2!

h

− 4δ(0)

io

+ 2 ω2

+

.

(10.229)

The remaining diagrams have either the form of three bubble in a chain, or of a watermelon, each with all possible combinations of the three line types (10.201)– (10.204). The sum of the three-bubbles diagrams is (3)

βF2 = −

η2 h 4 2!

+2

+2

+8 ω 2

+ 8ω 2

+ 8ω 4

i

,

(10.230)

while the watermelon-like diagrams contribute (4) βF2

η2 =− 4 2!

2 4 ω 3

+

+4

+

+ 4ω

2

.

(10.231)

Since the equal-time expectation value hq(τ ˙ ) q(τ )i vanishes according to Eq. (10.210), diagrams with a local contraction of a mixed line (10.202) vanish trivially, and have been omitted. We now show that if we calculate all Feynman integrals in d = 1 − ε time dimensions and take the limit ε → 0 at the end, we obtain unique finite results. These have the desired property that the sum of all Feynman diagrams contributing to each order η n vanishes, thus ensuring invariance of the perturbative expressions (10.195) and (10.198) under coordinate transformations.

730


10.7.2

Diagrammatic Expansion in d Time Dimensions

As a first step, we extend the dimension of the τ -axis to d, assuming τ to be a vector τ ≡ (τ 0 , . . . , τ d ), in which the zeroth component is the physical imaginary time, the others are auxiliary coordinates to be eliminated at the end. Then we replace the harmonic action (10.214) by Aω =

1 2

Z

h

i

dd τ ∂α x(τ )∂α x(τ ) + ω 2 x2 (τ ) ,

(10.232)

and the terms q˙2 in the transformed action (10.221) accordingly by ∂a q(τ )∂a q(τ ). The correlation functions (10.205), (10.207), and (10.211) are replaced by twopoint functions (2)

′

′

G (τ, τ ) = hq(τ )q(τ )i

′

= ∆(τ − τ )

=

Z

′

dd k eik(τ −τ ) , (2π)d k 2 + ω 2

(10.233)

and its derivatives dd k ikα ik(τ −τ ′ ) e , (10.234) (2π)d k 2 + ω 2 Z dd k kα kβ ik(τ −τ ′ ) (2) ′ ′ ′ Gαβ (τ, τ ) = h∂α q(τ )∂β q(τ )i = ∆αβ (τ − τ ) = e . (10.235) (2π)d k 2 + ω 2

′ ′ G(2) α (τ, τ ) = h∂α q(τ )q(τ )i

= ∆α (τ − τ ′ ) =

Z

The configuration space is still one-dimensional so that the indices µ, ν and the corresponding tensors in Eqs. (10.205), (10.207), and (10.211) are absent. The analytic continuation to d = 1 − ε time dimensions is most easily performed if the Feynman diagrams are calculated in momentum space. The three types of lines represent the analytic expressions =

1 , p2 + ω 2

=i

pα , p2 + ω 2

=

pα pβ . p2 + ω 2

(10.236)

Most diagrams in the last section converge in one-dimensional momentum space, thus requiring no regularization to make them finite, as we would expect for a quantum-mechanical system. Trouble arises, however, in some multiple momentum integrals, which yield different results depending on the order of their evaluation. As a typical example, take the Feynman integral Z

= − dd τ1 ∆(τ1 − τ2 )∆α (τ1 − τ2 )∆β (τ1 − τ2 )∆αβ (τ1 − τ2 ).

(10.237)

For the ordinary one-dimensional Euclidean time, a Fourier transformation yields the triple momentum space integral X=

Z

dk dp1 dp2 k 2 (p1 p2 ) . (10.238) 2π 2π 2π (k 2 + ω 2 )(p21 + ω 2)(p22 + ω 2)[(k + p1 + p2 )2 + ω 2 ] H. Kleinert, PATH INTEGRALS

731

10.8 Calculating Loop Diagrams

Integrating this first over k, then over p1 and p2 yields 1/32ω. In the order first p1 , then p2 and k, we find −3/32ω, whereas the order first p1 , then k and p2 , gives again 1/32ω. As we shall see below in Eq. (10.283), the correct result is 1/32ω. The unique correct evaluation will be possible by extending the momentum space to d dimensions and taking the limit d → 1 at the end. The way in which the ambiguity will be resolved may be illustrated by a typical Feynman integral Yd =

Z

d d k d d p1 d d p2 k 2 (p1 p2 ) − (kp1 )(kp2 ) ,(10.239) (2π)d (2π)d (2π)d (k 2 + ω 2 )(p21 + ω 2 )(p22 + ω 2 )[(k + p1 + p2 )2 + ω 2]

whose numerator vanished trivially in d = 1 dimensions. Due to the different contractions in d dimensions, however, Y0 will be seen to have the nonzero value Y0 = 1/32ω − (−1/32ω) in the limit d → 1, the result being split according to the two terms in the numerator [to appear in the Feynman integrals (10.281) and (10.283); see also Eq. (10.354)]. The diagrams which need a careful treatment are easily recognized in configuration space, where the one-dimensional correlation function (10.233) is the continuous function (10.206). Its first derivative (10.207) which has a jump at equal arguments is a rather unproblematic distribution, as long as the remaining integrand does not contain δ-functions or their derivatives. These appear with second derivatives of ∆(τ, τ ′ ), where the d-dimensional evaluation must be invoked to obtain a unique result.

10.8

Calculating Loop Diagrams

The loop integrals encountered in d dimensions are based on the basic one-loop integral I≡

Z

d¯d k ω d−2 1 , = Γ (1 − d/2) = 2 2 d/2 d=1 2ω k +ω (4π)

(10.240)

where we have abbreviated d¯d k ≡ dd k/(2π)d by analogy with h ¯ ≡ h/2π. The integral exists only for ω 6= 0 since it is otherwise divergent at small k. Such a divergence is called infrared divergence (IR-divergence) and ω plays the role of an infrared (IR) cutoff. By differentiation with respect to ω 2 we can easily generalize (10.240) to Iαβ

≡

Z

d¯d k (k 2 )β ω d+2β−2α Γ (d/2 + β) Γ (α − β − d/2) = . (k 2 + ω 2 )α (4π)d/2 Γ (d/2) Γ(α)

(10.241)

Note that for consistency of dimensional regularization, all integrals over a pure power of the momentum must vanish: I0β =

Z

d¯d k (k 2 )β = 0.

We recognize Veltman’s rule of Eq. (2.500).

(10.242)

732


With the help of Eqs. (10.240) and (10.241) we calculate immediately the local expectation values (10.233) and (10.235) and thus the local diagrams in (10.226) and (10.227): =

D

q

=

D

q2

E2

=

D

q2

E3

=

D

2

q

2

E

E

Z

= =

Z

=

h∂q ∂qi =

d¯d k 1 = , 2 2 d=1 k +ω 2ω !2 Z d¯d k 1 = , 2 2 d=1 4ω 2 k +ω

Z

d¯d k k 2 +ω 2

= hqqi h∂q∂qi =

Z

d¯d k k 2 +ω 2

Z

2

d¯d k k 2 +ω 2

!3

(10.243) (10.244)

1 , d=1 8ω 3

(10.245)

d¯d p p2 1 = − , 2 2 p +ω d=1 4

(10.246)

!2Z

=

d¯d p p2 1 = − . p2 +ω 2 d=1 8ω

(10.247)

The two-bubble diagrams in Eq. (10.229) can also be easily computed d¯d p 1 = , (p2 +ω 2)2 d=1 4ω 3 Z Z d¯d k Z d¯d p p2 1 d 2 = d τ1 ∆(τ1 − τ1 )∆α (τ1 − τ2 ) = , 2 2 2 2 2 k +ω (p +ω ) d=1 8ω 2 Z Z Z 1 d¯d k k 2 d¯d p = dd τ1 ∆αα (τ1 − τ1 )∆2 (τ1 − τ2 ) = − 2, 2 2 2 2 2 k +ω (p + ω ) d=1 8ω Z Z Z d d¯ k d¯d p 1 = dd τ1 ∆(τ1 − τ1 )∆2 (τ1 − τ2 ) = . k2 + ω2 (p2 + ω 2)2 d=1 8ω 4 =

Z

Z

dd τ1 ∆2 (τ1 − τ2 )

(10.248) (10.249) (10.250) (10.251)

For the three-bubble diagrams in Eq. (10.230) we find =

Z

Z

= =

dd τ1 ∆(τ1 − τ1 )∆2αα (τ1 − τ2 )∆(τ2 − τ2 )

Z

=

"Z

=

Z

=

Z

=

Z

d¯d q q 2 +ω 2

!2 Z

3 d¯d p(p2 )2 = − , 2 2 2 (p +ω ) d=1 16ω

(10.252)

dd τ1 ∆αα (τ1 − τ1 )∆2 (τ1 − τ2 )∆ββ (τ2 − τ2 ) d¯d q (q 2 )2 q 2 +ω 2

#2 Z

d¯d k 1 = , 2 2 2 d=1 (k +ω ) 16ω

(10.253)

dd τ1 ∆(τ1 − τ1 )∆2α (τ1 − τ2 )∆ββ (τ2 − τ2 ) d¯d k k2 + ω2

Z

d¯d p p2 (p2 + ω 2)2

Z

d¯d q q 2 1 = − , 2 2 d=1 q +ω 16ω

(10.254)

dd τ1 ∆αα (τ1 − τ1 )∆2 (τ1 − τ2 )∆(τ2 − τ2 )


733

10.8 Calculating Loop Diagrams Z

=

Z

=

Z

=

Z

=

Z

=

Z d¯d k k 2 Z d¯d p d¯d q 1 = − , 2 2 2 2 2 2 2 k +ω (p +ω ) q +ω d=1 16ω 3

(10.255)

dd τ1 ∆(τ1 − τ1 )∆2α (τ1 − τ2 )∆(τ2 − τ2 ) d¯d k k 2 +ω 2

d¯d p p2 (p2 +ω 2)2

Z

Z

d¯d q 1 = , q 2 +ω 2 d=1 16ω 3

(10.256)

dd τ1 ∆(τ1 − τ1 )∆2 (τ1 − τ2 )∆(τ2 − τ2 ) d¯d k k2 + ω2

Z

d¯d p (p2 + ω 2 )2

d¯d q 1 = . 2 2 q + ω d=1 16ω 5

Z

(10.257)

In these diagrams, it does not make any difference if we replace ∆2αα by ∆2αβ . We now turn to the watermelon-like diagrams in Eq. (10.231) which are more tedious to calculate. They require a further basic integral [26]: 2

J(p ) =

Z

d¯d k = (k 2 + ω 2 )[(k + p)2 + ω 2 ] p2 + 4ω 2 4

Γ (2 − d/2) = (4π)d/2

!d/2−2

Z

1 0

dx

Z

d¯d k [k 2 + p2 x(1 − x) + ω 2 ]2

d 1 3 p2 F 2− , ; ; 2 , 2 2 2 p + 4ω 2 !

(10.258)

where F (a, b; c; z) is the hypergeometric function (1.450). For d = 1, the result is simply J(p2 ) =

ω(p2

1 . + 4ω 2 )

(10.259)

We also define the more general integrals Jα1 ...αn (p) =

Z

d¯d k kα1 · · · kαn , (k 2 + ω 2 )[(k + p)2 + ω 2 ]

(10.260)

and further Jα1 ...αn ,β1 ...βm (p) =

Z

d¯d k kα1 · · · kαn (k + p)β1 · · · (k + p)βm . (k 2 + ω 2 )[(k + p)2 + ω 2]

(10.261)

The latter are linear combinations of momenta and the former, for instance Jα,β (p) = Jα (p) pβ + Jαβ (p).

(10.262)

Using Veltman’s rule (10.242), all integrals (10.261) can be reduced to combinations of p, I, J(p2 ). Relevant examples for our discussion are Jα (p) =

Z

d¯d k kα 1 = − pα J(p2 ), (k 2 + ω 2 )[(k + p)2 + ω 2 ] 2

(10.263)

734


and Z

Jαβ (p) =

d¯d k kα kβ pα pβ I δ + (d − 2) = αβ (k 2 + ω 2 )[(k + p)2 + ω 2 ] p2 2(d − 1) " # pα pβ 2 J(p2 ) 2 2 2 + −δαβ (p + 4ω ) + 2 d p + 4ω , (10.264) p 4(d − 1) "

#

whose trace is Jαα (p) =

Z

d¯d k k 2 = I − ω 2 J(p2 ). 2 2 2 2 (k + ω )[(k + p) + ω ]

(10.265)

Similarly we expand Jααβ (p) =

Z

1 d¯d k k 2 kβ = pβ [−I + ω 2 J(p2 )]. 2 2 2 2 (k + ω )[(k + p) + ω ] 2

(10.266)

The integrals appear in the following subdiagrams = J(p2 ),

= J,α (p),

= J,αβ (p),

= Jα (p),

= Jα,β (p),

= Jα,βγ (p),

= Jαβ (p),

= Jαβ,γ (p),

= Jαβ,γδ (p).

(10.267)

All two- and three-loop integrals needed for the calculation can be brought to the generic form K(a, b) =

Z

d¯d p (p2 )a J b (p2 ), a ≥ 0, b ≥ 1, a ≤ b,

(10.268)

and evaluated recursively as follows [27]. From the Feynman parametrization of the first line of Eq. (10.258) we observe that the two basic integrals (10.240) and (10.258) satisfy the differential equation J(p2 ) = −

2 ∂I 1 2 ∂J(p2 ) 2 ∂J(p ) + p − 2p . ∂ω 2 2 ∂ω 2 ∂p2

(10.269)

Differentiating K(a + 1, b) from Eq. (10.268) with respect to ω 2 , and using Eq. (10.269), we find the recursion relation 2b(d/2 − 1) I K(a − 1, b − 1) − 2ω 2 (2a − 2 − b + d)K(a − 1, b) K(a, b) = , (10.270) (b + 1)d/2 − 2b + a which may be solved for increasing a starting with K(0, 0) = 0, K(0, 2) =

Z

K(0, 1) =

Z

d¯d p J(p2 ) = I 2 ,

d¯d p J 2 (p2 ) = A, . . . ,


735


where A is the integral Z

A≡

d¯d p 1 d¯d p2 d¯d k . (p21 +ω 2)(p22 +ω 2)(k 2 +ω 2)[(p1 +p2 +k)2 +ω 2]

(10.272)

This integral will be needed only in d = 1 dimensions where it can be calculated directly from the configuration space version of this integral. For this we observe that the first watermelon-like diagram in (10.231) corresponds to an integral over the product of two diagrams J(p2 ) in (10.267): =

Z

Z

d

d τ1 ∆(τ1 − τ2 )∆(τ1 − τ2 )∆(τ1 − τ2 )∆(τ1 − τ2 ) = d¯d k J 2 (k) = A.(10.273)

Thus we find A in d = 1 dimensions from the simple τ -integral A=

Z

∞

−∞

4

dτ ∆ (τ, 0) =

Z

∞ −∞

dx

1 −ω|x| e 2ω

4

=

1 . 32ω 5

(10.274)

Since this configuration space integral contains no δ-functions, the calculation in d = 1 dimension is without subtlety. With the help of Eqs. (10.270), (10.271), and Veltman’s rule (10.242), according to which K(a, 0) ≡ 0,

(10.275)

we find further the integrals Z

d¯d p p2 J(p2 ) = K(1, 1) = −2ω 2 I 2 ,

4 d¯d p p2 J 2 (p2 ) = K(1, 2) = (I 3 − ω 2 A), 3 Z (6 − 5d)I 3 + 2dω 2A . d¯d p (p2 )2 J 2 (p2 ) = K(2, 2) = −8ω 2 3(4 − 3d) Z

(10.276) (10.277) (10.278)

We are thus prepared to calculate all remaining three-loop contributions from the watermelon-like diagrams in Eq. (10.231). The second is an integral over the product of subdiagrams Jαβ in (10.267) and yields =

Z

dd τ1 ∆2 (τ1 − τ2 )∆2αβ (τ1 − τ2 )

(pk)2 (p2 + ω 2 )(k 2 + ω 2 )(q 2 + ω 2 )[(p + k + q)2 + ω 2 ] Z Z 1 d = d¯ q Jαβ (q)Jαβ (q) = d¯d k (k 2 )2 J 2 (k) q→q−p 16 Z h i 1 1 2 d 2 2 2 2 2 2 + d¯ k dI + (d − 2)k −4ω I J(k)+ (k +4ω ) J (k) 4(d−1) 4 2 3 2 2 h i ω (6 − 5d)I + 2dω A ω =− − (6 − 5d)I 3 + 2dω 2A (10.279) 2 3(4 − 3d) 6(4 − 3d) h i ω2 ω2 3 =− (8 − 7d)I 3 + (d + 4)ω 2A = − (I 3 + 5ω 2A) = − . d=1 3(4 − 3d) 3 32ω =

Z

d¯d p d¯d k d¯d q

736


The third diagram contains two mixed lines. It is an integral over a product of the diagrams Jα (p) and Jβ,αβ in (10.267) and gives Z

= − dd τ1 ∆(τ1 − τ2 )∆α (τ1 − τ2 )∆β (τ1 − τ2 )∆αβ (τ1 − τ2 ) =

Z

d¯d k d¯d p1 d¯d p2

=

p2 →p2 −k

Z

1 =− 8

=−

(kp1 )(kp2 ) 2 2 2 2 (k +ω )(p1 +ω )(p22 +ω 2 ) [(k+p1 +p2 )2 +ω 2]

d¯d p [pβ Jα (p)Jαβ (p) + Jα (p)Jβαβ (p)] Z

h

d¯d p p2 J(p2 ) (p2 + 2ω 2 )J(p2 ) − 2I

i

(10.280)

h i ω2 1 ω2 (8 − 5d) I 3 − 2(4 − d)ω 2A = − (I 3 − 2ω 2 A) = − . d=1 6(4 − 3d) 2 32ω

The fourth diagram contains four mixed lines and is evaluated as follows: = −

Z

dd τ1 ∆α (τ1 − τ2 )∆α (τ1 − τ2 )∆β (τ1 − τ2 )∆β (τ1 − τ2 ). (10.281)

Since the integrand is regular and vanishes at infinity, we can perform a partial integration and rewrite the configuration space integral as =

Z

+ 2

dd τ1 ∆(τ1 − τ2 )∆αα (τ1 − τ2 )∆β (τ1 − τ2 )∆β (τ1 − τ2 ) Z

dd τ1 ∆(τ1 − τ2 )∆α (τ1 − τ2 )∆β (τ1 − τ2 )∆αβ (τ1 − τ2 ). (10.282)

The second integral has just been evaluated in (10.281). The first is precisely the integral Eq. (10.238) discussed above. It is calculated as follows: Z

dd τ1 ∆(τ1 − τ2 )∆αα (τ1 − τ2 )∆β (τ1 − τ2 )∆β (τ1 − τ2 )

k 2 (p1 p2 ) (k 2 +ω 2)(p21 +ω 2)(p22 +ω 2)[(k+p1 +p2 )2 +ω 2 ] Z ω2 Z d 2 2 2 d = d¯ p [pα Jα (p)Jββ + Jα (p)Jβαβ (p)] = d¯ p p J (p ) 4 ω2 1 = − (I 3 − ω 2 A) = . (10.283) d=1 32ω 3 =

Z

d¯d k d¯d p1 d¯d p2

Hence we obtain =

1 . 32ω

(10.284)

The fifth diagram in (10.231) is an integral of the product of two subdiagrams Jα (p) in (10.267) and yields =

Z

dd τ1 ∆(τ1 − τ2 )∆2α (τ1 − τ2 )∆(τ1 − τ2 )


737

10.8 Calculating Loop Diagrams Z

= − d¯d k d¯d p1 d¯d p2 =

k→k−p2

p1 p2 2 2 2 2 2 (k +ω )(p1 +ω )(p2 +ω 2) [(k+p1 +p2 )2 +ω 2]

Z

− d¯d k d¯d p1 d¯d p2

p1 p2 2 2 2 [(k−p2 ) +ω ] (p1 +ω 2)(p22 +ω 2) [(k+p1)2 +ω 2]

p1α d¯ k d¯ p1 2 = 2 p2 →−p2 (p1 +ω ) [(p1 +k)2 +ω 2 ] Z Z 1 = d¯d k Jα2 (k) = d¯d k k 2 J 2 (k 2 ) 4 14 3 1 . = (I − ω 2 A) = d=1 32ω 3 43 Z

d

Z

d

d¯d p2

p2α 2 2 (p2 +ω ) [(p2 +k)2 +ω 2]

(10.285)

We can now sum up all contributions to the free energy in Eqs. (10.226)–(10.231). An immediate simplification arises from the Veltman’s rule (10.242). This implies that all δ-functions at the origin are zero in dimensional regularization: (d)

δ (0) =

Z

dd k = 0. (2π)d

(10.286)

The first-order contribution (10.226) to the free energy is obviously zero by Eqs. (10.244) and (10.246). (1) The first second-order contribution βF2 becomes, from (10.245) and (10.247): (1) F2

=η

2

1 +a 3 2

1 1 a + − + 15ω 2 8ω 18 5

=−

η2 . 12ω

(10.287)

The parameter a has disappeared from this equation. (2) The second second-order contribution βF2 vanishes trivially, by Veltman’s rule (10.286). (3) The third second-order contribution βF2 in (10.230) vanishes nontrivially using (10.252)–(10.257): (3)

F2

= −

η2 2!

1 3 1 +2 − +2 16ω 16ω 16ω 1 1 1 2 2 4 + 8ω − + 8ω + 8ω = 0. (10.288) 16ω 3 16ω 3 16ω 5 4

−

The fourth second-order contribution, finally, associated with the watermelonlike diagrams in (10.231) yield via (10.280), (10.281), (10.284), (10.285), and (10.273): (4)

βF2

=−

η2 3 1 1 1 2 4 1 − +4 − + + 4ω 2 + ω 2! 32ω 32ω 32ω 32ω 3 3 32ω 5

η2 , 12ω (10.289)

=

canceling (10.287), and thus the entire free energy. This proves the invariance of the perturbatively defined path integral under coordinate transformations.

738

10.8.1


Reformulation in Configuration Space

The Feynman integrals in momentum space in the last section corresponds in τ -space to integrals over products of distributions. For many applications it is desirable to do the calculations directly in τ -space. This will lead to an extension of distribution theory which allows us to do precisely that. In dimensional regularization, an important simplification came from Veltman’s rule (10.286), according to which the delta function at the origin vanishes. In the more general calculations to come, we shall encounter generalized δ-functions, which are multiple derivatives of the ordinary δ-function: Z (d) δα(d) (τ ) ≡ ∂ δ (τ ) = d¯d k(ik)α1 . . . (ik)αn eikx , (10.290) α1 ...αn 1 ...αn with ∂α1 ...αn ≡ ∂α1 . . . ∂αn and d¯d k ≡ dd k (2π)d . By Veltman’s rule (10.242), all these vanish at the origin: Z (0) = d¯d k(ik)α1 . . . (ik)αn = 0. δα(d) (10.291) 1 ...αn In the extended coordinate space, the correlation function ∆(τ, τ ′ ) in (10.233), which we shall also write as ∆(τ − τ ′ ), is at equal times given by the integral [compare (10.240)] ∆(0) =

Z

ω d−2 d 1 d¯d k = Γ 1 − =I = . d=1 2ω k2 + ω2 2 (4π)d/2

The extension (10.234) of the time derivative (10.207), Z ikα ∆α (τ ) = d¯d k 2 eikτ k + ω2

(10.292)

(10.293)

vanishes at equal times, just like (10.210): ∆α (0) = 0.

(10.294)

This follows directly from a Taylor series expansion of 1/(k 2 + ω 2 ) in powers of k 2 , after imposing (10.291). The second derivative of ∆(τ ) has the Fourier representation (10.235). Contracting the indices yields Z k2 ∆αα (τ ) = − d¯d k 2 eikx = −δ (d) (τ ) + ω 2 ∆(τ ) . (10.295) k + ω2 This equation is a direct consequence of the definition of the correlation function as a solution to the inhomogeneous field equation (−∂α2 + ω 2 )q(τ ) = δ (d) (τ ).

(10.296)

Inserting Veltman’s rule (10.286) into (10.295), we obtain ω . d=1 2

∆αα (0) = ω 2 ∆(0) =

(10.297)

This ensures the vanishing of the first-order contribution (10.226) to the free energy F1 = −g η −∆αα (0) + ω 2 ∆(0) ∆(0) = 0. (10.298) H. Kleinert, PATH INTEGRALS

739


The same equation (10.295) allows us to calculate immediately the second-order contribution (10.227) from the local diagrams 1 1 a (1) F2 = −η 2 3g 2 + a ∆αα (0) − 5 + ω 2 ∆(0) ∆2 (0) 2 18 5 2 2 η . (10.299) = −η 2 ω 2 ∆3 (0) = − d=1 3 12ω The other contributions to the free energy in the expansion (10.225) require rules for calculating products of two and four distributions, which we are now going to develop.

10.8.2

Integrals over Products of Two Distributions

The simplest integrals are Z

δ (d) (k + p) (p2 + ω 2 )(k 2 + ω 2 ) ! Z d¯d k ω d−4 d (2 − d) = Γ 2− = ∆(0), = 2 2 2 d/2 (k + ω ) (4π) 2 2ω 2

dd τ ∆2 (τ ) =

Z

d¯d p d¯d k

(10.300)

and Z

d

d τ

∆2α (τ )

= − =

Z

h

d

d τ ∆(τ ) −δ

d ∆(0). 2

(d)

i

2

(τ ) + ω ∆(τ ) = ∆(0) − ω

2

Z

dd τ ∆2 (τ ) (10.301)

To obtain the second result we have performed a partial integration and used (10.295). In contrast to (10.300) and (10.301), the integral Z

(kp)2 δ (d) (k + p) (k 2 + ω 2)(p2 + ω 2 ) Z Z (k 2 )2 = d¯d k 2 = dd τ ∆2αα (τ ) (k + ω 2)2 Z

dd τ ∆2αβ (τ ) =

d¯d p d¯d k

(10.302)

diverges formally in d = 1 dimension. In dimensional regularization, however, we may decompose (k 2 )2 = (k 2 + ω 2 )2 − 2ω 2 (k 2 + ω 2 ) + ω 4 , and use (10.291) to evaluate Z

d

d τ

∆2αα (τ )

=

Z

(k 2 )2 d¯ k 2 = −2ω 2 2 2 (k + ω ) d

2

= −2 ω ∆(0)+ ω

4

Z

Z

d¯d k d¯d k 4 +ω (k 2 + ω 2 ) (k 2 + ω 2)2 Z

dd τ ∆2 (τ ).

(10.303)

Together with (10.300), we obtain the relation between integrals of products of two distributions Z

d

d τ

∆2αβ (τ )

=

Z

d

d τ

∆2αα (τ )

2

= −2ω ∆(0) + ω

= − (1 + d/2) ω 2 ∆(0) .

4

Z

dd τ ∆2 (τ ) (10.304)

740


An alternative way of deriving the equality (10.302) is to use partial integrations and the identity ∂α ∆αβ (τ ) = ∂β ∆αα (τ ),

(10.305)

which follows directly from the Fourier representation (10.293). Finally, from Eqs. (10.300), (10.301), and (10.304), we observe the useful identity Z

h

i

dd τ ∆2αβ (τ ) + 2ω 2 ∆2α (τ ) + ω 4 ∆2 (τ ) = 0 ,

(10.306)

which together with the inhomogeneous field equation (10.295) reduces the calculation of the second-order contribution of all three-bubble diagrams (10.230) to zero: (3)

F2

10.8.3

= −g 2 ∆2 (0)

Z

h

i

dd τ ∆2αβ (τ ) + 2ω 2 ∆2α (τ ) + ω 4 ∆2 (τ ) = 0 .

(10.307)

Integrals over Products of Four Distributions

Consider now the more delicate integrals arising from watermelon-like diagrams in (10.231) which contain products of four distributions, a nontrivial tensorial structure, and overlapping divergences. We start from the second to fourth diagrams: =

Z

dd τ ∆2 (τ )∆2αβ (τ ),

(10.308)

Z

= 4 dd τ ∆(τ )∆α (τ )∆β (τ )∆αβ (τ ),

4

=

Z

dd τ ∆α (τ )∆α (τ )∆β (τ )∆β (τ ).

(10.309) (10.310)

To isolate the subtleties with the tensorial structure exhibited in Eq. (10.239), we introduce the integral Yd =

Z

h

i

dd τ ∆2 (τ ) ∆2αβ (τ ) − ∆2αα (τ ) .

(10.311)

In d = 1 dimension, the bracket vanishes formally, but the limit d → 1 of the integral is nevertheless finite. We now decompose the Feynman diagram (10.308), into the sum Z

dd τ ∆2 (τ )∆2αβ (τ ) =

Z

dd τ ∆2 (τ )∆2αα (τ ) + Yd .

(10.312)

To obtain an analogous decomposition for the other two diagrams (10.309) and (10.310), we derive a few useful relations using the inhomogeneous field equation (10.295), partial integrations, and Veltman’s rules (10.286) or (10.291). From the inhomogeneous field equation, there is the relation −

Z

dd τ ∆αα (τ )∆3 (τ ) = ∆3 (0) − ω 2

Z

dd τ ∆4 (τ ).

(10.313)


741


By a partial integration, the left-hand side becomes Z

dd τ ∆αα (τ )∆3 (τ ) = −3

Z

dd τ ∆2α (τ )∆2 (τ ),

(10.314)

leading to Z

d

d τ

∆2α (τ )∆2 (τ )

1 1 = ∆3 (0) − ω 2 3 3

Z

dd τ ∆4 (τ ).

(10.315)

Invoking once more the inhomogeneous field equation (10.295) and Veltman’s rule (10.286), we obtain the integrals Z

d

d τ

∆2αα (τ )∆2 (τ )

−ω

4

Z

dd τ ∆4 (τ ) + 2ω 2∆3 (0) = 0,

(10.316)

and Z

dd τ ∆αα (τ )∆2β (τ )∆(τ ) = ω 2

Z

dd τ ∆2β (τ )∆2 (τ ).

(10.317)

Using (10.315), the integral (10.317) takes the form Z

d

d τ

1 2 3 1 = ω ∆ (0) − ω 4 3 3

∆αα (τ )∆2β (τ )∆(τ )

Z

dd τ ∆4 (τ ).

(10.318)

Partial integration, together with Eqs. (10.316) and (10.318), leads to Z

dd τ ∂β ∆αα (τ )∆β (τ )∆2 (τ ) = − =

Z

dd τ ∆2αα (τ )∆2 (τ ) − 2

4 2 3 1 ω ∆ (0) − ω 4 3 3

Z

Z

dd τ ∆αα (τ )∆2β (τ )∆(τ )

dd τ ∆4 (τ ).

(10.319)

A further partial integration, and use of Eqs. (10.305), (10.317), and (10.319) produces the decompositions of the second and third Feynman diagrams (10.309) and (10.310): 4

Z

d

d τ ∆(τ )∆α (τ )∆β (τ )∆αβ (τ ) = 4ω

2

Z

dd τ ∆2 (τ )∆2α (τ ) − 2 Yd ,

(10.320)

and Z

d

d τ

∆2α (τ )∆2β (τ )

= −3ω

2

Z

dd τ ∆2 (τ )∆2α (τ ) + Yd .

(10.321)

We now make the important observation that the subtle integral Yd of Eq. (10.311) appears in Eqs. (10.312), (10.320), and (10.321) in such a way that it drops out from the sum of the watermelon-like diagrams in (10.231): +4

+

=

Z

dd τ ∆2 (τ )∆2αα (τ ) + ω 2

Z

dd τ ∆2 (τ )∆2α (τ ).

(10.322)

Using now the relations (10.315) and (10.316), the right-hand side becomes a sum of completely regular integrals involving only products of propagators ∆(τ ).

742


We now add to this sum the first and last watermelon-like diagrams in Eq. (10.231) 2 4 ω 3

2 Z = ω 4 dd τ ∆4 (τ ), 3

(10.323)

and 4ω

2

= 4ω

2

Z

dd τ ∆2 (τ )∆2α (τ ),

(10.324)

and obtain for the total contribution of all watermelon-like diagrams in (10.231) the simple expression for η = 1: (4)

F2

= −2η 2 g 2

Z

dd τ ∆2 (τ )

2 4 2 ω ∆ (τ ) + ∆2αα (τ ) + 5ω 2 ∆2α (τ ) 3

2 η2 = η 2 ω 2 ∆3 (0) = . d=1 12ω 3

(10.325)

This cancels the finite contribution (10.299), thus making also the second-order free energy in (10.221) vanish, and confirming the invariance of the perturbatively defined path integral under coordinate transformations up to this order. Thus we have been able to relate all diagrams involving singular time derivatives of correlation functions to integrals over products of the regular correlation function (10.233), where they can be replaced directly by their d = 1 -version (10.206). The disappearance of the ambiguous integral Yd in the combination of watermelon-like diagrams (10.322) has the pleasant consequence that ultimately all calculations can be done in d = 1 dimensions after all. This leads us to expect that the dimensional regularization may be made superfluous by a more comfortable calculation procedure. This is indeed so and the rules will be developed in Section 10.11. Before we come to this it is useful, however, to point out a pure x-space way of finding the previous results.

10.9

Distributions as Limits of Bessel Function

In dimensional regularization it is, of course, possible to perform the above configuration space integrals over products of distributions without any reference to momentum space integrals. For this we express all distributions explicitly in terms of modified Bessel functions Kα (y).

10.9.1

Correlation Function and Derivatives

The basic correlation function in d-dimension is obtained from the integral in Eq. (10.233), as ∆(τ ) = cd y 1−d/2 K1−d/2 (y),

(10.326)

where y ≡ m |τ | is reduced length of τα , with the usual Euclidean norm |x| = K1−d/2 (y) is the modified Bessel function. The constant factor in front is cd =

ω d−2 . (2π)d/2

p τ12 + . . . + τd2 , and (10.327)


743

10.9 Distributions as Limits of Bessel Function

In one dimension, the correlation function (10.326) reduces to (10.201). The short-distance properties of the correlation functions is governed by the small-y behavior of Bessel function at origin4 Kβ (y) ≈

y≈0

1 Γ(β)(y/2)∓β , 2

Re β

>
0, such that the antisymmetry ∆α (−x) = −∆α (τ ) makes ∆α (0) = 0, as observed after Eq. (10.293). Explicitly, the small-τ behavior of the correlation function and its derivative is ∆(τ ) ∝ const.,

∆α (τ ) ∝ |τ |ε ∂α |τ |.

(10.330)

In contrast to these two correlation functions, the second derivative ∆αβ (τ ) = ∆(τ ) (∂α y)(∂β y) +

cd y d/2 Kd/2 (y) ∂αβ y 2−d , (d − 2)

(10.331)

is singular at short distance. The singularity comes from the second term in (10.331): ∂αβ y

2−d

ω 2−d = (2 − d) |y|d

yα yβ δαβ − d 2 , y

(10.332)

which is a distribution that is ambiguous at origin, and defined up to the addition of a δ (d) (τ )function. It is regularized in the same way as the divergence in the Fourier representation (10.291). Contracting the indices α and β in Eq. (10.332), we obtain ∂ 2 y 2−d = (2 − d)ω 2−d Sd δ (d) (τ ),

(10.333)

where Sd = 2π d/2 /Γ(d/2) is the surface of a unit sphere in d dimensions [recall Eq. (1.555)]. As a check, we take the trace of ∆αβ (τ ) in Eq. (10.331), and reproduce the inhomogeneous field equation (10.295): ∆αα (τ ) = ω 2 ∆(τ )−cd m2−d Sd

1 Γ (d/2) 2d/2 δ (d) (τ ) 2

= ω 2 ∆(τ ) − δ (d) (τ ).

(10.334)

Since δ (d) (τ ) vanishes at the origin by (10.291), we find once more Eq. (10.297). A further relation between distributions is found from the derivative h i ∂α ∆αβ (τ ) = ∂β −δ (d) (τ ) + ω 2 ∆(τ ) + ωSd ∆(τ )|y|d−1 (∂β y) δ (d) (τ ) = ∂β ∆λλ (τ ).

(10.335)

4


744


10.9.2

Integrals over Products of Two Distributions

Consider now the integrals over products of such distributions. If an integrand f (|x|) depends only on |x|, we may perform the integrals over the directions of the vectors Z Z ∞ dd τ f (τ ) = Sd dr rd−1 f (r), r ≡ |x|. (10.336) 0

Using the integral formula5 Z

0

∞

dy y Kβ2 (y) =

we can calculate directly: Z Z dd τ ∆2 (τ ) = ω −d c2d Sd

1 πβ 1 = Γ(1 + β)Γ(1 − β), 2 sin πβ 2

∞ 0

(10.337)

2 dy y K1−d/2 (y)

1 2−d = ω −d c2d Sd (1−d/2) Γ(1−d/2) Γ(d/2) = ∆(0), 2 2ω 2

(10.338)

and Z

dd τ ∆2α (τ )

=

ω 2−d c2d Sd

=

ω 2−d c2d Sd

Z

0

∞

2 dy yKd/2 (y)

d 1 Γ (1 + d/2) Γ (1 − d/2) = ∆(0), 2 2

(10.339)

in agreement with Eqs. (10.300) and (10.301). Inserting ∆(0) = 1/2ω from (10.292), these integrals give us once more the values of the Feynman diagrams (10.248), (10.251), (10.252), (10.255), and (10.257). Note that due to the relation6 i d h 1−d/2 Kd/2 (y) = −y d/2−1 y K1−d/2 (y) , (10.340) dy the integral over y in Eq. (10.339) can also be performed by parts, yielding Z Z ∞ d 2 2−d 2 d/2 1−d/2 2 d τ ∆α (τ ) = −ω cd Sd y Kd/2 y K1−d/2 − ω dd τ ∆2 (τ ) 0 Z 2 = ∆(0) − ω dd τ ∆2 (τ ). (10.341) The upper limit on the right-hand side gives zero because of the exponentially fast decrease of the Bessel function at infinity. This was obtained before in Eq. (10.301) from a partial integration and the inhomogeneous field equation (10.295). Using the explicit representations (10.326) and (10.331), we calculate similarly the integral Z Z Z dd τ ∆2αα (τ ) = dd τ ∆2αβ (τ ) = ω 4 dd τ ∆2 (τ ) − ω 4−d c2d Γ (d/2) Γ (1−d/2) Sd Z 4 = ω dd τ ∆2 (τ ) − 2ω 2 ∆(0) = − (1 + d/2) ω 2 ∆(0). (10.342) 5 6

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 6.521.3 ibid., Formulas 8.485 and 8.486.12 H. Kleinert, PATH INTEGRALS

10.9 Distributions as Limits of Bessel Function

745

The first equality follows from partial integrations. In the last equality we have used (10.338). We have omitted the integral containing the modified Bessel functions Z ∞ Z d ∞ −1 2 (d − 1) dzKd/2 (z)K1−d/2 (z) + dz z Kd/2 (z) , (10.343) 2 0 0 since this vanishes in one dimension as follows: π − Γ (1 − ε/2) [Γ (ε/2) + Γ (−ε/2)] ε2 Γ(ε) = 0. ε→0 4 Inserting into (10.342) ∆(0) = 1/2ω from (10.292), we find once more the value of the right-hand Feynman integral (10.249) and the middle one in (10.252). By combining the result (10.302) with (10.338) and (10.339), we can derive by proper integrations the fundamental rule in this generalized distribution calculus that the integral over the square of the δ-function vanishes. Indeed, solving the inhomogeneous field equation (10.295) for δ (d) (τ ), and squaring it, we obtain Z Z Z Z h i2 dd τ δ (d) (τ ) = ω 4 dd τ ∆2 (τ ) + 2ω 2 dd τ ∆2α (τ ) + dd τ ∆2ββ (τ ) = 0. (10.344) Thus we may formally calculate Z dd τ δ (d) (τ )δ (d) (τ ) = δ (d) (0) = 0,

(10.345)

pretending that one of the two δ-functions is an admissible smooth test function f (τ ) of ordinary distribution theory, where Z dd τ δ (d) (τ )f (τ ) = f (0). (10.346)

10.9.3

Integrals over Products of Four Distributions

The calculation of the configuration space integrals over products of four distributions in d = 1 dimension is straightforward as long as they are unique. Only if they are ambiguous, they require a calculation in d = 1 − ε dimension, with the limit ε → 0 taken at the end. A unique case is Z Z ∞ 4 dd τ ∆4 (τ ) = c4d ω −d Sd dy y 3−d K1−d/2 (y) 0 π2 3 d 1 = c41 ω −1 S1 4 Γ4 − Γ(d) = , (10.347) d=1 2 2 2 32ω 5 where we have set y ≡ ωτ. Similarly, we derive by partial integration Z Z dd τ ∆2 (τ )∆2α (τ ) = ω 2−d c4d Sd

0

= + =

∞

(10.348)

2 2 dyy 3−d Kd/2 (y)K1−d/2 (y)

1 2−d 4 ω cd Sd 2−d−1 Γ(d/2) Γ3 (1−d/2) 3 Z ∞ 3 d 1−d/2 d/2 dy y K1−d/2 y Kd/2 dy 0 Z 1 1 ∆3 (0) − ω 2 dd τ ∆4 (τ ) = . d=1 3 32ω

(10.349)

746


Using (10.326), (10.329), and (10.331), we find for the integral in d = 1 − ε dimensions Z Z 1 dd τ ∆(τ )∆α (τ )∆β (τ )∆αβ (τ ) = ω 2 dd τ ∆2 (τ )∆2α (τ ) − Yd , 2

(10.350)

where Yd is the integral Yd

=

−2(d − 1)ω 4−d c4d Sd

Z

0

∞

3 dyy 2−d K1−d/2 (y)Kd/2 (y).

(10.351)

In spite of the prefactor d − 1, this has a nontrivial limit for d → 1, the zero being compensated by a pole from the small-y part of the integral at y = 0. In order to see this we use the integral representation of the Bessel function [28]: Z ∞ 1 Kβ (y) = π −1/2 (y/2)−β Γ +β dt(cosh t)−2β cos(y sinh t). (10.352) 2 0 p In one dimension where β = 1/2, this becomes simply K1/2 (y) = π/2ye−y . For β = d/2 and β = 1 − d/2 written as β = (1 ∓ ε)/2, it is approximately equal to ε K(1∓ε)/2 (y) = π −1/2 (y/2)−(1∓ε)/2 Γ 1 ∓ 2 Z ∞ π −y −1 × e ±ε dt(cosh t) ln(cosh t) cos(y sinh t) , (10.353) 2 0 where the t-integral is regular at y = 0.7 After substituting (10.353) into (10.351), we obtain the finite value Yd

≈

ε≈0

=

ε→0

π2 2 ω 4−d c4d Sd ε Γ (1 + ε/2) Γ3 (1 − ε/2) × 2−5ε Γ(2ε) 4 2 1 π 1 = . 2 2ωπ 4 8ω

(10.354)

The prefactor d − 1 = −ε in (10.351) has been canceled by the pole in Γ(2ε). This integral coincides with the integral (10.311) whose subtle nature was discussed in the momentum space formulation (10.239). Indeed, inserting the Bessel expressions (10.326) and (10.331) into (10.311), we find Z dd τ ∆2 (τ ) ∆2αβ (τ ) − ∆2αα (τ ) Z ∞ h i2 d = −(d − 1)ω 4−d c4d Sd dy y 1−d/2 K1−d/2 (y) K 2 (y), (10.355) dy d/2 0 and a partial integration Z ∞ h Z ∞ i2 d 1−d/2 2 3 dy y (y) K1−d/2 (y) K (y) = 2 dy y 2−d K1−d/2 (y) Kd/2 (y) dy d/2 0 0

(10.356)

establishes contact with the integral (10.351) for Yd . Thus Eq. (10.350) is the same as (10.320). Knowing Yd , we also determine, after integrations by parts, the integral Z Z d 2 2 2 d τ ∆α (τ )∆β (τ ) = −3ω dd τ ∆2 (τ )∆2α (τ ) + Yd , (10.357) 7

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 3.511.1 and 3.521.2. H. Kleinert, PATH INTEGRALS

747

10.10 Simple Rules for Calculating Singular Integrals

which is the same as (10.321). It remains to calculate one more unproblematic integral over four distributions: Z Z 7 d 2 2 2 3 4 d 4 d τ ∆ (τ )∆λλ (τ ) = −2ω ∆ (0) + ω d τ ∆ (τ ) =− . (10.358) 32ω d=1 Combining this with (10.354) and (10.357) we find the Feynman diagram (10.280). The combination of (10.350) and (10.357) with (10.354) and (10.349), finally, yields the diagrams (10.324) and (10.323), respectively. Thus we see that there is no problem in calculating integrals over products of distributions in configuration space which produce the same results as dimensional regularization in momentum space.

10.10

Simple Rules for Calculating Singular Integrals

The above methods of calculating the Feynman integrals in d time dimensions with a subsequent limit d → 1 are obviously quite cumbersome. It is preferable to develop a simple procedure of finding the same results directly working with a one-dimensional time. This is possible if we only keep track of some basic aspects of the d-dimensional formulation [37]. Consider once more the ambiguous integrals coming from the first two watermelon diagrams in Eq. (10.231), which in the one-dimensional formulation represent the integrals I1 = I2 =

Z

∞

−∞ ∞

Z

−∞

¨ 2 (τ )∆2 (τ ) , dτ ∆

(10.359)

¨ )∆ ˙ 2 (τ )∆(τ ), dτ ∆(τ

(10.360)

evaluated before in the d-dimensional equations (10.283) and (10.281). Consider first the integral (10.359) which contains a square of a δ-function. We separate this out by writing I1 =

Z

∞

−∞

¨ 2 (τ )∆2 (τ ) = I1div + I1R , dτ ∆

(10.361)

with a divergent and a regular part I1div = ∆2 (0)

Z

∞

−∞

dτ δ 2 (τ ) ,

I1R =

Z

∞ −∞

¨ 2 (τ ) − δ 2 (τ ) . (10.362) dτ ∆2 (τ ) ∆ h

i

All other watermelon diagrams (10.231) lead to the well-defined integrals 1 ∆3 (0), 4ω 2 −∞ Z ∞ ˙ 2 (τ )∆2 (τ ) = 1 ∆3 (0), dτ ∆ 4 −∞ Z ∞ ˙ 4 (τ ) = 1 ω 2 ∆3 (0), dτ ∆ 4 −∞ Z

∞

dτ ∆4 (τ ) =

(10.363) (10.364) (10.365)

748


whose D-dimensional versions are (10.273), (10.285), and (10.281). Substituting these and (10.360), (10.361) into (10.231) yields the sum of all watermelon diagrams −

4 2!

¨ 2 (τ ) + 4∆(τ )∆ ˙ 2 (τ )∆(τ ¨ )+ ∆ ˙ 4 (τ )+ 4ω 2∆2 (τ )∆ ˙ 2 (τ )+ 2 ω 4 ∆4 (τ ) dτ ∆2 (τ )∆ 3 −∞ Z ∞ 17 2 3 = − 2∆2 (0) dτ δ 2 (τ ) − 2 I1R + 4I2 − (10.366) ω ∆ (0) . 6 −∞

Z

∞

Adding these to (10.229), (10.230), we obtain the sum of all second-order connected diagrams

Σ (all) = 3 δ(0) −

Z

∞

−∞

2

dτ δ (τ ) ∆2 (0) − 2 I1R + 4I2 −

7 2 3 ω ∆ (0) , (10.367) 2

where the integrals I1R and I2 are undefined. The sum has to vanish to guarantee coordinate independence. We therefore equate to zero both the singular and finite contributions in Eq. (10.367). The first yields the rule for the product of two δfunctions: δ 2 (τ ) = δ(0) δ(τ ) . This equality should of course be understood in the distributional sense: it holds after multiplying it with an arbitrary test function and integrating over τ . Z

dτ δ 2 (τ )f (τ ) ≡ δ(0)f (0).

(10.368)

The equation leads to a perfect cancellation of all powers of δ(0) arising from the expansion of the Jacobian action, which is the fundamental reason why the heuristic Veltman rule of setting δ(0) = 0 is applicable everywhere without problems. The vanishing of the regular parts of (10.367) requires the integrals (10.360) and (10.361) to satisfy 7 7 I1R + 4I2 = − ω 2 ∆3 (0) = − . 4 32ω

(10.369)

At this point we run into two difficulties. First, this single equation (10.369) for the two undefined integrals I1R and I2 is insufficient to specify both integrals, so that the requirement of reparametrization invariance alone is not enough to fix all ambiguous temporal integrals over products of distributions. Second, and more seriously, Eq. (10.369) leads to conflicts with standard integration rules based on the use of partial integration and equation of motion, and the independence of the order in which these operations are performed. Indeed, let us apply these rules to the calculation of the integrals I1R and I2 in different orders. Inserting the equation of motion (10.213) into the finite part of the integral (10.361) and making use of the regular integral (10.363), we find immediately I1R

=

Z

∞

−∞

¨ 2 (τ ) − δ 2 (τ ) dτ ∆2 (τ ) ∆ h

2

3

= −2ω ∆ (0) + ω

4

i

7 7 dτ ∆4 (τ ) = − ω 2 ∆3 (0) = − . (10.370) 4 32ω −∞

Z

∞


749


The same substitution of the equation of motion (10.213) into the other ambiguous integral I2 of (10.360) leads, after performing the regular integral (10.364), to I2 = − = −

Z

∞

−∞

˙ 2 (τ ) ∆(τ ) δ(τ ) + ω 2 dτ ∆

Z

∞ −∞

˙ 2 (τ ) ∆2 (τ ) dτ ∆

1 1 1 1 Z∞ dτ ǫ2 (τ ) δ(τ ) + ω 2∆3 (0) = −Iǫ2 δ + , (10.371) 8ω −∞ 4 8ω 4

where Iǫ2 δ denotes the undefined integral over a product of distributions Iǫ2 δ =

Z

∞

−∞

dτ ǫ2 (τ )δ(τ ) .

(10.372)

The integral I2 can apparently be fixed by applying partial integration to the integral (10.360) which reduces it to the completely regular form (10.365): I2 =

1 3

Z

∞

−∞

dτ ∆(τ )

Z d h˙3 i 1 ∞ ˙ 4 (τ ) = − 1 ω 2∆3 (0) = − 1 . (10.373) ∆ (τ ) = − dτ ∆ dτ 3 −∞ 12 96ω

There are no boundary terms due to the exponential vanishing at infinity of all functions involved. From (10.371) and (10.373) we conclude that Iǫ2 δ = 1/3. This, however, cannot be correct since the results (10.373) and (10.370) do not obey Eq. (10.369) which is needed for coordinate independence of the path integral. This was the reason why previous authors [32, 35] added a noncovariant correction term ¯ and thus ∆V = −g 2 (q 2 /6) to the classical action (10.185), which is proportional to h violates Feynman’s basic postulate that the phase factors eiA/¯h in a path integral should contain only the classical action along the paths. We shall see below that the correct value of the singular integral I in (10.372) is Iǫ2 δ =

Z

∞

−∞

dτ ǫ2 (τ )δ(τ ) = 0.

(10.374)

From the perspective of the previous sections where all integrals were defined in d = 1 − ǫ dimensions and continued to ǫ → 0 at the end, the inconsistency of Iǫ2 δ = 1/3 is obvious: Arbitrary application of partial integration and equation of motion to one-dimensional integrals is forbidden, and this is the case in the calculation (10.373). Problems arise whenever several dots can correspond to different contractions of partial derivatives ∂α , ∂β , . . ., from which they arise in the limit d → 1. The different contractions may lead to different integrals. In the pure one-dimensional calculation of the integrals I1R and I2 , all ambiguities can be accounted for by using partial integration and equation of motion (10.213) only according to the following integration rules: Rule 1. We perform a partial integration which allows us to apply subsequently the equation of motion (10.213). Rule 2. If the equation of motion (10.213) leads to integrals of the type (10.372), they must be performed using naively the Dirac rule for the δ-function and the

750


property ǫ(0) = 0. Examples are (10.374) and the trivially vanishing integrals for all odd powers of ǫ(τ ): Z

dτ ǫ2n+1 (τ ) δ(τ ) = 0,

n = integer,

(10.375)

which follow directly from the antisymmetry of ǫ2n+1 (τ ) and the symmetry of δ(τ ) contained in the regularized expressions (10.329) and (10.331). Rule 3. The above procedure leaves in general singular integrals, which must be treated once more with the same rules. Let us show that calculating the integrals I1R and I2 with these rules is consistent with the coordinate independence condition (10.369). In the integral I2 of (10.360) we first apply partial integration to find h i ˙ )d ∆ ˙ 2 (τ ) dτ ∆(τ ) ∆(τ dτ −∞ Z Z 1 ∞ 1 ∞ 4 ˙ ˙ 2 (τ ) ∆(τ ¨ ), = − dτ ∆ (τ ) − dτ ∆(τ ) ∆ 2 −∞ 2 −∞

I2 =

1 2

Z

∞

(10.376)

with no contributions from the boundary terms. Note that the partial integration (10.373) is forbidden since it does not allow for a subsequent application of the equation of motion (10.213). On the right-hand side of (10.376) it can be applied. This leads to a combination of two regular integrals (10.364) and (10.365) and the singular integral I, which we evaluate with the naive Dirac rule to I = 0, resulting in Z Z Z 1 ∞ 1 ∞ 1 2 ∞ 4 2 ˙ ˙ ˙ 2 (τ ) ∆2 (τ ) dτ ∆ (τ ) + dτ ∆ (τ ) ∆(τ ) δ(τ ) − ω dτ ∆ I2 = − 2 −∞ 2 −∞ 2 −∞ 1 1 2 3 1 = I − ω ∆ (0) = − . (10.377) 16ω 4 32ω If we calculate the finite part I1R of the integral (10.361) with the new rules we obtain a result different from (10.370). Integrating the first term in brackets by parts and using the equation of motion (10.213), we obtain I1R

= = =

Z

∞

−∞ ∞

Z

−∞ ∞

Z

−∞

¨ 2 (τ ) − δ 2 (τ ) dτ ∆2 (τ ) ∆ h

i

˙ ) ∆2 (τ ) − 2∆(τ ¨ )∆ ˙ 2 (τ ) ∆(τ ) − ∆2 (τ ) δ 2 (τ ) dτ − ˙˙ ∆˙ (τ ) ∆(τ h

˙ ) − ∆2 (τ ) δ 2 (τ ) −2I2 −ω 2 ˙ ) ∆2 (τ ) δ(τ dτ ∆(τ h

i

Z

∞

−∞

i

˙ 2 (τ ) ∆2 (τ ) . (10.378) dτ ∆

The last two terms are already known, while the remaining singular integral in brackets must be subjected once more to the same treatment. It is integrated by parts so that the equation of motion (10.213) can be applied to obtain Z

∞

−∞

˙ ) − ∆2 (τ ) δ 2 (τ ) = − ˙ ) ∆2 (τ ) δ(τ dτ ∆(τ h

−

Z

∞

−∞

i

Z

∞

−∞

¨ ) ∆2 (τ ) + 2∆ ˙ 2 (τ ) ∆(τ ) δ(τ ) dτ ∆(τ h

dτ ∆2 (τ ) δ 2 (τ ) = −ω 2 ∆3 (0) −

1 I. 4ω

i

(10.379)


751


Inserting this into Eq. (10.378) yields I1R

=

h i ¨ 2 (τ ) − δ 2 (τ ) = −2I2 − 5 ω 2 ∆3 (0) − 1 I = − 3 , (10.380) dτ ∆2 (τ ) ∆ 4 4ω 32ω −∞

Z

∞

the right-hand side following from I = 0, which is a consequence of Rule 3. We see now that the integrals (10.377) and (10.380) calculated with the new rules obey Eq. (10.369) which guarantees coordinate independence of the path integral. The applicability of Rules 1–3 follows immediately from the previously established dimensional continuation [23, 24]. It avoids completely the cumbersome calculations in 1 − ε-dimension with the subsequent limit ε → 0. Only some intermediate steps of the derivation require keeping track of the d-dimensional origin of the rules. For this, we continue the imaginary time coordinate τ to a d-dimensional spacetime vector τ → τ µ = (τ 0 , τ 1 , . . . , τ d−1 ), and note that the equation of motion (10.213) becomes a scalar field equation of the Klein-Gordon type

−∂α2 + ω 2 ∆(τ ) = δ (d) (τ ) .

(10.381)

In d dimensions, the relevant second-order diagrams are obtained by decomposing the harmonic expectation value Z

D

E

dd τ qα2 (τ ) q 2 (τ ) qβ2 (0) q 2(0)

(10.382)

into a sum of products of four two-point correlation functions according to the Wick rule. The fields qα (τ ) are the d-dimensional extensions qα (τ ) ≡ ∂α q(τ ) of q(τ ˙ ). Now the d-dimensional integrals, corresponding to the integrals (10.359) and (10.360), are defined uniquely by the contractions I1d

I2d

*

+

=

Z

d τ

=

Z

dd τ ∆2 (τ ) ∆2αβ (τ ) ,

=

Z

dd τ

=

Z

dd τ ∆(τ ) ∆α (τ ) ∆β (τ ) ∆αβ (τ ) .

d

*

qα (τ )qα (τ )q(τ )q(τ )qβ (0)qβ (0)q(0)q(0)

(10.383) +

qα (τ )qα (τ )q(τ )q(τ )qβ (0)qβ (0)q(0)q(0)

(10.384)

The different derivatives ∂α ∂β acting on ∆(τ ) prevent us from applying the field equation (10.381). This obstacle was hidden in the one-dimensional formulation. It can be overcome by a partial integration. Starting with I2d , we obtain I2d

1 = − 2

Z

h

i

dd τ ∆2β (τ ) ∆2α (τ ) + ∆(τ ) ∆αα (τ ) .

(10.385)

Treating I1d likewise we find I1d = −2I2d +

Z

dd τ ∆2 (τ ) ∆2αα (τ ) + 2

Z

dd τ ∆(τ ) ∆2β (τ ) ∆αα (τ ) . (10.386)

752


In the second equation we have used the fact that ∂α ∆αβ = ∂β ∆λλ . The right-hand sides of (10.385) and (10.386) contain now the contracted derivatives ∂α2 such that we can apply the field equation (10.381). This mechanism works to all orders in the perturbation expansion which is the reason for the applicability of Rules 1 and 2 which led to the results (10.377) and (10.380) ensuring coordinate independence. The value Iǫ2 δ = 0 according to the Rule 2 can be deduced from the regularized equation (10.385) in d = 1 − ε dimensions by using the field equation (10.334) to rewrite I2d as I2d

1 = − 2

Z

h

dd τ ∆2β (τ ) ∆2α (τ )+ω 2∆2 (τ )−∆(τ )δ (d) (τ )

1 1 ≈ − + d≈1 32ω 2

Z

dd τ ∆2β (τ )∆2α (τ )∆(τ )δ (d) (τ ) .

i

Comparison with (10.371) yields the regularized expression for Iǫ2 δ IǫR2 δ

=

Z

∞

−∞

2

dτ ǫ (τ )δ(τ )

R

= 8ω

Z

dd τ ∆2β (τ ) ∆(τ )δ (d) (τ ) = 0,

(10.387)

the vanishing for all ε > 0 being a consequence of the small-τ behavior ∆(τ ) ∆2α (τ ) ∝ |τ |2ε , which follows directly from (10.330). Let us briefly discuss an alternative possibility of giving up partial integration completely in ambiguous integrals containing ǫ- and δ-function, or their time derivatives, which makes unnecessary to satisfy Eq. (10.373). This yields a freedom in the definition of integral over product of distribution (10.372) which can be used to fix Iǫ2 δ = 1/4 from the requirement of coordinate independence [25]. Indeed, this value of I would make the integral (10.371) equal to I2 = 0, such that (10.369) would be satisfied and coordinate independence ensured. In contrast, giving up partial integration, the authors of Refs. [31, 33] have assumed the vanishing ǫ2 (τ ) at τ = 0 so that the integral Iǫ2 δ should vanish as well: Iǫ2 δ = 0. Then Eq. (10.371) yields I2 = 1/32ω which together with (10.370) does not obey the coordinate independence condition (10.369), making yet an another noncovariant quantum correction ∆V = g 2 (q 2 /2) necessary in the action, which we reject since it contradicts Feynman’s original rules of path integration. We do not consider giving up partial integration as an attractive option since it is an important tool for calculating higher-loop diagrams.

10.11

Perturbative Calculation on Finite Time Intervals

The above calculation rules can be extended with little effort to path integrals of time evolution amplitudes on finite time intervals. We shall use an imaginary time interval with τa = 0 and τb = β to have the closest connection to statistical mechanics. The ends of the paths will be fixed at τa and τb to be able to extract quantum-mechanical time evolution amplitudes by a mere replacement τ → −it. The extension to a finite time interval is nontrivial since the Feynman integrals in frequency space become sums over discrete frequencies whose d-dimensional generalizations can usually not be evaluated with standard formulas. The above ambiguities of the integrals, however, will appear in the sums in precisely the same way as before. The reason is that they stem from ordering ambiguities between q and q˙ in the perturbation expansions. These are properties of small H. Kleinert, PATH INTEGRALS

753

10.11 Perturbative Calculation on Finite Time Intervals

time intervals and thus of high frequencies, where the sums can be approximated by integrals. In fact, we have seen in the last section, that all ambiguities can be resolved by a careful treatment of the singularities of the correlation functions at small temporal spacings. For integrals on a time axis it is thus completely irrelevant whether the total time interval is finite or infinite, and the ambiguities can be resolved in the same way as before [38]. This can also be seen technically by calculating the frequency sums in the Feynman integrals of finite-time path integrals with the help of the Euler-Maclaurin formula (2.586) or the equivalent ζ-function methods described in Subsection 2.15.6. The lowest approximation involves the pure frequency integrals whose ambiguities have been resolved in the preceeding sections. The remaining correction terms in powers of the temperature T = 1/β are all unique and finite [see Eq. (2.590) or (2.550)]. The calculations of the Feynman integrals will most efficiently proceed in configuration space as described in Subsection 10.8.1. Keeping track of certain minimal features of the unique definition of all singular integrals in d dimensions, we shall develop reduction rules based on the equation of motion and partial integration. These will allow us to bring all singular Feynman integrals to a regular form in which the integrations can be done directly in one dimension. The integration rules will be in complete agreement with much more cumbersome calculations in d dimensions with the limit d → 1 taken at the end.

10.11.1

Diagrammatic Elements

The perturbation expansion for an evolution amplitude over a finite imaginary time proceeds as described in Section 10.6, except that the free energy in Eq. (10.200) becomes [recall (2.518)] βFω

= =

D X D Tr log(−∂ 2 + ω 2 ) = log(ωn2 + ω 2 ) 2β 2β n

D log [2 sinh (¯ hβω/2)] . β

(10.388)

As before, the diagrams contain four types of lines representing the correlation functions (10.201)– (10.190). Their explicit forms are, however, different. It will be convenient to let the frequency ω in the free part of the action (10.186) go to zero. Then the free energy (10.388) diverges logarithmically in ω. This divergence is, however, trivial. As explained in Section 2.9, the divergence is removed by replacing ω by the length of the q-axis according to the rule (2.353). For finite time intervals, the correlation functions are no longer given by (10.206) which would not have a finite limit for ω → 0. Instead, they satisfy Dirichlet boundary conditions, where we can go to ω = 0 without problem. The finiteness of the time interval removes a possible infrared divergence for ω → 0. The Dirichlet boundary conditions fix the paths at the ends of the time interval (0, β) making the fluctuations vanish, and thus also their correlation functions: ′ (2) ′ G(2) µν (0, τ ) = Gµν (β, τ ) = 0,

(2) G(2) µν (τ, 0) = Gµν (τ, β) = 0.

(10.389)

The first correlation function corresponding to (10.205) is now ′ ′ G(2) µν (τ, τ ) = δµν ∆(τ, τ ) =

,

(10.390)

where ∆(τ, τ ′ ) = ∆(τ ′ , τ ) =

1 1 ττ′ (β − τ> ) τ< = [−ǫ(τ −τ ′ )(τ −τ ′ ) + τ + τ ′ ] − , β 2 β

(10.391)

abbreviates the Euclidean version of G0 (t, t′ ) in Eq. (3.39). Being a Green function of the free equation of motion (10.212) for ω = 0, this satisfies the inhomogeneous differential equations ¨∆(τ, τ ′ ) = ∆¨(τ, τ ′ ) = −δ(τ − τ ′ ),

(10.392)

754


by analogy with Eq. (10.213) for ω = 0. In addition, there is now an independent equation in which the two derivatives act on the different time arguments: ˙∆˙(τ, τ ′ ) = δ(τ − τ ′ ) − 1/β.

(10.393)

For a finite time interval, the correlation functions (10.202) (10.203) differ by more than just a sign [recall (10.209)]. We therefore must distinguish the derivatives depending on whether the left or the right argument are differentiated. In the following, we shall denote the derivatives with respect to τ or τ ′ by a dot on the left or right, respectively, writing ˙∆ (τ, τ ′ ) ≡

d ∆(τ, τ ′ ), dτ

∆˙(τ, τ ′ ) ≡

d ∆(τ, τ ′ ). dτ ′

(10.394)

Differentiating (10.391) we obtain explicitly 1 1 τ′ ˙∆ (τ, τ ′ ) = − ǫ(τ − τ ′ ) + − , 2 2 β

1 1 τ ǫ(τ − τ ′ ) + − = ˙∆ (τ ′ , τ ) . 2 2 β

∆˙(τ, τ ′ ) =

(10.395)

The discontinuity at τ = τ ′ which does not depend on the boundary condition is of course the same as before, The two correlation functions (10.207) and (10.209) and their diagrammatic symbols are now ′ ′ ′ ∂τ G(2) µν (τ, τ ) ≡ hq˙µ (τ )qν (τ )i = δµν ˙∆ (τ, τ ) =

′ ′ ′ ∂τ ′ G(2) µν (τ, τ ) ≡ hqµ (τ )q˙ν (τ )i = δµν ∆˙(τ, τ ) =

,

(10.396)

.

(10.397)

The fourth correlation function (10.211) is now ′ ′ ∂τ ∂τ ′ G(2) µν (τ, τ ) = δµν ˙∆˙(τ, τ ) =

,

(10.398)

with ˙∆˙(τ, τ ′ ) being given by (10.393). Note the close similarity but also the difference of this with respect to the equation of motion (10.392).

10.11.2

Cumulant Expansion of D-Dimensional Free-Particle Amplitude in Curvilinear Coordinates

We shall now calculate the partition function of a point particle in curved space for a finite time interval. Starting point is the integral over the diagonal amplitude of a free point particle of unit mass (xa β|xa 0) in flat D-dimensional space Z Z = dD xa (xa β|xa 0), (10.399) with the path integral representation (xa β|xa 0)0 =

Z

(0)

DD x e−A

[x]

,

(10.400)

where A(0) [x] is the free-particle action A(0) [x] =

1 2

Z

β

dτ x˙ 2 (τ ).

(10.401)

0

Performing the Gaussian path integral leads to (xa β|xa 0)0 = e−(D/2)Tr log(−∂

2

)

= [2πβ]

−D/2

,


755


where the trace of the logarithm is evaluated with Dirichlet boundary conditions. The result is of course the D-dimensional imaginary-time version of the fluctuation factor (2.120) in natural units. A coordinate transformation xi (τ ) = xi (q µ (τ )) mapping xa to qaµ brings the action (10.401) to the form (10.185) with V (q(τ )) = 0: Z 1 β ∂xi (q) ∂xi (q) A[q] = dτ gµν (q(τ ))q˙µ (τ )q˙ν (τ ), with gµν (q) ≡ . (10.403) 2 0 ∂q µ ∂q ν In the formal notation (10.188), the measure transforms as follows: Z Z YZ YZ p D D D D x(τ ) ≡ d x(τ ) = J d q(τ ) ≡ J DD q g(qa ) , τ

(10.404)

τ

where g(q) ≡ det gµν (q) and J is the Jacobian of the coordinate transformation generalizing (10.219) and (10.218) "s ,s # " # Z β Y ∂xi (q(τ )) ∂xi (qa ) 1 g(q(τ )) = exp δ(0) dτ log . (10.405) J= δq µ (τ ) δq0µ 2 g(qa ) 0 τ Thus we may write the transformed path integral (10.400) in the form Z (xa β|xa 0)0 ≡ (qa β|qa 0)0 = DD q e−Atot [x] , with the total action in the exponent Z β 1 1 g(q(τ )) Atot [q] = dτ gµν (q(τ ))q˙µ (τ )q˙ν (τ ) − δ(0) log . 2 2 g(qa ) 0

(10.406)

(10.407)

Following the rules described in Subsection 10.6.1 we expand the action in powers of δq µ (τ ) = q µ (τ ) − qaµ . The action can then be decomposed into a free part Z 1 β (0) A [qa , δq] = dτ gµν (qa )δ q˙µ (τ )δ q˙ν (τ ) (10.408) 2 0 and an interacting part written somewhat more explicitly than in (10.193) with (10.187) and (10.192): Z β 1 int Atot [qa , δq] = dτ [gµν (q) − gµν (qa )]δ q˙µ δ q˙ν 2 0 ( ) 2 Z β g(qa + δq) 1 g(qa + δq) 1 − dτ δ(0) −1 − − 1 + ... . (10.409) 2 g(qa ) 2 g(qa ) 0 For simplicity, we assume the coordinates to be orthonormal at qaµ , i.e., gµν (qa ) = δµν . The path integral (10.406) is now formally defined by a perturbation expansion similar to (10.198): Z Z 1 int 2 D A(0) [q]−Aint [q] D −A(0) [q] int tot (qa β|qa 0) = D qe = D qe 1 − Atot + Atot − . . . 2

1 int 2 = (2πβ)−D/2 1 − Aint Atot − . . . , tot + 2

int 1 int 2 −D/2 Atot c − . . . ≡ e−βf (q) , (10.410) = (2πβ) exp − Atot c + 2 with the harmonic expectation values

h. . .i = (2πβ)D/2

Z

(0)

DD q(τ )(. . .)e−A

[q]

(10.411)

756


int 2 int 2 2 and their cumulants Aint − Atot , . . . [recall (3.482), (3.483)], containing only tot c = Atot connected diagrams. To emphasize the analogy with the cumulant expansion of the free energy in (10.198), we have defined the exponent in (10.410) as −βf (q). This q-dependent quantity f (q) is closely related to the alternative effective classical potential discussed in Subsection 3.25.4, apart from a normalization factor: 1 ˜ eff cl e−β Vω (q) . e−βf (q) = q 2π¯h2 /M kB T

(10.412)

If our calculation procedure respects coordinate independence, all expansion terms of βf (q) must vanish to yield the trivial exact results (10.400).

10.11.3

Propagator in 1 – ε Time Dimensions

In the dimensional regularization of the Feynman integrals on an infinite time interval in Subsection 10.7.2 we have continued all Feynman diagrams in momentum space to d = 1 − ε time dimensions. For the present Dirichlet boundary conditions, this standard continuation of quantum field theory is not directly applicable since the integrals in momentum space become sums over discrete frequencies νn = πn/β [compare (3.64)]. For such sums one has to set up completely new rules for a continuation, and there are many possibilities for doing this. Fortunately, it will not be necessary to make a choice since we can use the method developed in Subsection 10.10 to avoid continuations altogether. and work in a single physical time dimension. For a better understanding of the final procedure it is, however, useful to see how a dimensional continuation could proceed. We extend the imaginary time coordinate τ to a d-dimensional spacetime vector whose zeroth component is τ : z µ = (τ, z 1 , . . . , z d−1). In d = 1 − ε dimensions, the extended correlation function reads Z dε k ik(z−z′ ) ′ ′ ∆(τ, z; τ , z ) = e ∆ω (τ, τ ′ ), where ω ≡ |k|. (10.413) (2π)ε Here the extra ε-dimensional space coordinates z are assumed to live on infinite axes with translational invariance along all directions. Only the original τ -coordinate lies in a finite interval 0 ≤ τ ≤ β, with Dirichlet boundary conditions. The Fourier component in the integrand ∆ω (τ, τ ′ ) is the usual one-dimensional correlation function of a harmonic oscillator with the k-dependent frequency ω = |k|. It is the Green function which satisfies on the finite τ -interval the equation of motion −¨∆ ω (τ, τ ′ ) + ω 2 ∆ω (τ, τ ′ ) = δ(τ − τ ′ ),

(10.414)

with Dirichlet boundary conditions ∆ω (0, τ ) = ∆ω (β, τ ) = 0.

(10.415)

The explicit form was given in Eq. (3.36) for real times. Its obvious continuation to imaginary-time is ∆ω (τ, τ ′ ) =

sinh ω(β − τ> ) sinh ωτ< , ω sinh ωβ

(10.416)

where τ> and τ< denote the larger and smaller of the imaginary times τ and τ ′ , respectively. In d time dimensions, the equation of motion (10.392) becomes a scalar field equation of the Klein-Gordon type. Using Eq. (10.414) we obtain µµ ∆(τ, z;

τ ′ , z′ )

= ∆µµ (τ, z; τ ′ , z′ ) = ¨∆ (τ, z; τ ′ , z′ ) + zz ∆(τ, z; τ ′ , z′ ) Z dε k ik(z−z′ ) ′ 2 ′ = e ¨ ∆ (τ, τ ) − ω ∆ (τ, τ ) = ω ω (2π)ε = − δ(τ − τ ′ ) δ (ε) (z − z′ ) ≡ − δ (d) (z − z ′ ).

(10.417)



757

The important observation is now that for d spacetime dimensions, perturbation expansion of the path integral yields for the second correlation function ˙∆˙(τ, τ ′ ) in Eqs. (10.514) and (10.515) the extension µ ∆ν (z, z ′ ). This function differs from the contracted function µ ∆µ (z, z ′ ), and from ′ ′ µµ ∆(z, z ) which satisfies the field equation (10.417). In fact, all correlation functions ˙∆˙(τ, τ ) encountered in the diagrammatic expansion which have different time arguments always turn out to have the d-dimensional extension µ ∆ν (z, z ′ ). An important exception is the correlation function at equal times ˙∆˙(τ, τ ) whose d-dimensional extension is always µ ∆µ (z, z), which satisfies the right-hand equation (10.392) in the ε → 0 -limit. Indeed, it follows from Eq. (10.413) that Z dε k ˙∆˙ ω (τ, τ ) + ω 2 ∆ω (τ, τ ) . (10.418) µ ∆µ (z, z) = ε (2π) With the help of Eq. (10.416), the integrand in Eq. (10.418) can be brought to ˙∆˙ ω (τ, τ ) + ω 2 ∆ω (τ, τ ) = δ(0) −

ω cosh ω(2τ − β) . sinh ωβ

(10.419)

Substituting this into Eq. (10.418), we obtain µ ∆µ (z, z)

= δ (d) (z, z) − I ε .

The integral I ε is calculated as follows Z Z ∞ 1 Sε cosh z(1 − 2τ /β) dε k ω cosh ω(2τ − β) Iε = = dzz ε (2π)ε sinh ωβ β (2πβ)ε 0 sinh z 1 Sε Γ(ε + 1) = [ζ (ε + 1, 1 − τ /β) + ζ (ε + 1, τ /β)] , β (2πβ)ε 2ε+1

(10.420)

(10.421)

where Sε = 2π ε/2 /Γ(ε/2) is the surface of a unit sphere in ε dimension [recall Eq. (1.555)], and Γ(z) and ζ(z, q) are gamma and zeta functions, respectively. For small ε → 0, they have the limits ζ(ε + 1, q) → 1/ε − ψ(q), and Γ(ε/2) → 2/ε, so that I ε → 1/β, proving that the d-dimensional equation (10.420) at coinciding arguments reduces indeed to the one-dimensional equation (10.392). The explicit d-dimensional form will never be needed, since we can always treat µ ∆µ (z, z) as onedimensional functions ˙∆˙(τ, τ ), which can in turn be replaced everywhere by the right-hand side δ(0) − 1/β of (10.393).

10.11.4

Coordinate Independence for Dirichlet Boundary Conditions

Before calculating the path integral (10.410) in curved space with Dirichlet boundary conditions, let us first verify its coordinate independence following the procedure in Section 10.7. Thus we consider the perturbation expansion of the short-time amplitude of a free particle in one general coordinate. The free action is (10.220), and the interactions (10.221) and (10.223), all with ω = 0. Taking the parameter a = 1, the actions are Z 1 β (0) A [q] = dτ q˙2 (τ ), (10.422) 2 0 (10.423) and Aint tot

Z β 3 2 4 1 2 4 2 2 2 = dτ −ηq (τ ) + η q (τ ) q˙ (τ ) − δ(0) −ηq (τ ) + η q (τ ) . 2 2 0

(10.424)

int int 2

int 2 2 We calculate the cumulants Aint Atot c = Aint − Atot , . . . [recall tot c = Atot , tot (3.482), (3.483)] contributing to th quantity βf in Eq. (10.410) order by order in η. For a better

758


comparison with the previous expansion in Subsection 10.7.1 we shall denote the diagrammatic (m) contributions which are analogous to the different free energy terms βFn of order n by corre(m) sponding symbols βfn . There are two main differences with respect to Subsection 10.7.1: All diagrams with a prefactor ω are absent, and there are new diagrams involving the correlation functions at equal times hq˙µ (τ )q ν (τ )i and hq µ (τ )q˙ν (τ )i which previously vanished because of (10.210). Here they have the nonzero value ˙∆ (τ, τ ) = ∆˙(τ, τ ) = 1/2 − τ /β by Eq. (10.395). To first order in η, the quantity f (q) in Eq. (10.410) receives a contribution from the first cumulant of the linear terms in η of the interaction (10.424): Z

βf1 = Aint tot c = η

β

dτ

0

D

E − q 2 (τ )q˙2 (τ ) + δ(0)q 2 (τ ) + O(η 2 ).

(10.425)

There exists only three diagrams, two originating from the kinetic term and one from the Jacobian action: βf1 = − η

−2η

+ η δ(0)

.

(10.426)

Note the difference with respect to the diagrams (10.226) for infinite time interval with ω 2 -term in the action. The omitted η 2 -terms in (10.425) yield the second-order contribution (1) βf2

=η

2

Z

β

dτ

0

1 4 3 4 2 q (τ )q˙ (τ ) − δ(0) q (τ ) . 2 2 c

The associated local diagrams are [compare (10.227)]: " 9 (1) βf2 = η 2 + 18 2

(10.427)

#

3 − δ(0) 2

.

(10.428)

The second cumulant to order η 2 reads 1 − η2 2!

Z

β

dτ

0

Z

β 0

dτ ′ −q 2 (τ )q˙2 (τ ) + δ(0)q 2 (τ ) −q 2 (τ ′ )q˙2 (τ ′ ) + δ(0)q 2 (τ ′ ) c ,

leading to diagrams containing δ(0): η2 (2) βf2 = − 2 δ 2 (0) 2!

− 4 δ(0)

+4

+4

+4

+

.

(10.429)

The remaining diagrams are either of the three-bubble type, or of the watermelon type, each with all possible combinations of the four line types (10.390) and (10.396)–(10.398). The three-bubbles diagrams yield [compare (10.230)] (3)

βf2 = −

η2 h 4 2!

+2

−8

+2

−8

The watermelon-type diagrams contribute the same diagrams as in (10.231) for ω = 0: 2 η (4) βf2 = − 4 +4 + . 2!

i .

(10.430) (10.431)

For coordinate independence, the sum of the first-order diagrams (10.426) has to vanish. Analytically, this amounts to the equation βf1 = −η

Z

0

β

dτ ∆(τ, τ )˙∆˙(τ, τ ) + 2˙∆ 2 (τ, τ ) − δ(0)∆(τ, τ ) = 0.

(10.432)


759


In the d-dimensional extension, the correlation function ˙∆˙(τ, τ ) at equal times is the limit d → 1 of the contracted correlation function µ ∆µ (x, x) which satisfies the d-dimensional field equation (10.417). Thus we can use Eq. (10.393) to replace ˙∆˙(τ, τ ) by δ(0) − 1/β. This removes the infinite factor δ(0) in Eq. (10.432) coming from the measure. The remainder is calculated directly: Z β 1 2 (10.433) dτ − ∆(τ, τ ) + 2 ˙∆ (τ, τ ) = 0. β 0 This result is obtained without subtleties, since by Eqs. (10.391) and (10.395) ∆(τ, τ ) = τ −

τ2 , β

˙∆ 2 (τ, τ ) =

1 ∆(τ, τ ) − , 4 β

(10.434)

whose integrals yield 1 2β

Z

0

β

dτ ∆(τ, τ ) =

Z

β

dτ ˙∆ 2 (τ, τ ) =

0

β . 12

(10.435)

(i)

Let us evaluate the second-order diagrams in βf2 , i = 1, 2, 3, 4. The sum of the local diagrams in (10.428) consists of the integrals by (1)

βf2

=

3 2 η 2

Z

0

β

dτ 3∆2 (τ, τ )˙∆˙(τ, τ ) + 12∆(τ, τ )˙∆ 2 (τ, τ ) − δ(0)∆2 (τ, τ ) .

(10.436)

Replacing ˙∆˙(τ, τ ) in Eq. (10.436) again by δ(0) − 1/β, on account of the equation of motion (10.393), and taking into account the right-hand equation (10.434), " # Z β β3 (1) βf2 = η 2 3δ(0) dτ ∆2 (τ, τ ) = η 2 δ(0). (10.437) 10 0 We now calculate the sum of bubble diagrams (10.429)–(10.431), beginning with (10.429) whose analytic form is Z Z η2 β β dτ dτ ′ 2δ 2 (0)∆2 (τ, τ ′ ) (10.438) 2 0 0 −4 δ(0) ∆(τ, τ )˙∆ 2 (τ, τ ′ ) + 4˙∆ (τ, τ )∆(τ, τ ′ )˙∆ (τ, τ ′ ) + ∆2 (τ, τ ′ )˙∆˙(τ, τ ) .

(2)

βf2

=−

Inserting Eq. (10.393) into the last equal-time term, we obtain (2)

βf2

Z Z η2 β β dτ dτ ′ −2δ 2 (0)∆2 (τ, τ ′ ) 2 0 0 −4δ(0) ∆(τ, τ )˙∆ 2 (τ, τ ′ ) + 4 ˙∆ (τ, τ )∆(τ, τ ′ )˙∆ (τ, τ ′ ) − ∆2 (τ, τ ′ )/β .

=−

(10.439)

As we shall see below, the explicit evaluation of the integrals in this sum is not necessary. Just for completeness, we give the result: 3 η2 β4 β β3 β3 (2) βf2 = 2δ 2 (0) + 4δ(0) +4 − 2 90 45 180 90 4 3 β 2 β = η2 δ (0) + δ(0) . (10.440) 90 15 We now turn to the three-bubbles diagrams (10.431). Only three of these contain the correlation function µ ∆ν (x, x′ ) → ˙∆˙(τ, τ ′ ) for which Eq. (10.393) is not applicable: the second, fourth,

760


and sixth diagram. The other three-bubble diagrams in (10.431) containing the generalization µ ∆µ (x, x) of the equal-time propagator ˙∆˙(τ, τ ) can be calculated using Eq. (10.393). Consider first a partial sum consisting of the first three three-bubble diagrams in the sum (10.431). This has the analytic form

(3) βf2

1,2,3

Z Z η2 β β dτ dτ ′ 4 ∆(τ, τ ) ˙∆ 2 (τ, τ ′ ) ˙∆˙(τ ′ , τ ′ ) (10.441) =− 2 0 0 + 2 ˙∆˙(τ, τ )∆2 (τ, τ ′ )˙∆˙(τ ′ , τ ′ ) + 16 ˙∆ (τ, τ )∆(τ, τ ′ ) ˙∆ (τ, τ ′ ) ˙∆˙(τ ′ , τ ′ ) .

Replacing ˙∆˙(τ, τ ) and ˙∆˙(τ ′ , τ ′ ) by δ(0) − 1/β, according to of (10.393), we see that Eq. (10.441) contains, with opposite sign, precisely the previous sum (10.438) of all one-and two-bubble diagrams. Together they give Z Z η2 β β 4 (2) (3) dτ dτ ′ − ∆(τ, τ )˙∆ 2 (τ, τ ′ ) βf2 + βf2 =− 2 0 0 β 1,2,3 2 16 + 2 ∆2 (τ, τ ′ ) − ˙∆ (τ, τ )∆(τ, τ ′ ) ˙∆ (τ, τ ′ ) , (10.442) β β and can be evaluated directly to (2) βf2

+

(3) βf2

1,2,3

η2 = 2

4 β2 2 β4 16 β 3 − 2 + β 45 β 90 β 180

=

η2 7 2 β. 2 45

(10.443)

By the same direct calculation, the Feynman integral in the fifth three-bubble diagram in (10.431) yields Z βZ β β2 : I5 = dτ dτ ′ ˙∆ (τ, τ )˙∆ (τ, τ ′ ) ∆˙(τ, τ ′ ) ∆˙(τ ′ , τ ′ ) = − . (10.444) 720 0 0 The explicit results (10.443) and (10.444) are again not needed, since the last term in Eq. (10.442) is equal, with opposite sign, to the partial sum of the fourth and fifth three-bubble diagrams in Eq. (10.431). To see this, consider the Feynman integral associated with the sixth three-bubble diagram in Eq. (10.431): : I4

=

Z

0

β

Z

β

dτ dτ ′ ˙∆ (τ, τ )∆(τ, τ ′ )˙∆˙(τ, τ ′ ) ∆˙(τ ′ , τ ′ ),

whose d-dimensional extension is Z βZ β I4d = dd τ dd τ ′ α ∆(τ, τ )∆(τ, τ ′ ) α ∆β (τ, τ ′ )∆β (τ ′ , τ ′ ). 0

(10.445)

0

(10.446)

0

Adding this to the fifth Feynman integral (10.444) and performing a partial integration, we find in one dimension Z Z η2 β β 16 η2 (3) βf2 = − 16 (I4 + I5 ) = − dτ dτ ′ ˙∆ (τ, τ )˙∆ (τ, τ ′ )∆(τ, τ ′ ) 2 2 0 0 β 4,5 4 = −η 2 β2, (10.447) 45 where we have used d [˙∆ (τ, τ )] /dτ = −1/β obtained by differentiating (10.434). Comparing (10.447) with (10.442), we find the sum of all bubbles diagrams, except for the sixth and seventh three-bubble diagrams in Eq. (10.431), to be given by ′ (2) (3) βf2 + βf2

6,7

=

η2 2 2 β . 2 15

(10.448)


761


The prime on the sum denotes the exclusion of the diagrams indicated by subscripts. The correlation function ˙∆˙(τ, τ ′ ) in the two remaining diagrams of Eq. (10.431), whose d-dimensional extension is α ∆β (x, x′ ), cannot be replaced via Eq. (10.393), and the expression can only be simplified by applying partial integration to the seventh diagram in Eq. (10.431), yielding Z βZ β : I7 = dτ dτ ′ ∆(τ, τ )˙∆ (τ, τ ′ )˙∆˙(τ, τ ′ ) ∆˙(τ ′ , τ ′ ) 0

Z

→

0

β

0

1 2

=

Z

→

1 2

=

1 2β

Z

β

Z

0

dd τ dd τ ′ ∆(τ, τ ) α ∆(τ, τ ′ )α ∆β (τ, τ ′ )∆β (τ ′ , τ ′ )

0 βZ β 0

β 0

Z

dd τ dd τ ′ ∆(τ, τ )∆β (τ ′ , τ ′ )∂β′ [α ∆(τ, τ ′ )]

β

dτ dτ ′ ∆(τ, τ )˙∆ (τ ′ , τ ′ )

0

β

Z

0

Z

β

d 2 ˙∆ (τ, τ ′ ) ′ dτ

dτ dτ ′ ∆(τ, τ )˙∆ 2 (τ, τ ′ ) =

0

2

β2 . 90

(10.449)

The sixth diagram in the sum (10.431) diverges linearly. As before, we add and subtract the divergence Z βZ β : I6 = dτ dτ ′ ∆(τ, τ )˙∆˙ 2 (τ, τ ′ )∆(τ ′ , τ ′ ) 0

=

Z

0

β

0

+

Z

β

Z

0

β

0

Z

β

0

dτ dτ ′ ∆(τ, τ ) ˙∆˙ 2 (τ, τ ′ ) − δ 2 (τ − τ ′ ) ∆(τ ′ , τ ′ )

dτ dτ ′ ∆2 (τ, τ ) δ 2 (τ − τ ′ ).

(10.450)

In the first, finite term we go to d dimensions and replace δ(τ − τ ′ ) → δ (d) (τ − τ ′ ) = −∆ββ (τ, τ ′ ) using the field equation (10.417). After this, we apply partial integration and find Z βZ β I6R → dd τ dd τ ′ ∆(τ, τ ) α ∆2β (τ, τ ′ ) − ∆2γγ (τ, τ ′ ) ∆(τ ′ , τ ′ ) 0

=

0

Z

β

Z

β

0

→

0

Z

β

Z

β

0

0

dd τ dd τ ′ {−∂α [∆(τ, τ )] ∆β (τ, τ ′ ) α ∆β (τ, τ ′ )∆(τ ′ , τ ′ )

+ ∆(τ, τ )∆β (τ, τ ′ )∆γγ (τ, τ ′ )∂β′ [∆(τ ′ , τ ′ )]

dτ dτ ′ 2 {−˙∆ (τ, τ ) ∆˙(τ, τ ′ )˙∆˙(τ, τ ′ )∆(τ ′ , τ ′ )+

∆(τ, τ ) ∆˙(τ, τ ′ )˙∆ (τ ′ , τ ′ ) ∆¨(τ, τ ′ )} .

(10.451)

In going to the last line we have used d[∆(τ, τ )]/dτ = 2 ˙∆ (τ, τ ) following from (10.434). By interchanging the order of integration τ ↔ τ ′ , the first term in Eq. (10.451) reduces to the integral (10.449). In the last term we replace ∆¨(τ, τ ′ ) using the field equation (10.392) and the trivial equation (10.375). Thus we obtain I6 = I6R + I6div

(10.452)

with I6R I6div

β2 1 7β 2 β2 − = − , = 2 − 90 120 2 90 Z βZ β = dτ dτ ′ ∆2 (τ, τ ) δ 2 (τ − τ ′ ). 0

0

(10.453) (10.454)

762


With the help of the identity for distributions (10.368), the divergent part is calculated to be Z β β3 (10.455) I6div = δ(0) dτ ∆2 (τ, τ ) = δ(0) . 30 0 Using Eqs. (10.449) and (10.452) yields the sum of the sixth and seventh three-bubble diagrams in Eq. (10.431): η2 η2 β3 β2 (3) βf2 = − (2I6 + 16I7 ) = − 2δ(0) + . (10.456) 2 2 30 10 6,7 Adding this to (10.448), we obtain the sum of all bubble diagrams η2 β3 β2 (2) (3) βf2 + βf2 = − 2δ(0) + . 2 30 30

(10.457)

The contributions of the watermelon diagrams (10.431) correspond to the Feynman integrals Z βZ β (4) 2 βf2 = −2η dτ dτ ′ ∆2 (τ, τ ′ )˙∆˙ 2 (τ, τ ′ ) 0 0 + 4 ∆(τ, τ ′ ) ˙∆ (τ, τ ′ ) ∆˙(τ, τ ′ )˙∆˙(τ, τ ′ ) + ˙∆ 2 (τ, τ ′ ) ∆˙ 2 (τ, τ ′ ) . (10.458)

The third integral is unique and can be calculated directly: Z β Z β β2 : I10 = dτ dτ ′ ˙∆ 2 (τ, τ ′ ) ∆˙ 2 (τ, τ ′ ) = . 90 0 0

(10.459)

The second integral reads in d dimensions Z Z : I9 = dd τ dd τ ′ ∆(τ, τ ′ ) α ∆(τ, τ ′ )∆β (τ, τ ′ ) α ∆β (τ, τ ′ ). This is integrated partially to yield, in one dimension, Z Z 1 1 1 I9 = − I10 + I9′ ≡ − I10 − dτ dτ ′ ∆(τ, τ ′ ) ∆˙ 2 (τ, τ ′ )¨∆ (τ, τ ′ ). 2 2 2 The integral on the right-hand side is the one-dimensional version of Z Z 1 I9′ = − dd τ dd τ ′ ∆(τ, τ ′ )∆2β (τ, τ ′ ) αα ∆(τ, τ ′ ). 2

(10.460)

(10.461)

(10.462)

Using the field equation (10.417), going back to one dimension, and inserting ∆(τ, τ ′ ), ∆˙(τ, τ ′ ), and ¨∆ (τ, τ ′ ) from (10.391), (10.395), and (10.392), we perform all unique integrals and obtain Z 1 1 2 2 I9′ = β dτ ǫ (τ ) δ(τ ) + . (10.463) 48 240 According to Eq. (10.374), the integral over the product of distributions vanishes. Inserting the remainder and (10.459) into Eq. (10.461) gives: I9 = −

β2 . 720

(10.464)

We now evaluate the first integral in Eq. (10.458). Adding and subtracting the linear divergence yields Z βZ β Z βZ β ′ 2 ′ 2 ′ : I8 = dτ dτ ∆ (τ, τ )˙∆˙ (τ, τ ) = dτ dτ ′ ∆2 (τ, τ )δ 2 (τ −τ ′ ) 0

0

0

+

Z βZ 0

0

β

0

dτ dτ ′ ∆2 (τ, τ ′ ) ˙∆˙ 2 (τ, τ ′ )−δ 2 (τ −τ ′ ) .

(10.465)


763

10.11 Perturbative Calculation on Finite Time Intervals The finite second part of the integral (10.465) has the d-dimensional extension Z Z I8R = dd τ dd τ ′ ∆2 (τ, τ ′ ) α ∆2β (τ, τ ′ ) − ∆2γγ (τ, τ ′ ) ,

(10.466)

which after partial integration and going back to one dimension reduces to a combination of integrals Eqs. (10.464) and (10.463): I8R = −2I9 + 2I9′ = −

β2 . 72

(10.467)

The divergent part of I8 coincides with I6div in Eq. (10.454): I8div =

Z

0

β

Z

0

β

dτ dτ ′ ∆2 (τ, τ )δ 2 (τ − τ ′ ) = I6div = δ(0)

β3 . 30

(10.468)

Inserting this together with (10.459) and (10.464) into Eq. (10.458), we obtain the sum of watermelon diagrams η2 β2 β2 (4) βf2 = −2η 2 (I8 + 4I9 + I10 ) = − 4δ(0) − . (10.469) 2 30 30 For a flat space in curvilinear coordinates, the sum of the first-order diagrams vanish. To second order, the requirement of coordinate independence implies a vanishing sum of all connected diagrams (10.428)–(10.431). By adding the sum of terms in Eqs. (10.437), (10.457), and (10.469), we find indeed zero, thus confirming coordinate independence. It is not surprising that the integration rules for products of distributions derived in an infinite time interval τ ∈ [0, ∞) are applicable for finite time intervals. The singularities in the distributions come in only at a single point of the time axis, so that its total length is irrelevant. The procedure can easily be continued to higher-loop diagrams to define integrals over higher singular products of ǫ- and δ-functions. At the one-loop level, the cancellation of δ(0)s requires Z Z dτ ∆(τ, τ ) δ(0) = δ(0) dτ ∆(τ, τ ). (10.470) The second-order gave, in addition, the rule Z βZ β Z dτ dτ ′ ∆2 (τ, τ ) δ 2 (τ − τ ′ ) = δ(0) dτ ∆2 (τ, τ ), 0

(10.471)

0

To n-order we can derive the equation Z Z dτ1 . . . dτn ∆(τ1 , τ2 ) δ(τ1 , τ2 ) · · · ∆(τn , τ1 ) δ(τn , τ1 ) = δ(0) dτ ∆n (τ, τ ),

(10.472)

which reduces to Z Z

n

2

dτ1 dτn ∆ (τ1 , τ1 ) δ (τ1 − τn ) = δ(0)

Z

dτ ∆n (τ, τ ),

(10.473)

which is satisfied due to the integration rule (10.368). See Appendix 10C for a general derivation of (10.472).

10.11.5

Time Evolution Amplitude in Curved Space

The same Feynman diagrams which we calculated to verify coordinate independence appear also in the perturbation expansion of the time evolution amplitude in curved space if this is performed in normal or geodesic coordinates.

764


The path integral in curved space is derived by making the mapping from xi to qaµ in Subsection 10.11.2 nonholonomic, so that it can no longer be written as xi (τ ) = xi (q µ (τ )) but only as dxi (τ ) = ei µ (q)dq µ (τ ). Then the q-space may contain curvature and torsion, and the result of the path integral will not longer be trivial one in Eq. (10.402) but depend on Rµνλ κ (qa ) and Sµν λ (qa ). For simplicity, we shall ignore torsion. Then the action becomes (10.403) with the metric gµν (q) = ei µ (q)ei ν (q). It was shown in Subsection 10.3.2 that under nonholonomic coordinate transformations, of a timeQ Q the measure √ ¯n /6). sliced path integral transforms from the flat-space form n dD xn to n dD q gn exp(∆tR This had the consequence, in Section 10.4, that the time evolution amplitude for a particle on the surface of a sphere has an energy (10.164) corresponding to the Hamiltonian (1.414) which governs the Schrödinger equation (1.420). It contains a pure Laplace-Beltrami operator in the kinetic part. There is no extra R-term, which would be allowed if only covariance under ordinary coordinate transformations is required. This issue will be discussed in more detail in Subsection 11.1.1. Below we shall see that for perturbatively defined path integrals, the nonholonomic transformation must carry the flat-space measure into curved space as follows: ! Z β √ ¯ dτ R/8 . (10.474) DD x → DD q g exp 0

For a D-dimensional space with a general metric gµν (q) we can make use of the above proven coordinate invariance to bring the metric to the most convenient normal or geodesic coordinates (10.98) around some point qa . The advantage of these coordinates is that the derivatives and thus the affine connection vanish at this point. Its derivatives can directly be expressed in terms of the curvature tensor: ¯ τ κ µ (qa ) = − ∂κ Γ

1¯ µ ¯ σκτ µ (qa ) , Rτ κσ (qa ) + R 3

for normal coordinates.

(10.475)

Assuming qa to lie at the origin, we expand the metric and its determinant in powers of normal coordinates ∆ξ µ around the origin and find, dropping the smallness symbols ∆ in front of q and ξ in the transformation (10.98): 1 ¯ 2 ¯ δ ¯ λ κ σ τ Rµλνκ ξ λ ξ κ + η 2 R λνκ Rσµτ δ ξ ξ ξ ξ + . . . , 3 45 1 ¯ 1 ¯ µ ν 2 1 τ ¯ σ ¯ ¯ g(ξ) = 1− η R ξ ξ + η R R + R R ξµξν ξλξκ + . . . . µν µν λκ µσν λτ κ 3 18 5

gµν (ξ) = δµν + η

(10.476) (10.477)

These expansions have obviously the same power content in ξ µ as the previous one-dimensional expansions (10.222) had in q. The interaction (10.409) becomes in normal coordinates, up to order η2 : Aint tot [ξ]

=

Z

0

β

1¯ λ κ 2 1 ¯ δ ¯ λ κ σ τ ˙µ ˙ν dτ η Rµλνκ ξ ξ + η Rλµκ Rσντ δ ξ ξ ξ ξ ξ ξ 6 45 1 µ ν 2 1 σ ¯ δ µ ν λ κ ¯ ¯ + η δ(0)Rµν ξ ξ + η δ(0)Rµδν Rλσκ ξ ξ ξ ξ . 6 180

(10.478)

This has again the same powers in ξ µ as the one-dimensional interaction (10.424), leading to the same Feynman diagrams, differing only by the factors associated with the vertices. In one dimension, with the trivial vertices of the interaction (10.424), the sum of all diagrams vanishes. ¯ µνκλ and R ¯ µν , the result In curved space with the more complicated vertices proportional to R ¯ µνκλ . The dependence is easily is nonzero but depends on contractions of the curvature tensor R identified for each diagram. All bubble diagrams in (10.429)–(10.431) yield results proportional to 2 ¯ µν ¯2 R , while the watermelon-like diagrams (10.431) carry a factor R µνκλ . H. Kleinert, PATH INTEGRALS

765


When calculating the contributions of the first expectation value Aint tot [ξ] to the time evolution amplitude it is useful to reduce the D-dimensional expectation values of (10.478) to one-dimensional ones of (10.424) as follows using the contraction rules (8.63) and (8.64): hξ µ ξ ν i

λ κ µ ν ξ ξ ξ ξ D E ξ λ ξ κ ξ˙µ ξ˙ν D E ξ λ ξ κ ξ σ ξ τ ξ˙µ ξ˙ν

= = = = + +

δ µν hξ ξi ,

(10.479) κµ

δ λκ δ µν + δ λµ δ νκ + δ λν δ hξ ξ ξ ξi , (10.480) D E D ED E δ λκ δ µν hξ ξi ξ˙ ξ˙ + δ λµ δ κν + δ λν δ κµ ξ ξ˙ ξ ξ˙ , (10.481) D E δ λκ δ στ + δ λσ δ κτ + δ λτ δ κσ δ µν hξ ξ i hξ ξ i ξ˙ ξ˙ λµ κν στ δ (δ δ + δ σν δ τ κ + δ τ ν δ κσ ) + δ κµ δ σν δ λτ + δ λν δ τ σ + δ τ ν δ σλ δ σµ δ τ ν δ λκ + δ λν δ κτ + δ κν δ τ λ + δ τ µ δ λν δ κσ + δ κν δ σλ + δ σν δ λκ D ED E × hξ ξ i ξ ξ˙ ξ ξ˙ . (10.482)

Inserting these into the expectation value of (10.478) and performing the tensor contractions, we obtain Z β n D E D ED E i

int 1¯h − hξ ξi ξ˙ ξ˙ + ξ ξ˙ ξ ξ˙ + η δ(0) hξ ξi Atot [ξ] = dτ η R 6 0 D E D E2 2 ˙ ˙ 2 1 2 µνλκ λνµκ ¯ ¯ ¯ ¯ ¯ + η Rµν + Rµνλκ R + Rµνλκ R hξ ξi ξ ξ − hξ ξ i ξ ξ˙ 45 o 1 ¯2 + R ¯ µνλκ R ¯ µνλκ + R ¯ µνλκ R ¯ λνµκ hξ ξi hξ ξi . + η2 δ(0) R (10.483) µν 180 Individually, the four tensors in the brackets of (10.482) contribute the tensor contractions, using ¯ µνλκ in µν and the contraction to the Ricci tensor R ¯ λµκ δ δ µκ = R ¯λ δ : the antisymmetry of R

¯ λµκ δ R ¯ σντ δ δ λµ (δ κν δ στ + δ σν δ τ κ + δ τ ν δ κσ ) = 0, R ¯ λµκ δ R ¯ σντ δ δ κµ δ σν δ λτ + δ λν δ τ σ + δ τ ν δ σλ = R ¯ λ δ −R ¯λδ + R ¯ λ δ = 0, R ¯ λµκ δ R ¯ σντ δ δ σµ δ τ ν δ λκ + δ λν δ κτ + δ κν δ τ λ = R ¯ λµκ δ R ¯ σ δ δ λκ + R ¯ σλκδ + R ¯ σκλδ R 2 ¯ µν ¯ µνλκ R ¯ µνλκ + R ¯ µνλκ R ¯ λνµκ =− R +R ¯ λµκ δ R ¯ σντ δ δ τ µ δ λν δ κσ + δ κν δ σλ + δ σν δ λκ = R ¯ λτ κ δ (Rκλτ δ + Rλκτ δ ) = 0. R (10.484)

We now use the fundamental identity of Riemannian spaces

¯ µνλκ + R ¯ µλκν + R ¯ µκνλ = 0. R

(10.485)

By expressing the curvature tensor (10.31) in Riemannian space in terms of the Christoffel symbol (1.70) as Rµνλκ =

1 ¯ µ, Γ ¯ ν ]λκ , (∂µ ∂λ gνκ − ∂µ ∂κ gνλ − ∂ν ∂λ gµκ + ∂µ ∂κ gµλ ) − [Γ 2

(10.486)

we see that the identity (10.485) is a consequence of the symmetry of the metric and the singlevaluedness of the metric expressed by the integrability condition8 (∂λ ∂κ − ∂κ ∂λ )gµν = 0. Indeed, due to the symmetry of gµν we find ¯ µνλκ + R ¯ µλκν + R ¯ µκνλ = 1 [(∂ν ∂κ −∂κ ∂ν ) gµλ −(∂ν ∂λ −∂λ ∂ν ) gµκ −(∂λ ∂κ −∂κ ∂λ ) gµν ] = 0. R 2 The integrability has also the consequence that Rµνλκ = −Rµνκλ , 8

Rµνλκ = Rλκµν .

For the derivation see p. 1353 in the textbook [2].

(10.487)

766


Using (10.485) and (10.487) we find that ¯ µνλκ R ¯ µνλκ , ¯ µνλκ R ¯ λνµκ = 1 R R 2

(10.488)

2 ¯ µν so that the contracted curvature tensors in the parentheses of (10.483) can be replaced by R + 3 ¯ µνλκ ¯ R R . 2 µνλκ We now calculate explicitly the contribution of the first-order diagrams in (10.483) [compare (10.426)]: 1¯ βf1 = R −η . (10.489) +η + η δ(0) 6

corresponding to the analytic expression [compare (10.432)]: 1¯ βf1 = −η R 6

Z

β 0

dτ ∆(τ, τ )˙∆˙(τ, τ ) −˙∆2 (τ, τ ) − δ(0)∆(τ, τ ) .

(10.490)

Note that the combination of propagators in the brackets is different from the previous one in (10.432). Using the integrals (10.435) we find, setting η = 1: 1¯ βf1 = − R 6

Z

β 0

1 2 dτ − ∆(τ, τ ) − ˙∆ (τ, τ ) . β

(10.491)

Using Eq. (10.435), this becomes βf1 =

1¯ R 6

Z

0

β

dτ

3 β ¯ ∆(τ, τ ) = R. 2β 24

(10.492)

Adding to this the similar contribution coming from the nonholonomically transformed measure (10.474), we obtain the first-order expansion of the imaginary-time evolution amplitude 1 β ¯ (qa β|qa 0) = √ exp . (10.493) R(q ) + . . . a D 12 2πβ We now turn to the second-order contributions in η. The sum of the local diagrams (10.428) reads now # " 1 1 3 (1) ¯ µν R ¯ µν + R ¯ µνλκ R ¯ µνλκ βf2 = η 2 − + δ(0) R . (10.494) 45 2 4 In terms of the Feynman integrals, the brackets are equal to [compare (10.436)] Z

β

0

1 dτ ∆2 (τ, τ )˙∆˙(τ, τ ) − ∆(τ, τ )˙∆ 2 (τ, τ ) + δ(0)∆2 (τ, τ ) . 4

(10.495)

Inserting the equation of motion (10.393) and the right-hand equation (10.434), this becomes Z

β 0

5 5 5 1 dτ − ∆2 (τ, τ ) + δ(0)∆2 (τ, τ ) = [1 − δ(0)] . 4β 4 4 30

(10.496)

Thus we find (1) βf2

β2 = −η 1080 2

3 ¯2 2 ¯ Rµν + Rµνκλ [1 − δ(0)] . 2

(10.497)


767


Next we calculate the nonlocal contributions of order h ¯ 2 to βf coming from the cumulant * Z β Z β h 1 int 2 η2 1 ′ ¯ µν R ¯ µ′ ν ′ ξ µ (τ )ξ ν (τ ) ξ µ′ (τ ′ )ξ ν ′ (τ ′ ) dτ dτ δ 2 (0)R − Atot c = − 2 2 36 0 0 ¯ µλνκ R ¯ µ′ ν ′ ξ λ (τ )ξ κ (τ )ξ˙µ (τ )ξ˙ν (τ ) ξ µ′ (τ ′ )ξ ν ′ (τ ′ ) + 2 δ(0)R

(10.498) + i ¯ µλνκ R ¯ µ′ λ′ ν ′ κ′ ξ λ (τ )ξ κ (τ )ξ˙µ (τ )ξ˙ν (τ ) ξ λ′ (τ ′ )ξ κ′ (τ ′ )ξ˙µ′ (τ ′ )ξ˙ν ′ (τ ′ ) +R . c

The first two terms yield the connected diagrams [compare (10.429)] (2)

βf2

=−

¯ µν R ¯ µν η2 R 2 δ 2 (0) 2 36

− 4 δ(0)

−2

+

and the analytic expression diagrams [compare (10.438)]

(10.499)

,

¯ µν R ¯ µν Z β Z β η2 R =− dτ dτ ′ 2δ 2 (0)∆2 (τ, τ ′ ) (10.500) 2 36 0 0 −4 δ(0) ∆(τ, τ )˙∆ 2 (τ, τ ′ ) − 2˙∆ (τ, τ )∆(τ, τ ′ )˙∆ (τ, τ ′ ) + ∆2 (τ, τ ′ )˙∆˙(τ, τ ) .

(2) βf2

The third term in (10.498) leads to the three-bubble diagrams [compare (10.438)] (3)

βf2 = −

¯ µν R ¯ µν h η2 R 4 2 36

+2

−8

+4

+4

+2

−8

i .

(10.501)

The analytic expression for the diagrams 1,2,3 is [compare (10.441)]

(3) βf2

1,2,3

¯ µν R ¯ µν Z β Z β η2 R =− dτ dτ ′ 4 ∆(τ, τ ) ˙∆ 2 (τ, τ ′ ) ˙∆˙(τ ′ , τ ′ ) (10.502) 2 36 0 0 + 2 ˙∆˙(τ, τ )∆2 (τ, τ ′ )˙∆˙(τ ′ , τ ′ ) − 8 ˙∆ (τ, τ )∆(τ, τ ′ ) ˙∆ (τ, τ ′ ) ˙∆˙(τ ′ , τ ′ ) .

and for 4 and 5 [compare (10.447)] :

(3) βf2

¯ µν R ¯ µν ¯ µν R ¯ µν Z β Z β η2 R η2 R 4 =− (I4 +I5 ) = − dτ dτ ′ ˙∆ (τ, τ )˙∆ (τ, τ ′ )∆(τ, τ ′ ) 2 36 2 36 β 4,5 0 0 ¯ µν R ¯ µν 1 η2 R = − β2. (10.503) 2 36 45

For the diagrams 6 and 7, finally, we obtain [compare (10.457)] ¯ µν R ¯ µν ¯ µν R ¯ µν η2 R η2 R β3 β2 (3) βf2 = − (2I6 − 8I7 ) = − 2δ(0) − . 2 36 2 36 30 6 6,7

(10.504)

The sum of all bubbles diagrams (10.500) and (10.502) is therefore (2)

βf2

(3)

+ βf2

= η2

β2 ¯2 β3 ¯2 Rµν − η 2 δ(0) R . 432 1080 µν

(10.505)

¯ 2 in Eq. (10.497), leaving only a finite This compensates exactly the δ(0)-term proportional to R µν second-order term (2)

(2)

βf1 + βf2

(3)

+ βf2

= η2

β2 ¯2 β3 ¯2 2 ¯2 Rµν − R + η δ(0) R . µνκλ 720 1080 µνκλ

(10.506)

768


Finally we calculate the second-order watermelon diagrams (10.431) which contain the initially ambiguous Feynman integrals we make the following observation. Their sum is [compare (10.431)] η2 1 ¯ (4) ¯ µνλκ + R ¯ µνλκ R ¯ µλνκ βf2 = − 2 Rµνλκ R −2 + , (10.507) 2 36 corresponding to the analytic expression Z βZ β η2 3 1 ¯ 2 (3) Rµνκλ dτ dτ ′ ∆2 (τ, τ ′ )˙∆˙ 2 (τ, τ ′ ) βf2 = − 2 2 2 36 0 0 −2 ∆(τ, τ ′ ) ˙∆ (τ, τ ′ ) ∆˙(τ, τ ′ )˙∆˙(τ, τ ′ ) + ˙∆ 2 (τ, τ ′ ) ∆˙ 2 (τ, τ ′ ) =−

η2 ¯ 2 R (I8 − 2I9 + I10 ) , 24 µνκλ

(10.508)

where the integrals I8 , I9 , and I10 were evaluated before in Eqs. (10.468), (10.467), (10.464), and (10.459). Substituting the results into Eq. (10.508) and using the rules (10.368) and (10.374), we obtain Z βZ β β3 ¯2 η2 ¯ 2 (3) dτ dτ ′ ∆2 (τ, τ )δ 2 (τ − τ ′ ) = −η 2 R δ(0). (10.509) βf2 = − Rµνκλ 24 720 µνκλ 0 0 Thus the only role of the watermelon diagrams is to cancel the remaining δ(0)-term proportional ¯2 to R µνκλ in Eq. (10.506). It gives no finite contribution. The remaining total sum of all second-order contribution in Eq. (10.506). changes the diagonal time evolution amplitude (10.493) to 1 β ¯ β2 ¯2 2 ¯ (qa β|qa 0) = √ exp R(q ) + R − R + . . . . (10.510) a µν D 12 720 µνκλ 2πβ

In Chapter 11 we shall see that this expression agrees with what has been derived in Schrödinger ˆ = −∆/2 which contains only is the Laplacequantum mechanics from a Hamiltonian operator H Beltrami operator ∆ = g −1/2 ∂µ g 1/2 g µν (q)∂ν of Eq. (1.377) and no extra R-term: D E (qa β | qa 0) ≡ eβ∆/2 = (qa | eβ∆/2 | qa ) (10.511) β ¯ β2 1 ¯2 1 ¯ µνκλ ¯ 1 ¯ µν R ¯ µν +. . . . 1+ R+ = √ R + R Rµνκλ − R D 12 2 144 360 2πβ

This expansion due to DeWitt and Seeley will be derived in Section 11.6, the relevant equation being (11.110). Summarizing the results we have found that for one-dimensional q-space as well as for a Ddimensional curved space in normal coordinates, our calculation procedure on a one-dimensional τ -axis yields unique results. The procedure uses only the essence of the d-dimensional extension, together with the rules (10.368) and (10.374). The results guarantee the coordinate independence of path integrals. They also agree with the DeWitt-Seeley expansion of the short-time amplitude to be derived in Eq. (11.110). The agreement is ensured by the initially ambiguous integrals I8 and I9 satisfying the equations I8R + 4I9 + I10 = − I8R − 2I9 + I10 = 0,

β2 , 120

(10.512) (10.513)

as we can see from Eqs. (10.469) and (10.508). Since the integral I10 = β 2 /90 is unique, we must have I9 = −β 2 /720 and I8R = −β 2 /72, and this is indeed what we found from our integration rules. The main role of the d-dimensional extension of the τ -axis is, in this context, to forbid the application of the equation of motion (10.393) to correlation functions ˙∆˙(τ, τ ′ ). This would H. Kleinert, PATH INTEGRALS


769

fix immediately the finite part of the integral I8 to the wrong value I8R = −β 2 /18, leaving the integral I9 which fixes the integral over distributions (10.374). In this way, however, we could only satisfy one of the equations (10.512) and (10.513), the other would always be violated. Thus, any regularization different from ours will ruin immediately coordinate independence.

10.11.6

Covariant Results for Arbitrary Coordinates

It must be noted that if we were to use arbitrary rather than Riemann normal coordinates, we would find ambiguous integrals already at the two-loop level: Z βZ β : I14 = dτ dτ ′ ˙∆ (τ, τ ′ ) ∆˙(τ, τ ′ )˙∆˙(τ, τ ′ ), (10.514) 0

:

I15 =

Z

0

0

β

Z

β

dτ dτ ′ ∆(τ, τ ′ )˙∆˙ 2 (τ, τ ′ ).

(10.515)

0

Let us show that coordinate independence requires these integrals to have the values I14 = β/24,

R I15 = −β/8,

(10.516)

where the superscript R denotes the finite part of an integral. We study first the ambiguities arising in one dimension. Without dimensional extension, the values (10.516) would be incompatible with partial integration and the equation of motion (10.392). In the integral (10.514), we use the symmetry ¨∆ (τ, τ ′ ) = ∆¨(τ, τ ′ ), apply partial integration twice taking care of nonzero boundary terms, and obtain on the one hand Z Z Z Z 1 β β d 2 1 β β I14 = dτ dτ ′ ˙∆ (τ, τ ′ ) ∆˙ (τ, τ ′ ) = − dτ dτ ′ ∆˙ 2 (τ.τ ′ ) ¨∆ (τ, τ ′ ) 2 0 0 dτ 2 0 0 Z Z Z 1 β 3 β d 1 β β (10.517) =− dτ dτ ′ ′ ∆˙ 3 (τ, τ ′ ) = dτ ∆˙ (τ, 0)− ∆˙ 3 (τ, β) = . 6 0 0 dτ 6 0 12 On the other hand, we apply Eq. (10.393) and perform two regular integrals, reducing I14 to a form containing an undefined integral over a product of distributions: Z βZ β Z Z 1 β β dτ dτ ′ ˙∆ (τ, τ ′ ) ∆˙(τ, τ ′ )δ(τ − τ ′ ) − dτ dτ ′ ˙∆ (τ, τ ′ ) ∆˙(τ, τ ′ ) I14 = β 0 0 0 0 Z β Z βZ β 1 β = dτ dτ ′ − ǫ2 (τ − τ ′ )δ(τ − τ ′ ) + dτ ˙∆2 (τ, τ ) + 4 12 0 0 0 Z 1 1 = β − dτ ǫ2 (τ )δ(τ ) + . (10.518) 4 6

A third, mixed way of evaluating I14 employs one partial integration as in the first line of Eq. (10.517), then the equation of motion (10.392) to reduce I14 to yet another form Z Z 1 β β I14 = dτ dτ ′ ∆˙ 2 (τ, τ ′ )δ(τ − τ ′ ) = 2 0 0 Z Z Z 1 β β 1 β = dτ dτ ′ ǫ2 (τ − τ ′ )δ(τ − τ ′ ) + dτ ˙∆2 (τ, τ ) 8 0 0 2 0 Z 1 1 2 = β dτ ǫ (τ )δ(τ ) + . (10.519) 8 24 We now see that if we set [compare with the correct equation (10.374)] Z 1 dτ [ǫ(τ )]2 δ(τ ) ≡ , (false), 3

(10.520)

770


the last two results (10.519) and (10.518) coincide with the first in Eq. (10.517). The definition (10.520) would obviously be consistent with partial integration if we insert δ(τ ) = ǫ(τ ˙ )/2: Z Z Z 1 1 d 1 dτ [ǫ(τ )]2 δ(τ ) = dτ [ǫ(τ )]2 ǫ(τ ˙ )= dτ [ǫ(τ )]3 = . (10.521) 2 6 dτ 3 In spite of this consistency with partial integration and the equation of motion, Eq. (10.520) is incompatible with the requirement of coordinate independence. This can be seen from the discrepancy between the resulting value I14 = β/12 and the necessary (10.516). In earlier work on the subject by other authors [31]– [36], this discrepancy was compensated by adding the above-mentioned (on p. 726) noncovariant term to the classical action, in violation of Feynman’s construction rules for path integrals. A similar problem appears with the other Feynman integral (10.515). Applying first Eq. (10.393) we obtain I15 =

Z

0

β

β

Z

2 dτ dτ ∆(τ, τ )δ (τ − τ ) − β ′

0

′

2

′

Z

0

β

1 dτ ∆(τ, τ ) + 2 β

Z

0

β

Z

β

dτ dτ ′ ∆(τ, τ ′ ).

0

(10.522)

For the integral containing the square of the δ-function we must postulate the integration rule (10.368) to obtain a divergent term div I15 = δ(0)

Z

0

β

dτ ∆(τ, τ ) = δ(0)

β2 , 6

(10.523)

which is proportional to δ(0), and compensates a similar term from the measure. The remaining R integrals in (10.522) are finite and yield the regular part I15 = −β/4, which we shall see to be inconsistent with coordinate invariance. In another calculation of I15 , we first add and subtract the UV-divergent term, writing Z βZ β β2 I15 = dτ dτ ′ ∆(τ, τ ′ ) ˙∆˙ 2 (τ, τ ′ ) − δ 2 (τ − τ ′ ) + δ(0) . (10.524) 6 0 0 Replacing δ 2 (τ − τ ′ ) by the square of the left-hand side of the equation of motion (10.392), and integrating the terms in brackets by parts, we obtain Z βZ β R I15 = dτ dτ ′ ∆(τ, τ ′ ) ˙∆˙ 2 (τ, τ ′ ) − ∆¨ 2 (τ, τ ′ ) 0

=

Z

0

β

0

− =

Z

0

Z

β

dτ dτ ′ [−˙∆ (τ, τ ′ ) ∆˙(τ, τ ′ )˙∆˙(τ, τ ′ ) − ∆(τ, τ ′ ) ∆˙(τ, τ ′ )¨∆ d(τ, τ ′ )]

0

β

Z

β

0

−I14 +

dτ dτ ′ − ∆˙ 2 (τ, τ ′ ) ∆¨(τ, τ ′ ) − ∆(τ, τ ′ ) ∆˙(τ, τ ′ ) ∆¨˙(τ, τ ′ )

Z

0

β

Z

0

β

dτ dτ ′ ∆˙ 2 (τ, τ ′ ) ∆¨(τ, τ ′ ) = −I14 − β/6.

(10.525)

The value of the last integral follows from partial integration. For a third evaluation of I15 we insert the equation of motion (10.392) and bring the last integral in the fourth line of (10.525) to Z Z βZ β 1 1 − dτ dτ ′ ∆˙ 2 (τ, τ ′ )δ(τ − τ ′ ) = −β dτ ǫ2 (τ )δ(τ ) + . (10.526) 4 12 0 0 All three ways of calculation lead, with the assignment (10.520) to the singular integral, to the R same result I15 = −β/4 using the rule (10.520). This, however, is again in disagreement with the H. Kleinert, PATH INTEGRALS


771

R coordinate-independent value in Eq. (10.516). Note that both integrals I14 and I15 are too large by a factor 2 with respect to the necessary (10.516) for coordinate independence. How can we save coordinate independence while maintaining the equation of motion and partial integration? The direction in which the answer lies is suggested by the last line of Eq. (10.519): R 2 we must find a consistent way to have an integral dτ [ǫ(τ )] δ(τ ) = 0, as in Eq. (10.374), instead of the false value (10.520), which means that we need a reason for forbidding the application of partial integration to this singular integral. For the calculation at the infinite time interval, this problem was solved in Refs. [23]–[25] with the help of dimensional regularization. In dimensional regularization, we would write the Feynman integral (10.514) in d dimensions as Z Z d dd x dd x′ µ ∆(x, x′ )∆ν (x, x′ ) µ ∆ν (x, x′ ), (10.527) I14 =

and see that the different derivatives on µ ∆ν (x, x′ ) prevent us from applying the field equation (10.417), in contrast to the one-dimensional calculation. We can, however, apply partial integration as in the first line of Eq. (10.517), and arrive at Z Z 1 d I14 = − dd x dd x′ ∆2ν (x, x′ )∆µµ (x, x′ ). (10.528) 2 In contrast to the one-dimensional expression (10.517), a further partial integration is impossible. Instead, we may apply the field equation (10.417), go back to one dimension, and apply the integration rule (10.374) as in Eq. (10.519) to obtain the correct result I14 = β/24 guaranteeing coordinate independence. The Feynman integral (10.515) for I15 is treated likewise. Its d-dimensional extension is Z Z 2 d I15 = dd x dd x′ ∆(x, x′ ) [µ ∆ν (x, x′ )] . (10.529) The different derivatives on µ ∆ν (x, x′ ) make it impossible to apply a dimensionally extended version of equation (10.393) as in Eq. (10.522). We can, however, extract the UV-divergence as in Eq. (10.524), and perform a partial integration on the finite part which brings it to a dimensionally extended version of Eq. (10.525): Z R I15 = −I14 + dd x dd x′ ∆2ν (x, x′ )∆µµ (x, x′ ). (10.530) On the right-hand side we use the field equation (10.417), as in Eq. (10.526), return to d = 1, and R use the rule (10.374) to obtain the result I15 = −I14 −β/12 = −β/8, again guaranteeing coordinate independence. Thus, by keeping only track of a few essential properties of the theory in d dimensions we indeed obtain a simple consistent procedure for calculating singular Feynman integrals. All results obtained in this way ensure coordinate independence. They agree with what we would obtain using the one-dimensional integration rule (10.374) for the product of two ǫ- and one δ-distribution. Our procedure gives us unique rules telling us where we are allowed to apply partial integration and the equation of motion in one-dimensional expressions. Ultimately, all integrals are brought to a regular form, which can be continued back to one time dimension for a direct evaluation. This procedure is obviously much simpler than the previous explicit calculations in d dimensions with the limit d → 1 taken at the end. The coordinate independence would require the equations (10.516). Thus, although the calculation in normal coordinates are simpler and can be carried more easily to higher orders, the perturbation in arbitrary coordinates help to fix more ambiguous integrals. Let us see how the integrals I14 and I15 arise in the perturbation expansion of the time evolution amplitude in arbitrary coordinates up to the order η, and that the values in (10.516) are necessary

772


to guarantee a covariant result. We use arbitrary coordinates and expand the metric around the origin. Dropping the increment symbol δ in front of δq µ , we write: gµν (q)

= δµν +

√ 1 η(∂λ gµν )q λ + η (∂λ ∂κ gµν )q λ q κ , 2

(10.531)

with the expansion parameter η keeping track of the orders of the perturbation series. At the end it will be set equal to unity. The determinant has the expansion to order η: √

log g(q) =

η g µν (∂λ gµν )q λ + η

1 µν g [(∂λ ∂κ gµν ) − g στ (∂λ gµσ )(∂κ gντ )]q λ q κ . 2

(10.532)

The total interaction (10.409) becomes Z β nh i 1√ 1 int Atot [q] = dτ η (∂κ gµν )q κ + η (∂λ ∂κ gµν )q λ q κ q˙µ q˙ν (10.533) 2 4 0 h i o 1 1√ η δ(0)g µν (∂κ gµν )q κ − η δ(0) g µν (∂λ ∂κ gµν ) −g στ (∂λ gµσ )(∂κ gντ ) q λ q κ . − 2 4 Using the relations following directly from the definition of the Christoffel symbols (1.70) and (1.71), ∂κ gµν g

µν

=

∂λ ∂κ gµν = (∂τ ∂λ gµν ) = =

−gµσ gντ ∂κ g στ = Γκµν + Γκνµ = 2Γκ{µν} ,

g µν (∂κ gµν ) = 2Γκµ µ ,

∂λ Γκµν + ∂λ Γκνµ = ∂λ Γκ{µν} , 2g µν ∂κ Γσµν = 2∂κ (g µν Γσµν ) − 2Γσµν ∂κ g µν 2 (∂κ Γσµ µ + g µν Γσµτ Γλν τ + Γσµ ν Γλν µ ) ,

(10.534)

this becomes Aint tot [q] =

β

η √ η Γκµν q κ + ∂λ Γκµν q λ q κ q˙µ q˙ν 2 0 h√ i η − δ(0) η Γµκ µ q κ + ∂λ Γτ µ µ q λ q τ . 2

Z

dτ

(10.535)

The derivative of the Christoffel symbol in the last term can also be written differently using the identity ∂λ g µν = −g µσ g ντ ∂λ gστ as follows: ∂λ Γτ µ µ

∂λ g µν Γτ µν = g µν ∂λ Γτ µν = g µν ∂λ Γτ µν − g µσ g ντ (∂λ gστ ) Γτ µν ∂λ Γτ µν − (Γλµν + Γλνµ ) Γτ µν .

To first order in η, we obtain from the first cumulant Aint tot [q] c : = =

(1)

βf1

=η

Z

0

β

dτ

D1

E 1 ∂λ Γκµν q λ q κ q˙µ q˙ν − δ(0) ∂λ Γτ µ µ q λ q τ , 2 2 c

(10.536)

(10.537)

the diagrams (10.489) corresponding to the analytic expression [compare (10.490)] Z β n o η (1) βf1 = ∂λ Γκ{µν} dτ g µνg κλ ˙∆˙(τ , τ )∆(τ , τ )+2g µκg νλ ˙∆2 (τ , τ )−δ(0)g µνg κλ ∆(τ , τ ) 2 0 Z β η − g λκ (Γλν µ Γµκ ν + g τ µ Γτ κ ν Γµλν ) δ(0) dτ ∆(τ, τ ). (10.538) 2 0 Replacing ˙∆˙(τ, τ ) by δ(0) − 1/β according to (10.393), and using the integrals (10.435), the δ(0)terms in the first integral cancel and we obtain η g µν g κλ g λκ (1) µ ν τµ ν βf1 = −β (∂λ Γµκν −∂µ Γκλν )− (Γλν Γµκ + g Γτ κ Γµλν )δ(0) . (10.539) 4 6 3 H. Kleinert, PATH INTEGRALS

773

10.11 Perturbative Calculation on Finite Time Intervals In addition, there are contributions of order η from the second cumulant −

η 2!

Z

β

dτ

0

Z

β 0

D dτ ′ [Γκµν q κ (τ )q˙µ (τ )q˙ν (τ ) + δ(0)Γµκ µ q κ (τ )]

× [Γκµν q κ (τ ′ )q˙µ (τ ′ )q˙ν (τ ′ ) + δ(0)Γµκ µ q κ (τ ′ )]

E

c

,

(10.540)

These add to the free energy (2)

βf1

(3)

βf1

(4)

βf1

(5)

βf1

(6)

βf1

h η = − g λκ Γλµ µ Γκν ν 2

= − ηΓλµ µ (g λκ Γκν ν

+ δ 2 (0)

− 2δ(0) h + g νκ Γνκ λ )

− δ(0)

η = − (g µλ g κτ Γµλ ν Γκτ ν + g µν Γµκ κ Γνλ λ + 2g µν Γµν κ Γκλ λ ) 2 η µκ νλ = − (g g Γµλ τ Γκβ + 3g µκ Γµλ τ Γτ κ λ ) , 2 η = − g λκ (Γλν µ Γµκ ν + g µτ Γτ κ ν Γµλν ) . 2

i ,

i , ,

(10.541)

The Feynman integrals associated with the diagrams in the first and second lines are Z Z I11 = dτ dτ ′ ˙∆˙(τ, τ )∆(τ, τ ′ )˙∆˙(τ ′ , τ ′ )−2δ(0) ˙∆˙(τ, τ ) ∆(τ, τ ′ )+δ 2 (0) ∆(τ, τ ′ )

(10.542)

and I12 =

Z Z

dτ dτ ′ {˙∆ (τ, τ )˙∆ (τ, τ ′ )˙∆˙(τ ′ , τ ′ ) − δ(0) ˙∆ (τ, τ )˙∆ (τ, τ ′ )} ,

(10.543)

respectively. Replacing in Eqs. (10.542) and (10.543) ˙∆˙(τ, τ ) and ˙∆˙(τ ′ , τ ′ ) by δ(0) − 1/β leads to cancellation of the infinite factors δ(0) and δ 2 (0) from the measure, such that we are left with I11

1 = 2 β

Z

β

dτ

0

Z

β

dτ ′ ∆(τ, τ ′ ) =

0

β 12

(10.544)

and I12

1 =− β

Z

0

β

dτ

Z

0

β

dτ ′ ˙∆ (τ, τ )˙∆ (τ, τ ′ ) = −

β . 12

(10.545)

The Feynman integral of the diagram in the third line of Eq. (10.541) has d-dimensional extension Z Z I13 = dτ dτ ′ ˙∆ (τ, τ ) ∆˙(τ ′ , τ ′ )˙∆˙(τ, τ ′ ) Z Z → dd x dd x′ µ ∆(x, x)∆ν (x′ , x′ )µ ∆ν (x, x′ ). (10.546) Integrating this partially yields I13 =

1 β

Z Z

dτ dτ ′ ∆˙(τ, τ ′ )˙∆ (τ ′ , τ ′ ) =

1 β

Z

0

β

dτ

Z

0

β

dτ ′ ˙∆ (τ, τ )˙∆ (τ, τ ′ ) =

β , 12 (10.547)

774


where we have interchanged the order of integration τ ↔ τ ′ in the second line of Eq. (10.547) and used d[˙∆ (τ, τ )]/dτ = −1/β. Multiplying the integrals (10.544), (10.545), and (10.547) by corresponding vertices in Eq. (10.541) and adding them together, we obtain (2)

βf1

(3)

(4)

+ βf1 + βf1

=−

ηβ µν κλ g g Γµν τ Γκλτ . 24 (5)

(10.548)

(6)

The contributions of the last three diagrams in βf1 and βf1 of (10.541) are determined by the initially ambiguous integrals (10.514) and (10.515) to be equal to I14 = −β/24 and I15 = −β/8 + δ(0)β 2 /6, respectively. Moreover, the δ(0)-part in the latter, when inserted into the last (6) line of Eq. (10.541) for f1 , is canceled by the contribution of the local diagram with the factor (1) δ(0) in f1 of (10.539). We see here an example that with general coordinates, the divergences containing powers of δ(0) no longer cancel order by order in h ¯ , but do so at the end. R Thus only the finite part I15 = −β/24 remains and we find o ηn (5) (6)R R R βf1 +βf1 = − g µκ g νλ Γµλ τ Γκντ I14 +I15 + g λκ Γλν µ Γµκ ν 3I14 +I15 2 ηβ µν κλ = g g Γµκ σ Γνλσ . (10.549) 24 By adding this to (10.548), we find the sum of all diagrams in (10.541) as follows 6 X i=2

(i)

βf1 = −

ηβ µν κλ g g Γµν σ Γκλσ − Γµκ σ Γjl, n ). 24

(10.550)

Together with the regular part of (10.539) in the first line, this yields the sum of all first-order diagrams 6 X i=1

(i)

βf1 = −

ηβ ¯ ηβ µν κλ g g Rλµκν = R. 24 24

(10.551)

The result is covariant and agrees, of course, with Eq. (10.491) derived with normal coordinate. Note that to obtain this covariant result, the initially ambiguous integrals (10.514) and (10.515) over distributions appearing ni Eq. (10.549) mast satisfy R I14 + I15 R 3I14 + I15

β , 12 = 0, = −

(10.552)

which leaves only the values (10.516).

10.12

Effective Classical Potential in Curved Space

In Chapter 5 we have seen that the partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor exp[−βV eff cl (x0 )], where B(x0 ) = V eff cl (x0 ) is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of R integration is the temporal path average x0 ≡ β −1 0β dτ x(τ ). In this section we generalize this concept to curved space, and show how to calculate perturbatively the high-temperature expansion of V eff cl (q0 ). The requirement of independence under coordinate transformations q µ (τ ) → q ′µ (τ ) introduces subtleties into the definition H. Kleinert, PATH INTEGRALS

10.12 Effective Classical Potential in Curved Space

775

and treatment of the path average q0µ , and covariance is achieved only with the help of a procedure invented by Faddeev and Popov [49] to deal with gauge freedoms in quantum field theory. In the literature, attempts to introduce an effective classical potential in curved R −1 β space around a fixed temporal average q0 ≡ q¯(τ ) ≡ β 0 dτ q(τ ) have so far failed and produced a two-loop perturbative result for V eff cl (q0 ) which turned out to deviate from the covariant one by a noncovariant total derivative [34], in contrast to the covariant result (10.493) obtained with Dirichlet boundary conditions. For this reason, perturbatively defined path integrals with periodic boundary conditions in curved space have been of limited use in the presently popular first-quantized worldline approach to quantum field theory (also called the string-inspired approach reviewed in Ref. [50]). In particular, is has so far been impossible to calculate with periodic boundary conditions interesting quantities such as curved-space effective actions, gravitational anomalies, and index densities, all results having been reproduced with Dirichlet boundary conditions [46, 52]. The development in this chapter cures the problems by exhibiting a manifestly covariant integration procedure for periodic paths [51]. It is an adaption of similar procedures used before in the effective action formalism of two-dimensional sigmamodels [46]. Covariance is achieved by expanding the fluctuations in the neighborhood of any given point in powers of geodesic coordinates, and by a covariant definition of a path average different from the naive temporal average. As a result, we shall find the same locally covariant perturbation expansion of the effective classical potential as in Eq. (10.493) calculated with Dirichlet boundary conditions. All problems encountered in the literature occur in the first correction terms linear in β in the time evolution amplitude. It will therefore be sufficient to consider only to lowest-order perturbation expansion. For this reason we shall from now on drop the parameter of smallness η used before.

10.12.1

Covariant Fluctuation Expansion

We want to calculate the partition function from the functional integral over all periodic paths Z =

I

q

D D q g(q)e−A[q] ,

(10.553)

where the symbol indicates the periodicity of the paths. By analogy with (2.435), we split the paths into a time-independent and a time-dependent part: H

q µ = q0µ + η µ (τ ),

(10.554)

with the goal to express the partition function as in Eq. (3.808) by an ordinary integral over an effective classical partition function Z=

Z

dD q0 q −βV eff cl (q0 ) , √ D g(q0 ) e 2πβ

(10.555)

776


where V eff cl (q0 ) is the curved-space version of the effective classical partition function. For a covariant treatment, we parametrize the small fluctuations η µ (τ ) in terms of the prepoint normal coordinates ∆ξ µ (τ ) of the point q0µ introduced in Eq. (10.98), which are here geodesic due to the absence of torsion. Omitting the smallness symbols ∆, there will be some nonlinear decomposition q µ (τ ) = q0µ + η µ (q0 , ξ),

(10.556)

where η µ (q0 , ξ) = 0 for ξ µ = 0. Inverting the relation (10.98) we obtain 1¯ µ 1¯ µ σ τ σ τ κ η µ (q0 , ξ) = ξ µ − Γ στ (q0 )ξ ξ − Γστ κ (q0 )ξ ξ ξ − . . . , 2 6

(10.557)

¯ στ ...κ µ (q0 ) with more than two subscripts are defined simiwhere the coefficients Γ larly to covariant derivatives with respect to lower indices (they are not covariant quantities): ¯ στ κ µ (q0 ) = ∇κ Γ ¯ στ µ = ∂κ Γ ¯ στ µ − 2Γ ¯ κσ ν Γ ¯ ντ µ , . . . . Γ

(10.558)

¯ στ ...κ µ (q0 ) If the initial coordinates q µ are themselves geodesic at q0µ , all coefficients Γ in Eq. (10.557) are zero, so that η µ (τ ) = ξ µ (τ ), and the decomposition (10.556) is linear. In arbitrary coordinates, however, η µ (τ ) does not transform like a vector under coordinate transformations, and we must use the nonlinear decomposition (10.556). We now transform the path integral (10.553) to the new coordinates ξ µ (τ ) using Eqs. (10.556)–(10.558). The perturbation expansion for the transformed path integral over ξ µ (τ ) is constructed for any chosen q0µ by expanding the total action (10.407) including the measure factor (10.405) in powers of small linear fluctuations ξ µ (τ ). Being interested only in the lowest-order contributions we shall from now on drop the parameter of smallness η counting the orders in the earlier perturbation expansions. This is also useful since the similar symbol η µ (τ ) is used here to describe the path fluctuations. The action relevant for the terms to be calculated here consists of a free action, which we write after a partial integration as A(0) [q0 , ξ] = gµν (q0 )

Z

0

β

1 dτ ξ µ (τ )(−∂τ2 )ξ ν (τ ), 2

(10.559)

and an interaction which contains only the leading terms in (10.478): Aint tot [q0 ; ξ] =

Z

0

β

dτ

1¯ 1 ¯ µν ξ µ ξ ν . Rµλνκ ξ λξ κ ξ˙µ ξ˙ν + δ(0)R 6 6

(10.560)

The partition function (10.553) in terms of the coordinates ξ µ (τ ) is obtained from the perturbation expansion Z=

I

q

(0) [q

D D ξ(τ ) g(q0) e−A

int 0 ,ξ]−Atot [q0 ,ξ]

.


777


The path integral (10.561) cannot immediately be calculated perturbatively in the standard way, since the quadratic form of the free action (10.559) is degenerate. The spectrum of the operator −∂τ2 in the space of periodic functions ξ µ (τ ) has a zero mode. The zero mode is associated with the fluctuations of the temporal average of ξ µ (τ ): ξ0µ

= ξ¯µ ≡ β −1

Z

0

β

dτ ξ µ (τ ).

(10.562)

Small fluctuations of ξ0µ have the effect of moving the path as a whole infinitesimally through the manifold. The same movement can be achieved by changing q0µ infinitesimally. Thus we can replace the integral over the path average ξ0µ by an integral over q0µ , provided that we properly account for the change of measure arising from such a variable transformation. Anticipating such a change, the path average (10.562) can be set equal to zero eliminating the zero mode in the fluctuation spectrum. The basic free correlation function hξ µ (τ )ξ ν (τ ′ )i can then easily be found from its spectral representation as shown in Eq. (3.251). The result is q ¯ τ ′ ), hξ µ (τ )ξ ν (τ ′ )i 0 = g µν (q0 )(−∂τ2 )−1 δ(τ − τ ′ ) = g µν (q0 )∆(τ,

(10.563)

¯ τ ′ ) is a short notation for the translationally invariant periodic Green where ∆(τ, function Gp0,e′ (τ ) of the operator −∂τ2 without the zero mode in Eq. (3.251) (for a plot see Fig. 3.4): ′ 2 ′ ¯ τ ′ ) = ∆(τ ¯ − τ ′ ) ≡ (τ − τ ) − |τ − τ | + β , ∆(τ, 2β 2 12

¯ β]. τ, τ ′ ∈ [0, h

(10.564)

This notation is useful since we shall have to calculate Feynman integrals of precisely the same form as previously with the Dirichlet-type correlation function Eq. (10.391). In contrast to (10.392) and (10.393) for ∆(τ, τ ′ ), the translational invariance of the ¯ τ ′ ) = ∆(τ ¯ − τ ′ ), so that the first time periodic correlation function implies that ∆(τ, ¯ τ ′ ) have opposite signs: derivatives of ∆(τ, τ − τ ′ ǫ(τ − τ ′ ) ¯ τ ′ ) = −∆˙(τ, ¯ ˙∆(τ, τ ′) ≡ − , β 2

τ, τ ′ ∈ [0, h ¯ β],

(10.565)

and the three possible double time derivatives are equal, up tp a sign: ¯ ¯ ¯ τ ′ ) = −∆¨(τ, −¨∆(τ, τ ′ ) = ˙∆˙(τ, τ ′ ) = δ(τ − τ ′ ) − 1/β.

(10.566)

The right-hand side contains an extra term on the right-hand side due to the missing zero eigenmode in the spectral representation of the δ-function: 1 X −iωm (τ −τ ′ ) 1 e = δ(τ − τ ′ ) − . β m6=0 β

(10.567)

The third equation in (10.566) happens to coincide with the differential equation (10.393) in the Dirichlet case. All three equations have the same right-hand side ¯ τ ′ ). due to the translational invariance of ∆(τ,

778

10.12.2


Arbitrariness of q0µ

We now take advantage of an important property of the perturbation expansion of the partition function (10.561) around q µ (τ ) = q0µ : the independence of the choice of q0µ . The separation (10.556) into a constant q0µ and a time-dependent ξ µ (τ ) paths must lead to the same result for any nearby constant q0′ µ on the manifold. The result must therefore be invariant under an arbitrary infinitesimal displacement µ q0µ → q0ε = q0µ + εµ ,

|ε| ≪ 1.

(10.568)

In the path integral, this will be compensated by some translation of fluctuation coordinates ξ µ (τ ), which will have the general nonlinear form ξ µ → ξεµ = ξ µ − εν Qµν (q0 , ξ).

(10.569)

The transformation matrix Qµν (q0 , ξ) satisfies the obvious initial condition Qµν (q0 , 0) = δνµ . The path q µ (τ ) = q µ (q0 , ξ(τ )) must remain invariant under simultaneous transformations (10.568) and (10.569), which implies that δq µ ≡ qεµ − q µ = εν Dν q µ (q0 , ξ) = 0 ,

(10.570)

where Dµ is the infinitesimal transition operator Dµ =

∂ ∂ ν . µ − Qµ (q0 , ξ) ∂q0 ∂ξ ν

(10.571)

Geometrically, the matrix Qµν (q0 , ξ) plays the role of a locally flat nonlinear connection [46]. It can be calculated as follows. We express the vector q µ (q0 , ξ) in terms of the geodesic coordinates ξ µ using Eqs. (10.556), (10.557), and (10.558), and substitute this into Eq. (10.570). The coefficients of εν yield the equations δνµ +

∂η µ (q0 , ξ) ∂η µ (q0 , ξ) κ − Q (q , ξ) = 0, ν 0 ∂q0ν ∂ξ κ

(10.572)

where by Eq. (10.557): ∂η µ (q0 , ξ) 1 ¯ µ σ τ = − ∂ν Γ (στ ) (q0 )ξ ξ − . . . , ν ∂q0 2

(10.573)

and ∂η µ (q0 , ξ) ¯ (νσ) µ (q0 )ξ σ − 1 Γ ¯ (νστ ) µ (q0 )ξ σ ξ τ −. . . = δνµ − Γ ν ∂ξ 2 1 ¯ µ ¯ κ¯ µ ¯ κ¯ µ σ τ µ σ 1 µ µ ¯ ¯ = δν − Γνσ ξ − ∂σ Γντ + ∂ν Γστ −2Γτ ν Γκσ − Γτ σ Γκν ξ ξ −. . . . (10.574) 3 2 µ To find Qν (q0 , ξ), we invert the expansion (10.574) to 

∂η(q0 , ξ)  ∂ξ

!−1µ ¯µ  = δµ + Γ ν

σ νσ ξ +

ν

=

∂ξ µ (q0 , η) ∂η ν

1 σ τ ¯ µ + 1 ∂ν Γ ¯µ + Γ ¯κ Γ ¯µ ¯κ ¯µ ∂σ Γ ντ στ τ ν κσ− Γτ σ Γκν ξ ξ +. . .(10.575) 3 2

!

, η=η(q0 ,ξ)


779


the last equality indicating that the result (10.575) can also be obtained from the original expansion (10.98) in the present notation [compare (10.557)]: 1˜ µ 1˜ µ σ τ σ τ κ ξ µ (q0 , η) = η µ + Γ στ (q0 )η η + Γστ κ (q0 )η η η + . . . , 2 6

(10.576)

with coefficients ¯ στ µ , ˜ στ µ (q0 ) = Γ Γ ˜ στ κ µ (q0 ) = Γ ¯ στ κ µ + 3Γ ¯ κσ ν Γ ¯ ντ µ = ∂κ Γ ¯ στ µ + Γ ¯ κσ ν Γ ¯ ντ µ , Γ .. . .

(10.577)

Indeed, differentiating (10.576) with respect to η ν , and re-expressing the result in terms of ξ µ via Eq. (10.557), we find once more (10.575). Multiplying both sides of Eq. (10.572) by (10.575), we express the nonlinear connection Qµν (q0 , ξ) by means of geodesic coordinates ξ µ (τ ) as ¯ νσ µ (q0 )ξ σ + 1 R ¯ σντ µ (q0 )ξ σ ξ τ + . . . . Qµν (q0 , ξ) = δνµ + Γ 3

(10.578)

The effect of simultaneous transformations (10.568), (10.569) upon the fluctuation function η µ = η µ (q0 , ξ) in Eq. (10.557) is ¯ µ (q0 , η), η µ → η ′µ = η µ − εν Q ν

¯ µ (q0 , 0) = δ µ , Q ν ν

(10.579)

¯ µν (q0 , η) is related to Qµν (q0 , ξ) as follows where the matrix Q ¯ µ (q0 , η) Q ν

=

"

∂η Qκν (q0 , ξ)

µ

(q0 , ξ) ∂η µ (q0 , ξ) − ∂ξ κ ∂q0ν

#

.

(10.580)

ξ=ξ(q0 ,η)

¯ µ (q0 , η) = δ µ , Applying Eq. (10.572) to the right-hand side of Eq. (10.580) yields Q ν ν as it should to compensate the translation (10.568). The above independence of q0µ will be essential for constructing the correct perturbation expansion for the path integral (10.561). For some special cases of the Riemannian manifold, such as a surface of sphere in D + 1 dimensions which forms a homogeneous space O(D)/O(D − 1), all points are equivalent, and the local independence becomes global. This will be discussed further in Section 10.12.6.

10.12.3

Zero-Mode Properties

We are now prepared to eliminate the zero mode by the condition of vanishing average ξ¯µ = 0. As mentioned before, the vanishing fluctuation ξ µ (τ ) = 0 is obviously a classical saddle-point for the path integral (10.561). In addition, because of the symmetry (10.569) there exist other equivalent extrema ξεµ (τ ) = −εµ = const. The D components of εµ correspond to D zero modes which we shall eliminate in favor of

780


a change of q0µ . The proper way of doing this is provided by the Faddeev-Popov procedure. We insert into the path integral (10.561) the trivial unit integral, rewritten with the help of (10.568): 1=

Z

D

d q0 δ

(D)

(q0ε − q0 ) =

Z

dD q0 δ (D) (ε),

(10.581)

and decompose the measure of path integration over all periodic paths ξ µ (τ ) into a product of an ordinary integral over the temporal average ξ0µ = ξ¯µ , and a remainder containing only nonzero Fourier components [recall (2.440)] dD ξ0 I ′D D ξ. (10.582) √ D 2πβ According to Eq. (10.569), the path average ξ¯µ is translated under εµ as follows I

Z

DD ξ =

1 ξ¯µ → ξ¯εµ = ξ¯µ − εν β

β

Z

0

dτ Qµν (q0 , ξ(τ )).

(10.583)

Thus we can replace Z

dD ξ0 √ D → 2πβ

Z

dD ε 1 √ D det β 2πβ "

Z

β

0

dτ

Qµν (q0 , ξ(τ ))

#

.

(10.584)

Performing this replacement in (10.582) and performing the integral over εµ in the inserted unity (10.581), we obtain the measure of path integration in terms of q0µ and geodesic coordinates of zero temporal average I

dD ξ0 (D) (ε) D ′D ξ √ Dδ 2πβ " Z # Z I D d q0 1 β ′D µ D ξ det dτ Qν (q0 , ξ(τ )) . = √ D β 0 2πβ Z

DD ξ =

dD q0

I

I

(10.585)

The factor on the right-hand side is the Faddeev-Popov determinant ∆[q0 , ξ] for the change from ξ0µ to q0µ . We shall write it as an exponential: "

1 ∆[q0 , ξ] = det β

Z

β

0

dτ

Qµν (q0 , ξ)

#

FP [q

= e−A

0 ,ξ]

,

(10.586)

where AFP [q0 , ξ] is an auxiliary action accounting for the Faddeev-Popov determinant " Z # 1 β FP µ A [q0 , ξ] ≡ −tr log dτ Qν (q0 , ξ) , (10.587) β 0 which must be included into the interaction (10.560). Eq. (10.587), we find explictly FP

A [q0 , ξ] = −tr log 1 = 3β

Z

0

β

"

δνµ

+ (3β)

−1

Z

0

β

Inserting (10.578) into #

¯ σντ (q0 )ξ (τ )ξ (τ ) + . . . dτ R

¯ µν (q0 )ξ µ ξ ν + . . . . dτ R

µ

σ

τ


781


The contribution of this action will crucial for obtaining the correct perturbation expansion of the path integral (10.561). With the new interaction int FP Aint tot,FP [q0 , ξ] = Atot [q0 , ξ] + A [q0 , ξ]

(10.589)

the partition function (10.561) can be written as a classical partition function dD q0 q −βV eff cl (q0 ) , √ D g(q0 ) e 2πβ

Z

Z=

(10.590)

where V eff cl (q0 ) is the curved-space version of the effective classical partition function of Ref. [53]. The effective classical Boltzmann factor B(q0 ) ≡ e−βV

eff cl (q

0)

(10.591)

is given by the path integral B(q0 ) =

I

D ′D ξ

q

(0) [q

g(q0 )e−A

int 0 ,ξ]−Atot,FP [q0 ,ξ]

.

(10.592)

Since the zero mode is absent in the fluctuations on the right-hand side, the perturbation expansion is now straightforward. We expand the path integral (10.592) in powers of the interaction (10.589) around the free Boltzmann factor I

B0 (q0 ) =

D ′D ξ

q

g(q0 )e−

Rβ 0

dτ

1 g (q )ξ˙µ ξ˙ν 2 µν 0

(10.593)

as follows:

B(q0 ) = B0 (q0 ) 1 −

D

Eq0 Aint [q , ξ] tot,FP 0

Eq 0 1 D int + Atot,FP [q0 , ξ]2 − . . . , 2

(10.594)

where the q0 -dependent correlation functions are defined by the Gaussian path integrals h. . .iq0 = B −1 (q0 )

I

(0) [q

D ′D ξ [. . .]q0 e−A

0 ,ξ]

.

(10.595)

By taking the logarithm of (10.593), we obtain directly a cumulant expansion for the effective classical potential V eff cl (q0 ). For a proper normalization of the Gaussian path integral (10.593) we diagonalize the free action in the exponent by going back to the orthonormal components ∆xi in (10.97). Omitting again the smallness symbols ∆, the measure of the path integral becomes simply: I

D ′D ξ µ

q

g(q0 ) =

I

D ′D xi ,

(10.596)

and we find B0 (q0 ) =

I

D ′D xi e−

Rβ 0

dτ

1 ˙2 ξ 2

.

(10.597)

782


If we expand the fluctuations xi (τ ) into the eigenfunctions e−iωm τ of the operator −∂τ2 for periodic boundary conditions xi (0) = xi (β), xi (τ ) =

X

xim um (τ ) = xi0 +

m

X

m6=0

xim um (τ ), xi−m = xim∗ , m > 0,

(10.598)

and substitute this into the path integral (10.597), the exponent becomes −

1 2

Z

0

β

h

dτ xi (τ )

i2

=−

X β X 2 i 2 ωm x−m xim = −β ωm xim∗ xim . 2 m6=0 m>0

(10.599)

Remembering the explicit form of the measure (2.439) the Gaussian integrals in (10.597) yield the free-particle Boltzmann factor B0 (q0 ) = 1,

(10.600)

corresponding to a vanishing effective classical potential in Eq. (10.591). The perturbation expansion (10.594) becomes therefore simply Eq0

D

B(q0 ) = 1 − Aint tot,FP [q0 , ξ]

+

Eq 0 1 D int Atot,FP [q0 , ξ]2 − . . . . 2

(10.601)

The expectation values on the right-hand side are to be calculated with the help of Wick contractions involving the basic correlation function of ξ a (τ ) associated with the unperturbed action in (10.597): D

xi (τ )xj (τ ′ )

Eq 0

¯ τ ′) , = δ ij ∆(τ,

(10.602)

which is, of course, consistent with (10.563) via Eq. (10.97).

10.12.4

Covariant Perturbation Expansion

We now perform all possible Wick contractions of the fluctuations ξ µ (τ ) in the expectation values (10.601) using the correlation function (10.563). We restrict our attention to the lowest-order terms only, since all problems of previous treatments arise already there. Making use of Eqs. (10.564) and (10.566), we find for the interaction (10.560): D

Eq0 Aint tot [q0 , ξ]

Z

=

β

0

D Eq 0 i 1 h¯ ¯ µν (q0 ) hξ µ ξ ν iq0 Rµλνκ (q0 ) ξ λξ κ ξ˙µ ξ˙ν + δ(0)R 6

dτ

1 ¯ = R(q0 )β, 72

(10.603)

and for (10.588): D

FP

Eq0

A [q0 , ξ]

=

Z

0

β

dτ

1 ¯ 1 ¯ Rµν (q0 ) hξ µ ξ ν iq0 = R(q 0 )β. 3β 36

(10.604)


783


The sum of the two contributions yields the manifestly covariant high-temperature expansion up to two loops: Eq0

D

B(q0 ) = 1 − Aint tot,FP [q0 , ξ]

+ ... = 1 −

1 ¯ R(q0 )β + . . . 24

(10.605)

in agreement with the partition function density (10.493) calculated from Dirichlet boundary conditions. The associated partition function P

Z =

Z

dD q0 q √ D g(q0 ) B(q0 ) 2πβ

(10.606)

coincides with the partition function obtained by integrating over the partition function density (10.493). Note the crucial role of the action (10.588) coming from the Faddeev-Popov determinant in obtaining the correct two-loop coefficient in Eq. (10.605) and the normalization in Eq. (10.606). The intermediate transformation to the geodesic coordinates ξ µ (τ ) has made our calculations rather lengthy if the action is given in arbitrary coordinates, but it guarantees complete independence of the coordinates in the result (10.605). The entire derivation simplifies, of course, drastically if we choose from the outset geodesic coordinates to parametrize the curved space.

10.12.5

Covariant Result from Noncovariant Expansion

Having found the proper way of calculating the Boltzmann factor B(q0 ) we can easily set up a procedure for calculating the same covariant result without the use of the geodesic fluctuations ξ µ (τ ). Thus we would like to evaluate the path integral (10.593) by a direct expansion of the action in powers of the noncovariant fluctuations η µ (τ ) in Eq. (10.556). In order to make q0µ equal to the path average, q¯(τ ), we now require η µ (τ ) to have a vanishing temporal average η0µ = η¯µ = 0. Instead of (10.559), the free action reads now Z β 1 A(0) [q0 , η] = gµν (q0 ) dτ η µ (τ )(−∂τ2 )η ν (τ ) , (10.607) 2 0 and the small-β behavior of the path integral (10.606) is governed by the interaction Aint tot [q] fo Eq. (10.535) with unit smallness parameter η. In a notation as in (10.560), the interaction (10.535) reads Z β 1 κ λ κ Aint [q ; η] = dτ Γ η + ∂ Γ η η η˙ µ η˙ ν κµν λ κµν tot 0 2 0 1 − δ(0) Γµκ µ η κ + ∂λ Γτ µ µ η λ η τ . (10.608) 2 We must deduce the measure of functional integration over η-fluctuations without zero mode η0µ = η¯µ from the proper measure in (10.585) of ξ-fluctuations without zero mode: I I D ′D ξ J(q0 , ξ)∆FP [q0 , ξ] ≡ DD ξ(τ )J(q0 , ξ)δ (D) (ξ0 )∆FP [q0 , ξ]. (10.609) This is transformed to coordinates η µ (τ ) via Eqs. (10.576) and (10.577) yielding I I ¯ FP [q0 , η] , D ′D ξ J(q0 , ξ)∆FP [q0 , ξ] = D ′D η ∆

(10.610)

784


¯ FP [q0 , η] is obtained from the Faddeev-Popov determinant ∆FP [q0 , ξ] of (10.586) by exwhere ∆ pressed the coordinates η µ (τ ) in terms of ξ µ (τ ), and multiplying the result by a Jacobian accounting for the change of the δ-function of ξ0 to a δ-function of η0 via the transformation Eq. (10.576): µ ¯ FP [q0 , η] = ∆FP [q0 , ξ(q0 , η)] × det ∂ η¯ (q0 , ξ) . (10.611) ∆ ∂ξ ν ξ=ξ(q0 ,η) The last determinant has the exponential form ( " Z #) µ µ ∂ η¯ (q0 , ξ) 1 β ∂η (q0 , ξ) det = exp trlog dτ , ∂ξ ν β 0 ∂ξ ν ξ=ξ(q0 ,η) ξ=ξ(q0 ,η) where the matrix in the exponent has small-η expansion µ ∂η (q0 , ξ) ¯ νσ µ η σ = δνµ − Γ ∂ξ ν ξ=ξ(q0 ,η) 1 1 ¯ µ 1 ¯ κ¯ µ σ τ µ κ¯ µ ¯ ¯ − ∂σ Γντ + ∂ν Γστ −2Γτ ν Γκσ + Γτ σ Γκν η η + ... . 3 2 2

(10.612)

(10.613)

The factor (10.611) in (10.610) leads to a new contribution to the interaction (10.607), if we rewrite it as ¯ FP [q0 , η] = e−A¯FP [q0 ,η] . ∆

(10.614)

Combining Eqs. (10.586) and (10.612), we find a new Faddeev-Popov type action for η µ -fluctuations at vanishing η0µ : " Z # µ 1 β ∂η (q0 , ξ) FP FP ¯ A [q0 , η] = A [q0 , ξ(q0 , η)] − trlog dτ β 0 ∂ξ ν ξ=ξ(q0 ,η) Z β 1 = dτ Tστ (q0 )η σ η τ + . . . , (10.615) 2β 0 where ¯ στ µ − 2Γ ¯ σκ µ Γ ¯ µτ κ + Γ ¯ κµ µ Γ ¯ στ κ . Tστ (q0 ) = ∂µ Γ

(10.616)

The unperturbed correlation functions associated with the action (10.607) are: q0

hη µ (τ )η ν (τ ′ )i

¯ τ ′) = g µν (q0 )∆(τ,

(10.617)

and the free Boltzmann factor is the same as in Eq. (10.600). The perturbation expansion of the interacting Boltzmann factor is to be calculated from an expansion like (10.601):

q0 1 D int 2 Eq0 B(q0 ) = 1 − Aint + Atot,FP [q0 , η] − ... , tot,FP [q0 , η] 2

(10.618)

where the interaction is now

int ¯FP [q0 , η] . Aint tot,FP [q0 , η] = Atot [q0 , η] + A

(10.619)

As before in the Dirichlet case, the divergences containing powers of δ(0) no longer cancel order by order, but do so at the end. The calculations proceed as in the Dirichlet case, except that the correlation functions are now given by (10.564) which depend only on the difference of their arguments. H. Kleinert, PATH INTEGRALS

785

10.12 Effective Classical Potential in Curved Space (1)

The first expectation value contributing to (10.618) is given again by f1 of Eq. (10.538), ¯ τ ) of except that the integrals have to be evaluated with the periodic correlation function ∆(τ, Eq. (10.564), which has the properties ¯ τ) = 1 , ∆(τ, 12

¯ τ ) = ∆˙(τ, ¯ ˙∆(τ, τ) = 0 .

(10.620)

¯ Using further the common property ˙∆˙(τ, τ ′ ) = δ(τ − τ ′ ) − 1/β of Eq. (10.566), we find directly from (10.538):

q0 (1) f¯1 = Aint tot [q0 , η]

= −

β στ ¯ τ µ µ + g µν Γ ¯ τ µκ Γ ¯ σν κ + Γ ¯ τ ν µΓ ¯ σµ ν g ∂σ Γ 24 β2 ¯ τ µκ Γ ¯ σν κ + Γ ¯ τ µν Γ ¯ σν µ . δ(0) g στ g µν Γ 24

(10.621)

To this we must the expectation value of the Faddeev-Popov action:

FP q0 A¯ [q0 , η]

= −

β στ ¯ στ µ − 2Γ ¯ σν µ Γ ¯ µτ ν + Γ ¯ µκ µ Γ ¯ στ κ . g ∂µ Γ 24

(10.622)

The divergent term with the factor δ(0) in (10.621) is canceled by the same expression in the second-order contribution to (10.601) which we calculate now, evaluating the second cumulant ¯ τ ) of Eq. (10.564). The diagrams are the same (10.540) with the periodic correlation function ∆(τ, as in (10.542), but their evaluation is much simpler. Due to the absence of the zero modes in η µ (τ ), (2) (3) (4) all one-particle reducible diagrams vanish, so that the analogs of f1 , f1 , and f1 in (10.541) (5) (6) are all zero. Only those of f1 and f1 survive, which involve now the Feynman integrals I14 and ¯ τ ) which are I15 evaluated with ∆(τ, Z βZ β β2 β ¯ (τ, τ ′ ) ∆˙(τ, ¯ ¯ : I¯14 = dτ dτ ′ ˙∆ τ ′ )˙∆˙(τ, τ ′ ) = − + δ(0) , (10.623) 24 12 0 0 Z βZ β ¯ ¯ τ ′ )˙∆˙ ¯ 2 (τ, τ ′ ) = − β . : I15 = dτ dτ ′ ∆(τ, (10.624) 24 0 0 This leads to the second cumulant 2 Eq0 1 D int Atot,FP [q0 , η] = 2 c

− +

β στ µν ¯ ¯ σν κ + 2 Γ ¯ τ ν µΓ ¯ σµ ν g g Γτ µκ Γ 24 β2 ¯ τ µκ Γ ¯ σν κ + Γ ¯τ µν Γ ¯ σν µ . δ(0) g στ g µν Γ 24

(10.625)

¯ The sum of Eqs. (10.622) and (10.625) is finite and yields the same covariant result β R/24 as in Eq. (10.551), so that we re-obtain the same covariant perturbation expansion of the effective classical Boltzmann factor as before in Eq. (10.605). Note the importance of the contribution (10.622) from the Faddeev-Popov determinant in producing the curvature scalar. Neglecting this, as done by other authors in Ref. [34], will produce in the effective classical Boltzmann factor (10.618) an additional noncovariant term g στ Tστ (q0 )/24. This may be rewritten as a covariant divergence of a nonvectorial quantity ¯ µστ (q0 ). g στ Tστ = ∇µ V µ , V µ (q0 ) = g στ (q0 )Γ

(10.626)

As such it does not contribute to the integral over q0µ in Eq. (10.606), but it is nevertheless a wrong noncovariant result for the Boltzmann factor (10.605). The appearance of a noncovariant term in a treatment where q0µ is the path average of q µ (τ ) is not surprising. If the time dependence of a path shows an acceleration, the average of a path is not an invariant concept even for an infinitesimal time. One may covariantly impose the condition of a vanishing temporal average only upon fluctuation coordinates which have no acceleration. This is the case of geodesic coordinates ξ a (τ ) since their equation of motion at q0µ is ξä (τ ) = 0.

786


10.12.6

Particle on Unit Sphere

A special treatment exists for particle in homogeneous spaces. As an example, consider a quantum particle moving on a unit sphere in D + 1 dimensions. The partition function is defined by Eq. (10.553) with the Euclidean action (10.407) and the invariant measure (10.405), where the metric and its determinant are gµν (q) = δµν +

qµ qν , 1 − q2

1 . 1 − q2

g(q) =

(10.627)

It is, of course, possible to calculate the Boltzmann factor B(q0 ) with the procedure of Section 10.12.3. Instead of doing this we shall, however, exploit the homogeneity of the sphere. The invariance under reparametrizations of general Riemannian space becomes here an isometry of the metric (10.627). Consequently, the Boltzmann factor B(q0 ) in Eq. (10.606) becomes independent of the choice of q0µ , and the integral over q0µ in (10.590) yields simply the total surface of the sphere times the Boltzmann factor B(q0 ). The homogeneity of the space allows us to treat paths q µ (τ ) themselves as small quantum fluctuations around the origin q0µ = 0, which extremizes the path integral (10.553). The possibility of this expansion is due to the fact that Γµστ = q µ gστ vanishes at q µ (τ ) = 0, so that the movement is at this point free of acceleration, this being similar to the situation in geodesic coordinates. As before we now take account of the fact that there are other equivalent saddle-points due to isometries of the metric (10.627) on the sphere (see, e.g., [54]). The infinitesimal transformations of a small vector q µ : q

qεµ = q µ + εµ 1 − q 2 , εµ = const, µ = 1, . . . , D

(10.628)

move the origin q0µ = 0 by a small amount on the surface of the sphere. Due to the rotational symmetry of the system in the D-dimensional space, these fluctuations have a vanishing action. There are also D(D − 1)/2 more isometries consisting of the rotations around the origin q µ (τ ) = 0 on the surface of the sphere. These are, however, irrelevant in the present context since they leave the origin unchanged. The transformations (10.628) of the origin may be eliminated from the path integral (10.553) by including a factor δ (D) (¯ q ) to enforce the vanishing of the temporal R −1 β FP path average q¯ = β 0 dτ q(τ ). The associated Faddeev-Popov determinant ∆ [q] is determined by the integral FP

∆ [q]

Z

D

d εδ

(D)

FP

(¯ qε ) = ∆ [q]

Z

D

d εδ

(D)

ε

µ1

β

Z

β

0

dτ

q

!

1 − q 2 = 1 . (10.629)

The result has the exponential form FP

∆ [q] =

!D 1Zβ q FP dτ 1 − q 2 = e−A [q] , β 0

where AFP [q] must be added to the action (10.407): 1 A [q] = −D log β FP

Z

0

β

dτ

q

1−

q2

!

.

(10.630)

(10.631)


787


The Boltzmann factor B(q0 ) ≡ B is then given by the path integral without zero modes I Y

B =

µ

dq (τ ) g(q(τ )) δ (D) (¯ q )∆FP [q]e−A[q]

µ,τ

I

=

q

D ′D q

q

g(q(τ ))∆FP [q]e−A[q] ,

(10.632)

where the measure D ′D q is defined as in Eq. (2.439). This can also be written as B=

I

J [q]−AFP [q]

D ′D q e−A[q]−A

,

(10.633)

where AJ [q] is a contribution to the action (10.407) coming from the product Yq τ

J [q]

g(q(τ )) ≡ e−A

.

(10.634)

By inserting (10.627), this becomes AJ [q] = −

Z

0

β

Z β 1 1 dτ δ(0) log g(q) = dτ δ(0) log(1 − q 2 ) . 2 2 0

(10.635)

The total partition function is, of course, obtained from B by multiplication with the surface of the unit sphere in D + 1 dimensions 2π (D+1)/2 /Γ(D + 1)/2). To calculate B from (10.633), we now expand A[q], AJ [q] and AFP [q] in powers of q µ (τ ). The metric gµν (q) and its determinant g(q) in Eq. (10.627) have the expansions g(q) = 1 + q 2 + . . . ,

gµν (q) = δµν + qµ qν + . . . ,

(10.636)

and the unperturbed action reads (0)

A [q] =

Z

β

0

1 dτ q˙2 (τ ). 2

(10.637)

In the absence of the zero eigenmodes due to the δ-function over q¯ in Eq. (10.632), we find as in Eq. (10.600) the free Boltzmann factor B0 = 1.

(10.638)

The free correlation function looks similar to (10.602): ¯ τ ′ ). hq µ (τ )q ν (τ ′ )i = δ µν ∆(τ,

(10.639)

The interactions coming from the higher expansions terms in Eq. (10.636) begin with int J Aint tot [q] = A [q] + A [q] =

Z

0

β

dτ

i 1h (q q) ˙ 2 − δ(0)q 2 . 2

(10.640)

788


To the same order, the Faddeev-Popov interaction (10.631) contributes D A [q] = 2β FP

Z

β

dτ q 2 .

0

(10.641)

This has an important effect upon the two-loop perturbation expansion of the Boltzmann factor Eq 0

D

B(q0 ) = 1 − Aint tot [q]

D

Eq 0

− AFP [q]

+ . . . = B(0) ≡ B.

(10.642)

Performing the Wick contractions with the correlation function (10.639) with the properties (10.564)–(10.566), we find from Eqs. (10.640), (10.641) D

Eq 0

Aint tot [q]

o 1Z β n ¯ ¯ τ ) + D(D + 1) ˙∆ ¯ 2 (τ, τ )−δ(0)D ∆(τ, ¯ τ) dτ D ˙∆˙(τ, τ )∆(τ, 2 0 ( ) Z 1 β D¯ D 2 ¯ = dτ − ∆(τ, τ ) + D(D + 1) ˙∆ (τ, τ ) = − β, (10.643) 2 0 β 24

=

and D

FP

Eq 0

A [q]

D = 2β

Z

0

β

2 ¯ τ ) = D β. dτ D ∆(τ, 24

(10.644)

Their combination in Eq. (10.642) yields the high-temperature expansion B =1−

D(D − 1) β + ... . 24

(10.645)

This is in perfect agreement with Eqs. (10.493) and (10.605), since the scalar curva¯ = D(D −1). It is remarkable how the ture for a unit sphere in D +1 dimensions is R contribution (10.644) of the Faddeev-Popov determinant has made the noncovariant result (10.643) covariant.

10.13

Covariant Effective Action for Quantum Particle with Coordinate-Dependent Mass

The classical behavior of a system is completely determined by the extrema of the classical action. The quantum-mechanical properties can be found from the extrema of the effective action (see Subsection 3.22.5). This important quantity can in general only be calculated perturbatively. This will be done here for a particle with a coordinate-dependent mass. The calculation [44] will make use of the background method of Subsection 3.23.6 combined with the techniques developed earlier in this chapter. From the one-particle-irreducible (1PI) Feynman diagrams with no external lines we obtain an expansion in powers of the Planck constant h ¯ . The result will be applicable to a large variety of interesting physical systems, for instance compound nuclei, where the collective Hamiltonian, commonly derived from a microscopic description via time-dependent Hartree-Fock theory [45], contains coordinate-dependent mass parameters. H. Kleinert, PATH INTEGRALS

10.13 CovariantEffectiveActionforQuantumParticlewithCoordinate-DependentMass 789

10.13.1

Formulating the Problem

Consider a particle with coordinate-dependent mass m(q) moving as in the onedimensional potential V (q). We shall study the Euclidean version of the system where the paths q(t) are continued to an imaginary times τ = −it and the Lagrangian for q(τ ) has the form L(q, q) ˙ =

1 m(q)q˙2 + V (q). 2

(10.646)

The dot stands for the derivative with respect to the imaginary time. The qdependent mass may be written as mg(q) where g(q) plays the role of a onedimensional dynamical metric. It is the trivial 1 × 1 Hessian metric of the system [recall the definition (1.12) and Eq. (1.384)]. In D-dimensional configuration space, the kinetic term would read mgµν (q)q˙µ q˙ν /2, having the same form as in the curved-space action (10.185). Under an arbitrary single-valued coordinate transformation q = q(qe), the potential V (q) is assumed to transform like a scalar whereas the metric m(q) is a one-dimensional tensor of rank two: f(qe) [dqe(q)/dq]2 . m(q) = m

V (q) = V (q(qe)) ≡ Ve (qe),

(10.647)

This coordinate transformation leaves the Lagrangian (10.646) and thus also the classical action Z

A[q] =

∞

−∞

dτ L(q, q) ˙

(10.648)

invariant. Quantum theory has to possess the same invariance, exhibited automatically by Schrödinger theory. It must be manifest in the effective action. This will be achieved by combining the background technique in Subsection 3.23.6 with the techniques of Sections 10.6–10.10. In the background field method [46] we split all paths into q(τ ) = Q(τ ) + δq(τ ), where Q(τ ) is the final extremal orbit and δq describes the quantum fluctuations around it. At the one-loop level, the covariant effective action ˙ and a correction term ∆L. Γ[Q] becomes a sum of the classical Lagrangian L(Q, Q) It is defined by the path integral [recall (3.773)] e−Γ[Q]/¯h =

Z

Dµ(δq)e−(1/¯h){A[Q+δq]−

R

dτ δq δΓ[Q]/δQ}

,

(10.649)

where the measure of functional integration Dµ(δq) is obtained from the initial q −1 Q invariant measure Dµ(q) = Z τ dq(τ ) m(q) and reads Dµ(δq) = Z −1

Y τ

dδq(τ )

q

m(Q) e(1/2)δ(0)

R

dτ log[m(Q+δq)/m(Q)]

,

(10.650)

with Z being some normalization factor. The generating functional (10.649) possesses the same symmetry under reparametrizations of the configuration space as the classical action (10.648).

790


We now calculate Γ[Q] in Eq. (10.649) perturbatively as a power series in h ¯: Γ[Q] = A[Q] + h ¯ Γ1 [Q] + h ¯ 2 Γ2 [Q] + . . . .

(10.651)

The quantum corrections to the classical action (10.648) are obtained by expanding A[Q + δq] and the measure (10.650) covariantly in powers of δq: 1 DA D2A δx(τ ) + dτ dτ ′ δx(τ ) δx(τ ′ ) A[Q + δq] = A[Q] + dτ δQ(τ ) 2 δQ(τ )δQ(τ ′ ) Z Z Z 1 D3A + dτ dτ ′ dτ ′′ δx(τ ) δx(τ ′ ) δx(τ ′′ ) + . . . . (10.652) 6 δQ(τ )δQ(τ ′ )δQ(τ ′′ ) Z

Z

Z

The expansion is of the type (10.101), i.e., the expressions δx are covariant fluctuations related to the ordinary variations δq in the same way as the normal coordinates ∆xµ are related to the differences ∆q µ in the expansion (10.98). The symbol D/δQ denotes the covariant functional derivative in one dimension. To first order, this is the ordinary functional derivative 1 DA[Q] δA[Q] ¨ ). = = V ′ (Q) − m′ (Q) Q˙ 2 (τ ) − m(Q) Q(τ δQ(τ ) δQ(τ ) 2

(10.653)

This vanishes for the classical orbit Q(τ ). The second covariant derivative is [compare (10.100)] D 2 A[Q] δ 2 A[Q] δA[Q] = − Γ(Q(τ )) , ′ ′ δQ(τ )δQ(τ ) δQ(τ )δQ(τ ) δQ(τ ′ )

(10.654)

where Γ(Q) = m′ (Q)/2 m(Q) is the one-dimensional version of the Christoffel symbol for the metric gµν = δµν m(Q). More explicitly, the result is δ 2 A[Q] ˙ τ + m′ (Q)Q+ ¨ 1 m′′ (Q)Q˙ 2 −V ′′ (Q) δ(τ −τ ′ ). = − m(Q)∂τ2 +m′ (Q)Q∂ ′ δQ(τ )δQ(τ ) 2 (10.655)

The validity of the expansion (10.652) follows from the fact that it is equivalent by a coordinates transformation to an ordinary functional expansion in Riemannian coordinates where the Christoffel symbol vanishes for the particular background coordinates. The inverse of the functional matrix (10.654) supplies us with the free correlation function G(τ, τ ′ ) of the fluctuations δx(τ ). The higher derivatives define the interactions. The expansion terms Γn [Q] in (10.651) are found from all one-particleirreducible vacuum diagrams (3.781) formed with the propagator G(τ, τ ′ ) and the interaction vertices. The one-loop correction to the effective action is given by the simple harmonic path integral e−Γ1 [Q] =

Z

q

(2) [Q,δx]

Dδx m(Q) e−A

,


10.13 CovariantEffectiveActionforQuantumParticlewithCoordinate-DependentMass 791

with the quadratic part of the expansion (10.652): 1 A [Q, δx] = 2 (2)

Z

∞

−∞

dτ dτ ′ δx(τ )

D 2 A[Q] δx(τ ′ ) . δQ(τ )δQ(τ ′ )

(10.657)

The presence of m(Q) in the free part of the covariant kinetic term (10.657) and in the measure in Eq. (10.656) suggests exchanging the fluctuation δx by the new q coordinates δ˜ x = h(Q)δx, where h(Q) ≡ m(Q) is the one-dimensional version of q

the triad (10.12) e(Q) = m(Q) associated with the metric m(Q). The fluctuations δ˜ x correspond to the differences ∆xi in (10.97). The covariant derivative of e(Q) vanishes DQ e(Q) = ∂Q e(Q) −Γ(Q) e(Q) ≡ 0 [recall (10.40)]. Then (10.657) becomes A(2) [Q, δx] =

1 Z∞ D 2 A[Q] dτ dτ ′ δ˜ x(τ ) δ˜ x(τ ′ ) , ˜ )δ Q(τ ˜ ′) 2 −∞ δ Q(τ

(10.658)

where D 2 A[Q] D 2 A[Q] d2 −1 = e−1 (Q) e (Q) = − + ω 2 (Q(τ )) δ(τ −τ ′ ), (10.659) ′) 2 ′ ˜ ˜ δQ(τ )δQ(τ dτ δ Q(τ )δ Q(τ ) "

#

and ω 2 (Q) = e−1 (Q) D 2 V (Q) e−1 (Q) = e−1 (Q) DV ′ (Q) e−1 (Q) 1 = [V ′′ (Q) − Γ(Q) V ′ (Q)] . m(Q)

(10.660)

Note that this is the one-dimensional version of the Laplace-Beltrami operator (1.377) applied to V (Q): " !# V ′ (Q) d q ω (Q) = ∆V (Q) = q m(Q) . m(Q) m(Q) dQ

1

2

(10.661)

Indeed, e−2 D 2 is the one-dimensional version of g µν Dµ Dν [recall (10.38)] Since V (Q) is a scalar, so is ∆V (Q). Equation (10.659) shows that the fluctuations δ˜ x behave like those of a harmonic oscillator with the time-dependent frequency ω 2 (Q). The functional measure of integration in Eq. (10.656) simplifies in terms of δ˜ x: Y τ

q

dδx(τ ) m(Q) =

Y

dδ˜ x(τ ) .

(10.662)

τ

This allows us to integrate the Gaussian path integral (10.656) trivially to obtain the one-loop quantum correction to the effective action h i 1 Γ1 [Q] = Tr log −∂τ2 + ω 2 (Q(τ )) . 2

(10.663)

Due to the τ -dependence of ω 2 , this cannot be evaluated explicitly. For sufficiently slow motion of Q(τ ), however, we can resort to a gradient expansion which yields asymptotically a local expression for the effective action.

792


10.13.2

Gradient Expansion

The gradient expansion of the one-loop effective action (10.663) has the general form 1 Γ1 [Q] = dτ V1 (Q) + Z1 (Q) Q˙ 2 + · · · 2 −∞ Z

∞

.

(10.664)

It is found explicitly by recalling the gradient expansion of the trace of the logarithm derived in Eq. (4.302): [∂τ ω 2(τ )]2 Tr log −∂τ2 + ω 2 (τ ) ≡ dτ ω(τ ) + + ... . 32ω 5 (τ ) −∞ h

Z

i

∞

(

)

(10.665)

Inserting Ω(τ ) = ω(Q(τ )), we identify V1 (Q) = h ¯ ω(Q)/2,

Z1 (Q) =

(Dω 2)2 (Q) , 32 ω 5(Q)

(10.666)

and obtain the effective action to order h ¯ for slow motion 1 eff Γ [Q] = dτ m (Q) Q˙ 2 + V eff (Q) , 2 −∞ eff

Z

∞

(10.667)

where the bare metric meff (Q) and the potential V eff (Q) are related to the initial classical expressions by (Dω 2)2 (Q) , 32 ω 5(Q) ω(Q) V eff (Q) = V (Q) + h ¯ . 2

meff (Q) = m(Q) + h ¯

(10.668) (10.669)

For an Q-independent mass m(Q), the result reduces to a known result [48]. The range of validity of the expansion is determined by the characteristic time scale 1/ω. Within this time, the particle has to move only little.

Appendix 10A

Nonholonomic Gauge Transformations in Electromagnetism

To introduce the subject, let us first recall the standard treatment of magnetism. Since there are no magnetic monopoles, a magnetic field B(x) satisfies the identity ∇ · B(x) = 0, implying that only two of the three field components of B(x) are independent. To account for this, one usually expresses a magnetic field B(x) in terms of a vector potential A(x), setting B(x) = ∇ × A(x). Then Ampère’s law, which relates the magnetic field to the electric current density j(x) by ∇×B = 4πj(x), becomes a second-order differential equation for the vector potential A(x) in terms of an electric current ∇× [∇× A(x)] = j(x).

(10A.1)

The vector potential A(x) is a gauge field . Given A(x), any locally gauge-transformed field A(x) → A′ (x) = A(x) + ∇Λ(x)

(10A.2) H. Kleinert, PATH INTEGRALS

Nonholonomic Gauge Transformations in Electromagnetism 793

Appendix 10A

yields the same magnetic field B(x). This reduces the number of physical degrees of freedom in the gauge field A(x) to two, just as those in B(x). In order for this to hold, the transformation function must be single-valued, i.e., it must have commuting derivatives (∂i ∂j − ∂j ∂i )Λ(x) = 0.

(10A.3)

The equation for absence of magnetic monopoles ∇ · B = 0 is ensured if the vector potential has commuting derivatives (∂i ∂j − ∂j ∂i )A(x) = 0.

(10A.4)

This integrability property makes ∇ · B = 0 the Bianchi identity in this gauge field representation of the magnetic field. In order to solve (10A.1), we remove the gauge ambiguity by choosing a particular gauge, for instance the transverse gauge ∇· A(x) = 0 in which ∇× [∇× A(x)] = −∇2 A(x), and obtain Z j(x′ ) . (10A.5) A(x) = d3 x′ |x − x′ | The associated magnetic field is B(x) =

Z

d3 x′

j(x′ ) × R′ , R′3

R′ ≡ x′ − x.

(10A.6)

This standard representation of magnetic fields is not the only possible one. There exists another one in terms of a scalar potential Λ(x), which must, however, be multivalued to account for the two physical degrees of freedom in the magnetic field.

10A.1

Gradient Representation of Magnetic Field of Current Loops

Consider an infinitesimally thin closed wire carrying an electric current I along the line L. It corresponds to a current density j(x) = I (x; L), where (x; L) is the δ-function on the line L: Z (x; L) = dx′ δ (3) (x − x′ ).

(10A.7)

(10A.8)

L

For a closed line L, this function has zero divergence: ∇· (x; L) = 0.

(10A.9)

This follows from the property of the δ-function on an arbitrary line L connecting the points x1 and x2 : ∇· (x; L) = δ(x2 ) − δ(x1 ). For closed loops, the right-hand side vanishes. From Eq. (10A.5) we obtain the associated vector potential Z 1 A(x) = I dx′ , |x − x′ | L

(10A.10)

(10A.11)

yielding the magnetic field B(x) = −I

Z

L

dx′ × R′ , R′3

R′ ≡ x′ − x.

(10A.12)

794

10 Spaces with Curvature and Torsion B(x) = ∇Ω(x; L)

Figure 10.5 Infinitesimally thin closed current loop L. The magnetic field B(x) at the point x is proportional to the solid angle Ω(x) under which the loop is seen from x. In any single-valued definition of Ω(x), there is some surface S across which Ω(x) jumps by 4π. In the multivalued definition, this surface is absent. Let us now derive the same result from a scalar field. Let Ω(x; S) be the solid angle under which the current loop L is seen from the point x (see Fig. 10.5). If S denotes an arbitrary smooth surface enclosed by the loop L, and dS′ a surface element, then Ω(x; S) can be calculated from the surface integral Z dS′ · R′ Ω(x; S) = . (10A.13) R′3 S The argument S in Ω(x; S) emphasizes that the definition depends on the choice of the surface S. The range of Ω(x; S) is from −2π to 2π, as can be most easily be seen if L lies in the xy-plane and S is chosen to lie in the same place. Then we find for Ω(x; S) the value 2π for x just below S, and −2π just above. Let us calculate from (10A.13) the vector field B(x; S) = I∇Ω(x; S).

(10A.14)

For this we rewrite ∇Ω(x; S) =

Z

S

R′ k dSk′ ∇ ′3 R

=−

Z

S

dSk′ ∇′

R′ k , R′3

(10A.15)

which can be rearranged to Z Z ′ ′ ′ ′ ′ Rk ′ ′ Rk ′ ′ Rk ∇Ω(x; S) = − dSk ∂i ′3 − dSi ∂k ′3 + dSi ∂k ′3 . R R R S S With the help of Stokes’ theorem Z Z (dSk ∂i − dSi ∂k )f (x) = ǫkil dxl f (x), S

(10A.16)

(10A.17)

L

and the relation ∂k′ (Rk′ /R′3 ) = 4πδ (3) (x − x′ ), we obtain Z Z dx′ × R′ ′ (3) ′ ∇Ω(x; S) = − + 4π dS δ (x − x ) . R′3 L S

(10A.18)


Appendix 10A


Multiplying the first term by I, we reobtain the magnetic field (10A.12) of the current I. The second term yields the singular magnetic field of an infinitely thin magnetic dipole layer lying on the arbitrarily chosen surface S enclosed by L. The second term is a consequence of the fact that the solid angle Ω(x; S) was defined by the surface integral (10A.13). If x crosses the surface S, the solid angle jumps by 4π. It is useful to re-express Eq. (10A.15) in a slightly different way. By analogy with (10A.8) we define a δ-function on a surface as Z (x; S) = dS′ δ (3) (x − x′ ), (10A.19) S

and observe that Stokes’ theorem (10A.17) can be written as an identity for δ-functions: ∇× (x; S) = (x; L),

(10A.20)

where L is the boundary of the surface S. This equation proves once more the zero divergence (10A.9). Using the δ-function (10A.19) on the surface S, Eq. (10A.15) can be rewritten as Z Rk′ , (10A.21) ∇Ω(x; S) = − d3 x′ δk (x′ ; S)∇′ ′3 R and if we used also Eq. (10A.20), we find from (10A.18) a magnetic field Z R′ 3 ′ ′ Bi (x; S) = −I d x [∇ × (x; S)] × ′3 + 4π (x ; S) . R

(10A.22)

Stokes theorem written in the form (10A.20) displays an important property. If we move the surface S to S ′ with the same boundary, the δ-function δ(x; S) changes by (x; S) → (x; S ′ ) = (x; S) + ∇δ(x; V ),

(10A.23)

where δ(x; V ) ≡

Z

d3 x′ δ (3) (x − x′ ),

(10A.24)

and V is the volume over which the surface has swept. Under this transformation, the curl on the left-hand side of (10A.20) is invariant. Comparing (10A.23) with (10A.2) we identify (10A.23) as a novel type of gauge transformation.9 The magnetic field in the first term of (10A.22) is invariant under this, the second is not. It is then obvious how to find a gauge-invariant magnetic field: we simply subtract the singular S-dependent term and form B(x) = I [∇Ω(x; S) + 4π (x; S)] .

(10A.25)

This field is independent of the choice of S and coincides with the magnetic field (10A.12) derived in the usual gauge theory. Hence the description of the magnetic field as a gradient of field Ω(x; S) is completely equivalent to the usual gauge field description in terms of the vector potential A(x). Both are gauge theories, but of a completely different type. The gauge freedom (10A.23) can be used to move the surface S into a standard configuration. One possibility is the make gauge fixing. choose S so that the third component of (x; S) vanishes. This is called the axial gauge. If (x; S) does not have this property, we can always shift S by a volume V determined by the equation Z z δ(V ) = − δz (x; S), (10A.26) −∞

9

For a discussion of this gauge freedom, which is independent of the electromagnetic one, see Ref. [56].

796


and the transformation (10A.23) will produce a (x; S) in the axial gauge, to be denoted by ax (x; S) Equation (10A.25) suggests defining a solid angle Ω(x) which is independent of S and depends only on the boundary L of S: ∇Ω(x; L) ≡ ∇Ω(x; S) + 4π (x; S).

(10A.27)

This is is the analytic continuation of Ω(x; S) through the surface S which removes the jump and produces a multivalued function Ω(x; L) ranging from −∞ to ∞. At each point in space, there are infinitely many Riemann sheets starting from a singularity at L. The values of Ω(x; L) on the sheets differ by integer multiples of 4π. From this multivalued function, the magnetic field (10A.12) can be obtained as a simple gradient: B(x) = I ∇Ω(x; L).

(10A.28)

Ampère’s law (10A.1) implies that the multivalued solid angle Ω(x; L) satisfies the equation (∂i ∂j − ∂j ∂i )Ω(x; L) = 4πǫijk δk (x; L).

(10A.29)

Thus, as a consequence of its multivaluedness, Ω(x; L) violates the Schwarz integrability condition as the coordinate transformations do in Eq. (10.19). This makes it an unusual mathematical object to deal with. It is, however, perfectly suited to describe the physics. To see explicitly how Eq. (10A.29) is fulfilled by Ω(x; L), let us go to two dimensions where the loop corresponds to two points (in which the loop intersects a plane). For simplicity, we move one of them to infinity, and place the other at the coordinate origin. The role of the solid angle Ω(x; L) is now played by the azimuthal angle ϕ(x) of the point x: ϕ(x) = arctan

x2 . x1

(10A.30)

The function arctan(x2 /x1 ) is usually made unique by cutting the x-plane from the origin along some line C to infinity, preferably along a straight line to x = (−∞, 0), and assuming ϕ(x) to jump from π to −π when crossing the cut. The cut corresponds to the magnetic dipole surface S in the integral (10A.13). In contrast to this, we shall take ϕ(x) to be the multivalued analytic continuation of this function. Then the derivative ∂i yields xj . (10A.31) ∂i ϕ(x) = −ǫij 1 2 (x ) + (x2 )2 With the single-valued definition of ∂i ϕ(x), there would have been a δ-function ǫij δj (C; x) across the cut C, corresponding to the second term in (10A.18). When integrating the curl of (10A.31) across the surface s of a small circle c around the origin, we obtain by Stokes’ theorem Z Z d2 x(∂i ∂j − ∂j ∂i )ϕ(x) = dxi ∂i ϕ(x), (10A.32) s

c

which is equal to 2π in the multivalued definition of ϕ(x). This result implies the violation of the integrability condition as in (10A.41): (∂1 ∂2 − ∂2 ∂1 )ϕ(x) = 2πδ (2) (x),

(10A.33)

whose three-dimensional generalization is (10A.29). In the single-valued definition with the jump by 2π across the cut, the right-hand side of (10A.32) would vanish, making ϕ(x) satisfy the integrability condition (10A.29). On the basis of Eq. (10A.33) we may construct a Green function for solving the corresponding differential equation with an arbitrary source, which is a superposition of infinitesimally thin linelike currents piercing the two-dimensional space at the points xn : X j(x) = In δ (2) (x − xn ), (10A.34) n


Appendix 10A


where In are currents. We may then easily solve the differential equation (∂1 ∂2 − ∂2 ∂1 )f (x) = j(x),

(10A.35)

with the help of the Green function G(x, x′ ) =

1 ϕ(x − x′ ) 2π

(10A.36)

which satisfies (∂1 ∂2 − ∂2 ∂1 )G(x − x′ ) = δ (2) (x − x′ ).

(10A.37)

The solution of (10A.35) is obviously f (x) =

Z

d2 x′ G(x, x′ )j(x).

(10A.38)

The gradient of f (x) yields the magnetic field of an arbitrary set of line-like currents vertical to the plane under consideration. It is interesting to realize that the Green function (10A.36) is the imaginary part of the complex function (1/2π) log(z −z ′ ) with z = x1 +ix2 , whose real part (1/2π) log |z −z ′ | is the Green function G∆ (x − x′ ) of the two dimensional Poisson equation: (∂12 + ∂22 )G∆ (x − x′ ) = δ (2) (x − x′ ).

(10A.39)

It is important to point out that the superposition of line-like currents cannot be smeared out into a continuous distribution. The integral (10A.38) yields the superposition of multivalued functions f (x) =

1 X x2 − x2n In arctan 1 , 2π n x − x1n

(10A.40)

which is properly defined only if one can clearly continue it analytically into the all parts of the Riemann sheets defined by the endpoints of the cut at the origin. If we were to replace the sum by an integral, this possibility would be lost. Thus it is, strictly speaking, impossible to represent arbitrary continuous magnetic fields as gradients of superpositions of scalar potentials Ω(x; L). This, however, is not a severe disadvantage of this representation since any current can be approximated by a superposition of line-like currents with any desired accuracy, and the same will be true for the associated magnetic fields. The arbitrariness of the shape of the jumping surface is the origin of a further interesting gauge structure which has interesting physical consequences discussed in Subsection 10A.5.

10A.2

Generating Magnetic Fields by Multivalued Gauge Transformations

After this first exercise in multivalued functions, we now turn to another example in magnetism which will lead directly to our intended geometric application. We observed before that the local gauge transformation (10A.2) produces the same magnetic field B(x) = ∇× A(x) only, as long as the function Λ(x) satisfies the Schwarz integrability criterion (10A.29) (∂i ∂j − ∂j ∂i )Λ(x) = 0.

(10A.41)

Any function Λ(x) violating this condition would change the magnetic field by ∆Bk (x) = ǫkij (∂i ∂j − ∂j ∂i )Λ(x),

(10A.42)

798


thus being no proper gauge function. The gradient of Λ(x) A(x) = ∇Λ(x)

(10A.43)

would be a nontrivial vector potential. By analogy with the multivalued coordinate transformations violating the integrability conditions of Schwarz as in (10A.29), the function Λ(x) will be called nonholonomic gauge function. Having just learned how to deal with multivalued functions we may change our attitude towards gauge transformations and decide to generate all magnetic fields approximately in a field-free space by such improper gauge transformations Λ(x). By choosing for instance Λ(x) = ΦΩ(x; L),

(10A.44)

we find from (10A.29) that this generates a field Bk (x) = ǫkij (∂i ∂j − ∂j ∂i )Λ(x) = Φδk (x; L).

(10A.45)

This is a magnetic field of total flux Φ inside an infinitesimal tube. By a superposition of such infinitesimally thin flux tubes analogous to (10A.38) we can obviously generate a discrete approximation to any desired magnetic field in a field-free space.

10A.3

Magnetic Monopoles

Multivalued fields have also been used to describe magnetic monopoles [10, 13, 14]. A monopole charge density ρm (x) is the source of a magnetic field B(x) as defined by the equation ∇· B(x) = 4πρm (x).

(10A.46)

If B(x) is expressed in terms of a vector potential A(x) as B(x) = ∇× A(x), equation (10A.46) implies the noncommutativity of derivatives in front of the vector potential A(x): 1 ǫijk (∂i ∂j − ∂j ∂i )Ak (x) = 4πρm (x). 2

(10A.47)

Thus A(x) must be multivalued. Dirac in his famous theory of monopoles [15] made the field singlevalued by attaching to the world line of the particle a jumping world surface, whose intersection with a coordinate plane at a fixed time forms the Dirac string, along which the magnetic field of the monopole is imported from infinity. This world surface can be made physically irrelevant by quantizing it appropriately with respect to the charge. Its shape in space is just as irrelevant as that of the jumping surface S in Fig. 10.5. The invariance under shape deformations constitute once more a second gauge structure of the type mentioned earlier and discussed in Refs. [2, 6, 7, 10, 12]. Once we allow ourselves to work with multivalued fields, we may easily go one step further and express also A(x) as a gradient of a scalar field as in (10A.43). Then the condition becomes ǫijk ∂i ∂j ∂k Λ(x) = 4πρm (x).

(10A.48)

Let us construct the field of a magnetic monopole of charge g at a point x0 , which satisfies (10A.46) with ρm (x) = g δ (3) (x − x0 ). Physically, this can be done only by setting up an infinitely thin solenoid along an arbitrary line L↑0 whose initial point lies at x0 and the final anywhere at infinity. The superscript ↑ indicates that the line has a strating point x0 . Inside this solenoid, the magnetic field is infinite, equal to Binside (x; L) = 4πg (x; L↑0 ), where (x; L↑0 ) is the open-ended version of (10A.8) Z (x; L↑0 ) = d3 x′ δ (3) (x − x′ ).

(10A.49)

(10A.50)

L↑ 0 , x0


Appendix 10A


The divergence of this function is concentrated at the starting point: ∇· (x; L↑0 ) = δ (3) (x − x0 ).

(10A.51)

This follows from (10A.10) by moving the end point to infinity. By analogy with the curl relation (10A.20) we observe a further gauge invariance. If we deform the line L, at fixed initial point x, the δ-function (10A.50) changes as follows: ↑ (x; L↑0 ) → (x; L′↑ 0 ) = (x; L0 ) + ∇× (x; S),

(10A.52)

where S is the surface over which L↑ has swept on its way to L′↑ . Under this gauge transformation, the relation (10A.51) is obviously invariant. We shall call this monopole gauge invariance. The flux (10A.49) inside the solenoid is therefore a monopole gauge field . It is straightforward to construct from it the ordinary gauge field A(x) of the monopole. First we define the L↑0 -dependent field A(x; L↑0 ) = −g

Z

d3 x′

∇′ × (x′ ; L↑0 ) =g R′

Z

d3 x′ (x′ ; L↑0 ) ×

R′ . R′3

(10A.53)

The curl of the first expression is ∇× A(x; L↑0 ) = −g

Z

d3 x′

∇′ × [∇′ × (x′ ; L↑0 )] , R′

and consists of two terms Z Z ∇′ [∇′ · (x′ ; L↑0 )] ∇′ 2 (x′ ; L↑0 ) −g d3 x′ + g d3 x′ . ′ R R′

(10A.54)

(10A.55)

After an integration by parts, and using (10A.51), the first term is L↑0 -independent and reads Z 1 x − x0 g d3 x′ δ (3) (x − x0 )∇′ ′ = g . (10A.56) R |x − x0 |3 The second term becomes, after two integration by parts, −4πg (x′ ; L↑0 ).

(10A.57)

The first term is the desired magnetic field of the monopole. Its divergence is δ(x − x0 ), which we wanted to archive. The second term is the monopole gauge field, the magnetic field inside the solenoid. The total divergence of this field is, of course, zero. By analogy with (10A.25) we now subtract the latter term and find the magnetic field of the monopole B(x) = ∇× A(x; L↑0 ) + 4πg (x; L↑0 ).

(10A.58)

This field is independent of the string L↑0 . It depends only on the source point x0 and satisfies ∇· B(x) = 4πg δ (3) (x − x0 ). Let us calculate the vector potential for some simple choices of x0 and L↑0 , for instance x0 = 0 and L↑0 along the positive z-axis, so that (x; L↑0 ) becomes ˆz Θ(z)δ(x)δ(y), where zˆ is the unit vector in z-direction. Inserting this into the second expression in (10A.53) yields Z ∞ zˆ × x ↑ (g) A (x; L0 ) = −g dz ′ p 3/2 0 x2 + y 2 + (z ′ − z)2 ˆz × x (y, −x, 0) = −g =g . (10A.59) r(r − z) r(r − z)

800


If L↑0 runs to −∞, so that (x; L↑0 ) is equal to −ˆz Θ(−z)δ(x)δ(y), we obtain Z 0 ˆz × x ↑ (g) A (x; L0 ) = g dz ′ p 3/2 −∞ x2 + y 2 + (z ′ − z)2 zˆ × x (y, −x, 0) = −g . =g r(r + z) r(r + z)

(10A.60)

The vector potential has only azimuthal components. If we parametrize (x, y, z) in terms of spherical angles θ, ϕ) as r(sin θ cos ϕ, sin θ sin ϕ, cos θ), these are ↑ A(g) ϕ (x; L0 ) =

g g ↑ (1 − cos θ) or A(g) (1 + cos θ), ϕ (x; L0 ) = − r sin θ r sin θ

(10A.61)

respectively. The shape of the line L↑0 can be brought to a standard form, which corresponds to fixing a gauge of the field (x; L↑0 ). For example, we may always choose L↑0 to run from x0 into the z-direction. Also here, there exists an equivalent formulation in terms of a multivalued A-field with infinitely many Riemann sheets around the line L. For a detailed discussion of the physics of multivalued fields see Refs. [2, 6, 7, 10, 12]. An interesting observation is the following: If the gauge function Λ(x) is considered as a nonholonomic displacement in some fictitious crystal dimension, then the magnetic field of a current loop which gives rise to noncommuting derivatives (∂i ∂j −∂j ∂i )Λ(x) 6= 0 is the analog of a dislocaton [compare (10.23)], and thus implies torsion in the crystal. A magnetic monopole, on the other hand, arises from noncommuting derivatives (∂i ∂j − ∂j ∂i )∂k Λ(x) 6= 0 in Eq. (10A.48). It corresponds to a disclination [see (10.57)] and implies curvature. The defects in the multivalued description of magnetism are therefore similar to those in a crystal where dislocations much more abundantly observed than disclinations. They are opposite to those in general relativity which is governed by curvature alone, with no evidence for torsion so far [11].

10A.4

Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations

Multivalued gauge transformations are the perfect tool to minimally couple electromagnetism to any type of matter. Consider for instance a free nonrelativistic point particle with a Lagrangian L=

M 2 x˙ . 2

(10A.62)

The equations of motion are invariant under a gauge transformation ˙ L → L′ = L + ∇Λ(x) x, Rt since this changes the action A = tab dtL merely by a surface term:

(10A.63)

A′ → A = A + Λ(xb ) − Λ(xa ).

(10A.64)

The invariance is absent if we take Λ(x) to be a multivalued gauge function. In this case, a nontrivial vector potential A(x) = ∇Λ(x) (working in natural units with e = 1) is created in the field-free space, and the nonholonomically gauge-transformed Lagrangian corresponding to (10A.63), L′ =

M 2 ˙ x˙ + A(x) x, 2

(10A.65)

describes correctly the dynamics of a free particle in an external magnetic field. H. Kleinert, PATH INTEGRALS

Appendix 10A


The coupling derived by multivalued gauge transformations is automatically invariant under additional ordinary single-valued gauge transformations of the vector potential A(x) → A′ (x) = A(x) + ∇Λ(x),

(10A.66)

since these add to the Lagrangian (10A.65) once more the same pure derivative term which changes the action by an irrelevant surface term as in (10A.64). The same procedure leadsRin quantum mechanics to the minimal coupling of the Schrödinger field ψ(x). The action is A = dtd3 x L with a Lagrange density (in natural units with h ¯ = 1) 1 L = ψ ∗ (x) i∂t + ∇2 ψ(x). (10A.67) 2M The physics described by a Schrödinger wave function ψ(x) is invariant under arbitrary local phase changes ψ(x, t) → ψ ′ (x) = eiΛ(x) ψ(x, t),

(10A.68)

called local U(1) transformations. This implies that the Lagrange density (10A.67) may equally well be replaced by the gauge-transformed one 1 2 ∗ (10A.69) D ψ(x, t), L = ψ (x, t) i∂t + 2M where −iD ≡ −i∇ − ∇Λ(x) is the operator of physical momentum. We may now go over to nonzero magnetic fields by admitting gauge transformations with multivalued Λ(x) whose gradient is a nontrivial vector potential A(x) as in (10A.43). Then −iD turns into the covariant momentum operator ˆ = −iD = −i∇ − A(x), P

(10A.70)

and the Lagrange density (10A.69) describes correctly the magnetic coupling in quantum mechanics. As in the classical case, the coupling derived by multivalued gauge transformations is automatically invariant under ordinary single-valued gauge transformations under which the vector potential A(x) changes as in (10A.66), whereas the Schrödinger wave function undergoes a local U(1)-transformation (10A.68). This invariance is a direct consequence of the simple transformation behavior of Dψ(x, t) under gauge transformations (10A.66) and (10A.68) which is Dψ(x, t) → Dψ ′ (x, t) = eiΛ(x) Dψ(x, t).

(10A.71)

Thus Dψ(x, t) transforms just like ψ(x, t) itself, and for this reason, D is called gauge-covariant derivative. The generation of magnetic fields by a multivalued gauge transformation is the simplest example for the power of the nonholonomic mapping principle. After this discussion it is quite suggestive to introduce the same mathematics into differential geometry, where the role of gauge transformations is played by reparametrizations of the space coordinates. This is precisely what is done in Subsection (10.2.2).

10A.5

Gauge Field Representation of Current Loops and Monopoles

In the previous subsections we have given examples for the use of multivalued fields in describing magnetic phenomena. The nonholonomic gauge transformations by which we created line-like nonzero field configurations were shown to be the natural origin of the minimal couplings to the classical actions as well as to the Schrödinger equation. It is interesting to observe that there exists a fully fledged theory of magnetism (which is easily generalized to electromagnetism) with these

802


multivalued fields, if we properly handle the freedom in choosing the jumping surfaces S whose boundary represents the physical current loop in Eq. (10A.12). To understand this we pose ourselves the problem of setting up an action formalism for calculating the magnetic energy of a current loop in the gradient representation of the magnetic field. In this Euclidean field theory, the action is the field energy: Z 1 d3 x B2 (x). (10A.72) E= 8π Inserting the gradient representation (10A.28) of the magnetic field, we can write Z I2 E= d3 x [∇Ω(x)]2 . 8π

(10A.73)

This holds for the multivalued solid angle Ω(x) which is independent of S. In order to perform field theoretic calculations, we must go over to the single-valued representation (10A.25), so that Z I2 E= d3 x [∇Ω(x; S) + 4π (x; S)]2 . (10A.74) 8π The δ-function removes the unphysical field energy on the artificial magnetic dipole layer on S. Let us calculate the magnetic field energy of the current loop from the action (10A.74). For this we rewrite the action (10A.74) in terms of an auxiliary vector field B(x) as Z 1 I E = d3 x − B2 (x) − B(x) · [∇Ω(x; S) + 4π (x; S)] . (10A.75) 8π 4π A partial integration brings the second term to Z I d3 x ∇· B(x) Ω(x; S). 4π Extremizing this in Ω(x) yields the equation ∇· B(x) = 0,

(10A.76)

implying that the field lines of B(x) form closed loops. This equation may be enforced identically (as a Bianchi identity) by expressing B(x) as a curl of an auxiliary vector potential A(x), setting B(x) ≡ ∇× A(x).

(10A.77)

With this ansatz, the equation which brings the action (10A.75) to the form Z 1 3 2 E = d x − [∇× A(x)] − I [∇× A(x)] · (x; S) . 8π A further partial integration leads to Z 1 E = d3 x − [∇× A(x)]2 − I A(x) · [∇× (x; S)] , 8π

(10A.78)

(10A.79)

and we identify in the linear term in A(x) the auxiliary current j(x) ≡ I ∇× (x; S) = I (x; L),

(10A.80)

due to Stoke’s law (10A.20). According to Eq. (10A.9), this current is conserved for closed loops L. H. Kleinert, PATH INTEGRALS

Appendix 10B

Comparison of Multivalued Basis Tetrads with Vierbein Fields803

By extremizing the action (10A.78), we obtain Ampère’s law (10A.1). Thus the auxiliary quantities B(x), A(x), and j(x) coincide with the usual magnetic quantities with the same name. If we insert the explicit solution (10A.5) of Ampère’s law into the energy, we obtain the Biot-Savart energy for an arbitrary current distribution Z 1 1 E= d3 x d3 x′ j(x) j(x′ ). (10A.81) 2 |x − x′ | Note that the action (10A.78) is invariant under two mutually dual gauge transformations, the usual magnetic one in (10A.2), by which the vector potential receives a gradient of an arbitrary scalar field, and the gauge transformation (10A.23), by which the irrelevant surface S is moved to another configuration S ′ . Thus we have proved the complete equivalence of the gradient representation of the magnetic field to the usual gauge field representation. In the gradient representation, there exists a new type of gauge invariance which expresses the physical irrelevance of the jumping surface appearing when using single-valued solid angles. The action (10A.79) describes magnetism in terms of a double gauge theory [8], in which both the gauge of A(x) and the shape of S can be changed arbitrarily. By setting up a grandcanonical partition function of many fluctuating surfaces it is possible to describe a large family of phase transitions mediated by the proliferation of line-like defects. Examples are vortex lines in the superfluid-normal transition in helium and dislocation and disclination lines in the melting transition of crystals [2, 6, 7, 10, 12]. Let us now go through the analogous calculation for a gas of monopoles at xn from the magnetic energy formed with the field (10A.58): 1 E= 8π

Z

3

"

d x ∇ × A + 4πg

X n

#2

(x; L↑n )

.

As in (10A.75) we introduce an auxiliary magnetic field and rewrite (10A.82) as ( #) " Z X 1 2 1 3 ↑ E = d x − B (x) − (x; Ln ) . B(x) · ∇ × A + g 8π 4π n Extremizing this in A yields ∇ × B = 0, so that we may set B = ∇Λ, and obtain ( ) Z X 1 2 3 ↑ [∇Λ(x)] + gΛ(x) ∇ · (x; Ln ) . E= d x − 8π n

(10A.82)

(10A.83)

(10A.84)

Recalling (10A.51), the extremal Λ field is Λ(x) = −

X 4πg X 1 δ(x − xn ) = g , 2 |x − xn | ∇ n n

(10A.85)

which leads, after reinsertion into (10A.84), to the Coulomb interaction energy E=

Appendix 10B

g2 X 1 . 2 |x − xn′ | n ′

(10A.86)

n,n

Comparison of Multivalued Basis Tetrads and Vierbein Fields

The standard tetrads or vierbein fields were introduced a long time ago in gravitational theories of spinning particles both in purely Riemann [16] as well as in Riemann-Cartan spacetimes [17, 18,

804


19, 20, 2]. Their mathematics is described in detail in the literature [21]. Their purpose was to define at every point a local Lorentz frame by means of another set of coordinate differentials dxα = h αλ (q)dq λ ,

α = 0, 1, 2, 3,

(10B.1)

which can be contracted with Dirac matrices γ α to form locally Lorentz invariant quantities. Local Lorentz frames are reached by requiring the induced metric in these coordinates to be Minkowskian: gαβ = hα µ (q)hβ ν (q)gµν (q) = ηαβ ,

(10B.2)

where ηαβ is the flat Minkowski metric (10.30). Just like ei µ (q) in (10.12), these vierbeins possess reciprocals hα µ (q) ≡ ηαβ g µν (q)hβ ν (q),

(10B.3)

and satisfy orthonormality and completeness relations as in (10.13): hα µ hβ µ = δα β ,

hα µ hα ν = δµ ν .

(10B.4)

They also can be multiplied with each other as in (10.14) to yield the metric gµν (q) = hα µ (q)hβ ν (q)ηαβ .

(10B.5)

Thus they constitute another “square root” of the metric. The relation between these square roots is some linear transformation ea µ (q) = ea α (q)hα µ (q),

(10B.6)

which must necessarily be a local Lorentz transformation Λa α (q) = ea α (q),

(10B.7)

since this matrix connects the two Minkowski metrics (10.30) and (10B.2) with each other: ηab Λa α (q)Λb β (q) = ηαβ .

(10B.8)

The different local Lorentz transformations allow us to choose different local Lorentz frames which distinguish fields with definite spin by the irreducible representations of these transformations. The physical consequences of the theory must be independent of this local choice, and this is the reason why the presence of spinning fields requires the existence of an additional gauge freedom under local Lorentz transformations, in addition to Einstein’s invariance under general coordinate transformations. Since the latter may be viewed as local translations, the theory with spinning particles are locally Poincaré invariant. The vierbein fields hα µ (q) have in common with ours that both violate the integrability condition as in (10.22), thus describing nonholonomic coordinates dxα for which there exists only a differential relation (10B.1) to the physical coordinates q µ . However, they differ from our multivalued tetrads ea µ(q) by being single-valued fields satisfying the integrability condition (∂µ ∂ν − ∂ν ∂µ )h αλ (q) = 0,

(10B.9)

in contrast to our multivalued tetrads e iλ (q) in Eq. (10.23). In the local coordinate system dxα , curvature arises from a violation of the integrability condition of the local Lorentz transformations (10B.7). The simple equation (10.24) for the torsion tensor in terms of the multivalued tetrads e iλ (q) must be contrasted with a similar-looking, but geometrically quite different, quantity formed from the vierbein fields hα λ (q) and their reciprocals, the objects of anholonomy [21]: Ωαβγ (q) =

1 µ hα (q)hβ ν (q) ∂µ hγν (q) − ∂ν hγµ (q) . 2

(10B.10)


Cancellation of Powers of δ(0)

Appendix 10C

805

Figure 10.6 Coordinate system q µ and the two sets of local nonholonomic coordinates dxα and dxa . The intermediate coordinates dxα have a Minkowski metric only at the point q, the coordinates dxa in an entire small neighborhood (at the cost of a closure failure). A combination of these similar to (10.28), h

γ γ γ γ K αβ (q) = Ωαβ (q) − Ωβ α (q) + Ω αβ (q),

(10B.11)

appears in the so-called spin connection h

Γαβ γ = hγ λ hα µ hβ ν (Kµν λ − K µνλ ),

(10B.12)

which is needed to form a covariant derivative of local vectors vα (q) = vµ (q)hα µ (q),

v α (q) = v µ (q)hα µ (q).

(10B.13)

Dα v β (q) = ∂α v β (q) + Γαγ β (q)v γ (q).

(10B.14)

These have the form Dα vβ (q) = ∂α vβ (q) − Γαβγ (q)vγ (q),

For details see Ref. [2, 6]. In spite of the similarity between the defining equations (10.24) and h

(10B.10), the tensor Ωαβγ (q) bears no relation to torsion, and K αβγ (q) is independent of the contortion Kαβ γ . In fact, the objects of anholonomy Ωαβγ (q) are in general nonzero in the absence of torsion [22], and may even be nonzero in flat spacetime, where the matrices hα µ (q) degenerate h

to local Lorentz transformations. The quantities K αβγ (q), and thus the spin connection (10B.12), characterize the orientation of the local Lorentz frames. The nonholonomic coordinates dxα transform the metric gµν (q) to a Minkowskian form ηab at any given point q µ . They correspond to a small “falling elevator” of Einstein in which the gravitational forces vanish precisely at the center of mass, the neighborhood still being subject to tidal forces. In contrast, the nonholonomic coordinates dxa flatten the spacetime in an entire neighborhood of the point. This is at the expense of producing defects in spacetime (like those produced when flattening an orange peel by stepping on it), as will be explained in Section IV. The affine connection Γab c (q) in the latter coordinates dxa vanishes identically. The difference between our multivalued tetrads and the usual vierbeins is illustrated in the diagram of Fig. 10.6.

Appendix 10C

Cancellation of Powers of δ(0)

There is a simple way of proving the cancellation of all UV-divergences δ(0). Consider a free particle whose mass depends on the time with an action Atot [q] =

Z

0

β

1 1 2 Z(τ )q˙ (τ ) − δ(0) log Z(τ ) , dτ 2 2

(10C.1)

806


where Z(τ ) is some function of τ but independent now of the path q(τ ). The last term is the simplest nontrivial form of the Jacobian action in (10.407). Since it is independent of q, it is conveniently taken out of the path integral as a factor Rβ (1/2)δ(0) dτ log Z(τ ) 0 J =e . (10C.2) We split the action into a sum of a free and an interacting part A(0) =

Z

0

β

dτ

1 2 q˙ (τ ), 2

Aint =

Z

β

dτ 0

1 [Z(τ ) − 1] q˙2 (τ ), 2

(10C.3)

and calculate the transition amplitude (10.410) as a sum of all connected diagrams in the cumulant expansion Z Z 1 2 −A(0) [q]−Aint [q] −A(0) [q] (0 β|0 0) = J Dq(τ )e =J Dq(τ )e 1 − Aint + Aint − . . . 2

1 A2int − . . . = (2πβ)−1/2 J 1 − hAint i + 2 2 1 = (2πβ)−1/2 J e−hAint ic + 2 hAint ic −... . (10C.4) We now show that the infinite series of δ(0)-powers appearing in a Taylor expansion of the exponential (10C.2) is precisely compensated by the sum of all terms in the perturbation expansion (10C.4). Being interested only in these singular terms, we may extend the τ -interval to the entire time axis. Then Eq. (10.393) yields the propagator ˙∆˙(τ, τ ′ ) = δ(τ − τ ′ ), and we find the first-order expansion term Z Z 1 1 hAint ic = dτ [Z(τ ) − 1] ˙∆˙(τ, τ ) = − δ(0) dτ [1 − Z(τ )]. (10C.5) 2 2 To second order, divergent integrals appear involving products of distributions, thus requiring an intermediate extension to d dimensions as follows Z Z

2 1 1 Aint c = dτ1 dτ2 (Z − 1)1 (Z − 1)2 2 ˙∆˙(τ1 , τ2 ) ˙∆˙(τ2 , τ1 ) 2 2 Z Z 1 1 → dd x1 dd x2 (Z − 1)1 (Z − 1)2 2 µ ∆ν (x1 , x2 ) ν ∆µ (x2 , x1 ) 2 2 Z Z 1 1 d d = d x1 d x2 (Z − 1)1 (Z − 1)2 2 ∆µµ (x2 , x1 ) ∆νν (x1 , x2 ) , (10C.6) 2 2 the last line following from partial integrations. For brevity, we have abbreviated [Z(τi ) − 1] by (Z − 1)i . Using the field equation (10.417) and going back to one dimension yields, with the further abbreviation (Z − 1)i → zi : Z Z

2 1 Aint c = dτ1 dτ2 z1 z2 δ 2 (τ1 , τ2 ). (10C.7) 2 To third order we calculate ZZZ

3 1 1 1 Aint c = dτ1 dτ2 dτ3 z1 z2 z3 8 ˙∆˙(τ1 , τ2 ) ˙∆˙(τ2 , τ3 ) ˙∆˙(τ3 , τ1 ) 2 2 2 ZZZ 1 1 1 d d d → d x1 d x2 d x3 z1 z2 z3 8 µ ∆ν (x1 , x2 ) ν ∆σ (x2 , x3 ) σ ∆µ (x3 , x1 ) 2 2 2 ZZZ 1 1 1 =− dd x1 dd x2 dd x3 z1 z2 z3 8 ∆µµ (x3 , x1 ) ∆νν (x1 , x2 ) ∆σσ (x2 , x3 ). 2 2 2

(10C.8)


807


Applying again the field equation (10.417) and going back to one dimension, this reduces to Z Z Z

3 Aint c = dτ1 dτ2 dτ3 z1 z2 z3 δ(τ1 , τ2 ) δ(τ2 , τ3 ) δ(τ3 , τ1 ). (10C.9)

Continuing to n-order and substituting Eqs. (10C.5), (10C.7), (10C.9), etc. into (10C.4), we obtain in the exponent of Eq. (10C.4) a sum − hAint ic +

∞ 1 3 1 X cn 1 2 Aint c − Aint c + . . . = (−1)n , 2 3! 2 1 n

(10C.10)

with cn =

Z

dτ1 . . . dτn C(τ1 , τ2 ) C(τ2 , τ3 ) . . . C(τn , τ1 )

(10C.11)

where C(τ, τ ′ ) = [Z(τ ) − 1] δ(τ, τ ′ ).

(10C.12)

Substituting this into Eq. (10C.11) and using the rule (10.368) yields Z Z Z n 2 cn = dτ1 dτn [Z(τ1 ) − 1] δ (τ1 − τn ) = δ(0) dτ [Z(τ ) − 1]n .

(10C.13)

Inserting these numbers into the expansion (10C.10), we obtain 1 2 1 3 − hAint ic + Aint c − Aint c + . . . 2 3!

=

1 δ(0) 2

=

1 − δ(0) 2

Z

dτ

∞ X 1

Z

(−1)n

[Z(τ ) − 1]n n

dτ log Z(τ ),

(10C.14)

which compensates precisely the Jacobian factor J in (10C.4).

Notes and References Path integrals in spaces with curvature but no torsion have been discussed by B.S. DeWitt, Rev. Mod. Phys. 29, 377 (1957). We do not agree with the measure of path integration in this standard work since it produces a ¯ physically incorrect h ¯ 2 R/12M -term in the energy. This has to be subtracted from the Lagrangian before summing over all paths to obtain the correct energy spectrum on surfaces of spheres and on group spaces. Thus, DeWitt’s action in the short-time amplitude of the path integral is nonclassical, in contrast to the very idea of path integration. A similar criticism holds for K.S. Cheng, J. Math. Phys. 13, 1723 (1972), ¯ who has an extra h ¯ 2 R/6M -term. See also related problems in H. Kamo and T. Kawai, Prog. Theor. Phys. 50, 680 (1973); M.B. Menskii, Theor. Math. Phys 18, 190 (1974); T. Kawai, Found. Phys. 5, 143 (1975); J.S. Dowker, Functional Integration and Its Applications, Clarendon, 1975; M.M. Mizrahi, J. Math. Phys. 16, 2201 (1975); C. Hsue, J. Math. Phys. 16, 2326 (1975); J. Hartle and S. Hawking, Phys. Rev. D 13, 2188 (1976); M. Omote, Nucl. Phys. B 120, 325 (1977); H. Dekker, Physica A 103, 586 (1980);

808


G.M. Gavazzi, Nuovo Cimento A 101, 241 (1981); T. Miura, Prog. Theor. Phys. 66, 672 (1981); C. Grosche and F. Steiner, J. Math. Phys. 36, 2354 (1995). A review on the earlier variety of ambiguous attempts at quantizing such systems is given in the article by M.S. Marinov, Physics Reports 60, 1 (1980). ¯ A measure in the phase space formulation of path integrals which avoids an R-term was found by K. Kuchar, J. Math. Phys. 24, 2122 (1983). The nonholonomic mapping principle is discussed in detail in Ref. [6]. The classical variational principle which yields autoparallels rather than geodesic particle trajectories was found by P. Fiziev and H. Kleinert, Europhys. Lett. 35, 241 (1996) (hep-th/9503074). The new principle made it possible to derive the Euler equation for the spinning top from within the body-fixed reference frame. See P. Fiziev and H. Kleinert, Berlin preprint 1995 (hep-th/9503075). The development of perturbatively defined path integrals after Section 10.6 is due to Refs. [23, 24, 25]. The individual citations refer to [1] H. Kleinert, Mod. Phys. Lett. A 4, lin.de/~kleinert/199).

2329 (1989) (http://www.physik.fu-ber-

[2] H. Kleinert, Gauge Fields in Condensed Matter, Vol. II, Stresses and Defects, World Scientific, Singapore, 1989 (ibid.http/b2). [3] P. Fiziev and H. Kleinert, Berlin preprint 1995 (hep-th/9503075). [4] H. Kleinert und A. Pelster, Gen. Rel. Grav. 31, 1439 (1999) (gr-qc/9605028). [5] H. Kleinert, Phys. Lett. B 236, 315 (1990) (ibid.http/202). [6] H. Kleinert, Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion, Gen. Rel. Grav. 32, 769 (2000) (ibid.http/258); Act. Phys. Pol. B 29, 1033 (1998) (gr-qc/9801003). [7] H. Kleinert, Theory of Fluctuating Nonholonomic Fields and Applications: Statistical Mechanics of Vortices and Defects and New Physical Laws in Spaces with Curvature and Torsion, in: Proceedings of NATO Advanced Study Institute on Formation and Interaction of Topological Defects, Plenum Press, New York, 1995, S. 201–232. [8] H. Kleinert, Double Gauge Theory of Stresses and Defects, Phys. Lett. A 97, 51 (1983) (ibid.http/107). [9] The theory of multivalued functions developed in detail in the textbook [12] was in 1991 so unfamiliar to field theorists that the prestigious Physical Review Letters accepted, amazingly, a Comment paper on Eq. (10A.33) by C.R. Hagen, Phys. Rev. Lett. 66, 2681 (1991), and a reply by R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 66, 2682 (1991). [10] H. Kleinert, Int. J. Mod. Phys. A 7, 4693 (1992) (ibid.http/203). [11] H. Kleinert, Gravity and Defects, World Scientific, Singapore, 2007 (in preparation). [12] H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, Superflow and Vortex Lines, World Scientific, Singapore, 1989 (ibid.http/b1). [13] H. Kleinert, Phys. Lett. B 246, 127 (1990) (ibid.http/205). H. Kleinert, PATH INTEGRALS


809

[14] H. Kleinert, Phys. Lett. B 293, 168 (1992) (ibid.http/211). [15] P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931); Phys. Rev. 74, 817 (1948), Phys. Rev. 74, 817 (1948). See also J. Schwinger, Particles, Sources and Fields, Vols. 1 and 2, Addison Wesley, Reading, Mass., 1970 and 1973. [16] S. Weinberg, Gravitation and Cosmology – Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972. [17] R. Utiyama, Phys. Rev. 101, 1597 (1956). [18] T.W.B. Kibble, J. Math. Phys. 2, 212 (1961). [19] F.W. Hehl, P. von der Heyde, G.D. Kerlick, and J.M. Nester, Rev. Mod. Phys. 48, 393 (1976). [20] F.W. Hehl, J.D. McCrea, E.W. Mielke, and Y. Ne’eman, Phys. Rep. 258, 1 (1995). [21] J.A. Schouten, Ricci-Calculus, Springer, Berlin, Second Edition, 1954. [22] These differences are explained in detail in pp. 1400–1401 of [2]. [23] H. Kleinert and A. Chervyakov, Phys. Lett. B 464, 257 (1999) (hep-th/9906156). [24] H. Kleinert and A. Chervyakov, Phys. Lett. B 477, 373 (2000) (quant-ph/9912056). [25] H. Kleinert and A. Chervyakov, Eur. Phys. J. C 19, 743 (2001) (quant-ph/0002067). [26] J.A. Gracey, Nucl. Phys. B 341, 403 (1990); Nucl. Phys. B 367, 657 (1991). [27] N.D. Tracas and N.D. Vlachos, Phys. Lett. B 257, 140 (1991). [28] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin, Heidelberg, New York, 1966, p. 85, or H. Bateman and A. Erdelyi, Higher transcendental functions, v.2, McGraw-Hill Book Company, Inc., 1953, p. 83, Formula 27. [29] G. ’t Hooft and M. Veltman, Nucl. Phys. B 44, 189 (1972). [30] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories, World Scientific, Singapore, 2001 (ibid.http/b8). [31] J.L. Gervais and A. Jevicki, Nucl. Phys. B 110, 93 (1976). [32] P. Salomonson, Nucl. Phys. B 121, 433 (1977). [33] J. de Boer, B. Peeters, K. Skenderis and P. van Nieuwenhuizen, Nucl. Phys. B 446, 211 (1995) (hep-th/9504097); Nucl. Phys. B 459, 631 (1996) (hep-th/9509158). See in particular the extra terms in Appendix A of the first paper required by the awkward regularization of these authors. [34] K. Schalm, and P. van Nieuwenhuizen, Phys. Lett. B 446, 247 (1998) (hep-th/9810115); F. Bastianelli and A. Zirotti, Nucl. Phys. B 642, 372 (2002) (hep-th/0205182). [35] F. Bastianelli and O. Corradini, Phys. Rev. D 60, 044014 (1999) (hep-th/9810119). [36] A. Hatzinikitas, K. Schalm, and P. van Nieuwenhuizen, Trace and chiral anomalies in string and ordinary field theory from Feynman diagrams for nonlinear sigma models , Nucl. Phys. B 518, 424 (1998) (hep-th/9711088). [37] H. Kleinert and A. Chervyakov, Integrals over Products of Distributions and Coordinate Independence of Zero-Temperature Path Integrals, Phys. Lett. A 308, 85 (2003) (quantph/0204067). [38] H. Kleinert and A. Chervyakov, Int. J. Mod. Phys. A 17, 2019 (2002) (ibid.http/330). [39] H. Kleinert, Gen. Rel. Grav. 32, 769 (2000) (ibid.http/258).

810


[40] B.S. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach, New-York, London, Paris, 1965. [41] R.T. Seeley, Proc. Symp. Pure Math. 10, 589 (1967); H.P. McKean and I.M. Singer, J. Diff. Geom. 1, 43 (1967). [42] This is a slight modification of the discussion in H. Kleinert, Phys. Lett. A 116, 57 (1986) (ibid.http/129). See Eq. (27). [43] N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields, Interscience, New York, 1959. [44] H. Kleinert and A. Chervyakov, Phys. Lett. A 299, 319 (2002) (quant-ph/0206022). [45] K. Goeke and P.-G. Reinhart, Ann. Phys. 112, 328 (1978). [46] L. Alvarez-Gaumé, D.Z. Freedman, and S. Mukhi, Ann. of Phys. 134, 85 (1981); J. Honerkamp, Nucl. Phys. B 36, 130 (1972); G. Ecker and J. Honerkamp, Nucl. Phys. B 35 481 (1971); L. Tataru, Phys. Rev. D 12, 3351 (1975); G.A. Vilkoviski, Nucl. Phys. B 234, 125 (1984); E.S. Fradkin and A.A. Tseytlin, Nucl. Phys. B 234, 509 (1984); A.A. Tseytlin, Phys. Lett. B 223, 165 (1989); E. Braaten, T.L. Curtright, and C.K. Zachos, Nucl. Phys. B 260, 630 (1985); P.S. Howe, G. Papadopoulos, and K.S. Stelle, Nucl. Phys. B 296, 26 (1988); P.S. Howe and K.S. Stelle, Int. J. Mod. Phys. A 4, 1871 (1989); V.V. Belokurov and D.I. Kazakov, Particles & Nuclei 23, 1322 (1992). [47] C.M. Fraser, Z. Phys. C 28, 101 (1985). See also: J. Iliopoulos, C. Itzykson, and A. Martin, Rev. Mod. Phys. 47, 165 (1975); K. Kikkawa, Prog. Theor. Phys. 56, 947 (1976); H. Kleinert, Fortschr. Phys. 26, 565 (1978); R. MacKenzie, F. Wilczek, and A. Zee, Phys. Rev. Lett. 53, 2203 (1984); I.J.R. Aitchison and C.M. Fraser, Phys. Lett. B 146, 63 (1984). [48] F. Cametti, G. Jona-Lasinio, C. Presilla, and F. Toninelli, Comparison between quantum and classical dynamics in the effective action formalism, Proc. of the Int. School of Physics ”Enrico Fermi”, CXLIII Ed. by G. Casati, I. Guarneri, and U. Smilansky, Amsterdam, IOS Press, 2000, pp. 431-448 (quant-ph/9910065). See also B.R. Frieden and A. Plastino, Phys. Lett. A 287, 325 (2001) (quant-ph/0006012). [49] L.D. Faddeev and V.N. Popov, Phys. Lett. B 25, 29 (1967). [50] C. Schubert, Phys. Rep. 355, 73 (2001) (hep-th/0101036). [51] H. Kleinert and A. Chervyakov, Perturbatively Defined Effective Classical Potential in Curved Space, (quant-ph/0301081) (ibid.http/339). [52] L. Alvarez-Gaumé and E. Witten, Nucl. Phys. B 234, 269 (1984). [53] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (ibid.http/159). [54] S. Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity, (John Wiley & Sons, New York, 1972). There are D(D + 1)/2 independent Killing vectors lεµ (q) describing the isometries. [55] See Part IV in the textbook [2] dealing with the differential geometry of defects and gravity with torsion, pp. 1427–1431. [56] H. Kleinert, Phys. Lett. B 246 , 127 (1990) (ibid.http/205); Int. J. Mod. Phys. A 7 , 4693 (1992) (ibid.http/203); and Phys. Lett. B 293 , 168 (1992) (ibid.http/211). See also the textbook [2].



Felix est quantulumcunque temporis contigit, bene collocatus est. Happy is he who has well employed the time, however small the time slices may be. Seneca

11 Schr¨ odinger Equation in General Metric-Affine Spaces We now use the path integral representation of the last chapter to find out which Schrödinger equation is obeyed by the time evolution amplitude in a space with curvature and torsion. If there is only curvature, the result establishes the connection with the operator quantum mechanics described in Chapter 1. In particular, it will properly reproduce the energy spectra of the systems in Sections 1.14 and 1.15 — a particle on the surface of a sphere and a spinning top — which were quantized there via group commutation rules. If the space carries torsion also, the Schrödinger operator emerging from our formulation will be a prediction. Its correctness will be verified in Chapter 13 by an application to the path integral of the Coulomb system which can be transformed into a harmonic oscillator by a nonholonomic mapping involving curvature and torsion.

11.1

Integral Equation for Time Evolution Amplitude

Consider the time-sliced path integral Eq. (10.146) 1 ′ ˆ hq|e−i(t−t )H/¯h |q ′ i = q D 2πi¯ hǫ/M



NY +1 n=2

 

Z

q

g(qn )



i dD ∆qn q D e 2πiǫ¯ h/M



PN+1 n=1

(Aǫ +AǫJ )/¯ h

, (11.1)

with the integrals over ∆qn to be performed successively from n = N down to n = 1. Let us study the effect of the last ∆qn -integration upon the remaining product of integrals. We denote the entire product briefly by ψ(qN +1 , tN +1 ) ≡ ψ(q, t) and the product without the last factor by ψ(qN , tN ) = ψ(qN +1 − ∆qN +1 , tN +1 − ǫ) ≡ ψ(q − ∆q, t − ǫ). Since the initial coordinate q0 and time t0 of the amplitude are kept fixed in the sequel, they are not shown in the arguments. We assume N to be so large that the amplitude has had time to develop from the initial state localized at q ′ to a smooth 811

812

11 Schr¨ odinger Equation in General Metric-Affine Spaces

function of ψ(q − ∆q, t − ǫ), smooth q compared to the width of the last short-time amplitude, which is of the order h ¯ ǫ tr (gµν )/M. From Eq. (11.1) we deduce the recursion relation ψ(q, t) =

q

g(q)

Z

dD ∆q q

2πiǫ¯ h/M

D

i exp (Aǫ + AǫJ ) ψ(q − ∆q, t − ǫ). h ¯

(11.2)

This is an integral equation ψ(q, t) =

Z

dD ∆q K ǫ (∆q) ψ(q − ∆q, t − ǫ),

(11.3)

with an integral kernel q

g(q) i ǫ ǫ K ǫ (∆q) = q exp + A ) . (A J D h ¯ 2πiǫ¯ h/M

(11.4)

The integral equation (11.3) will now be turned into a Schrödinger equation. This will be done in two ways, a short way which gives direct insight into the relevance of the different terms in the mapping (10.96), and a historic more tedious way, which is useful for comparing our path integral with previous alternative proposals in the literature (cited at the end).

11.1.1

From Recursion Relation to Schr¨ odinger Equation

The evaluation of (11.3) is much easier if we take advantage of the simplicity of the integral kernel K ǫ (∆q) and the measure when expressed in terms of the variables ∆xi . Thus we introduce into (11.3) the integration variables ∆ξ µ ≡ ∆xi ei µ , with ei µ evaluated at the postpoint q. The explicit relation between ∆ξ µ and ∆q µ follows directly from (10.96). In terms of ∆ξ µ , we rewrite (11.3) as ψ(q, t) =

Z

dD ∆ξ K0ǫ (∆ξ)ψ(q − ∆q(∆ξ), t − ǫ),

with the zeroth-order kernel q

g(q) iM µ ν K0ǫ (∆ξ) = q exp g (q)∆ξ ∆ξ µν D h ¯ 2ǫ 2πiǫ¯ h/M

(11.5)

of unit normalization

Z

dD ∆ξ K0ǫ (∆ξ) = 1.

(11.6)

To perform the integrals in (11.2), we expand the wave function as 1 ψ(q − ∆q, t − ǫ) = 1 − ∆q ∂µ + ∆q µ ∆q ν ∂µ ∂ν + . . . 2

µ

ψ(q, t − ǫ),

(11.7)


813

11.1 Integral Equation for Time Evolution Amplitude

and the coordinate differences ∆q µ in powers of ∆ξ by inverting Eq. (10.96): 1 1 ∆q = ∆ξ + Γµν λ ∆ξ µ ∆ξ ν − (∂σ Γµν λ −Γµν τ Γ{στ } λ )∆ξ µ ∆ξ ν ∆ξ σ +. . . .(11.8) 2! 3!

λ

λ

All affine connections are evaluated at the postpoint q. Including in (11.2) only the relevant expansion terms, we find the integral equation ψ(q, t) = ×

Z

dD ∆ξ K0ǫ (∆ξ)

1 − ∆ξ µ +

(11.9)

1 1 Γνλ µ ∆ξ ν ∆ξ λ ∂µ + ∆ξ µ ∆ξ ν ∂µ ∂ν + . . . 2! 2

ψ(q, t − ǫ).

The evaluation requires only the normalization integral (11.6) and the two-point correlation function µ

ν

h∆ξ ∆ξ i =

Z

dD ∆ξ K0ǫ (∆ξ)∆ξ µ ∆ξ ν =

i¯ hǫ µν g (q). M

(11.10)

The result is h ¯ 2 µν ψ(q, t) = 1 + iǫ (g ∂µ ∂ν − Γν νµ ∂µ ) + . . . 2M "

#

ψ(q, t − ǫ).

(11.11)

The differential operator in parentheses is proportional to the covariant Laplacian of the field ψ(q, t − ǫ): Dµ D µ ψ ≡ g µν Dµ Dν ψ = g µν Dµ ∂ν ψ = (g µν ∂µ ∂ν − Γν νµ ∂µ )ψ.

(11.12)

In a space with no torsion, this is equal to the Laplace-Beltrami operator applied to the field ψ: 1 √ ∆ψ = √ ∂µ gg µν ∂ν ψ. g

(11.13)

In a more general space, the relation between the two operators is obtained by working out the derivatives !

1 √ ∆ = g ∂µ ∂ν + √ ∂µ g g µν ∂ν + (∂µ g µν )∂ν . g µν

(11.14)

Using 1 √ √ ∂µ g g

!

=

1 στ ¯ ν, g ∂µ gστ = Γ µν 2

∂µ g σν = −g σλ g νκ ∂µ gλκ , ¯ µ µν −Γ ¯ ν µµ, ∂µ g µν = −Γ

(11.15)

814


we see that 1 √ ¯ µ µν , √ (∂µ g µν g) = −Γ g

(11.16)

¯ µ µν ∂ν )ψ = D ¯ µD ¯ µ ψ. ∆ψ = (g µν ∂µ ∂ν − Γ

(11.17)

and hence

Thus, the relation between the Laplacian and the Laplace-Beltrami operator is given by ¯ µD ¯ µ − Kµ µν ∂ν )ψ = (D ¯ µD ¯ µ − 2S ν ∂ν )ψ, Dµ D µ ψ = (D

(11.18)

¯µ denotes the covariant derivative formed with the Riemannian affine conwhere D ¯ µν λ , and Sµ is the contracted torsion nection, the Christoffel symbol Γ Sµ ≡ Sµν ν .

(11.19)

As a result, the amplitude ψ(q, t) in (11.2) satisfies the equation !

iǫ¯ h Dµ D µ ψ(q, t − ǫ) + O(ǫ2 ). ψ(q, t) = 1 + 2M

(11.20)

In the limit ǫ → 0, this leads to the Schrödinger equation ˆ 0 ψ(q, t), i¯ h∂t ψ(q, t) = H

(11.21)

ˆ 0 is the free-particle Schrödinger operator where H ¯2 ˆ0 = − h H Dµ D µ . 2M

(11.22)

It is the naively expected generalization of the flat-space operator ¯2 2 ˆ0 = − h H ∂i , 2M

(11.23)

from which (11.22) arises by transforming the derivatives with respect to Cartesian coordinates ∂i to the general coordinate derivatives ∂µ via the nonholonomic transformation ∂i = ei µ ∂µ .

(11.24)

∂i 2 = ei µ ∂µ eiν ∂ν = g µν ∂µ ∂ν − Γµ µν ∂ν ,

(11.25)

The result is

which coincides with the Laplacian Dµ D µ when applied to a scalar field. Note that the operator (11.22) contains no extra term proportional to the scalar curvature R allowed by other theories. H. Kleinert, PATH INTEGRALS

815


11.1.2

Alternative Evaluation

For completeness, we also present an alternative evaluation of the q-integrals in Eq. (11.3) which is more tedious but facilitates comparison with previous work. First, the postpoint action Aǫ is conveniently split into the leading term Aǫ0 =

M gµν (q)∆q µ ∆q ν 2ǫ

(11.26)

and a remainder ∆Aǫ ≡ Aǫ − Aǫ0 .

(11.27)

Correspondingly, we introduce as in (11.5) the zeroth-order kernel K0ǫ (∆q) with the unit normalization

q

g(q)

i ǫ =q A , D exp h ¯ 0 2πiǫ¯ h/M

Z

(11.28)

dD ∆q K0ǫ (∆q) = 1,

(11.29)

and expand K ǫ (∆q) around K0ǫ (∆q) with a series of correction terms of higher order in ∆q: ǫ

K (∆q) =

K0ǫ (∆q)[1

+ C(∆q)] ≡

K0ǫ (∆q)

"

1+

∞ X

cn (∆q)

n=1

n

#

.

(11.30)

Under the smoothness assumptions above, the wave function ψ(q − ∆q, t − ǫ) can be expanded into a Taylor series around the endpoint q, so that the integral equation (11.2) reads ψ(q, t) =

Z

D

d ∆q

K0ǫ (∆q)

"

1+

∞ X

cn (∆q)

n=1

n

#

(11.31)

1 × 1 − ∆q ∂µ + ∆q µ ∆q ν ∂µ ∂ν + . . . ψ(q, t − ǫ) . 2

µ

Due to the normalization property (11.6), the leading term simply reproduces ψ(q, t − ǫ). To calculate the correction terms cn (∆q), we expand C(∆q) = exp

i (∆Aǫ + AǫJ ) − 1 h ¯

(11.32)

in powers of ∆q µ . After inserting here ∆Aǫ from (11.27) with Aǫ0 from (11.26), we expand Aǫ as in (10.107) [recalling (10.121)]. By separating the expansion for C into even and odd powers of ∆q, C = C e + C ø,

(11.33)

816


we find for the odd terms C ø = −Γ{µν} ν ∆q µ −

iM Γµνλ ∆q µ ∆q ν ∆q λ + . . . , h ¯ 2ǫ

(11.34)

4 X

(11.35)

and the even terms Ce =

Cae + . . . ,

a=1

with 1 C1e = [∂{µ Γνλ} λ + Γ{νκ σ Γ{σ|µ}} κ + Γ{µσ} σ Γ{νλ} λ − Γ{νκ} σ Γ{µσ} κ ]∆q µ ∆q ν , 2 iM e C2 = Γ{µν} ν Γσλκ ∆q µ ∆q σ ∆q λ ∆q κ , 2¯ hǫ iM 1 1 τ σ τ σ e C3 = gκτ (∂λ Γµν + Γµν Γ{λσ} ) + Γµν Γλκσ ∆q µ ∆q ν ∆q λ ∆q κ , 2¯ hǫ 3 4 2 1 M µ ν λ σ τ κ C4e = − (11.36) 2 2 Γµνλ Γστ κ ∆q ∆q ∆q ∆q ∆q ∆q . 2 4¯ he The dots denote terms of higher order in ∆q µ which do not contribute to the limit ǫ → 0. The evaluation now proceeds perturbatively and requires the harmonic expectation values hO(∆q)i0 ≡

Z

dD ∆q K0ǫ (∆q) O(∆q).

(11.37)

The relevant correlation functions are i¯ hǫ µν g , M !2 i¯ hǫ = g µνλκ , M

h∆q µ ∆q ν i0 = h∆q µ ∆q ν ∆q λ ∆q κ i0 µ

ν

λ

κ

σ

τ

h∆q ∆q ∆q ∆q ∆q ∆q i0 =

i¯ hǫ M

!3

(11.38) (11.39)

g µνλκστ .

(11.40)

The tensor g µνλκ in the second expectation (11.39) collects three Wick contractions [recall (3.302)] and reads g µνλκ ≡ g µν g λκ + g µλ g νκ + g µκ g νλ .

(11.41)

The tensor g µνλκστ in the third expectation (11.40) collecting 15 Wick contractions is obtained recursively following the rule (3.303) by expanding g µνλκστ = g µν g λκστ + g µλ g νκστ + g µκ g νλστ + g µσ g νλκτ + g µτ g νλκσ .

(11.42)

A product of 2n factors ∆q results in (2n − 1)!! pair contractions. H. Kleinert, PATH INTEGRALS


817

Let us collect all contributions in (11.31) relevant to order ǫ. Obviously, the highest derivative term of ψ(q, t − ǫ) is 12 ∆q µ ∆q ν ∂µ ∂ν ψ(q, t − ǫ). It receives only a leading contribution from K0ǫ (∆q), h ¯ 2 µν (11.43) iǫ g (q)∂µ ∂ν ψ(q, t − ǫ), 2M with no more corrections from C(∆q). The term with one derivative ∂µ on ψ(q, t−ǫ) in (11.31) becomes Aµ ∂µ ψ(q, t − ǫ),

(11.44)

Aµ = −hC ø ∆q µ i0 .

(11.45)

where Aµ is the expectation involving the odd correction terms Using the rules (11.38) and (11.39), we find iM Aµ = Γστ λ ∆q σ ∆q τ ∆q λ ∆q µ + . . . Γ{λν} ν ∆q λ + 2¯ hǫ 0 h ¯ 1 µν {µν} νµ νµ = iǫ Γ ν − (Γ ν + Γ ν + Γν ) + . . . M 2 h ¯ = −iǫ Γν νµ + . . . . (11.46) 2M In combination with (11.43), this produces the Laplacian Dµ D µ of the field ψ(q, t−ǫ) as in Eq. (11.11). We now turn to the remaining contributions in (11.31) which contain no more derivatives of ψ(q, t − ǫ). They are all due to the expectation value hC e i0 of the even correction terms. Let us define i¯ h i¯ h Veff ≡ hCi0 = hC e i0 , (11.47) ǫ ǫ to be called the effective potential caused by the correction terms hCi0 of (11.32). Using the expectation values (11.38)–(11.40) we find Veff =

h ¯2 h ¯2 X B v≡ vA , M M A,B

where the sum runs over the six terms 1 v2 1 = − (Γ{µσ} σ Γ{νλ} λ − Γ{νκ} σ Γ{µσ} κ )g µν , 2 1 v2 2 = Γ{µν} τ Γλστ g µνσλ , 8 1 v2 3 = Γ{µκ} κ Γντ λ g µντ λ , 2 1 v2 4 = − Γµνλ Γστ κ g µνλστ κ , 8 1 v3 1 = − (∂{µ Γνλ} λ + Γ{νκ σ Γ{σ|µ}} κ )g µν , 2 1 v3 2 = gµτ (∂κ Γλν τ + Γλν σ Γ{κσ} τ )g µνλκ . 6

(11.48)

(11.49)

818


The subscripts 2 and 3 distinguish contributions coming from the quadratic and the cubic terms in the expansion (10.96) of ∆xi . By inserting on the right-hand sides the explicit expansions (11.42) and (11.41), we find after some algebra that the sum of all vA B -terms is zero. In fact, the v2 B - and v3 B -terms disappear separately. A simple structural reason for this is given in Appendix 11A. Explicitly, the cancellation is rather obvious for v3 B after inserting (11.41). For v2 B , the proof requires more work which is relegated to Appendix 11A. Note that in a space without curvature and torsion, the above manipulations are equivalent to a direct transformation of the flat-space integral equation ψ(x, t) =

Z

dD ∆x q

2πiǫ¯ h/M

D

M exp iǫ (∆xi )2 2

× (1 − ∆xi ∂xi + 12 ∆xi ∆xj ∂xi ∂xj + . . .)ψ(x, t − ǫ) " # iǫ¯ h 2 2 = 1+ ∂ + O(ǫ ) ψ(x, t − ǫ), 2M i

(11.50)

to the variable ∆q by a coordinate transformation. In a general metric-affine space, the wave function ψ(q, t) has no counter image in x-space so that (11.50) cannot be used as a starting point for a nonholonomic transformation.

11.2

Equivalent Path Integral Representations

From the derivation of the Schrödinger equation in Subsection 11.1.1 we learn an important lesson. When deriving the transformation law (10.96) between the finite coordinate differences ∆xi and ∆q µ by evaluating the integral equation (10.60) along the autoparallel, the cubic terms in ∆q, which make the action and measure lengthy, can be dropped altogether. A completely equivalent path integral representation of the time evolution amplitude is obtained by transforming the flat-space path integral (10.89) into the general metric-affine one (10.146) with the help of the shortened transformation i

i

∆x = e λ

1 ∆q − Γµν λ ∆q µ ∆q ν . 2! λ

(11.51)

This has the simple Jacobian J=

∂(∆x) = det (eiκ ) det (δ κ µ − ei κ ei{µ,ν} ∆q ν ), ∂(∆q)

(11.52)

whose effective action reads i ǫ 1 AJ = −ei κ ei κ,ν ∆q ν − ei µ ei{κ,ν} ej κ ej{µ,λ} ∆q ν ∆q λ + . . . . h ¯ 2

(11.53)

With the help of (10.16), this is expressed in terms of the connection yielding i ǫ 1 AJ = −Γ{νµ} µ ∆q ν − Γ{νκ} σ Γ{µ,σ} κ ∆q ν ∆q µ + . . . . h ¯ 2

(11.54)


819

11.2 Equivalent Path Integral Representations

The mapping (11.51) has, however, an unattractive feature: The short-time action following from (11.51) Aǫ =

M M (∆xi )2 = gµν ∆q µ ∆q ν −Γµνλ ∆q µ ∆q ν ∆q λ 2ǫ 2ǫ 1 σ µ ν λ κ + Γλκ Γµνσ ∆q ∆q ∆q ∆q + . . . 4

(11.55)

is no longer equal to the classical action (10.107) [recall the convention (10.121)] evaluated along the autoparallel. This was also a feature of another mapping which is the most convenient for calculations. Instead of deriving the relation between ∆xi and ∆q µ by evaluating (10.60) along the autoparallel, one may assume, for the moment, the absence of curvature and torsion and expand ∆xi = xi (q) − xi (q − ∆q) in powers of ∆q: 1 1 ∆xi = ei µ ∆q µ − ei µ,ν ∆q µ ∆q ν + ei µ,νλ ∆q µ ∆q ν ∆q λ + . . . . 2 3!

(11.56)

After this, curvature and torsion are introduced by allowing the functions x(q) and ∂µ x(q) to be nonintegrable in the sense of the Schwartz criterion, i.e., the second derivatives of x(q) and ei µ (q) need not commute with each other [implying that the right-hand side of (11.56) can no longer be written as xi (q) − xi (q − ∆q)]. The expansion (11.56) is then a definition of the transformation from ∆xi to ∆q µ . Using the identities (10.16), (10.131), and (10.132), the transformation (11.56) turns into

1 Γµν λ ∆q µ ∆q ν 2! 1 + (∂σ Γµν λ + Γµν τ Γστ λ )∆q µ ∆q ν ∆q σ + . . . . 3!

∆xi = ei λ ∆q λ −

(11.57)

This differs from the correct one (10.96) by the third-order term ∆′ xi =

1 i 1 e [τ,σ] ek τ ek ν,µ ∆q µ ∆q ν ∆q σ = ei λ Sστ λ Γµν τ ∆q µ ∆q ν ∆q σ , 3! 3!

(11.58)

which vanishes if the q-space has no torsion. The Jacobian associated with (11.56) is ∂(∆x) 1 J= = det (eiκ ) det δ κ µ − ei κ ei{µ,ν} ∆q ν + eκi ei{µ,νλ} ∆q ν ∆q λ + . . . , (11.59) ∂(∆q) 2

and corresponds to the effective action i ǫ 1 AJ = −ei κ ei κ,ν ∆q ν + [ei µ ei{µ,νλ} −ei µ ei{κ,ν} ej κ ej{µ,λ} ]∆q ν ∆q λ + . . . . (11.60) h ¯ 2 With (10.16), (10.131), and (10.132), this becomes i ǫ 1 AJ = − Γ{νµ} µ ∆q ν + (∂{µ Γν,κ} κ + Γ{ν,κ σ Γµ},σ κ − Γ{νκ} σ Γ{µ,σ} κ )∆q ν ∆q µ +. . . , h ¯ 2 (11.61)

820


which differs from the proper Jacobian action (10.145) only by one index symmetrization. To find the short-time action following from the mapping (11.56), we form Aǫ =

M Mh gµν ∆q µ ∆q ν −ei µ ei ν,λ ∆q µ ∆q ν ∆q λ (11.62) (∆xi )2 = 2ǫ 2ǫ i 1 i i 1 + e µ e ν,λκ + ei µ,ν ei λ,κ ∆q µ ∆q ν ∆q λ ∆q κ +. . . , 3 4

and use the identities (10.16), (10.131), and (10.132) to obtain M A = gµν ∆q µ ∆q ν − Γµνλ ∆q µ ∆q ν ∆q λ (11.63) 2ǫ 1 1 + gµτ (∂κ Γλν τ + Γλν δ Γκδ τ ) + Γλκ σ Γµνσ ∆q µ ∆q ν ∆q λ ∆q κ + . . . . 3 4 ǫ

This differs from the proper short-time action (10.107) [recall the convention (10.107)] only by the absence of the symmetrization in the indices κ and δ in the fourth term, the difference vanishing if the q-space has no torsion. The equivalent path integral representation in which (11.2) contains the shorttime action (11.63) and the Jacobian action (11.61) will be useful in Chapter 13 when solving the path integral of the Coulomb system. The equivalence of different time-sliced path integral representations manifests itself in certain moment properties of the integral kernel (11.4). The derivations of the Schrödinger equation in Subsections. 11.1.1 and 11.1.2 have made use only of the following three moment properties of the kernel: Z

dD ∆q K ǫ (∆q) = 1 + . . . ,

(11.64)

h ¯ Γµ µν + . . . , 2M Z h ¯ dD ∆q K ǫ (∆q)∆q µ ∆q ν = iǫ g µν + . . . , M Z

dD ∆q K ǫ (∆q)∆q ν = −iǫ

(11.65) (11.66)

evaluated at fixed postpoint q. The omitted terms indicated by the dots and all higher moments contribute to higher orders in ǫ which are irrelevant for the derivation of the differential equation obeyed by the amplitude. Any kernel K ǫ (∆q) with these properties leads to the same Schrödinger equation. If a kernel is written as K ǫ (∆q) = K0ǫ (∆q)[1 + C(∆q)],

(11.67)

where K0ǫ (∆q) is the free-particle postpoint kernel q

g(q) i µ ν K0ǫ (∆q) = q exp g (q) ∆q ∆q , µν D h ¯ 2πiǫ¯ h/M

(11.68)


821

11.2 Equivalent Path Integral Representations

the moment properties (11.64)–(11.66) are equivalent to hCi0 = 0 + . . . , h ¯ hC ∆q µ i0 = iǫ Γµ µν + . . . , 2M

(11.69) (11.70)

where the expectation values are taken with respect to the kernel K0ǫ (∆q), the dots on the right-hand side indicating terms of the order ǫ2 . Note that the third of the three moment properties is trivially true since it receives only a contribution from the leading part of the kernel K ǫ (∆q), i.e., from K0ǫ (∆q). The verification of the other two requires some work, in particular the first, which is the normalization condition. Two kernels K1ǫ , K2ǫ are equivalent if their correction terms C1 , C2 have expectations which are small of the order ǫ2 : hC1 i0 = hC2 i0 = O(ǫ2 ), h(C1 − C2 )∆q µ i0 = O(ǫ2 ).

(11.71) (11.72)

These are necessary and sufficient conditions for the equivalence. Many possible correction terms C lead to the same moment integrals. All of them are physically equivalent, being associated with the same Schrödinger equation. The simplest possibilities are 1 K ǫ (∆q) = K0ǫ (∆q) 1 + Γµ µ ν ∆q ν , 2

(11.73)

or ǫ

K (∆q) =

K0ǫ (∆q)

i M µ 1− Γµ ν ∆q ν gλκ ∆q λ ∆q κ , D + 2 2¯ hǫ

(11.74)

where D is the space dimension. The zero-order kernel satisfies automatically the first and third moment condition, (11.64) and (11.66), while the additional term enforces precisely the second condition, (11.65), without changing the others. The equivalent kernels can also be considered as the result of working with Jacobian actions i ǫ 1 µ 1 AJ = Γµ ν ∆q ν − (Γµ µ ν ∆q ν )2 , h ¯ 2 8 i ǫ i M µ A = − Γµ ν ∆q ν gλκ ∆q λ ∆q κ , h ¯ J D + 2 2¯ hǫ

(11.75) (11.76)

instead of the original one (10.145). Indeed, the second term in (11.75) can further be reduced by perturbation theory to 1 1 − (Γµ µ ν ∆q ν )2 → − Γµ µ ν Γλ λ κ h∆q ν ∆q κ i0 8 8 h ¯ = −iǫ (Γµ µ ν )2 , 8M

(11.77)

822


yielding an alternative and most useful form for the Jacobian action i ǫ 1 µ h ¯ AJ = Γµ ν ∆q ν − iǫ (Γµ µ ν )2 . h ¯ 2 8M

(11.78)

Remarkably, this expression involves only the connection contracted in the first two indices: Γµ µν = g µλ Γµλ ν .

11.3

(11.79)

Potentials and Vector Potentials

It is straightforward to find the effect of external potentials and vector potentials upon the Schrödinger equation. For this, we merely observe that the time-sliced potential term e e Aǫpot = Aµ ∆q µ − ∂ν Aµ ∆q µ ∆q ν − ǫV (q) + . . . c 2c

(11.80) ǫ

derived in Eq. (10.181) appears in the kernel K ǫ (∆q) via a factor eiApot /¯h . This factor can be combined with the postpoint expansion of the wave function in the integral equation (11.31), which becomes ψ(q, t) =

Z

dD ∆q K0ǫ (∆q) [1 + C(∆q)]

1 ×e 1 − ∆q ∂µ + ∆q µ ∆q ν ∂µ ∂ν ψ(q, t − ǫ) + . . . 2 " Z e = dD ∆q K0ǫ (∆q) [1 + C(∆q)] 1 − ∆q µ ∂µ − i Aµ (11.81) h ¯c # e e 1 µ ν + ∆q ∆q ∂µ − i Aµ ∂ν − i Aν − iǫV (q) ψ(q, t − ǫ) + . . . . 2 h ¯c h ¯c iAǫpot /¯ h

µ

By going through the steps of Subsection 11.1.1 or 11.1.2, we obtain the same Schrödinger equation as in (11.21), ˆ i¯ h∂t ψ(q, t) = Hψ(q, t).

(11.82)

ˆ differs from the free operator H ˆ 0 of (11.22), The Hamiltonian operator H ¯2 ˆ0 = − h H Dµ D µ , 2M

(11.83)

in two ways. First, a potential energy V (q) is added. Second, the covariant derivatives Dµ are replaced by DµA ≡ Dµ − i

e Aµ . h ¯c

(11.84)

This is the Schrödinger version of the minimal substitution in Eq. (2.636). H. Kleinert, PATH INTEGRALS

823

11.4 Unitarity Problem

The minimal substitution extends the covariance of Dµ with respect to coordinate changes to a covariance with respect to gauge transformations of the vector potential Aµ . The subtraction of iAµ /¯ h on the right-hand side of (11.84) reflects the fact that Pµ = pµ − Aµ rather than pµ is the gauge-invariant physical momentum of a particle in the presence of an electromagnetic field. Only the use of Pµ guarantees the gauge invariance of the electromagnetic interaction, just as in the flat-space action (2.635). Let us briefly verify that the Schrödinger equation (11.82) with the covariant derivative (11.84) is invariant under gauge transformations. If the amplitude is multiplied by a space-dependent phase ψ(q) → e−i(e/¯hc)Λ(q) ψ(q),

(11.85)

the covariant derivative (11.84) is multiplied by precisely the same phase: Dµ ψ(q) → e−i(e/¯hc)Λ(q) Dµ ψ(q),

(11.86)

if the vector potential is gauge tranformed as follows: Aµ → Aµ + ∂µ Λ(q).

(11.87)

Under these joint transformations, the Schrödinger equation (11.82) is multiplied by an overall phase factor e−iΛ(q) , and thus gauge invariant. Adding a potential V (q), the Hamilton operator in the Schrödinger equation (11.82) is therefore ¯2 A A µ ˆ =− h D D + V (q). H 2M µ

(11.88)

Observe that the mixed terms containing derivative and vector potential appears in the symmetrized form −

h ¯ (ˆ pµ Aµ + Aµ pˆµ ) . 2Mc

(11.89)

This corresponds to a symmetric time slicing of the interaction term q˙µ Aµ which was derived in Section 10.5 by using the equation of motion in calculation of the short-time action. Here we see that this time slicing guarantees the gauge invariance of the Schrödinger equation. Note further that there is no extra R-term in the Schrödinger equation (11.88).

11.4

Unitarity Problem

The appearance of the Laplace operator Dµ D µ in the free-particle Schrödinger equation (11.82) is in conflict with the traditional physical scalar product between two wave functions ψ1 (q) and ψ2 (q): hψ2 |ψ1 i ≡

Z

q

dD q g(q)ψ2∗ (q)ψ1 (q).

(11.90)

824


In such a scalar product, only the Laplace-Beltrami operator (11.13), 1 √ ∆ = √ ∂µ gg µν ∂ν , g

(11.91)

is a Hermitian operator, not the Laplacian Dµ D µ . The bothersome term is the contracted torsion term (Dµ D µ − ∆)ψ = −2S µ ∂µ ψ.

(11.92)

This term ruins the Hermiticity and thus also the unitarity of the time evolution operator of a particle in a space with curvature and torsion. For presently known physical systems in spaces with curvature and torsion, the unitarity problem is fortunately absent. Consider first field theories of gravity with torsion. There, the torsion field Sµν λ is generated by the spin current density of the fundamental matter fields. The requirement of renormalizability restricts these fields to carry spin 1/2. However, the spin current density of spin-1/2 particles happens to be a completely antisymmetric tensor.1 This property carries over to the torsion tensor. Hence, the torsion field in the universe satisfies S µ = 0. This implies that for a particle in a universe with curvature and torsion, the Laplacian always degenerates into the Laplace-Beltrami operator, assuring unitarity after all. In Chapter 13 we shall witness another way of escaping the unitarity problem. The path integral of the three-dimensional Coulomb system is solved by a multivalued transformation to a four-dimensional space with torsion where the physical scalar product is hψ2 |ψ1 iphys ≡

Z

√ dD q g w(q)ψ2∗(q)ψ1 (q),

(11.93)

with some scalar weight function w(q). This scalar product is different from the naive scalar product (11.90). It is, however, reparametrization-invariant, and w(q) makes the Laplacian Dµ D µ a Hermitian operator. The characteristic property of torsion in the transformed Coulomb system is that Sµ (q) = Sµν ν can be written as a gradient of a scalar function: Sµ (q) = ∂µ σ(q) [see Eq. (13.138)]. Such torsion fields admit a Hermitian Laplace operator of a scalar field in a scalar product (11.93) with the weight w(q) = e−2σ(q) .

(11.94)

Thus, the physical scalar product can be expressed in terms of the naive one as follows: hψ2 |ψ1 iphys ≡

Z

q

dD q g(q)e−2σ(q) ψ2∗ (q)ψ1 (q).

(11.95)

1

See, for example, H. Kleinert, Gauge Fields in Condensed Matter , op. cit., Vol. II, Part IV, Differential Geometry of Defects and Gravity with Torsion, p. 1432 (http://www.physik.fu-berlin.de/~kleinert/b8). H. Kleinert, PATH INTEGRALS

825

11.4 Unitarity Problem

To prove the Hermiticity, we observe that within the naive scalar product (11.93), a partial integration changes the covariant derivative −Dµ into Dµ∗ ≡ (Dµ + 2Sµ ).

(11.96)

Consider, for example, the scalar product Z

√ dD q gU µν1 ...νn Dµ Vν1 ...νn .

(11.97)

A partial integration of the derivative term ∂µ in Dµ gives surface term −

√ dD dq[(∂µ gU µν1 ...νn )Vν1 ...νn X√ − gU µν1 ...νi ...νn Γµνi λi Vν1 ...λi ...νn ].

Z

(11.98)

i

Now we use √ ¯ ν √ √ ∂µ g = g Γ g(2Sµ + Γµν ν ), µν =

(11.99)

to rewrite (11.98) as surface term − −

Z

√ dD q g [(∂µ U µν1 ...νn )Vν1 ...νn

X i

Γµνi λi U µν1 ...νi ...νn Vν1 ...λi ...νn −2Sµ U µν1 ...νn Vν1 ...νn ] ,

(11.100)

which is equal to surface term −

Z

√ dD q g(Dµ∗ U µν1 ...νn )Vν1 ...νn .

(11.101)

In the physical scalar product (11.95), the corresponding operation is Z

√ dD q ge−2σ(q) U µν1 ...νn Dµ Vν1 ...νn = = surface term − = surface term −

Z Z

√ dD q g(Dµ∗ e−2σ(q) U µν1 ...νn )Vν1 ...νn √ √ dD q ge−2σ(q) (Dµ gU µν1 ...νn )Vν1 ...νn .

(11.102)

Hence, iDµ is a Hermitian operator, and so is the Laplacian Dµ D µ . For spaces with an arbitrary torsion, the correct scalar product has yet to be found. Thus the quantum equivalence principle is so far only applicable to spaces with arbitrary curvature and gradient torsion.

826

11.5


Alternative Attempts

Our procedure has to be contrasted with earlier proposals for constructing path integrals in spaces with curvature, in which torsion was always assumed to be absent. In the notable work of DeWitt,2 the measure is taken to be proportional to N Z Y

n=1

q

dqn g(qn−1) =

N Z Y

n=1

q

dqn g(qn − ∆q)

(11.103)

so that the expansion in powers ∆q gives a Jacobian action AǫJ¯0 of Eq. (10.133). If one uses this action instead of the correct expression AǫJ of Eq. (10.145), the amplitude obeys a Schrödinger equation ˆ 0 + Veff )ψ(q, t), i¯ h∂t ψ(q, t) = (H

(11.104)

¯2 ˆ0 = − h ∆ H 2M

(11.105)

where

contains the Laplace-Beltrami operator ∆, and Veff is an additional effective potential

Veff =

h ¯2 ¯ R. 6M

(11.106)

The Schrödinger equation (11.104) differs from ours in Eq. (11.21) derived by the nonholonomic mapping procedure by the extra R-term. The derivation is reviewed ¯ is such that the surface of a in Appendix 11A. Note that our sign convention for R 2 ¯ sphere of radius R has R= 2/R . There is definite experimental evidence for the absence which will of such a term In the amplitude proposed by DeWitt, the short-time amplitude carries an extra semiclassical prefactor, a curved-space version of the Van Vleck-Pauli-Morette deter¯ which reduces minant in Eq. (4.119). It contributes another term proportional to R 2 ¯ (see Appendix 11B). Other path integral prescriptions lead (11.106) to (¯ h /12M)R even to additional noncovariant terms.3 All such nonclassical terms proportional to h ¯ 2 have to be subtracted from the classical action to arrive at the correct amplitude which satisfies the Schrödinger equation (11.104) without an extra Veff . There are two compelling arguments in favor of our construction principle: On the one hand, if the space has only curvature and no torsion, our path integral gives, as we have seen in Sections 8.7–8.9 and 10.4, the correct time evolution amplitude of a particle on the surface of a sphere in D dimensions and on group spaces. In contrast to other proposals, only the classical action appears in the short-time amplitude. In spaces with constant curvature, just as in flat space, our amplitude agrees with 2 3

See Section 11.5. See the review article by M.S. Marinov quoted at the end of the previous chapter. H. Kleinert, PATH INTEGRALS

827

11.6 DeWitt-Seeley Expansion of Time Evolution Amplitude

that obtained in operator quantum mechanics by quantizing the theory via the commutation rules of the generators of the group of motion. In the presence of torsion the result is new. It will be tested by the integration of the path integral of the Coulomb system in Chapter 13. This requires a coordinate transformation to an auxiliary space with curvature and torsion, which reduces the system to a harmonic oscillator. The new quantum equivalence principle leads to the correct result.

11.6

DeWitt-Seeley Expansion of Time Evolution Amplitude

An important tool for comparing the results of path integrals in curved space with operator results of Schrödinger theory is the short-time expansion of the imaginarytime evolution amplitude (q, β | q ′ , 0). In Schrödinger theory, the amplitude is given ˆ by the matrix elements of the evolution operator e−β H = eβ∆β/2 , with the LaplaceBeltrami operator (11.13). This expansion has first been given by DeWitt and by Seeley4 and reads (q β | q ′ 0) = (q | eβ∆/2 | q ′ ) = √

1 2πβ

D

e−gµν ∆q

µ ∆q ν /2β

∞ X

β k ak (q, q ′ ).

(11.107)

k=0

The expansion coefficients are, up to fourth order in ∆q µ , 1 ¯ 1 ¯ ¯ 1 ¯µ ν ¯ Rµν ∆q µ ∆q ν + R κ λ Rµσντ + Rκλ Rστ ∆q κ ∆q λ ∆q σ ∆q τ, 12 360 288 1 ¯ 1 ¯¯ 1 ¯ κλ ¯ 1 ¯ κλσ ¯ 1 ¯κ ¯ ′ a1 (q, q ) ≡ R+ RRµν+ R Rκµλν + R µ Rκλσν− R µ Rκν ∆q µ ∆q ν, 12 144 360 360 180 1 ¯2 1 ¯ µνκλ ¯ 1 ¯ µν ¯ ′ a2 (q, q ) ≡ R + R Rµνκλ − R Rµν , (11.108) 288 720 720

a0 (q, q ′ ) ≡ 1+

where ∆q µ ≡ (q − q ′ )µ and all curvature tensors are evaluated at q. For ∆q µ = 0 this simplifies to β ¯ β 2 1 ¯2 1 ¯ µνκλ ¯ µν ¯ ¯ 1 + R + R + R R − R R + ... . (q β | q 0) = √ µνκλ µν D 12 2 144 360 2πβ (11.109) This can also be written in the cumulant form as (

1

(q β | q 0) = √ 4

1 2πβ

D

)

β ¯ β 2 ¯ µνκλ ¯ ¯ µν R ¯ µν + . . . . (11.110) exp 1 + R + R Rµνκλ − R 12 720 "

#

B.S. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach, New-York, 1965. R.T. Seeley, Proc. Symp. Pure Math. 10 1967 589. See also H.P. McKean and I.M. Singer. J. Diff. Geom. 1 , 43 (1967).

828


The derivation goes as follows. In a neighborhood of some arbitrary point q0µ we expand the Laplace-Beltrami operator in normal coordinates where the metric and its determinant have the expansions (10.476) and (10.477) as 1¯ 2¯ k1 k2 µ ∆ = ∂2 − R ik1 jk2 (q0 )(q − q0 ) (q − q0 ) ∂µ ∂ν − Rµν (q0 )(q − q0 ) ∂ν . (11.111) 3 3 ˆ = −∆/2 in the exponent of Eq. (11.107) is now The time evolution operator H ˆ 0 and an interaction part H ˆ int as follows separated into a free part H ˆ 0 = − 1 ∂2, H (11.112) 2 ˆ int = 1 R ¯ ik1 jk2 (q − q0 )k1 (q − q0 )k2 ∂µ ∂ν + 1 R ¯ µν (q − q0 )µ ∂ν . H (11.113) 6 3 We now recall Eq. (1.294) and see that the transition amplitude (11.107) satisfies the integral equation ˆ 0 +H ˆ int ) −β(H

′

(q β | q 0) = hq | e

= (q β | q ′ 0)0 − where

Z

ˆ0 −β H

′

| q i = hq | e β

0

"

1−

Z

0

β

ˆ0 σH

dσe

ˆ −σH

ˆ int e H

Z

ˆ int (¯ dσ dD q¯ (q β − σ | q¯ 0)0 H q ) (¯ q σ | q 0), ˆ

(q β | q ′ 0)0 = hq | e−β H0 | q ′ i = √

1 −(∆q)2 /2β . n e 2πβ

#

| q′i (11.114)

(11.115)

ˆ int we can replace H ˆ in the last exponential of Eq. (11.114) by H ˆ0 To first order in H and obtain ′

′

(q β | q 0) ≈ (q β | q 0)0 −

Z

β

0

Z

ˆ int (¯ dσ dD q¯ (q β − σ | q¯ 0)0 H q ) (¯ q σ | q 0)0 . (11.116)

Inserting (11.113) and choosing q0 = q ′ , we find ′

′

(

(q β | q 0) = (q β | q 0)0 1 + ×

"

Z

0

1¯ − R q κ ∆¯ qλ µκνλ ∆¯ 6

β

dD (∆¯ q ) −[∆¯q−(σ/β) ∆q]2 /2a (11.117) √ D e 2πa ! #) δ µν ∆¯ q µ ∆¯ qν 1 ¯ ∆¯ q µ ∆¯ qν − + + Rµν , σ σ2 3 σ

dσ

Z

where we have replaced the integrating variable q¯ by ∆¯ q = q¯ − q ′ and introduced the variable a ≡ (β − σ)σ/β. There is initially also a term of fourth order in ∆¯ q ¯ µκνλ in µκ and νλ. The which vanishes, however, because of the antisymmetry of R remaining Gaussian integrals are performed after shifting ∆¯ q → ∆¯ q + σ ∆q/β, and we obtain ( " #) 1Z β σ ¯ a¯ ′ ′ ′ ′ µ ν (q β | q 0) = (q β | q 0)0 1 + dσ 2 Rµν (q )∆q ∆q + R(q ) 6 0 β σ # " 1 ¯ β ¯ ′ ′ µ ν = (q β | q ′ 0)0 1 + R R(q ) . (11.118) µν (q )∆q ∆q + 12 12 H. Kleinert, PATH INTEGRALS

829

11.6 DeWitt-Seeley Expansion of Time Evolution Amplitude

Note that all geometrical quantities are evaluated at the initial point q ′ . They can be re-expressed in power series around the final position q using the fact that in normal coordinates 1¯ k1 k2 gµν (q ′ ) = gµν (q) + R + ... , (11.119) ik jk (q)∆q ∆q 3 1 2 gµν (q ′ )∆q µ ∆q ν = gµν (q)∆q µ ∆q ν , (11.120) the latter equation being true to all orders in ∆q due to the antisymmetry of the ¯ µνκλ in all terms of the expansion (11.119), which is just another form of tensors R writing the expansion (10.476) up to the second order in ∆q µ . Going back to the general coordinates, we obtain all coefficients of the expansion (11.107) linear in the curvature tensor "

#

1 1 ¯ β ¯ −gµν (q)∆q µ ∆q ν /2β µ µ (q β | q 0) ≃ √ 1+ R R(q) . (11.121) µν (q)∆q ∆q + n e 12 12 2πβ ′

The higher terms in (11.107) can be derived similarly, although with much more effort. A simple cross check of the expansion (11.107) to high orders is possible if we restrict the space to the surface of a sphere of radius r in D dimensions which has D − 1 dimensions. Then ¯ µνκλ = 1 (gµλ gνκ − gµκ gνλ ) , µ, ν = 1, 2, . . . , D − 1. R (11.122) r2 Contractions yield Ricci tensor and scalar curvature ¯ µν = R ¯ κµν κ = D − 2 gµν , R r2 and further:

¯=R ¯ µ µ = (D − 1)(D − 2) R r2

(11.123)

2 2(D − 1)(D − 2) ¯ 2 = (D − 1)(D − 2) . ¯2 = , R (11.124) R µνκλ µν r4 r4 Inserting these into (11.108), we obtain the DeWitt-Seeley short-time expansion of the diagonal amplitude for any q, up to order β 2 :

(q β | q 0) = √

1 2πβ

D−1

"

1 + (D−1)(D−2)

β 12r 2

β2 + (D−1)(D−2)(5D − 17D + 18) + . . . . (11.125) 1440r 4 2

#

This expansion may easily be reproduced by a simple direct calculation5 of the partition function for a particle on the surface of a sphere Z(β) =

∞ X

dl exp[−l(l + D−2)β/2r 2] ,

(11.126)

l=0

5

H. Kleinert, Phys. Lett. A 116 , 57 (1986) (http://www.physik.fu-berlin.de/~kleinert/ 129). See Eq. (27).

830


where −l(l +D −2) are the eigenvalues of the Laplace-Beltrami operator on a sphere [recall (10.164)] and dl = (2l + D − 2)(l + D − 3)!/l!(D − 2)! their degeneracies [recall (8.113)]. Since the space is homogeneous, the amplitude (q β | q 0) is obtained from this by dividing out the constant surface of a sphere: (q β | q 0) =

Γ(D/2) Z(β). 2π D/2 r D−1

(11.127)

For any given D, the sum in (11.126) easily be expanded in powers of β. As an example, take D = 3 where Z(β) =

∞ X

(2l + 1) exp[−l(l + 1)β/2r 2] .

(11.128)

l=0

In the small-β limit, the sum (11.128) is evaluated as follows Z

Z(β) =

0

∞

2

d [l(l + 1)] exp[−l(l + 1)β/2r ] +

∞ X l=0

h

(2l+1) 1 − l(l+1)β/2r 2 + . . .

i

.

(11.129)

The integral is immediately done and yields Z

∞

0

dz exp(−zβ/2r 2 ) =

2r 2 . β

(11.130)

The sums are divergent but can be evaluated by analytic continuation from negative P −z powers of l to positive ones with the help of Riemann zeta functions ζ(z) = ∞ n=1 n , which vanishes for all even negative arguments. Thus we find ∞ X

(2l + 1) = 1 +

l=0

∞ X l=1

(2l + 1) = 1 + 2ζ(−1) −

1 1 = , 2 3

(11.131)

∞ ∞ β X β β β X (2l + 1)l(l + 1) = − (2l3 + l) = − 2 [2ζ(−3) + ζ(−1)] = . 2 2 2r l=0 2r l=1 2r 30r 2 (11.132) Substituting these into (11.129) yields

−

2r 2 Z(β) = β

β β2 1+ 2 + + ... . 6r 60r 4 !

(11.133)

Dividing out the constant surface of a sphere 4πr 2 as required by Eq. (11.127), we obtain indeed the expansion (11.125) for the surface of a sphere in three dimensions.

Appendix 11A

Cancellations in Effective Potential

Here we demonstrate the cancellation of the terms v2 B and v3 B in formula (11.48) for the effective potential. First we give a simple reason why the cancellation occurs separately for the contributions H. Kleinert, PATH INTEGRALS

Appendix 11A

Cancellations in Effective Potential

831

stemming from the second and third terms in the expansion (10.96) of ∆xi . Consider the model integral Z d∆x (∆x)2 √ , (11A.1) exp − 2ǫ 2πǫ and assume that ∆x has an expansion of the type (10.96): ∆x = ∆q[1 + a2 ∆q + a3 (∆q)2 + . . .].

(11A.2)

The integral transforms into Z d∆q (∆q)2 √ [1 + 2a2 ∆q + 2a3 (∆q)2 + a22 (∆q)2 + . . .] , [1 + 2a2 ∆q + 3a3 (∆q)2 + . . .] exp − 2ǫ 2πǫ (11A.3) and is evaluated perturbatively via the expansion Z d∆q (∆q)2 (∆q)3 (∆q)4 (∆q)4 √ 1 − a2 − a3 − a22 exp − 2ǫ ǫ ǫ 2ǫ 2πǫ 6 4 (∆q) (∆q) + a22 − 2a22 + 3a3 (∆q)2 + . . . . (11A.4) 2ǫ ǫ If hOi0 denotes the harmonic expectation value Z d∆q √ O exp[−(∆q)2 /2ǫ], hOi0 ≡ 2πiǫ

(11A.5)

one has h(∆q)2 i0 = ǫ, h(∆q)4 i0 = 3!ǫ2 ,

h(∆q)6 i0 = 5!ǫ3 , . . . .

(11A.6)

Using these values we find that the a2 - and a3 -terms cancel separately. Precisely this cancellation mechanism is active in the separate cancellation of the more complicated expressions v1 B , v2 B in Eq. (11.49). To demonstrate the cancellations explicitly, consider first the derivative terms in v3 B . They are 1 1 v3 ∂Γ = − g µν ∂{µ Γνλ} λ + gµτ ∂κ Γλν τ (g µν g λκ + g µλ g νκ + g µκ g νλ ). 2 6

(11A.7)

Due to the symmetrization of the first term in µνλ, this gives zero. The cancellation of the remaining terms in v3 B which are quadratic in Γ is most easily shown by writing all indices as subscripts, inserting gµνλκ from (11.41), and working out the contractions. To calculate the v2 B -terms, it is useful to introduce the notation Γ1µ ≡ Γµν ν , Γ2µ ≡ ν ˜ 1µ =(Γ ˜ T1µ =(Γ ˜ 2µ =Γ ˜ T2µ =Γ Γνµ , Γ3µ ≡ Γν ν µ and similarly the matrices Γ ˆ µ )λκ , Γ ˆ µ )κλ , Γ ˆ λµκ , Γ ˆ κµλ , T ˜ ˜ Γ3µ =Γ ˆ λκµ , Γ3µ =Γ ˆ κλµ . For contractions such as Γ1µ Γ1µ we write Γ1 Γ1 , and for Γµνλ Γµνλ we write ˜ 1Γ ˜1 = Γ ˜ 2Γ ˜2 = Γ ˜3Γ ˜ 3 , whichever is most convenient. Similarly, Γµνλ Γλµν = Γ ˜ 1Γ ˜ 2T = Γ ˜2Γ ˜3T = Γ ˜ 3Γ ˜ 1 . Then we work out Γ v2 1

=

v2 2

=

v2 3

=

v2 4

=

1 ˜ 3 (Γ ˜1 + Γ ˜T + Γ ˜2 + Γ ˜ T )], − [(Γ1 + Γ2 )2 − Γ 1 2 8 1 ˜ 3 (Γ ˜3 + Γ ˜ T )], [Γ3 Γ3 + Γ 3 8 1 [(Γ1 + Γ2 )2 + Γ3 (Γ1 + Γ2 )], 4 1 − [Γ1 2 + Γ2 2 + Γ3 2 + 2(Γ1 Γ2 + Γ2 Γ3 + Γ3 Γ1 ) 8 ˜ 3 (Γ ˜1 + Γ ˜ T1 + Γ ˜2 + Γ ˜ T2 + Γ3 + Γ ˜ T3 )]. +Γ

(11A.8)

832


It is easy to check that the sum of these v2 B -terms vanishes. Incidentally, if the symmetrizations in (11.49) following from our Jacobian action had been absent, we would find the additional terms ∆v3 ∂Γ 2

∆v3 Γ

= =

1¯ 2 1 ˜ ˜T R − ∂µ S µ + (Γ 3 Γ2 − Γ3 Γ2 ), 6 3 6 1 ˜ ˜ 1˜ ˜ ˜ ˜T − Γ 3 Γ2 + (Γ3 Γ2 + Γ3 Γ2 + Γ3 Γ2 ), 2 6

(11A.9) (11A.10)

whose sum yields the additional contribution to the v3 B -terms ∆v3 =

1¯ 2 2˜ ˜ R − ∂µ S µ + Γ 3 S1 , 6 3 3

(11A.11)

˜ 2 = −Γ ˜ 3 S˜1 . S˜3 Γ

(11A.12)

after having used the identity

The first term in (11A.11) is the R-term derived by K.S. Cheng6 as an effective potential in the Schrödinger equation. For v2 B , we would find the extra terms 1 ˜ 3Γ ˜ 2 ) + 1 [(Γ1 + Γ2 )2 − Γ ˜ 3 (Γ ˜1 + Γ ˜T + Γ ˜2 + Γ ˜ T )], ∆v2 1 = − (Γ1 Γ1 − Γ 1 2 2 8

∆v2 3 =

1 (Γ1 − Γ2 )(Γ1 + Γ2 + Γ3 ), 4

(11A.13)

(11A.14)

which add up to 1 1 1˜ ˜ 1˜ ˜ ∆v2 = − S1 S1 + Γ3 S1 − Γ 3 S1 + S1 S3 , 2 2 3 2

(11A.15)

where we have written S1 for Sµ and used some trivial identities such as ˜3Γ ˜ 2T = Γ ˜ 3Γ ˜ 1. Γ

(11A.16)

Thus we would obtain an additional effective potential Veff = (¯ h2 /M )v with v=

1 2 1 1 1˜ ˜ R − g µν ∂µ Sν − (S12 − Γ3 S1 ) + S˜1 S˜3 − Γ 3 S1 . 6 3 2 2 3

(11A.17)

The second and the fourth term can be combined to 2 1 − Dµ S µ − Γ3 S1 . 3 6

(11A.18)

Due to the presence of Γ’s in v, this is a noncovariant expression which cannot possibly be physically correct. In the absence of torsion, however, v happens to be reparametrization-invariant, and this ¯ is the reason why the resulting effective potential Veff = h ¯ 2 R/6M appeared acceptable in earlier works. A procedure which has no reparametrization-invariant extension to spaces with torsion cannot be correct. 6

K.S. Cheng, J. Math. Phys. 13 , 1723 (1972). H. Kleinert, PATH INTEGRALS

Appendix 11B

DeWitt’s Amplitude

Appendix 11B

DeWitt’s Amplitude

833

DeWitt, in his frequently quoted paper,7 attempted to quantize the motion of a particle in a curved space using the naive measure of path integration as in Eq. (10.152), but with the shorttime amplitude (with q ≡ qn , q ′ ≡ qn−1 ) p −1/2 g(q)g(q ′ ) (ǫ/M )D/2 D1/2 i ǫ A , exp p D ¯h 2πiǫ¯h/M

(11B.1)

where D is the curved-space analog of the Van Vleck-Pauli-Morette determinant [defined in flat space after Eq. (4.118)]: D = detD [−∂qµ ∂q′ν Aǫ (q, q ′ )] .

(11B.2)

After taking the Jacobian action AǫJ0 into account, this leads to an integral kernel K ǫ (∆q) which p −1/2 differs from our correct one in Eq. (11.4) by an extra factor g(q)g(q ′ ) (ǫ/M )D/2 D1/2 . This has 1 ¯ µ ν the postpoint expansion 1 + 12 Rµν ∆q ∆q + . . . . When treated perturbatively, the extra term is ¯ equivalent to ǫ¯hR/12M , reducing the effective potential (11.106) by a factor 1/2. In order to obtain the correct amplitude, DeWitt had to add a nonclassical term to the action in the path integral. This term is proportional to h ¯ 2 and removes the unwanted term Veff , i.e., ∆DW Aǫ = ǫVeff . Such a correction procedure must be rejected on the grounds that it runs contrary to the very essence of the entire path integral approach, in which the contribution of each path is controlled entirely by the phase eiA/¯h with the classical action in the exponent. The short-time kernel proposed by Cheng is the same as DeWitt’s, except that it does not include the extra Van Vleck-Pauli-Morette determinant in (11B.1). In this case, the result is the full effective potential (11.106), which must be artificially subtracted from the classical action to obtain the correct amplitude.

Notes and References The first path integral in a curved space was written down by B.S. DeWitt, Rev. Mod. Phys. 29, 377 (1957), making use of previous work by C. DeWitt-Morette, Phys. Rev. 81, 848 (1951). A modified ansatz is due to K.S. Cheng, J. Math. Phys. 13, 1723 (1972). For recent discussions with results different from ours see H. Kamo and T. Kawai, Prog. Theor. Phys. 50, 680, (1973); T. Kawai, Found. Phys. 5, 143 (1975); H. Dekker, Physica A 103, 586 (1980); G.M. Gavazzi, Nuovo Cimento 101A, 241 (1981). A good survey of most attempts is given by M.S. Marinov, Phys. Rep. 60, 1 (1980). In a space without torsion, C. DeWitt-Morette, working with stochastic differential equations, ¯ postulates a Hamiltonian operator without an extra R-term. See her lectures presented at the 1989 Erice Summer School on Quantum Mechanics in Curved Spacetime, ed. by V. de Sabbata, Plenum Press, 1990, and references quoted therein, in particular C. DeWitt-Morette, K.D. Elworthy, B.L. Nelson, and G.S. Sammelman, Ann. Inst. H. Poincaré A 32, 327 (1980). 7

B.S. DeWitt, Rev. Mod. Phys. 29 , 377 (1957).

834


¯ A path measure in a phase space path integral which does not produce any R-term was proposed by K. Kuchar, J. Math. Phys. 24, 2122 (1983). The short derivation of the Schrödinger equation in Subsection 11.1.1 is due to P. Fiziev and H. Kleinert, J. Phys. A 29, 7619 (1996) (hep-th/9604172).



To reach a depth profounder still, and still Profounder, in the fathomless abyss. William Cowper (1731-1800), The Winter Morning Walk

12 New Path Integral Formula for Singular Potentials In Chapter 8 we have seen that for systems with a centrifugal barrier, the Euclidean form of Feynman’s original time-sliced path integral formula diverges for certain attractive barriers. This happens even if the quantum statistics of the systems is well defined. The same problem arises for a particle in an attractive Coulomb potential, and thus in any atomic system. In this chapter we set up a new and more flexible path integral formula which is free of this problem for any singular potential. This has recently turned out to be the key for a simple solution of many other path integrals which were earlier considered intractable.

12.1

Path Collapse in Feynman’s formula for the Coulomb System

The attractive Coulomb potential V (r) = −e2 /r has a singularity at the coordinate origin r = 0. This singularity is weaker than that of the centrifugal barrier, but strong enough to cause a catastrophe in the Euclidean path integral. Recall that an attractive centrifugal barrier does not even possess a classical partition function. The same thing is true for the attractive Coulomb potential where formula (2.344) reads Zcl =

Z

d3 x

e2 exp β . q 3 r 2π¯ h2 β/M !

The integral diverges near the origin. In addition, there is a divergence at large r. The leading part of the latter can be removed by subtracting the free-particle partition function and forming Zcl′

≡ Zcl − Zcl |e=0 =

Z

d3 x q

2

2π¯ h β/M 835

3

e2 exp β r

"

!

#

−1 ,

(12.1)

836

12 New Path Integral Formula for Singular Potentials

leaving only a quadratic divergence. In a realistic many-body system with an equal number of oppositely charged particles, this disappears by screening effects. Thus we shall not worry about it any further and concentrate only on the remaining small-r divergence. In a real atom, this singularity is not present since the nucleus is not a point particle but occupies a finite volume. However, this “physical regularization” of the singularity is not required for quantum-mechanical stability. The Schrödinger equation is perfectly solvable for the singular pure −e2 /r potential. We should therefore be able to recover the existing Schrödinger results from the path integral formalism without any short-distance regularization. On the basis of Feynman’s original time-sliced formula, this is impossible. If a path consists of a finite number of straight pieces, its Euclidean action A=

Z

τb

τa

e2 M ′2 x (τ ) − dτ 2 r(τ ) "

#

(12.2)

can be lowered indefinitely by a path with an almost stretched configuration which corresponds to a slowly moving particle sliding down into the −e2 /r abyss. We call this phenomenon a path collapse. In nature, this catastrophe is prevented by quantum fluctuations. In order to understand how this happens, it is useful to reinterpret the paths in the Euclidean path integral as random lines parametrized by τ ∈ (τa , τb ). Their distribution is governed by the “Boltzmann factor” e−A/¯h , whose effective “quantum” temperature is Teff ≡ h ¯ /kB (τb − τa ).1 The logarithm of the Euclidean amplitude (xb τb |xa τa ) multiplied by −Teff defines a free energy F = E − kB T S of the random line with fixed endpoints. The quantum fluctuations equip the path with a configurational entropy S. This must be sufficiently singular to produce a regular free energy bounded from below. Obviously, such a mechanism can only work if the exact path integral contains an infinite number of infinitesimally small sections. Only these can contain enough configurational entropy near the singularity to halt the collapse. The variational approach in Section 5.10 has shown an important effect of the configurational entropy of quantum fluctuations. It smoothes the singular Coulomb potential producing an effective classical potential that is finite at the origin. A path collapse was avoided by defining the path integral as an infinite product of integrals over all Fourier coefficients. The infinitely high-frequency components were integrated out and this produced the desired stability. These high-frequency components are absent in a finitely time-sliced path with a finite number of pieces, where frequencies Ωm are bounded by twice the inverse slice thickness 1/ǫ [recall Eqs. (2.101), (2.102)]. Unfortunately, the path measure used in the variational approach is unsuitable for exact calculations of nontrivial path integrals. Except for the free particle and 1

This amounts to viewing the path as a polymer with configurational fluctuations in space, a possibility which is a major topic in Chapters 15 and 16. H. Kleinert, PATH INTEGRALS

837

12.1 Path Collapse in Feynman’s formula for the Coulomb System

the harmonic oscillator, these are all based on solving a finite number of ordinary integrals in a time-sliced formula. We therefore need a more powerful time-sliced path integral formula which avoids a collapse in singular potentials. For the Coulomb system, such a formula has been found in 1979 by Duru and Kleinert.2 It has become the basis for solving the path integral of many other nontrivial systems. Here we describe the most general extension of this formula which will later be applied to a number of systems. For the attractive Coulomb potential and other singular potentials, such as attractive centrifugal barriers, it will not only halt the collapse, but also be the key to an analytic solution. The derived stabilization is achieved by introducing a path-independent width of the time slices. If the path approaches an abyss, the widths decrease and the number of slices increases. This enables the configurational entropy of Eq. (12.3) to grow large enough to cancel the singularity in the energy. To see the cancellation mechanism, consider a random line with n links which has, on a simple cubic lattice in D dimensions, (2D)n configurations with an entropy S = n log(2D).

(12.3)

If the number of time slices n increases near the −e2 /r singularity like const/r, then the entropy is proportional to 1/r. A path section which slides down into the abyss must stretch itself to make the kinetic energy small. But then it gives up a certain entropy S, and this raises the free energy by kB Teff S according to (12.3). This compensates for the singularity in the potential and halts the collapse. The purpose of this chapter is to set up a path integral formula in which this stabilizing mechanism is at work. It should be pointed out that no instability problem would certainly arise if we were to define the imaginary-time path integral for the time evolution amplitude in the continuum (xb τb |xa τa ) ≡

Z

D D x(τ )

Z

1 D D p(τ ) exp (2π¯ h)D h ¯

Z

τb

τa

dτ [ipx˙ − H(p, x)]

(12.4)

without any time slicing as the solution of the Schrödinger differential equation ˆ τ |xa τa ) = h ¯ δ(τ − τa )δ (D) (x − xa ) (¯ h∂τ + H)(x

(12.5)

ˆ n (x) = En ψn (x), [compare Eq. (1.304)]. After solving the Schrödinger equation Hψ the spectral representation (1.319) renders directly the amplitude (12.4). All subtleties described above are due to the finite number of time slices in the path integral. As explained at the end of Section 2.1, the explicit sum over all paths is an essential ingredient of Feynman’s global approach to the phenomena of 2

I.H. Duru and H. Kleinert, Phys. Lett. B 84, 30 (1979) (http://www.physik.fu-berlin.de/~kleinert/65); Fortschr. Phys. 30 , 401 (1982) (ibid.http/83). See also the historical remarks in the preface.

838


quantum fluctuations. Within this approach, the finite time slicing is essential for being able to perform this sum in any nontrivial system. We now present a general solution to the stability problem of time-sliced quantum-statistical path integrals.

12.2

Stable Path Integral with Singular Potentials

Consider the fixed-energy amplitude (9.1) which is the local matrix element ˆ ai (xb |xa )E = hxb |R|x

(12.6)

of the resolvent operator (1.315): ˆ= R

i¯ h . ˆ + iη E−H

(12.7)

Recall that the iη-prescription ensures the causality of the Fourier transform of (12.6), making it vanish for tb < ta [see the discussion after Eq. (1.323)]. The fixed-energy amplitude has poles of the form (xb |xa )E =

X n

i¯ h ψn (xb )ψn∗ (xa ) + . . . E − En + iη

at the bound-state energies, and a cut along the continuum part of the energy spectrum. The energy integral over the discontinuity across the singularities yields the completeness relation (1.326). The new path integral formula is based on the following observation. If the system possesses a Feynman path integral for the time evolution amplitude, it does so also for the fixed-energy amplitude. This is seen after rewriting the latter as an integral (xb |xa )E =

Z

∞

ta

Ê (tb − ta )|xa i dtb hxb |U

(12.8)

involving the modified time evolution operator ˆ UÊ (t) ≡ e−it(H−E)/¯h ,

(12.9)

that is associated with the modified Hamiltonian Ê ≡ H ˆ − E. H

(12.10)

Obviously, as long as the matrix elements of the ordinary time evolution operator ˆ Uˆ (t) = e−itH/¯h can be represented by a time-sliced Feynman path integral, the ˆ same is true for the matrix elements of the modified operator UÊ (t) = e−itHE /¯h . Its H. Kleinert, PATH INTEGRALS

839

12.2 Stable Path Integral with Singular Potentials

explicit form is obtained, as in Section 2.1, by slicing the t-variable into N +1 pieces, ˆ E /¯ factorizing exp(−itH h) into the product of N + 1 factors, ˆ

ˆ

ˆ

e−itHE /¯h = e−iǫHE /¯h · · · e−iǫHE /¯h ,

(12.11)

and inserting a sequence of N completeness relations N Z Y

dD xn |xn ihxn | = 1

(12.12)

n=1

(omitting the continuum part of the spectrum). In this way, we have arrived at the path integral for the time-sliced amplitude with tb − ta = ǫ(N + 1) N Z Y

ˆ N (tb − ta )|xa i = hxb |U E

n=1

D

d xn

" NY +1 Z

d D pn i N exp A , D (2π¯ h) h ¯ E #

n=1

(12.13)

where AN E is the sliced action N +1 X

AN E =

{pn (xn − xn−1 ) − ǫ[H(pn , xn ) − E]} .

(12.14)

n=1

In the limit of large N at fixed tb − ta = ǫ(N + 1), this defines the path integral Ê (t)|xa i = hxb |U

Z

D D x(t′ )

D D p(t′ ) i exp (2π¯ h)D h ¯

Z

Z

0

t

˙ ′ ) − HE (p(t′ ), x(t′ ))] . dt′ [px(t

(12.15) It is easy to derive a finite-N approximation also for the fixed-energy amplitude (xb |xa )E of Eq. (12.8). The additional integral over tb > ta can be approximated at the level of a finite N by an integral over the slice thickness ǫ: Z

∞

dtb = (N + 1)

ta

Z

0

∞

dǫ.

(12.16)

The resulting finite-N approximation to the fixed-energy amplitude, Z

∞

0

dǫhxb |UÊN (ǫ(N + 1))|xa i =

Z

∞

ÊN (tb − ta )|xa i, dtb hxb |U (12.17) converges against the correct limit (xb |xa )E . As an example, take the free-particle case where (xb |xa )N E ≡ (N + 1)

(xb |xa )N E = (N + 1) "

Z

0

∞

ta

1 dǫ q D 2πi(N + 1)ǫ¯ h/M

#

M × exp i (xb − xa )2 + iE(N + 1)ǫ . 2(N + 1)ǫ

(12.18)

After a trivial change of the integration variable, this is the same integral as in (1.339) whose result was given in (1.344) and (1.351), depending on the sign of

840


the energy E. The N-dependence happens to disappear completely as observed in Section 2.2.4. In the general case of an arbitrary smooth potential, the convergence is still assured by the dominance of the kinetic term in the integral measure. The time-sliced path integral formula for the fixed-energy amplitude (xb |xa )E given by (12.17), (12.13), (12.14) has apparently the same range of validity as the original Feynman path integral for the time evolution amplitude (xb tb |xa ta ). Thus, so far nothing has been gained. However, the new formula has an important advantage over Feynman’s. Due to the additional time integration it possesses a new functional degree of freedom. This can be exploited to find a path-integral formula without collapse at imaginary times. The starting point is the observation that the ˆ in Eq. (12.7) may be rewritten in the following three ways: resolvent operator R ˆ= R

i¯ h fˆ , ˆ + iη) l fˆl (E − H

(12.19)

i¯ h , ˆ + iη)fˆr (E − H

(12.20)

i¯ h fˆl , ˆ ˆ fl (E − H + iη)fˆr

(12.21)

or ˆ = fˆr R or, more generally, ˆ = fˆr R

ˆ and x where fˆl , fˆr are arbitrary operators which may depend on p ˆ. They are called regulating functions. In the subsequent discussion, we shall avoid operator-ordering ˆ , although the general case can subtleties by assuming fˆl , fˆr to depend only on x also be treated along similar lines. Moreover, in the specific application to follow in Chapters 13 and 14, the operators fˆl , fˆr to be assumed consist of two different powers of one and the same operator fˆ, i.e., fˆl = fˆ1−λ ,

fˆr = fˆλ ,

(12.22)

whose product is ˆ fˆl fˆr = f.

(12.23)

Taking the local matrix elements of (12.21) renders the alternative representations for the fixed-energy amplitude ˆ a i = (xb |xa )E = hxb |R|x

Z

∞

sa

dsb hxb |UÊ (sb − sa )|xa i,

(12.24)

where UÊ (s) is the generalization of the modified time evolution operator (12.9), to be called the pseudotime evolution operator , ˆ UÊ (s) ≡ fr (x)e−isfl (x)(H−E)fr (x) fl (x).


841

12.2 Stable Path Integral with Singular Potentials

The operator in the exponent, ˆ − E)fr (x), ˆ E ≡ fl (x)(H H

(12.26)

may be considered as an auxiliary Hamiltonian which drives the state vectors |xi ˆ of the system along a pseudotime s-axis, with the operator e−isHE (p,x)/¯h . Note that ˆ E is in general not Hermitian, in which case UÊ (s) is not unitary. H As usual, we convert the expression (12.24) into a path integral by slicing the ˆ E /¯ pseudotime interval (0, s) into N +1 pieces, factorizing exp(−isH h) into a product of N + 1 factors, and inserting a sequence of N completeness relations. The result is the approximate integral representation for the fixed-energy amplitude, Z

(xb |xa )E ≈ (N + 1)

∞

0

dǫs hxb |UÊN (ǫs (N + 1)) |xa i,

(12.27)

with the path integral for the pseudotime-sliced amplitude hxb |UÊN

(ǫs (N + 1)) |xa i = fr (xb )fl (xa )

N Z Y

D

d xn

" NY +1 Z n=1

n=1

d D pn N eiAE /¯h , (12.28) D (2π¯ h) #

whose time-sliced action reads AN E =

N +1 X

{pn (xn − xn−1 ) − ǫs fl (xn )[H(pn , xn ) − E]fr (xn )} .

(12.29)

n=1

These equations constitute the desired generalization of the formulas (12.13)– (12.17). In the limit of large N, we can write the fixed-energy amplitude as an integral (xb |xa )E =

Z

∞

0

dShxb |UÊ (S)|xa i

(12.30)

over the amplitude (

Dp(s) i hxb |UÊ (S)|xa i = fr (xb )fl (xa ) Dx(s) exp 2π¯ h h ¯ Z

Z

Z

0

S

)

ds[px′ −HE (p, x)] . (12.31)

The prime on x(s) denotes the derivative with respect to the pseudotime s. For a standard Hamiltonian of the form H = T (p) + V (x),

(12.32)

with the kinetic energy T (p) =

p2 , 2M

(12.33)

842


the momenta pn in (12.28) can be integrated out and we obtain the configuration space path integral fr (xb )fl (xa ) hxb |UÊN (ǫs (N + 1)) |xa i = q D 2πiǫs fl (xb )fr (xa )¯ h/M ×

N Y

n=1

with the sliced action AN E

=

N +1 X n=1

  

D

d xn

Z

q

2πiǫs f (xn )¯ h/M

(12.34) 

 iAN h E /¯ , De

(

)

M (xn − xn−1 )2 − ǫs fl (xn )[V (xn ) − E]fr (xn−1 ) . (12.35) 2ǫs fl (xn )fr (xn−1 )

In the limit of large N, this may be written as a path integral (

i hxb |UÊ (S)|xa i = fr (xb )fl (xa ) Dx(s)exp h ¯ Z

Z

0

S

"

M ′2 ds x − fl (V −E)fr 2fl fr

#)

, (12.36)

with the slicing specification (12.35). The path integral formula for the fixed-energy amplitude based on Eqs. (12.30) and (12.36) is independent of the particular choice of the functions fl (x), fr (x), just like the most general operator expression for the resolvent (12.21). Feynman’s original time-sliced formula is, of course, recovered with the special choice fl (x) ≡ fr (x) ≡ 1. When comparing (12.25) with (12.9), we see that for each infinitesimal pseudotime slice, the thickness of the true time slices has the space-dependent value dt = dsfl (xn )fr (xn−1 ).

(12.37)

The freedom in choosing f (x) amounts to an invariance under path-dependent time reparametrizations of the fixed-energy amplitude (12.30). Note that the invariance is exact in the general operator formula (12.21) for the resolvent and in the continuum path integral formula based on (12.30) and (12.36). However, the finite pseudotime slicing in (12.34), (12.35) used to define the path integral, destroys this invariance. At a finite value of N, different choices of f (x) produce different approximations to ˆ the matrix element of the operator UÊ (s) = fr (ˆ x)e−isHE (ˆp,ˆx)/¯h fl (ˆ x). Their quality can vary greatly. In fact, if the potential is singular and the regulating functions fr (x), fl (x) are not suitably chosen, the Euclidean pseudotime-sliced expression may not exist at all. This is what happens in the Coulomb system if the functions fl (x) and fr (x) are both chosen to be unity as in Feynman’s path integral formula. The new reparametrization freedom gained by the functions fl (x), fr (x) is therefore not just a luxury. It is essential for stabilizing the Euclidean time-sliced orbital fluctuations in singular potentials. In the case of the Coulomb system, any choice of the regulating functions fl (x), fr (x) with f (x) = r leads to a regular auxiliary Hamiltonian HE , and the H. Kleinert, PATH INTEGRALS

843

12.3 Time-Dependent Regularization

path integral expressions (12.27)–(12.36) are all well defined. This was the important discovery of Duru and Kleinert in 1979, to be described in detail in Chapter 13, which has made a large class of previously non-existing Feynman path integrals solvable. By a similar Duru-Kleinert transformation with fl (x), fr (x) with q f (x) = fl (x)fr (x) = r 2 , the earlier difficulties with the centrifugal barrier are resolved, as will be seen in Chapter 14.

12.3

Time-Dependent Regularization

Before treating specific cases, let us note that there exists a further generalization of the above path integral formula which is useful in systems with a time-dependent Hamiltonian H(p, x, t). There we introduce an auxiliary Hamiltonian ˆ = fl (x, t)[H(ˆ ˆ r (x, t), H p, x, t) − E]f

(12.38)

where Eˆ is the differential operator for the energy which is canonically conjugate to the time t: Eˆ ≡ i¯ h∂t .

(12.39)

The auxiliary Hamiltonian acts on an extended Hilbert space, in which the states are localized in space and time. These states will be denoted by |x, t}. They satisfy the orthogonality and completeness relations {x t|x′ t′ } = δ (D) (x − x′ )δ(t − t′ ),

(12.40)

and Z

dD x

Z

dt|x t}{x t| = 1,

(12.41)

respectively. By construction, the Hamiltonian H does not depend explicitly on the pseudotime s. The pseudotime evolution operator is therefore obtained by a simple exponentiation, as in (12.25), ˆ ˆ ˆ U(s) ≡ fr (x, t)e−isfl(x,t)(H−E)fr (x,t) fl (x, t).

(12.42)

The derivation of the path integral is then completely analogous to the timeindependent case. The operator (12.42) is sliced into N + 1 pieces, and N completeness relations (12.41) are inserted to obtain the path integral {xb tb |Uˆ N (s)|xa ta } = fr (xb , tb )fl (xa , ta ) ×

N Z Y

n=1

D

d xn dtn

" NY +1 Z n=1

dD pn dEn iAN /¯h e , (2π¯ h)D 2π¯ h #

(12.43)

with the pseudotime-sliced action A

N

=

N +1 X

{pn (xn − xn−1 ) − En (tn − tn−1 )

n=1

− fl (xn , tn ) [H(pn , xn , tn ) − En ] fr (xn−1 , tn−1 )},

(12.44)

844


where xb = xN +1 , tb = tN +1 ; xa = x0 , ta = t0 . This describes orbital fluctuations in the phase space of spacetime which contains fluctuating worldlines x(s), t(s) and their canonically conjugate spacetime p(s), E(s). In the limit N → ∞ we write this as ˆ {xb tb |U(S)|x a ta } = fr (pb , xb , tb )fl (pa , xa , ta ) Z Z D D p(s) DE(s) iA/¯h D D x(s)Dt(s) , × e (2π¯ h)D 2π¯ h

(12.45)

with the continuous action A[p, x, E, t] =

Z

S

0

ds{p(s)x′ (s) − E(s)t′ (s)

−fl (p(s), x(s), t(s)) [H(p(s), x(s), t(s)) − E(s)] fr (p(s), x(s), t(s))}. (12.46) In the pseudotime-sliced formula (12.43), we can integrate out all intermediate energy variables En and obtain ˆ {xb tb |U(S)|x a ta } =

N Z Y

D

d xn

n=1

× δ tb − ta − ǫs

" NY +1 Z

n=1 N +1 X

d D pn (2π¯ h)D

#

(12.47) !

Ñ /¯ h

fl (pn , xn , tn )fr (pn−1 , xn−1 , tn−1 ) eiA

n=1

with the action AÑ =

N +1 X

[pn (xn − xn−1 ) − ǫs fl (pn xn , tn )H(pn , xn , tn )fr (pn−1 , xn−1 , tn−1 )].

n=1

(12.48) This looks just like an ordinary time-sliced action with a time-dependent Hamiltonian. The constant width of the time slices ǫ = (tb − ta )/(N + 1), however, has now become variable and depends on phase space and time: ǫ → ǫs fl (pn xn , tn )fr (pn−1 , xn−1 , tn−1 ).

(12.49)

The δ-function in (12.47) ensures the correct relation between the pseudotime s and the physical time t. In the continuum limit we may write (12.47) as ˆ {xb tb |U(S)|x a ta } =

Z

D D x(s)

Z

D D p(s) δ(tb − ta − (2π¯ h)D

Z

0

S

˜

ds f (x, t))eiA/¯h , (12.50)

with the pseudotime action ˜ x, t] = A[p,

Z

0

S

ds [px′ − fl (x, t)H(p, x, t)fr (x, t)] ,

(12.51)

which is a functional of the s-dependent paths x(s), p(s), t(s). Note that in the continuum formula, the splitting of the regulating function f (x, t) into fl (x, t) and H. Kleinert, PATH INTEGRALS

845

12.4 Relation to Schr¨ odinger Theory. Wave Functions

fr (x, t) according to the parameter λ in Eq. (12.22) cannot be expressed properly since fl , H, and fr are commuting c-number functions. We have written them in a way indicating their order in the time-sliced expressions (12.47), (12.48). The integral over S yields the original time evolution amplitude (xb ta |xa ta ) =

Z

∞

0

ˆ dS{xb tb |U(S)|x a ta } =

(

) i¯ h xb tb x t . ˆ −E ˆ a a H

(12.52)

Indeed, by Fourier decomposing the scalar products {xb tb |xa ta }, {xb tb |xa ta } =

dD p Z dE ip(xb −xa )/¯h−iE(tb −ta )/¯h e , (2π¯ h)D 2π¯ h

Z

(12.53)

we see that the right-hand side satisfies the same Schrödinger equation as the lefthand side: [H(−i¯ h∂x , x, t) − i¯ h∂t ] (x t|xa ta ) = −i¯ hδ (D) (x − xa )δ(t − ta )

(12.54)

[recall (1.304) and (12.5)]. If the δ-function in (12.47) is written as a Fourier integral, we obtain a kind of spectral decomposition of the amplitude (12.52), (xb ta |xa ta ) =

Z

∞

−∞

−iE(tb −ta )/¯ h

dEe

Z

0

∞

dS{xbtb |UÊ (S)|xa ta },

(12.55)

with the pseudotime evolution amplitude: ˆ UÊ (s) ≡ fr (ˆ p, x, t)e−isfl(ˆp,x,t)(H−E)fr (ˆp,x,t) fl (ˆ p, x, t).

12.4

(12.56)

Relation to Schr¨ odinger Theory. Wave Functions

For completeness, consider also the ordinary Schrödinger quantum mechanics deˆ This operator is the generator of translascribed by the pseudo-Hamiltonian H. tions of the system along the pseudotime axis s. Let φ(x, t, s) be a solution of the pseudotime Schrödinger equation ˆ t)φ(x, t, s) = i¯ H(ˆ p, x, E, h∂s φ(x, t, s),

(12.57)

written more explicitly as fl (x, t) [H(ˆ p, x, t) − i¯ h∂t ] fr (x, t)φ(x, t, s) = i¯ h∂s φ(x, t, s).

(12.58)

Since the left-hand side is independent of s, the s-dependence of φ(x, t, s) can be factored out: φ(x, t, s) = φE (x, t)e−iEs/¯h .

(12.59)

If H is independent of the time t, it is always possible to stabilize the path integral by a time-independent reparametrization function f (x). Then we remove an oscillating factor e−iEt/¯h from φE (x, t) and factorize φE (x, t) = φE,E (x)e−iEt/¯h .

(12.60)

846


This leaves us with the time- and pseudotime-independent equation H(ˆ p, x, E)φE,E (x) = fl (x) [H(ˆ p, x) − E] fr (x)φE,E (x) = EφE,E (x).

(12.61)

For each value of E, there will be a different spectrum of eigenvalues En . This is indicated by writing the eigenvalues En as En (E) and the associated eigenstates φEn ,E(x) as φEn (E) . Suppose that we possess a complete set of such eigenstates at a fixed energy E labeled by a quantum number n (which is here assumed to take discrete values although it may include continuous values, as usual). We can then write down a spectral representation for the local matrix elements of the pseudotime evolution amplitude (12.25): X hxb |UÊ (S)|xa i = fr (xb )fl (xa ) φEn (E) (xb )φ∗ (xa )e−iSEn (E)/¯h . (12.62) En (E)

n

From this we find the expansion for the fixed-energy amplitude (12.24): X i¯ h . φEn (E) (xb )φ∗En (E) (xa ) (xb |xa )E = fr (xb )fl (xa ) En (E) n

The time evolution amplitude is given by the Fourier transform Z ∞ i¯ h dE iE(tb −ta )/¯h X (xb tb |xa ta ) = fr (xb )fl (xa ) e φEn (E) (xb )φ∗En (E) (xa ) . (12.63) h En (E) −∞ 2π¯ n This is to be compared with the usual spectral representation of this amplitude for ˆ the time-independent Hamiltonian H (xb tb |xa ta ) =

X


(12.64)

n

where ψn (x) are the solutions of the ordinary time-independent Schrödinger equation: H(ˆ p, x)ψn (x) = En ψn (x).

(12.65)

The relation between the two representations (12.63) and (12.64) is found by observing that for the energy E coinciding with the energy En , the eigenvalue En (E) vanishes, i.e., i¯ h/En (E) has poles at E = En of the form i¯ h 1 i¯ h ≈ ′ . (12.66) En (E) En (En ) E − En + iη These contribute to the energy integral in (12.63) with a sum (xb tb |xa ta ) ∼ fr (xb )fl (xa )

X

φEn (En ) (xb )φ∗En (En ) (xa )e−iEn (tb −ta )/¯h .

(12.67)

n

A comparison with (12.64) shows the relation between the bound-state wave functions of the ordinary and the pseudotime Schrödinger equation. In general, the function i¯ h/En (E) also has cuts whose discontinuities contain the continuum wave functions of the Schrödinger equation (12.65). These observations will become more transparent in Section 13.8 when treating in detail the bound and continuum wave functions of the Coulomb system. H. Kleinert, PATH INTEGRALS


847

Notes and References The general path integral formula with time reparametrization was introduced by H. Kleinert, Phys. Lett. A 120, 361 (1987) (http://www.physik.fu-berlin.de/~kleinert/163). The stability aspects are discussed in H. Kleinert, Phys. Lett. B 224, 313 (1989) (ibid.http/195).

H. Kleinert, PATH INTEGRALS June 16, 2007 ( /home/kleinert/kleinert/books/pathis/pthic13.tex) Et modo quae fuerat semita, facta via est. What was only a path is now made a high road. Martial, Epig., Book 7, 60

13 Path Integral of Coulomb System One of the most important successes of Schrödinger quantum mechanics is the explanation of the energy levels and transition amplitudes of the hydrogen atom. Within the path integral formulation of quantum mechanics, this fundamental system has resisted for many years all attempts at a solution. An essential advance was made in 1979 when Duru and Kleinert [1] recognized the need to work with a generalized pseudotime-sliced path integral of the type described in Chapter 12. After an appropriate coordinates transformation the path integral became harmonic and solvable. A generalization of this two-step transformation has meanwhile led to the solution of many other path integrals to be presented in Chapter 14. The final solution of the problem turned out to be quite subtle due to the nonholonomic nature of the subsequent coordinate transformation which required the development of a correct path integral in spaces with curvature and torsion [2], as done in Chapter 10. Only this made it possible to avoid unwanted fluctuation corrections in the Duru-Kleinert transformation of the Coulomb system, a problem in all earlier attempts. The first consistent solution was presented in the first edition of this book in 1990.

13.1

Pseudotime Evolution Amplitude

Consider the path integral for the time evolution amplitude of an electron-proton system with a Coulomb interaction. If me and mp denote the masses of the two particles whose reduced mass is M = me mp /(me + mp ), and if e is the electron charge, the system is governed by the Hamiltonian H=

p2 e2 − . 2M r

(13.1)

The formal continuum path integral for the time evolution amplitude reads (xb tb |xa ta ) =

Z

i D x(t) exp h ¯ 3

848

Z

ta

tb

dt(px˙ − H) .

(13.2)

849

13.1 Pseudotime Evolution Amplitude

As observed in the last chapter, its Euclidean version cannot be time-sliced into a finite number of integrals since the paths would collapse. The paths would stretch out into a straight line with x˙ ≈ 0 and slide down into the 1/r-abyss. A path integral whose Euclidean version is stable can be written down using the pseudotime evolution amplitude (12.28). A convenient family of regulating functions is fl (x) = f (x)1−λ ,

fr (x) = f (x)λ ,

(13.3)

whose product satisfies fl (x)fr (x) = f (x) = r. Since the path integral represents the general resolvent operator (12.21), all results must be independent of the splitting parameter λ after going to the continuum limit. This independence is useful in checking the calculations. Thus we consider the fixed-energy amplitude (xb |xa )E =

∞

Z

0

dShxb |UÊ (S)|xa i,

(13.4)

with the pseudotime-evolution amplitude Z

hxb |UÊ (S)|xa i = rbλ ra1−λ D D x(s)

iZS D D p(s) exp ds[px′ − r 1−λ (H − E)r λ ] , (2π¯ h)D h ¯ 0 (13.5) (

Z

)

where the prime denotes the derivative with respect to the pseudotime argument s. For the sake of generality, we have allowed for a general dimension D of orbital motion. After time slicing and with the notation ∆xn ≡ xn − xn−1 , ǫs ≡ S/(N + 1), the amplitude (13.5) reads hxb |UÊ (S)|xa i ≈

rbλ ra1−λ

NY +1 Z ∞

−∞

n=2

D

d ∆xn

" NY +1 Z n=1

∞

−∞

d D pn N eiAE /¯h , D (2π¯ h) #

(13.6)

where the action is AN E [p, x]

=

N +1 X n=1

"

pn ∆xn −

pn 2 − E + ǫs e2 . 2M !

λ ǫs rn1−λ rn−1

#

(13.7)

The term ǫs e2 carries initially a factor (rn−1 /rn )λ which is dropped, since it is equal to unity in the continuum limit. When integrating out the momentum variables, λ N + 1 factors 1/(rn1−λ rn−1 )D/2 appear. After rearranging these, the configuration space path integral becomes rbλ ra1−λ

hxb |UÊ (S)|xa i ≈ q D 2π iǫs h ¯ rb1−λ raλ /M

NY +1 n=2

with the pseudotime-sliced action ′ AN E [x, x ]

2

= (N + 1)ǫs e +

N +1 X n=1

"

 Z   q

D

d ∆xn

2πiǫs h ¯ rn−1 /M



′ h  iAN E [x,x ]/¯ , (13.8) De

M (∆xn )2 + ǫs rn E . λ 2 ǫs rn1−λ rn−1 #

(13.9)

850

13 Path Integral of Coulomb System

λ In the last term, we have replaced rn1−λ rn−1 by rn without changing the continuum limit. The limiting action can formally be written as ′

2

AE [x, x ] = e S +

Z

0

S

M ′2 ds x + Er . 2r

(13.10)

We now solve the Coulomb path integral first in two dimensions, assuming that the movement of the electron is restricted to a plane while the electric field extends into the third dimension. Afterwards we proceed to the physical three-dimensional system. The case of an arbitrary dimension D will be solved in Chapter 14.

13.2

Solution for the Two-Dimensional Coulomb System

First we observe that the kinetic pseudoenergy has a scale dimension [rp2 ] ∼ [r −1 ] which is precisely opposite to that of the potential term [r +1 ]. The dimensional situation is similar to that of the harmonic oscillator, where the dimensions are [p2 ] = [r −2 ] and [r +2 ], respectively. It is possible make the correspondence perfect by describing the Coulomb system in terms of “square root coordinates”, i.e., by transforming r → u2 . In two dimensions, the appropriate square root is given by the Levi-Civita transformation x1 = (u1 )2 − (u2 )2 , x2 = 2u1 u2.

(13.11)

If we imagine the vectors x and u to move in the complex planes parametrized by x = x1 + ix2 and u = u1 + iu2 , the transformed variable u corresponds to the complex square root: √ (13.12) u = x. Let us also introduce the matrix A(u) =

u1 −u2 u2 u1

!

,

(13.13)

and write (13.11) as a matrix equation: x = A(u)u.

(13.14)

The Levi-Civita transformation is an integrable coordinate transformation which carries the flat xi -space into a flat uµ -space. We mention this fact since in the later treatment of the three-dimensional hydrogen atom, the transition to the “square root coordinates” will require a nonintegrable (nonholonomic) coordinate transformation defined only differentially. As explained in Chapter 10, such mappings change, in general, a flat Euclidean space into a space with curvature and torsion. The generation of torsion is precisely the reason why the three-dimensional system remained unsolved until 1990. In two dimensions, this phenomenon happens to be absent. H. Kleinert, PATH INTEGRALS

13.2 Solution for the Two-Dimensional Coulomb System

851

If we write the transformation (13.11) in terms of a basis dyad eiµ (u) as dxi = ei µ (u) duµ, this is given by ∂xi e µ (u) = µ (u) = 2Ai µ (u), ∂u

(13.15)

1 1 ei µ (u) = (A−1 )T i µ (u) = 2 Ai µ (u). 2 2u

(13.16)

i

with the reciprocal dyad

The associated affine connection Γµν λ = ei λ ∂µ ei ν =

1 [(∂µ A)T A]νλ u2

(13.17)

has the matrix elements (Γµ )ν λ = Γµν λ : (Γ1 )µ

ν

1 = u2 1 u2

(Γ2 )µ ν =

u1 −u2 u2 u1 2

1

u u −u1 u2

! !

ν

= µ

1 A(u)µ ν , 2u2

ν

.

(13.18)

µ

The affine connection satisfies the important identity Γµ µν ≡ 0,

(13.19)

which follows from the defining relation Γµ µν ≡ g µν ei λ ∂µ ei ν ,

(13.20)

by inserting the obvious special property of ei µ ∂µ ei µ = ∂u2 xi (u) = 0,

(13.21)

using the diagonality of g µν = δ µν /4r. The identity (13.19) will be shown in Section 13.6 to be the essential geometric reason for the absence of the time slicing corrections. The torsion and the Riemann-Cartan curvature tensor vanish identically, the former because of the specific form of the matrix elements (13.18), the latter due to the linearity of the basis dyads ei µ (u) in u which guarantees trivially the integrability conditions, i.e., ei λ (∂µ ei ν − ∂ν ei µ ) ≡ 0, ei κ (∂µ ∂ν − ∂ν ∂µ ) ei λ ≡ 0, and thus Sµν λ ≡ 0, Rµνλ κ ≡ 0.

(13.22) (13.23)

852


In the continuum limit, the Levi-Civita transformation (13.10) into that of a harmonic oscillator. With 2

x′2 = 4u2 u′ = 4r u′

converts the action

2

(13.24)

we find A[x] = e2 S +

Z

S

0

ds

4M ′2 u + Eu2 . 2

(13.25)

Apart from the trivial term e2 S, this is the action of a harmonic oscillator Aos [u] =

S

Z

0

µ ds (u′2 − ω 2 u2 ), 2

(13.26)

which oscillates in the pseudotime s with an effective mass µ = 4M,

(13.27)

and a pseudofrequency ω=

q

−E/2M.

(13.28)

Note that ω has the dimension 1/s corresponding to [ω] = [r/t] (in contrast to a usual frequency whose dimension is [1/t]). The path integral is well defined only as long as the energy E of the Coulomb system is negative, i.e., in the bound-state regime. The amplitude in the continuum regime with positive E will be obtained by analytic continuation. In the regularized form, the pseudotime-sliced amplitude is calculated as follows. Choosing a splitting parameter λ = 1/2 and ignoring for the moment all complications due to the finite time slicing, we deduce from (13.14) that dx = 2A(u)du,

(13.29)

d2 xn = 4un 2 d2 un .

(13.30)

and hence

Since the x- and the u-space are both Euclidean, the integrals over ∆xn in (13.8) can be rewritten as integrals over xn , and transformed directly to un variables. The result is 1 2 hxb |UÊ (S)|xa i = eie S/¯h [(ub S|ua 0) + (−ub S|ua 0)], (13.31) 4 where (ub S|ua 0) denotes the time-sliced oscillator amplitude N Y 1 (ub S|ua 0) ≈ 2πi¯ hǫs /µ n=1

"Z

d2 u n 2πi¯ hǫs /µ

#

(13.32)

N i X µ 1 × exp ∆un 2 − ǫs ω 2 un 2 h ¯ n=1 2 ǫs

(

)

.


853

13.2 Solution for the Two-Dimensional Coulomb System

Figure 13.1 Illustration of associated final points in u-space, to be summed in the oscillator amplitude. In x-space, the paths run from xa to xb once directly and once after √ crossing the cut into the second sheet of the complex function u = x.

The evaluation of the Gaussian integrals yields, in the continuum limit, µω i µω exp [(ub 2 + ua 2 ) cos ωS − 2ub ua ] . (13.33) (ub S|ua 0) = 2πi¯ h sin ωS h ¯ sin ωS

The symmetrization in ub in Eq. (13.31) is necessary since for each path from xa to xb , there are two paths in the square root space, one from ua to ub and one from ua to −ub (see Fig. 13.1). The fixed-energy amplitude is obtained by the integral (13.4) over the pseudotime evolution amplitude (12.18): (xb |xa )E =

Z

0

∞

dS eie

2 S/¯ h

1 [(ub S|ua 0) + (−ub S|ua 0)]. 4

(13.34)

By inserting (13.33), this becomes (xb |xa )E

1Z ∞ = dS exp(ie2 S/¯ h)F 2 (S) 2 0h i h i × exp −πF 2 (S)(u2b + u2a ) cos ωS cosh 2πF 2 (S)ub ua , (13.35)


F (S) =

q

µω/2πi¯ h sin ωS,

(13.36)

for the one-dimensional fluctuation factor [recall (2.164)]. The coordinates ub and ua on the right-hand side are related to xb xa on the left-hand side by u2a,b = ra,b ,

ub ua =

q

(rb ra + xb xa )/2.

(13.37)

When performing the integral over S, we have to pass around the singularities in F (S) in accordance with the iη-prescription, replacing ω → ω − iη. Equivalently, we

854


can rotate the contour of S-integration to make it run along the negative imaginary semi-axis, S = −iσ, σ ∈ (0, ∞). This amounts to going over to the Euclidean amplitude of the harmonic oscillator in which the singularities are completely avoided. The amplitude is rewritten in a more compact form by introducing the variables

Then

̺ ≡ e−2iωS = e−2ωσ , 2Mω q µω κ ≡ = = −2ME/¯ h2 , 2¯ h h ¯ s e2 e4 M ν ≡ = . 2ω¯ h −2¯ h2 E

(13.38) (13.39) (13.40)

√ 2 ̺ , πF (S) = κ 1−̺ 2

eie

2 s/¯ h

F 2 (S) =

(13.41)

2 ̺1/2−ν κ , π 1−̺

(13.42)

and the fixed-energy amplitude of the two-dimensional Coulomb system takes the form " # √ Z 2 ̺q M 1 ̺−1/2−ν (xb |xa )E = −i d̺ cos 2κ (rb ra + xb xa )/2 π¯ h 0 1−̺ 1−̺ " # 1+̺ × exp −κ (rb + ra ) . (13.43) 1−̺ This can be used to find the energy spectrum and the wave functions as shown in Section 13.8. Note that the integral converges only for ν < 1/2. It is possible to write down another integral representation converging for all ν 6= 1/2, 3/2, 5/2, . . . . To do this we change the variable of integrations to ζ≡

1+̺ , 1−̺

(13.44)

so that d̺ 1 = dζ, 2 (1 − ̺) 2

̺=

ζ −1 . ζ +1

(13.45)

This leads to M 1Z ∞ dζ(ζ − 1)−ν−1/2 (ζ + 1)ν−1/2 π¯ h 2 1

(xb |xa )E = −i

q

q

× cos 2κ ζ 2 − 1 (rb ra + xb xa )/2 e−κζ(rb +ra ) .

(13.46)


13.3 Absence of Time Slicing Corrections for D = 2

855

The integrand has a cut in the complex ζ-plane extending from z = −1 to −∞ and from ζ = 1 to ∞. The integral runs along the right-hand cut. The integral is transformed into an integral along a contour C which encircles the right-hand cut in the clockwise sense. Since the cut is of the type (ζ − 1)−ν−1/2 , we may replace Z

1

∞

1 πeiπ(ν+1/2) sin[π(ν + 1/2)] 2πi

dζ(ζ − 1)−ν−1/2 . . . →

Z

C

dζ(ζ − 1)−ν−1/2 . . . ,

(13.47)

and the fixed-energy amplitude reads (xb |xa )E = −i

M 1 πeiπ(ν+1/2) Z dζ (ζ − 1)−ν−1/2 (ζ + 1)ν−1/2 π¯ h 2 sin[π(ν + 1/2)] C 2πi

q

× cos 2κ

13.3

ζ2

q

− 1 (rb ra + xb xa )/2 e−κζ(rb +ra ) .

(13.48)

Absence of Time Slicing Corrections for D = 2

We now convince ourselves that the finite thickness of the pseudotime slices in the intermediate formulas does not change the time evolution amplitude obtained in the last section [4]. The reader who is unaware of the historic difficulties which had to be overcome may not be interested in the upcoming technical discussion. He may skip this section and be satisfied with a brief argument given in Section 13.6. The potential term in the action (13.9) can be ignored since it is of order ǫs and the time slicing can only produce higher than linear correction terms in ǫs which do not contribute in the continuum limit ǫs → 0. The crucial point where corrections might enter is in the transformation of the measure and the pseudotime-sliced kinetic terms in (13.8), (13.9). In vector notation, the coordinate transformation reads, at every time slice n, xn = A(un )un .

(13.49)

Among the equivalent possibilities offered in Section 11.2 to transform a time-sliced path integral we choose the Taylor expansion (11.56) to map ∆x into ∆u. After inserting (13.15) into (11.56) we find ∆xi = 2Ai µ (u)∆uµ − ∂ν Ai µ (u)∆uµ ∆uν .

(13.50)

Since the mapping x(u) is quadratic in u, there are no higher-order expansion terms. Note that due to the absence of curvature and torsion in the u-space, the coordinate transformation is holonomic and ∆x can also be calculated directly from x(u) − x(u − ∆u). Indeed, using the linearity of A(u) in u, we find from (13.49) ∆xn = A(un )un − A(un−1 )un−1 = 2A(un − 21 ∆un )∆un = 2A(¯ un )∆un ,

(13.51)

¯ n is average across the slice. The Taylor expansion of A(un − 21 ∆un ) has only two terms where u and leads to (13.50). Using (13.51) we can write (∆xn )2 ¯n u

= 4¯ u2n (∆un )2 ,

(13.52)

≡ (un + un−1 )/2.

(13.53)

856


The kinetic term of the short-time action in the nth time slice of Eq. (13.9) therefore becomes Aǫ =

M (∆xn )2 M 4¯ u2n (∆un )2 . = λ 2ǫs rn1−λ rn−1 2ǫs (u2n )1−λ (u2n−1 )λ

(13.54)

This is expanded around the postpoint and yields ¯n u

=

un−1 2 ¯n u (u2n )1−λ (u2n−1 )λ

= =

1 un − ∆un , 2 un − ∆un , un ∆un 1 ∆un 2 1 + (2λ − 1) + − λ u2n 4 u2n 2 un ∆un . + 2λ2 u2n

(13.55) (13.56)

(13.57)

It is useful to separate the short-time action into a leading term Aǫ0 (∆un ) = 4M

(∆un )2 , 2ǫs

(13.58)

plus correction terms ∆Aǫ = 4M "

(∆un )2 2ǫs

(13.59)

un ∆un × (2λ − 1) + u2n

1 ∆un 2 un ∆un 2 −λ + 2λ 4 u2n u2n

2 #

,

which will be treated perturbatively. In order to perform the transformation of the measure of integration in (13.8), we expand ∆x accordingly: ∆xi

1 = 2Ai µ (u − ∆u)∆uµ 2 = 2Ai µ (u)∆uµ − ∂ν Ai µ (u)∆uµ ∆uν .

(13.60)

The indices n have been suppressed, for brevity. This expression has, of course, the general form (10.96), after inserting there ei µ (u) = 2Ai µ (u).

(13.61)

Since the transformation matrix Ai µ (u) in (13.15) is linear in u, the matrix ei µ (u) has no second derivatives, and the Jacobian action (11.60) and (10.145) reduce to i ǫ A ¯ J h

1 −ei µ ei {µ,ν} ∆uν − ei µ ei {κ,ν} ej κ ej µ,λ ∆uν ∆uλ 2 1 = −Γ{νµ} µ ∆uν − Γ{νκ} µ Γ{µλ} κ ∆uν ∆uλ . 2 The expansion coefficients are easily calculated using the reciprocal basis dyad =

ei κ =

1 i e κ, 2u2

(13.62)

(13.63)

as Γνµ µ

= ei µ ∂ν ei µ =

Γµν µ

= −ei ν ∂µ ei µ

Γνκ µ Γλµ κ

2uν , u2 2uν = 2 , u

= −∂λ ei κ ∂ν ei κ = −

(13.64)

2 νλ 2 (δ u − 2uν uλ ). u4 H. Kleinert, PATH INTEGRALS


857

The second equation is found directly from −∂µ ei µ = −∂µ (2u2 )−1 ei µ = ei µ 2uµ u−2 ,

(13.65)

which, in turn, follows from the obvious identity ∂µ ei µ = 0. Note that the third expression in (13.64) is automatically equal to Γ{νκ} σ Γ{λσ} κ , i.e., of the form required in (13.62), since the uµ space has no torsion and Γνκ σ = Γκν σ . After inserting Eqs. (13.64) into the right-hand side of (13.62), we find the postpoint expansion " # 2 i ǫ un ∆un ∆un 2 u∆un A =− 2 − +2 + ... . (13.66) ¯h J u2n u2n u2n The measure of integration in (13.8) contains additional factors rb , rn , ra which require a further treatment. First we rewrite it as N Z (rb /ra )2λ−1 Y d2 ∆xn 2πiǫs ¯h n=1 2πiǫs rn−1 /M

≈ =

NY 2λ N Z +1 Y 1 d2 ∆xn rn 2πiǫs ¯h n=1 2πiǫs ¯hrn /M n=1 rn−1 Z N +1 Y N d2 ∆xn 1 eiAf /¯h . (13.67) 2πiǫs ¯h n=2 2πiǫs ¯hrn /M

On the left-hand side, we have shifted the n by one Q unit making use of the fact that with R Qlabels N +1 R N ∆xn = xn − xn−1 we can certainly write n=2 d2 ∆xn = n=1 d2 ∆xn . In the first expression on the right-hand side we have further shifted the subscripts of the factors 1/rn−1 in the integral QN +1 measure from n − 1 to n and compensated for this by an overall factor n=1 (rn /rn−1 ). Together Q +1 2λ with the prefactor (rb /ra )2λ−1 , this can be expressed as a product N n=1 (rn /rn−1 ) . There is 2 only a negligible error of order ǫs at the upper end [this being the reason for writing the symbol ≈ rather than = in (13.67)]. In the last part of the equation we have introduced an effective action AN f ≡

N +1 X n=1

Aǫf

(13.68)

due to the (rn /rn−1 )2λ factors, with i ǫ r2 u2 Af = 2λ log 2 n = 2λ log 2 n . ¯h rn−1 un−1

(13.69)

The subscript f indicates that the general origin of this term lies in the rescaling factors fl (xb ), fr (xa ). We now go over from ∆xn - to ∆un -integrations using the relation i ǫ 2 2 2 d ∆x = 4u d ∆u exp A . (13.70) ¯h J The measure becomes N Z Y 1 4 4d2 ∆un i N × exp (AJ + AN ) , f 2 2 · 2πiǫs ¯h n=1 2 · 2πiǫs ¯h/M ¯h

(13.71)

ǫ where AN J is the sum over all time-sliced Jacobian action terms AJ of (13.66):

AN J ≡

N +1 X n=1

AǫJ .

(13.72)

858


The extra factors 2 in the measure denominators of (13.71) are introduced to let the un -integrations run over the entire u-space, in which case the x-space is traversed twice. The time-sliced expression (13.71) has an important feature which was absent in the continuous formulation. It receives dominant contributions not only from the region neighborhood un ∼ un−1 , in which case (∆un )2 is of order ǫs , but also from un ∼ −un−1 where (¯ un )2 is of order ǫs . This is understandable since both configurations correspond to xn being close to xn−1 and must be included. Fortunately, for symmetry reasons, they give identical contributions so that we need to discuss only the case un ∼ un−1 , the contribution from the second case being simply included by dropping the factors 2 in the measure denominators. To process the measure further, we expand the action Aǫf (13.71) around the postpoint and find " # 2 2 i ǫ un un ∆un ∆un 2 u∆un A = 2λ log = 2λ 2 − +2 + ... . (13.73) ¯h f u2n−1 u2n u2n u2n A comparison with (13.66) shows that adding (i/¯h)Aǫf and (i/¯h)AǫJ merely changes 2λ in Aǫf into 2λ − 1. Thus, altogether, the time-slicing produces the short-time action Aǫ = Aǫ0 + ∆corr Aǫ ,

(13.74)

with the leading free-particle action Aǫ0 (∆un ) = 4M

(∆un )2 , 2ǫs

(13.75)

and the total correction term i i ∆corr Aǫ ≡ (∆Aǫ + AǫJ + Aǫf ) h ¯ ¯h " 2 # i ∆un 2 un ∆un 1 ∆un 2 un ∆un 2 = 4M (2λ − 1) + −λ + 2λ ¯h 2ǫs u2n 4 u2n u2n " # 2 ∆un 2 un ∆un un ∆un + (2λ − 1) 2 − +2 + ... . u2n u2n u2n

(13.76)

We now show that the action ∆corr Aǫ is equivalent to zero by proving that the kernel associated with the short-time action 4 i ǫ K ǫ (∆u) = exp (A0 + ∆corr Aǫ ) (13.77) 2 · 2πiǫs ¯h/M ¯h is equivalent to the zeroth-order free-particle kernel K0ǫ (∆u) =

4 i ǫ exp A0 . 2 · 2πiǫs ¯h/M ¯ h

(13.78)

The equivalence is established by checking the equivalence relations (11.71) and (11.72). For the kernel (13.77), the correction (11.71) is i ǫ C1 = C ≡ exp ∆corr A − 1. (13.79) ¯h It has to be compared with the trivial factor of the kernel (13.78): C2 = 0.



859

Thus, the equivalence requires showing that hCi0

hC (p∆u)i0 The basic correlation functions due to

K0ǫ (∆u)

=

0,

=

0.

are

i¯hǫs µν δ , 4M n i¯hǫs δ µ1 ...µ2n , = 4M

h∆uµ ∆uν i0

(13.81)

≡

h∆uµ1 · · · ∆uµ2n i0

(13.82) n > 1,

(13.83)

where the contraction tensors δ µ1 ...µ2n of Eq. (8.64), determined recursively from δ µ1 ...µ2n ≡ δ µ1 µ2 δ µ3 µ4 ...µ2n + δ µ1 µ3 δ µ2 µ4 ...µ2n + . . . + δ µ1 µ2n δ µ2 µ3 ...µ2n−1 .

(13.84)

They consist of (2n − 1)!! products of pair contractions δ µi µj . More specifically, we encounter, in calculating (13.81), expectations of the type k+l i¯hǫs [D + 2(k + l − 1)]!! h(∆u)2k (u∆u)2l i0 = (2l − 1)!!(u2 )l , (13.85) 4M (D + 2l − 2)!! and

h(∆u)2k (u∆u)2l (u∆u)(p∆u)i0 =

i¯hǫs 4M

k+l+1

[D + 2(k + l)]!! (2l − 1)!!(up), (D + 2l)!!

(13.86)

where we have allowed for a general u-space dimension D. Expanding (13.81) we now check that, up to first order in ǫs , the expectations hCi0 and hC (p∆u)i0 vanish: 2 i 1 i ǫ hCi0 = h∆corr A i0 + h(∆corr Aǫ )2 i0 = 0, (13.87) ¯h 2! ¯h i hC (p∆u)i0 = h∆corr Aǫ (p∆u)i0 = 0. (13.88) ¯h Indeed, the first term in (13.87) becomes ¯hǫs 1 (D + 2)D i ǫ 2D +2 h∆corr A i0 = 2i − −λ − 2λ , (13.89) ¯h M 4 16 16 and reduces for D = 2 to i ¯hǫs h∆corr Aǫ i0 = −i ¯h M

2 1 λ− . 2

(13.90)

This is canceled identically in λ by the second term in (13.87), which is equal to 2 1 i i ¯hǫs (D + 4)(D + 2) 1 D+2 h(∆corr Aǫ )2 i0 = 4(2λ − 1)2 + 4(2λ − 1)2 − 8(2λ − 1)2 , 2! ¯h 2 M 64 4 16 (13.91) and reduces for D = 2 to 2 2 1 i ¯hǫs 1 ǫ 2 h(∆corr A ) i0 = i λ− . (13.92) 2! ¯h M 2 Similarly, the expectation (13.88), ¯ 2 ǫs h [(2λ − 1)(D + 2)/4 − (2λ − 1)], (13.93) 4M is seen to vanish identically in λ for D = 2. Thus there is no finite time slicing correction to the naive transformation formula (13.34) for the Coulomb path integral in two dimensions. h∆corr Aǫ (p∆u)i0 = −

860

13.4


Solution for the Three-Dimensional Coulomb System

We now turn to the physically relevant Coulomb system in three dimensions. The first problem is to find again some kind of “square root” coordinates to convert the potential −Er in the pseudotime Hamiltonian in the exponent of Eq. (13.5) into a harmonic potential. In two dimensions, the answer was a complex square root. Here, it is a “quaternionic square root” known as the Kustaanheimo-Stiefel transformation, which was used extensively in celestial mechanics [5]. To apply this transformation, the three-vectors x must first be mapped into a four-dimensional uµ -space (µ = 1, 2, 3, 4) via the equations xi = z¯σ i z,

r = z¯z.

(13.94)

Here σ i are the Pauli matrices (1.445), and z, z¯ the two-component objects z1 z2

z=

!

,

z¯ = (z1∗ , z2∗ ),

(13.95)

called “spinors”. Their components are related to the four-vectors uµ by z1 = (u1 + iu2 ),

z2 = (u3 + iu4 ).

(13.96)

The coordinates uµ can be parametrized in terms of the spherical angles of the three-vector x and an additional arbitrary angle γ as follows: √ r cos(θ/2)e−i[(ϕ+γ)/2] , z1 = √ z2 = r sin(θ/2)ei[(ϕ−γ)/2] . In Eqs. (13.94), the angle γ obviously cancels. Each point in x-space corresponds to an entire curve in uµ -space along which the angle γ runs through the interval [0, 4π]. We can write (13.94) also in a matrix form   

1





x   x2  = A(~ u )    3 x

u1 u2 u3 u4



  , 

(13.97)

with the 3 × 4 matrix u3 u4 u1 u2  4 u1  A(~u) =  u −u3 −u2 . u1 u2 −u3 −u4

(13.98)

r = (u1)2 + (u2 )2 + (u3 )2 + (u4 )2 ≡ (~u)2 ,

(13.99)



Since



this transformation certainly makes the potential −Er in the pseudotime Hamiltonian harmonic in ~u. The arrow on top indicates the four-vector nature of uµ . H. Kleinert, PATH INTEGRALS

861

13.4 Solution for the Three-Dimensional Coulomb System

Consider now the kinetic term ds(M/2r)(dx/ds)2 in the action (13.10). Each path x(s) is associated with an infinite set of paths ~u(s) in ~u-space, depending on the choice of a dummy path γ(s) in parameter space. The mapping of the tangent vectors duµ into dxi is given by R

  

1



3



4

1

2





dx u u u u    dx2  u1   = 2  u4 −u3 −u2   dx3 u1 u2 −u3 −u4

du1 du2 du3 du4



  . 

(13.100)

To make the mapping unique we must prescribe at least some differential equation for the dummy angle dγ. This is done most simply by replacing dγ by a parameter which is more naturally related to the components dxi on the left-hand side. We embed the tangent vector (dx1 , dx2 , dx3 ) into a fictitious four-dimensional space and define a new, fourth component dx4 by an additional fourth row in the matrix A(~u), thereby extending (13.29) to the four-vector equation d~x = 2A(~u)d~u.

(13.101)

The arrow on top of x indicates that x has become a four-vector. For symmetry reasons, we choose the 4 × 4 matrix A(~u) as    

A(~u) = 

u3 u4 u1 u2 u4 −u3 −u2 u1 u1 u2 −u3 −u4 2 u −u1 u4 −u3



  . 

(13.102)

The fourth row implies the following relation between dx4 and dγ: dx4 = 2(u2 du1 − u1 du2 + u4 du3 − u3 du4 ) = r(cos θ dϕ + dγ).

(13.103)

Now we make the important observation that this relation is not integrable since ∂x4 /∂u1 = 2u2, ∂x4 /∂u2 = −2u1 , and hence (∂u1 ∂u2 − ∂u2 ∂u1 )x4 (uµ ) = −4,

(∂u3 ∂u4 − ∂u4 ∂u3 )x4 (uµ ) = −4,

(13.104)

implying that x4 (uµ ) does not satisfy the integrability criterion of Schwarz [recall (10.19)]. The mapping is nonholonomic and changes the Euclidean geometry of the four-dimensional ~x-space into a non-Euclidean ~u-space with curvature and torsion. This will be discussed in detail in the next section. The impossibility of finding a unique mapping between the points of ~x- and ~u-space has the consequence that the mapping between paths is multivalued with respect to the initial point. After having chosen a specific image for the initial point, the mapping (13.102) determines the image path uniquely.

862


We now incorporate the dummy fourth dimension into the action by replacing x in the kinetic term by the four-vector ~x and extending the kinetic action to AN kin ≡

N +1 X

M (~xn − ~xn−1 )2 . λ 2 ǫs rn1−λ rn−1

n=1

(13.105)

The additional contribution of the fourth components x4n − x4n−1 can be eliminated trivially from the final pseudotime evolution amplitude by integrating each time slice over dx4n−1 with the measure NY +1 Z ∞ n=1

−∞

d(∆x4 )n q

λ 2πiǫs h ¯ rn1−λ rn−1 /M

.

(13.106)

Note that in these integrals, the radial coordinates rn are fixed numbers. In contrast to the spatial integrals d3 xn−1 , the fourth coordinate must be integrated also over the initial auxiliary coordinate x40 = x4a . Thus we use the trivial identity NY +1 n=1

 

Z



+1 i NX M (∆x4n )2  exp q = 1. λ λ h ¯ n=1 2 ǫs rn1−λ rn−1 2πiǫs h ¯ rn1−λ rn−1 /M

d(∆x4 )n

∞

−∞

"

#

(13.107)

Hence the pseudotime evolution amplitude of the Coulomb system in three dimensions can be rewritten as the four-dimensional path integral hxb |UÊ (S)|xa i = ×

Z

dx4a

NY +1 n=2

rbλ ra1−λ (2πiǫs h ¯ rb1−λ raλ /M)2

"Z

∞

−∞

d4 ∆xn i N exp A , 2 (2πiǫs h ¯ rn−1 /M) h ¯ E #

(13.108)

where AN E is the action (13.9) in which the three-vectors xn are replaced by the fourvectors ~xn , although r is still the length of the spatial part of ~x. By distributing the factors rb , rn , ra evenly over the intervals, shifting the subscripts n of the factors 1/rn in the measure to n + 1, and using the same procedure as in Eq. (13.67), we arrive at the pseudotime evolution amplitude hxb |UÊ (S)|xa i = ×

" NY +1 Z n=2

1 (2π iǫs h ¯ /M)2

Z

∞

−∞

dx4a ra

d4 ∆~xn i exp (AN + AN f ) , 2 2 (2πiǫs h ¯ rn /M) h ¯ E #

(13.109)

with the sliced action AN x, ~x′ ] E [~

2

= (N + 1)ǫs e +

N +1 X n=1

"

M (∆~xn )2 λ + ǫs rn1−λ rn−1 E . λ 2 ǫs rn1−λ rn−1 #

(13.110)


863


The action AN f accounts for all remaining factors in the integral measure. The Q +1 (rn /rn−1)3λ−2 . prefactor is now (rb /ra )3λ−2 and can be written as a product N 1 The index shift in the factor 1/r changes the power 3λ − 2 to 3λ N +1 X i N ~u2 Af = 3λ log 2 n h ¯ ~un−1 n=1

!

(13.111)

[compare with (13.68)]. As in the two-dimensional case we shall at first ignore the subtleties due to the time slicing. Thus we set λ = 0 and apply the transformation formally to the continuum limit of the action AN E , which has the form (13.10), except that x is replaced by ~x. Using the properties of the matrix (13.102) AT = 4~u2 A−1 , det A =

q

det (AAT ) = 16r 2 ,

(13.112)

we see that ~x′2 = 4~u2~u′2 = 4r~u′2 , d4 x = 16r 2 d4 u.

(13.113) (13.114)

In this way, we find the formal relation 1 2 hxb |UÊ (S)|xa i = eie S/¯h 16

Z

dx4a (~ub S|~ua 0) ra

(13.115)

to the time evolution amplitude of the four-dimensional harmonic oscillator (~ub S|~ua 0) =

Z

i D u(s) exp Aos , h ¯

(13.116)

µ ds (~u′2 − ω 2~u2 ). 2

(13.117)

4

with the action Aos =

Z

0

S

The parameters are, as in (13.27) and (13.28), µ = 4M,

ω=

q

−E/2M .

(13.118)

The relation (13.115) is the analog of (13.31). Instead of a sum over the two images R of each point x in u-space, there is now an integral dx4a /ra over the infinitely many images in the four-dimensional ~u-space. This integral can be rewritten as an integral over the third Euler angle γ using the relation (13.103). Since x and thus the Rpolar R angles θ, ϕ remain fixed during the integration, we have directly dx4a /ra = dγa . As far as the range of integration is concerned, we observe that it may be restricted to a single period γa ∈ [0, 4π]. The other periods can be included in the oscillator amplitude. By specifying a four-vector u~b , all paths are summed which run either to

864


the final Euler angle γb or to all its periodic repetitions [which by (13.97) have the same u~b]. This was the lesson learned in Section 6.1. Equation (13.115) contains, instead, a sum over all initial periods which is completely equivalent to this. Thus the relation (13.115) reads, more specifically, 1 4π 2 hxb |UÊ (S)|xa i = eie S/¯h dγa (~ub S|~ua0). (13.119) 16 0 The reason why the other periods in (13.115) must be omitted can best be understood by comparison with the two-dimensional case. There we observed a two-fold degeneracy of contributions to the time-sliced path integral which cancel all factors 2 in the measure (13.71). Here the same thing happens except with an infinite degeneracy: When integrating over all images d4 un of d4 xn in the oscillator path integral we cover the original x-space once for γn ∈ [0, 4π] and repeat doing so for all periods γn ∈ [4πl, 4π(l + 1)]. This suggests that each volume element d4 un must be divided by an infinite factor to remove this degeneracy. However, this is not necessary since the gradient term produces precisely the same infinite factor. Indeed, Z

(~un + ~un−1 )2 (~un − ~un−1 )2

(13.120)

is small for ~xn ≈ ~xn−1 at infinitely many places of γn − γn−1 , once for each periodic repetition of the interval [0, 4π]. The infinite degeneracy cancels the infinite factor in the denominator of the measure. The only place where this cancellation does not R occur is in the integral dx4a /ra . Here the infinite factor in the denominator is still present, but it can be removed by restricting the integration over γa in (13.119) to a single period [6]. Note that a shift of γa by a half-period 2π changes ~u to −~u and thus corresponds to the two-fold degeneracy in the previous two-dimensional system. The time-sliced path integral for the harmonic oscillator can, of course, be done immediately, the amplitude being the four-dimensional version of (13.33): # " "Z # N N Y d4 ∆un i X µ 1 1 2 2 2 exp ∆~un − ǫs ω ~un (~ubS|~ua 0) = (2πi¯ hǫs /µ)2 n=1 2πi¯ hǫs /µ h ¯ n=1 2 ǫs

ω2 i µω [(~ub 2 +~u2a) cos ωS −2~ub~ua ] . (13.121) = exp 2 (2πi¯ h sin ωS /µ) h ¯ sin ωS

To find the fixed-energy amplitude we have to integrate this over S: 1 4π dγa (~ubS|~ua 0). (13.122) 16 0 0 Just like (13.35), the integral is written most conveniently in terms of the variables (13.38), (13.40), so that we obtain the fixed-energy amplitude of the threedimensional Coulomb system (xb |xa )E =

Z

∞

dSeie

2 S/¯ h

Z

4π 1 ∞ 2 (xb |xa )E = dSeie S/¯h dγa (~ub s|~ua 0) 16 0 0 " # ! √ Z Z 2 ̺ ωM 2 ∞ dx4a 1 ̺−ν 1+̺ = −i 2 2 d̺ exp 2κ ~ub~ua exp −κ (rb + ra ) . (1 − ̺)2 1−̺ 1−̺ 2π h ¯ −∞ ra 0

Z

Z



865

In order to perform the integral over dx4b , we now express ~ub~ua in terms of the polar angles √ ~ub~ua = rb ra {cos(θb /2) cos(θa /2) cos[(ϕb − ϕa + γb − γa )/2] + sin(θb /2) sin(θa /2) cos[(ϕb − ϕa − γb + γa )/2]} . (13.123) A trigonometric rearrangement brings this to the form √ ~ub~ua = rb ra {cos[(θb − θa )/2] cos[(ϕb − ϕa )/2] cos[(γb − γa )/2] − cos[(θb + θa )/2] sin[(ϕb − ϕa )/2] sin[(γb − γa )/2]} ,

(13.124)

and further to ~ub~ua = where β is defined by tan or cos

q

(rb ra + xb xa )/2 cos[(γb − γa + β)/2],

β cos[(θb + θa )/2] sin[(ϕb − ϕa )/2] = , 2 cos[(θb − θa )/2] cos[(ϕb − ϕa )/2]

β θb − θa ϕb − ϕa rb ra q = cos cos . 2 2 2 (rb ra + xb xa )/2

(13.125)

(13.126)

(13.127)

The integral 04π dγa can now be done at each fixed x. This gives the fixed-energy amplitude of the Coulomb system [1, 7, 9, 8]. ! √ Z 2 ̺q Mκ 1 ̺−ν (xb |xa )E = −i d̺ I0 2κ (rb ra + xb xa )/2 π¯ h 0 (1 − ̺)2 1−̺ " # 1+̺ (rb + ra ) , × exp −κ (13.128) 1−̺ R

where κ and ν are the same parameters as in Eqs. (13.40). The integral converges only for ν < 1, as in the two-dimensional case. It is again possible to write down another integral representation which converges for all ν 6= 1, 2, 3, . . . by changing the variables of integration to ζ ≡ (1 + ̺)/(1 − ̺) and transforming the integral over ζ into a contour integral encircling the cut from ζ = 1 to ∞ in the clockwise sense. Since the cut is now of the type (ζ − 1)−ν , the replacement rule is πeiπν dζ(ζ − 1) . . . → sin πν 1 This leads to the representation Z

∞

−ν

Z

C

dζ (ζ − 1)−ν . . . . 2πi

M κ πeiπν dζ (xb |xa )E = −i (ζ − 1)−ν (ζ + 1)ν π¯ h 2 sinq πν C 2πi q × I0 (2κ ζ 2 − 1 (rb ra + xb xa )/2)e−κζ(rb +ra ) .

(13.129)

Z

(13.130)

866

13.5


Absence of Time Slicing Corrections for D = 3

Let us now prove that for the three-dimensional Coulomb system also, the finite time-slicing procedure does not change the formal result of the last section. The reader not interested in the details is again referred to the brief argument in Section 13.6. The action AN E in the time-sliced path integral has to be supplemented, in each slice, by the Jacobian action [as in (13.62)] i ǫ A ¯ J h

=

−eµ ei {µ,ν} ∆uν − ei µ ei {κ,ν} ej κ ei {µ,λ} ∆uν ∆uλ

=

1 −Γ{νµ} µ ∆uν − Γ{νκ} σ Γ{λσ} κ ∆uν ∆uλ . 2

(13.131)

The basis tetrad ei µ = ∂xi /∂uµ = 2Ai µ (~u),

i = 1, 2, 3, 4,

(13.132)

is now given by the 4 × 4 matrix (13.102), with the reciprocal tetrad ei µ =

1 i e µ. 2~u2

From this we find the matrix components of the  u1  1  −u2 (Γ1 )µ ν = ~u2  u3 u4  u2 1  1  u4 (Γ2 )µ ν =  2 −u ~u u3  u3 4 1  −u1 (Γ3 )µ ν = ~u2  −u −u2  u4  1  u3 (Γ4 )µ ν = ~u2  u2 −u1

(13.133)

connection [compare (13.18)]  ν u2 −u3 −u4 u1 −u4 u3   , 4 1 u u u2  −u3 −u2 u1 µ  ν −u1 u4 −u3 u2 −u3 −u4   , u3 u2 −u1  u4 u1 u2 µ  ν u4 u1 u2 u3 u2 −u1   , 2 −u u3 u4  u1 −u4 u3 µ  ν −u3 −u2 u1 u4 u1 u2   . 1 4 −u u −u3  −u2 u3 u4 µ

(13.134)

As in the two-dimensional case [see Eq. (13.19)], the connection satisfies the important identity Γµ µν ≡ 0,

(13.135)

which is again a consequence of the relation [compare (13.21)] ∂µ ei µ = 0.

(13.136)

In Section 13.6, this will be shown to be the essential reason for the absence of the time slicing corrections being proved in this section. However, there is now an important difference with respect to the two-dimensional case. The present mapping dxi = ei µ (u)duµ is not integrable. Taking the antisymmetric part of Γµν λ we find the uµ -space to carry a torsion Sµν λ whose only nonzero components are S12 λ = S34 λ =

1 (−u2 , u1 , −u4 , u3 )λ . ~u2

(13.137)



867

The once-contracted torsion is Sµ = Sµν ν =

uµ . ~u2

(13.138)

For this reason, the contracted connections 4uν , ~u2 2uν = 2 ~u

Γνµ µ

=

ei µ ∂ν ei µ =

Γµν µ

=

−ei ν ∂µ ei µ

(13.139)

are no longer equal, as they were in (13.64). Symmetrization in the lower indices gives Γ{νµ} µ =

3uν . ~u2

(13.140)

Due to this, the ∆uν ∆uλ -terms in (13.131) are, in contrast to the two-dimensional case, not given directly by Γνκ σ Γλσ κ = −

4 νλ 2 (δ ~u − 2uν uλ ). ~u4

(13.141)

The symmetrization in the lower indices is necessary and yields Γ{νκ} σ Γ{λσ} κ

= =

Γνκ σ Γλσ κ − 2Γνκ σ Sλσ κ + Sνκ σ Sλσ κ σ

κ

2

4

(13.142) 4

Γνκ Γλσ − 2(−δνλ ~u + 2uν uλ )/~u + uν uλ /~u .

Collecting all terms, the Jacobian action (13.131) becomes " # 2 i ǫ ~un ∆~un ∆~un 2 5 ~un ∆~un A =− 3 − + + ... . ¯h J ~u2n ~u2n 2 ~u2n

(13.143)

In contrast to the two-dimensional equation (13.66), this cannot be incorporated into Aǫf . Although the two expressions contain the same terms, their coefficients are different [see (13.111)]: 2 ~un i ǫ A = 3λ log (13.144) ¯h f ~u2n−1 " # 2 ~un ∆~un ∆~un 2 ~u∆~un = 3λ log 2 − +2 + ... . ~u2n ~u2n ~u2n It is then convenient to rewrite (omitting the subscripts n) ~u2 ~u∆~u i ǫ AJ = −2 log + 2 ¯h (~u − ∆~u)2 ~u 2 2 ∆~u 3 ~u∆~u − 2 + + ... , ~u 2 ~u2

(13.145)

and absorb the first term into Aǫf , which changes 3λ to (3λ − 2). Thus, we obtain altogether the additional action [to be compared with (13.76)] " 2 # i i ∆~u2 ~u∆~u 1 ∆~u2 ~u∆~u ǫ 2 ∆corr A = 4M (2λ − 1) 2 + ( − λ) 2 + 2λ ¯h ¯h 2ǫ ~u 4 ~u ~u2 " # 2 ~u∆~u ∆~u2 ~u∆~u +(3λ − 2) 2 2 − 2 + 2 ~u ~u ~u2 2 ~u∆~u (∆~u)2 3 ~u∆~u + 2 − + + ... . (13.146) ~u ~u2 2 ~u2

868


Using this we now show that the expansion of the correction term i ∆corr Aǫ − 1 C = exp ¯h

(13.147)

has the vanishing expectations hCi0

hC (~ p∆~u)i0

=

0,

=

0,

(13.148)

i.e., i 1 h∆corr Aǫ i0 + ¯h 2

2 i h(∆corr Aǫ )2 i0 ¯h i h∆corr A (~ p∆~u)i0 ¯h

= 0,

(13.149)

= 0,

(13.150)

as in (13.87), (13.88). In fact, using formula (13.86) the expectation (13.150) is immediately found to be proportional to D+2 1 1 i −2(2λ − 1) + 2(3λ − 2) + , (13.151) 16 4 4 which vanishes identically in λ for D = 4. Similarly, using formula (13.85), the first term in (13.149) has an expectation proportional to 1 (D + 2)D D+2 D 2 D 3 i −2 −λ − 4λ2 − (3λ − 2) − − − , (13.152) 4 16 16 4 4 4 8 i.e., for D = 4, 3 −i (2λ − 1)2 , 8

(13.153)

to which the second term adds 1 2 (D + 4)(D + 2) 21 2D + 2 i 4(2λ − 1) + 9(2λ − 1) − 12(2λ − 1) , 2 64 4 16

(13.154)

which cancels (13.154) for D = 4. Thus the sum of all time slicing corrections vanishes also in the three-dimensional case.

13.6

Geometric Argument for Absence of Time Slicing Corrections

As mentioned before, the basic reason for the absence of the time slicing corrections can be shown to be the property of the connection Γµ µλ = g µν ei λ ∂µ ei ν = 0,

(13.155)

which, in turn, follows from the basic identity ∂µ eiµ = 0 satisfied by the basis tetrad, and from the diagonality of the metric g µν ∝ δ µν . Indeed, it is possible to apply the techniques of Sections 10.1, 10.2 to the general pseudotime evolution amplitude (12.28) with the regulating functions fl = f (x),

fr ≡ 1.


869

13.7 Comparison with Schr¨ odinger Theory

Since this regularization affects only the postpoints at each time slice, it is straightforward to repeat the derivation of an equivalent short-time amplitude given in Section 11.3. The result can be expressed in the form (dropping subscripts n) p g(q) i ǫ ǫ ǫ exp (A + A ) , (13.157) K (∆q) = p J D ¯h 2πiǫ¯hf /M where f abbreviates the postpoint value f (xn ) and Aǫ is the short-time action Aǫ =

M gµν (q)∆q µ ∆q ν . 2ǫf

(13.158)

There exists now a simple expression for the Jacobian action. Using formula (11.75), it becomes simply i ǫ 1 ¯hf A = Γµ µ ν ∆q ν − iǫ (Γµ µ ν )2 . ¯ J h 2 8M

(13.159)

In the postpoint formulation, the measure needs no further transformation. This can be seen directly from the time-sliced expression (13.8) for λ = 0 or, more explicitly, from the vanishing of the extra action Aǫf in (13.69) for D = 2 and in (13.144) for D = 3. As a result, the vanishing contracted connection appearing in (13.159) makes all time-slicing corrections vanish. Only the basic short-time action (13.158) survives: ∆Aǫ = 4M

(∆u)2 . 2ǫs

(13.160)

Thanks to this fortunate circumstance, the formal solution found in 1979 by Duru and Kleinert happens to be correct.

13.7

Comparison with Schr¨ odinger Theory

For completeness, let us also show the significance of the geometric property Γµ µλ = 0 within the Schrödinger theory. Consider the Schrödinger equation of the Coulomb system

−

1 2 2 e2 h ¯ ∇ − E ψ(x) = ψ(x), 2M r

(13.161)

to be transformed to that of a harmonic oscillator. The postpoint regularization of the path integral with the functions (13.156) corresponds to multiplying the Schrödinger equation with fl = r from the left. This gives

−

1 2 2 h ¯ r∇ − Er ψ(x) = e2 ψ(x). 2M

(13.162)

We now go over to the square root coordinates uµ transforming −Er into the harmonic potential −E(uµ )2 and the Laplacian ∇2 into g µν ∂µ ∂ν − Γµ µλ ∂λ . The geometric property Γµ µλ = 0 ensures now the absence of the second term and the result is simply g µν ∂µ ∂ν . Since g µν = δ µν /4r, the Schrödinger equation (13.162) takes the simple form

1 2 2 − h ¯ ∂µ − E(uµ )2 ψ(uµ ) = e2 ψ(uµ ). 8M

(13.163)

870


Due to the factor (uµ )2 accompanying the energy E, the physical scalar product in which the states of different energies are orthogonal to each other is given by ′

hψ |ψi =

Z

d4 uψ ′ (uµ )(uµ )2 ψ(u).

(13.164)

This corresponds precisely to the scalar product given in Eq. (11.95) with the purpose of making the Laplace operator (here ∆ = (1/4u2 )∂u2 ) hermitian in the uµ -space with torsion. Indeed, the once-contracted torsion tensor Sµ = Sµν ν can be written as a gradient of a scalar function: Sµ = ∂µ σ(~u),

σ(~u) =

1 log ~u2 . 2

(13.165)

Quite generally, we have shown in Eq. (11.104) that if Sµ (q) is a partial derivative of a scalar field σ(q), the physical scalar product is given by (11.95): Z

hψ2 |ψ1 iphys ≡

q

dD q g(q)e−2σ(q) ψ2∗ (q)ψ1 (q).

(13.166)

√

(13.167)

From (13.132), we have g = 16~u4 ,

so that the physical scalar product is hψ2 |ψ1 iphys =

Z

4

√

−2σ

d u ge

ψ2∗ (~u)ψ1 (~u)

=

Z

d4 u 16~u2 ψ2∗ (~u)ψ1 (~u).

(13.168)

The Laplace operator obtained from ∂~x2 by the nonholonomic KustaanheimoStiefel transformation is ∆ = (1/4~u2)∂µ2 . This is Hermitian in the physical scalar product (13.168), but not in the naive one (11.90) with the integral measure R 4 d u 16~u4. In two dimensions, the torsion vanishes and the physical scalar product reduces to the naive one: Z Z 2 √ ∗ hψ2 |ψ1 iphys = d u gψ2 (u)ψ1 (u) = d2 u 4u2ψ2∗ (u)ψ1 (u). (13.169) With µ = 4M and −E = µω 2 /2, Eq. (13.163) is the Schrödinger equation of a harmonic oscillator: "

#

1 2 2 µ 2 µ 2 − h ¯ ∂µ + ω (u ) ψ(uµ ) = Eψ(uµ). 2µ 2

(13.170)

The eigenvalues of the pseudoenergy E are EN = h ¯ ω(N + Du /2),

(13.171)

where Du = 4 is the dimension of uµ -space, N=

Du X

ni

(13.172)

i=1


871


sums up the integer principal quantum numbers of the factorized wave functions in each direction of the uµ -space. The multivaluedness of the mapping from x to uµ allows only symmetric wave functions to be associated with Coulomb states. Hence N must be even and can be written as N = 2(n − 1). The pseudoenergy spectrum is therefore En = h ¯ ω 2(n + Du /4 − 1),

n = 1, 2, 3, . . . .

(13.173)

According to (13.170), the Coulomb wave functions must all have a pseudoenergy En = e2 .

(13.174)

The two equations are fulfilled if the oscillator frequency has the discrete values ω = ωn ≡

e2 , 2(n + Du /4 − 1)

n = 1, 2, 3 . . . .

(13.175)

With ω 2 = −E/2M and Du = 4, this yields the En = −2Mωn2 = −

Me4 1 , h ¯ 2 2n2

(13.176)

showing that the number N/2 = η corresponds to the usual principal quantum number of the Coulomb wave functions. Let us now focus our attention upon the three-dimensional Coulomb system where Du = 4. In this case, not all even oscillator wave functions correspond to Coulomb bound-state wave functions. This follows from the fact that the Coulomb wave functions do not depend on the dummy fourth coordinate x4 (or the dummy angle γ). Thus they satisfy the constraint ∂x4 ψ = 0, implying in uµ -space [recall (13.132)] −ir∂x4 ψ(x) = −ire4 µ ∂µ ψ(uµ ) = −i = −i∂γ ψ(uµ ) = 0.

i 1h 2 (u ∂1 − u1 ∂2 ) + (u4 ∂3 − u3 ∂4 ) ψ(uµ ) 2 (13.177)

The explicit construction of the oscillator and Coulomb bound-state wave functions is most conveniently done in terms of the complex coordinates (13.96). In terms of these, the constraint (13.177) reads 1 [¯ z ∂z¯ − z∂z ] ψ(z, z ∗ ) = 0. 2

(13.178)

This will be used below to select the Coulomb states. To solve the Schrödinger equation (13.170) we simplify the notation by going over to atomic natural units, where h ¯ = 1, M = 1, e2 = 1, µ = 4M = 4. All lengths are measured in units of the Bohr radius (4.339), whose numerical value is aH = h ¯ 2 /Me2 = 5.2917 × 10−9 cm, all energies in units of EH ≡ e2 /aH =

872


Me4 /¯ h2 = 4.359 × 10−11 erg = 27.210 eV, and all frequencies ω in units of ωH ≡ Me4 /¯ h3 = 4.133 × 1016 /sec(= 4π× Rydberg frequency νR ). Then (13.170) reads (after multiplication by 4M/¯ h2 ) i 1h 2 µ ˆ )≡ −∂µ + 16ω 2(uµ )2 ψ(uµ ) = 4ψ(uµ ). hψ(u 2

(13.179)

ˆ is obviously 4ω(N + 2) = 8ωn. To satisfy the The spectrum of the operator h equation, the frequency ω has to be equal to ωn = 1/2n. ˆ can be brought to the standard form We now observe that the operator h h i ˆ s = 1 −∂ 2 + 4(uµ )2 , h µ 2

(13.180)

with the help of the ω-dependent transformation ˆ = 4ωeiϑDˆ h ˆ s e−iϑDˆ , h

(13.181)

ˆ is an infinitesimal dilation operator which in this context in which the operator D is called tilt operator [10, 11]: ˆ ≡ − 1 iuµ ∂µ , D 2

(13.182)

ϑ = log(2ω).

(13.183)

and ϑ is the tilt angle

The Coulomb wave functions are therefore given by the rescaled solutions of the standardized Schrödinger equation (13.180): √ ˆ ψ(uµ ) = eiϑD ψ s (uµ ) = ψ s ( 2ωuµ ). (13.184) Note that for a solution with a principal quantum number n the scale parameter √ 2ω depends on n: √ ψn (uµ ) = ψns (uµ / n). (13.185) The standardized wave functions ψns (uµ ) are constructed most conveniently by means of four sets of creation and annihilation operators a ˆ†1 , a ˆ†2 , ˆb†1 , ˆb†2 , and a ˆ1 , a ˆ2 , ˆb1 , ˆb2 . They are combinations of z1 , z2 , their complex-conjugates, and the associated differential operators ∂z1 , ∂z2 , ∂z1∗ , ∂z2∗ . The combinations are the same as in (9.127), (9.128), written down once for z1 and once for z2 . In addition, we choose the indices so that ai and bi transform by the same spinor representation of the rotation group. If cij is the 2 × 2 matrix 2

c = iσ =

0 1 −1 0

!

,

(13.186)


873


then cij zj transforms like zi∗ . We therefore define the creation operators 1 a ˆ†1 ≡ − √ (−∂z2∗ + z2 ), 2 1 a ˆ†2 ≡ √ (−∂z1∗ + z1 ), 2

ˆb†1 ≡ √1 (−∂z + z ∗ ), 1 1 2 ˆb†2 ≡ √1 (−∂z + z ∗ ), 2 2 2

(13.187)

and the annihilation operators 1 a ˆ1 ≡ − √ (∂z2 + z2∗ ), 2 1 a ˆ2 ≡ √ (∂z1 + z1∗ ), 2

ˆb1 ≡ √1 (∂z ∗ + z1 ), 2 1 ˆb2 ≡ √1 (∂z ∗ + z2 ). 2 2

(13.188)

Note that ∂z† = −∂z ∗ . The standardized oscillator Hamiltonian is then ˆ s = 2(ˆ h a† aˆ + ˆb†ˆb + 2),

(13.189)

where we have used the same spinor notation as in (13.94). The ground state of the ˆ2 and ˆb1 , ˆb2 . It has therefore the four-dimensional oscillator is annihilated by aˆ1 , a wave function 1 1 ∗ ∗ µ 2 hz, z ∗ |0i = ψs,0000 (z, z ∗ ) = √ e−z1 z1 −z2 z2 = √ e−(u ) . (13.190) π π The complete set of oscillator wave functions is obtained, as usual, by applying the creation operators to the ground state, a

a

b

b

†n †n2 ˆ†n1 ˆ†n2 ˆ2 b1 b2 |0i, |na1 , na2 , nb1 , nb2 i = Nna1 ,na2 ,nb1 ,nb2 aˆ1 1 a

(13.191)

with the normalization factor 1 Nna1 ,na2 ,nb1 ,nb2 = q . na1 !na2 !nb1 !nb2 !

(13.192)

ˆ s are obtained from the sum of the number of a- and b-quanta The eigenvalues of h as 2(na1 + na2 + nb1 + nb2 + 2) = 2(N + 2) = 4n.

(13.193)

The Coulomb bound-state wave functions are in one-to-one correspondence with those oscillator wave functions which satisfy the constraint (13.178), which may now be written as ˆ 05 = − 1 (ˆ L ˆ − ˆb†ˆb)ψ s = 0. (13.194) a† a 2 These states carry an equal number of a- and b-quanta. They diagonalize the (mutually commuting) a- and b-spins â ≡ 1 a L ˆ† σi a ˆ, i 2

ˆ a ≡ 1 ˆb† σiˆb, L i 2

(13.195)

874


with the quantum numbers la = (na1 + na2 )/2, lb = (nb1 + nb2 )/2,

ma = (na1 − na2 )/2, mb = (nb1 − nb2 )/2,

(13.196)

ˆ 2, L ˆ 3 . By defining where l, m are the eigenvalues of L na1 ≡ n1 + m, na2 ≡ n2 , nb1 = n2 + m, nb2 = n1 , na1 ≡ n1 , na2 ≡ n2 − m, nb1 = n2 , nb2 = n1 − m,

for m ≥ 0, for m ≤ 0,

(13.197)

we establish contact with the eigenstates |n1 , n2 , mi which arise naturally when diagonalizing the Coulomb Hamiltonian in parabolic coordinates. The relation between these states and the usual Coulomb wave function of a given angular momentum ˆ i is equal to the sum of a|nlmi is obvious since the angular momentum operator L and b-spins. The rediagonalization is achieved by the usual vector coupling coefficients (see the last equation in Appendix 13A). Note that after the tilt transformation (13.184), the exponential behavior of µ 2 the oscillator wave functions ψns (uµ ) ∝ polynomial(uµ ) × e−(u ) goes correctly over into the exponential r-dependence of the Coulomb wave functions ψ(x) ∝ polynomial(x) × e−r/n . ˆ is Hermitian It is important to realize that although the dilation operator D ˆ iϑD and the operator e at a fixed angle ϑ is unitary, the Coulomb bound states ψn , ˆ arising from the complete set of oscillator states ψns by applying eiϑD , do not span the Hilbert space. Due to the n-dependence of the tilt angle ϑn = log(1/n), a section of the Hilbert space is not reached. The continuum states of the Coulomb system, which are obtained by tilting another complete set of states, precisely fill this section. Intuitively we can understand this incompleteness simply as follows. The wave functions ψns (uµ ) have for increasing n spatial oscillations with shorter and P shorter wavelength. These allow the completeness sum n ψns (uµ )ψns∗ (uµ ) to build up a δ-function which is necessary to span the Hilbert space. In contrast, when forming the sum of the dilated wave functions X √ √ ψns (uµ / n)ψns∗ (uµ / n), n

the terms of larger n have increasingly stretched spatial oscillations which are not sufficient to build up an infinitely narrow distribution. A few more algebraic properties of the creation and annihilation operator representation of the Coulomb wave functions are collected in Appendix 13A.

13.8

Angular Decomposition of Amplitude, and Radial Wave Functions

Let us also give an angular decomposition of the fixed-energy amplitude. This serves as a convenient starting point for extracting the radial wave functions of the H. Kleinert, PATH INTEGRALS

13.8 Angular Decomposition of Amplitude, and Radial Wave Functions

875

Coulomb system which will, in Chapter 14, enable us to find the Coulomb amplitude to D dimensions. We begin with the expression (13.128), (xb |xa )E

! √ 2 ̺q Mκ Z 1 ̺−ν = −i d̺ I0 2κ (rb ra + xb xa )/2 π¯ h 0 (1 − ̺)2 1−̺ ( ) 1+̺ × exp −κ (13.198) (rb + ra ) , 1−̺

and rewrite the Bessel function as I0 (z cos(θ/2)), where θ is the relative angle between xa and xb , and √ 2 ̺ √ z ≡ 2κ rb ra . (13.199) 1−̺ Now we make use of the expansion1 1 kz 2

µ−ν

∞ X

1 Γ(l + µ) (2l + µ)F (−l, l + µ; 1 + ν; k 2 )(−)l I2l+µ (z). l! Γ(1 + ν) l=0 (13.200) Setting k = cos(θ/2), ν = 2q > 0, µ = 1 + 2q, and using formulas (1.442), (1.443) for the rotation functions, this becomes

Iν (kz) = k µ

∞ 2 X (2l + 1)dlqq (θ)I2l+1 (z), z l=|q|

(13.201)

∞ 2X (2l + 1)Pl (cos θ)I2l+1 (z). z l=0

(13.202)

I2q (z cos(θ/2)) = reducing for q = 0 to I0 (z cos(θ/2)) =

After inserting this into (13.128) and substituting y = − 12 log ̺, so that ̺ = e−2y ,

√ z = 2κ rb ra

1 , sinh y

(13.203)

we expand the fixed-energy amplitude into spherical harmonics

(xb |xa )E =

∞ 1 X 2l + 1 (rb |ra )E,l Pl (cos θ) rb ra l=0 4π

∞ l X 1 X ∗ = (rb |ra )E,l Ylm (ˆ xb )Ylm (ˆ xa ), rb ra l=0 m=−l 1

(13.204)

G.N. Watson, Theory of Bessel Functions, Cambridge University Press, London, 1966, 2nd ed., p.140, formula (3).

876


with the radial amplitude Z √ 2M ∞ 1 dy (rb |ra )E,l = −i rb ra e2νy h ¯ 0 sinh y

× exp [−κ coth y(rb + ra )] I2l+1

(13.205) √ 2κ rb ra

!

1 . sinh y

Now we apply the integral formula (9.29) and find (rb |ra )E,l = −i

M Γ(−ν + l + 1) Wν,l+1/2 (2κrb ) Mν,l+1/2 (2κra ) . h ¯ κ (2l + 1)!

On the right-hand side the energy E is contained in the parameters κ = q

(13.206)

q

−2ME/¯ h2

and ν =e2 /2ω¯ h= −e4 M /2¯ h2 E. The Gamma function has poles at ν = n with n = l + 1, l + 2, l + 3, . . . . These correspond to the bound states of the Coulomb system. Writing κ=

1 1 , aH ν

(13.207)

h ¯2 Me2

(13.208)

with the Bohr radius aH ≡

(for the electron, aH ≈ 0.529 × 10−8cm), we have the approximations near the poles at ν ≈ n, (−)nr 1 , nr ! ν − n 1 2h ¯ 2 κ2 1 ≈ , ν−n n 2M E − En 1 1 κ ≈ , aH n

Γ(−ν + l + 1) ≈ −

where nr = n − l − 1. Hence −iΓ(−ν + l + 1)

(13.209)

M (−)nr 1 i¯ h ≈ 2 . h ¯κ n nr ! aH E − En

(13.210)

Let us expand the pole parts of the spectral representation of the radial fixed-energy amplitude in the form (rb |ra )E,l =

∞ X

i¯ h Rnl (rb )Rnl (ra ) + . . . . n=l+1 E − En

(13.211)

The radial wave functions defined by this expansion correspond to the normalized bound-state wave functions 1 ψnlm (x) = Rn,l (r)Ylm (ˆ x). (13.212) r H. Kleinert, PATH INTEGRALS

13.8 Angular Decomposition of Amplitude, and Radial Wave Functions

877

By comparing the pole terms of (13.206) and (13.211) [using (13.210) and formula (9.48) for the Whittaker functions, together with (9.50)], we identify the radial wave functions as Rnl (r) =

1 1/2 aH n (2l

1 + 1)!

v u u t

(n + l)! (n − l − 1)!

× (2r/naH )l+1 e−r/naH M(−n + l + 1, 2l + 2, 2r/naH ) v u

1 u (n − l − 1)! −r/naH = 1/2 t e (2r/naH )l+1 L2l+1 n−l−1 (2r/naH ). (13.213) (n + l)! aH n To obtain the last expression we have used formula (9.53).2 It must be noted that the normalization integrals of the wave functions Rnl (r) differ by a factor z/2n = (2r/naH )/2n from those of the harmonic oscillator (9.54), which are contained in integral tables. However, due to the recursion relation for the Laguerre polynomials zLµn (z) = (2n + µ + 1)Lµn (z) − (n + µ)Lµn−1 (z) − (n + 1)Lµn+1 (z),

(13.214)

the factor z/2n leaves the values of the normalization integrals unchanged. The orthogonality of the wave functions with different n is much harder to verify since the two Laguerre polynomials in the integrals have different arguments. Here the group-theoretic treatment of Appendix 13A provides the simplest solution. The orthogonality is shown in Eq. (13A.28). We now turn to the continuous wave functions. The fixed-energy amplitude has a cut in the energy plane for positive energy where κ = −ik and ν = i/aH k are imaginary. In this case we write ν = iν ′ . From the discontinuity we can extract the scattering wave functions. The discontinuity is given by disc (rb |ra )E,l = (rb |ra )E+iη,l − (rb |ra )E−iη,l " # M Γ(−iν ′ + l + 1) ′ ′ Wiν ′ ,l+1/2 (−2ikrb ) Miν ′ ,l+1/2 (−2ikra ) + (ν → −ν ) .(13.215) = h ¯k (2l + 1)! In the second term, we replace Miν ′ ,l+1/2 (−2ikr) = e−iπ(l+1) M−iν ′ ,l+1/2 (2ikr),

(13.216)

and use the relation, valid for arg z ∈ (−π/2, 3π/2), 2µ 6= −1, −2, −3, . . . , Mλ,µ (z) =

2

Γ(2µ + 1) Γ(2µ + 1) eiπλ e−iπ(µ+1/2) Wλ,µ (z) + eiπλ W−λ,µ (eiπ z), Γ(µ + λ + 1/2) Γ(µ − λ + 1/2) (13.217)

Compare L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965, p. 119. Note the different definition of our Laguerre polynomials Lµn =[(−)µ /(n + µ)!]Ln+µ µ |L.L. .

878


to find M |Γ(−iν ′ + l + 1)|2 πν ′ e Miν ′ ,l+1/2 (−2ikrb ) M−iν ′ ,l+1/2 (2ikra ) . h ¯k (2l + 1)!2 (13.218) The continuum states enter the completeness relation as disc (rb |ra )E,l =

Z

∞

0

∞ X dE ∗ disc (rb |ra )E,l + Rnl (rb )Rnl (ra ) = δ(rb − ra ) 2π¯ h n=l+1

(13.219)

[compare (1.326)]. Inserting (13.215) and replacing the continuum integral R∞ R∞ dE/2π¯ h by the momentum integral h/2πM, the continuum part of the 0 −∞ dkk¯ completeness relation becomes Z

∞

−∞

∗ dkRkl (rb )Rkl (ra ),

(13.220)

with the radial wave functions Rkl (r) =

s

1 |Γ(−iν ′ + l + 1)| πν ′ /2 Miν ′ ,l+1/2 (−2ikr). e 2π (2l + 1)!

(13.221)

By expressing the Whittaker function Mλ,µ (z) in terms of the confluent hypergeometric functions, the Kummer functions M(a, b, z), as Mλ,µ (z) = z µ+1/2 e−z/2 M(µ − λ + 1/2, 2µ + 1, z),

(13.222)

we recover the well-known result of Schrödinger quantum mechanics:3 Rkl (r) =

13.9

s

1 |Γ(−iν ′ + l + 1)| πν ′ /2 ikr e e (−2ikr)l+1 M(−iν ′ + l + 1, 2l + 2, −2ikr). 2π (2l + 1)! (13.223)

Remarks on Geometry of Four-Dimensional uµ-Space

A few remarks are in order on the Riemann geometry of the ~u-space in four dimensions with the metric gµν = 4~u2δµν . As in two dimensions, the Cartan curvature tensor Rµνλ κ vanishes trivially since ei µ (~u) is linear in ~u: (∂µ ∂ν − ∂ν ∂µ )ei λ (~u) = 0.

(13.224)

¯ µνλ κ is In contrast to two dimensions, however, the Riemann curvature tensor R nonzero. The associated Ricci tensor [see (10.41)], has the matrix elements ¯ µνλ µ ¯ νλ = R R 3 = − 6 (δνλ~u2 − ~uν ~uλ ), 2~u 3

(13.225)

L.D. Landau and E.M. Lifshitz, op. cit., p. 120. H. Kleinert, PATH INTEGRALS

13.9 Remarks on Geometry of Four-Dimensional uµ -Space

879

yielding the scalar curvature ¯ = g νλ R ¯ νλ = − 9 . R 2~u4

(13.226)

In general, a diagonal metric of the form gµν (q) = Ω2 (q)δµν

(13.227)

is called conformally flat since it can be obtained from a flat space with a unit metric gµν = δµν by a conformal transformation a la Weyl gµν (q) → Ω2 (q)gµν (q).

(13.228)

Under such a transformation, the Christoffel symbol changes as follows: ¯ µν λ → Γ ¯ λ + Ω,µ δν λ + Ω,ν δµ λ − gµν g λκ Ω,κ , Γ µν

(13.229)

the subscript separated by a comma indicating a differentiation, i.e., Ω,µ ≡ ∂µ Ω. In D dimensions, the Ricci tensor changes according to ¯ µν → Ω−2 R ¯ µν − (D − 2)(Ω−3 Ω;µν − 2Ω−4 Ω,µ Ω,ν ) R h

i

−gµν g λκ (D − 3)Ω−4 Ω,λ Ω,κ + Ω−3 Ω;λκ ¯ µν + (D − 2)Ω−1 (Ω−1 );µν − gµν (D − 2)−1 Ω−D (ΩD−2 );λκ g λκ. (13.230) = Ω−2 R

A subscript separated by a semicolon denotes the covariant derivative formed with Riemann connection, i.e., ¯ µν λ Ω,λ . Ω;µν = Dν Ω,µ = Ωµν − Γ

(13.231)

The curvature scalar goes over into h

i

¯→R ¯ Ω = Ω−2 R ¯ − 2(D − 1)Ω−1 Ω;µν g µν − (D − 1)(D − 4)Ω−2 Ω,µ Ω,ν g µν .(13.232) R The metric gµν = 4~u2δµν in the ~u-space description of the hydrogen atom is conformally flat, so that we can use the above relations to obtain all geometric ¯ µν = 0 by inserting quantities from the initially trivial metric gµν = δµν with R Ω = 2|~u|, so that Ω,µ = 2

uµ , |~u|

Ω,µν =

2 µν 2 (δ ~u − uµ uν ). |~u|3

(13.233)

From the right-hand sides of (13.230) and (13.232) we obtain ¯ µν = −3(D − 2) 1 (δµν ~u2 − ~uµ~uν ), R 4~u 6 ¯ = −3(D − 1)(D − 2) 1 . R 4~u4

(13.234)

880


For D = 4, these agree with (13.225) and (13.226). For D = 2, they vanish. In the Coulomb system, the conformally flat metric (13.228) arose from the nonholonomic coordinate transformation (13.101) with the basis tetrads (13.132) and their inverses (13.133), which produced the torsion tensor (13.137). In the notation (13.228), the torsion tensor has a contraction Sµ (q) ≡ Sµν λ (q) =

1 ∂µ Ω2 (q). 2Ω2 (q)

(13.235)

Note that although Sµ (q) is the gradient of a scalar field, the torsion tensor (13.137) is not a so-called gradient torsion, which is defined by the general form Sµν λ (q) =

i 1h λ δµ ∂ν s(q) − δν λ ∂µ s(q) . 2

(13.236)

For a gradient torsion, Sµ (q) is also a gradient: Sµ (q) = ∂µ σ(q) where σ(q) = (D − 1)s(q)/2. But it is, of course, not the only tensor, for which Sµ (q) is a gradient. Note that under a conformal transformation a la Weyl, a massless scalar field φ(q) is transformed as φ(q) → Ω1−D/2 (q)φ(q).

(13.237)

¯ 2 applied to φ(q) goes over into The Laplace-Beltrami differential operator ∆ = D 1−D/2 Ω Ω (q)∆ φ(q) where 1 1 ∆Ω = Ω−2 ∆ − (D − 2)Ω−1 Ω;µν g µν − (D − 2)(D − 4)Ω−2 Ω,µ Ω,ν g µν . (13.238) 2 4

¯ 2 and the Comparison with (13.234) shows that there exists a combination of ∆ = D ¯ which may be called Weyl-covariant Laplacian. This Riemann curvature scalar R combination is ∆−

1D −2 ¯ R. 4D −1

(13.239)

When applied to the scalar field it transforms as 1D−2 ¯ 1D −2 ¯ ∆− R φ(q) − − −→ Ω−1−D/2 ∆ − R φ(q). 4D−1 4D −1

(13.240)

Thus we can define a massless scalar field in a conformally invariant way by requiring the vanishing of (13.240) as a wave equation. This symmetry property has made the combination (13.239) a favorite Laplacian operator in curved spaces [12].

13.10

Solution in Momentum Space

The path integral for a point particle in a Coulomb potential can also be solved in momentum space. The solution is so far the only indirect evidence for the question H. Kleinert, PATH INTEGRALS

881

13.10 Solution in Momentum Space

first raised by Bryce DeWitt in his fundamental 1957 paper [13], whether the Hamiltonian operator for a particle in curved space contains merely the Laplace-Beltrami operator ∆ in the kinetic energy, or whether there exists an additional term proportional to h ¯ 2 R. Recall the various older path integral literature on the subject cited in Chapter 10. From the measure generated by the nonholonomic mapping principle in Subsection 10.3.2 it follows that there is no extra h ¯ 2 R-term. See the discussion in Section 11.5. It would, of course, be more satisfactory to have a direct experimental evidence, but so far all experimentally accessible systems in curved space have either a very small R caused by gravitation, whose detection is presently impossible, or a constant R which does not change level spacings, an example for the latter being the spinning symmetric and asymmetric top discussed in the context of Eq. (1.468). We show now that if we assume the presence of an extra h ¯ 2 R in the momentum space formulation of the path integral of the Coulomb system, such an extra term would cause experimentally wrong level spacings in the hydrogen atom [14].

13.10.1

Gauge-Invariant Canonical Path Integral

Starting point for our treatment is the path integral formulation for the matrix ˆ ≡ i/(E − H) ˆ rewritten in elements in momentum space of the resolvent operator R the form (12.19): ˆ= R

i fˆ ˆ ˆ f (E − H)

(13.241)

where f is an arbitrary function of space, momentum, and some parameter s. From Eq. (12.31) we find the canonical Euclidean path integral (in the atomic units specified on p. 871): hpb |UÊ (S)|pa i =

Z

3

D x(s) (

D 3 p(s) 2π¯ " h

Z

iZS × exp ds p · x′ − f h ¯ 0

p2 α −E +f 2 r !

#)

fa , (13.242)

and the fixed-energy amplitude (pb |pa )fE

=

Z

∞ 0

dShpb |UÊ (S)|pa i.

(13.243)

The left-hand side carries a superscript f to remind us of the presence of f on the right-hand side, although the amplitude does not really depend on f . This freedom of choice may be viewed as a gauge invariance [15]. of (13.257) under f → f ′ . Such an invariance permits us to subject (13.257) to an additional path integration over f , as long as a gauge-fixing functional Φ[f ] ensures that only a specific “gauge” contributes. Thus we shall calculate the amplitude as a path integral (pb |pa )E =

Z

Df Φ[f ] (pb |pa )fE .

(13.244)

882


The only condition on Φ[f ] is that it must be normalized to have a unit integral: Df Φ[f ] = 1. The choice which leads to the desired solution of the path integral is

R

Φ[f ] =

Y s



 1 i p2 exp − 2 f − r 2 −E  2r r 2 "

!#2  

.

(13.245)



With this, the total action in the path integral (13.244) becomes A[p, x, h] =

Z

0

S



r2 ds −p′ · x − 2

p2 −E 2

!2



1 f − 2 f 2 + α . 2r r

(13.246)

The path integrals over f and x in (13.257) are Gaussian and can be done, in this order, yielding a new action 1Z S 4p′2 2 ds , (13.247) A[p] = 2 +α 2 2 2 0 (p + pE ) √ where we have introduced pE ≡ −2E, assuming E to be negative. The positive regime can be obtained by analytic continuation. Now, a stereographic projection "

≡

#

2pE p , p2 + p2E

π4 ≡

p2 − p2E p2 + p2E

(13.248)

!

(13.249)

transforms (13.247) to the form 1 A[~π] = 2

Z

0

S

1 2 ds 2 ~π ′ + α2 , pE

where ~π denotes the four-dimensional unit vectors ( , π4 ). This describes a point particle with pseudomass µ = 1/p2E moving on a four-dimensional unit sphere. The pseudotime evolution amplitude of this system is 2

(~πb S|~πa 0) = e−iS pE

Z

D~π eiA[~π] . (2π)3/2 p3E

(13.250)

There is an exponential prefactor arising from the transformation of the functional measure in (13.257) to the unit sphere. Let us see how this comes about. When integrating out the spatial fluctuations in going from (13.246) to (13.247), the canonical measure in each time slice d3 pn d3 xn /(2π)3 becomes d3 pn 8/(2π)3/2 (p2n + p2E ). From the stereographic projection (13.248) we see that this is equal to d~πn /(2π)3/2 p3E , where d~πn denotes the product of integrals over the solid angle on the surface of the R unit sphere in four dimensions, with the integral d~π yielding the total surface 2π 2 . R R 4 Alternatively we may also write d~πn = d πn δ(πn2 − 1), or use an explicit angular form of the type (8.119) or (8.123). From Chapter 10 we know that in a curved space, the time-slicedqmeasure R of path integration is given by the product of invariant integrals dqn g(qn ) at H. Kleinert, PATH INTEGRALS

883

13.10 Solution in Momentum Space

each time slice, multiplied by an effective action contribution exp(iAǫeff (qn )) = ¯ n )/6µ), where R ¯ is the scalar curvature. For a sphere of radius r in D exp(iǫR(q ¯ = (D − 1)(D − 2)/r2, implying here for D = 4 that exp(iAǫ ) = dimensions, R eff exp(iǫ/µ) = exp(iǫ p2E ). Thus, when transforming the time-sliced measure in the original path integral (13.244) to the time-sliced measure on the sphere in (13.250), 2 we generate the factor e−iS pE in (13.250) [compare (10.152) and (10.153)]: N Z Y

n=1

N Z N Z Y Y d3 pn d3 xn d 3 pn 8 d~πn = = 2 3 3/2 2 (2π) (2π) (pn + pE ) n=1 (2π)3/2 p3E n=1 N Y

=

2

e−iǫpE

n=1

Z

d~πn 2 2 eiǫpE = e−iSpE 3 3/2 (2π) pE

Z

D~π . (13.251) (2π)3/2 p3E

A complete set of orthonormal hyperspherical functions on this sphere is Ynlm (~π ), where n, l, m are the quantum numbers of the hydrogen atom with the well-known ranges (n = 1, 2, 3, . . . , l = 0, . . . , n − 1, m = −l, . . . , l). They can be expressed j (u) of the SU(2) matrices in terms of the three-dimensional representation Dm 1 m2 1 2 3 u = ~π~σ with the Pauli matrices ~σ ≡ (1, σ , σ , σ ) as Y2j+1,l,m(~π ) =

s

X 2j + 1 j (j, m1 ; j, m2 |l, m) Dm (u). 1 m2 2π 2 m1 ,m2 =−j,...,j

(13.252)

The orthonormality and completeness relations are Z

d~π Yn∗′ l′ m′ (~π )Ynlm (~π ) = δnn′ δll′ δmm′ ,

X

n,l,m

Ynlm(~π ′ )Ynlm(~π ) = δ (4) (~π ′ − ~π ), (13.253)

where the δ-function satisfies d~π δ (4) (~π ′ − ~π ) = 1. When restricting the complete sum to l and m only we obtain the four-dimensional analog of the Legendre polynomial: R

n2 Ynlm (~π )Ynlm (~π ) = 2 Pn (cos ϑ), 2π l,m ′

X

Pn (cos ϑ) =

sin nϑ , n sin ϑ

(13.254)

where ϑ is the angle between the four-vectors ~πb and ~πa : cos ϑ = ~πb~πa =

(p2b − p2E )(p2a − p2E ) + 4p2E pb · pa . (p2b + p2E )(p2a + p2E )

(13.255)

The path integral for a particle on the surface of a sphere was solved Sections 8.7 and 10.4. The solution of (13.250) reads (~πb S|~πa 0) =

(2π)3/2 p3E

∞ X

h i S n2 2 2 2 P (cos ϑ) exp −i(pE n − α ) . (13.256) 2 n 2 n=1 2π

For the path integral itself in (13.250), the exponential contains the eigenvalues of ˆ 2 /2µ which in D dimensions are l(l + the squared angular-momentum operator L

884


D − 2)/2µ, l = 0, 1, 2, . . . . In our system with D = 4, l = n − 1, the eigenvalues ˆ 2 are n2 − 1, leading to an exponential e−i[p2E (n2 −1)−α2 ]S/2 . Together with the of L exponential prefactor in (13.250), this leads to the exponential in (13.256). The integral over S in (13.257) with (13.257) can now be done yielding the amplitude at zero fixed pseudoenergy (~πb |~πa )0 = (2π)3/2 p3E

∞ X

n2 2i P (cos ϑ) . n 2 2En2 + α2 n=1 2π

This has poles displaying the hydrogen spectrum at energies: 1 En = − 2 , n = 1, 2, 3, . . . . 2n

13.10.2

(13.257)

(13.258)

Another Form of Action

Consider the following generalization of the final action (13.247) with an arbitrary function h depending on p and s: 1 Ae [p] = 2

Z

0

S

1 4p˙ 2 2 ds 2 −α h . 2 2 h (p + pE ) "

#

(13.259)

This action is invariant under reparametrizations s → s′ if one transforms simultaneously h → hds/ds′. The path integral with the action (13.247) in the exponent may thus be viewed as a path integral with the gauge-invariant action (13.259) and R an additional path integral dh Φ[h] with an arbitrary gauge-fixing functional Φ[h]. Going back to a real-pseudotime parameter s = iτ , the action corresponding to the Euclidean expression (13.259) describes the dynamics of the point particle in the Coulomb potential reads 1 Z τb 1 4p˙ 2 2 A[p] = dτ 2 +α h . 2 2 2 τa h (p + pE ) At the extremum in h, this action reduces to "

A[p] = 2α

Z

τb

τa

#

v u u dτ t

p˙ 2 2. (p2 + p2E )

(13.260)

(13.261)

This is the manifestly reparametrization invariant form of an action in a curved 2 space with a metric g µν = δ µν / (p2 + p2E ) . In fact, this action coincides with the classical eikonal in momentum space: S(pb , pa ; E) = −

Z

pb pa

dτ p˙ · x.

(13.262)

Observing that the central attractive force makes p˙ point in the direction −x, and inserting r = α(p2 + p2E )/2, we find precisely the action (13.261). In fact, the canonical quantization of a system with the action (13.261) à la Dirac leads directly to a path integral with action (13.260) (see also the discussion in Chapter 19). The eikonal (13.262), and thus the action (13.261), determines the classical orbits via the first extremal principle of theoretical mechanics found in 1744 by Maupertius (see p. 377). H. Kleinert, PATH INTEGRALS

Appendix 13A

13.10.3

Dynamical Group of Coulomb States

885

Absence of Extra R-Term

Since the Coulomb path integral in momentum space is equivalent to that of a point particle on a sphere, we can use it to give experimental limits on the possible ˆ = −µ¯ presence of an extra R-term in the Hamiltonian operator H h2 ∆/2 + c µ¯ h2 R/2 of the Schrödinger equation in curved space, which is not excluded by Einstein’s covariance arguments. For this purpose we assuming the R-term to be universal for all spinless point particles. Otherwise every such particle would be characterized by two parameters, the mass m and the extra parameter c. We further assume that the complete equivalence of coordinate and momentum space formulations of quantum mechanics persists in curved space. In the exponent of (13.256), the extra term c µ¯ h2 R/2 would appear as an extra constant 3c added to n2 . The hydrogen spectrum would then have the energies En = −1/2(n2 + 3c). The only theoretically proposed candidates for c are 1/24, 1/12, and 1/8.4 These parameters would imply a strong distortion of the hydrogen spectrum (13.258) which would certainly have been noticed experimentally a long time ago. In fact, a prediction of the distorted spectra would have led to discarding Schrödinger theory right from the beginning as a possible quantum theory of atomic physics. At fundamental level, the present discussion confirms the validity of the nonholonomic mapping principle of Chapter 10 which requires the extra factor R ¯ exp(iAeff /¯ h) = exp(i¯ h dsR/6µ) in the measure of path integration in curved space [recall Eq. (10.152)]. Without this factor, the spectrum would be En = −α/2(n2 −1) rather than −α/2(n2 − 1), which would definitely be wrong, since the energy would be infinite in the ground state where n = 1!

Appendix 13A

Dynamical Group of Coulomb States

√ The subspace of oscillator wave functions ψ s (uµ / n) in the standardized form (13.184), which do not depend on x4 (i.e., on γ), is obtained by applying an equal number of creation operators a† and b† to the ground state wave function (13.190). They are equal to the scalar products between the localized bra states hz, z ∗ | and the ket states |na1 , na2 , nb1 , nb2 i of (13.191). These ket states form an irreducible representation of the dynamical group O(4,2), the orthogonal group of six-dimensional flat space whose metric gAB has four positive and two negative entries (1, 1, 1, 1, −1, −1). ˆ AB ≡ −L ˆ BA , A, B = 1, . . . 6, of this group are constructed from the The 15 generators L spinors a ˆ≡

a ˆ1 a ˆ2

,

ˆb ≡

ˆb1 ˆb2

,

(13A.1)

and their Hermitian-adjoints, using the Pauli σ-matrices and c ≡ iσ 2 , as follows (since Lij carry subscripts, we define σi ≡ σ i ):

4

ˆ ij = 1 a L ˆ† σk a ˆ + ˆb† σk ˆb 2 ˆ i4 = 1 a L ˆ† σi a ˆ − ˆb† σiˆb , 2 See Notes and References in Chapter 10.

i, j, k = 1, 2, 3 cyclic,

886

13 Path Integral of Coulomb System ˆ i5 = 1 a L ˆ† σi cˆb† − a ˆcσiˆb , 2 ˆ i6 = i a L ˆ† σi cˆb† + a ˆcσiˆb , 2 ˆ 45 = 1 a L ˆ† cˆb† − a ˆcˆb , 2i ˆ 46 = 1 a L ˆ† cˆb† + a ˆcˆb , 2 1 ˆ 56 = L a ˆ† a ˆ + ˆb†ˆb + 2 . 2

(13A.2)

ˆ 56 on the states with an equal number of a- and b-quanta are obviously The eigenvalues of L 1 a n1 + na2 + nb1 + nb2 + 2 = n. 2

(13A.3)

The commutation rules between these operators are

ˆ AB , L ˆ AC ] = igAA L ˆ BC , [L

(13A.4)

where n is the principal quantum number [see (13.193)]. It can be verified that the following combinations of position and momentum operators in a three-dimensional Euclidean space are elements of the Lie algebra of O(4,2): r xi

= =

−i(x∂x + 1) = −ir∂xi =

ˆ 56 − L ˆ 46 , L ˆ i5 − L ˆ i4 , L ˆ 45 , L ˆ i6 . L

(13A.5)

The last equation follows from the transformation formula [recall (13.133)] ∂xi =

1 i e µ ∂µ 2~u2

(13A.6)

together with 1 1 (z1 + z1∗ ), u2 = (z1 − z1∗ ), 2 2i 1 1 u3 = (z2 + z2∗ ), u4 = (z2 − z2∗ ), 2 2i u1 =

(13A.7)

and ∂1 = (∂z1 + ∂z1∗ ), ∂3 = (∂z2 + ∂z2∗ ),

∂2 = i(∂z1 − ∂z1∗ ), ∂4 = i(∂z2 − ∂z2∗ ).

(13A.8)

Hence i z σi ∂z¯ + ∂z σi z). −ir∂xi = − (¯ 2

(13A.9)

ˆ AB can be expressed in terms of the z, z ∗-variables as By analogy with (13A.2), the generators L follows: ˆ ij L ˆ i4 L

1 (¯ z σk ∂z¯ − ∂z σk z), 2 1 = − (¯ z σi z − ∂z σi ∂z¯), 2 =


Appendix 13A

Dynamical Group of Coulomb States ˆ i5 L

=

ˆ i6 L

=

ˆ 45 L

=

ˆ 46 L

=

ˆ 56 L

=

1 (¯ z σi z + ∂z σi ∂z¯), 2 i − (¯ z σi ∂z¯ + ∂z σi z), 2 i − (¯ z ∂z¯ + ∂z z), 2 1 − (¯ z z + ∂z ∂z¯), 2 1 (¯ z z − ∂z ∂z¯). 2

887

(13A.10)

Going over to the operators xi , ∂xi , they become ˆ ij L ˆ i4 L ˆ i5 L ˆ i6 L ˆ 45 L ˆ 46 L ˆ 56 L

i = − (xi ∂xj − xj ∂xi ), 2 1 = −xi ∂x2 − xi + 2∂xi x∂x , 2 1 = −xi ∂x2 + xi + 2∂xi x∂x , 2 = −ir∂xi , = −i(xi ∂xi + 1), 1 = (−r∂x2 − r), 2 1 = (−r∂x2 + r), 2

(13A.11)

where the purely spatial operators ∂x2 and x∂x are equal to ∂x2µ and xµ ∂xµ because of the constraint (13.177). The Lie algebra of the differential operators (13A.11) is isomorphic to the Lie algebra of the conformal group in four spacetime dimensions, which is an extension of the inhomogeneous Lorentz group or Poincaré group, defined by the commutators in Minkowski space (µ, ν = 0, 1, 2, 3), whose metric has the diagonal elements (+1, −1, −1, −1), [Pµ , Pν ] = 0,

(13A.12)

[Lµν , Pλ ] = −i(gµλ Pν − gνλ Pµ ), [Lµν , Lλκ ] = −i(gµλ Lνκ − gνλ Lµκ − gµκ Lνλ − gνκ Lµλ ).

(13A.13) (13A.14)

The extension involves the generators D of dilatations xµ → ρxµ and Kµ of special conformal transformations 5 xµ →

xµ − cµ x2 , 1 − 2cx + c2 x2

(13A.15)

with the additional commutation rules [D, Pµ ] = −iPµ , [D, Kµ ] = iKµ , [D, Lµν ] = 0, [Kµ , Kν ] = 0, [Kµ , Pν ] = −2i(gµν D+Lµν ), [Kµ , Lνλ ] = i(gµν Kλ − gµλ Kν ).

(13A.16) (13A.17)

The commutation rules can be represented by the differential operators ˆ µν = i(xµ ∂ν − xν ∂µ ), D ˆ = ixµ ∂µ , Pˆµ = i∂µ , M ˆ µ = i(2xµ xν ∂ν − x2 ∂µ ). K 5

(13A.18) (13A.19)

Note the difference with respect to the conformal transformations à la Weyl in Eq. (13.228). They correspond to local dilatations.

888


Their combinations Jµν ≡ Lµν , Jµ5 ≡

1 1 (Pµ − Kµ ), Jµ6 ≡ (Pµ + Kµ ), J56 ≡ D, 2 2

(13A.20)

satisfy the commutation relation of O(4,2): [JAB , JCD ] = −i(¯ gAC JBD − g¯BC JAD − g¯AD JBC − g¯BD JAC ),

(13A.21)

where the metric g¯AB has the diagonal values (+1, −1, −1, −1, −1, +1). When working with oscillator wave functions which are factorized in the four uµ -coordinates, the most convenient form of the generators is ˆ 12 L ˆ 13 L ˆ 14 L ˆ 15 L

= i(u1 ∂2 − u2 ∂1 − u3 ∂4 + u4 ∂3 )/2, = i(u1 ∂3 + u2 ∂4 − u3 ∂1 − u4 ∂2 )/2,

ˆ 16 L ˆ 23 L ˆ 24 L

= −i(u1 ∂3 + u2 ∂4 + u3 ∂1 + u4 ∂2 )/2, = i(u1 ∂4 − u2 ∂3 + u3 ∂2 − u4 ∂1 )/2,

ˆ 25 L ˆ L26 ˆ 34 L ˆ 35 L

= (u1 u4 + u2 u3 ) + (∂1 ∂4 − ∂2 ∂3 )/4, = −i(u1 ∂4 − u2 ∂3 − u3 ∂2 + u4 ∂1 )/2,

ˆ 36 L ˆ 45 L ˆ 46 L

= −i(u1 ∂1 + u2 ∂2 − u3 ∂3 − u4 ∂4 )/2, = −i(u1 ∂1 + u2 ∂2 + u3 ∂3 + u4 ∂4 + 2)/2,

ˆ 56 L

= −(u1 u3 + u2 u4 ) + (∂1 ∂3 + ∂2 ∂4 )/4, = (u1 u3 + u2 u4 ) + (∂1 ∂3 + ∂2 ∂4 )/4,

= −(u1 u4 − u2 u3 ) + (∂1 ∂4 − ∂2 ∂3 )/4,

= [(u1 )2 + (u2 )2 − (u3 )2 − (u4 )2 ]/2 + (∂12 + ∂22 − ∂32 − ∂42 )/8, = −[(u1 )2 + (u2 )2 − (u3 )2 − (u4 )2 ]/2 + (∂12 + ∂22 − ∂32 − ∂42 )/8,

= −(uµ )2 /2 − ∂µ2 /8, = (uµ )2 /2 − ∂µ2 /8.

(13A.22)

The commutation rules (13A.4) between these generators make the solution of the Schrödinger equation very simple. Rewriting (13.162) as E r aH r∇2 − − 1 ψ(x) = 0, (13A.23) − 2 EH aH ˆ 46 , L ˆ 56 and going to atomic natural units with aH = 1, EH = 1, we express r∂x2 and r in terms of L via (13A.11). This gives 1 ˆ ˆ ˆ ˆ (L56 + L46 ) − E(L56 − L46 ) − 1 ψ = 0. (13A.24) 2 With the help of Lie’s expansion formula 2 ˆ ˆ −iA ˆ ˆ B] ˆ + i [A, ˆ [A, ˆ B]] ˆ + ... eiA Be = 1 + i[A, 2!

ˆ 45 and B ˆ=L ˆ 56 and the commutators [L ˆ 45 , L ˆ 56 ] = iL45 and [L ˆ 45 , L ˆ 46 ] = iL ˆ 56 , this can for Aˆ = L be rewritten as h i ˆ ˆ ˆ 45 −iϑL eϑ eiϑL45 L e − 1 ψ = 0, (13A.25) 56 with

ϑ=

1 log(−2E). 2

(13A.26)


889


ˆ 56 with an eigenvalue n, the solutions of (13A.25) are obviously If ψn denotes the eigenstates of L ˆ 56 whose eigenvalues are n = 1, 2, 3, . . . given by the tilted eigenstates eiϑL45 ψn of the generator L [as follows directly from the representation (13A.2)]. For these states, the parameter ϑ takes the values ϑ = ϑn = − log n,

(13A.27)

with the energies En = −1/2n2 . ˆ 46 − L ˆ 56 , Since the energy E in the Schrödinger equation (13A.24) is accompanied by a factor L the physical scalar product between Coulomb states is s H ′s ˆ ˆ hψ ′H n′ |ψn iphys ≡ hψ n′ |(L56 − L46 )|ψn i = δn′ n .

(13A.28)

Within this scalar product, the Coulomb wave functions √ 1 1 ˆ ψnH (x) = √ eiϑn D ψns (uµ ) = √ ψns (uµ / n) n n

(13A.29)

are orthonormal. The physical scalar product (13A.28) agrees of course with the scalar product (13.164) and with the scalar product (11.95) derived for a space with torsion in Section 11.4, apart from a trivial constant factor. It is now easy to calculate the physical matrix elements of the dipole operator xi and the momentum operator −i∂xi using the representations (13A.5). Only operations within the Lie algebra of the group O(4,2) have to be performed. This is why O(4,2) is called the dynamical group of the Coulomb system [11]. For completeness, let us state the relation between the states in the oscillator basis |n1 n2 mi and the eigenstates of a fixed angular momentum |nlmi of (13.213): |nlmi = (−)m ×

X

√ 2l + 1

n1 +n2 +m=(n−1)/2

1 2 (n − 1) 1 2 (n2 − n1

+ m)

1 2 (n − 1) 1 2 (n1 − n2

l + m) −m

|n1 n2 mi.

(13A.30)

The coefficients are the standard Wigner 3j-symbols [16].

Notes and References For remarks on the history of the solution of the path integral of the Coulomb system see the preface. The advantages of four-dimensional uµ -space in describing the three-dimensional Coulomb system was first exploited by P. Kustaanheimo and E. Stiefel, J. Reine Angew. Math. 218, 204 (1965). See also the textbook by E. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics, Springer, Berlin, 1971. Within Schrödinger’s quantum mechanics, an analog transformation was introduced in E. Schrödinger, Proc. R. Irish Acad. 46, 183 (1941). See also L. Infeld and T.E. Hull, Rev. Mod. Phys. 23, 21 (1951). Among the numerous applications of the transformation to the Schrödinger equation, see M. Boiteux, Physica 65, 381 (1973); A.O. Barut, C.K.E. Schneider, and R. Wilson, J. Math. Phys. 20, 2244 (1979); J. Kennedy, Proc. R. Irish Acad. A 82, 1 (1982). The individual citations refer to

890


[1] I.H. Duru and H. Kleinert, Phys. Lett. B 84, 30 (1979) (http://www.physik.fu-berlin.de/~kleinert/65); Fortschr. Phys. 30, 401 (1982) (ibid.http/83). See also the historical remarks in the preface. [2] H. Kleinert, Mod. Phys. Lett. A 4, 2329 (1989) (ibid.http/199). [3] H. Kleinert, Gen. Rel. Grav. 32, 769 (2000) (ibid.http/258); Act. Phys. Pol. B 29, 1033 (1998) (gr-qc/9801003). [4] H. Kleinert, Phys. Lett. B 189, 187 (1987) (ibid.http/162). [5] For the interpretation of this transformation as a quaternionic square root, see F.H.J. Cornish, J. Phys. A 17, 323, 2191 (1984). [6] This restriction was missed in a paper by R. Ho and A. Inomata, Phys. Rev. Lett. 48, 231 (1982). [7] Within Schrödinger’s quantum mechanics, this expression had earlier been obtained by L.C. Hostler, J. Math. Phys. 5, 591 (1964). [8] Note also a calculation of the time evolution amplitude of the Coulomb system by S.M. Blinder, Phys. Rev. A 43, 13 (1993). His result is given in terms of an infinite P series which is, unfortunately, as complicated as the well-known spectral representation n ψn (xb )ψn∗ (xb )e−iEn (tb −ta )/¯h . [9] There exists an interesting perturbative solution of the path integral of the integrated R Coulomb amplitude d3 x (xb tb |xa ta ) by M.J. Goovaerts and J.T. Devreese, J. Math. Phys. 13, 1070 (1972). There exists a related perturbative solution for the potential δ(x): M.J. Goovaerts, A. Babcenco, and J.T. Devreese, J. Math. Phys. 14, 554 (1973).

[10] For the introduction and extensive use of the tilt operator in calculating transition amplitudes, see H. Kleinert, Group Dynamics of the Hydrogen Atom, Lectures presented at the 1967 Boulder Summer School, in Lectures in Theoretical Physics, Vol. X B, pp. 427–482, ed. by W.E. Brittin and A.O. Barut, Gordon and Breach New York, 1968 (ibid.http/4). [11] H. Kleinert, Fortschr. Phys. 6, 1 (1968) (ibid.http/1). [12] N.D. Birell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge, 1982. [13] B.S. DeWitt, Rev. Mod. Phys. 29, 377 (1957). [14] H. Kleinert, Phys. Lett A 252, 277 (1999) (quant-ph/9807073). [15] K. Fujikawa, Prog. Theor. Phys. 96 863 (1996) (hep-th/9609029); (hep-th/9608052). [16] A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1960.



Acribus, ut ferme talia, initiis, incurioso fine. As is usual in such matters, keen in commencing, negligent at the end. Tacitus, Annales, Book 6, 17

14 Solution of Further Path Integrals by Duru-Kleinert Method The combination of a path-dependent time reparametrization and a compensating coordinate transformation, used by Duru and Kleinert to transform the Coulomb path integral into a harmonic-oscillator path integral, can be generalized to relate a variety of path integrals to each other. In this way, many unknown path integrals can be solved by their relation to known path integrals. In this chapter, the method is explained for a typical sample of one-dimensional path integrals as well as for a more involved three-dimensional system. The latter describes a generalization of the Coulomb system consisting of two particles which carry both electric and magnetic charges. It is commonly referred to as the dionium atom (by analogy with the positronium atom, the bound state between electron and positron). We also discuss further possible generalizations of the solution method.

14.1

One-Dimensional Systems

In one space dimension, the general relation to be established is the following: Let ˆ be a Hamiltonian operator H ˆ = Tˆ + Vˆ , H

(14.1)

with the kinetic term Tˆ = pˆ2 /2M, and define the auxiliary Hamiltonian operator Ê = H ˆ − E, H

(14.2)

with the associated time evolution amplitude ˆ hxb |UÊ (t)|xa i ≡ hxb |e−itHE /¯h |xa i.

(14.3)

An integration of this amplitude over all t > 0 yields the fixed-energy amplitude (xb |xa )E =

Z

∞

ta

Ê (tb − ta )|xa i dtb hxb |U 891

(14.4)

892

14 Solution of Further Path Integrals by Duru-Kleinert Method

[recall (12.8)]. This can formally be written as a path integral (xb |xa )E =

Z

∞

ta

dtb

Z

Dx(t)eiAE [x]/¯h ,

(14.5)

with an action AE [x] =

Z

tb

ta

M 2 dt x˙ (t) − V (x(t)) + E . 2

(14.6)

As in the Coulomb system, another path integral representation is found for the amplitude (14.5) by making use of the more general representation (12.21) of the resolvent operator. By choosing two arbitrary regulating functions fl (x), fr (x) whose product is f (x), we introduce the modified auxiliary Hamiltonian operator ˆ E = fl (x)(H ˆ − E)fr (x). H

(14.7)

The associated pseudotime evolution amplitude ˆ

hxb |UÊ (S)|xa i ≡ fr (xb )fl (xa )hxb |e−iS HE /¯h |xa i

(14.8)

yields upon integration over all S > 0 the same fixed-energy amplitude as (14.4): (xb |xa )E =

Z

0

∞

dShxb |UÊ (S)|xa i

(14.9)

[recall (12.30)]. The amplitude can therefore be calculated from the path integral (xb |xa )E =

Z

0

∞

dS fr (xb )fl (xa )

Z

iAfE [x]/¯ h

Dx(s)e

,

(14.10)

with the modified action AfE [x]

=

Z

S 0

(

)

M ds x′2 (s) − f (x(s))[V (x(s)) − E] . 2f (x(s))

(14.11)

As observed in (12.37), this action is obtained from (14.6) by a path-dependent time reparametrization satisfying dt = ds f (x(s)).

(14.12)

The introduction of f (x) has brought the kinetic term to an inconvenient form containing a space-dependent mass M/f (x). This space dependence is removed by a coordinate transformation x = h(q).

(14.13)

Since the coordinate differentials are related by dx = h′ (q)dq,


893

14.1 One-Dimensional Systems

we require the function h(q) to satisfy h′2 (q) = f (h(q)).

(14.15)

Then the action (14.11) reads, in terms of the new coordinate q, Af,q E

=

Z

S 0

M ′2 ds q (s) − f (q(s))[V (q(s)) − E] , 2

(14.16)

with the obvious notation f (q) ≡ f (h(q)),

V (q) ≡ V (h(q)).

(14.17)

In the transformed action (14.16), the kinetic term has the usual form. The important fact to be proved and exploited in the sequel is the following: The initial fixed-energy amplitude (14.5) can be related to the fixed-pseudoenergy amplitude associated with the transformed action (14.16), if this action is extended by an effective potential 

h ¯ 2 1 h′′′ 3 Veff (q) = −  − M 4 h′ 8

h′′ h′

!2  .

(14.18)

The effective potential is caused by time slicing effects and will be derived in the next section. Thus, instead of the naively transformed action (14.16), the fixedpseudoenergy amplitude (qb |qa )E is obtained from the extended action ADK E,E [q] =

Z

0

S

ds

M ′2 q (s) − f (q(s))[V (q(s)) − E] − Veff (q(s)) + E , 2

(14.19)

by calculating the path integral (qb |qa )E =

Z

∞

0

dS

Z

DK

Dq(s)eiAE,E [q] .

(14.20)

The relation to be derived which leads to a solution of many nontrivial path integrals is (xb |xa )E = [f (xb )f (xa )]1/4 (qb |qa )E=0 .

(14.21)

The procedure is an obvious generalization of the Duru-Kleinert transformation of the Coulomb path integral in Section 13.1, as indicated by the superscript DK on the transformed actions. Correspondingly, the actions AE [x] and ADK E,E [q], whose path integrals (14.5) and (14.20) producing the same fixed-energy amplitude (xb |xa )E via the relation (14.21), are called DK-equivalent. The prefactor on the right-hand side has its origin in the normalization properties of the states. With dx = dq h′ (q) = dq f (h(q))1/2 , the completeness relation Z

dx|xihx| = 1

(14.22)

894


goes over into Z

q

dq f (q)|h(q)ihh(q)| = 1.

(14.23)

We want the transformed states |qi to satisfy the completeness relation Z

dq|qihq| ≡ 1.

(14.24)

This implies the relation between new and old states: |xi = f (q)−1/4 |qi.

(14.25)

At first sight it appears as though the normalization factor in (14.21) should have the opposite power −1/4, but the sign is correct as it is. The reason lies, roughly speaking, in a factor [f (xb )f (xa )]1/2 by which the pseudotimes dt and ds in the integrals (14.4) and (14.20) differ from each other. This causes the fixed-energy amplitude to be no longer proportional to the dimensions of the states, in which case Eq. (14.21) would have indeed carried a factor [f (xb )f (xa )]−1/4 . The extra factor [f (xb )f (xa )]1/2 arising from the pseudotime integration inverts the naively expected prefactor. In applications, the situation is usually as follows: There exists a solved path integral for a system with a singular potential. The time-sliced action is not the naively sliced classical action, but a more complicated regularized one which is free of path collapse problems. The most important example is the radial path integral (8.36) which involves a logarithm of a Bessel function rather than a centrifugal barrier. Further examples are the path integrals (8.173) and (8.206) of a particle near the surface of a sphere in D = 3 and D = 4 dimensions, where angular barriers are regulated by Bessel functions. In these examples, the explicit form of the timesliced path integral without collapse as well as its solution are obtained from an angular momentum projection of a simple Euclidean path integral. In the first step of the solution procedure, the introduction of a path-dependent new time s via dt = ds f (x(s)) removes the dangerous singularities by an appropriate choice of the regulating function f (x). The transformed system has a regular potential and possesses a time-sliced path integral, but it has an unconventional kinetic term. In the second step, the coordinate transformation brings the kinetic term to the conventional form. The final fixed-pseudoenergy amplitude (qb |qa )E evaluated at E = 0 coincides with the known amplitude of the initial system, apart from the above-discussed factor which is inversely related to the normalization of the states. Note that with (14.15), the relation (14.21) can also be written as (xb |xa )E = [h′ (qb )h′ (qa )]1/2 (qb |qa )E=0 .

(14.26)

This transformation formula will be used to find a number of path integrals. First, however, we shall derive the effective potential (14.18) as promised. H. Kleinert, PATH INTEGRALS

895

14.2 Derivation of the Effective Potential

14.2

Derivation of the Effective Potential

In order to derive the effective potential (14.18), we consider the pseudotime-sliced path integral associated with the regularized pseudotime evolution operator (14.8): hxb |UÊ (S)|xa i

fr (xb )fl (xa )

N Y

 Z  q



i N A , h ¯

(14.27)

M (∆xn )2 + ǫs [E − V (xn )]fl (xn )fr (xn−1 ) . 2ǫs fl (xn )fr (xn−1 )

(14.28)

≈q 2πiǫs h ¯ fl (xb )fr (xa )/M

n=1

dxn

2πiǫs h ¯ fn /M

 exp

where N

A =

N +1 X n=1

(

)

In the measure, we have used the abbreviation fn ≡ f (xn ) = fl (xn )fr (xn ). From now on, the potential V (x) is omitted as being inessential to the discussion. By shifting the product index and the subscripts of fn by one unit, and by compensating for this with a prefactor, the integration measure in (14.27) acquires the postpoint form [f (xb )f (xa )]1/4 fr (xa ) q fr (xb ) 2πiǫs h ¯ /M "

#−5/4 "

fl (xa ) fl (xb )

#1/4 N +1 Z Y n=2

d∆xn q

2πiǫs h ¯ fn /M

,

(14.29)

where the integrals over ∆xn = xn − xn−1 are done successively from high to low n, each at a fixed postpoint position xn . We now go over to the new coordinate q with a transformation function x = h(q) satisfying (14.15) which makes the leading kinetic term simple: AN 0

=

N +1 X n=1

M (∆qn )2 . 2ǫs

(14.30)

The postpoint expansion of ∆xn reads at each n (omitting the subscripts) 1 1 ∆x = x(q) − x(q − ∆q) = e1 ∆q − e2 (∆q)2 + e3 (∆q)3 + . . . , 2 6

(14.31)

with the expansion coefficients e1 ≡ h′ = f 1/2 , e2 ≡ h′′ , e3 ≡ h′′′ , . . .

(14.32)

evaluated at the postpoint qn . The expansion (14.31) is the one-dimensional analog of the expansion (11.56), the coefficients corresponding to the basis triads ei µ in Eq. (10.12) and their derivatives (e1 =e ˆ i µ , e2 =e ˆ i µ,ν , . . .). Let us also introduce the µ analog of the reciprocal triad ei defined in (10.12): e¯ ≡ 1/e1 = 1/h′ = 1/f 1/2 .

(14.33)

896


With it, we expand the kinetic term in (14.28) as (∆q)2 1 (∆xn )2 1 1 − e¯e2 ∆q + e¯e3 + (¯ = ee2 )2 (∆q)2 + . . . 2ǫs fl (xn )fr (xn−1 ) 2ǫs 3 4  

× 1+ 

fr′ fr



∆q + 

fr′ fr

!2

−



1 fr′′  2 fr

 

(∆q)2 + . . . ,

(14.34)



where fr′ ≡ dfr /dq. From Eq. (14.31) we see that the transformation of the measure has the Jacobian ∂∆x 1 J= = f 1/2 1 − e¯e2 ∆q + e¯e3 (∆q)2 + . . . ∂∆q 2

(14.35)

[this being a special case of (11.59)]. Since the subsequent algebra is tedious, we restrict the regulating functions fl (x) and fr (x) somewhat as in Eq. (13.3) by assuming them to be different powers fl (x) = f (x)1−λ and fr (x) = f (x)λ of a single function f (x), where λ is an arbitrary splitting parameter. Then the measure (14.29) becomes 3λ/2 (1−3λ)/2 N +1 Z Y fa

fb

q

2πiǫs h ¯ /M

n=2

d∆xn q

2πiǫs h ¯ fn /M

,

(14.36)

with the obvious notation fb ≡ f (xb ), fa ≡ f (xa ). We now distribute the prefactor 3λ/2 fb fa(1−3λ)/2 evenly over the time interval by writing 3λ/2 fb fa(1−3λ)/2

=

1/4 fb fa1/4

NY +1 n=1

fn−1 fn

!1/4−3λ/2

.

(14.37)

Then the path integral (14.27) becomes 1/4

fb fa1/4

hxb |UÊ (s)|xa i ≈ q 2πiǫs h ¯ /M

N Y

n=1

 Z  q

d∆qn

2πiǫs h ¯ /M

 

(14.38)

+1 i NX M × exp (∆qn )2 + ǫs f (qn )[E − V (qn )] + . . . h ¯ n=1 2ǫs

(

"

#)

[1 + C(qn , ∆qn )],

where 1 + C is a correction factor arising from the three-step transformation 1 + C ≡ (1 + Cmeas )(1 + Cf )(1 + Cact ).

(14.39)

Dropping irrelevant higher orders in ∆q, the three contributions on the right-hand side have the following origins: The transformation of the measure (14.35) gives rise to the time slicing correction 1 Cmeas = −¯ ee2 ∆q + e¯e3 (∆q)2 + . . . . 2


897

14.2 Derivation of the Effective Potential

The rearrangement of the f -factors in (14.37) produces 1 3λ − 4 2

Cf =

f′ 1 f ′′ − ∆q + (∆q)2 f 2 f

!"

3 3λ + 4 2

1 − 2

!

1 3λ − 4 2

!

f′ f

#

!2

(∆q)2 + . . . .

(14.41)

The transformation of the pseudotime-sliced kinetic term (14.34) yields Cact

i (∆q)2 f′ = M − e¯e2 − λ ∆q h ¯ 2ǫs f (

!





1 f′ 1 1 f ′′ +  e¯e3 + (¯ ee2 )2 + −λ + λ(λ + 1) 3 4 2 f f

  !2  ′  f  − λ¯ ee2  (∆q)2  f

M 2 (∆q)4 f′ − 2 e ¯ e − λ 2 f 2¯ h 4ǫ2s

!2

(∆q)2 + . . . .

(14.42)

We now calculate an equivalent kernel according to Section 11.2. The correction terms are evaluated perturbatively using the expectation values i¯ h M

2n

h(∆q) i0 =

!n

(2n − 1)!!.

(14.43)

First we find the expectation value (11.70). Listing only the relevant terms of order ǫs , we obtain hC∆qi0 = i¯ hǫs

"

1 3λ − −¯ ee2 − 4 2

!

f′ f′ 3 + e¯e2 − λ f 2 f

!#

.

(14.44)

The λ-terms cancel each other identically. The remainder vanishes upon using the relation (14.15), which reads in the present notation e21 = f,

(14.45)

implying that 2e1 e2 = f ′ ,

2¯ ee2 = f ′ /f,

(14.46)

and yielding indeed hC∆qi0 ≡ 0. We now turn to the expectation hCi0 which determines the effective potential via Eq. (11.47). By differentiating the second equation in (14.46), we see that h

i

ee2 )2 + e¯e3 . f ′′ /f = 2 (¯

(14.47)

By expressing f and f ′′ in Eqs. (14.41) and (14.42) in terms of the e-functions, we obtain ! o 1 3λ n Cf = − −2¯ ee2 ∆q + [(¯ ee2 )2 + e¯e3 ] (∆q)2 4 2 ! ! 3 3λ 1 3λ −2 + − (¯ ee2 )2 (∆q)2 + . . . , (14.48) 4 2 4 2

898


M (∆q)2 Cact = i − (1 − 2λ)¯ ee2 ∆q h ¯ 2ǫs 1 1 + e¯e3 + (¯ ee2 )2 − λ(¯ ee2 + e¯e3 ) + 2λ(λ + 1)(¯ ee2 )2 − 2λ(¯ ee2 )2 (∆q)2 3 4 M 2 (∆q)4 − 2 (1 − 2λ)2 (¯ ee2 )2 (∆q)2 + . . . . (14.49) 2¯ h 4ǫ2s

After forming the product (14.39), the total correction reads 1 iM C = e¯e2 − λ ∆q − (∆q)2 − 3 2 h ¯ ǫs 7 M 1 1 7 2 9 2 2 +(¯ ee2 ) λ− λ− (∆q) + i 4λ − λ + (∆q)4 2 6 2 2 8 h ¯ ǫs # 2 1 1 M2 6 − λ− (∆q) 2 2 h ¯ 2 ǫ2s 3 1 1 1 M 2 4 +¯ ee3 − λ − (∆q) − λ− i (∆q) + . . . . (14.50) 2 2 2 3 h ¯ ǫs

Using (14.43), we find the expectation to the relevant order ǫs : ǫs h ¯ 1 3 hCi0 = − e¯e3 − (¯ ee2 )2 . M 4 8

(14.51)

It amounts to an effective potential Veff

i¯ h2 1 3 e¯e3 − (¯ ee2 )2 . = − M 4 8

(14.52)

By inserting (14.32) and (14.33), this turns into the expression (14.18) which we wanted to derive. In summary we have shown that the kernel in (14.38) +1 i NX M K (∆q) = q exp (∆qn )2 + ǫs Ef (qn ) h ¯ 2ǫ s 2πiǫs h ¯ /M n=1 ǫs

1

(

)

[1 + C] (14.53)

can be replaced by the simpler equivalent kernel +1 i NX M K (∆q) = q exp (∆qn )2 + ǫs Ef (qn ) − ǫs Veff h ¯ 2ǫ s 2πiǫs h ¯ /M n=1 ǫs

1

(

)

, (14.54)

in which the correction factor 1 + C is accounted for by the effective potential Veff of Eq. (14.18). This result is independent of the splitting parameter λ [1]. The same result emerges, after a lengthier algebra, for a completely general splitting of the regulating function f (x) into a product fl (x)fr (x). H. Kleinert, PATH INTEGRALS

899

14.3 Comparison with Schr¨ odinger Quantum Mechanics

14.3

Comparison with Schr¨ odinger Quantum Mechanics

The DK transformation of the action (14.11) into the action (14.19) has of course a correspondence in Schrödinger quantum mechanics. In analogy with the introduction of the pseudotime evolution amplitude (14.8), we multiply the Schrödinger equation h ¯2 2 − h∂t ψ(x, t) ∂ − E ψ(x, t) = i¯ 2M x

"

#

(14.55)

from the left by an arbitrary regulating function fl (x), and obtain h ¯2 − h∂t ψf (x, t), fl (x)∂x2 fr (x) − Ef (x) ψf (x, t) = f (x)i¯ 2M

"

#

(14.56)

with the transformed wave function ψf (x, t) ≡ fr (x)−1 ψ(x, t). After the coordinate transformation (14.14), we arrive at 

h ¯ 1 − fl (q) ′ ∂q 2M h (q)

!2



fr (q) − Ef (q) ψf (q, t) = f (q)i¯ h∂t ψf (x, t),

(14.57)

having used the notation f (q) ≡ f (h(q)) as in (14.17). Inserting (14.15), the Schrödinger equation becomes h ¯ 2 −1 h′′ 2 − f (q) ∂q − ′ ∂q fr (q) − Ef (q) ψf (q, t) = f (q)i¯ h∂t ψf (q, t). (14.58) 2M r h

"

!

#

After going from ψf (q, t) to a new wave function −1/4

φ(q, t) = fr3/4 (q)fl

(q)ψf (q, t)

related to the initial one by ψ(x, t) ≡fr (q)ψf (q, t)= f 1/4 (q)φ(q, t), the Schrödinger equation takes the form h ¯ 2 ′ −1/2 2 h′′ − h (q) ∂q − ′ ∂q h′ (q)1/2 − Ef (q) φ(q, t) 2M h 1 2 = − ∂ + Veff − Ef (q) φ(q, t) = f (q)i¯ h∂t φ(q, t), 2M q

"

!

#

(14.59)

where Veff is precisely the effective potential (14.18). The operator f (q)∂t on the right-hand side is equal to the pseudotime derivative ∂s .

900

14.4


Applications

We now present some typical solutions of path integrals via the DK method. The initial fixed-energy amplitudes will all have the generic action AE =

Z

M 2 dt x˙ (t) − V (x) + E , 2

(14.60)

with different potentials V (x) which usually do not allow for a naive time slicing. The associated path integrals are known from certain projections of Euclidean path integrals. In the sequel, we omit the subscript E for brevity (since we want to use its place for another subscript referring to the potential under consideration). The solution follows the general two-step procedure described in Section 14.4.

14.4.1

Radial Harmonic Oscillator and Morse System

Consider the action of a harmonic oscillator in D dimensions with an angular momentum lO at a fixed energy EO : AO =

Z

M 2 µ2 − 1/4 M 2 2 dt r˙ − h ¯2 O − ω r + EO . 2 2Mr 2 2 "

#

(14.61)

Here µO is an abbreviation for µO = DO /2 − 1 + lO

(14.62)

[recall (8.137)], DO denotes the dimension, and lO the orbital angular momentum of the system. The subscript O indicates that we are dealing with the harmonic oscillator. A free particle is described by the ω → 0 -limit of this action. Due to the centrifugal barrier, the time evolution amplitude possesses only a complicated time-sliced path integral involving Bessel functions. According to the rule (8.139), the centrifugal barrier requires the regularization ǫh ¯

2 2 µO

− 1/4 M − − −→ i¯ h log I˜µO rn rn−1 . 2 2Mrn i¯hǫ

(14.63)

This smoothens the small-r fluctuations and prevents a path collapse in the Euclidean path integral with µO = 0. The time-sliced path integral can then be solved using the formula (8.14). The final amplitude is obtained most simply, however, by solving the harmonic oscillator in DO Cartesian coordinates, and by projecting the result into a state of fixed angular momentum lO . The result was given in Eq. (9.32), and reads for rb > ra 1 1 Γ((1 + µ)/2−ν) Mω 2 Mω 2 (rb |ra )EO ,lO = −i √ Wν,µ/2 rb Mν,µ/2 r , (14.64) ω rb ra Γ(µ + 1) h ¯ h ¯ a

where the parameters on the right-hand side are ν = νO ≡

EO , 2ω¯ h

µ = µO .


901

14.4 Applications

A stable pseudotime evolution amplitude exists after a path-dependent time transformation with the regulating function f (r) = r 2 .

(14.66)

The time-transformed Hamiltonian HO = r 2

p2 µ2 − 1/4 M 2 4 +h ¯2 O + ω r − EO r 2 M 2M 2

(14.67)

is free of the barrier singularity. Thus, when time-slicing the action 2 AfO=r

=

Z

∞

0

M r ′ 2 µ2O − 1/4 M 2 4 ds − − ω r + EO r 2 2 r2 2M 2

!

(14.68)

associated with HO , no Bessel functions are needed. Note that the factor 1/r2 accompanying r ′2 = [dr(s)/ds]2 does not produce additional problems. It merely diminishes the fluctuations at small r. However, the r-dependence of the kinetic term is undesirable for an evaluation of the time-sliced path integral. We therefore go over to a new coordinate x via the transformation r = h(x) ≡ ex ,

(14.69)

the transformation function h(x) being related to the regulating function f (x) by (14.15): h′2 = e2x = f (r) = r 2 .

(14.70)

The resulting effective potential (14.18) happens to be a constant: Veff



h ¯ 2  1 h′′′ 3 = − − M 4 h′ 8

h′′ h′

!2  

h ¯2 . = 8M

(14.71)

Together with this constant, the DK-transformed radial oscillator action becomes ADK O

=

Z

0

∞

M ′2 µ2 Mω 2 4x ds x − O − e + EO e2x . 2 2M 2 "

#

(14.72)

The effective potential (14.71) has changed the initial centrifugal barrier term from (µ2O − 1/4)/2M to µ2O /2M. We have omitted the pseudoenergy E since it is set equal to zero in the final DK relation (14.26). With the identifications M 2 ω , 2 B = −EO , h ¯ 2 µ2O C = + EM , 2M A =

(14.73)

902


the action (14.72) goes over into AM =

Z

∞

0

ds

M ′2 x − (VM − EM ) . 2

(14.74)

This is the action for the so-called Morse potential VM (x) = Ae4x + Be2x + C.

(14.75)

Its fixed-energy amplitude (xb |xa )EM =

Z

∞

0

dS

Z

Dx(s)eiAM /¯h

(14.76)

is therefore equivalent to the radial amplitude of the oscillator (14.64) via the DK relation (14.26), which now reads (rb |ra )EO ,l = e(xb +xa )/2 (xb |xa )EM ,

(14.77)

where r = ex .

14.4.2

Radial Coulomb System and Morse System

By a similar argument, the completely different path integral of the radial Coulomb system can be shown to be DK-equivalent to the path integral of the Morse potential. The action is Z

AC =

M 2 µ2 − 1/4 e2 dt r˙ − h ¯2 C + + EC , 2 2Mr 2 r "

#

(14.78)

where µC = DC /2 − 1 + lC .

(14.79)

For e2 = 0, the action describes a free particle moving in a centrifugal barrier potential. As in the previous example, the action (14.78) does not lead to a timesliced amplitude of the Feynman type, but involves Bessel functions. We must again remove the barrier via a path-dependent time transformation with f (r) = r 2

(14.80)

by introducing the pseudotime s satisfying dt = ds r 2(s). This leads to the timetransformed action 2 AfC =r

=

Z

0

∞

2 M r ′2 2 µC − 1/4 ds − h ¯ + e2 r + EC r 2 . 2 r2 2M

"

#

(14.81)

To bring the kinetic term to the standard form, we change the variable r to x via r = ex .


903

14.4 Applications

This introduces the same effective potential as in (14.71), h ¯2 1 Veff = , (14.83) M8 canceling the 1/4-term in the former centrifugal barrier. Thus we arrive at the DK transform of the radial Coulomb action # " Z ∞ 2 M µ ADK = ds x′2 − h ¯ 2 C + e2 ex + EC e2x . (14.84) C 2 2M 0 A trivial change of variables x = 2¯ x, ¯ /4, M = M µC = 2¯ µ,

(14.85)

brings this to the form ADK C

=

Z

0

∞

¯ µ ¯2 M x¯′2 − h ¯ 2 ¯ + e2 e2¯x + EC e4¯x , ds 2 2M #

"

(14.86)

and establishes contact with the Morse action (14.74). Upon replacing x¯ by x we see that 1 (rb |ra )EC ,lC = e(xb +xa ) (xb |xa )EM , (14.87) 2 with r = e2x . The factor 1/2 accounts for the fact that the normalized states are related by |xi = |¯ xi/2. The identification of the parameters is now A = −EC , B = −e2 , µ ¯2 C = h ¯ 2 ¯ + EM . 2M

14.4.3

(14.88)

Equivalence of Radial Coulomb System and Radial Oscillator

Since the radial oscillator and the radial Coulomb system are both DK-equivalent to a Morse system, they are DK-equivalent to each other. The relation between the parameters is MO = 4MC , µO = 2µC , EO = e2 ,

−

MO 2 ω = EC , 2 √ rO = rC .

(14.89)

904


We have added subscripts O, C also to the masses M to emphasize the systems to which they belong. The relation µO = 2µC implies DO /2 − 1 + lO = 2(DC /2 − 1 + lC )

(14.90)

for all dimensions and angular momenta of the two systems. Due to the square root √ relation rO = rC , the orbital angular momenta satisfy lO = 2lC .

(14.91)

DO = 2DC − 2.

(14.92)

For the dimensions, this implies

In the cases DC = 2 and 3, there is complete agreement with Chapter 13 where the dimensions of the DK-equivalent oscillators were 2 and 4, respectively. To relate the amplitudes with each other we find it useful to keep the notation as close as possible to that of Chapter 13 and denote the radial coordinate of the radial oscillator by u. Then the DK relation for the pseudotime evolution amplitudes states that (rb |ra )EC ,µC =

1√ ub ua (ub |ua )EO ,µO , 2

(14.93)

with the right-hand side given by (14.64) (after replacing r by u, MO by 4MC , and MO ωu2/¯ h by 2κr). Note once more that the prefactor on the right-hand side has a dimension opposite to what one might have expected from the quantum-mechanical completeness relation Z

∞

0

dr|rihr| = 1,

(14.94)

Z

(14.95)

whose u-space version reads Z

∞

0

du 2u|rihr| =

du|uihu| = 1.

As explained in Section 14.1, the reason lies in the different dimensions (by a factor r) of the pseudotimes over which the evolution amplitudes are integrated when going to the fixed-energy amplitudes. A further factor 1/4 contained in (14.93) is due to the mass relation MO = 4MC . Let us check the relation (14.93) for DC = 3. The fixed-energy amplitude of the Coulomb system has the partial-wave expansion (xb |xa )EC =

∞ X

lC X

lC =0 m=−lC

1 (rb |ra )EC ,lC YlC m (θb , ϕb )Yl∗C m (θa , ϕa ). rb ra

(14.96)


905

14.4 Applications

The four-dimensional oscillator, on the other hand, has (~ub |~ua)EO =

∞ X

(ub |ua)EO ,lO

(14.97)

lO =0

l /2

O X lO + 1 lO /2 ∗ × D lO /2 (ϕn , θn , γn )Dm (ϕn−1 , θn−1 , γn−1). 1 m2 2π 2 m1 ,m2 =−lO /2 m1 m2

We now take Eq. (13.122), (xb |xa )EC =

∞

Z

and observe that the integral l /2

0

dSeie

R 4π 0

2 S/¯ h

1 16

Z

0

4π

dγa (~ubS|~ua 0),

(14.98)

dγa over the sum of angular wave functions

O X lO + 1 dlO /2 (θb )dlmO1/2m2 (θa )eim1 (ϕb −ϕa )+im2 (γb −γa ) 2π 2 m1 ,m2 =−lO /2 m1 m2

(14.99)

produces a sum lO /2

8

X

YlO /2,m,0 (θb , φb )Yl∗O /2,m,0 (θa , φa ),

(14.100)

m

with the spherical harmonics YlO /2,m (θ, φ) =

s

lO + 1 imφ lO /2 e dm,0 (θ). 4π

(14.101)

Only even lO -values survive the integration, and we identify lC = lO /2 Recalling the radial amplitude of the harmonic oscillator (9.32), we find from (14.93) the radial amplitude of the Coulomb system in any dimension DC for rb > ra : (rb |ra )EC ,lC = −i

MC Γ(−ν + lC + (DC − 1)/2) Wν,lC +DC /2−1 (2κrb )Mν,lC +DC /2−1 (2κra ), h ¯κ (2lC + DC − 2)! (14.102)

q

q

2

where κ = −2ME/¯ h and ν = −e4 MC /2¯ h2 EC as in (13.39) and (13.40). For DC = 3, this agrees with (13.206). The full DC -dimensional amplitude is given by the sum over partial waves (xb |xa )EC ,lC =

1 (rb ra )

(DC −1)/2

∞ X l=0

(rb |ra )EC ,lC

X

∗ Ylm (ˆ xb )Ylm (ˆ xa ),

m

which becomes with (8.125) (xb |xa )EC ,lC =

1 (rb ra )

(DC −1)/2

∞ X

lC =0

(rb |ra )EC ,lC

2lC + DC − 2 1 (DC /2−1) C (cos ∆ϑn ). DC − 2 SDC lC

(14.103)

906


It is easy to perform the sum if we make use of an integral representation of the radial amplitude obtained by DK-transforming the integral representation (9.25) of the radial oscillator amplitude. Replacing the imaginary time by the new variable of integration ̺ = e−2ω(τb −τa ) , the radial variables r by u, and the oscillator mass M by MO to match the notation of Chapter 13, the amplitude (9.25) can be rewritten as √ ! Z 1 1+̺ 2 ̺ d̺ −ν −κ(u2b +u2a ) 1−̺ MO √ (ub |ua )EO ,lO = −i ub ua ̺ e IlO +DO /2−1 2κub ua ,(14.104) h ¯ 1−̺ 0 2̺ with κ≡

MO ω , 2¯ h

ν ≡ EO /2¯ hω.

(14.105)

From the DK relation (14.93) we obtain (rb |ra )lC ,EC and insert it into (14.103). Then we recall the summation formula (suppressing all subscripts C)

1 kz 2

D/2−1/2

ID/2−3/2 (kz) = k

D−2

∞ X

1 Γ(l + D − 2) (2l + D − 2) l=0 l! Γ(D/2 − 1/2)

×F (−l, l + D − 2; D/2 − 1/2; (1 + k 2 )/2)(−)l I2l+D−2 (z),

(14.106)

which follows from Eq. (13.200) for ν = D/2 − 3/2 and µ = D − 2. After expressing D/2−1 with the help of the right-hand side in terms of the Gegenbauer polynomial Cl (8.105), the summation formula becomes 1 1 D/2−1/2 2 (2π)

D−2

z ID/2−3/2 (kz)/(kz)D/2−3/2 2 ∞ X 2lC + D − 2 1 (D/2−1) = ClC ((1 + k 2 )/2)I2lC +D−2 (z). D − 2 SD lC =0

(14.107)

Setting √ 2 ̺ z ≡ 2κub ua , 1−̺

k ≡ cos(ϑ/2),

(14.108)

the sum over the partial waves in (14.103) is easily performed, and we obtain for the fixed-energy amplitude of the Coulomb system in D dimensions the generalization of the integral representations (13.43) and (13.128) in two and three dimensions: 1 M κD−2 d̺ ̺−ν h ¯ (2π)(D−1)/2 0 (1 − ̺)2 √ !(D−3)/2 1+̺ 2 ̺ e−κ 1−̺ (rb +ra ) ID/2−3/2 (kz) /(kz)D/2−3/2 , 1−̺

(xb |xa )E = − i × where

Z

√ 2 ̺q kz = 2κ (rb ra + xb xa )/2, 1−̺

(14.109)


907

14.4 Applications

and κ, ν are the Coulomb parameters (13.40). By changing the integration variable to ζ = (1 + ̺)/(1 − ρ) as in (13.44), the integral in (14.109) is transformed into a contour integral encircling the cut from ζ = 1 to ∞ in the clockwise sense. Then the amplitude reads [2] (xb |xa )E = −i ×

Z

C

κD−2 M πeiπ(ν−D/2+3/2) 2¯ h (2π)(D−1)/2 sin[π(ν − D/2 + 3/2)]

(14.111)

dζ (ζ − 1)−ν+D/2−3/2 (ζ + 1)ν+D/2−3/2 e−κζ(rb +ra ) ID/2−3/2 (z) /z D/2−3/2 . 2πi

This expression generalizes the integral representations (13.48) and (13.130) for DC = 2 and DC = 3, respectively. It is worth emphasizing that due to the catastrophic centrifugal barriers, there is no way of establishing this relation for the time-sliced radial amplitudes without the intermediate Morse potential. This has been attempted in the literature [3]by using the DK transformation with the regulating function f (r) = r and a pseudotime s satisfying dt = ds r(s) (which were successful in two and three dimensions). Although this transformation removes the Coulomb singularity, it weakens the centrifugal barrier insufficiently to a still catastrophic 1/r-singularity. Let us exhibit the place where such an attempt fails. The starting point is the pseudotime-sliced amplitude (13.8), rbλ ra1−λ

hxb |UÊ (s)|xa i ≈ q D/2 2πǫs h ¯ r 1−λ r λ /M

with the action AN E

2

= −(N + 1)ǫs e +

N +1 X n=1

"

 Z N Y   q

n=1

D

d ∆xn

2πǫs h ¯ rn /M



 −AN h E /¯ , De

M (xn − xn−1 )2 λ + ǫs Ern1−λ rn−1 . λ 2ǫs rn1−λ rn−1 #

(14.112)

(14.113)

In contrast to Chapter 13, we work here conveniently with an imaginary-time. In any dimension D, the amplitude has the angular decomposition X 1 ∗ hrb |UÊ (s)|ra il Ylm (ˆ xb )Ylm (ˆ xa ). (rb ra )D−1/2

hxb |UÊ (s)|xa i =

(14.114)

The action for the radial amplitude is obtained by decomposing 2 AN E = −(N + 1)ǫs e +

N +1 X n=1

"

M λ 2ǫs rn1−λ rn−1

#

2 (rn2 +rn−1 −2rn rn−1 cos ϑn ) + ǫs Ern ,(14.115)

λ where ϑn is the angle between xn and xn−1 . We have replaced Ern1−λ rn−1 by Ern 2 since the difference is of order ǫs and thus negligible. We now go through the same steps as in Section 8.5. For an individual time slice, the ϑn -part of the exponential is expanded as ∞ X X M λ 1−λ ∗ exp rn rn−1 cos ϑn = eh a ˜l (h) Ylm (ˆ xb )Ylm (ˆ xa ), ǫs m l=0

(14.116)

908


with 2π al (h) = h

(D−1)/2

I˜D/2−1+l (h),

h=

M λ 1−λ r r h ¯ ǫs n n−1

(14.117)

[recall (8.129) and (8.101)]: The radial part of the propagator is then rb λ ra1−λ

hrb |UÊ (s)|ra il ≈ q 2π¯ hǫs rb 1−λ raλ /M

NY +1 n=2

with the radial action AN E

2

= −(N +1)ǫs e +

N +1 X n=1

"

 

Z

−1/2



d∆rn rn−1  −AN /¯h q e E , 2π¯ hǫs /M

(14.118)

M (rn − rn−1 )2 M λ 1−λ −h ¯ log I˜D/2−1+l r r −ǫs Ern . λ 1−λ 2ǫs rn rn−1 h ¯ ǫs n n−1 (14.119)

#

At this place we simplify the calculation by choosing the symmetric splitting parameter λ = 1/2. Going over to square root coordinates √ u n = rn , (14.120) we calculate ∆rn = = ∂∆rn = ∂∆un −1/2 rn−1 =

(un + un−1 )∆un , 2un (1 − ∆un /2un )∆un , 2un (1 − ∆un /un ), −1 u−1 n (1 − ∆un /un ) ,

transforming the measure of integration into √ N Z ub ua Y d∆un q

2π¯ hǫs /M

n=1

q

2π¯ hǫs /M

(14.121)

.

(14.122)

Note that there are no higher ∆un correction terms. The kinetic energy is 4¯ un2 (∆un )2 4 1 (∆un )4 = (∆un )2 + + ... . 2ǫs un un−1 2ǫs 4 un2 "

#

(14.123)

The (∆un )4 -term can be replaced right away by its expectation value and renders an effective potential Veff (un2 ) = ǫs h ¯2

1 3 . 2 · 4M 4un2

(14.124)

The radial amplitude becomes simply   √ NY +1 Z ∞ u u du 2 N b a n   e−AE /¯h , q hrb |UÊ (s)|ra il ≈ q 2π¯ hǫs /M n=2 0 2π¯ hǫs /M

(14.125)


909

14.4 Applications

with AN E

= −(N + 1)ǫs +

N +1 X n=1

"

4M (∆un )2 M + Veff (un2 ) − h ¯ log I˜D/2−1+l un un−1 . 2 2ǫs h ¯ ǫs (14.126)

#

Due to the 1/un2 -singularity in Veff (un2 ), the time-sliced path integral does not exist. Apart from the ǫs /un2 -term, there should be infinitely many terms of increasing order of the type (ǫs /un2 )2 , . . . , whose resummation is needed to obtain the correct threshold small-un behavior of the amplitude as discussed in Section 8.2. To have the usual kinetic term of the harmonic oscillator MO (∆un )2 /2ǫs , we must identify 4M with the oscillator mass MO [called µ in (13.27); see also (14.89) with MC ≡ M], MO = 4M.

(14.127)

The centrifugal barrier in (14.126) resides in −¯ h log I˜D/2−1+l

!

MO /4 3 un un−1 + ǫs h ¯2 + ... , h ¯ ǫs 8MO u2n

(14.128)

and is given by h ¯2 4 ǫs 2MO un un−1

"

D −1+l 2

2

#

1 3 − + ǫs h ¯2 + ... . 4 8MO u2n

(14.129)

This can be rewritten more explicitly as h ¯ 1 1 ǫs (DC − 2 + 2lC )2 − , 2MO un un−1 4

(14.130)

where we have added the subscript C to D to record its being the dimension of the Coulomb system. The expression in parentheses is identified with the parameter µO of the harmonic oscillator, which appears in the subscript of the Bessel function in (8.139). This implies µO = 2µC ,

(14.131)

in agreement with the relation (14.90). Indeed, the higher terms in the expansion (14.129) must all conspire to sum up to the Bessel-regulated centrifugal barrier in the time-sliced radial amplitude of the harmonic oscillator MO un un−1 . −¯ h log I˜D−2+2l h ¯ ǫs

(14.132)

This is quite hard to verify term by term, although it must happen. Using the stronger regulating function f = r 2 , these difficulties are avoided. Instead of the pseudotime evolution amplitude (14.112), we have 2

rb ra2−2λ

hxE |Uê (s)|xa i ≈ q D/2 2πǫsh ¯ rb 2−2λ ra2λ /M

N Y

n=1

 Z   q

D

d ∆xn

2πǫs h ¯ rn2 /M



 −AfEN /¯h , (14.133) De

910


with the time-sliced transformed action AfEN

2

= −(N + 1)ǫs e +

N +1 X n=1

"

M

(rn2 2λ 2ǫs rn2−2λ rn−1

+

2 rn−1

− 2rn rn−1 cos ϑn ) −

ǫs Ern2

#

.

(14.134)

For λ = 1/2, the cos ∆ϑn -term is now free of the radial variables rn , rn−1 , rn , rn−1, and the angular decomposition of the amplitude as in (14.114)–(14.119) gives the radial amplitude with a time-sliced action AfEN

2

= −(N +1)ǫs e +

N +1 X n=1

M (rn −rn−1 )2 M −¯ h log I˜D/2−1+l −ǫs Ern2 . (14.135) 2ǫs rn rn−1 h ¯ ǫs

"

#

Since rn , rn−1 are absent in the Bessel function, the limit of small ǫs is now uniform in the integration variables rn and the logarithmic term in the energy can directly be replaced by ǫs

i h ¯ h (DC /2 − 1 + lC )2 − 1/4 , 2MC

(14.136)

where we have added the subscripts C, for clarity. To perform the integration over the rn variables, one goes over to new coordinates x with r = h(x) = ex .

(14.137)

The measure of integration is √

rb ra

q

2π¯ hǫs /M

N +1

NY +1 Z n=2

d∆rn . rn−1

(14.138)

Expanding 1/rn−1 around the postpoint rn gives 1 rn−1

1 = rn

rn rn−1

!

=

e∆xn . exn

(14.139)

We now write (dropping subscripts n) ∆r = ex − ex−∆x = ex (1 − e−∆x ),

(14.140)

∂∆r = ex e−∆x . ∂∆x

(14.141)

and find the Jacobian

In the x-coordinates, the measure becomes simply e(xb +xa )/2 q

2π¯ hǫs /M

N +1

NY +1 Z

d∆xn .

(14.142)

n=2


911

14.4 Applications

The kinetic term in the action turns into AN E =

N +1 X n=1

M (1 − cos ∆xn ), ǫs

(14.143)

and has the expansions AN E =

N +1 X n=1

1 M (∆x)2 − (∆xn )4 + . . . . 2ǫs 12

(14.144)

The higher-order terms contribute with higher powers of ǫs uniformly in x. They can be treated as usual. This is why the path-dependent time transformation of the radial Coulomb system to a radial oscillator with the regulating function f = r 2 is free of problems.

14.4.4

Angular Barrier near Sphere, and Rosen-Morse Potential

For another application of the solution method, consider the path integral for a mass point near the surface of a sphere in three dimensions, projected into a state of fixed azimuthal angular momentum m = 0, ±1, ±2, . . . . The projection generates an angular barrier ∝ (m2 −1/4)/ sin2 θ which is a potential of the Pöschl-Teller type. With µ = Mr 2 , the real-time action is APT =

Z

µ h ¯2 h ¯ 2 “ m2 − 1/4 ” − dt θ˙2 + + EPT . 2 8µ 2µ sin2 θ "

#

(14.145)

The quotation marks are defined in analogy with those of the centrifugal barrier in Eq. (8.139). The precise meaning is given by the proper time-sliced expression in Eq. (8.174) whose limiting form for narrow time slices is (8.176). After an analytic continuation of the parameter m to arbitrary real numbers µ, the resulting amplitude was given in (8.186). In the sequel we refrain from using the symbol µ for the noninteger m-values to avoid confusion with the mass parameter µ. The spectral representation of the associated fixed-energy amplitude is easily written down; it arises by simply integrating (8.186) over −idτb and reads (θb |θa )m,EPT =

q

∞ X

i¯ h (14.146) ¯ 2 L2 /2µ n=0 EPT − h 2n + 2m + 1 (n + 2m)! −m −m × Pn+m (cos θb )Pn+m (cos θ a ), 2 n!

sin θb sin θa

where L2 = l(l + 1) with l = n + m [recall (8.224) for D = 3]. The sum over n can be done using the so-called Sommerfeld-Watson transformation [4]. The sum is re-expressed as a contour integral in the complex n-plane and deformed in such a way that only the Regge poles at 1 n + m = l = l(EPT ) ≡ − + 2

s

1 2µEPT + 4 h ¯2

(14.147)

912


contribute, with both signs of the square root. The result for θb > θa is [5] (θb |θa )m,EPT =

q

sin θb sin θa

−iµ Γ(m − l(EPT ))Γ(l(EPT ) + m + 1) h ¯ −m −m (−cos θb )Pl(E (cos θa ). × Pl(E PT ) PT )

(14.148)

Here we shall consider m as a free parameter characterizing the interaction strength of the Pöschl-Teller potential [6] VPT (θ) =

h ¯ 2 m2 . 2µ sin2 θ

(14.149)

The regulating function removing the angular barrier is f (θ) = sin2 θ,

(14.150)

and the time-transformed action reads with dt = ds sin2 θ(s) =sin2 θ AfPT

=

Z

0

∞

µ h ¯2 2 h ¯2 2 ′2 ds sin θ − (m − 1/4) + EPT sin2 θ . (14.151) 2 θ + 8µ 2µ 2 sin θ "

#

We now bring the kinetic term to the conventional form by the variable change sin θ =

1 , cosh x

cos θ = − tanh x,

(14.152)

which maps the interval θ ∈ (0, π) into x ∈ (−∞, ∞). Then we have h′ (x) = sin θ =

1 . cosh x

(14.153)

1 1 − 2 tanh2 x , cosh x

(14.154)

Forming the higher derivatives h′′ (x) = −

tanh x , cosh x

h′′′ (x) = −

the effective potential is found to be Veff

h ¯2 1 = 1+ . 8µ cosh2 x

(14.155)

The DK-transformed action is therefore ADK PT

=

Z

0

∞

µ ′2 h ¯ 2 m2 1 ds x − + EPT . 2 2µ cosh2 x "

#

(14.156)

It describes the motion of a mass point in a smooth potential well known as the Rosen-Morse potential (also called the modified Pöschl-Teller potential ) [7]. The standard parametrization is h ¯ 2 s(s + 1) VRM (x) = − . 2µ cosh2 x


913

14.4 Applications

This corresponds to l(EPT ) in (14.147) having the value s. The energy of the RosenMorse potential determines the parameter m in the action (14.156), and we identify m = m(ERM ) =

q

−2µERM /¯ h2 .

(14.158)

It is obvious that the time-sliced amplitude of the Rosen-Morse potential has no path collapse problems. Its fixed-energy amplitude is thus DK-equivalent to the Pöschl-Teller amplitude (14.148), with the precise relation being (θb |θa )m,EPT =

q

sin θb sin θa (xb |xa )m,ERM ,

(14.159)

where tanh x = − cos θ, θ ∈ (0, π), x ∈ (−∞, ∞). Inserting (14.148), the amplitude of the Rosen-Morse system reads explicitly −iµ Γ(m(ERM ) − s)Γ(s + m(ERM ) + 1) h ¯ × Ps−m(ERM ) (tanh xb )Ps−m(ERM ) (− tanh xa ).

(xb |xa )m(ERM ) =

(14.160)

The bound states lie at the poles of the first Gamma function where m(ERM ) = s − n,

n = 0, 1, 2, . . . , [s],

(14.161)

with [s] denoting the largest integer number ≤ s. From the residues we extract the normalized wave functions [5] ψn (x) =

q

Γ(2s − n + 1)(s − n)/nPsn−s (tanh x).

(14.162)

For noninteger values of s, these are not polynomials. However, the identity between hypergeometric functions (1.450) F (a, b; c; z) = (1 − z)c−a−b F (c − a, c − b; c; z)

(14.163)

permits relating them to polynomials: Psn−s (tanh x) =

2n−s 1 F (−n,1 + 2s − n;s − n + 1;(1−tanh x)/2) . Γ(s − n + 1) coshs−n x (14.164)

The continuum wave functions are obtained from (14.162) by an appropriate analytic continuation of m to −ik. This amounts to replacing n by s + ik.

14.4.5

Angular Barrier near Four-Dimensional Sphere, and General Rosen-Morse Potential

Let us extend the previous path integral of a mass point moving near the surface of a sphere from D = 3 to D = 4 dimensions. By projecting the amplitude into a state of fixed azimuthal angular momenta m1 and m2 , an angular barrier is generated in

914


the Euler angle β proportional to (m21 + m22 − 1/4 − 2m1 m2 cos θ)/ sin2 β. This is again a potential of the Pöschl-Teller type, although of a more general form to be denoted by a subscript PT ′ . The action (8.211) is, with µ = Mr 2 /4, APT ′ =

Z

µ h ¯2 h ¯ 2 “ m21 + m22 − 2m1 m2 cos β − 1/4 ” dt β˙ 2 + + EPT ′ , (14.165) − 2 32µ 2µ sin2 β "

#

where the quotation marks indicate the need to regularize the angular barrier via Bessel functions as specified in (8.207). The projected amplitude was given in Eq. (8.202) and continued to arbitrary real values of m1 = µ1 , m2 = µ2 with µ1 ≥ µ2 ≥ 0 in (8.212). As in subsection 14.4.4, we shall also use the parameters m1 , m2 when they have noninteger values. The most general Pöschl-Teller potential h ¯ 2 s1 (s1 + 1) s2 (s2 + 1) VPT ′ (β) = + 2µ sin2 (β/2) cos2 (β/2) "

#

(14.166)

can easily be mapped onto the above angular barrier, up to a trivial additive constant. The fixed-energy amplitude is obtained directly from Eq. (8.212) by an integration over −idτb . It reads for m1 ≥ m2 (θb |θa )m1 ,m2 ,EPT ′ =

q

sin θb sin θa i¯ h 2n + 2m1 + 1 n+m1 1 dm1 ,m2 (θb )dn+m × m1 ,m2 (θa ), (14.167) 2 ′ 2 E − h ¯ L /8µ 2 n=0 PT ∞ X

where L2 is given by L2 = (l + 1)2 − 1/4 with l = 2n + 2m1 [recall (8.219) with (8.224)]. As in Eq. (14.146), the sum over n can be performed with the help of a Sommerfeld-Watson transformation by rewriting the sum as a contour integral in the complex n-plane. After deforming the contour in such a way that only the Regge poles at 2n + 2m1 = l = l(EPT ′ ) ≡ −1 + 2

s

1 2µEPT ′ + 16 h ¯2

(14.168)

contribute, with both signs of the square root, we find for θb > θa : (θb |θa )m1 ,m2 ,EPT ′ =

q

sin θb sin θa

−2iµ h ¯

(14.169)

1 l(EPT ′ )/2 PT ′ )/2 (θ ), × Γ(m1 −l(EPT ′ )/2)Γ(l(EPT ′ )/2−m1 +1) dm1 ,−m (θb − π)dl(E a m1 ,m2 2 2

with arbitrary real parameters m1 , m2 characterizing the interaction strength. The regulating function which removes the angular barrier is f (β) = sin2 β,


915

14.4 Applications

and the time-transformed action reads, with dt = ds sin2 β(s), =sin2 β AfPT ′

=

Z

0

∞

h ¯2 µ ′2 ds sin2 β 2 β + 32µ 2 sin β # 2 h ¯ 2 2 2 − (m + m2 − 1/4 − 2m1 m2 cos β) + EPT ′ sin β . 2µ 1 "

(14.171)

We now bring the kinetic term to the conventional form by the variable change sin β = ±

1 , cos β = − tanh x. cosh x

(14.172)

As in the previous case, this leads to the effective potential Veff

h ¯2 1 = 1+ . 8µ cosh2 x

(14.173)

The DK-transformed action is then µ h ¯2 2 3¯ h2 ds x′2 − (m1 + m22 + 2m1 m2 tanh x) + EPT ′ − 2 2µ 32µ "

!

#

1 = . 0 cosh2 x (14.174) It contains a smooth potential well near the origin known as the general Rosen-Morse potential [7]. A convenient general parametrization is ADK PT ′

Z

∞

h ¯2 s(s + 1) − VRM′ (x) = + 2c tanh(x) , 2µ cosh2 (x) "

#

(14.175)

which amounts to choosing EPT ′ =

h ¯2 [s(s + 1) + 3/32], 2µ

m1 m2 = c,

in (14.174). Inserting this into (14.168) makes l(EPT ′ )/2 equal to s. The energy of the general Rosen-Morse potential fixes the third parameter to ERM′ =

h ¯2 2 (m + c2 /m21 ). 2µ 1

(14.176)

The solution of this equation will be a function m1 (EPM′ ). Correspondingly, we define m2 (EPM′ ) ≡ c/m1 (EPM′ ). Feynman’s time-sliced amplitude certainly exists for this potential, and the fixedenergy amplitude is determined in terms of the angular-projected amplitude (14.169) of a mass point near the surface of a sphere which describes the motion in a general Pöschl-Teller potential. The relation is [8] (θb |θa )m1 ,m2 ,EPT ′ =

q

sin θb sin θa (xb |xa )m1 ,m2 ,ERM′ ,

(14.177)

916


with tanh x = − cos β, β ∈ (0, π), x ∈ (−∞, ∞). Explicitly we have −2iµ Γ(m1 (ERM′ ) − s)Γ(s − m1 (ERM′ ) + 1) (14.178) h ¯ 1 × dsm1 (ERM′ ),−m2 (ERM′ ) (θb (xb ) − π)dsm1 (ERM′ ),m2 (ERM′ ) (θa (xa )). 2

(xb |xa )EPT ′ =

The bound states lie at the poles of the first Gamma function. With the energydependent function m1 (ERM′ ) defined by (14.176), they are given by the solutions of the equation m1 (ERM′ ) = s − n,

n = 0, 1, . . . , [s] .

(14.179)

The residues in (14.178) render the normalized wave functions Ψn

v u 2 um (x) = t 1

− m22 Γ(s + 1 − m1 )n! m1 Γ(s + 1 − m2 )Γ(s + 1 + m2 )

(14.180)

× [ 12 (1 + tanh x)](m1 −m2 )/2 [ 12 (1 − tanh x)](m1 +m2 )/2 Pn(m1 −m2 ,m1 +m2 ) (− tanh x),

or, expressed in terms of hypergeometric functions, Ψn (x) =

v u 2 u m1 t

− m22 Γ(s + 1 + m1 )Γ(s + 1 − m2 ) m1 n!Γ(1 + m1 − m2 )2 Γ(s + 1 + m2 )

× [ 12 (1 + tanh x)](m1 −m2 )/2 [ 12 (1 − tanh x)](m1 +m2 )/2 × F (2s − n + 1, −n; 1 + m1 − m2 ; 12 (1 + tanh x)) ,

(14.181)

with m1 = s − n and m2 = c/m1 [9]. The continuum wave functions are obtained from these by an appropriate analytic continuation of m1 to complex values −ik satisfying the relation k 2 = −(m21 + c2 /m21 ) [compare (14.176)].

14.4.6

Hulth´ en Potential and General Rosen-Morse Potential

For a further application of the solution method, consider the path integral of a particle moving along the positive r-axis with the singular Hulthén potential VH (r) = g

1 , −1

(14.182)

er/a

where g and a are energy and length parameters. Note that this potential contains the Coulomb system in the limit a → ∞ at ag = e2 = fixed. The fixed-energy amplitude is controlled by the action AH =

Z

M 2 dt r˙ − VH (r) + EH . 2


917

14.4 Applications

The potential is singular at r = 0, and for g < 0, the Euclidean time-sliced amplitude does not exist due to path collapse. A regulating function which stabilizes the fluctuations is f (r) = 4(1 − e−r/a )2 .

(14.184)

The time-transformed action is therefore AfH

=

Z

r ′2 M ds − g 4e−r/a (1 − e−r/a )+ EH 4(1 − e−r/a )2 . (14.185) 2 4(1−e−r/a )2

∞

0

#

"

The coordinate transformation leading to a conventional kinetic energy in terms of the new variable x is found by solving the differential equation dr = h′ (x), dx

(14.186)

with h′ = The solution is

q

f = 2(1 − e−r/a ).

(14.187)

r = x + a log[2 cosh(x/a)] = log(e2x/a + 1), a

(14.188)

so that h′ (x) = 2

e2x/a ex/a = . e2x/a + 1 cosh(x/a)

(14.189)

The semi-axis r ∈ (0, ∞) is mapped into the entire x-axis. To find the effective potential we calculate the derivatives 1 1 1 ex/a [1 − tanh(x/a)], = a cosh2 (x/a) a cosh(x/a) 2 sinh x 2 ex/a h′′′ (x) = − 2 = − [tanh(x/a) − tan2 (x/a)], (14.190) 3 2 a cosh x a cosh(x/a) h′′ (x) =

and obtain 1 e−x/a h′′ 1 = = [1 − tanh(x/a)], (14.191) ′ h a cosh(x/a) a h′′′ 2 e−x/a sinh(x/a) 2 = − = − 2 tanh(x/a)[1 − tanh(x/a)], 2 ′ 2 h a a cosh (x/a) so that the effective potential becomes Veff

h ¯2 4 = 2 − 2 tanh(x/a) − . 2 2 8Ma cosh (x/a) "

#

(14.192)

918


After adding this to the time-transformed potential, the DK-transformed action is found to be ADK H

=

Z

∞

0

M ′2 h ¯2 ds x − g + EH − 2 2Ma2 (

!

1 cosh (x/a)

h ¯2 + 2EH + tanh(x/a) + 4Ma2 !

2

h ¯2 2EH − 4Ma2

!)

.

(14.193)

This is the action governing the fixed-energy amplitude of the general Rosen-Morse potential h ¯2 s(s + 1) − VRM′ (x) = + c tanh x . 2µ cosh2 x "

#

(14.194)

Since this potential is smooth, there exists a time-sliced path integral of the Feynman type. The relation between the fixed-energy amplitudes is (rb |ra )EH = e(xb +xa )/2a [cosh(xb /a) cosh(xa /a)]−1/2 (xb |xa )ERM′ ,

(14.195)

with r/a = log(e2x/a + 1) ∈ (0, ∞), x ∈ (−∞, ∞). The amplitude on the right-hand side is known from the last section; it is related to the amplitude for the motion of a mass point on the surface of a sphere in four dimensions, projected into a state of fixed azimuthal angular momenta m1 and m2 . Only a simple rescaling of x/a to x is necessary to make the relation explicit. In the literature, a solution of the time-sliced path integral with the action (14.183) has been attempted using a regulating function [10] f = a2 (er/a − 1).

(14.196)

This implies going to the new variables r = −2 log cos(β/2), a

(14.197)

so that 2

2

2

f = a tan (β/2) = a

"

#

1 −1 . cos2 (β/2)

(14.198)

Note that this does not lead to a solution of the time-sliced path integral, since the transformed potential is still singular. Indeed, with h′ = a tan(β/2), h′′ = a/[2 cos2 (β/2)], h′′′ = a sin(β/2)/[2 cos3 (β/2)], we would find the effective potential h ¯2 1 h ¯2 3 1 Veff (β) = (1 + 2 cos β) = − , (14.199) 2 2 8Ma2 sin β 32Ma2 sin (β/2) cos2 (β/2) "

#

and a transformed action A˜DK H =

Z

0

∞

Ma4 ′ 2 1 (ds/a ) β − g + EH − 1 + Veff (β) , (14.200) 2 2 cos (β/2) 2

(

"

#

)


14.5 D -Dimensional Systems

919

which is of the general Pöschl-Teller type (14.166). Due to the presence of the 1/ cos2 (β/2)-term, the Euclidean time evolution amplitude cannot be time-sliced. Only by starting from the particle near the surface of a sphere with the particular Bessel function regularization of (8.207), can a well-defined time-sliced amplitude be written down whose action looks like (14.200) in the continuum limit. It would be impossible, however, to invent this regularization when starting from the continuum action (14.200).

14.4.7

Extended Hulth´ en Potential and General Rosen-Morse Potential

The alert reader will have noticed that the regulating function (14.182) overkills the ga/r singularity of the Hulthén potential (14.182). In fact, we may add to the potential a term g′ (er/a − 1)2

∆VH =

(14.201)

without loosing the stability of the path integral. In the limit a → ∞, the extended potential contains the radial Coulomb system plus a centrifugal barrier, if we set ¯ 2 l(l + 1)/2M. The potential (14.201) adds to the ga = −e2 = const and g ′a2 = h time-transformed action (14.185) a term ∆AfH

=−

Z

∞

0

ds g ′4e−2r/a ,

(14.202)

which winds up in the final DK-transformed action as ∆ADK H

=−

Z

0

∞

"

#

1 ds g 2 − 2 tanh (x/a) − . 2 cosh (x/a) ′

2

(14.203)

Therefore, the extended Hulthén potential is again DK-equivalent to the general Rosen-Morse potential with the same relation (14.195) between the amplitudes, but with different relations between the constants.

14.5

D-Dimensional Systems

Let us now perform the path-dependent time transformation in D dimensions. The fixed-energy amplitude is given by the integral (xb |xa )E =

Z

0

∞

dShxb |UÊ (S)|xa i,

(14.204)

with the pseudotime evolution amplitude i ˆ − E)fr (x) |xa i. (14.205) hxb |UÊ (S)|xa i = fr (xb )fl (xa )hxb | exp − Sfl (x)(H h ¯

920


It has the time-sliced path integral hxb |UÊ (S)|xa i ≈

(14.206)

fr (xb )fl (xa )

q

2πiǫs h ¯ fl (xb )fr (xa )/M

D

N Y

n=1

 

Z

with the action N

A =

N +1 X n=1

(

dxn q

2πiǫs h ¯ fn /M



 exp

i N A , h ¯

M (∆xn )2 + ǫs [E − V (xn )]fl (xn )fr (xn−1 ) , 2ǫs fl (xn )fr (xn−1 ) )

(14.207)

where the integration measure contains the abbreviation fn ≡ f (rn ) = fl (xn )fr (xn ). The time-transformed measure of path integration reads fr (xb )fl (xa ) q


D

N Z Y

dD xn q

2πiǫs h ¯ fn /M

n=1

D.

(14.208)

By shifting the product index and the subscripts of fn by one unit, and by compensating for this with a prefactor, the integration measure in (14.27) acquires the postpoint form fr (xb )fl (xa ) q


D

v D u +1 Z u f (xb ) NY t

f (xa )

n=2

dD ∆xn q

2πiǫs h ¯ fn /M

D.

(14.209)

The integrals over each coordinate difference ∆xn = xn − xn−1 are done at fixed postpoint positions xn . To simplify the subsequent discussion, it is preferable to work only with the postpoint regularization in which fl (x) = f (x) and fr (x) ≡ 1. Then the measure becomes simply f (xa ) q

2πiǫs f (xa )¯ h/M

D

NY +1 Z n=2

dD ∆xn q

2πiǫs h ¯ fn /M

D.

(14.210)

We now introduce the coordinate transformation. In D dimensions it is given by xi = hi (q).

(14.211)

The differential mapping may be written as in Chapter 10 as dxi = ∂µ hi (q) = ei µ (q)dq µ .

(14.212)

The transformation of a single time slice in the path integral can be done following the discussion in Sections 10.3 and 10.4. This leads to the path integral (xb |xa )E ≈ q

f (qa ) 2πiǫs f (qa )¯ h/M

Z

D

0

∞

dS

  Z D 1/2 d ∆q g (q ) n n  iAtot /¯  h ,  q D e

NY +1 n=2

(14.213)

2πiǫs h ¯ fn /M


921

14.6 Path Integral of the Dionium Atom

with the total time-sliced action Atot =

N +1 X n=1

Aǫtot .

(14.214)

Each slice contains three terms Aǫtot = Aǫ + AǫJ + Aǫpot .

(14.215)

In the postpoint form, the first two terms were given in (13.158) and (13.159). They are equal to Aǫ + AǫJ =

M h ¯ h ¯2 gµν (q)∆q µ ∆q ν − i Γµ µ ν ∆q ν − ǫs f (Γµ µ ν )2 . 2ǫf 2 8M

(14.216)

The third term contains the effect of a potential and a vector potential as derived in (10.182). After the DK transformation, it reads Aǫpot = Aµ ∆q µ − iǫs f

14.6

h ¯ (Aν Γµ µν + Dµ Aµ ) − ǫs f V (q). 2M

(14.217)

Path Integral of the Dionium Atom

We now apply the generalized D-dimensional Duru-Kleinert transformation to the path integral of a dionium atom in three dimensions. This is a system of two particles with both electric and magnetic charges (e1 , g1 ) and (e2 , g2 ) [11]. Its Lagrangian for the relative motion reads L=

M 2 x˙ + A(x)x˙ − V (x), 2

(14.218)

where x is the distance vector pointing from the first to the second article, M the reduced mass, V (x) a Coulomb potential e2 V (x) = − , r

(14.219)

and A(x) the vector potential ˆz × x 1 1 A(x) = h ¯q − r r−z r+z

=h ¯q

(xˆ y − yˆ x)z . 2 r(x + y 2 )

(14.220)

The coupling constants are q ≡ −(e1 g2 − e2 g1 )/¯ hc and e2 ≡ −(e1 e2 + g1 g2 ). The vector potential (14.220) implies an obvious generalization of the magnetic monopole interaction (8.299) with an electric charge [recall Appendix 10A.3] The potenial If we take the coupling as and e2 ≡ −e1 e2 − g1 g2 in (14.220) we allow for the two particles to carry both electric and magnetic charges of the two particles, if we take for V (x) the potential V (x) = −

e2 . r

(14.221)

922


The hydrogen atom is a special case of the dionium atom with e1 = −e2 = e and q = 0, l0 = 0. An electron around a pure magnetic monopole has e1 = e, g2 = g, e2 = g1 = 0. In the vector potential (14.220) we have made use of the gauge freedom A → A(x) + ∇Λ(x) to enforce the transverse gauge ∇A(x) = 0. In addition, we have taken advantage of the extra monopole gauge invariance which allows us to choose the shape of the Dirac string that imports the magnetic flux to the monopoles. The field A(x) in (14.220) has two strings of equal strength importing the flux, one along the positive x3 -axis from minus infinity to the origin, the other along the negative x3 -axis from plus infinity to the origin. It is the average of the vector potentials (10A.59) and (10A.60). For the sake of generality, we shall assume the potential V (x) to contain an extra 2 1/r -potential: V (x) = −

h ¯ 2 l02 e2 . + r 2Mr 2

(14.222)

The extra potential is parametrized as a centrifugal barrier with an effective angular momentum h ¯ l0 . At the formal level, i.e., without worrying about path collapse and time slicing corrections, the amplitude has been derived in Ref. [12]. Here we reproduce the derivation and demonstrate, in addition, that the time slicing produces no corrections.

14.6.1

Formal Solution

We extend the action of the type (14.11) by a dummy fourth coordinate as in the Coulomb system and go over to ~u-coordinates depending on the radial coordinate √ u = r and the Euler angles θ, ϕ, γ as given in Eq. (13.97). Then the action reads M 2 2 M 4 ˙2 h ¯q e2 h ¯ 2 l02 A = dt 4u u˙ + u θ + ϕ˙ 2 + γ˙ 2 +2 γ˙ + ϕ ˙ cos θ − − +E . 2 2 Mu4 u2 2Mu4 (14.223) (

Z

"

!

#

)

By performing the Duru-Kleinert time reparametrization dt = ds r(s) and changing the mass to µ = 4M, the action takes the form A

DK

=

Z

0

∞

µ ′2 u2 ′2 4¯ hq 4¯ h2 l02 ds u + θ + ϕ′2 + γ ′2 +2 γ ′ + 2 ϕ′ cos θ − + Eu2 . 2 4 µu 2µu2 (14.224) (

"

!

#

)

This can be rewritten in a canonical form A=

Z

0

∞

ds(pu u′ + pθ θ + pϕ ϕ′ + pγ γ ′ − H),

(14.225)

with the Hamiltonian H. Kleinert, PATH INTEGRALS

923


H = +

1 4 1 2 2 p2u + 2 p2θ + p + (p + h ¯ q) − 2(p + h ¯ q)p cos θ γ γ ϕ 2µ u sin2 θ ϕ

i 4 h 2 2 2 −2¯ h qp + h ¯ (l − q ) . γ 0 2µu2

(14.226)

In the canonical path integral, the momenta are dummy integration variables so that we can replace pγ + h ¯ q by pγ . Then the action becomes A=

Z

0

∞

¯ ds[pu u′ + pθ θ + pϕ ϕ′ + (pγ − h ¯ q)γ ′ − H],

(14.227)

with the Hamiltonian ¯ = H +

1 4 1 2 2 p2u + 2 p2θ + p + p − 2p p cos θ γ ϕ γ 2µ u sin2 θ ϕ

i 4 h 2 2 2 −2¯ h q(p − h ¯ q) + h ¯ (l − q ) . γ 0 2µu2

(14.228)

This differs from the pure Coulomb case in three ways: First, the Hamiltonian has an extra centrifugal barrier proportional to the charge parameter 4q: V (r) =

−8¯ hq(pγ − h ¯ q) . 2 2µu

(14.229)

Second, there is an extra centrifugal barrier

V (r) =

2 h ¯ 2 lextra , 2µu2

(14.230)

whose effective quantum number of angular momentum is given by 2 lextra ≡ 4(l02 − q 2 ).

(14.231)

Third, the action (14.227) contains an additional term Z

∞

dsγ ′ .

(14.232)

∆A = −¯ hq(γb − γa ).

(14.233)

∆A = −¯ hq

0

Fortunately, this is a pure surface term

In the case q 2 = l02 , the extra centrifugal barrier vanishes, making it straightforward to write down the fixed-energy amplitude (xb |xa )E of the system. It is given by a simple modification of the relation (13.122) that expresses the fixed-energy

924


amplitude of the Coulomb system (xb |xa )E in terms of the four-dimensional harmonic oscillator amplitude (~ub S|~ua 0). Due to (14.232), the modification consists of a simple extra phase factor e−iq(γb −γa ) in the integral over γa so that Z

(xb |xa )E =

∞

dSeie

0

2 S/¯ h

1 Z 4π dγa e−iq(γa −γb ) (~ub S|~ua 0). 16 0

(14.234)

The integral over γa forces the momentum pγ in the canonical action (14.227) to take the value h ¯ q. This eliminates the term proportional to pγ − h ¯ in (14.228). In the general case l0 6= q, the amplitude becomes (xb |xa )E =

Z

∞

0

ie2 S/¯ h

dSe

1 16

Z

0

4π

dγa e−iq(γa −γb ) (~ub S|~ua0)lextra ,

(14.235)

where the subscript lextra indicates the presence of the extra centrifugal barrier potential in the harmonic oscillator amplitude. This amplitude was given for any dimension D in Eqs. (8.132) with (8.143). In the present case of D = 4, it has the partial-wave expansion [compare (8.161)] (~ub S|~ua0)lextra

∞ X lO + 1 1 (ub S|ua 0)˜lO = 3/2 (ub ua ) lO =0 2π 2

(14.236)

lO /2

×

dlmO1/2m2 (θb )dlmO1/2m2 (θa )eim1 (ϕb −ϕa )+im2 (γb −γa ) ,

X

m1 ,m2 =−lO /2

with the radial amplitude √ MO ω ub ua i(MO ω/2¯h)(u2 +u2a ) cot ωS MO ωubua b e (ub S|ua 0)˜lO = I˜lO +1 . i¯ h sin ωS i¯ h sin ωS

(14.237)

This differs from the pure oscillator amplitude [compare (8.141) for D = 4] by having the index lO + 1 of the Bessel function replaced by the square root of the “shifted square” as in (8.145): ˜lO + 1 ≡

q

2 (lO + 1)2 + lextra =

q

q

(lO + 1)2 + 4(l02 − q 2 ) = 2 (jD + 1/2)2 + l02 − q 2 . (14.238)

The expansion (14.236) is inserted into (14.235) with the variables ub , ua replaced √ √ by rb , ra . Just as in the Coulomb case in (14.100) and (14.101), the integral R 4π −iq(γb −γa ) over the sum of angular wave functions 0 dγa e lO /2 X lO + 1 dlO /2 (θb )dlmO1/2m2 (θa )eim1 (ϕb −ϕa )+im2 (γb −γa ) 2π 2 m1 ,m2 =−l/2 m1 m2

(14.239)

can immediately be done, resulting in lO /2

8

X m

lO /2 lO /2 ∗ Ym,q (θb , ϕb )Ym,q (θa , ϕa ),

(14.240)


925


lO /2 where Ym,q (θ, ϕ) are the monopole spherical harmonics (8.277). They coincide with the wave functions of a spinning symmetric top which possesses a spin q along the body axis. Physically, this spin is caused by the field’s momentum density = (E × B)/4πc encircling the radial distance vector x. The Poynting vector yielding the energy density is S = E × B/4π. If a magnetically charged particle lies at the origin and electrically charged particle orbits around it at x, the total angular momentum carried by the fields is [13]

J=

Z

1 d x x × (x ) = 4πc 3 ′

′

′

Z

g x′ e(x′ − x) eg ˆ. dx x × × = x ′ 3 ′ 3 |x | |x − x| c 3 ′

"

′

#

(14.241)

The quantization of the angular momentum eg h ¯ =n , c 2

n = integer

(14.242)

is Dirac’s famous charge quantization condition [15] [see also Eq. (8.303)]. Thus we arrive at the fixed-energy amplitude of the dionium atom, labeled by the subscript D, (xb |xa )ED =

jD X 1 X jD jD ∗ (rb |ra )ED ,jD Ym,q (θb , ϕb )Ym,q (θa , ϕa ), rb ra j D m=−jD

(14.243)

where the sum over jD = lO /2 runs over integer or half-integer values depending on q, and with the radial amplitude given by the pseudotime integral q over the radial oscillator amplitude of mass MO = 4MC and frequency ω = −E/2MC [recall (13.118)]: ! √ √ Z MO ω rb ra 1 ∞ ie2 S/¯ h MO ω rb ra dSe (rb |ra )ED ,jD = I˜ 2 0 i¯ h sin ωS lO +1 i¯ h sin ωS iMO ω × exp (rb + ra ) cot ωS , (14.244) 2¯ h q

2

q

where κ = −2MED /¯ h and ν = −e4 MD /2¯ h2 ED as in (13.39) and (13.40). Note that the dionium atom can be a fermion, even if the constituent particles are both bosons (or both fermions). After the variable changes e2 /¯ h = 2ων, ωS = −iy, we do the S-integral as in (9.29) and find for rb > ra the radial amplitude of the dionium atom (rb |ra )ED ,jD = −i

MD Γ(−ν + ˜jD + 1) Wν, ˜jD +1/2 (2κrb )Mν, ˜jD +1/2 (2κra ), (14.245) h ¯κ (2˜jD + 1)!

q

where ˜jD = ˜lO /2 = (jD + 12 )2 + l02 − q 2 − 12 . For q = 0 and l0 = 0, thus reduces properly to the three-dimensional Coulomb amplitude (14.102). The energy eigenvalues are obtained from the poles of the Gamma function at ν = νn ≡ ˜jD + n,

n = 1, 2, 3, . . . ,

(14.246)

926


which yield 1 En = −Mc2 α2 h q i2 . 2 1 1 2 2 2 n − 2 + (jD + 2 ) + l0 − q

(14.247)

From the residues at the poles and the discontinuity across the cut at E > 0 in (14.245), we can extract the bound and continuum radial wave functions by the same method as in Section 13.8 from Eqs. (13.211)–(13.223).

14.6.2

Absence of Time Slicing Corrections

Let us now show that the above formal manipulations receive no correction in a proper time-sliced treatment [14]. Due to the presence of centrifugal and angular barriers, a path collapse can be avoided only after an appropriate regularization of both singularities. This is achieved by the path-dependent time transformation dt = ds f (x(s)) with the postpoint regulating functions fl (x) = f (x) = r2 sin2 θ,

fr (x) ≡ 1.

(14.248)

After the extension of the path integral by an extra dummy dimension x4 , the time-sliced timetransformed fixed-energy amplitude to be studied is [see (14.204)–(14.210)] Z ∞ Z 1 1 4 dS (xb |xa )E ≈ dxa 2 2 2 (2πi¯ h ǫ ra sin θa 0 s /M ) Z N +1 Y N d4 ∆xn × eiA /¯h , (14.249) 4 2 4 (2πi¯hǫs /M ) rn sin θn n=2 with the sliced postpoint action N +1 X M (∆xin )2 ¯h 2 N 2 i A = Ai,i (xn ) . − ǫs rn sin θn [V (xn ) − E] +Ai (xn )∆x − ǫs 2ǫs rn2 sin2 θn 2M n=1

(14.250)

On this action, we now perform the coordinate transformation in two steps. First we go through the nonholonomic Kustaanheimo-Stiefel transformation as in Section 13.4 and express the fourdimensional ~u-space in terms of r = |x| and the Euler angles θ, ϕ, γ. Explicitly, x1 x2

= =

r sin θ cos ϕ, r sin θ sin ϕ,

x3 dx4

= =

r cos θ, r cos θdϕ + rdγ. µ

Only the last equation is nonholonomic. If q = 1, 2, 3, 4 denotes the transformation matrix reads  sin θ cos ϕ r cos θ cos ϕ −r sin θ sin ϕ i  ∂x sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ ei µ = µ =   cos θ −r sin θ 0 ∂q 0 0 r cos θ

It has the metric

gµν = ei µ eiν



1  0 =  0 0

0 r2 0 0

0 0 r2 r2 cos θ

 0  0 , r2 cos θ  r2

(14.251) components r, β, ϕ, γ, the  0 0  . 0  r

(14.252)


927

14.6 Path Integral of the Dionium Atom with an inverse g µν



1  0 =  0 0

0 1/r2 0 0

0 0 1/r2 sin2 θ − cos θ/r2 sin2 θ

 0  0 . 2 2 − cos θ/r sin θ  1/r2 sin2 θ

(14.253)

The regulating function f (x) = r2 sin2 θ removes the singularities in g µν and thus in the free part of the Hamiltonian (1/2M )g µν pµ pν ; there is no more danger of path collapse in the Euclidean amplitude. In a second step we go to new coordinates r = eξ ,

sin θ = 1/ cosh β,

cos θ = − tanh β,

(14.254)

as in the treatment of the angular barriers for D = 3 and D = 4 in Section 14.4. With q µ = 1, 2, 3, 4 denoting the coordinates ξ, β, ϕ, γ, respectively, the combined transformation matrix reads   sinh β −eξ cosh−1 β sin ϕ 0 eξ cosh−1 β cos ϕ −eξ cosh 2 β cos ϕ   ξ sinh β eξ cosh−1 β cos ϕ 0   e cosh−1 β sin ϕ −eξ cosh 2 β sin ϕ (14.255) ei µ =  , −2  −eξ tanh β −eξ cosh β 0 0  0 0 −eξ tanh β eξ with the metric

gµν

and the determinant

e2ξ  0 =  0 0 

0 e2ξ cosh−2 β 0 0

0 0 e2ξ −e2ξ tanh β

 0  0 , 2ξ −e tanh β  e2ξ

g = e8ξ / cosh4 β. The inverse metric is completely regular:  −2ξ e 0 0 2 −2ξ  0 e cosh β 0 g µν =   0 0 e−2ξ cosh2 β 0 0 e−2ξ sinh β cosh β

(14.256)

(14.257)

 0  0 . e−2ξ sinh β cosh β  e−2ξ cosh2 β

(14.258)

We now calculate the transformed actions (14.216) and (14.217). The relevant quantities which could contain time slicing corrections are Dµ Aµ and Γµ µν . The former quantity, being equal to ∂i Ai , vanishes in the transverse gauge under consideration. The calculation of Γµ µν is somewhat tedious (see Appendix 14A) but yields a surprisingly simple result: Γµµν = (−1, 0, 0, 0).

(14.259)

Because of this simplicity, the transformed sliced action is easily written down. It is split into two parts, Aǫtot = Aǫϕβ + Aǫϕγ ,

(14.260)

one containing only the coordinates ξ, β, Aǫξβ =

i¯h M [(∆ξ)2n cosh2 βn + (∆βn )2 ] + ∆ξn 2ǫs 2 2 2 2 ξn e e ¯h (lextra + 1/4) Ee2ξn − ǫs − + − , cosh2 βn 2M cosh2 βn cosh2 βn

(14.261)

928


the other dealing predominantly with ϕ, γ, M cosh2 βn [(∆ϕn )2 + (∆γn )2 − 2∆γn ∆ϕn tanh βn ] 2ǫs ¯h2 q 2 +h ¯ q tanh βn ∆ϕn − ǫs . 2M cosh2 βn

Aǫϕγ =

(14.262)

Hence the fixed-energy amplitude becomes (xb |xa )E

≈

Z

∞

dS 0

Z

N Z Y 1 drn dβn dγa 2πi¯hǫs ra2 /M cosh βa n=1 2πi¯hǫs /M cosh2 βn+1 ! N +1 i X ǫ × exp A (ϕb γb S|ϕa γa 0)[β] . (14.263) ¯h n=1 ξβ

4π

0

The last factor is a pseudotime evolution amplitude in the angles ϕ, γ which is still a functional of β(t), as indicated by the subscript [β], X X (ϕb γb S|ϕa γa 0)[β] ≈ (14.264) ϕN +1 =ϕb +2πlϕ γN +1 =γb +4πlγ b b

N Z Y dϕn dγn 1 exp × 2πi¯hǫs /M cosh βb n=1 2πi¯hǫs /M cosh βn

N +1 i X ǫ A ¯h n=1 ϕγ

!

.

The sums over lbϕ , lbγ account for the cyclic properties of the angles ϕ and γ with the periods 2π and 4π, respectively, at a fixed coordinate xb (as in the examples in Section 6.1). γ We now introduce auxiliary momentum variables pϕ n , pn and go over to the canonical form of the amplitude (14.264): (ϕb γb S|ϕa γa 0)[β] ≈ × exp

"

i ¯ h

N Z Y

n=1

N +1 X

dϕn

NY +1 Z n=1

Y NY N Z +1 Z dpϕ dpγn n dγn 2π¯h n=1 2π¯h n=1

γ pϕ n ∆ϕn + pn ∆γn

(14.265)

n=1

ǫs pγn ¯hq 2 γ γ − [(pϕ ¯ q)2 + 2pϕ ¯ q) tanh βn ] + ǫs n ) + (pn + h n (pn + h 2M M cosh2 βn

.

The momenta pγn are dummy integration variables and can be replaced by pγn − ¯hq. The dϕn , dγn integrals run over the full extended zone schemes ϕn , γn ∈ (−∞, ∞) and enforce the equality of all pγn . At the end, only the integrals over a common single momentum pϕ , pγ remain and we arrive at (ϕb γb S|ϕa γa 0)[β] ∞ X ≈ e−iq(γb −γa )

(14.266)

Z Z ∞ X dpϕ dpγ ipϕ (ϕb +2πlϕ −ϕa )/¯h ipγ (γb +4πlγ −γa )/¯h b b e e 2π¯h 2π¯h ϕ γ lb =−∞ lb =−∞ ( ) N +1 i X 1 (pγ − ¯hq)¯ hq 2 2 × exp − (p + pγ + 2pϕ pγ tanh βn ) − ǫs . ¯h n=1 2M ǫs ϕ M cosh2 βn

We can now do the sums over lbϕ , lbγ which force the momenta pϕ to integer values and pγ to half-integer values by Poisson’s formula, so that X 1 1 (ϕb γb S|ϕa γa 0)[β] = e−iq(γb −γa ) eim1 (ϕb −ϕa ) eim2 (γb −γa ) (14.267) 2π 4π m ,m 1

2


929

14.7 Time-Dependent Duru-Kleinert Transformation ) N +1 i X ¯h2 ǫs ¯h2 (m2 − q)q 2 2 (m + m2 + 2m1 m2 tanh βn ) − × exp − . ¯h n=1 2M ǫs 1 M cosh2 βn (

With this, the expression for the fixed-energy amplitude (14.263) of the dionium atom contains the magnetic charge only at three places: the extra centrifugal barrier in Aǫξβ , the phase factor of the remaining integral over γa , and the last term in (14.267). This last term, however, can be dropped since the integral over γa forces the half-integer number m2 to become equal to ¯hq. The γb -integral over the remaining functional of β(t) gives, incidentally, Z 4π X 1 dγa (ϕb γb S|ϕa γa 0)[β] = eim1 (ϕb −ϕa ) 2π 0 m1 ( ) N +1 X i ¯h2 × exp − [m2 + q 2 + 2m1 q tanh βn . (14.268) ¯h n=1 2M ǫs 1 The time-sliced expression has the parameter q at precisely the same places as the previous formal one. This proves that formula (14.235) with (14.236) is unchanged by time slicing corrections, thus completing the solution of the path integral of the dionium atom. Note that after inserting (14.268), the time-sliced path integral (14.263) is a combination of a general Rosen-Morse system in β and a Morse system in ξ. Let us end this discussion by the remark that like the Coulomb system, the dionium atom can be treated in a purely group-theoretic way, using only operations within the dynamical group O(4,2). This is explained in Appendix 14B.

14.7

Time-Dependent Duru-Kleinert Transformation

By generalizing the above transformation method to time-dependent regulating functions, we can derive further relations between amplitudes of different physical systems. In the path-dependent time transformation dt = ds fl (x)fr (x) regularizing the path integrals, we may allow for functions fl (p, x, t) and fr (p, x, t) depending on positions, momenta, and time. Such functions complicate the subsequent transformation to new coordinates q, in which the kinetic term of the amplitude (12.47) with respect to the pseudotime s has the standard form (M/2)q′2 (s). In particular, a momentum dependence of fl and fr leads to involved formulas, which is the reason why this case has not yet been investigated (just like the even more general case where the right-hand side of the transformation dt = ds f contains terms proportional to dx). If one restricts the transformation to depend only on time and uses the special splitting with the regulating functions fl = f and fr = 1, or fl = 1 and fr = f , the result in one spatial dimension is relatively simple. On the basis of Section 12.3, the following relation is found [instead of (14.26)] between the time evolution amplitude of an initial system and a fixed-energy amplitude of the transformed systems at E = 0: (xb tb |xa ta ) = g(qb , tb )g(qa , ta ){qb tb |qa ta }E=0 ,

(14.269)

where {qb tb |qa ta }E=0 denotes the spacetime extension of the fixed-pseudoenergy amplitude. It is calculated by time-slicing the expression Z

0

∞

dS{xb tb |UÊ (S)|xa ta }

(14.270)

930


on the right-hand side of (12.55), transforming the coordinates x to q, and adapting the normalization to the completeness relation of the states Z

dx

Z

dt|q t}{q t| = 1.

(14.271)

This leads to the path integral representation {qb tb |qa ta }E =

Z

∞

0

dS

Z

dEe−iE(tb −ta )/¯h

Z

DK

Dq(s)eiAE,E /¯h ,

(14.272)

with the DK-transformed action ADK E,E

=

Z

0

S

ds

nM

2

q ′2 (s) − f (q(s), t(s))[V (q(s), t) − E] + E o

−Veff (q(s), t(s)) − ∆Veff (q(s), t(s)) .

(14.273)

Note that the initial potential may depend explicitly on time. The function t(s) is now given by the time-dependent differential equation dt = f (x, t). ds

(14.274)

The coordinate transformation also depends on time, x = h(q, t),

(14.275)

h′2 (q, t) = f (h(q, t), t),

(14.276)

and satisfies the equation

where h′ (q, t) ≡ ∂q h(q, t) [compare (14.15)]. The function f (q(s), t(s)) used in (14.273) is an abbreviation for f (h(q, t), t) evaluated at the time t = t(s). In addition to the effective potential Veff determined by Eq. (14.18), there is now a further contribution which is due to the time dependence of h(q, t) [16]: ′2

∆Veff = Mh

Z

¨ ∓ i¯ dq h′ h hh˙ ′ h′ .

(14.277)

The upper sign must be used if the relation between t and s is calculated from the time-sliced postpoint recursion relation tn+1 − tn = ǫs f (qn+1 , tn+1 ).

(14.278)

The lower sign holds when solving the prepoint relation tn+1 − tn = ǫs f (qn , tn ).

(14.279)

Note that the first term of ∆Veff contributes even at the classical level. If a function h(q, t) is found satisfying Newton’s equation of motion ¨=− Mh

∂V (h, t) , ∂h


931

14.7 Time-Dependent Duru-Kleinert Transformation

with V (h) ≡ V (x)|x=h , then the first term eliminates the potential in the action (14.273), and the transformed system is classically free. This happens if the new coordinate q(t) associated with x, t is identified with the initial value, at some time t0 , of the classical orbit running through x, t. These are trivially time-independent and therefore behave like the coordinates of a free particle (see the subsequent example). The normalization factor g(q, t) is determined by the differential equation g′ 1 h′′ M ˙ + i h′ h. = ′ g 2h h ¯

(14.281)

The solution reads q

g(q, t) = eiΛ(q,t) h′ (q, t),

(14.282)

M Λ(q, t) = ± h ¯

(14.283)

with

Z

q

˙ dq h′ h.

Thus, in addition to the normalization factor in (14.26), the time-dependent DK relation (14.269) also contains a phase factor. As an example [17], we transform the amplitude of a harmonic oscillator to that of a free particle. The classical orbits are given by x(t) = x0 cos ωt, so that the transformation x(t) = h(q, t) = q cos ωt leads to a coordinate q(t) which moves without acceleration. For brevity, we write cos ωt as c(t). Obviously, f (q, t) = c2 (t) is a pure function of the time [18], and the differential relation between the time t and the pseudotime s is integrated to S=

1 sin ω(tb − ta ) . ω c(tb )c(ta )

(14.284)

This equation can be solved for ta (S) at fixed tb , or for tb (S) at fixed ta . The solution ta (S) is obtained from the time-sliced postpoint recursion relation (14.278), while tb (S) arises from the prepoint recursion (14.279). The DK action (14.273) is simplified in these two cases to ADK E,E

M (qb − qa )2 c(tb ) [tb − ta (S)] = ± i¯ h log +E + ES. 2 S c(ta ) [tb (S) − ta ] (

)

(14.285)

The E-integration in (14.272) yields in the first case {qb tb |qa ta }E =

Z

0

∞

dSδ(tb − ta (S))

2

c(ta ) e(i/¯h)M (qb −qa ) /2S q , c(tb ) 2π¯ hiS/M

(14.286)

and the integration over S using −dta (S)/dS = c2 (ta ) results in 2

1 e(i/¯h)M (qb −qa ) /2S q {qb tb |qa ta }E = . c(tb )c(ta ) 2π¯ hiS/M

(14.287)

932


The same amplitude is obtained for the lower sign in (14.285) with dtb (S)/dS = c2 (tb ). After inserting this together with (18.596) and (14.283) into (14.269) [the integration there gives Λ(q, t) = (M/¯ h)q 2 c(t)c(t)/2], ˙ we obtain 2

1 e(i/¯h)M (qb −qa ) /2S 2 ˙ b )−qa2 c(ta )c(t ˙ a )]/2 q q (xb tb |xa ta ) = e(i/¯h)M [qb c(tb )c(t . c(tb )c(ta ) 2π¯ hiS/M

(14.288) Since qb and qa are equal to xb /c(tb ) and xa /c(ta ), respectively, a few trigonometric identities lead to the well-known expression (2.168) for the amplitude of the harmonic oscillator. It is obvious that a combination of this transformation with a time-independent Duru-Kleinert transformation makes it possible to reduce also the path integral of the Coulomb system to that of a free particle. It will be interesting to find out which hitherto unsolved path integrals can be integrated by means of such generalized DK transformations.

Appendix 14A

Affine Connection of Dionium Atom

From the transformation matrices (14.255), we calculate the derivatives [with q µ = (ξ, β, ϕ, γ) and the abbreviation f,ξ ≡ ∂ξ f ] ei µ,ξ = ei µ ,

ei µ,β

ei µ,φ



(14A.1)

2

sinh β β −eξ cosh −eξ 1−sinh cos φ 2 β cos φ cosh3 β

 sinh β  −eξ cosh 2 β sin φ =   −eξ cosh−2 β 0 ξ

−1

2

β −eξ 1−sinh sin φ cosh3 β sinh β ξ 2e cosh3 β 0

−e cosh β sin φ  eξ cosh−1 β cos φ =   0 0 

sinh β eξ cosh 2 β sin φ ξ sinh β −e cosh2 β cos φ

0 0

sinh β eξ cosh 2 β sin φ sinh β −eξ cosh 2 β cos φ 0 −eξ cosh−2 β ξ

−1

0

 0  , 0  0

−e cosh β cos φ −eξ cosh−1 β sin φ 0 0

ei µ,α = 0.



0 0  , 0  0

(14A.2)



(14A.3) (14A.4)

From these we find Γµνλ = ei λ ei ν,µ by contraction with ei λ : Γξµν = gµν ,

Γβµν

Γφµν

0  −e2ξ cosh−2 =   0 0  0  0  =   −e2ξ cosh−2 0 

(14A.5)

e2ξ cosh−2 β sinh β −e2ξ cosh 3β 0 0 0 0 sinh β e2ξ cosh 3β 0

0 0 0 0

 0  0 , −2 2ξ −e cosh β  0  2ξ e cosh−2 β 0 sinh β −e2ξ cosh 0  3β  , 0 0  0 0

(14A.6)

(14A.7)


Appendix 14B

Algebraic Aspects of Dionium States Γαµν = 0.

933 (14A.8)

A contraction with the inverse metric g µν yields Γµ µ ν = (−1, 0, 0, 0), as stated in Eq. (14.259).

Appendix 14B

Algebraic Aspects of Dionium States

In Appendix 13A we have shown that certain combinations of xµ and ∂µ operators satisfy the commutation rules of the Lie algebra of the group O(4,2) [see Eqs. (13A.11)]. This permits solving all dynamical problems via group operations. In the case l0 = q (i.e., lextra = 0), the grouptheoretic approach can be extended to include the dionium atom. In fact, it is easy to see [19] that the Lie algebra of O(4,2) remains the same if the generators LAB (A, B = 1, . . . , 6) of Eq. (13A.11) are extended to (xi ≡ xi ) ˆ ij L

=

ˆ i4 L

=

ˆ i5 L

=

ˆ i6 L

=

ˆ 45 L

=

ˆ 46 L

=

ˆ 56 L

=

i r − (xi ∂xj − xj ∂xi ) + q 2 xk⊥ , 2 x⊥ 1 r i 2 i δi3 2 xi −x ∂x − x + 2∂xi x∂x + 2iq 2 (x⊥ × ∇)i − (−) q 2 , 2 x⊥ x⊥ 1 r i 2 i δi3 2 xi −x ∂x + x + 2∂xi x∂x + 2iq 2 (x⊥ × ∇)i − (−) q 2 , 2 x⊥ x⊥ q −ir∂xi − 2 (x × x⊥ )i , x⊥ −i(x∂x + 1), 1 z r −r∂x2 − r + 2iq 2 (x × ∇)3 + q 2 2 , 2 x⊥ x⊥ 1 z r −r∂x2 + r + 2iq 2 (x × ∇)3 + q 2 2 . 2 x⊥ x⊥

(14B.1)

ˆ 05 = −ir∂x4 = The representation space is now characterized by the eigenvalue of the operator L 1 † †ˆ ˆ −i∂γ = − 2 (ˆ a a ˆ − b b) being equal to q. The wave functions are generalizations of the q = 0 -wave functions of the Coulomb system.

Notes and References [1] The special case λ = 1/2 has also been treated by N.K. Pak and I. Sokmen, Phys. Rev. A 30, 1692 (1984). [2] L.C. Hostler, J. Math. Phys. 11, 2966 (1970). [3] F. Steiner, Phys. Lett. A 106, 256, 363 (1984). [4] A. Sommerfeld, Partial Differential Equations in Physics, Lectures in Theoretical Physics, Vol. 6, Academic, New York, 1949; T. Regge, Nuovo Cimento 14, 951 (1959); F. Calogero, Nuovo Cimento 28, 761 (1963); A.O. Barut, The Theory of the Scattering Matrix , MacMillan, New York, 1967, p. 140; P.D.B. Collins and E.J. Squires, Regge Poles in Particle Physics, Springer Tracts in Modern Physics, Vol. 49, Springer, Berlin 1968. [5] H. Kleinert and I. Mustapic, J. Math. Phys. 33, 643 (1992) (http://www.physik.fuberlin.de/ kleinert/207). [6] G. Pöschl and E. Teller, Z. Phys. 83, 1439 (1933). See also S. Fl¨ ugge, Practical Quantum Mechanics, Springer, Berlin, 1974, p. 89.

934


[7] N. Rosen, P.M. Morse, Phys. Rev. 42, 210 (1932); S. Fl¨ ugge, op. cit., p. 94. See also L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965, §23. [8] This DK relation between these amplitudes was first given by G. Junker and A. Inomata, in Path Integrals From meV to MeV , edited by M.C. Gutzwiller, A. Inomata, J.R. Klauder, and L. Streit, World Scientific, Singapore, 1986, p. 315, and by M. Böhm and G. Junker, J. Math. Phys. 28, 1978 (1987). Watch out for mistakes. For instance, in Eq. (3.28) of the first paper, the authors claim to have calculated the fixed-energy amplitude, but give only its imaginary part restricted to the bound-state poles.Their result (3.33) lacks the continuum states. Further errors in their Section V have been pointed out in Footnote 20 of Chapter 8. [9] For the explicit extraction of the wave functions see Ref. [5], Section IV B. [10] J.M. Cai, P.Y. Cai, and A. Inomata, Phys. Rev. A 34, 4621 (1986). [11] J. Schwinger, A Magnetic Model of Matter, Science, 165, 717 (1969). See also the review article K.A. Milton, Theoretical and Experimental Status of Monopoles, (hep-ex/0602040). [12] H. Kleinert, Phys. Lett. A 116, 201 (1989). [13] J.J. Thomson, On Momentum in the Electric field , Philos. Mag. 8, 331 (1904). [14] This proof was done in collaboration with my undergraduate student J. Zaun. [15] P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931); Phys. Rev. 74, 817 (1948), Phys. Rev. 74, 817 (1948). [16] A. Pelster and A. Wunderlin, Zeitschr. Phys. B 89, 373 (1992). See also the similar generalization of the DK transformation in stochastic differential equations by S.N. Storchak, Phys. Lett. A 135, 77 (1989). [17] For other examples see C. Grosche, Phys. Lett. A 182, 28 (1952). [18] This special case was treated by P.Y. Cai, A. Inomata, and P. Wang, Phys. Lett. 91, 331 (1982). Note that the transformation to free particles is based on a general observation in H. Kleinert, Phys. Lett. B 94, 373 (1980) (http://www.physik.fu-berlin.de/~kleinert/71). [19] A.O. Barut, C.K.E. Schneider, and R. Wilson, J. Math. Phys. 20, 2244 (1979).


H. Kleinert, PATH INTEGRALS June 16, 2007 ( /home/kleinert/kleinert/books/pathis/pthic15.tex) Thou com’st in such a questionable shape, That I will speak to thee. William Shakespeare, Hamlet

15 Path Integrals in Polymer Physics The use of path integrals is not confined to the quantum-mechanical description point particles in spacetime. An important field of applications lies in polymer physics where they are an ideal tool for studying the statistical fluctuations of linelike physical objects.

15.1

Polymers and Ideal Random Chains

A polymer is a long chain of many identical molecules connected with each other at joints which allow for spatial rotations. A large class of polymers behaves approximately like an idealized random chain. This is defined as a chain of N links of a fixed length a, whose rotational angles occur all with equal probability (see Fig. 15.1). The probability distribution of the end-to-end distance vector xb − xa of such an object is given by PN (xb − xa ) =

N Z Y

d3 ∆xn

n=1

N X 1 (3) δ(|∆x | − a) δ (x − x − ∆xn ). (15.1) n b a 4πa2 n=1

The last δ-function makes sure that the vectors ∆xn of the chain elements add up correctly to the distance vector xb − xa . The δ-functions under the product enforce the fixed length of the chain elements. The length a is also called the bond length of the random chain. The angular probabilities of the links are spherically symmetric. The factors 1/4πa2 ensure the proper normalization of the individual one-link probabilities P1 (∆x) =

1 δ(|∆x| − a) 4πa2

(15.2)

in the integral Z

d3 xb P1 (xb − xa ) = 1.

(15.3)

The same normalization holds for each N: Z

d3 xb PN (xb − xa ) = 1. 935

(15.4)

936

15 Path Integrals in Polymer Physics

Figure 15.1 Random chain consisting of N links ∆xn of length a connecting xa = x0 and xb = xN .

If the second δ-function in (15.1) is Fourier-decomposed as δ

(3)

xb − xa −

N X

!

Z

∆xn =

n=1

d3 k ik(xb −xa )−ik PN ∆xn n=1 e , (2π)3

(15.5)

we see that PN (xb − xa ) has the Fourier representation PN (xb − xa ) =

Z

d3 k ik(xb −xa ) ˜ e PN (k), (2π)3

(15.6)

with PÑ (k) =

N Z Y

d3 ∆xn

n=1

1 δ(|∆xn | − a)e−ik∆xn . 4πa2

(15.7)

Thus, the Fourier transform PÑ (k) factorizes into a product of N Fouriertransformed one-link probabilities: h

PÑ (k) = P˜1 (k)

.

(15.8)

1 sin ka δ(|∆x| − a)e−ik∆x = . 2 4πa ka

(15.9)

These are easily calculated: P˜1 (k) =

Z

d3 ∆x

iN

The desired end-to-end probability distribution is then given by the integral PN (R) =

Z

d3 k ˜ [P1 (k)]N eikR (2π)3

1 = 2π 2 R

Z

0

∞

"

sin ka dkk sin kR ka

#N

,

(15.10)


937

15.2 Moments of End-to-End Distribution

where we have introduced the end-to-end distance vector R ≡ xb − xa .

(15.11)

The generalization of the one-link distribution to D dimensions is P1 (∆x) =

1 δ(|∆x| − a), SD aD−1

(15.12)

with SD being the surface of a unit sphere in D dimensions [see Eq. (1.555)]. To calculate P˜1 (k) =

Z

dD ∆x

1 δ(|∆x| − a)e−ik∆x , SD aD−1

(15.13)

we insert for√e−ik∆x = e−ik|∆x| cos ∆ϑ the expansion (8.129) with (8.101), and use Y0,0 (ˆ x) = 1/ SD to perform the integral as in (8.249). Using the relation between modified and ordinary Bessel functions Jν (z)1 Iν (e−iπ/2 z) = e−iπ/2 Jν (z),

(15.14)

this gives P˜1 (k) =

Γ(D/2) JD/2−1 (ka), (ka/2)D/2−1

(15.15)

where Jµ (z) is the Bessel function.

15.2

Moments of End-to-End Distribution

The end-to-end distribution of a random chain is, of course, invariant under rotations so that the Fourier-transformed probability can only have even Taylor expansion coefficients: PÑ (k) = [P˜1 (k)]N =

∞ X

PN,2l

l=0

(ka)2l . (2l)!

(15.16)

The expansion coefficients provide us with a direct measure for the even moments of the end-to-end distribution. These are defined by hR2l i ≡

Z

dD R R2l PN (R).

(15.17)

The relation between hR2l i and PN,2l is found by expanding the exponential under the inverse of the Fourier integral (15.6): PÑ (k) = 1

Z

dD R e−ikR PN (R) =

∞ Z X

n=0

dD R

(−ikR)n PN (R), n!


(15.18)

938


and observing that the angular average of (kR)n is related to the average of Rn in D dimensions by   

0 , (n − 1)!!(D − 2)!! h(kR) i = k hR i  ,  (D + n − 2)!! n

n

n

n = odd, n = even.

(15.19)

The three-dimensional result 1/(n+1) follows immediately from the angular average R1 n (1/2) −1 d cos θ cos θ being 1/(n + 1) for even n. In D dimensions, it is most easily derived by assuming, for a moment, that the vectors R have a Gaussian distribution 2 2 (0) PN (R) = (D/2πNa2 )3/2 e−R D/2N a . Then the expectation values of all products of Ri can be expressed in terms of the pair expectation value hRi Rj i(0) =

1 δij a2 N D

(15.20)

via Wick’s rule (3.302). The result is hRi1 Ri2 · · · Rin i(0) =

1 δi i i ...i an N n/2 , D n/2 1 2 3 n

(15.21)

with the contraction tensor δi1 i2 i3 ...in of Eqs. (8.64) and (13.84), which has the recursive definition δi1 i2 i3 ...in = δi1 i2 δi3 i4 ...in + δi1 i3 δi2 i4 ...in + . . . δi1 in δi2 i3 ...in−1 .

(15.22)

A full contraction of the indices gives, for even n, the Gaussian expectation values: hRn i(0) =

(D + n − 2)!! n n/2 Γ(D/2 + n/2) 2n/2 n n/2 a N = a N , (D − 2)!!D n/2 Γ(D/2) D n/2

(15.23)

for instance D

R4

E(0)

=

(D + 2) 4 2 aN , D

D

R6

E(0)

=

(D + 2) (D + 4) 6 3 aN . D2

(15.24)

By contracting (15.21) with ki1 ki2 · · · kin we find h(kR)n i(0) = (n − 1)!!

1 (ka)n N n/2 = k n h(R)n i(0) dn , n/2 D

(15.25)

with dn =

(n − 1)!!(D − 2)!! . (D + n − 2)!!

(15.26)

Relation (15.25) holds for any rotation-invariant size distribution of R, in particular for PN (R), thus proving Eq. (15.17) for random chains. Hence, the expansion coefficients PN,2l are related to the moments hR2l iN by PN,2l = (−1)l d2l hR2l i,


939

15.2 Moments of End-to-End Distribution

and the moment expansion (15.16) becomes PÑ (k) =

∞ X l=0

(−1)l (k)2l d2l hR2l i. (2l)!

(15.28)

Let us calculate the even moments hR2l i of the polymer distribution PN (R) explicitly for D = 3. We expand the logarithm of the Fourier transform PÑ (k) as follows: sin ka log PÑ (k) = N log P˜1 (k) = N log ka

!

=N

∞ X

22l (−1)l B2l (ka)2l , (15.29) (2l)!2l l=1

where Bl are the Bernoulli numbers B2 = 1/6, B4 = −1/30, . . . . Then we note that for a Taylor series of an arbitrary function y(x) y(x) =

∞ X

an n x , n=1 n!

(15.30)

the exponential function ey(x) has the expansion ey(x) =

∞ X

bn n x , n=1 n!

(15.31)

with the coefficients n X Y bn 1 ai = n! {mi } i=0 mi ! i!

mi

.

(15.32)

The sum over the powers mi = 0, 1, 2, . . . obeys the constraint n=

n X i=1

i · mi .

(15.33)

Note that the expansion coefficients an of y(x) are the cumulants of the expansion coefficients bn of ey(x) as defined in Section 3.17. For the coefficients an of the expansion (15.29), an =

(

−N22l (−1)l B2l /2l 0

for n = 2l, for n = 2l + 1,

(15.34)

we find, via the relation (15.27), the moments l X Y

1 N22i (−1)i B2i hR2l i = a2l (−1)l (2l + 1)! mi ! (2i)!2i {mi } i=1 with the sum over mi = 0, 1, 2, . . .

"

constrained by

#mi

,

(15.35)

940


l=

l X i=1

i · mi .

(15.36)

For l = 1 and 2 we obtain the moments 2

5 2 hR i = a4 N 2 1 − . 3 5N

2

4

hR i = a N,

(15.37)

In the limit of large N, the leading behavior of the moments is the same as in (15.20) and (15.24). The linear growth of hR2 i with the number of links N is characteristic for a random chain. In the presence of interactions, there will be a different power behavior expressed as a so-called scaling law hR2 i ∝ a2 N 2ν .

(15.38)

The number ν is called the critical exponent of this scaling law. It is intuitively obvious that ν must be a number between ν = 1/2 for a random chain as in (15.37), and ν = 1 for a completely stiff chain. Note that the knowledge of all moments of the end-to-end distribution determines completely the shape of the distribution by an expansion ∞ n X 1 n (−1) ∂Rn δ(R). hR i PL (R) = D−1 SD R n! n=0

(15.39)

This can easily be verified by calculating the integrals (15.17) using the integrals formula Z

15.3

dz z n ∂zn δ(z) = (−1)n n! .

(15.40)

Exact End-to-End Distribution in Three Dimensions

Consider the Fourier representation (15.10), rewritten as i PN (R) = 2 2 4π a R

Z

∞

−∞

−iηR/a

dη η e

sin η η

!N

,

(15.41)

with the dimensionless integration variable η ≡ ka. By expanding N 1 X N sin η = (−1)n exp [i(N − 2n)η] , N (2i) n=0 n

!

N

(15.42)

we find the finite series PN (R) =

1

N X

n

(−1) 2N +2 iN −1 π 2 a2 R n=0

!

N IN (N − 2n − R/a), n

(15.43)


941

15.3 Exact End-to-End Distribution in Three Dimensions

where IN (x) are the integrals IN (x) ≡

Z

∞

−∞

dη

eiηx . η N −1

(15.44)

For N ≥ 2, these integrals are all singular. The singularity can be avoided by noting that the initial integral (15.41) is perfectly regular at η = 0. We therefore replace the expression (sin η/η)N in the integrand by [sin(η − iǫ)/(η − iǫ)]N . This regularizes each term in the expansion (15.43) and leads to well-defined integrals: IN (x) =

Z

∞

−∞

dη

eix(η−iǫ) . (η − iǫ)N −1

(15.45)

For x < 0, the contour of integration can be closed by a large semicircle in the lower half-plane. Since the lower half-plane contains no singularity, the residue theorem shows that IN (x) = 0,

x < 0.

(15.46)

For x > 0, on the other hand, an expansion of the exponential function in powers of η −iǫ produces a pole, and the residue theorem yields IN (x) =

2πiN −1 N −2 x , (N − 2)!

x > 0.

(15.47)

Hence we arrive at the finite series !

X 1 N PN (R) = N +1 (−1)n (N − 2n − R/a)N −2 . (15.48) 2 n 2 (N − 2)!πa R 0≤n≤(N −R/a)/2

The distribution √ is displayed for various values of N in Fig. 15.2, where √ we have plotted 2πR2 N PN (R) against the rescaled distance variable ρ = R/ N a. With this N-dependent rescaling all curves have the same unit area. Note that they converge rapidly towards a universal zero-order distribution (0) PN (R)

=

s

3

3 3R2 exp − 2πNa2 2Na2 (

)

→

(0) PL (R)

=

s

D

D 2 e−DR /2aL . (15.49) 2πaL

In the limit of large N, the length L will be used as a subscript rather than the diverging N. The proof of the limit is most easily given in Fourier space. For large N at finite k 2 a2 N, the Nth power of the quantity P˜1 (k) in Eq. (15.15) can be approximated by [P˜1 (k)]N ∼ e−N k

2 a2 /2D

.

(15.50)

Then the Fourier transform (15.10) is performed with the zero-order result (15.49). √ In Fig. 15.2 we see that this large-N limit is approached uniformly in ρ = R/ N a. The approach to this limit is studied analytically in the following two sections.

942


Figure 15.2 √ End-to-end distribution PN (R)√of random chain with N links. The func2 tions 2πR N PN (R) are plotted against R/ N a which gives all curves the same unit area. Note the fast convergence for growing N . The dashed curve is the continuum distri(0) bution PL (R) of Eq. (15.49). The circles on the abscissa mark the maximal end-to-end distance.

15.4

Short-Distance Expansion for Long Polymer

At finite N, we expect corrections which are expandable in the form PN (R) =

(0) PN (k)

∞ X

"

#

1 1+ C (R2 /Na2 ) , n n N n=1

(15.51)

where the functions Cn (x) are power series in x starting with x0 . Let us derive this expansion. In three dimensions, we start from (15.29) and separate the right-hand side into the leading k 2 -term and a remainder ∞ X

22l (−1)l B2l 2 2 l ˜ (k a ) . C(k) ≡ exp N (2l)!2l l=2 "

#

(15.52)

Exponentiating both sides of (15.29), the end-to-end probability factorizes as 2 2 ˜ PÑ (k) = e−N a k /6 C(k).

(15.53)

˜ The function C(k) is now expanded in a power series ˜ C(k) =1+

X

Cñ,l N n (a2 k 2 )l ,

(15.54)

n=1,2,... l=2n,2n+1,...


943

15.4 Short-Distance Expansion for Long Polymer

with the lowest coefficients 1 C˜1,2 = − , 180 1 C˜1,4 = − , 37800

1 C˜1,3 = − , 2835 1 C˜2,4 = , 64800

... .

(15.55)

For any dimension D, we factorize 2 2 ˜ PÑ (k) = e−N a k /2D C(k)

(15.56)

˜ and find C(k) by expanding (15.15) in powers of k and proceeding as before. This gives the coefficients C˜1,2 C˜1,3 C˜1,4 C˜2,4

= = = =

−1/4D 2 (D + 2), −1/3D 3 (D + 2)(D + 4), −(5D + 12)/8D 4(D + 2)2 (D + 4)(D + 6), 1/32D 4(D + 2)2 .

(15.57)

˜ We now Fourier-transform (15.56). The leading term in C(k) yields the zero-order distribution (15.49) in D dimensions, (0) PN (R)

=

s

D

D 2 2 e−DR /2N a , 2 2πNa

(15.58)

or, written in terms of the reduced distance variable ρ = R/Na, (0) PN (R)

=

s

D

D 2 e−DN ρ /2 . 2 2πNa

(15.59)

˜ To account for the corrections in C(k) we take the expansion (15.54), emphasize the 2 2 dependence on k a by writing ˜ ¯ 2 a2 ), C(k) = C(k and observe that in the Fourier transform PN (R) =

Z

Z d3 k ikR d3 k ikR −N k2 a2 /2D ¯ 2 2 e P (k) = e e C(k a ), N (2π)3 (2π)3

(15.60)

the series can be pulled out of the integral by replacing each power (k 2 a2 )p by (−2D∂N )p . The result has the form ¯ PN (R) = C(−2D∂ N)

Z

d3 k ikR −N k2 a2 /2D (0) ¯ e e = C(−2D∂ N )PN (R). 3 (2π)

(15.61)

Going back to coordinate space, we obtain an expansion D PN (R) = 2πNa2

D/2

2 /2

e−DN ρ

C(R).

(15.62)

944


For D = 3, the function C(R) is given by the series ∞ X

C(R) = 1 +

Cn,l N −n (Nρ2 )l ,

(15.63)

n=1 l=0,...,2n

with the coefficients C1,l

9 3 3 = − , ,− , 4 2 20

C2,l

29 69 981 1341 81 = ,− , ,− , . 160 40 400 1400 800

(15.64)

For any D, we find the coefficients C1,l = C2,l =

15.5

D2 D D − , ,− , 4 2 4(D + 2) (3D 2 − 2D + 8)D (D 2 + 2D + 8)D (3D 2 + 14D + 40)D 2 ,− , , 96(D + 2) 8(D + 2) 16(D + 2)2 ! (3D 2 + 22D + 56)D 3 D4 , . − 24(D + 2)2 (D + 4) 32(D + 2)2 !

(15.65)

Saddle Point Approximation to Three-Dimensional End-to-End Distribution

Another study of the approach to the limiting distribution (15.49) proceeds via the saddle point approximation. For this, the integral in (15.41) is rewritten as Z

∞

−∞

dη η e−N f (η) ,

(15.66)

with !

R sin η f (η) = i η − log . Na η

(15.67)

The extremum of f lies at η = η¯, where η¯ solves the equation coth(i¯ η) −

R 1 = . i¯ η Na

(15.68)

The function on the left-hand side is known as the Langevin function: 1 L(x) ≡ coth x − . x

(15.69)

The extremum lies at the imaginary position η¯ ≡ −i¯ x with x¯ being determined by the equation L(¯ x) =

R . Na


945

15.6 Path Integral for Continuous Gaussian Distribution

The extremum is a minimum of f (η) since f ′′ (¯ η ) = L′ (¯ x) = −

1 1 + 2 > 0. 2 sinh x¯ x¯

(15.71)

By shifting the integration contour vertically into the complex η plane to make it run through the minimum at −i¯ x, we obtain ∞ 1 N −N f (¯ η) dη (−i¯ x + η) exp − f ′′ (¯ η )η 2 e 2 2 4iπ a R 2 −∞ s η¯ 2π = e−N f (¯η ) . 2 2 4π a R Nf ′′ (¯ x)

Z

PN (R) ≈ −

(15.72)

When expressed in terms of the reduced distance ρ ≡ R/Na ∈ [0, 1], this reads 1 Li (ρ)2 PN (R) ≈ (2πNa2 )3/2 ρ {1 − [Li (ρ)/ sinh Li (ρ)]2 }1/2

(

sinh Li (ρ) Li (ρ) exp[ρLi (ρ)]

)N

. (15.73)

Here we have introduced the inverse Langevin function Li (ρ) since it allows us to express x¯ as x¯ = Li (ρ)

(15.74)

[inverting Eq. (15.70)]. The result (15.73) is valid in the entire interval ρ ∈ [0, 1] corresponding to R ∈ [0, Na]; it ignores corrections of the order 1/N. By expanding the right-hand side in a power series in ρ, we find PN (R) = N

3 2πNa2

3/2

3R2 exp − 2Na2

!

3R2 9R4 1+ − + . . . , (15.75) 2N 2 a2 20N 3 a4 !

with some normalization constant N . At each order of truncation, N is determined R 3 in such a way that d R PN (R)= 1. As a check we take the limit ρ2 ≪ 1/N and find powers in ρ which agree with those in (15.62), for D = 3, with the expansion (15.63) of the correction factor.

15.6

Path Integral for Continuous Gaussian Distribution

The limiting end-to-end distribution (15.49) is equal to the imaginary-time amplitude of a free particle in natural units with h ¯ = 1: M (xb − xa )2 (xb τb |xa τa ) = q . D exp − 2 τb − τa 2π(τb − τa )/M 1

"

#

(15.76)

We merely have to identify xb − xa ≡ R,

(15.77)

946


and replace τb − τa → Na, M → D/a.

(15.78) (15.79)

Thus we can describe a polymer with R2 ≪ Na2 by the path integral PL (R) =

Z

(

D D D x exp − 2a

L

Z

0

)

ds [x′ (s)]2 =

s

D −DR2 /2La . e 2πa

(15.80)

The number of time slices is here N [in contrast to (2.64) where it was N + 1), and the total length of the polymer is L = Na. Let us calculate the Fourier transformation of the distribution (15.80): P˜L (q) =

Z

dD R e−i q·R PL (R).

(15.81)

After a quadratic completion the integral yields 2 P˜L (q) = e−Laq /2D ,

(15.82)

with the power series expansion P˜L (q) =

∞ X

lq

(−1)

l=0

2l

l!

La 2D

l

.

(15.83)

Comparison with the moments (15.23) shows that we can rewrite this as P˜L (q) =

∞ X

(−1)l

l=0

D E q 2l Γ(D/2) 2l R . l! 22l Γ(D/2 + l)

(15.84)

This is a completely general relation: the expansion coefficients of the Fourier transform yield directly the moments of a function, up to trivial numerical factors specified by (15.84). The end-to-end distribution determines rather directly the structure factor of a dilute solution of polymers which is observable in static neutron and light scattering experiments: 1 S(q) = 2 L

Z

0

L

ds

Z

0

L

D

′

E

ds′ eiq·[x(s)−x(s )] .

(15.85)

The average over all polymers running from x(0) to x(L) can be written, more explicitly, as D

′

E

eiq·[x(s)−x(s )] =

Z

dD x(L)

Z

dD (x(s′ )−x(s)) ′

Z

dD x(0)

(15.86)

×PL−s′ (x(L)−x(s′ ))e−iq·x(s ) Ps′−s (x(s′ )−x(s))eiq·x(s) Ps−0 (x(s)−x(0)). H. Kleinert, PATH INTEGRALS

947

15.6 Path Integral for Continuous Gaussian Distribution

The integrals over initial and final positions give unity due to the normalization (15.4), so that we remain with D

iq·[x(s)−x(s′ )]

e

E

=

Z

dD R e−iq·R Ps′ −s (R).

(15.87)

Since this depends only on L′ ≡ |s′ − s| and not on s + s′ , we decompose the double R L integral in over s and s′ in (15.85) into 2 0 dL′ (L − L′ ) and obtain Z 2 ZL ′ ′ ′ S(q) = 2 dL (L − L ) dD R eiq·R(L ) PL′ (R), L 0

(15.88)

or, recalling (15.81), 2 ZL ′ dL (L − L′ )P˜L′ (q). L2 0

S(q) =

(15.89)

Inserting (15.82) we obtain the Debye structure factor of Gaussian random paths: S Gauss (q) =

2 −x x − 1 + e , x2

x≡

q 2 aL . 2D

(15.90)

This function starts out like 1 − x/3 + x2 /12 + . . . for small q and falls of like q −2 for q 2 ≫ 2D/aL. The Taylor coefficients are determined by the moments of the end-to-end distribution. By inserting (15.84) into (15.89) we obtain: ∞ X

Γ(D/2) 2 S(q) = (−1) q 2l 2 l!Γ(l + D/2) L2 l=0 l 2l

Z

0

L

D

E

dL′ (L − L′ ) R2l .

(15.91)

Although the end-to-end √ distribution (15.80) agrees with the true polymer distribution (15.1) for R ≪ Na, it is important to realize that the nature of the fluctuations in the two expressions is quite different. In the polymer expression, the length of each link ∆xn is fixed. In the sliced action of the path integral Eq. (15.80), AN = a

N X

M (∆xn )2 , a2 n=1 2

(15.92)

on the other hand, each small section fluctuates around zero with a mean square h(∆xn )2 i0 =

a a2 = . M D

(15.93)

Yet, if the end-to-end distance of the polymer is small compared to the completely stretched configuration, the distributions are practically the same. There exists a qualitative difference only if the polymer is almost completely stretched. While the polymer distribution vanishes for R > Na, the path integral (15.80) gives a nonzero value for arbitrarily large R. Quantitatively, however, the difference is insignificant since it is exponentially small (see Fig. 15.2).

948

15.7


Stiff Polymers

The end-to-end distribution of real polymers found in nature is never the same as that of a random chain. Usually, the joints do not allow for an equal probability of all spherical angles. The forward angles are often preferred and the polymer is stiff at shorter distances. Fortunately, if averaged over many links, the effects of the stiffness becomes less and less relevant. For a very long random chain with a finite stiffness one finds the same linear dependence of the square end-to-end distance on the length L = Na as for ideal random chains which has, according to Eq. (15.23), the Gaussian expectations: hR2 i = aL,

hR2l i =

(D + 2l − 2)!! (aL)l . l (D − 2)!!D

(15.94)

For a stiff chain, the expectation value hR2 i will increase aL to aeff L, where aeff is the effective bond length In the limit of a very large stiffness, called the rod limit, the law (15.94) turns into hR2 i ≡ L2 ,

hR2l i ≡ L2l ,

(15.95)

i.e., the effective bond length length aeff increases to L. This intuitively obvious statement can easily be found from the normalized end-to-end distribution, which coincides in the rod limit with the one-link expression (15.12): PLrod(R) =

1 δ(R − L), SD RD−1

(15.96)

and yields [recall (15.17)] n

hR i =

Z

D

d RR

n

PLrod (R)

=

Z

0

∞

dR Rn δ(R − L) = Ln .

(15.97)

By expanding PLrod (R) in powers of L, we obtain the series PLrod (R) =

∞ n X 1 n n (−1) L hR i ∂ n δ(R). SD RD−1 n=0 n! R

(15.98)

An expansion of this form holds for any stiffness: the moments of the distribution are the Taylor coefficients of the expansion of PL (R) into a series of derivatives of δ(R)-functions. Let us also calculate the Fourier transformation (15.81) of this distribution. Recalling (15.15), we find P˜Lrod(q) = P˜ rod (qL) ≡

Γ(D/2) JD/2−1 (qL). (qL/2)D/2−1

(15.99)

For an arbitrary rotationally symmetric PL (R) = PL (R), we simply have to superimpose these distributions for all R: P˜L (q) = SD

Z

∞ 0

dR RD−1 P˜ rod (qL)PL (R).


949

15.7 Stiff Polymers

This is simply proved by decomposing and performing the Fourier transformation R∞ R∞ ′ ′ rod ′ (15.81) on PL (R) = 0 dR δ(R − R )PR′ (R)=SD 0 dR′ R′D−1 PRrod ′ (R)PL (R ), and performing the Fourier transformation (15.81) on PRrod In D = 3 dimensions, ′ (R). (15.100) takes the simple form: P˜L (q) = 4π

Z

∞

0

dR R2

sin qR PL (R). qR

(15.101)

Inserting the power series expansion for the Bessel function2 Jν (z) =

ν X ∞ z

2

l=0

(−1)k (z/2)2l l!Γ(ν + l + 1)

(15.102)

into (15.99), and this into (15.100), we obtain P˜L (q) =

∞ X

l

(−1)

l=0

2l

q 2

Γ(D/2) SD l!Γ(D/2 + l)

Z

0

∞

dR RD−1 R2l PL (R),

(15.103)

in agreement with the general expansion (15.84). The same result is obtained by inserting into (15.103) the expansion (15.98) and using the integrals R∞ m n n 0 dR R ∂R δ(R) = δmn (−1) n!, which are proved by n partial integrations. The structure factor of a completely stiff polymer (rod limit) is obtained by inserting (15.99) into (15.89). The resulting S rod (q) depends only on qL: S

rod

!D/2

4−2D 2 (qL)= 2 2 + q L qL

Γ(D/2)JD/2−2 (qL) + 2F (1/2; 3/2, D/2;−q 2L2/4),(15.104)

where F (a; b, c; z) is the hypergeometric function (1.450). For D = 3, the integral R (15.89) reduces to (2/L2 ) 0L dL′ (L − L′ )(sin qL′ )/qL′ , as in the similar equation (15.9), and the result is simply S

rod

2 (z) = 2 [cos z − 1 + z Si(z)] , z

Si(z) ≡

Z

0

z

dt sin t. t

(15.105)

This starts out like 1 − z 2 /36 + z 4 /1800 + . . . . For large z we use the limit of the sine integral3 Si(z) → π/2 to find S rod (q) → π/qL. For of an arbitrary rotationally symmetric end-to-end distribution PL (R), the structure factor can be expressed, by analogy with (15.100), as a superposition of rod limits: SL (q) = SD

Z

0

∞

dR RD−1 S rod (qR)PL (R).

(15.106)

When passing from long to short polymers at a given stiffness, there is a crossover between the moments (15.94) and (15.95) and the behaviors of the structure function. Let us study this in detail. 2 3

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.440. ibid., Formulas 8.230 and 8.232.

950


15.7.1

Sliced Path Integral

The stiffness of a polymer pictured in Fig. 15.1 may be parameterized by the bending energy N κ X (un − un−1 )2 , 2a n=1

N Ebend =

(15.107)

where un are the unit vectors specifying the directions of the links. The initial and final link directions of the polymer have a distribution −1 1 NY (ub L|ua 0) = A n=1

"Z

N dun κ X exp − (un − un−1 )2 , A 2akB T n=1

#

"

#

(15.108)

where A is some normalization constant, which we shall choose such that the measure of integration coincides with that of a time-sliced path integral near a unit sphere in Eq. (8.150). Comparison if the bending energy (15.107) with the Euclidean action (8.151) we identify Mr 2 κ = , h ¯ǫ akB T and see that we must replace N → N − 1 and set A=

D−1

q

2πakB T /κ

(15.109)

.

(15.110)

The result of the integrations in (15.108) is then known from Eq. (8.155): (ub L|ua 0) =

∞ h X

I˜l+D/2−1 (h)

iN X

∗ Ylm (ub )Ylm (ua ),

m

l=0

h≡

κ , akB T

(15.111)

with the modified Bessel function I˜l+D/2−1 (z) of Eq. (8.11). The partition function of the polymer is obtained by integrating over all final and averaging over all initial link directions [1]: N N κ X d ua Y dun ZN = exp − (un − un−1 )2 SD n=1 A 2akB T n=1 Z Z d ua = d ub (ub L|ua 0). SD Inserting here the spectral representation (15.111) we find

"Z

Z

ZN = I˜D/2−1

κ akB T

N

#

=

"s

"

2πκ −κ/akB T κ e ID/2−1 akB T akB T

Knowing this we may define the normalized distribution function 1 PN (ub , ua ) = (ub L|ua 0), ZN whose integral over ua as well as over ua is unity: Z

dub PN (ub , ua ) =

Z

dua PN (ub , ua ) = 1.

#

#N

(15.112)

.

(15.113)

(15.114)

(15.115)


951

15.7 Stiff Polymers

15.7.2

Relation to Classical Heisenberg Model

The above partition function is closely related to the partition function of the onedimensional classical Heisenberg model of ferromagnetism which is defined by ZNHeis

≡

Z

N Z N d ua Y J X dun exp un · un−1 , SD n=1 kB T n=1 #

"

(15.116)

where J are interaction energies due to exchange integrals of electrons in a ferromagnet. This differs from (15.112) by a trivial normalization factor, being equal to s

ZNHeis = 

2πJ kB T

2−D

ID/2−1

J kB T

 N 

.

(15.117)

Identifying J ≡ κ/a we may use the Heisenberg partition functions for all calculations of stiff polymers. As an example take the correlation function between neighboring tangent vectors hun · un−1 i. In order to calculate this we observe that the partition function (15.116) can just as well be calculated exactly with a slight modification that the interaction strength J of the Heisenberg model depends on the link n. The result is the corresponding generalization of (15.117): N Y

s

2πJn  ZNHeis (J1 , . . . , JN ) = kB T n=1

2−D

ID/2−1



Jn  . kB T

(15.118)

This expression may be used as a generating function for expectation values hun · un−1 i which measure the degree of alignment of neighboring spin directions. Indeed, we find directly

ID/2 (J/kB T ) dZNHeis (J1 , . . . , JN ) hun · un−1 i = (kB T ) = . dJn ID/2−1 (J/kB T ) Jn ≡J

(15.119)

This expectation value measures directly the internal energy per link of te chain. Indeed, since the free energy is FN = −kB T log ZNHeis , we obtain [recall (1.545)] EN = N hun · un−1 i = N

ID/2 (J/kB T ) . ID/2−1 (J/kB T )

(15.120)

Let us also calculate the expectation value of the angle between next-to-nearest neighbors hun+1 · un−1 i. We do this by considering the expectation value

ID/2 (J/kB T ) d2 ZNHeis (J1 , . . . , JN ) h(un+1 · un )(un · un−1 )i = (kB T )2 = dJn+1 dJn ID/2−1 (J/kB T ) Jn ≡J "

#2

.

(15.121)

Then we prove that the left-hand side is in fact equal to the desired expectation value hun+1 · un−1 i. For this we decompose the last vector un+1 into a component

952


θn+1,n un+1

θn+1,n−1

un θn,n−1

un−1

Figure 15.3 Neighboring links for the calculation of expectation values.

parallel to un and a component perpendicular to it: un+1 = (un+1 ·un ) un +u⊥ n+1 . The corresponding decomposition of the expectation value hun+1 ·un−1 i is hun+1 ·un−1 i = h(un+1 ·un )(un ·un−1)i +hu⊥ n+1 ·un i. Now, the energy in the Boltzmann factor depends only on (un+1 · un ) + (un · un−1 ) = cos θn+1,n + cos θn,n−1 , so that the integral over u⊥ n+1 runs over a surface of a sphere of radius sin θn+1,n in D − 1 dimensions [recall (8.116)] and the Boltzmann factor does not depend on the angles. The integral ⊥ receives therefore equal contributions from u⊥ n+1 and −un+1 , and vanishes. This proves that "

ID/2 (J/kB T ) hun+1 · un−1 i = hun+1 · un i hun · un−1 i = ID/2−1 (J/kB T )

#2

,

(15.122)

and further, by induction, that "

ID/2 (J/kB T ) hul · uk i = ID/2−1 (J/kB T )

#|l−k|

.

(15.123)

For the polymer, this implies an exponential falloff hul · uk i = e−|l−k|a/ξ ,

(15.124)

where ξ is the persistence length "

#

ID/2 (κ/akB T ) ξ = −a/ log . ID/2−1 (κ/akB T )

(15.125)

For D = 3, this is equal to ξ=−

a log [coth(κ/akB T ) − akB T /κ]

.

(15.126)

Knowing the correlation functions it is easy to calculate the magnetic susceptibility. The total magnetic moment is M=a

N X

un ,

(15.127)

n=0

so that we find the total expectation value D

E

M2 = a2 (N + 1)

−(N +1)a/ξ 1 + e−a/ξ 2 −a/ξ 1 − e − 2a e . 1 − e−a/ξ (1 − e−a/ξ )2

(15.128)

The susceptibility is directly proportional to this. For more details see [2]. H. Kleinert, PATH INTEGRALS

953

15.7 Stiff Polymers

15.7.3

End-to-End Distribution

A modification of the path integral (15.108) yields the distribution of the end-to-end distance at given initial and final directions of the polymer links: R = xb − xa = a

N X

(15.129)

un

n=1

of the stiff polymer: −1 1 1 NY PN (ub , ua ; R) = ZN A n=2

N X dun (D) δ (R − a un ) A n=1

"Z

#

−1 κ NX × exp − (un+1 − un )2 , 2akB T n=1

"

#

(15.130)

whose integral over R leads back to the distribution PN (ub , ua ): Z

dD R PN (ub , ua ; R) = PN (ub , ua ).

(15.131)

If we integrate in (15.130) over all final directions and average over the initial ones, we obtain the physically more accessible end-to-end distribution PN (R) =

Z

dub

Z

dua PN (ub , ua ; R). SD

(15.132)

The sliced path integral, as it stands, does not yet give quite the desired probability. Two small corrections are necessary, the same that brought the integral near the surface of a sphere to the path integral on the sphere in Section 8.9. After including these, we obtain a proper overall normalization of PN (ub , ua ; R).

15.7.4

Moments of End-to-End Distribution

Since the δ (D) -function in (15.130) contains vectors un of unit length, the calculation of the complete distribution is not straightforward. Moments of the distribution, however, which are defined by the integrals 2l

hR i =

Z

dD R R2l PN (R),

(15.133)

are relatively easy to find from the multiple integrals 1 hR i = ZN 2l

× R

2l

Z

1 d R A D

"

Z

dub

κ exp − 2akB T

NY −1

n=2 N −1 X n=1

"Z

dun A

N X dua (D) δ (R − aun ) SD n=1

# "Z

(un+1 − un )

2

#

#

.

(15.134)

954


Performing the integral over R gives 1 1 hR2l i = ZN A

Z

dub

NY −1 n=2

"Z

dun A

# "Z

dua SD

#

a

N X

n=1

un

!2l

−1 κ NX × exp − (un+1 − un )2 . 2akB T n=1

"

#

(15.135)

Due to the normalization property (15.115), the trivial moment is equal to unity: h1i =

15.8

Z

D

d R PN (R) =

Z

dub

Z

dua PN (ub , ua |L) = 1. SD

(15.136)

Continuum Formulation

Some properties of stiff polymers are conveniently studied in the continuum limit of the sliced path integral (15.108), in which the bond length a goes to zero and the link number to infinity so that L = Na stays constant. In this limit, the bending energy (15.107) becomes Ebend

κ = 2

Z

L

0

ds (∂s u)2 ,

(15.137)

where u(s) =

d x(s), ds

(15.138)

is the unit tangent vector of the space curve along which the √ polymer runs. The parameter s is the arc length of the line elements, i.e., ds = dx2 .

15.8.1

Path Integral

If the continuum limit is taken purely formally on the product of integrals (15.108), we obtain a path integral (ub L|ua 0) =

Z

Du e−(κ/2kB T )

RL 0

ds [u′ (s)]2

,

(15.139)

This coincides with the Euclidean version of a path integral for a particle on the surface of a sphere. It is a nonlinear σ-model (recall p. 655). The result of the integration has been given in Section 8.9 where we found that it does not quite agree with what we would obtain from the continuum limit of the discrete solution (15.111) using the limiting formula (8.156), which is ∞ X

!

X kB T ∗ P (ub , ua |L) = exp −L L2 Ylm (ub )Ylm (ua ), 2κ m l=0

(15.140)


955

15.8 Continuum Formulation

with L2 = (D/2 − 1 + l)2 − 1/4.

(15.141)

For a particle on a sphere, the discrete expression (15.130) requires a correction since it does not contain the proper time-sliced action and measure. According the Section 8.9 the correction replaces L2 by the eigenvalues of the square angular ˆ 2, momentum operator in D dimensions, L ˆ 2 = l(l + D − 2). L2 → L

(15.142)

After this replacement the expectation of the trivial moment h1i in (15.136) is equal to unity since it gives the distribution (15.140) the proper normalization: Z

dub P (ub , ua |L) = 1.

(15.143)

This follows from the integral Z

dub

X

∗ Ylm (ub )Ylm (ua ) = δl0 ,

(15.144)

m

ˆ 2 in (15.140) instead of L2 , no extra which was derived in (8.249). Thus, with L ∗ normalization factor is required. The sum over Ylm (u2 )Ylm (u1 ) may furthermore be rewritten in terms of Gegenbauer polynomials using the addition theorem (8.125), so that we obtain ∞ X

!

kB T ˆ 2 1 2l + D − 2 (D/2−1) L Cl P (ub , ua |L) = exp −L (u2 u1 ). 2κ SD D − 2 l=0

15.8.2

(15.145)

Correlation Functions and Moments

We are now ready to evaluate the expectation values of R2l . In the continuum approximation we write 2l

R =

"Z

0

L

ds u(s)

#2l

.

(15.146)

The expectation value of the lowest moment hR2 i is given by the double-integral over the correlation function hu(s2 )u(s1 )i: hR2 i =

Z

0

L

ds2

Z

L

0

ds1 hu(s2 )u(s1 )i = 2

Z

0

L

ds2

Z

0

s2

ds1 hu(s2 )u(s1 )i.

(15.147)

The correlation function is calculated from the path integral via the composition law as in Eq. (3.298), which yields here dua du2 du1 SD × P (ub , u2 |L − s2 ) u2 P (u2 , u1 |s2 − s1 ) u1 P (u1 , ua |s1 ). (15.148)

hu(s2 )u(s1 )i =

Z

dub

Z

Z

Z

956


The integrals over ua and ub remove the initial and final distributions via the normalization integral (15.143), leaving hu(s2 )u(s1 )i =

Z

du2

Z

du1 u2 u1 P (u2 , u1 |s2 − s1 ). SD

(15.149)

Due to the manifest rotational invariance, the normalized integral over u1 can be omitted. By inserting the spectral representation (15.145) with the eigenvalues (15.142) we obtain hu(s2 )u(s1 )i = =

X

Z

du2 u2 u1 P (u2 , u1 |s2 − s1 )

ˆ 2 /2κ −(s2 −s1 )kB T L

e

l

"Z

#

1 2l + D − 2 (D/2−1) du2 u2 u1 (u2 u1 ) . (15.150) Cl SD D − 2

We now calculate the integral in the brackets with the help of the recursion relation (15.152) for the Gegenbauer functions4 (ν)

zCl (z) =

h i 1 (ν) ν (2ν + l − 1)Cl−1(z) + (l + 1)Cl+1 (z) . 2(ν + l)

(15.151)

Obviously, the integral over u2 lets only the term l = 1 survive. This, in turn, involves the integral Z

(D/2−1)

du2 C0

(cos θ) = SD .

(15.152)

The l = 1 -factor D/(D − 2) in (15.150) is canceled by the first l = 1 -factor in the recursion (15.151) and we obtain the correlation function "

#

kB T hu(s2 )u(s1 )i = exp −(s2 − s1 ) (D − 1) , 2κ

(15.153)

ˆ 2 = l(l + D − 2) at l = 1. The where D − 1 in the exponent is the eigenvalue of L correlation function (15.153) agrees with the sliced result (15.124) if we identify the continuous version of the persistence length (15.126) with ξ ≡ 2κ/kB T (D − 1).

(15.154)

Indeed, taking the limit a → 0 in (15.125), we find with the help of the asymptotic behavior (8.12) precisely the relation (15.154). After performing the double-integral in (15.147) we arrive at the desired result for the first moment: n

h

hR2 i = 2 ξL − ξ 2 1 − e−L/ξ 4

io

.

(15.155)



957

15.8 Continuum Formulation

This is valid for all D. The result may be compared with the expectation value of the squared magnetic moment of the Heisenberg chain in Eq. (15.128), which reduces to this in the limit a → 0 at fixed L = (N + 1)a. For small L/ξ, the second moment (15.155) has the large-stiffness expansion 

1L 1 hR2 i = L2 1 − + 3ξ 12

L ξ

!2

1 − 60

L ξ

!3



+ . . . ,

(15.156)

the first term being characteristic for a completely stiff chain [see Eq. (15.95)]. For large L/ξ, on the other hand, we find the small-stiffness expansion !

ξ hR i ≈ 2ξL 1 − + ... , L 2

(15.157)

where the dots denote exponentially small terms. The first term agrees with relation (15.94) for a random chain with an effective bond length aeff = 2ξ =

4 κ . D − 1 kB T

(15.158)

The calculation of higher expectations hR2l i becomes rapidly complicated. Take, for instance, the moment hR4 i, which is given by the quadruple integral over the four-point correlation function 4

hR i = 8

Z

0

L

ds4

Z

s4 0

Z

ds3

0

s3

ds2

Z

s2

0

ds1 δi4 i3 i2 i1 hui4 (s4 )ui3 (s3 )ui2 (s2 )ui1 (s1 )i, (15.159)

with the symmetric pair contraction tensor of Eq. (15.22): δi4 i3 i2 i1 ≡ (δi4 i3 δi2 i1 + δi4 i2 δi3 i1 + δi4 i1 δi3 i2 ) .

(15.160)

The factor 8 and the symmetrization of the indices arise when bringing the integral R4 =

Z

0

L

ds4

Z

0

L

ds3

Z

L 0

ds2

Z

L

0

ds1 (u(s4 )u(s3 ))(u(s2 )u(s1 ))

(15.161)

to the s-ordered form in (15.159). This form is needed for the s-ordered evaluation of the u-integrals which proceeds by a direct extension of the previous procedure for hR2 i. We write down the extension of expression (15.148) and perform the integrals over ua and ub which remove the initial and final distributions via the normalization integral (15.143), leaving δi4 i3 i2 i1 times an integral [the extension of (15.149)]: du1 (15.162) SD × ui4 ui3 ui2 ui1 P (u4 , u3 |s4 − s3 )P (u3 , u2 |s3 − s2 )P (u2 , u1 |s2 − s1 ) .

hui4 (s4 )ui3 (s3 )ui2 (s2 )ui1 (s1 )i =

Z

du4

Z

du3

Z

du2

Z

958


The normalized integral over u1 can again be omitted. Still, the expression is complicated. A somewhat tedious calculation yields 4(D + 2) 2 2 D 2 + 6D − 1 D − 7 −L/ξ − (15.163) hR i = L ξ − 8Lξ 3 e D D2 D+1 " # 3 2 (D + 5)2 −L/ξ (D − 1)5 −2DL/(D−1)ξ 4 D + 23D − 7D + 1 + 4ξ −2 e + 3 e . D3 (D + 1)2 D (D + 1)2 !

4

For small values of L/ξ , we find the large-stiffness expansion 

2L 25D − 17 hR4 i = L4 1 − + 3ξ 90(D − 1)

L ξ

!2

7D 2 − 8D + 3 −4 315(D − 1)2

L ξ

!3



+ . . .  ,(15.164)

the leading term being equal to (15.95) for a completely stiff chain. In the opposite limit of large L/ξ, the small-stiffness expansion is hR4 i = 4



2

3

2

D + 23D − 7D + 1 D+2 2 2 D + 6D − 1 ξ L ξ 1−2 + D D(D + 2) L D 2 (D + 2)

ξ L

!2 

+ ...

,

(15.165) where the dots denote exponentially small terms. The leading term agrees again with the expectation hR4 i of Eq. (15.94) for a random chain whose distribution is (15.49) with an effective link length aeff = 2ξ of Eq. (15.158). The remaining terms are corrections caused by the stiffness of the chain. It is possible to find a correction factor to the Gaussian distribution which maintains the unit normalization and ensures that the moment hR2 i has the small-ξ exact expansion (15.157) whereas hR2 i is equal to (15.165) up to the first correction term in ξ/L. This has the form PL (R) =

s

D

2D−1 ξ 3D−1 R2 D(4D−1) R4 D 2 e−DR /4Lξ 1− − . (15.166) + 4πLξ 4 L 4 L2 16(D+2) ξL3 (

)

In three dimensions, this was first written down by Daniels [3]. It is easy to match also the moment hR4 i by adding in the curly brackets the following terms 1−7D+23D 2 +D 3 D + 2 ξ 2 R2 1 R4 1 + + . D+1 8D L2 ξL 32 L4 "

!

#

(15.167)

These terms do not, however, improve the fits to Monte Carlo data for ξ > 1/10L, since the expansion is strongly divergent. From the approximation (15.166) with the additional term (15.167) we calculate the small-stiffness expansion of all even and odd moments as follows: 

where A1 = n

2n Γ(D/2 + n/2) n n  ξ ξ hRn i = L ξ 1 + A + A 1 2 D n/2 Γ(D/2) L L n − 2 − 2d2 − 4d (n − 1) , 4d (2 + d)

A2 = n(n − 2)

!2



+ ... ,

1 − 7d + 23d2 + d3 . 8d2 (1 + d)

(15.168)

(15.169)


959

15.9 Schrödinger Equation and Recursive Solution for Moments

15.9

Schr¨ odinger Equation and Recursive Solution for End-to-End Distribution Moments

The most efficient way of calculating the moments of the end-to-end distribution proceeds by setting up a Schrödinger equation satisfied by (15.112) and solving it recursively with similar methods as developed in 3.19 and Appendix 3C.

15.9.1

Setting up the Schr¨ odinger Equation

In the continuum limit, we write (15.112) as a path integral [compare (15.139)] Z

Z

Z

PL (R) ∝ dub dua D

D−1

uδ

(D)

Z

R−

0

L

!

ds u(s) e−(¯κ/2)

RL 0

ds [u′ (s)]2

,

(15.170)

where we have introduced the reduced stiffness κ ¯=

κ ξ = (D − 1) , kB T 2

(15.171)

for brevity. After a Fourier representation of the δ-function, this becomes PL (R) ∝

Z

i∞

−i∞

dD λ κ¯ e 2πi

·R/2

Z

dub

Z

dua (ub L|ua 0) ,

(15.172)

where (ub L|ua 0) ≡

Z

u(L)=ub

u(0)=ua

D D−1u e−(¯κ/2)

RL 0

ds {[u′ (s)]2 + ·u(s)}

(15.173)

describes a point particle of mass M = κ ¯ moving on a unit sphere. In contrast to the discussion in Section 8.7 there is now an additional external field which prevents us from finding an exact solution. However, all even moments hRi1 Ri2 · · · Ri2l i of the end-to-end distribution (15.172)Rcan be extracted from the expansion coefficients R in powers of λi of the integral dub dua over (15.173). The presence of these directional integrals permits us to assume the external electric field to point in the z-direction, = λˆ z , and the D E or the Dth direction in D-dimensions. Then R R moments R2l are proportional to the derivatives (2/¯ κ)2l ∂λ2l dub dua (ub L|ua 0) . The proportionality factors have been calculated in Eq. (15.84). It is unnecessary to know these since we can always use the rod limit (15.95) to normalize the moments. To find these derivatives, we perform a perturbation expansion of the path integral (15.173) around the solvable case λ = 0. In natural units with κ ¯ = 1, the path integral (15.173) solves obviously the imaginary-time Schrödinger equation 1 1 d − ∆u + ·u+ 2 2 dτ

!

(u τ |ua 0) = 0,

(15.174)

960


where ∆u is the Laplacian on a unit sphere. In the probability distribution (15.211), only the integrated expression ψ(z, τ ; λ) ≡

Z

dua (u τ |ua 0)

(15.175)

appears, which is a function of z = cos θ only, where θ is the angle between u and the electric field . For ψ(z, τ ; λ), the Schrödinger equation reads ˆ ψ(z, τ ; λ) = − d ψ(z, τ ; λ), H dτ

(15.176)

with the simpler Hamiltonian operator Î = − 1 ∆ + λ z ˆ ≡ H ˆ0 + λ H H 2 " # 2 1 d 1 2 d = − (1 − z ) 2 − (D − 1)z + λz. 2 dz dz 2

(15.177)

Now the desired moments (15.135) can be obtained from the coefficient of λ2l /(2l)! of the power series expansion of the z-integral over (15.175) at imaginary time τ = L: f (L; λ) ≡

15.9.2

1

Z

−1

dz ψ(z, L; λ).

(15.178)

Recursive Solution of Schr¨ odinger Equation.

The function f (L; λ) has a spectral representation f (L; λ) =

∞ X l=0

R1

R

(l)† (z) exp −E (l) L −1 dz ϕ

R1

−1

1 −1

dza ϕ(l) (za )

dz ϕ(l)† (z) ϕ(l) (z)

,

(15.179)

where ϕ(l) (z) are the solutions of the time-independent Schrödinger equation ˆ (l) (z) = E (l) ϕ(l) (z). Applying perturbation theory to this problem, we start Hϕ ˆ 0 = −∆/2, which are given from the eigenstates of the unperturbed Hamiltonian H D/2−1 (l) by the Gegenbauer polynomials Cl (z) with the eigenvalues E0 = l(l+D −2)/2. Following the methods explained in 3.19 and Appendix 3C we now set up a recursion scheme for the perturbation expansion of the eigenvalues and eigenfunctions [4].Starting point is the expansion of energy eigenvalues and wave-functions in powers of the coupling constant λ: E

(l)

=

∞ X

j=0

(l) ǫj

j

λ,

(l)

|ϕ i =

∞ X

l′ ,i=0

(l)

γl′ ,i λi αl′ |l′ i .

(15.180)

The wave functions ϕ(l) (z) are the scalar products hz|ϕ(l) (λ)i. We have inserted extra normalization constants αl′ for convenience which will be fixed soon. The unperturbed state vectors |li are normalized to unity, but the state vectors |ϕ(l) i of H. Kleinert, PATH INTEGRALS


961

the interacting system will be normalized in such a way, that hϕ(l) |li = 1 holds to all orders, implying that (l)

(l)

γl,i = δi,0

γk,0 = δl,k .

(15.181)

Inserting the above expansions into the Schrödinger equation, projecting the result onto the base vector hk|αk , and extracting the coefficient of λj , we obtain the equation (l) (k)

γk,i ǫ0 +

∞ X

i X αj (l) (l) (l) Vk,j γj,i−1 = ǫj γk,i−j , α j=0 k j=0

(15.182)

where Vk,j = λhk|z|ji are the matrix elements of the interaction between unperturbed states. For i = 0, Eq. (15.182) is satisfied identically. For i > 0, it leads to the following two recursion relations, one for k = l: (l)

ǫi =

X

(l)

γl+n,i−1Wn(l) ,

(15.183)

n=±1

the other one for k 6= l: (l)

γk,i =

i−1 X

j=1

(l) (l)

ǫj γk,i−j − (k) ǫ0

X

(l)

γk+n,i−1Wn(l)

n=±1 (l) − ǫ0

,

(15.184)

where only n = −1 and n = 1 contribute to the sums over n since Wn(l) ≡

αl+n hl| z |l + ni = 0, for n 6= ±1. αl

(15.185)

The vanishing of Wn(l) for n 6= ±1 is due to the band-diagonal form of the matrix of the interaction z in the unperturbed basis |ni. It is this property which makes the sums in (15.183) and (15.184) finite and leads to recursion relations with a finite (l) (l) number of terms for all ǫi and γk,i . To calculate Wn(l) , it is convenient to express hl| z |l + ni as matrix elements between unnormalized noninteracting states |n} as {l|z|l + n} hl| z |l + ni = q , {l|l}{l + n|l + n}

(15.186)

where expectation values are defined by the integrals Z

{k|F (z)|l} ≡

1 −1

D/2−1

Ck

D/2−1

(z) F (z) Cl

(z)(1 − z 2 )(D−3)/2 dz,

(15.187)

from which we find5 {l|l} = 5

24−D Γ(l + D − 2) π . l! (2l + D − 2) Γ(D/2 − 1)2

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 7.313.1 and 7.313.2.

(15.188)

962


Expanding the numerator of (15.186) with the help of the recursion relation (15.151) for the Gegenbauer polynomials written now in the form (l + 1)|l + 1} = (2l + D − 2) z |l} − (l + D − 3)|l − 1},

(15.189)

we find the only non-vanishing matrix elements to be l+1 {l + 1|l + 1}, 2l + D − 2 l+D−3 {l − 1|z|l} = {l − 1|l − 1} . 2l + D − 2 {l + 1|z|l} =

(15.190) (15.191)

Inserting these together with (15.188) into (15.186) gives hl|z|l − 1i =

v u u t

l(l + D − 3) , (2l + D − 2)(2l + D − 4)

(15.192)

and a corresponding result for hl|z|l + 1i. We now fix the normalization constants αl′ by setting (l)

W1 =

αl+1 hl| z |l + 1i = 1 αl

(15.193)

for all l, which determines the ratios v u

u (l + 1) (l + D − 2) αl . = hl| z |l + 1i = t αl+1 (2l + D) (2l + D − 2)

(15.194)

Setting further α1 = 1, we obtain 

αl = 

l Y

j=1

1/2

(2l + D − 2)(2l + D − 4)  l(l + D − 3)

.

(15.195)

Using this we find from (15.185) the remaining nonzero Wn(l) for n = −1: (l)

W−1 =

l(l + D − 3) . (2l + D − 2)(2l + D − 4)

(15.196) (l)

We are now ready to solve the recursion relations of (15.183) and (15.184) γk,i and (l) (l) ǫi order by order in i. For the initial order i = 0, the values of the γk,i are (l) given by Eq. (15.181). The coefficients ǫi are equal to the unperturbed energies (l) (l) ǫ0 = E0 = l(l + D − 2)/2. For each i = 1, 2, 3, . . . , there is only a finite number (l) (l) of non-vanishing γk,j and ǫj with j < i on the right-hand sides of (15.183) and (l) (l) (15.184) which allows us to calculate γk,i and ǫi on the left-hand sides. In this way it is easy to find the perturbation expansions for the energy and the wave functions to high orders. H. Kleinert, PATH INTEGRALS

963


Inserting the resulting expansions (15.180) into Eq. (15.179), only the totally symmetric parts in ϕ(l) (z) will survive the integration in the numerators, i.e., we may insert only ϕ(l) symm (z)

=

hz|ϕ(l) symm i

=

∞ X i=0

(l)

γ0,i λi hz|0i .

(15.197)

(l)

The denominators of (15.179) become explicitly l′ ,i |γl′ ,i αl′ |2 λ2i , where the sum over i is limited by power of λ2 up to which we want to carry the perturbation series; also l′ is restricted to a finite number of terms only, because of the band(l) diagonal structure of the γl′ ,i . Extracting the coefficients of the power expansion in λ from (15.179) we obtain all desired moments of the end-to-end distribution, in particular the second and fourth moments (15.155) and (15.164). Higher even moments are easily found with the help of a Mathematica program, which is available for download in notebook form [5]. The expressions are too lengthy to be written down here. We may, however, expand the even moments hRn i in powers of L/ξ to find a general large-stiffness expansion valid for all even and odd n: P

n L n (−13 − n + 5D (1 + n)) L2 L3 L4 hRn i = 1 − + − a + a 3 3 4 4 +. . . , Ln 6ξ 360 (D − 1) ξ2 ξ ξ

(15.198)

where 444−63n+15n2 +7D 2 (4+15n+5n2)+2D (−124−141n+7n2) , 45360(D − 1)2 n 2 3 a4 = D + D d + D d + D d , 0 1 2 3 5443200(d − 1)3

a3 = n

(15.199)

with

D0 = 3 −5610+2921n−822n2 +67n3 , D1 = 8490+12103n−3426n2 +461n3 ,

D2 = 45 −2−187n−46n2 +7n3 ,

The lowest odd moments are, up to order l4 , hR i = 1− L

3 R = 1− L3

15.9.3

D4 = 35 −6+31n+30n2 +5n3 .

(15.200)

l 5D−7 2 33 − 43D + 14D2 3 861 − 1469D + 855D2 − 175D3 4 + l − l − l ... , 2 3 6 180(D−1) 3780(D − 1) 453600 (D − 1)

l 5D−4 2 195−484D+329D2 3 609−2201D+2955D2 −1435D3 4 + l − l − l ... . 2 3 2 30 (D−1) 7560 (−1 + d) 151200 (D−1)

From Moments to End-to-End Distribution for D=3

We now use the recursively calculated moments to calculate the end-to-end distribution itself. It can be parameterized by an analytic function of r = R/L [4]: PL (R) ∝ r k (1 − r β )m ,

(15.201)

964


whose moments are exactly calculable:

3+k +2l 3+k Γ +m+1 Γ

2l β β . r = 3+k +2l Γ 3+k Γ + m + 1 β β

(15.202)

We now adjust the three parameters k, β, and m to fit the three most important moments of this distribution to the exact values, ignoring all others. If the distances were distributed uniformly over the interval r ∈ [0, 1], D the E moments would 2l unif 2l be hr i = 1/(2l + 2). Comparing our exact moments r (ξ) with those of the uniform distribution we find that hr 2l i(ξ)/hr 2liunif has a maximum for n close to nmax (ξ) ≡ 4ξ/L. We identify the most important moments as those with n = nmax (ξ) and n = nmax (ξ) ± 1. If nmax (ξ) ≤ 1, we choose the lowest even moments hr 2 i, hr 4 i, and hr 6 i. In particular, we have fitted hr 2i, hr 4 i and hr 6 i for small persistence length ξ < L/2. For ξ = L/2, we have started with hr 4 i, for ξ = L with hr 8 i and for ξ = 2L with hr 16 i, including always the following two higher even moments. After these adjustments, whose results are shown in Fig. 15.4, we obtain the distributions shown in Fig.15.6 for various persistence lengths ξ. They are in excellent agreement with the Monte Carlo data (symbols) and better than the one-loop perturbative results (thin curves) of Ref. [6], which are good only for very stiff polymers. The Mathematica program to do these fits are available from the internet address given in Footnote 5.

k

17.5 15 12.5 10 7.5 5 2.5

80

22 20 18 16 14 12

β

660

40 20 0.5

1

ξ/L

1.5

2

0.5

1

ξ/L

1.5

2

m

0.5

1

1.5

ξ/L

2

Figure 15.4 Paramters k, β, and m for a best fit of end-to-end distribution (15.201).

For small persistence lengths ξ/L = 1/400, 1/100, 1/30, the curves are well approximated by Gaussian random chain distributions on a lattice with lattice constant 2 aeff = 2ξ, i.e., PL (R) → e−3R /4Lξ [recall (15.75)]. This ensures that the lowest moment hR2 i = aeff L is properly fitted. In fact, we can easily check that our fitting program yields for the parameters k, β, m in the end-to-end distribution (15.201) the ξ → 0 behavior: k → −ξ, β → 2 + 2ξ, m → 3/4ξ, so that (15.201) tends to the correct Gaussian behavior. In the opposite limit of large ξ, we find that k → 10ξ−7/2, β → 40ξ+5, m → 10, which has no obvious analytic approach to the exact limiting behavior PL (R) → (1− r)−5/2 e−1/4ξ(1−r) , although the distribution at ξ = 2 is fitted numerically extremely well. H. Kleinert, PATH INTEGRALS


965

The distribution functions can be inserted into Eq. (15.89) to calculate the structure factors shown in Fig. 15.5. They interpolate smoothly between the Debye limit (15.90) and the stiff limit (15.105). 1 0.8 0.6 S(q) 0.4 0.2 5

ξ/L = 2 ξ/L = 1 ξ/L = 1/2 ξ/L = 1/5, . . . , 1/400 √ q ξ 20 30 10 15 25

Figure 15.5 Structure functions for different persistence lengths ξ/L = 1/400, 1/100, 1/30, 1/10, 1/5, 1/2, 1, 2, (from bottom to top) following from the end-to-end distributions in Fig. 15.6. The curves with low ξ almost coincide in this plot over the ξ-dependent absissa. The very stiff curves fall off like 1/q, the soft ones like 1/q 2 [see Eqs. (15.105) and (15.90)].

15.9.4

Large-Stiffness Approximation to End-to-End Distribution

The full end-to-end distribution (15.132) cannot be calculated exactly. It is, however, quite easy to find a satisfactory approximation for large stiffness [6]. We start with the expression (15.170) for the end-to-end distribution PL (R). In Eq. (3.230) we have shown that a harmonic path integral including the integrals over the end points can be found, up to a trivial factor, by summing over all paths with Neumann boundary conditions. These are satisfied if we expand the fields u(s) into a Fourier series of the form (2.444): u(s) = u0 + (s) = u0 +

∞ X

un cos νn s,

νn = nπ/L.

(15.203)

n=1

Let us parametrize the unit vectors u in D dimensions in terms of the first D − 1 -dimensional coordinates uµ ≡ q µ with µ = 1, . . . , D − 1. The Dth component is then given by a power series σ≡

q

1 − q 2 ≈ 1 − q 2 /2 − (q 2 )2 /8 + . . . .

(15.204)

The we approximate the action harmonically as follows: A = A κ ¯ ≈ 2

(0)

Z

0

+A L

int

κ ¯ = 2

Z

0

L

1 ds [u′ (s)]2 + δ(0) log(1 − q 2 ) 2

1 ds [q ′(s)]2 − δ(0) 2

Z

0

L

ds q 2.

(15.205)

966


√ The last term comes from the invariant measure of integration dD−1 q/ 1 − q 2 [recall (10.635) and (10.640)]. Assuming, as before, that R points into the z, or Dth, direction we factorize δ

(D)

R−

L

Z

0

ds u(s)

!

= δ R−L+ Z

× δ (D−1)

L

0

L

Z

0

ds

!

1 2 1 q (s) + [q 2 (s)]2 + . . . 2 8

!

ds q(s) ,

(15.206)

where R ≡ |R|. The second δ-function on the right-hand side enforces q¯ = L−1

Z

L

0

ds q µ (s) = 0 ,

µ = 1, . . . , d − 1 ,

(15.207)

and thus the vanishing of the zero-frequency parts q0µ in the first D − 1 components of the Fourier decomposition (15.203). It was shown in Eqs. (10.631) and (10.641) that the last δ-function has a distorting effect upon the measure of path integration which must be compensated by a Faddeev-Popov action AFP e =

D−1 2L

Z

L 0

ds q 2 ,

(15.208)

where the number D of dimensions of q µ -space (10.641) has been replaced by the present number D − 1. In the large-stiffness limit we have to take only the first harmonic term in the action (15.205) into account, so that the path integral (15.170) becomes simply PL (R) ∝

Z

NBC

D ′ D−1 q δ R − L +

Z

L

0

!

RL 1 ′ 2 ds q 2 (s) e−(¯κ/2) 0 ds [q (s)] . 2

(15.209)

The subscript of the integral emphasizes the Neumann boundary conditions. The prime on the measure of the path integral indicates the absence of the zero-frequency component of q µ (s) in the Fourier decomposition due to (15.207). Representing the remaining δ-function in (15.209) by a Fourier integral, we obtain PL (R) ∝ κ ¯

Z

i∞

−i∞

dω 2 κ¯ω2 (L−R) e 2πi

Z

NBC

D

′ D−1

"

κ ¯ q exp − 2

Z

0

L

′2

2 2

ds q + ω q

#

. (15.210)

The integral over all paths with Neumann boundary conditions is known from Eq. (2.448). At zero average path, the result is Z

NBC

D

′ D−1

"

κ ¯ q exp − 2

Z

0

L

′2

2 2

ds q + ω q

#

ωL ∝ sinh ωL

(D−1)/2

,

(15.211)

so that PL (R) ∝ κ ¯

Z

i∞

−i∞

dω 2 κ¯ω2 (L−R) e 2πi

ωL sinh ωL

(D−1)/2

.

(15.212)


967


The integral can easily be done in D = 3 dimensions using the original product representation (2.447) of ωL/ sinh ωL, where PL (R) ∝ κ ¯

i∞

Z

−i∞

∞ dω 2 κ¯ω2 (L−R) Y ω2 e 1+ 2 2πi νn n=1

!−1

.

(15.213)

If we shift the contour of integration to the left we run through poles at ω 2 = −νk2 with residues ∞ X

∞ Y

νk2

k=1

k2 1− 2 n

n(6=k)=1

!−1

.

(15.214)

The product is evaluated by the limit procedure for small ǫ:  " ∞ Y 

n=1

(k + ǫ)2 1− n2

#−1  " 

(k + ǫ)2 1−  k2

#

→

(k + ǫ)π −2ǫ −2ǫπ → sin(k + ǫ)π k sin(k + ǫ)π

→ −

2ǫπ → 2(−1)k+1. (15.215) cos kπ sin ǫπ

Hence we obtain [6] PL (R) ∝

∞ X

2

2(−1)k+1κ ¯ νk2 e−¯κνk (L−R) .

(15.216)

k=1

It is now convenient to introduce the reduced end-to-end distance r ≡ R/L and the flexibility of the polymer l ≡ L/ξ, and replace κ ¯ νk2 (L − R) → k 2 π 2 (1 − r)/l so that (15.216) becomes PL (R) = N L(R) = N

∞ X

(−1)k+1 k 2 π 2 e−k

2 π 2 (1−r)/l

,

(15.217)

dr r 2 PL (R) = 1.

(15.218)

k=1

where N is a normalization factor determined to satisfy Z

d3 R PL (R) = 4πL3

∞

Z

0

The sum must be evaluated numerically, and leads to the distributions shown in Fig. 15.6. The above method is inconvenient if D 6= 3, since the simple pole structure of (15.213) is no longer there. For general D, we expand

ωL sinh ωL

(D−1)/2

= (ωL)

(D−1)/2

∞ X

k

(−1)

k=0

!

−(D − 1)/2 −(2k+(D−1)/2)ωL e , (15.219) k

and obtain from (15.212): PL (R) ∝

∞ X

(−1)k

k=0

!

−(D − 1)/2 Ik (R/L), k

(15.220)

968


ξ/L=

Figure 15.6 Normalized end-to-end distribution of stiff polymer according to our analytic formula (15.201), plotted for persistence lengths ξ/L = 1/400, 1/100, 1/30, 1/10, 1/5, 1/2, 1, 2 (fat curves). They are compared with the Monte Carlo calculations (symbols) and with the large-stiffness approximation (15.217) (thin curves) of Ref. [6] which fits well for ξ/L = 2 and 1 but becomes bad small ξ/L < 1. For very small values such as ξ/L = 1/400, 1/100, 1/30, our theoretical curves are well approximated by Gaussian random chain distributions on a lattice with lattice constant aeff = 2ξ of Eq. (15.158) which ensures that the lowest moments hR2 i = aeff L are properly fitted. The Daniels approximation (15.166) fits our theoretical curves well up to larger ξ/L ≈ 1/10.

with the integrals Ik (r) ≡

Z

i∞

−i∞

d¯ ω (D+1)/2 −[2k+(D−1)/2]¯ω +(D−1)¯ω2 (1−r)/2l ω ¯ e , 2πi

(15.221)

where ω ¯ is the dimensionless variable ωL. The integrals are evaluated with the help of the formula6 Z

i∞

−i∞

q dx ν βx2 /2−qx 1 −q 2 /4β x e =√ e D q/ β , ν 2πi 2πβ (ν+1)/2

(15.222)

where Dν (z) is the parabolic cylinder functions which for integer ν are proportional to Hermite polynomials:7 √ 1 2 Dn (z) = √ n e−z /4 Hn (z/ 2). 2

(15.223)

Thus we find 6 7

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 3.462.3 and 3.462.4. ibid., Formula 9.253. H. Kleinert, PATH INTEGRALS

969

15.9 Schrödinger Equation and Recursive Solution for Moments "

1 l Ik (r) = √ 2π (D−1)(1−r)

#(D+3)/4

[2k+(D−1)/2]2

− 4(D−1)(1−r)/l

e





2k+(D−1)/2  D(D+1)/2  q , (D−1)(1−r)/l

(15.224)

which becomes for D = 3 



(2k+1)2 1 1 2k+1  − 4(1−r)/l Ik (r) = √ q e H2  q . 3 2 2π 2(1−r)/l 2 (1−r)/l

(15.225)

If the sum (15.220) is performed numerically for D = 3, and the integral over PL (R) is normalized to satisfy (15.218), the resulting curves fall on top of those in Fig. 15.6 which were calculated from (15.217). In contrast to (15.217), which converges rapidly for small r, the sum (15.220) convergent rapidly for r close to unity. Let us compare the low moments of the above distribution with the exact moments in Eqs. (15.156) and (15.164). in the large-stiffness expansion. We set ω ¯ ≡ ωL and expand 2

f (¯ ω )≡

s

ω ¯ sinh ω ¯

D−1

(15.226)

in a power series D − 1 2 (D−1)(5D−1) 4 (D−1)(15+14D+35D 2) 6 f (¯ ω )=1− 2 ω ¯ + ω ¯ − ω ¯ +. . . .(15.227) 2 ·3 25 ·32 ·5 27 ·34 ·5·7 Under the integral (15.212), each power of ω ¯ 2 may be replaced by a differential operator 2

2

ˆ¯ ≡ − ω ¯2 → ω

L d 2l d =− . κ ¯ dr D − 1 dr

(15.228)

2 The expansion f (ω ¯ˆ ) can then be pulled out of theintegral, which by itself yields a 2 ˆ¯ δ(r − 1), which is a series of δ-function, so that we obtain PL (R) in the form f ω derivatives of δ-functions of r − 1 staring with

l d (15+14 D+35 D 2) l3 d3 (−1+5 D) l2 d2 PL (R) ∝ 1+ + + + . . . δ(r − 1). 6 dr 360 (D−1) dr 2 dr 3 45360 (D−1)2 (15.229) "

#

From this we find easily the moments m

hR i =

Z

D

m

d RR PL (R) ∝

Z

0

∞

dr r D−1 r m PL (R).

(15.230)

Let us introduce auxiliary expectation values with respect to the simple integrals R hf (r)i1 ∝ dr f (r)PL, rather than to D-dimensional volume integrals. The unnormalized moments hr m i are then given by hr D−1+m i1 . Within these one-dimensional expectations, the moments of z = r − 1 are h(r − 1)n i1 ∝

Z

∞

0

dr (r − 1)n PL (R).

(15.231)

970


The moments hr m i h[1+(r −1)]m i = h[1+(r −1)]D−1+mR i1 are obtained by expanding the binomial in powers of r −1 and using the integrals dz z m δ (n) (z) = (−1)m m! δmn to find, up to the third power in L/ξ = l, D−1 (5D−1)(D−2) 2 (35D3 +14D+15)(D−2)(D−3) 3 1− l + l − l , 6 360 45360(D − 1) 2

2 D+1 (5D−1)D(D+1) 2 (35D +14D+ 15)D(D+1) l+ l − l3 , R = N L2 1 − 6 360(D − 1) 45360(D − 1)

4 D+3 (5D−1)(D+2)(D+3) 2 l+ l R = N L4 1 − 6 360(D − 1) (35D2 +14D+15)(D+1)(D+2)(D+3) 3 − l . 45360(D − 1)2

R0 = N

The zeroth moment determines the normalization factor N to ensure that hR0 i = 1. Dividing this out of the other moments yields "

1 R2 = L2 1 − l + 3 " D E 2 R4 = L4 1 − l + 3 D

E

#

13D−9 2 8 l − l3 + . . . , 180(D−1) 945 # 23D−11 2 123D 2 −98D+39 3 l − l + ... . 90(D−1) 1890(D−1)2

(15.232) (15.233)

These agree up to the l-terms with the exact expansions (15.156) and (15.164) [or with the general formula (15.198)]. For D = 3, these expansions become 1 R = L 1− l+ 3 D E 2 R4 = L4 1 − l + 3 D

2

E

2

1 2 l − 12 29 2 l − 90

8 3 l + ... , 945 71 3 l + ... . 630

(15.234) (15.235)

Remarkably, these happen to agree in one more term with the exact expansions (15.156) and (15.164) than expected, as will be understood after Eq. (15.295).

15.9.5

Higher Loop Corrections

Let us calculate perturbative corrections to the large-stiffness limit. For this we replace the harmonic path integral (15.210) by the full expression ! Z Z L p tot cor −1 (15.236) P (r; L) = SD D ′ D−1 q(s)δ r − L−1 ds 1 − q 2 (s) e−A [q]−A [qb ,qa ] , NBC

0

where 1 Atot [q] = 2ε

Z

0

L

ds [ gµν (q) q˙µ (s)q˙ν (s)−εδ(s, s) log g(q(s))] + AFP − εL

R . 8

(15.237)

For convenience, we have introduced here the parameter ε = kB T /κ = 1/¯ κ, the reduced inverse stiffness, related to the flexibility l ≡ L/ξ by l = εL(d − 1)/2. We also have added the correction term εL R/8 to have a unit normalization of the partition function Z ∞ Z = SD dr rD−1 P (r; L) = 1. (15.238) 0


971


The Faddeev-Popov action, whose harmonic approximation was used in (15.208), is now ! Z L p FP −1 A [q] = −(D − 1) log L ds 1 − q 2 (s) , (15.239) 0

To perform higher-order calculations we have added an extra action which corrects for the omission of fluctuations of the velocities at the endpoints when restricting the paths to Neumann boundary conditions with zero end-point velocities. The extra action contains, of course, only at the endpoints and reads [7] Acor [qb , qa ] = − log J[qb , qa ] = −[q 2 (0) + q 2 (L)]/4.

(15.240)

Thus we represent the partition function (15.238) by the path integral with Neumann boundary conditions Z Z= D ′ D−1 q(s) exp −Atot [q] − Acor [qb , qa ] − AFP [q] . (15.241) NBC

A similar path integral can be set up for the moments of the distribution. We express the square distance R2 in coordinates (15.207) as 2

R =

Z

L

ds 0

Z

L

0

′

′

ds u(s) · u(s ) =

Z

0

L

p ds 1 − q 2 (s)

!2

= R2 ,

(15.242)

we find immediately the following representation for all, even and odd, moments [compare (15.147)] 2 n

h (R ) i =

* "Z

L

0

Z

0

L

′

′

ds ds u(s) · u(s )

#n +

(15.243)

in the form h Rn i =

Z

NBC

D ′ D−1 q(s) exp −Atot [q] − Acor [qb , qa ] − AFP n [q] .

(15.244)

This differs from Eq. (15.241) only by in the Faddeev-Popov action for these moments, which is ! Z L p −1 FP ds 1 − q 2 (s) , An [q] = −(n + D − 1) log L (15.245) 0

rather than (15.239). There is no need to divide the path integral (15.244) by Z since this has unit normalization, as will be verified order by order in the perturbation expansion. The Green function of the operator d2 /ds2 with these boundary conditions has the form ∆′N (s, s′ ) =

L 3

−

| s − s′ | (s + s′ ) (s2 + s′2 ) − + . 2 2 2L

(15.246)

The zero temporal average (15.207) manifests itself in the property Z

0

L

ds ∆′N (s, s′ )

= 0.

In the following we shall simply write ∆(s, s′ ) for ∆′N (s, s′ ), for brevity.

(15.247)

972


Partition Function and Moments Up to Four Loops We are now prepared to perform the explicit perturbative calculation of the partition function and all even moments in powers of the inverse stiffness ε up to order ε2 ∝ l2 . This requires evaluating Feynman diagrams up to four loops. The associated integrals will contain products of distributions, which will be calculated unambiguously with the help of our simple formulas in Chapter 10. For a systematic treatment of the expansion parameter ε, we rescale the coordinates q µ → εq µ , and rewrite the path integral (15.244) as Z h Rn i = D ′ D−1 q(s) exp {−Atot,n [q; ε].} , (15.248) NBC

with the total action Atot,n [q; ε] = −

L

1 2 (q q) ˙2 1 2 ds q˙ + ε + δ(0) log(1 − εq ) 2 1 − εq 2 2 0 # " Z ε 2 1 L p R ds 1 − εq 2 − q (0) + q 2 (L) − εL . σn log L 0 4 8 Z

(15.249)

The constant σn is an abbreviation for

σn ≡ n + (d − 1).

(15.250)

For n = 0, the path integral (15.248) must yield the normalized partition function Z = 1. For the perturbation expansion we separate Atot,n [q; ε] = A(0) [q] + Aint n [q; ε],

(15.251)

with a free action (0)

A

1 [q] = 2

Z

L

ds q˙2 (s),

(15.252)

0

and a large-stiffness expansion of the interaction int1 2 int2 Aint n [q; ε] = εAn [q] + ε An [q] + . . . .

(15.253)

The free part of the path integral (15.248) is normalized to unity: Z Z RL 2 (0) −(1/2) ds q˙ (s) 0 Z (0) ≡ D ′ D−1 q(s) e−A [q] = D ′ D−1 q(s) e = 1. NBC

(15.254)

NBC

The first expansion term of the interaction (15.253) is Z 1 L R int1 2 An [q] = ds [q(s)q(s)] ˙ − ρn (s) q 2 (s) − L , 2 0 8

(15.255)

where

ρn (s) ≡ δn + [δ(s) + δ(s − L)] /2,

δn ≡ δ(0) − σn /L.

(15.256)

The second term in ρn (s) represents the end point terms in the action (15.249) and is important for canceling singularities in the expansion. The next expansion term in (15.253) reads Z 1 L 1h σn i 2 int2 2 ds [q(s)q(s)] ˙ − δ(0) − q (s) q 2 (s) An [q] = 2 0 2 2L Z L Z L σn + ds ds′ q 2 (s)q 2 (s′ ). (15.257) 8L2 0 0 H. Kleinert, PATH INTEGRALS


973

The perturbation expansion of the partition function in powers of ε consists of expectation values of the interaction and its powers to be calculated with the free partition function (15.254). For an arbitrary functional of q(s), these expectation values will be denoted by Z RL 2 −(1/2) ds q˙ (s) 0 h F [q] i0 ≡ D ′ D−1 q(s) F [q] e . (15.258) NBC

With this notation, the perturbative expansion of the path integral (15.248) reads 1 int 2 h Rn i/Ln = 1 − hAint n [q; ε]i0 + hAn [q; ε] i0 − . . . 2 1 int1 2 int1 2 int2 = 1 − ε hAn [q]i0 + ε −hAn [q]i0 + hAn [q] i0 − . . . . 2

(15.259)

For the evaluation of the expectation values we must perform all possible Wick contractions with the basic propagator h q µ (s)q ν (s′ ) i0 = δ µν ∆(s, s′ ) ,

(15.260)

where ∆(s, s′ ) is the Green function (15.246) of the unperturbed action (15.252). The relevant loop integrals Ii and Hi are calculated using the dimensional regularization rules of Chapter 10. They are listed in Appendix 15A and Appendix 15B. We now state the results for various terms in the expansion (15.259): (D−1) σn 1 1 R (D−1)n int1 hAn [q]i0 = I1 + DI2 − ∆(0, 0) − ∆(L, L) − L = L , 2 L 2 2 8 12 i (D−1)σ (D2 −1) h σn n hAint2 δ(0) + I3 +2(D+2)I4 + (D−1)I12 + 2I5 n [q]i0 = 4 2L 8L2 (D2 − 1) (D−1) = L3 δ(0) + L2 (25D2 +36D+23) + n(11D+5) , 120 1440 2 1 int1 2 L2 (D−1)L D 2 R hA [q] i0 = δ(0) + − δn + − 2 n 2 12 2L L 8 (D−1) n + [H1 − 2(H2n + H3n − H5 ) + H6 − 4D(H4n − H7 − H10 ) 4 (D−1) + H11 + 2D2 (H8 + H9 ) + [DH12 + 2(D + 2)H13 + DH14 ] 4 2 L2 (D−1)n (D−1) = + L3 δ(0) 2 12 120 (D−1) + L2 (25D2 − 22D + 25) + 4(n + 4D − 2) 1440 (D−1)D (D−1) + L3 δ(0) + L2 (29D − 1) . (15.261) 120 720 Inserting these results into Eq. (15.259), we find all even and odd moments up to order ε2 ∝ l2 h Rn i/Ln

(D−1)n (D−1)2 n2 (D−1)(4n + 5D − 13)n + ε2 L 2 + − O(ε3 ) 12 288 1440 2 n n (4n + 5D − 13)n 2 = 1− l+ + l − O(l3 ). (15.262) 6 72 360(D−1) = 1 − εL

For n = 0 this gives the properly normalized partition function Z = 1. For all n it reproduces the large-stiffness expansion (15.198) up to order l4 .

974


Correlation Function Up to Four Loops As an important test of the correctness of our perturbation theory we calculating the correlation function up to four loops and verify that it yields the simple expression (15.153), which reads in the present units ′

′

G(s, s′ ) = e−|s−s |/ξ = e−|s−s |l/L .

(15.263)

Starting point is the path integral representation with Neumann boundary conditions for the twopoint correlation function Z n o (0) G(s, s′ ) = hu(s) · u(s′ )i = D ′ D−1 q(s) f (s, s′ ) exp −Atot [q; ε] , (15.264) NBC

with the action of Eq. (15.249) for n = 0. The function in the integrand f (s, s′ ) ≡ f (q(s), q(s′ )) is an abbreviation for the scalar product u(s) · u(s′ ) expressed in terms of independent coordinates q µ (s): p p f (q(s), q(s′ )) ≡ u(s) · u(s′ ) = 1 − q 2 (s) 1 − q 2 (s′ ) + q(s) q(s′ ). (15.265) √ Rescaling the coordinates q → ε q, and expanding in powers of ε yields: f (q(s), q(s′ )) = 1 + εf1 (q(s), q(s′ )) + ε2 f2 (q(s), q(s′ )) + . . . ,

(15.266)

where 1 2 1 q (s) − q 2 (s′ ), 2 2 1 2 1 1 q (s) q 2 (s′ ) − [q 2 (s)]2 − [q 2 (s′ )]2 . 4 8 8

f1 (q(s), q(s′ )) =

q(s) q(s′ ) −

f2 (q(s), q(s′ )) =

(15.267) (15.268)

We shall attribute the integrand f (q(s), q(s′ )) to an interaction Af [q; ε] defined by f

f (q(s), q(s′ )) ≡ e−A

[q;ε]

,

(15.269)

which has the ε-expansion Af [q; ε] = − log f (q(s), q(s′ )) ′

= −εf1 (q(s), q(s ))+ε

2

1 2 ′ −f2 (q(s), q(s ))+ f1 (s, s ) −. . . , 2 ′

(15.270)

to be added to the interaction (15.253) with n = 0. Thus we obtain the perturbation expansion of the path integral (15.264) D E E 1 D int f G(s, s′ ) = 1− (Aint + (A0 [q; ε] + Af [q; ε])2 − . . . . 0 [q; ε] + A [q; ε]) 2 0 0

(15.271)

Inserting the interaction terms (15.253) and (15.271), we obtain G(s, s′ ) = 1+ εhf1(q(s), q(s′ ))i0+ε2 hf2 (q(s), q(s′ ))i0 −hf1 (q(s), q(s′ )) Aint1 0 [q]i0 +. . . ,

(15.272)

and the expectation values can now be calculated using the propagator (15.260) with the Green function (15.246). In going through this calculation we observe that because of translational invariance in the pseudotime, s → s + s0 , the Green function ∆(s, s′ ) + C is just as good a Green function satisfying Neumann boundary conditions as ∆(s, s′ ). We may demonstrate this explicitly by setting C = H. Kleinert, PATH INTEGRALS


975

L(a−1)/3 with an arbitrary constant a, and calculating the expectation values in Eq. (15.272) using the modified Green function. Details are given in Appendix 15C [see Eq. (15C.1)], where we list various expressions and integrals appearing in the Wick contractions of the expansion Eq. (15.272). Using these results we find the a-independent terms up to second order in ε: (D − 1) | s − s′ |, (15.273) 2 hf2 (q(s), q(s′ ))i0 − hf1 (q(s), q(s′ )) Aint1 0 [q]i0 1 1 (D − 1) 1 1 K3 + K4 + K5 = (D − 1) D1 − (D + 1)D22 − K1 − DK2 − 2 8 L 2 2 1 (15.274) = (D − 1)2 (s − s′ )2 . 8 hf1 (q(s), q(s′ ))i0 = −

This leads indeed to the correct large-stiffness expansion of the exact two-point correlation function (15.263): G(s, s′ ) = 1−ε

D−1 (D−1)2 | s−s′ | (s−s′ )2 | s−s′ | + ε2 (s−s′ )2 +. . .= 1− + − . . . . (15.275) 2 8 ξ 2ξ 2

Radial Distribution up to Four Loops We now turn to the most important quantity characterizing a polymer, the radial distribution function. We eliminate the δ-function in Eq. (15.236) enforcing the end-to-end distance by considering the Fourier transform Z P (k; L) = dr eik(r−1) P (r; L) . (15.276) This is calculated from the path integral with Neumann boundary conditions Z P (k; L) = D ′ D−1 q(s) exp −Atot k [q; ε] ,

(15.277)

NBC

where the action Atot k [q; ε] reads, with the same rescaled coordinates as in (15.249), L

1 2 (q q) ˙2 1 ik p 2 2−1 q˙ + ε + δ(0) log(1 − εq ) − 1 − εq 2 1 − εq 2 2 L 0 R 1 2 − ε q (0) + q 2 (L) − εL ≡ A0 [q] + Aint (15.278) k [q; ε]. 4 8

Atot k [q; ε] =

Z

ds

As before in Eq. (15.253), we expand the interaction in powers of the coupling constant ε. The first term coincides with Eq. (15.255), except that σn is replaced by ρk (s) = δk + [δ(s) + δ(s − L)] /2,

δk = δ(0) − ik/L,

(15.279)

so that Aint1 k [q]

=

Z

0

L

ds

o 1n R 2 [q(s)q(s)] ˙ − ρk (s) q 2 (s) − L . 2 8

(15.280)

The second expansion term Aint2 k [q] is simpler than the previous (15.257) by not containing the last nonlocal term: Z L 1 1 ik int2 2 2 Ak [q] = ds [q(s)q(s)] ˙ − δ(0) − q (s) q 2 (s). (15.281) 2 2 2L 0

976


Apart from that, the perturbation expansion of (15.277) has the same general form as in (15.259): 1 int1 2 int1 2 int2 P (k; L) = 1 − ε hAk [q]i0 + ε −hAk [q]i0 + hAk [q] i0 − . . . . (15.282) 2 The expectation values can be expressed in terms of the same integrals listed in Appendix 15A and Appendix 15B as follows: (D − 1) ik 1 1 R int1 hA, k [q]i0 = I1 + DI2 − ∆(0, 0) − ∆(L, L) − L 2 L 2 2 8 (D − 1) [(D − 1) − ik] = −L , (15.283) 12 (D2 − 1) ik δ(0) + I3 + 2(D + 2)I4 hAint2 = , k [q]i0 4 2L (D2 − 1) (D2 − 1) [7(D + 2) + 3ik] = L3 δ(0) + L2 , (15.284) 120 720 2 1 int1 2 L2 (D − 1)L D 2 R hA, k [q] i0 = δ(0) + − δk + − 2 2 12 2L L 8 (D − 1) k + H1 − 2(H2k + H3k − H5 ) + H6 − 4D(H4k − H7 − H10 ) 4 (D − 1) + H11 + 2D2 (H8 + H9 ) + [DH12 + 2(D + 2)H13 + DH14 ] 4 2 (D − 1)2 [(D − 1) − ik] = L2 2 · 122 (D − 1) (D − 1) + L3 δ(0) + L2 (13D2 − 6D + 21) + 4ik(2D + ik) 120 1440 2 (D − 1) 3 (D − 1)D + L δ(0) + L (29D − 1) . (15.285) 120 720 In this way we find the large-stiffness expansion up to order ε2 : (D − 1) (D − 1) P (k; L) = 1 + εL [(D − 1) − ik] + ε2 L2 (15.286) 12 1440 × (ik)2 (5D−1)−2ik(5D2 −11D+8)+(D−1)(5D2 −11D+14) +O(ε3 ).

This can also be rewritten as (D−1) (D−3) (5D2 −11D+14) 2 P (k; L) = P1 loop (k; L) 1+ l+ ik+ l +O(l3 ) , 6 180(D−1) 360 (15.287) where the prefactor P1 loop (k; L) has the expansion P1 loop (k; L) = 1 − εL

(D − 1) (D − 1)(5D − 1) (ik) + ε2 L2 (ik)2 − . . . . 22 · 3 25 · 32 · 5

(15.288)

With the identification ω ¯ 2 = ikεL, this is the expansion of one-loop functional determinant in (15.227). By Fourier-transforming (15.286), we obtain the radial distribution function l ′ l2 [δ (r−1) + (d − 1) δ(r−1)] + [(5d − 1) δ ′′ (r−1) 6 360(d − 1) + 2(5d2 −11d+8) δ ′(r−1) + (d−1)(5d2 −11d+14)δ(r−1) + O(l3 ). (15.289)

P (r; l) = δ(r−1) +



977

As a crosscheck, we can calculate from this expansion once more the even and odd moments Z h Rn i = Ln dr rn+(D−1) P (r; l), (15.290) and find that they agree with Eq. (15.262). Using the higher-order expansion of the moments in (15.198) we can easily extend the distribution (15.289) to arbitrarily high orders in l. Keeping only the terms up to order l4 , we find that the one-loop end-to-end distribution function (15.212) receives a correction factor: Z ∞ dˆ ω 2 −iˆω2 (r−1)(D−1)/2l ω ¯ (D−1)/2 −V (l,ˆω2 ) P (r; l) ∝ e e , (15.291) sinh ω ¯ −∞ 2π with ω ˆ2 ω ˆ4 ω ˆ6 + V2 (l) 2 + V3 (l) 3 + . . . . V (l, ω ˆ 2 ) ≡ V0 (l) + V¯ (l, ω ˆ 2 ) = V0 (l) + V1 (l) l l l

(15.292)

The first term d−1 d − 9 2 (d − 1) 32 − 13 d + 5 d2 3 V0 (l) = − l+ l + l 6 360 6480 2 3 4 34 − 272 d + 259 d − 110 d + 25 d 4 − l + ... 259200

(15.293)

contributes only to the normalization of P (r; l), and can be omitted in (15.291). The remainder has the expansion coefficients −455 + 431 d + 91 d2 + 5 d3 4 d−3 2 (−5 + 9 d) 3 V1 (l) = − l + l + l + ... , 2 360 7560 (−1 + d) 907200 (d − 1) −31 + 42 d + 25 d2 l4 (5 − 3 d) l3 V2 (l) = − − + ... , (15.294) 7560 907200 (d − 1) (d − 1) l4 V3 (l) = − + ... . 18900 In the physical most interesting case of three dimensions, the first nonzero correction arises to order l3 . This explains the remarkable agreement of the moments in (15.234) and (15.234) up to order l2 . The correction terms V¯ (l, ω ˆ 2 ) may be included perturbatively into the sum over k in Eq. (15.220) by noting that the expectation value of powers of ω ˆ 2 /l within the ω ˆ -integral (15.221) are

2 2k + (D − 1)/2 2 2 ω ˆ /l = a2k ≡ , ω ˆ /l = 3a4k , (D − 1)(1 − r)

so that we obtain an extra factor e−fk fk =

V1 (l)a2k

+

where up to order l4 :

3V2 (l)−V12 (l)

a4k

2 3 ω ˆ /l = 15a6k ,

4 3 + 15V3 (l)−12V1(l)V2 (l)+ V1 (l) a6k + . . . , 3

(15.295)

(15.296)

3D − 5 3 156 − 231D − 26D2 − 7D3 4 l + l + ... , 2520 907200 (D − 1) 4 D−1 4 l + ... . (15.297) 15V3 (l)−12V1 (l)V2 (l)+ V13 (l) = − 3 1260 3V2 (l)−V12 (l) =

978

15.10


Excluded-Volume Effects

A significant modification of these properties is brought about by the interactions between the chain elements. If two of them come close to each other, the molecular forces prevent them from occupying the same place. This is called the excludedvolume effect. In less than four dimensions, it gives rise to a scaling law for the expectation value hR2 i as a function of L: hR2 i ∝ L2ν ,

(15.298)

as stated in (15.38). The critical exponent ν is a number between the random-chain value ν = 1/2 and the stiff-chain value ν = 1. To derive this behavior we consider the polymer in the limiting path integral approximation (15.80) to a random chain which was derived for R2 /La ≪ 1 and which is very accurate whenever the probability distribution is sizable. Thus we start with the time-sliced expression 1 PN (R) = q D 2πa/M

NY −1 n=1

 Z   q

D

d xn

2πa/M



 D  exp

−AN /¯ h ,

(15.299)

with the action AN = a

N X

M (∆xn )2 , a2 n=1 2

(15.300)

and the mass parameter (15.79). In the sequel we use natural units in which energies are measured in units of kB T , and write down all expressions in the continuum limit. The probability (15.299) is then written as PL (R) =

Z

L [x]

D D x e−A

,

(15.301)

where we have used the label L = Na rather than N. From the discussion in the previous section we know that although this path integral represents an ideal random chain, we can also account for a finite stiffness by interpreting the number a as an effective length parameter aeff given by (15.158). The total Euclidean time in the path integral τb − τa = h ¯ /kB T corresponds to the total length of the polymer L. We now assume that the molecules of the polymer repel each other with a twobody potential V (x, x′ ). Then the action in the path integral (15.301) has to be supplemented by an interaction

Aint =

1 2

Z

0

L

dτ

Z

0

L

dτ ′ V (x(τ ), x(τ ′ )).


979

15.10 Excluded-Volume Effects

Note that the interaction is of a purely spatial nature and does not depend on the parameters τ , τ ′ , i.e., it does not matter which two molecules in the chain come close to each other. The effects of an interaction of this type are most elegantly calculated by making use of a Hubbard-Stratonovich transformation. Generalizing the procedure in Subsection 7.15.1, we introduce an auxiliary fluctuating field variable ϕ(x) at every space point x and replace Aint by Aϕint

=

L

Z

0

1 dτ ϕ(x(τ )) − 2

Z

dD xdD x′ ϕ(x)V −1 (x, x′ )ϕ(x′ ).

(15.303)

Here V −1 (x, x′ ) denotes the inverse of V (x, x′ ) under functional multiplication, defined by the integral equation Z

dD x′ V −1 (x, x′ )V (x′ , x′′ ) = δ (D) (x − x′′ ).

(15.304)

To see the equivalence of the action (15.303) with (15.302), we rewrite (15.303) as Aϕint =

Z

dD x ρ(x)ϕ(x) −

1 2

Z

dD xdD x′ ϕ(x)V −1 (x, x′ )ϕ(x′ ),

(15.305)

where ρ(x) is the particle density ρ(x) ≡

Z

L

0

dτ δ (D) (x − x(τ )).

(15.306)

Then we perform a quadratic completion to Aϕint

1 =− 2

Z

h

i

dD xdD x′ ϕ′ (x)V −1 (x, x′ )ϕ′ (x′ ) − ρ(x)V (x, x′ )ρ(x′ ) ,

(15.307)

with the shifted field ′

ϕ (x) ≡ ϕ(x) −

Z

dD x′ V (x, x′ )ρ(x′ ).

(15.308)

Now we perform the functional integral Z

ϕ

Dϕ(x) e−Aint

(15.309)

integrating ϕ(x) at each point x from −i∞ to i∞ along the imaginary field axis. −1/2 The result is a constant functional determinant [det V −1 (x, x′ )] . This can be ignored since we shall ultimately normalize the end-to-end distribution to unity. Inserting (15.306) into the surviving second term in (15.307), we obtain precisely the original interaction (15.302). Thus we may study the excluded-volume problem by means of the equivalent path integral PL (R) ∝

Z

D D x(τ )

Z

Dϕ(x) e−A ,

(15.310)

980


where the action A is given by the sum ˙ ϕ] + A[ϕ], A = AL [x, x,

(15.311)

of the line and field actions L

A [x, ϕ] ≡

Z

L

M 2 x˙ + ϕ(x(τ )) , dτ 2

(15.312)

1 2

Z

(15.313)

0

A[ϕ] ≡ −

dD xdD x′ ϕ(x)V −1 (x, x′ )ϕ(x′ ),

respectively. The path integral (15.310) over x(τ ) and ϕ(x) has the following physical interpretation. The line action (15.312) describes the orbit of a particle in a space-dependent random potential ϕ(x). The path integral over x(τ ) yields the endto-end distribution of the fluctuating polymer in this potential. The path integral over all potentials ϕ(x) with the weight e−A[ϕ] accounts for the repulsive cloud of the fluctuating chain elements. To be convergent, all ϕ(x) integrations in (15.310) have to run along the imaginary field axis. To evaluate the path integrals (15.310), it is useful to separate x(τ )- and ϕ(x)integrations and to write end-to-end distributions as an average over ϕ-fluctuations PL (R) ∝

Z

Dϕ(x) e−A[ϕ] PLϕ (R, 0),

(15.314)

where PLϕ (R, 0) =

Z

L [x,ϕ]

D D x(τ ) e−A

(15.315)

is the end-to-end distribution of a random chain moving in a fixed external potential ϕ(x). The presence of this potential destroys the translational invariance of PLϕ . This is why we have recorded the initial and final points 0 and R. In the final distribution PL (R) of (15.314), the invariance is of course restored by the integration over all ϕ(x). It is possible to express the distribution PLϕ (R, 0) in terms of solutions of an associated Schrödinger equation. With the action (15.312), this equation is obviously "

#

∂ 1 − ∂R 2 + ϕ(R) PLϕ (R, 0) = δ (D) (R − 0)δ(L). ∂L 2M

(15.316)

If ψEϕ (R) denotes the time-independent solutions of the Hamiltonian operator ˆ ϕ = − 1 ∂R 2 + ϕ(R), H 2M

(15.317)

the probability PLϕ (R) has a spectral representation of the form PLϕ (R, 0) =

Z

dEe−ELψEϕ (R)ψEϕ ∗ (0),

L > 0.

(15.318)


981


From now on, we assume the interaction to be dominated by the simplest possible repulsive potential proportional to a δ-function: V (x, x′ ) = vaD δ (D) (x − x′ ).

(15.319)

Then the functional inverse is V −1 (x, x′ ) = v −1 a−D δ (D) (x − x′ ),

(15.320)

and the ϕ-action (15.312) reduces to v −1 a−D A[ϕ] = − 2

Z

dD x ϕ2 (x).

(15.321)

The path integrals (15.314), (15.315) can be solved approximately by applying the semiclassical methods of Chapter 4 to both the x(τ )- and the ϕ(x)-path integrals. These are dominated by the extrema of the action and evaluated via the leading saddle point approximation. In the ϕ(x)-integral, the saddle point is given by the equation v −1 a−D ϕ(x) =

δ log PLϕ (R, 0). δϕ(x)

(15.322)

This is the semiclassical approximation to the exact equation −1 −D

v a

hϕ(x)i = hρ(x)i ≡

*Z

0

L

dτ δ

(D)

(x − x(τ ))

+

,

(15.323)

x

where h. . .ix is the average over all line fluctuations calculated with the help of the probability distribution (15.315). The exact equation (15.323) follows from a functional differentiation of the path integral for PLϕ with respect to ϕ(x): δ PLϕ (R) = δϕ(x)

Z

δ Dϕ δϕ(x)

Z

L [x,ϕ]−A[ϕ]

D D x e−A

= 0.

(15.324)

By anchoring one end of the polymer at the origin and carrying the path integral from there to x(τ ), and further on to R, the right-hand side of (15.323) can be expressed as a convolution integral over two end-to-end distributions: *Z

0

L

dτ δ

(D)

(x − x(τ ))

+

= x

Z

0

L

ϕ dL′ PLϕ′ (x)PL−L ′ (R − x).

(15.325)

With (15.323), this becomes −1 −D

v a

hϕ(x)ix =

which is the same as (15.323).

Z

0

L

ϕ dL′ PLϕ′ (x)PL−L ′ (R − x),

(15.326)

982


According to Eq. (15.322), the extremal ϕ(x) depends really on two variables, x and R. This makes the solution difficult, even at the semiclassical level. It becomes simple only for R = 0, i.e., for a closed polymer. Then only the variable x remains and, by rotational symmetry, ϕ(x) can depend only on r = |x|. For R 6= 0, on the other hand, the rotational symmetry is distorted to an ellipsoidal geometry, in which a closed-form solution of the problem is hard to find. As an approximation, we may use a rotationally symmetric ansatz ϕ(x) ≈ ϕ(r) also for R 6= 0 and calculate the end-to-end probability distribution PL (R) via the semiclassical approximation to the two path integrals in Eq. (15.310). The saddle point in the path integral over ϕ(x) gives the formula [compare (15.314)] PL (R) ∼

PLϕ (R, 0)

=

Z

( Z

D

D x exp −

L 0

)

M 2 x˙ + ϕ(r(τ )) . dτ 2

(15.327)

Thereby it is hoped that for moderate R, the error is small enough to justify this approximation. Anyhow, the analytic results supply a convenient starting point for better approximations. Neglecting the ellipsoidal distortion, it is easy to calculate the path integral over x(τ ) for PLϕ (R, 0) in the saddle point approximation. At an arbitrary given ϕ(r), we must find the classical orbits. The Euler-Lagrange equation has the first integral of motion M 2 x˙ − ϕ(r) = E = const. 2

(15.328)

At fixed L, we have to find the classical solutions for all energies E and all angular momenta l. The path integral reduces an ordinary double integral over E and l which, in turn, is evaluated in the saddle point approximation. In a rotationally symmetric potential ϕ(r), the leading saddle point has the angular momentum l = 0 corresponding to a symmetric polymer distribution. Then Eq. (15.328) turns into a purely radial differential equation dr dτ = q . 2[E + ϕ(r)]/M

(15.329)

For a polymer running from the origin to R we calculate L=

Z

0

dr

R

q

2[E + ϕ(r)]/M

.

(15.330)

This determines the energy E as a function of L. It is a functional of the yet unknown field ϕ(r): E = EL [ϕ].


983


The classical action for such an orbit can be expressed in the form M 2 x˙ + ϕ(r(τ )) 2 0 Z L Z L M 2 x˙ − ϕ(r(τ )) + = − dτ dτ M x˙ 2 2 0 0 Z

Acl [x, ϕ] =

L

dτ

= −EL +

Z

R

0

q

dr 2M[E + ϕ(r)].

(15.332)

In this expression, we may consider E as an independent variational parameter. The relation (15.330) between E, L, R, ϕ(r), by which E is fixed, reemerges when extremizing the classical expression Acl [x, ϕ]: ∂ ˙ ϕ] = 0. Acl [x, x, ∂E

(15.333)

The classical approximation to the entire action A[x, ϕ] + A[ϕ] is then Acl = −EL +

Z

0

r

dr

′

q

2M[E +

ϕ(r ′ )]

1 − v −1 a−D 2

Z

dD xϕ2 (r).

(15.334)

This action is now extremized independently in ϕ(r), E. The extremum in ϕ(r) is obviously given by the algebraic equation ′

ϕ(r ) =

(

0 q −1 ′ 1−D r / 2M[E + ϕ(r ′)] MvaD SD

for

r ′ > r, r ′ < r,

(15.335)

which is easily solved. We rewrite it as E + ϕ(r) = ξ 3 ϕ−2 (r),

(15.336)

ξ 3 = αr −2δ ,

(15.337)

δ ≡D−1>0

(15.338)


where

and α≡

M 2 2D −2 v a SD . 2

(15.339)

For large ξ ≫ 1/E, i.e., small r ≪ α2/δ E −6/δ , we expand the solution as follows ϕ(r) = ξ −

E E2 + + ... . 3 9

(15.340)

984


This expansion is reinserted into the classical action (15.334), making it a power series in E. A further extremization in E yields E = E(L, r). The extremal value of the action yields an approximate distribution function of a monomer in the closed polymer (which runs through the origin): PL (R) ∝ e−Acl (L,R) .

(15.341)

To see how this happens consider first the noninteracting limit where v = 0. Then the solution of (15.335) is ϕ(r) ≡ 0, and the classical action (15.334) becomes Acl = −EL +

√

2MER.

(15.342)

The extremization in E gives E=

M R2 , 2 L2

(15.343)

Acl =

M R2 . 2 L

(15.344)

yielding the extremal action

The approximate distribution is therefore 2 /2L

PN (R) ∝ e−Acl = e−M R

.

(15.345)

The interacting case is now treated in the same way. Using (15.336), the classical action (15.334) can be written as Acl = −EL +

s

M 2

Z

0

R

dr

′

"

#

1 1 ϕ + E − ξ 3/2 . 2 ϕ+E

q

(15.346)

By expanding this action in a power series in E [after having inserted (15.340) for ϕ], we obtain with ǫ(r) ≡ E/ξ = Eα−1/3 r 2δ/3 Acl = −EL +

s

M 1/6 α 2

Z

R 0

′ ′−δ/3

dr r

3 1 + ǫ(r ′ ) − ǫ2 (r ′ ) + . . . . 2 6

(15.347)

As long as δ < 3, i.e., for D < 4,

(15.348)

the integral exists and yields an expansion 1 Acl = −EL + a0 (R) + a1 (R)E − a2 (R)E 2 + . . . , 2

(15.349)


985


with the coefficients M a0 (R) = − 2 s

s

9 1−δ/3 1/6 1 R α , 2 δ−3

M 1+δ/3 −1/6 1 α R , 2 δ+3 s 1 M 1+δ −1/2 1 a2 (R) = R α . 3 2 δ+1 a1 (R) = 3

(15.350)

Extremizing Acl in E gives the action Acl = a0 (R) +

1 [L − a1 (R)]2 + . . . . 2a2 (R)

(15.351)

The approximate end-to-end distribution function is therefore (

)

1 PL (R) ≈ N exp −a0 (R) − [L − a1 (R)]2 , 2a2 (R)

(15.352)

where N is an appropriate normalization factor. The distribution is peaked around L=3

s

M 1+δ/3 −1/6 1 R . α 2 δ+3

(15.353)

This shows the most important consequence of the excluded-volume effect: The average value of R2 grows like 

D+2 hR2 i ≈ α1/(D+2)  3

s

6/(D+2)

2  L M

.

(15.354)

Thus we have found a scaling law of the form (15.298) with the critical exponent ν=

3 . D+2

(15.355)

The repulsion between the chain elements makes the excluded-volume chain reach out further into space than a random chain [although less than a completely stiff chain, which is always reached by the solution (15.354) for D = 1]. The restriction D < 4 in (15.348) is quite important. The value D uc = 4

(15.356)

is called the upper critical dimension. Above it, the set of all possible intersections of a random chain has the measure zero and any short-range repulsion becomes irrelevant. In fact, for D > Duc it is possible to show that the polymer behaves like a random chain without any excluded-volume effect satisfying hR2 i ∝ L.

986

15.11


Flory’s Argument

Once we expect a power-like scaling law of the form hR2 i ∝ L2ν ,

(15.357)

the critical exponent (15.355) can be derived from a very simple dimensional argument due to Flory. We take the action A=

Z

L 0

M 2 vaD dτ x˙ − 2 2

Z

L

0

dτ

Z

0

L

dτ ′ δ (D) (x(τ ) − x(τ ′ )),

(15.358)

with M = D/a, and replace the two terms by their dimensional content, L for the τ -variable and R- for each x-component. Then A∼

M R2 vaD L2 − . L 2 L2 2 RD

(15.359)

Extremizing this expression in R at fixed L gives R ∼ R−D−1 L2 , L

(15.360)

R2 ∼ L6/(D+2) ,

(15.361)

implying

and thus the critical exponent (15.355).

15.12

Polymer Field Theory

There exists an alternative approach to finding the power laws caused by the excluded-volume effects in polymers which is superior to the previous one. It is based on an intimate relationship of polymer fluctuations with field fluctuations in a certain somewhat artificial and unphysical limit. This limit happens to be accessible to approximate analytic methods developed in recent years in quantum field theory. According to Chapter 7, the statistical mechanics of a many-particle ensemble can be described by a single fluctuating field. Each particle in such an ensemble moves through spacetime along a fluctuating orbit in the same way as a random chain does in the approximation (15.80) to a polymer in Section 15.6. Thus we can immediately conclude that ensembles of polymers may also be described by a single fluctuating field. But how about a single polymer? Is it possible to project out a single polymer of the ensemble in the field-theoretic description? The answer is positive. We start with the result of the last section. The end-to-end distribution of the polymer in the excluded-volume problem is rewritten as an integral over the fluctuating field ϕ(x): PL (xb , xa ) =

Z

Dϕ e−A[ϕ] PLϕ (xb , xa ),


987

15.12 Polymer Field Theory

with an action for the auxiliary ϕ(x) field [see (15.312)] 1 A[ϕ] = − 2

Z

dD xdD x′ ϕ(x)V −1 (x, x′ )ϕ(x′ ),

(15.363)

and an end-to-end distribution [see (15.315)] PLϕ (xb , xa )

=

Z

(

Dx exp −

Z

L

0

M 2 dτ x˙ + ϕ(x(τ )) 2

)

,

(15.364)

which satisfies the Schrödinger equation [see (15.316)] "

#

∂ 1 − ∇2 + ϕ(x) PLϕ (x, x′ ) = δ (3) (x − x′ )δ(L). ∂L 2M

(15.365)

Since PL and PLϕ vanish for L < 0, it is convenient to go over to the Laplace transforms ∞ 1 2 P (x, x ) = dLe−Lm /2M PL (x, x′ ), 2M 0 Z ∞ 1 2 ϕ ′ Pm2 (x, x ) = dLe−Lm /2M PLϕ (x, x′ ). 2M 0 m2

′

Z

(15.366) (15.367)

The latter satisfies the L-independent equation [−∇2 + m2 + 2Mϕ(x)]Pmϕ2 (x, x′ ) = δ (3) (x − x′ ).

(15.368)

The quantity m2 /2M is, of course, just the negative energy variable −E in (15.332): −E ≡

m2 . 2M

(15.369)

The distributions Pmϕ2 (x, x′ ) describe the probability of a polymer of any length run2 ning from x′ to x, with a Boltzmann-like factor e−Lm /2M governing the distribution of lengths. Thus m2 /2M plays the role of a chemical potential. We now observe that the solution of Eq. (15.368) can be considered as the correlation function of an auxiliary fluctuating complex field ψ(x): Pmϕ2 (x, x′ ) = Gϕ0 (x, x′ ) = hψ ∗ (x)ψ(x′ )iϕ R Dψ ∗ (x)Dψ(x) ψ ∗ (x)ψ(x′ ) exp {−A[ψ ∗ , ψ, ϕ]} R ≡ , Dψ ∗ (x)Dψ(x) exp {−A[ψ ∗ , ψ, ϕ]}

(15.370)

with a field action Z

n

o

A[ψ ∗ , ψ, ϕ] = dD x ∇ψ ∗ (x)∇ψ(x) + m2 ψ ∗ (x)ψ(x) + 2Mϕ(x)ψ ∗ (x)ψ(x) .(15.371)

988


The second part of Eq. (15.370) defines the expectations h. . .iψ . In this way, we express the Laplace-transformed distribution Pm2 (xb , xa ) in (15.366) in the purely field-theoretic form Pm2 (x, x′ ) =

Z

Dϕ exp {−A[ϕ]} hψ ∗ (x)ψ(x′ )iϕ

1 dD ydD y′ ϕ(y)V −1 (y, y′ )ϕ(y′) 2 R Dψ ∗ Dψ ψ ∗ (x)ψ(x′ ) exp {−A[ψ ∗ , ψ, ϕ]} R R × . Dψ ∗ Dψ exp {−A[ψ ∗ , ψ, ϕ]} =

Z

Dϕ exp

Z

(15.372)

It involves only a fluctuating field which contains all information on the path fluctuations. The field ψ(x) is, of course, the analog of the second-quantized field in Chapter 7. Consider now the probability distribution of a single monomer in a closed polymer chain. Inserting the polymer density function ρ(R) ≡

Z

0

L

dτ δ (D) (R − x(τ ))

(15.373)

into the original path integral for a closed polymer PL (R) =

Z

0

L

dτ

Z

D

D x

Z

Dϕ exp {−AL − A[ϕ]} δ (D) (R − x(τ )),

(15.374)

the δ-function splits the path integral into two parts PL (R) =

Z

Dϕ exp {−A[ϕ]}

Z

0

L

ϕ dτ PL−τ (0, R)Pτϕ (R, 0).

(15.375)

When going to the Laplace transform, the convolution integral factorizes, yielding Pm2 (R) =

Z

Dϕ(x) exp {−A[ϕ]} Pmϕ2 (0, R)Pmϕ2 (R, 0).

(15.376)

With the help of the field-theoretic expression for Pm2 (R) in Eq. (15.370), the product of the correlation functions can be rewritten as Pmϕ2 (0, R)Pmϕ2 (0, R) = hψ ∗ (R)ψ(0)iϕ hψ ∗ (0)ψ(R)iϕ .

(15.377)

We now observe that the field ψ appears only quadratically in the action A[ψ ∗ , ψ, ϕ]. The product of correlation functions in (15.377) can therefore be viewed as a term in the Wick expansion (recall Section 3.10) of the four-field correlation function hψ ∗ (R)ψ(R)ψ ∗(0)ψ(0)iϕ .

(15.378)

This would be equal to the sum of pair contractions hψ ∗ (R)ψ(R)iϕ hψ ∗ (0)ψ(0)iϕ + hψ ∗ (R)ψ(0)iϕ hψ ∗ (0)ψ(R)iϕ .

(15.379)


989


There are no contributions containing expectations of two ψ or two ψ ∗ fields which could, in general, appear in this expansion. This allows the right-hand side of (15.377) to be expressed as a difference between (15.378) and the first term of (15.379): Pmϕ2 (0, R)Pmϕ2 (0, R) = hψ ∗ (R)ψ(R)ψ ∗(0)ψ(0)iϕ − hψ ∗ (R)ψ(R)iϕ hψ ∗ (0)ψ(0)iϕ . (15.380) The right-hand side only contains correlation functions of a collective field, the density field [8] ρ(R) = ψ ∗ (R)ψ(R),

(15.381)

Pmϕ2 (0, R)Pmϕ2 (0, R) = hρ(R)ρ(0)iϕ − hρ(R)iϕ hρ(0)iϕ .

(15.382)

in terms of which

Now, the right-hand side is the connected correlation function of the density field ρ(R): hρ(R)ρ(0)iϕ,c ≡ hρ(R)ρ(0)iϕ − hρ(R)iϕ hρ(0)iϕ .

(15.383)

In Section 3.10 we have shown how to generate all connected correlation functions: The action A[ψ, ψ ∗ , ϕ] is extended by a source term in the density field ρ(x) Asource [ψ ∗ , ψ, K] = −

Z

dD xK(x)ρ(x) =

Z

dD xK(x)ψ ∗ (x)ψ(x)),

(15.384)

and one considers the partition function Z[K, ϕ] ≡

Z

DψDψ ∗ exp {−A [ψ ∗ , ψ, ϕ] − Asource [ψ ∗ , ψ, K]} .

(15.385)

This is the generating functional of all correlation functions of the density field ρ(R) = ψ ∗ (R)ψ(R) at a fixed ϕ(x). They are obtained from the functional derivatives hρ(x1 ) · · · ρ(xn )iϕ = Z[K, ϕ]−1

δ δ ··· Z[K, ϕ] . K=0 δK(x1 ) δK(xn )

(15.386)

Recalling Eq. (3.556), the connected correlation functions of ρ(x) are obtained similarly from the logarithm of Z[K, ϕ]: hρ(x1 ) · · · ρ(xn )iϕ,c =

δ δ ··· log Z[K, ϕ] . K=0 δK(x1 ) δK(xn )

(15.387)

990


For n = 2, the connectedness is seen directly by performing the differentiations according to the chain rule: δ δ log Z[K, ϕ] K=0 δK(R) δK(0) δ δ = Z −1 [K, ϕ] Z[K, ϕ] K=0 δK(R) δK(0) = hρ(R)ρ(0)iϕ − hρ(R)iϕ hρ(0)iϕ .

hρ(R)ρ(0)iϕ,c =

(15.388)

This agrees indeed with (15.383). We can therefore rewrite the product of Laplacetransformed distributions (15.382) at a fixed ϕ(x) as δ δ log Z[K, ϕ] . K=0 δK(R) δK(0)

Pmϕ2 (0, R)Pmϕ2 (0, R) =

(15.389)

The Laplace-transformed monomer distribution (15.376) is then obtained by averaging over ϕ(x), i.e., by the path integral δ δ Pm2 (R) = δK(R) δK(0)

Z

Dϕ(x) exp {−A[ϕ]} log Z[K, ϕ]

K=0

.

(15.390)

Were it not for the logarithm in front of Z, this would be a standard calculation of correlation functions within the combined ψ, ϕ field theory whose action is Atot [ψ ∗ , ψ, ϕ] = A[ψ ∗ , ψ, ϕ] + A[ϕ] =

Z

1 − 2

dD x ∇ψ ∗ ∇ψ + m2 ψ ∗ ψ + 2Mϕψ ∗ ψ

Z

dD xdD x′ ϕ(x)V −1 (x, x′ )ϕ(x′ ).

(15.391)

To account for the logarithm we introduce a simple mathematical device called the replica trick [9]. We consider log Z[K, ϕ] in (15.388)–(15.390) as the limit 1 n (Z − 1) , n→0 n

log Z = lim

(15.392)

and observe that the nth power of the generating functional, Z n , can be thought of as arising from a field theory in which every field ψ occurs n times, i.e., with n identical replica. Thus we add an extra internal symmetry label α = 1, . . . , n to the fields ψ(x) and calculate Z n formally as Z n [K, ϕ] =

Z

Dψα∗ Dψα exp {−A[ψα∗ , ψα , ϕ]−A[ϕ] −Asource [ψα∗ , ψα , K]} , (15.393)

with the replica field action A[ψα , ψα∗ , ϕ] =

Z

dD x ∇ψα∗ ∇ψα + m2 ψα∗ ψα + 2Mϕ ψα∗ ψα ,

(15.394)


991


and the source term Asource [ψα∗ , ψα , K]

=−

Z

dD xψα∗ (x)ψα (x)K(x).

(15.395)

A sum is implied over repeated indices α. By construction, the action is symmetric under the group U(n) of all unitary transformations of the replica fields ψα . In the partition function (15.393), it is now easy to integrate out the ϕ(x)fluctuations. This gives Z n [K, ϕ] =

Z

Dψα∗ Dψα exp {−An [ψα∗ , ψα ] − Asource [ψα∗ , ψα , K]} ,

(15.396)

with the action A

n

[ψα∗ , ψα ]

=

Z

dD x ∇ψα∗ ∇ψα + m2 ψα∗ ψα

1 + (2M)2 2

Z

dD xdD x′ ψα ∗ (x)ψα (x)V (x, x′ )ψβ ∗ (x′ )ψβ (x′ ). (15.397)

It describes a self-interacting field theory with an additional U(n) symmetry. In the special case of a local repulsive potential V (x, x′ ) of Eq. (15.319), the second term becomes simply Aint [ψα∗ , ψα ]

Z 1 2 D = (2M) va dD x [ψα ∗ (x)ψα (x)]2 . 2

(15.398)

Using this action, we can find log Z[K, ϕ] via (15.392) from the functional integral log Z[K, ϕ] ≡ lim

n→0

1 n

Z

Dψα∗ Dψα exp {−An [ψα∗ , ψα ] − Asource [ψα∗ , ψα , K]} − 1 .

(15.399) This is the generating functional of the Laplace-transformed distribution (15.390) which we wanted to calculate. A polymer can run along the same line in two orientations. In the above description with complex replica fields it was assumed that the two orientations can be distinguished. If they are indistinguishable, the polymer fields Ψα (x) have to be taken as real. Such a field-theoretic description of a fluctuating polymer has an important advantage over the initial path integral formulation based on the analogy with a particle orbit. It allows us to establish contact with the well-developed theory of critical phenomena in field theory. The end-to-end distribution of long polymers at large L is determined by the small-E regime in Eqs. (15.330)–(15.349), which corresponds to the small-m2 limit of the system here [see (15.369)]. This is precisely the regime studied in the quantum field-theoretic approach to critical phenomena in many-body systems [10, 11]. It can be shown that for D larger than the upper critical dimension D uc = 4, the behavior for m2 → 0 of all Green functions coincides with the free-field behavior. For D = D uc , this behavior can be deduced from scale invariance arguments of the action, using naive dimensional counting arguments.

992


The fluctuations turn out to cause only logarithmic corrections to the scale-invariant power laws. One of the main developments in quantum field theory in recent years was the discovery that the scaling powers for D < Duc can be calculated via an expansion of all quantities in powers of ǫ = D uc − D,

(15.400)

the so-called ǫ-expansion. The ǫ-expansion for the critical exponent ν which rules the relation between R2 and the length of a polymer L, hR2 i ∝ L2ν , can be derived from a real φ4 -theory with n replica as follows [12]: ν −1 = 2 + 2

(n+2)ǫ n+8

n

−1 −

ǫ (13n 2(n+8)2

+ 44)

ǫ 3 2 + 8(n+8) 4 [3n − 452n − 2672n − 5312 ǫ3 + 32(n+8) 6

+ ζ(3)(n + 8) · 96(5n + 22)]

[3n5 + 398n4 − 12900n3 − 81552n2 − 219968n − 357120

+ ζ(3)(n + 8) · 16(3n4 − 194n3 + 148n2 + 9472n + 19488)

+ ζ(4)(n + 8)3 · 288(5n + 22) ǫ4

− ζ(5)(n + 8)2 · 1280(2n2 + 55n + 186)]

+ 128(n+8)8 [3n7 − 1198n6 − 27484n5 − 1055344n4

−5242112n3 − 5256704n2 + 6999040n − 626688

− ζ(3)(n + 8) · 16(13n6 − 310n5 + 19004n4 + 102400n3 − 381536n2 − 2792576n − 4240640)

− ζ 2 (3)(n + 8)2 · 1024(2n4 + 18n3 + 981n2 + 6994n + 11688)

+ ζ(4)(n + 8)3 · 48(3n4 − 194n3 + 148n2 + 9472n + 19488)

+ ζ(5)(n + 8)2 · 256(155n4 + 3026n3 + 989n2 − 66018n − 130608) − ζ(6)(n + 8)4 · 6400(2n2 + 55n + 186)

o

+ ζ(7)(n + 8)3 · 56448(14n2 + 189n + 526)] ,

(15.401)

where ζ(x) is Riemann’s zeta function (2.513). As shown above, the single-polymer properties must emerge in the limit n → 0. There, ν −1 has the ǫ-expansion ν −1 = 2 −

ǫ 11 2 − ǫ + 0.114 425 ǫ3 − 0.287 512 ǫ4 + 0.956 133 ǫ5. 4 128

(15.402)

This is to be compared with the much simpler result of the last section ν −1 =

D+2 ǫ = 2− . 3 3

(15.403)

A term-by-term comparison is meaningless since the field-theoretic ǫ-expansion has a grave problem: The coefficients of the ǫn -terms grow, for large n, like n!, so that the series does not converge anywhere! Fortunately, the signs are alternating and the series can be resummed [13]. A simple first approximation used in ǫ-expansions H. Kleinert, PATH INTEGRALS

993


is to re-express the series (15.402) as a ratio of two polynomials of roughly equal degree ν −1 |rat =

2. + 1.023 606 ǫ − 0.225 661 ǫ2 , 1. + 0.636 803 ǫ − 0.011 746 ǫ2 + 0.002 677 ǫ3

(15.404)

called a Padé approximation. Its Taylor coefficients up to ǫ5 coincide with those of the initial series (15.402). It can be shown that this approximation would converge with increasing orders in ǫ towards the exact function represented by the divergent series. In Fig. 15.7, we plot the three functions (15.403), (15.402), and (15.404), the

Figure 15.7 Comparison of critical exponent ν in Flory approximation (dashed line) with result of divergent ǫ-expansion obtained from quantum field theory (dotted line) and its Padé resummation (solid line). The value of the latter gives the best approximation ν ≈ 0.585 at ǫ = 1.

last one giving the most reliable approximation ν −1 ≈ 0.585.

(15.405)

Note that the simple Flory curve lies very close to the Padé curve whose calculation requires a great amount of work. There exists a general scaling relation between the exponent ν and another exponent appearing in the total number of polymer configurations of length L which behaves like S = Lα−2 .

(15.406)

α = 2 − Dν.

(15.407)

The relation is

Direct enumeration studies of random chains on a computer suggest a number 1 α∼ , 4

(15.408)

994


corresponding to ν = 7/12 ≈ 0.583, very close to (15.405). The Flory estimate for the exponent α reads, incidentally, α=

4−D . D+2

(15.409)

In three dimensions, this yields α = 1/5, not too far from (15.408). The discrepancies arise from inaccuracies in both treatments. In the first treatment, they are due to the use of the saddle point approximation and the fact that the δ-function does not completely rule out the crossing of the lines, as required by the true self-avoidance of the polymer. The field theoretic ǫ-expansion, on the other hand, which in principle can give arbitrarily accurate results, has the problem of being divergent for any ǫ. Resummation procedures are needed and the order of the expansion must be quite large (≈ ǫ5 ) to extract reliable numbers.

15.13

Fermi Fields for Self-Avoiding Lines

There exists another way of enforcing the self-avoiding property of random lines [14]. It is based on the observation that for a polymer field theory with n fluctuating complex fields Ψα and a U(n)-symmetric fourth-order self-interaction as in the action (15.397), the symmetric incorporation of a set of m anticommuting Grassmann fields removes the effect of m of the Bose fields. For free fields this observation is trivial since the functional determinant of Bose and Fermi fields are inverse to one-another. In the presence of a fourth-order self-interaction, where the replica action has the form (15.397), we can always go back, by a Hubbard-Stratonovich transformation, to the action involving the auxiliary field ϕ(x) in the exponent of (15.393). This exponent is purely quadratic in the replica field, and each path integral over a Fermi field cancels a functional determinant coming from the Bose field. This boson-destructive effect of fermions allows us to study theories with a negative number of replica. We simply have to use more Fermi than Bose fields. Moreover, we may conclude that a theory with n = −2 has necessarily trivial critical exponents. From the above arguments it is equivalent to a single complex Fermi field theory with fourth-order self-interaction. However, for anticommuting Grassmann fields, such an interaction vanishes: (θ† θ)2 = [(θ1 − iθ2 )(θ1 + iθ2 )]2 = [2iθ1 θ2 ]2 = 0.

(15.410)

Looking back at the ǫ-expansion for the critical exponent ν in Eq. (15.401) we can verify that up to the power ǫ5 all powers in ǫ do indeed vanish and ν takes the mean-field value 1/2.

Appendix 15A ∆(0, 0) =

Basic Integrals ∆(L, L) = L/3,


Appendix 15B

Loop Integrals

I1 I2 I3 I4 I5

Z

=

Z

=

Z

=

Z

=

Z

=

L

995

ds ∆(s, s) = L2 /6,

(15A.2)

ds ˙∆ 2 (s, s) = L/12,

(15A.3)

ds ∆2 (s, s) = L3 /30,

(15A.4)

ds ∆(s, s) ˙∆ 2 (s, s) = 7L2 /360,

(15A.5)

0 L 0 L 0 L 0 L

ds 0

Z

L

ds′ ∆2 (s, s′ ) = L4 /90,

(15A.6)

0

∆(0, L) = ∆(L, 0) = −L/6, Z L I6 = ds ∆2 (s, 0) + ∆2 (s, L) = 2L3 /45,

(15A.7) (15A.8)

0

I7

Z

=

Z

L

ds

0

Z

L

ds′ ∆(s, s)˙∆ 2 (s, s′ ) = L3 /45,

(15A.9)

0

L

Z

L

ds′ ˙∆ (s, s)∆(s, s′ )˙∆ (s, s′ ) = L3 /180,

(15A.10)

ds ∆(s, s) ˙∆ 2 (s, 0) + ˙∆ 2 (s, L) = 11L2/90,

(15A.11)

˙∆ (0, 0) = −˙∆ (L, L) = −1/2, Z L Z L I11 = ds ds′ ∆(s, s) ∆˙(s, s′ )˙∆ (s′ , s′ ) = L3 /360,

(15A.13)

I8 I9 I10

=

Z

=

Z

=

ds 0

0

L 0 L

ds ˙∆ (s, s) [∆(s, 0)˙∆ (s, 0) + ∆(s, L)˙∆ (s, L)] = 17L2 /360, (15A.12)

0

0

I12

Z

=

(15A.14)

0

L

ds 0

Appendix 15B

Z

L

ds′ ∆(s, s′ )˙∆ 2 (s, s′ ) = L3 /90.

(15A.15)

0

Loop Integrals

We list here the Feynman integrals evaluated with dimensional regularization rules whenever necessary. Depending whether they occur in the calculation of the moments from the expectations (15.261)–(15.261) of from the expectations (15.283)–(15.285) we encounter the integrals depending on ρn (s) ≡ δn + [δ(s) + δ(s − L)] /2 with δn = δ(0) − σn /L or ρk (s) = δk + [δ(s) + δ(s − L)] /2 with δk = δ(0) − ik/L: n(k)

H1

n(k)

H2

n(k)

H3

n(k)

H4

=

Z

0

L

ds

Z

L

ds′ ρn(k) (s)ρn(k) (s′ )∆2 (s, s′ ),

0

1 2 2 = δn(k) I5 + δn(k) I6 + ∆ (0, 0) + ∆2 (0, L) , 2 Z L Z L I9 = ds ds′ ρn(k) (s′ )∆(s, s)˙∆ 2 (s, s′ ) = δn(k) I7 + , 2 0 0 Z L Z L I6 ′ ′ 2 ′ δ(0), = ds ds ρn(k) (s )˙∆ d(s, s)∆ (s, s ) = δn(k) I5 + 2 0 0 Z L Z L I10 = ds ds′ ρn(k) (s′ )˙∆ (s, s)˙∆ (s, s′ )∆(s, s′ ) = δn(k) I8 + . 2 0 0

(15B.1) (15B.2) (15B.3) (15B.4)

996


When calculating h Rn i, we need to insert here δn = δ(0) − σn /L, thus obtaining L4 2 L3 L2 δ (0)+ (3−D−n) δ(0)+ (45−24D+4D2)−4n(6−2D−n) , 90 45 360 3 2 L L H2n = δ(0) + (15−4D−4n), 45 180 L3 L4 2 δ (0)+ (3−D−n)δ(0), H3n = 90 90 L3 L2 H4n = δ(0)+ (21−4D−4n) , 180 720 H1n =

(15B.5) (15B.6) (15B.7) (15B.8)

where the values for n = 0 correspond to the partition function Z = h R0 i. The substitution δk = δ(0) − ik/L required for the calculation of P (k; L) yields L4 2 L3 L2 5L2 δ (0) + [2 + (−ik)] δ(0) + (−ik) [4 + (−ik)] + , 90 45 90 72 L2 L3 δ(0) + [11 + 4(−ik)] , H2k = 45 180 L4 2 L3 H3k = δ (0) + [2 + (−ik)] δ(0), 90 90 L3 L2 H4k = δ(0) + [17 + 4(−ik)] . 180 720 H1k =

(15B.9) (15B.10) (15B.11) (15B.12)

The other loop integrals are H5 =

L

Z

ds

0

H6 =

ds

0

H7 =

ds

0

H8 =

ds

0

H9 =

ds

0

H10 =

Z

Z

Z

Z

Z

ds′ ˙∆ d(s, s)∆2 (s, s′ )˙∆ d(s′ , s′ ) = δ 2 (0)I5 =

L

L

ds′ ˙∆ (s, s)˙∆ (s, s′ ) ∆˙(s, s′ )˙∆ (s′ , s′ ) = −

0

L

ds′ ˙∆ (s, s)∆(s, s′ )˙∆ d(s, s′ )˙∆ (s′ , s′ ) =

L

ds L

ds

Z

Z

ds′ ∆(s, s)˙∆ (s, s′ )˙∆ d(s, s′ )˙∆ (s′ , s′ ) = L

L3 δ(0), 180

L2 , 720

(15B.14) (15B.15) (15B.16)

I8 7L2 I10 − − H8 = , 2 L 360

L 0

(15B.13)

L4 2 δ (0), 90

ds′ ˙∆ (s, s)˙∆ (s, s′ )∆(s, s′ )˙∆ d(s′ , s′ ) = δ(0)I8 =

0

0

H11 =

L

0

L

Z

Z

L3 δ(0), 45

0

L

Z

ds′ ∆(s, s)˙∆ 2 (s, s′ )˙∆ d(s′ , s′ ) = δ(0)I7 =

0

L

Z

L

0

L

Z

Z

I9 I7 7L2 − = , 4 2L 360

(15B.17) (15B.18)

I1 2 2(I3 − I11 ) ) − − 2I4 , L L L3 L2 δ(0) + , (15B.19) = 30 9

ds′ ∆(s, s)˙∆ d2 (s, s′ )∆(s′ , s′ ) = δ(0)I3 + ( 0

+ 2 ∆2 (L, L) ∆˙(L, L) − ∆2 (0, 0) ∆˙(0, 0) − 2H10 Z L Z L L2 H12 = ds ds′ ˙∆ 2 (s, s′ ) ∆˙ 2 (s, s′ ) = , (15B.20) 90 0 0 Z L Z L I4 I12 H12 L2 H13 = ds ds′ ∆(s, s′ )˙∆ (s, s′ ) ∆˙(s, s′ )˙∆ d(s′ , s) = − − =− , (15B.21) 2 2L 2 720 0 0 Z L Z L 2(I3 − I12 ) I5 H14 = ds ds′ ∆2 (s, s′ )˙∆ d2 (s, s′ ) = δ(0)I3 − − 2I4 + 2 , L L 0 0 2 L3 11L2 δ(0) + . (15B.22) + 2 ∆ (L, L) ∆˙(L, L) − ∆2 (0, 0) ∆˙(0, 0) − 2H13 = 30 72 H. Kleinert, PATH INTEGRALS

997

Appendix 15C

Integrals Involving Modified Green Function

Appendix 15C

Integrals Involving Modified Green Function

To demonstrate the translational invariance of results showed in the main text we use the slightly modified Green function ∆(s, s′ ) =

L | s − s′ | (s + s′ ) (s2 + s′2 ) a− − + , 3 2 2 2L

(15C.1)

containing an arbitrary constant a. The following combination yields the standard Feynman propagator for the infinite interval [compare (3.246)] 1 1 1 ∆F (s, s′ ) = ∆F (s−s′ ) = ∆(s, s′ )− ∆(s, s)− ∆(s′ , s′ ) = − (s−s′ ). 2 2 2 Other useful relations fulfilled by the Green function (15C.1) are, assuming s ≥ s′ , (s−s′ )(s+s′ )2 La ′ 2 ′ ′ ′ ′ s(L−s) D1 (s, s ) = ∆ (s, s )−∆(s, s)∆(s , s ) = (s−s ) + − , L 4L2 3 D2 (s, s′ ) = ∆(s, s)−∆(s′ , s′ ) =

′

(15C.2)

(15C.3)

′

(s−s )(s+s −L) . L

(15C.4)

The following integrals are needed: L

(s4 +s′4 ) (2s3 +s′3 ) − , 4L2 3L 0 ((a+3)s2 +as′2 ) sa (20a−9) 2 + − L+ L , 6 3 180 Z L (−2s4 +6s2 s′2 +3s′4 ) J2 (s, s′ ) = dt ˙∆ (t, t)∆(t, s)˙∆ (t, s′ ) = , 24L2 0 (3s3 −3s2 s′ −6ss′2 −s′3 ) (5s2 −12ss′ −4as′2 ) s′ a (20a−3) 2 + − − L+ L , 12L 24 6 720 J1 (s, s′ ) =

Z

dt ∆(t, t)˙∆ (t, s)˙∆ (t, s′ ) =

(15C.5)

(15C.6) L

4

2 ′2

′4

2

′2

(s +6s s +s ) (s +3s )s dt ∆(t, s)∆(t, s′ ) = − + , 60L 6 0 (s2 +s′2 ) (5a2 −10a+6) 3 − L+ L . 6 45

J3 (s, s′ ) =

Z

These are the building blocks for other relations: (d−1) (d+1) 2 (d−1)(s−s′) hf2 (q(s), q(s′ ))i = D1 (s, s′ )− D2 (s, s′ ) = − 2 4 4 ′ 2 ′2 ′ d(s−s )(s+s′ )2 2La (d−3)s−(d+1)s (d−1)s −(d+1)s × + − + , 3 2 L 2L2

(15C.7)

(15C.8)

and 1 1 K1 (s, s′ ) = J1 (s, s′ )− J1 (s, s)− J1 (s′ , s′ ), 2 2 La (s+s′ ) (s2 +ss′ +s′2 ) ′ = − (s−s ) − + , 6 4 6L K2 (s, s′ ) = J2 (s, s′ )+J2 (s′ , s)−J2 (s, s)−J2 (s′ , s′ ),

(15C.9)

998

15 Path Integrals in Polymer Physics 1 (5s+7s′ ) (s+s′ )2 − + , 4 12L 4L2 1 1 (s+2s′ ) (s+s′ )2 K3 (s, s′ ) = J3 (s, s′ )− J3 (s, s)− J3 (s′ , s′ ) = −(s−s′ )2 − , 2 2 6 8L 1 1 (s−s′)2 (s+s′ −2L)2 K4 (s, s′ ) = ∆(0, s)∆(0, s′ )− ∆2 (0, s)− ∆2 (0, s′ ) = − , 2 2 8L2 1 1 (s2 −s′2 )2 . K5 (s, s′ ) = ∆(L, s)∆(L, s′ )− ∆2 (L, s)− ∆2 (L, s′ ) = − 2 2 8L2 = − (s−s′ )2

(15C.10) (15C.11) (15C.12) (15C.13)

Notes and References Random chains were first considered by K. Pearson, Nature 77, 294 (1905) who studied the related problem of a random walk of a drunken sailor. A.A. Markoff, Wahrscheinlichkeitsrechnung, Leipzig 1912; L. Rayleigh, Phil. Mag. 37, 3221 (1919); S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). The exact solution of PN (R) was found by L.R.G. Treloar, Trans. Faraday Soc. 42, 77 (1946); K. Nagai, J. Phys. Soc. Japan, 5, 201 (1950); M.C. Wang and E. Guth, J. Chem. Phys. 20, 1144 (1952); C. Hsiung, A. Hsiung, and A. Gordus, J. Chem. Phys. 34, 535 (1961). The path integral approach to polymer physics has been advanced by S.F. Edwards, Proc. Phys. Soc. Lond. 85, 613 (1965); 88, 265 (1966); 91, 513 (1967); 92, 9 (1967). See also H.P. Gilles, K.F. Freed, J. Chem. Phys. 63, 852 (1975) and the comprehensive studies by K.F. Freed, Adv. Chem. Phys. 22,1 (1972); K. Itô and H.P. McKean, Diffusion Processes and Their Simple Paths, Springer, Berlin, 1965. An alternative model to stimulate stiffness was formulated by A. Kholodenko, Ann. Phys. 202, 186 (1990); J. Chem. Phys. 96, 700 (1991) exploiting the statistical properties of a Fermi field. Although the physical properties of a stiff polymer are slightly misrepresented by this model, the distribution functions agree quite well with those of the Kratky-Porod chain. The advantage of this model is that many properties can be calculated analytically. The important role of the upper critical dimension Duc = 4 for polymers was first pointed out by R.J. Rubin, J. Chem. Phys. 21, 2073 (1953). The simple scaling law hR2 i ∝ L6/(D+2) , D < 4 was first found by P.J. Flory, J. Chem. Phys. 17, 303 (1949) on dimensional grounds. The critical exponent ν has been deduced from computer enumerations of all self-avoiding polymer configurations by C. Domb, Adv. Chem. Phys. 15, 229 (1969); D.S. Kenzie, Phys. Rep. 27, 35 (1976). For a general discussion of the entire subject, in particular the field-theoretic aspects, see P.G. DeGennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, N.Y., 1979. The relation between ensembles of random chains and field theory is derived in detail in H. Kleinert, Gauge Fields in Condensed Matter , World Scientific, Singapore, 1989, Vol. I, Part I; Fluctuating Fields and Random Chains, World Scientific, Singapore, 1989 (http://www.physik.fu-berlin.de/~kleinert/b1).



999

The particular citations in this chapter refer to the publications [1] The path integral (15.112) for stiff polymers was proposed by O. Kratky and G. Porod, Rec. Trav. Chim. 68, 1106 (1949). The lowest moments were calculated by J.J. Hermans and R. Ullman, Physica 18, 951 (1952); N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Japan 22, 219 (1967), and up to order 6 by R. Koyama, J. Phys. Soc. Japan, 34, 1029 (1973). Still higher moments are found numerically by H. Yamakawa and M. Fujii, J. Chem. Phys. 50, 6641 (1974), and in a large-stiffness expansion in T. Norisuye, H. Murakama, and H. Fujita, Macromolecules 11, 966 (1978). The last two papers also give the end-to-end distribution for large stiffness. [2] H.E. Stanley, Phys. Rev. 179, 570 (1969). [3] H.E. Daniels, Proc. Roy. Soc. Edinburgh 63A, 29 (1952); W. Gobush, H. Yamakawa, W.H. Stockmayer, and W.S. Magee, J. Chem. Phys. 57, 2839 (1972). [4] B. Hamprecht and H. Kleinert, J. Phys. A: Math. Gen. 37, 8561 (2004) (ibid.http/345). Mathematica program can be downloaded from ibid.http/b5/prm15. [5] The Mathematica notebook can be obtained from ibid.http/b5/pgm15. The program needs less than 2 minutes to calculate R32 .

[6] J. Wilhelm and E. Frey, Phys. Rev. Lett. 77, 2581 (1996). See also: A. Dhar and D. Chaudhuri, Phys. Rev. Lett 89, 065502 (2002); S. Stepanow and M. Schuetz, Europhys. Lett. 60, 546 (2002); J. Samuel and S. Sinha, Phys. Rev. E 66 050801(R) (2002).

[7] H. Kleinert and A. Chervyakov, Perturbation Theory for Path Integrals of Stiff Polymers, Berlin Preprint 2005 (ibid.http/358). [8] For a detailed theory of such fields with applications to superconductors and superfluids see H. Kleinert, Collective Quantum Fields, Fortschr. Phys. 26, 565 (1978) (ibid.http/55). [9] S.F. Edwards and P.W. Anderson, J. Phys. F: Metal Phys. 965 (1975). Applications of replica field theory to the large-L behavior of hR2 i and other critical exponents were given by P.G. DeGennes, Phys. Lett. A 38, 339 (1972); J. des Cloizeaux, Phys. Rev. 10, 1665 (1974); J. Phys. (Paris) Lett. 39 L151 (1978). [10] See D.J. Amit, Renormalization Group and Critical Phenomena, World Scientific, Singapore, 1984; G. Parisi, Statistical Field Theory, Addison-Wesley, Reading, Mass., 1988. [11] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories, World Scientific, Singapore, 2000 (ibid.http/b8). [12] For the calculation of such quantities within the φ4 -theory in 4 − ǫ dimensions up to order ǫ5 , see H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin, and S.A. Larin, Phys. Lett. B 272, 39 (1991) (hep-th/9503230). [13] For a comprehensive list of the many references on the resummation of divergent ǫexpansions obtained in the field-theoretic approach to critical phenomena, see the textbook [11]. [14] A.J. McKane, Phys. Lett. A 76, 22 (1980).

H. Kleinert, PATH INTEGRALS June 16, 2007 ( /home/kleinert/kleinert/books/pathis/pthic16.tex) Take any shape but that, and my firm nerves Shall never tremble. William Shakespeare, Macbeth

16 Polymers and Particle Orbits in Multiply Connected Spaces In the previous chapter, the binary interaction potential between the polymer elements was approximated by a δ-function. Quantum-mechanically, this potential is not completely impenetrable. Correspondingly, a polymer with such an interaction has a finite probability of self-intersections. This is only a rough approximation to the situation in nature where the atomic potential is of the hard-core type and selfintersections are extremely rare. In a grand-canonical ensemble, a polymer is often entangled with itself and with others. It can be disentangled only if it has open ends. Macroscopic fluctuations are required to achieve this in the form of worm-like creeping processes. Compared with local fluctuations, these take a very long time. For closed polymers, disentangling is impossible without breaking bonds at the cost of large activation energies. In order to study such entanglement phenomena in their purest form, it is useful to idealize the strongly repulsive interaction potential, as in Section 15.10, to a topological constraint of the type discussed in Chapter 6. Entanglement phenomena play an important role also in quantum mechanics. Fluctuating particle orbits may get entangled with magnetic flux tubes or with other particle orbits. In fact, the statistical properties of Bose and Fermi particles may be viewed as entanglement phenomena, as will be shown in this chapter.

16.1

Simple Model for Entangled Polymers

Consider the simplest model system with a topological constraint producing entanglement phenomena: a fixed polymer stretched out along the z-axis and a fluctuating second polymer. Arbitrary entanglements with the straight polymer may occur. The possible entanglements of the fluctuating polymer with itself are ignored, for simplicity. Let us study the end-to-end distribution of the fluctuating second polymer. At first we neglect the third dimension, which can be trivially included at a later stage, imagining the movement to be confined entirely to the xy-plane. If the polymer along the z-axis is infinitely thin, the total end-to-end distribution of 1000

1001

16.1 Simple Model for Entangled Polymers

the fluctuating polymer in the plane is certainly independent of the presence of the central polymer. In the random-chain approximation it reads for not too large R PN (xb − xa ) =

s

2

2 2 e−(xb −xa ) /2La , 2πLa

(16.1)

where x is a planar vector. In the presence of the central polymer, an interesting new problem arises: How does the end-to-end distribution decompose with respect to the number of times by which the fluctuating polymer is wrapped around the central polymer? To define this number, we choose an arbitrary reference line from the origin to infinity, say the x-axis. For each path from xa to xb , we count how often it crosses this line, including a minus sign for opposite directions of the crossings. In this way, each path receives an integer-valued label n which depends on the position of the reference line. A property independent of the choice of the reference line exists for the pairs of paths with fixed ends. The difference path is closed. The number n of times by which a closed path encircles the origin is a topological invariant called the winding number . Let us find the decomposition of the probability distribution of a closed polymer PN (xb − xa ) with respect to n: PN (xb − xa ) =

∞ X

PNn (xb , xa ).

(16.2)

n=−∞

The topological constraint destroys the translational invariance of the total distribution on the left-hand side, so that the different fixed-n distributions on the right-hand side depend separately on both xb and xa . In a path integral it is easy to keep track of the number of crossings n. The angular difference between initial and final points xb and xa is given by the integral ϕb − ϕa =

Z

tb

ta

dt ϕ(t) ˙ =

Z

tb

ta

x1 x˙ 2 − x2 x˙ 1 dt = x21 + x22

Z

xb

xa

x × dx . x2

(16.3)

Given two paths C1 and C2 connecting xa and xb , this integral differs by an integer multiple of 2π. The winding number is therefore given by the contour integral over the closed difference path C: 1 I x × dx . n= 2π C x2

(16.4)

In order to decompose the end-to-end distribution (16.2) with respect to the winding number, we recall the angular decomposition of the imaginary-time evolution amplitude in Eqs. (8.9) and (8.17) of a free particle in two dimensions PL (xb − xa ) =

X m

√

1 1 (rb τb |ra τa )m eim(ϕb −ϕa ) , rb ra 2π

(16.5)

1002

16 Polymers and Particle Orbits in Multiply Connected Spaces

with the radial amplitude √

rb ra −(r2 +ra2 )/La rb ra Im 2 . (rb τb |ra τa )m = 2 e b La La

(16.6)

We have inserted the polymer parameters following the rules of Section 15.6, replacing M/2¯ h(τb − τa ) by 1/La, and using the label L = Na in PL rather than N, as in Eq. (15.301). We now recall that according to Section 6.1, an angular path integral consisting of a product of integrals N Z Y

π

n=1 −π

dϕn , 2π

(16.7)

whose conjugate momenta are integer-valued, can be converted into a product of ordinary integrals N Z Y

∞

n=1 −∞

dϕn , 2π

(16.8)

whose conjugate momenta are continuous. These become independent of the time slice n by momentum conservation, and the common momentum is eventually restricted to its proper integer values by a final sum over an integer number n occurring in the Poisson formula [see (6.9)] X

∞ X

eik(ϕb +2πn−ϕa ) =

n

m=−∞

δ(k − m)eim(ϕb −ϕa ) .

(16.9)

Obviously, the number n on the left-hand side is precisely the winding number by which we want to sort the end-to-end distribution. The desired restricted probability PLn (xb , xa ) for a given winding number n is therefore obtained by converting the sum over m in Eq. (16.5) into an integral over µ and another sum over n as in Eq. (1.205), and by omitting the sum over n. The result is: PLn (xb , xa )

2 Z∞ rb ra 2 2 = dµe−(rb +ra )/La I|µ| 2 La −∞ La

1 iµ(ϕb −ϕa +2πn) e . 2π

(16.10)

From this we find the desired probability of a closed polymer running through a point x with various winding numbers n around the central polymer: PLn (x, x)

2 = La

Z

∞

−∞

−2r 2 /La

dµe

I|µ|

r2 2 La

!

1 i2πµn e . 2π

(16.11)

Let us also calculate the partition function of a closed polymer with a given winding number n. To make the partition function finite, we change the system by H. Kleinert, PATH INTEGRALS

1003

16.1 Simple Model for Entangled Polymers

adding a harmonic oscillator potential centered at the origin.1 If ω is measured in units 1/length, the above probability becomes PLn (x, x)

2 ω = a sinh ωL

Z

∞ −∞

−2(r 2 /a)ω coth ωL

dµe

I|µ|

2 r2ω a sinh ωL

!

1 i2πµn . (16.12) e 2π

This can be integrated over the entire space using the formula (2.467). The result is Z Z ∞ 1 PLn ≡ d2 xPLn (x, x) = dµe−|µ|ωL e2πiµn . (16.13) 2 sinh ωL −∞ To check this formula, we sum both sides over all n. Then the integral over µ is reduced, via Poisson’s formula, to a sum over integers µ = m = 0, ±1, ±2, . . . , and we find PL ≡

Z

∞ X

d2 xPL (x, x) =

PLn

n=−∞

1 1 2 = −1 = . −ωL 2 sinh ωL 1 − e [2 sinh(ωL/2)]2

(16.14)

As we should have expected, this is the partition function of the two-dimensional harmonic oscillator. To find the contribution of the various winding numbers, we perform the integral over µ and obtain ωL 1 . sinh ωL 4π 2 n2 + ω 2 L2 The right-hand factor is recognized as a term arising in the expansion PLn =

1 coth(ωL/2) = 2ωL =

∞ X

4π 2 n2

n=−∞ ∞ X

1 L2

2 n=−∞ ωn

(16.15)

1 + (ωL)2 1 . + ω2

(16.16)

The quantities ωn ≡ 2πn/L are the polymer analogs of the Matsubara frequencies. Thus we may write PLn = PL · αn ,

(16.17)

where αn is the relative probability of finding the winding number n (with the normalization Σn αn = 1), αn

1 = 2 ωn + ω 2

"

∞ X

1 2 2 n=−∞ ωn + ω

#−1

1 1 1 ωL = coth 2 2 2 L ωn + ω 2ωL 2

1

−1

.

(16.18)

Alternatively, we may add a magnetic field with the Landau frequency ω = −eB/M c, as done in (16.33). Then the amplitude contains ω/2 instead of ω, and an extra factor emωL/2 .

1004

16.2


Entangled Fluctuating Particle Orbit: Aharonov-Bohm Effect

The entanglement of a fluctuating polymer around a straight central polymer has an interesting quantum-mechanical counterpart known as the Aharonov-Bohm effect. Consider a free nonrelativistic charged particle moving through a space containing an infinitely thin tube of finite magnetic flux along the z-direction: B3 =

g ǫ3jk ∂j ∂k ϕ = g δ (2) (x⊥ ), 2π

(16.19)

where x⊥ is the transverse vector x⊥ ≡ (x1 , x2 ). Let us study the associated path integral. The magnetic interaction is given by [recall Eq. (2.627)] e Z tb dt x˙ · A, c ta

Amag =

(16.20)

where e is the charge and A the vector potential. The flux tube (16.19) is obtained from the components in the xy-plane. Ai =

g ∂i ϕ, 2π

(i = 1, 2),

(16.21)

where ϕ is the azimuthal angle around the tube: ϕ(x) ≡ arctan(x2 /x1 ).

(16.22)

Note that the derivatives in front of ϕ in (16.19) commute everywhere, except at the origin where Stokes’ theorem yields Z

2

d x (∂1 ∂2 − ∂2 ∂1 )ϕ =

I

dϕ = 2π.

(16.23)

The total magnetic flux through the tube is defined by the integral Φ=

Z

d2 x B3 .

(16.24)

Inserting (16.19) we see that the total flux is equal to g: Φ = g.

(16.25)

With the vector potentoal (16.21), the interaction (16.20) takes the form Amag = −¯ hµ0

Z

tb

ta

dt ϕ, ˙

(16.26)

where µ0 is the dimensionless number µ0 ≡ −

eg . 2π¯ hc


1005

16.2 Entangled Fluctuating Particle Orbit: Aharonov-Bohm Effect

The minus sign is a matter of convention. Since the particle orbits are present at all times, their worldlines in spacetime can be considered as being closed at infinity, and the integral 1 n= 2π

Z

tb

ta

dt ϕ˙

(16.28)

is the topological invariant (16.4) with integer values of the winding number n. The magnetic interaction (16.26) is therefore a purely topological one, its value being hµ0 2πn. Amag = −¯

(16.29)

After adding this to the action of a free particle in the radial decomposition (8.9) of the quantum-mechanical path integral,we rewrite the sum over the azimuthal quantum numbers m via Poisson’s summation formula as in (16.10), and obtain (xb τb |xa τa ) =

1 dµ √ (rb τb |ra τa )µ rb ra −∞ ∞ X 1 i(µ−µ0 )(ϕb +2πn−ϕa ) e × . n=−∞ 2π

Z

∞

(16.30)

Since the winding number n is often not easy to measure experimentally, let us extract observable consequences which are independent of n. The sum over all n forces µ to be equal to µ0 modulo an arbitrary integer number m = 0, ±1, ±2, . . . . The result is (xb τb |xa τa ) =

∞ X

m=−∞

√

1 1 (rb τb |ra τa )m+µ0 eim(ϕb −ϕa ) , rb ra 2π

(16.31)

with the radial amplitude (rb τb |ra τa )m+µ0 =

√

M M rb2 + ra2 M rb ra 1 rb ra exp − I|m+µ0 | .(16.32) h ¯ (τb − τa ) 2¯ h τb − τa h ¯ τb − τa (

)

For the sake of generality, we allow for the presence of a homogeneous magnetic field B whose Landau frequency is ω = −eB/Mc. In analogy with the parameter µ0 in (16.27), it is defined with a minus sign. Using the radial amplitude (9.105), we see that (16.32) is simply generalized to (rb τb |ra τa )m+µ0 =

√

Mω η Mω rb ra exp − coth η(rb2 + ra2 ) 2¯ hη sinh η 2¯ h2 ! Mωrb ra × I|m+µ0 | e(m+µ0 )η , 2¯ h sinh η

(16.33)

where η ≡ ω(τb − τa )/2. At this point we can make an interesting observation: If µ0 is an integer number, i.e., if eg = integer, (16.34) 2π¯ hc

1006


the quantum-mechanical particle distribution function (x tb |x ta ) in (16.31) becomes independent of the magnetic flux tube along the z-axis. The condition implies that the magnetic flux is an integer multiple of the fundamental flux quantum Φ0 ≡ g0 ≡

2π¯ hc hc = . e e

(16.35)

We recognize this infinitely thin tube as a Dirac string. Such undetectable strings were used in Sections 8.12, Appendix 10A.3, and Section 14.6 to import the magnetic flux of a magnetic monopole from infinity to a certain point where the magnetic field lines emerge radially. In Appendix 10A.3 we have made the string invisible mathematically imposing monopole gauge invariance. The present discussion shows explicitly that the flux quantization makes Dirac strings indeed undetectable by any charged particle. This observation inspired Dirac his speculation on the existence of magnetic monopoles. In low-temperature physics, a quantization of magnetic flux is observable in typeII superconductors. Superconductors are perfect diamagnets which expel magnetic fields. Those of type II have the property that above a certain critical external field called Hc1 , the expulsion is not perfect but they admit quantized magnetic tubes of flux Φ0 (Shubnikov phase). For increasing fields, there are more and more such flux tubes. They are squeezed together and can form a periodically arranged bundle. When cut across in the xy-plane, the bundle looks like a hexagonal planar flux lattice [4]. If the central magnetic flux tube in (16.19) carries an amount of flux that is not an integer multiple of Φ0 , the amplitude of particles passing the tube displays an interesting interference pattern. This was initially a surprise since the space is free of magnetic fields. To calculate this pattern we consider the fixed-energy amplitude of a free particle in two dimensions (9.12), decomposed into partial waves via the addition theorem (9.14) for Bessel functions as in (9.15): (xb |xa )E = −

∞ 2iM X 1 Im (κr< )Km (κr> ) eim(ϕb −ϕa ) . h ¯ m=−∞ 2π

(16.36)

Comparing this with (16.30) and repeating the arguments leading to (16.31), (16.32), we can immediately write down the fixed-energy amplitude in the presence of a flux Φ0 : (xb |xa )E = −

∞ 2iM X 1 I|m+µ0 | (κr< )K|m+µ0 | (κr> ) eim(ϕb −ϕa ) . h ¯ m=−∞ 2π

(16.37)

The wave functions are now easily extracted. In the complex E-plane, the righthand side has a discontinuity across the positive real axis. By going through the same steps as in (9.15)–(9.22), we derive for the discontinuity Z

∞

−∞

∞ X dE disc(xb |xa )E = 2π¯ h m=−∞

Z

∞ 0

dkkJ|m+µ0 | (krb )J|m+µ0 | (kra )

1 im(ϕb −ϕb ) e . (16.38) 2π


1007


The integration measure (dE/2π¯ h)(2πM/¯ h) has been replaced by 0∞ dkk according to Eq. (9.23). In the absence of the flux tube, the amplitude (16.38) reduces to that of a free particle, which has the decomposition R

Z

∞

−∞

R

dE disc(xb |xa )E = 2π¯ h =

Z

d2 k ik(xb −xa ) 1 e = 2 (2π) 2π Z ∞ X

∞

m=−∞ 0

Z

∞

0

dkkJ0 (k|xb − xa |)

dkkJm (krb )Jm (kra )

1 im(ϕb −ϕa ) e . 2π

(16.39)

If a flux tube is present, a beam of incoming charged particles is deflected even though the space around the z-axis contains no magnetic field. Let us calculate the scattering amplitude and the ensuing cross section from the fixed-energy amplitude (16.37). Recall the results of the quantum-mechanical scattering theory due to Lippmann and Schwinger. In this theory one studies the effect of an interaction upon an incoming free-particle state ϕk of wave vector k. The result is the scattering state ψk obtained from the Lippmann-Schwinger integral equation ψk = ϕk +

1 Vˆ ψk ˆ 0 + iη E−H

i ˆ = ϕk − R(E) Vˆ ϕk , h ¯

(16.40)

ˆ where E is the energy of the incoming particle, Vˆ the potential, and R(E) the resolvent operator (1.315). The scattering states ψk are solutions of the Schrödinger equation ˆ k = (H ˆ 0 + Vˆ )ψk = Eψk . Hψ

(16.41)

In x-space, the Lippmann-Schwinger equation reads i h ¯

ψk (x) = ϕk (x) −

Z

dD x′ (x|x′ )E V (x′ )ϕk (x′ ).

(16.42)

The first term describes the impinging particles, the second the scattered ones. For the scattering amplitude, only the large-x behavior of the second term matters. One usually normalizes ϕk (x) to eikx and factorizes the second term asymptotically into a product of an outgoing spherical wave times a scattering amplitude. In three dimensions, the asymptotic behavior far away from the scattering center is |x|→∞

ikx

ψk (x) − −−→ e

ei|k||x| + f (θ, ϕ) + . . . , |x|

(16.43)

where θ and ϕ are the scattering angles of the outgoing beam and f is the scattering amplitude. Its square gives directly the differential cross section dσ = |f (θ, ϕ)|2. dΩ

(16.44)

1008


In two dimensions, the corresponding splitting is |x|→∞ ei|k||x| ψk (x) − −−→ eikx + q f (ϕ) + . . . . |x|

(16.45)

The scattering amplitude f (ϕ) which depends only on the azimuthal angle ϕ = arctan(x2 /x1 ) yields the differential cross section dσ = |f (ϕ)|2 . dϕ

(16.46)

To calculate f (ϕ), we observe that the most general solution Ψ(x) of the Schrödinger equation (16.41) is obtained by forming the convolution integral of the discontinuity of the resolvent with an arbitrary wave function φ(x): ψ(x) =

Z

dD x′ disc(x|x′ )E φ(x′ ).

(16.47)

Using (16.38), this becomes some linear combination of wave functions J|m+µ0 | (kr) ∞ X

ψ(x) =

am J|m+µ0 | (kr)eimϕ ,

(16.48)

m=−∞

which certainly satisfies the Schrödinger equation (16.41). The coefficients am have to be chosen to satisfy the scattering boundary condition at spatial infinity. Suppose that the incident particles carry a wave vector k = (−k, 0). In the incoming region x → ∞, they are described by a wave function lim ψ(x) = e−ikx e−iµ0 ϕ .

(16.49)

x→∞

The extra phase factor is necessary for the correct wave vector since in the presence of the gauge field eAi = −¯ hcµ0 ∂i ϕ,

(16.50)

the physical momentum p = h ¯ k is not given by the usual derivative operator −i¯ h∇ but by the gauge-invariant momentum operator e ˆ = −i¯ h(∇ + iµ0 ∇ϕ). P h∇ − A = −i¯ c

(16.51)

The corresponding incident gauge-invariant particle current is j(x) = −i

h ¯ † ↔ e ψ ∇ ψ(x) − A(x)ψ † ψ(x). 2M Mc

(16.52)

We demonstrate below that the correct choice for the coefficients am is am = (−i)|m+µ0 | ,



1009

leading to the scattering amplitude 1 e−iϕ/2 f (ϕ) = √ e−iπ/4 sin πµ0 , cos(ϕ/2) 2π

(16.54)

i.e., to the cross section 1 dσ 1 = sin2 πµ0 2 . dϕ 2π cos (ϕ/2)

(16.55)

It has a strong peak near the forward direction ϕ ≈ π. For µ0 = integer, there is no scattering at all and the flux tube becomes an invisible Dirac string. To derive (16.53) and (16.54) we may assume that µ0 ∈ (0, 1). Otherwise, we could simply shift the sum over m in (16.37) by an integer ∆m, and this would merely produce an overall factor ei∆m(ϕb −ϕa ) in disc(xb |xa )E . This would wind up as a factor ei∆m ϕ in ψ(x). For µ0 ∈ (0, 1), we split the wave function (16.48) into three parts: ψk = ψ (1) + ψ (2) + ψ (3) .

(16.56)

The first collects the terms with positive m, ψ

(1)

=

∞ X

(−i)m+µ0 Jm+µ0 eimϕ ,

(16.57)

m=1

the second those with negative m, ψ

(2)

= =

−1 X

(−i)m+µ0 J|m+µ0 | eimϕ

m=−∞ ∞ X

(−i)m−µ0 Jm−µ0 e−imϕ ,

(16.58)

m=1

and the third contains only the term m = 0, ψ (3) = (−i)|µ0 | J|µ0 | .

(16.59)

Obviously, ψ (2) may be obtained from ψ (1) via the identity ψ (2) (r, ϕ, µ0 ) = ψ (1) (r, −ϕ, −µ0 ).

(16.60)

Thus, the wave function (16.56) requires only a calculation of ψ (1) . As a first step we observe that the sum (16.57) has an integral representation 1 ψ (1) = (−i)µ0 e−iρ cos ϕ I(ρ), 2

(16.61)

with I(ρ) being the integral I(ρ) ≡

Z

0

ρ

′

dρ′ eiρ

cos ϕ

J1+µ0 − iJµ0 eiϕ .

(16.62)

1010


We have set kr ≡ ρ such that kx ≡ ρ cos ϕ. To prove the integral representation, we differentiate (16.61) and find the differential equation 1 (16.63) ∂ρ ψ (1) = −i cos ϕ ψ (1) + (−i)µ0 J1+µ0 − iJµ0 eiϕ , 2 with all functions depending only on kr ≡ ρ. Precisely the same equation is obeyed by the sum (16.57):

∂ρ ψ

(1)

=

∞ X

(−i)m+µ0 ∂ρ Jm+µ0 eimϕ

m=1 ∞ X

1 (−i)m+µ0 (Jm+µ0 −1 − Jm+µ0 +1 ) eimϕ (16.64) 2 m=1 ∞ i X 1 = − (−i)m+µ0 Jm+µ0 eimϕ eiϕ + e−iϕ + (−i)µ0 J1+µ0 − iJµ0 eie . 2 m=1 2 =

Thus, the two expressions (16.57) and (16.61) for ψ (1) can differ at most by an integration constant. However, the constant must be zero since both expressions vanish at ρ = 0. This proves the integral representation (16.61). In order to derive the scattering amplitude for the magnetic flux tube, we have (1) into two to find the asymptotic of the wave function. This is done by splitting ψ∞ (1) terms, a contribution ψ∞ in which the integral I is carried all the way to infinity, to be denoted by I∞ , and a remainder ∆ψ (1) which vanishes for r → ∞. The integral I∞ can be calculated analytically using the formula Z

∞

0

dρeiβρ Jα (kρ) =

(k 2

1 eiα arcsin(β/k) , − β 2 )1/2

0 < β < k, α > −2.

(16.65)

This gives i 1 h iµ0 (π/2−|ϕ|) e − ieiϕ ei(1+µ0 )(π/2−|ϕ|) | sin ϕ| 0 ( i 0, ϕ < 0, = eiµ0 (π/2−|ϕ|) e−i|ϕ| − eiϕ = (16.66) −iµ0 ϕ µ0 e 2i , ϕ > 0, | sin ϕ|

I∞ ≡

Z

∞

′

dρ′ eiρ

cos ϕ

J1+µ0 − iJµ0 eiϕ =

with ϕ ∈ (−π, π). Hence we have =

(

0, e−ikx e−iµ0 ϕ ,

ϕ < 0, ϕ > 0.

(16.67)

(2) ψ∞ =

(

e−ikx e−iµ0 ϕ , 0,

ϕ < 0, ϕ > 0.

(16.68)

(1) ψ∞

Using (16.60), we find

(1) (2) The sum ψ∞ + ψ∞ represents the incoming wave (16.49). The scattered wave must therefore reside in the remainder

ψsc = ∆ψ (1) + ∆ψ (2) + ψ (3) .


1011


√ For the scattering amplitude, only the leading 1/ r-behavior of the three terms is relevant. To find it for ∆ψ (1) , we take (16.61) and write the remainder of the integral (16.62) as ∆I(ρ) ≡ I(ρ) − I∞ =

Z

∞

ρ

′

dρ′ eiρ

cos ϕ

J1+µ0 − ieiϕ Jµ0 .

(16.70)

At large ρ, the asymptotic expansion Jα (ρ) ∼ renders

q

2/πρ cos(ρ − α/2 − π/4)

∆I(ρ) =

s

(16.71)

2 [A(ρ) + B(ρ)], π

(16.72)

with the integrals dρ′ ′ √ ′ eiρ cos ϕ cos [ρ′ − (1 + µ0 )/2 − π/4] , ρ ρ Z ∞ dρ′ ′ √ ′ eiρ cos ϕ cos [ρ′ − µ0 /2 − π/4] . B(ρ) = −ieiϕ ρ ρ′ Z

A(ρ) =

∞

(16.73)

Separating the cosine into exponentials and changing the variable ρ′ in the two terms to ρ′ = t2 /(1 ± cos ϕ), we find ∞ (−i)1/2+µ0 ∞ i3/2+µ0 2 2 A(ρ) = √ dteit + √ dte−it , √ √ ρ(1+cos ϕ) ρ(1−cos ϕ) 1 + cos θ 1 − cos θ (16.74) and a corresponding expression for B(ρ). The asymptotic expansion of the error function

"

Z

Z

∞ x

#

Z

2

dte±it = ±

i exp(±ix2 ) + ... 2 x

(16.75)

leads to 



1 1 1 eiρ e−iρ  eiρ cos ϕ , A(ρ) = (−i) 2 +µ0 q + i 2 +µ0 q 2 2 2 ρ(1 + cos ϕ) ρ(1 − cos ϕ)

eiϕ  2

B(ρ) = (−i)

(16.76) 

1 1 eiρ e−iρ  eiρ cos ϕ . + i− 2 +µ0 q × (−i)− 2 +µ0 q 2 2 ρ(1 + cos ϕ) ρ(1 − cos ϕ)

Adding the two terms together in (16.72) and inserting everything into (16.61) gives the asymptotic behavior " # √ iϕ iϕ −i (1) µ0 iρ 1 + e −iρ 1 − e ∆ψ = √ (−1) e + ie . (16.77) 2 2πρ 1 + cos ϕ 1 − cos ϕ

1012


Together with ∆ψ (2) found via (16.60), we obtain √ " # −i iρ cos(πµ0 − ϕ/2) (1) (2) −iρ e + e−i(ρ cos ϕ+µ0 ϕ) . ∆ψ + ∆ψ = √ + ie 2πρ cos(ϕ/2) (16.78) Adding further ψ (3) from (16.59) with the asymptotic limit given by (16.71), the total wave function is seen to behave like ψ(x) → e−i(ρ cos ϕ+µ0 ϕ) + ψsc (x),

(16.79)

1 sin πµ0 −iϕ/2 . eiρ e 2πiρ cos(ϕ/2)

(16.80)

with the scattered wave ψsc = √

This corresponds precisely to the scattering amplitude (16.54) with the cross section (16.55). Let us mention that for half-integer values of µ0 , the solution of the Schrödinger equation has the simple integral representation s √ i −i(ϕ/2+ρ cos ϕ) Z ρ(1+cos ϕ) 2 e ψ(x) = dteit . (16.81) 2 0 It vanishes on the line ϕ = π, i.e., directly behind the flux tube and is manifestly single-valued.

16.3

Aharonov-Bohm Effect and Fractional Statistics

It was noted in Section 7.5 and it is worth mentioning once more in this context that the amplitude for the relative motion of two fermion orbits can be obtained from the amplitude of the Aharonov-Bohm effect. For this, we take the amplitude with µ0 = 1 and sum it over the final states with ϕb , ϕb + π, to account for particle identity. The result is 2iM X I|m+1| (κr< )K|m+1| (κr> ) h ¯ m i 1 h im(ϕb −ϕa ) × e + (−)m eim(ϕb −ϕa ) 2π 4iM −i(ϕb −ϕa ) X 1 =− e I|m| (κr< )K|m| (κr> ) eim(ϕb −ϕa ) . h ¯ 2π m=odd

(xb |xa )E + (−xb |xa )E = −

(16.82)

The sum over the two identical final states selects only the odd wave functions, as in (7.267)–(7.268). When calculating observable quantities such as particle densities or partition functions which involve only the trace of the amplitude, the phase factor e−i(ϕb −ϕa ) has no observable consequences and can be omitted. H. Kleinert, PATH INTEGRALS

1013

16.3 Aharonov-Bohm Effect and Fractional Statistics

For µ0 6= 1, the resulting amplitudes may be interpreted as describing particles in two dimensions obeying an unusual fractional statistics. This interpretation has recently come to enjoy great popularity.2 since it has led to an understanding of the experimental data of the fractional quantum Hall effect. The data can be explained by the following assumption: The excitations of a gas of electrons with Coulomb interactions in a quasi-two-dimensional material traversed by a strong magnetic field can be viewed, to lowest approximation, as a gas of quasi-particles which has no Coulomb interactions, but a new effective pair interaction. Each pair behaves as if one partner were accompanied by a thin magnetic flux tube of a certain value of µ0 . While the quasi-particles of the ground state carry an integer-valued µ0 and act statistically like ordinary electrons, the elementary excitations carry a fractional value of µ0 and display fractional statistics (more in Section 16.11). To study the fundamental thermodynamic properties of an ensemble of such particles, we calculate the partition function of a particle running around a thin flux tube along the z-axis. For finiteness, we assume the presence of an additional homogeneous magnetic field in the z-direction. Ignoring the third dimension, we take the amplitude (16.33), integrate it over all space, and find Z =

Z

d2 x(xb τb |xa τa )

∞ 1 µ0 η X emη e = 2 m=−∞

Z

0

∞

dξe−ξ cosh η I|m+µ0 | (ξ),

(16.83)

where ξ ≡ Mωr 2 /2¯ h sinh η

(16.84)

[recall that ω = −eB/Mc and η = ω(τb − τa )/2]. To calculate the partition function, the difference between the Euclidean times τb , τa is set equal to τb −τa = h ¯ /kβ T = h ¯ β, so that η = ω¯ hβ/2. To deal with two identical particles, we also need the integral in which xb is exchanged by −xb : Zex =

Z

∞ 1 µ0 η X d x(−xb τb |xa τa ) = e (−)m emη 2 m=−∞ 2

Z

0

∞

dξe−ξ cosh η I|m+µ0 | (ξ). (16.85)

In order to facilitate writing joint equations for both expansions, let us denote Z and Zex by Z1 and Z1ex , respectively. The integrals are performed with the help of formula (2.466), yielding the sums Z1,1ex =

∞ 1 X 1 (±)m eη(m+µ0 ) e−η|m+µ0 | . 2 m=−∞ sinh η

(16.86)

These sums are obviously periodic under µ0 → µ0 + 2. Because of translational R invariance, the partition function Z ≡ Z1 diverges with the total area V = d2 x as 2

See Notes and References at the end of Chapter 7.

1014


an overall factor. To enforce convergence, we multiply the volume elements d2 x with an exponential regulating factor e−ǫξ . Then the area integrals can be extended over all space. In terms of the variable ξ of (16.84), the measure in the above rotationally symmetric integrals can be written as d2 x = le2 (T )

sinh η dξ, η

(16.87)

q

h2 /kB T M introduced in Eq. (2.346). The role with the thermal length le (T ) ≡ 2π¯ R 2 of the total area V = d x is now played by the finite quantity V ≡

Z

d2 xe−ǫξ =

le2 (T ) sinh η . ǫ η

(16.88)

Inserting the factor e−ǫξ into the integrals in Eqs. (16.83) and (16.85), and defining a variable η ′ slightly different from η by cosh η ′ ≡ ǫ + cosh η,

(16.89)

which has the expansion η′

e

= cosh η + = eη

q

cosh2 η ′ − 1 ! ǫ 1 −η ǫ2 1+ + ... , − e sinh η 2 sinh3 η ′

η > 0,

(16.90)

the regulated sums (16.86) for Z1 and Z1ex look almost the same as before: Z1,1ex

∞ 1 X 1 ′ = (±)m eη(m+µ0 ) e−η |m+µ0 | . ′ 2 m=−∞ sinh η

(16.91)

Separating positive and negative values of m + µ0 , the two sums can be done for µ0 ∈ (0, 1) and for µ0 ∈ (−1, 0). In the combined interval µ0 ∈ (−1, 1), we find Z1,1ex

1 ηµ0 1 = e 2 sinh η ′

(

′

′

e−η µ0 eη µ0 ′ + − eη |µ0 | , 1∓a 1∓b )

(16.92)

where ′

a ≡ eη−η ,

′

b ≡ e−η−η .

Two identical particles have the partition function 1 1 1 Z = (Z1 + Z1ex ) = eηµ0 2 2 sinh η ′

(

′

′

e−η µ0 eη µ0 η′ |µ0 | + − e . 1 − a2 1 − b2 )

(16.93)

It is symmetric under the simultaneous exchange µ0 → −µ0 , η → −η. Outside the interval µ0 ∈ (−1, 1), it is defined by periodic extension. H. Kleinert, PATH INTEGRALS

1015

16.3 Aharonov-Bohm Effect and Fractional Statistics

In the absence of the thin flux tube we may take Z1,1ex directly from the amplitude (2.660). For xb = xa and xb = −xa , this yields with the present variables (xa τb |xa τa ) = le−2 (T )

η , sinh η

(−xa τb |xa τa ) = le−2 (T )

η e−2 cosh η ξ . (16.94) sinh η

Their regulated spatial integrals are 1 ∞ 1 dξe−ǫξ = , 2 0 2ǫ Z 1 ∞ 1 . = dξe−(ǫ+2 cosh η)ξ = 2 0 2(ǫ + 2 cosh η)

Z1,0 = Z1ex ,0

Z

(16.95)

The subtracted partition functions ∆Z1,1ex ≡ Z1,1ex − 12 Z1,0 have a finite ǫ → 0 -limit, and ∆Z = Z − 12 Z1,0 becomes for µ0 ∈ (0, 2) "

#

1 1 ∆Z = − cothη + 2(µ0 − 1) − 2e2(µ0 +1)η + 4e2ηµ0 . 8 sinh η sinh 2η

(16.96)

These results can be used to calculate the second coefficient appearing in a virial expansion of the equation of state. For a dilute gas of particles with the above magnetic interactions it reads ∞ X pV Br nr−1 . =1+ NkB T r=2

(16.97)

Here n is the number density of the particles N/V . In many-body theory it is shown that the coefficient B2 depends on the two-body partition function Z2 as follows: B2 = V

!

1 Z2 − , 2 Z12

(16.98)

where Z1 is the two-dimensional single particle partition function of mass M. In the presence of the homogeneous magnetic field, Z1 is given by Z1,0 of Eq. (16.95). Without the regulating factor, the spatial integral over the imaginary-time amplitude in (16.94) gives directly Z1 = V

η le2 (T ) sinh η

.

(16.99)

Separating the center of mass from the relative motion, we see that Z2 = 2Z1 Zrel and obtain B2 =

V l2 (T ) sinh η (Z1 /2 − 2Zrel ) = e (Z1 − 4Zrel ). Z1 2η

(16.100)

The difference on the right-hand side is convergent for V → ∞. It can be evaluated using any regulator for the area integration, in particular the exponential regulator of Eq. (16.88).

1016


Figure 16.1 Second virial coefficient B2 as function of flux µ0 for various external magnetic field strengths parametrized by η = −(eB/2M c)¯ hβ. For a better comparison, each curve has been normalized to unity at µ0 = 0.

The partition function for the relative motion of two identical particles is obtained from Z by replacing M by the reduced mass, i.e., M → M/2. This renders a factor 1/2 via le−2 (T ). Hence Z1 /2 − 2Zrel becomes equal to −∆Z and (16.100) yields [5] l2 (T ) 1 B2 = e cothη +2(µ0 − 1) −2e2(µ0 +1)η +4e2ηµ0 . 4η sinh(2η) "

#

(16.101)

The behavior of B2 as a function of µ0 is shown in Fig. 16.1. In the absence of a magnetic field, it reduces to i le2 (T ) h B2 = 1 − 2(1 − |µ0 |2 )2 , µ0 ∈ (−1, 1). (16.102) 4 As µ0 grows from zero to infinity, B2 oscillates with a period 2 between B2 = −le2 (T )/4 for even values of µ0 , and B2 = le2 (T )/4 for odd values. These are the wellknown second virial coefficients of free bosons and fermions. They can, of course, be obtained in a simpler and more direct way from the symmetric and antisymmetric combinations of (16.95), Z0 = (1/2)(Z1,0 ± Z1ex ,0 ). Subtracting Z1,0 leaves ±Z1ex ,0 /2 which reduces for η = 0 to ±1/8. Accounting for the factor 2 in the reduced mass, this yields B2 = ∓le2 (T )/4. The expression (16.102) can be interpreted as the virial coefficient of particles which are neither bosons nor fermions, but obey the laws of fractional statistics. These particles are the anyons introduced in Section 7.5. Unfortunately, there are at present no experimental data for the virial coefficients to which the theoretical expressions (16.102) [or (16.101)] could be compared. At this point we should mention older, meanwhile discarded speculations that the phenomenon of high-temperature superconductivity might be explained by fractional statistics of some elementary excitations. Indeed, the change in statistics can H. Kleinert, PATH INTEGRALS

16.4 Self-Entanglement of Polymer

1017

Figure 16.2 Lefthanded trefoil knot in polymer.

be derived from the electromagnetic interaction between the electrons in a quasi-twodimensional layer of material moving in a strong magnetic field. Also, it is possible to construct a model of anyonic two-dimensional superconductivity in which topological effects lead to a Meissner screening of magnetic fields (see Section 16.13). A closer investigation, however, shows that the currents in the model show dissipation after all. It must be emphasized that the equality between electromagnetic and statistical interaction used in the above calculations is restricted to two-particle systems and cannot be extended to arbitrarily many particles as in Section 7.5. Although it is possible to distribute the magnetic flux in a many-particle system equally between the constituents producing the desired behavior under particle exchange, an equal distribution of the charges would create an unwanted additional Coulomb potential, and the purely topological character of the interaction would be destroyed. The problem does not arise for charged particles such as electrons. Nevertheless, there is a definite need for a better and universally applicable theoretical description of anyons. This will be presented in Section 16.7.

16.4

Self-Entanglement of Polymer

An interesting consequence of the excluded-volume properties of polymers is the possibility of a self-entanglement of a closed polymer. Since its line elements are forbidden to cross each other, the fluctuations are unable to explore all possible configurations. An initially circular polymer, for example, can never turn into a trefoil knot of the form shown in Fig. 16.2 without breaking a molecular bond. In the chemical formation process of a large number of polymers, many entangled configurations arise. It is an interesting problem to find the distribution of the various independent topology classes. Until recently, the lack of theoretical methods has made analytic work almost impossible, restricting it to classification questions. Only Monte Carlo methods have yielded quantitative insights. Since 1989, however, interesting new quantum field-theoretic methods have been developed promising significant progress in the near future. Here we survey these methods and indicate how to derive analytic results. First, we introduce the relevant topological concepts. A closed polymer will in general form a knot. A circular polymer represents a trivial knot. Two knots are called equivalent if they can be deformed into each

1018


Figure 16.3 Nonprime (compound) knot. The dashed line separates two pieces. After closing the open ends, the pieces form two prime knots.

Figure 16.4 Illustration of multiplication law a1 a2 ≈ a3 in knot group. The loops a1 and a2 are equivalent, a1 ≈ a2 , while a4 ≈ a−1 1 .

other without breaking any line. Such deformations are called isotopic. A first step towards the classification consists in separating the equivalence classes into irreducible and reducible ones, defining prime or simple knots and nonprime or compound knots, respectively. A compound knot is characterized by the existence of a plane which is intersected twice (after some isotopic deformation) (see Fig. 16.3). By closing the open ends on each side of the plane one obtains two new knots. These may or may not be reduced further in the same way until one arrives at simple knots. One important step towards distinguishing different equivalence classes of simple and compound knots is the knot group defined as follows. In the multiply connected space created by a certain knot, choose an arbitrary point P (see Fig. 16.4). Then consider all possible closed loops starting from P and ending again at P . Two such loops are said to be equivalent if they can be deformed into each other without crossing the lines of the knot under consideration. The loops are, however, allowed to have arbitrary self-intersections. The classes of equivalent knots form the knot group. Group multiplication is defined by running through any two loops of two equivalence classes successively. The class whose loops can be contracted into the H. Kleinert, PATH INTEGRALS


1019

Figure 16.5 Inequivalent (compound) knots possessing isomorphic knot groups. The upper is the granny knot, the lower the square knot. They are stereoisomers characterized by the same Alexander polynomial (t2 − t + 1)2 but different HOMFLY polynomials [see (16.126)].

point P is defined as the unit element e. Changing the orientation of the loops in a class corresponds to inverting the associated group element. In this way, the classification of knots can be related to the classification of all possible knot groups. Consider the trivial knot, the circle. Obviously, the closed loops through P fall into classes labeled by an integer number n. The associated knot group is the group of integers. Conversely, no nontrivial knot is associated with this group. Although this trivial example might at first suggest a one-to-one correspondence between the simple knots and their knot groups, there is none. Many examples are known where inequivalent knots have isomorphic knot groups. In particular, all mirror-reflected knots which usually are inequivalent to the original ones (such as the right- and left-handed trefoil), have the same knot group. Thus, the knot groups necessarily yield an incomplete classification of knots. An example is shown in Fig. 16.5. Fortunately, the degeneracies are quite rare. Only a small fraction of inequivalent knots cannot be distinguished by their knot groups. The easiest way of picturing a knot in 3 dimensions is by drawing its projection onto the paper plane. The lines in the projection show a number of crossings, and the drawing must distinguish the top from the bottom line. The knot is then deformed isotopically until the projection has the minimal number of crossings. In the projection, all isotopic deformations can be decomposed into a succession of three elementary types, the so-called Reidemeister moves shown in Fig. 16.6. A picture of all simple knots up to n = 8 is shown in Fig. 16.7. The numbers of inequivalent simple and compound knots with a given number n of minimal crossings are listed in Table 16.1. The projected pictures can be used to construct an important algebraic quantity characterizing the knot group, called the Alexander polynomial discovered in 1928. It reduces the classification of knot groups to that of polynomials. This type of work is typical of the field of algebraic topology.

1020


Figure 16.6 Reidemeister moves in projection image of knot which do not change its class of isotopy. They define the movements of ambient isotopy. For ribbons, only the second and third movements are allowed, defining the regular isotopy. The first movement is forbidden since it changes its writhe [for the definition see Eq. (16.110)]. Table 16.1 Numbers of simple and compound knots with n minimal crossings in a projected plane.

n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

simple knots 1 0 0 1 1 2 3 7 21 49 166 548 – – –

compound knots 0 0 0 0 0 0 1 1 3 5 10 37 154 484 1115

We explain the construction for the trefoil knot. Attaching a directional arrow to the polymer and selecting an arbitrary starting point, we follow the arrow until we run into a first underpass. This point is denoted by 1. Now we continue to the next underpass denoted by 2, etc., up to n (see Fig. 16.8). The polymer sections between two successive underpasses i and i + 1 are named xi+1 . At each underpass from xi to xi+1 , we record (see Table 16.2) whether the overpassing section xk runs from right to left (type r) or from left to right (type l). We now set up a matrix Aij . Each underpass with label i defines a row Aij , j = 1, 2, 3, . . . according to the H. Kleinert, PATH INTEGRALS


1021

Figure 16.7 Simple knots with up to 8 minimal crossings. The number of crossings under each picture carries a subscript enumerating the equivalence classes.

1022


Figure 16.8 Labeling of underpasses for construction of Alexander polynomial t2 − t + 1 of the left-handed trefoil. Table 16.2 Tables of underpasses (under) and directions (dir) r or l ofoverpassing lines (over), for trefoil knot 31 and knot 41 of Fig. 16.7.

31 :

under x1 x2 x3

over x3 x1 x2

dir r r r

41 :

under x1 x2 x3 x4

over x4 x1 x2 x3

dir r l r l

following rules: Let xk be the overpassing section. If xk coincides with xi or xi+1 the underpass is called trivial . In this, case the ith row of the matrix Aij has the elements Aii = −1,

Ai,i+1 = 1.

(16.103)

All other row elements Aij vanish. If the underpass is nontrivial, the nonvanishing row elements are Aii = 1, Aii+1 = −t, Aik = t − 1, Aii = −t, Aii+1 = 1, Aik = t − 1,

type r (right to left), type l (left to right).

(16.104) (16.105)

In this way, we find the matrix of the trefoil knot: 



1 −t t − 1  1 −t  Aij =  t − 1 . −t t − 1 1

(16.106)

As another example, the knot 41 in Fig. 16.7 has the matrix    

Aij = 

1 −t 0 t−1 t − 1 −t 1 0 0 t−1 1 −t 1 0 t − 1 −t



  . 

(16.107)


1023


Table 16.3 Alexander, Jones, and HOMFLY polynomials for smallest simple knots up to 8 crossings. The numbers specify the coefficients; for instance, the knot 71 has the Alexander polynomial A(t) = 1 − t + t2 − t3 + t4 − t5 + t6 and the knot 88 has the Jones polynomial J(t) = t−3 (1 − t + 2t2 − t3 + t4 ). For the HOMFLY polynomial H(t, α), see the explanation on p. 1029. A(t)

J(t)

H(t,α)

31

1−11

(0)1

([0]2−1)([0]1)

41

1−31

(−2)−1

(1[−1]1)([−1])

51

1−11−11

(0)1101

([0]03−2)([0]041)([0]01)

52

2−32

(0)101

([0]11−1)([0]11)

61

2−52

(−2)−10−1

(1[0]−11)([−1]−1)

62

1−33−31

(−1)−11−1

([2]−21)(1[−3]1)([0]1)

63

1−35−31

(−3)1−11

(−1[3]−1)(−1[3]−1)(1)

71

1−11−11−11

(0)1111101

([0]004−3)([0]0010−4)([0]006−1)([0]001)

72

3−53

(0)10101

([0]10−11)([0]111)

73

2−33−32

(0)110201

(−22−10[0])(−1330[0])(1100[0])

74

4−74

(0)10201

(−1020[0])(121[0])

75

2−45−42

(0)1102−11

([0]020−1)([0]032−1)([0]011)

76

1−57−51

(−1)−12−11

([1]−12−1)([1]−22)([0]−1)

77

1−59−52

(−3)1−21−1

(1−2[2])(−2[2]−1)([1])

81

3−73

(−2)−10−10−1

(1−10[0]1)(−1−1[−1])

82

1−33−33−31

(1)1−11−1

(1−33[0])(3−74[0])(1−51[0])(−10[0])

83

4−94

(−4)−10−20−1

(10[−1]01)(−1[−2]−1)

84

2−55−52

(−3)−10−21−1

(2[−2]01)(1[−3]−21)([−1]−1)

85

1−34−55−31

(1)1−21−1

(2−54[0])(3−84[0])(1−51[0])(−10[0])

86

2−67−62

(−1)−11−21−1

(1−1−1[2])(1−2−2[1])(−1−1[0])

87

1−35−55−31

(−2)1−12−11

([−1]4−2)([−3]8−3)([−1]5−1)([0]1)

88

2−69−62

(−3)1−12−11

(−1[2]1−1)(−1[2]2−1)([1]1)

s

s

89

1−35−75−31

(−4)−11−21−1

(2[−3]2)(3[−8]3)(1[−5]1)([−1])

810

1−36−76−31

(−2)1−13−11

([−2]6−3)([−3]9−3)([−1]5−1)([0]1)

s

811

2−79−72

(−1)−12−21−1

(1−21[1])(1−2−1[1])(−1−1[0])

812

1−7(13)−71

(−4)−11−31−1

(1−1[1]−11)(−2[1]−2)([1])

813

2−7(11)−72

(−3)1−22−11

([0]2−1)(−1[1]2−1)([1]1)

814

2−8(11)−82

(−1)−12−22−1

([1])(1−1−1[1])(1−1[0])

815

3−8(11)−83

(0)1103−22−1

(1−4310[0])(−3520[0])(210[0])

816

1−48−98−41

(−2)1−23−21

([0]2−1)([−2]5−2)([−1]4−1)([0]1)

817

1−48−(11)8−41

(−4)−12−32−1

(1[−1]1)(2[−5]2)(1[−4]1)([−1])

s

818

1−5(10)−(13)(10)−51

(−4)−13−33−1

(−1[3]−1)(1[−1]1)(1[−3]1)([−1])

s

819

1−1010−11

(0)11111

(1−5500[0])(−5(10)00[0])(−1600[0])(100[0])

820

1−23−21

(−1)101

([−1]4−2)([−1]4−1)([0]1)

821

1−45−41

(0)1−11−1

(1−33[0])(1−32[0])(−10[0])

s

1024


Figure 16.9 Exceptional knots found by Kinoshita and Terasaka (a), Conway (b), and Seifert (c), all with same Alexander polynomial as trivial knot [|A(t)| ≡ 1].

Having set up the n × n-matrix Aij , we choose an arbitrary subdeterminant (minor) of order n − 1. It is a polynomial in t with integer coefficients. This polynomial is divided by a suitable power of t to make it start out with a constant. The result is the Alexander polynomial A(t). It is independent of the choice of the subdeterminant. For the left-handed trefoil, the matrix (16.106) yields A(t) = t2 − t + 1.

(16.108)

For the knot 41 , we find from (16.107) A(t) = t2 − 3t + 1.

(16.109)

The Alexander polynomials of the simple knots in Fig. 16.7 are shown in Table 16.3. Note that the replacement t → 1/t leaves the Alexander polynomial invariant (after renormalizing it back by some power of t to start out with a constant). The Alexander polynomial of a composite knot factorizes into those of the simple knots it is composed of. If two knots are mirror images of each other, they have the same knot group and the same Alexander polynomials. Due to the factorization property, two composite knots whose simple parts differ by mirror reflection (stereoisomers; see Fig. 16.5) have the same polynomial. Thus the Alexander polynomial cannot render a complete classification of inequivalent knots. This is true even after removing degeneracies of the above type. In Fig. 16.9, we give the simplest examples of knots with an Alexander polynomial A(t) ≡ ±1 of the trivial knot. Up to 11 crossings, these are the only examples. Since the total number of simple knots up to n = 11 is 795, the exceptions are indeed very few. Recent years have witnessed the development of simpler construction procedures and more efficient polynomials for the classification of knots and links of several H. Kleinert, PATH INTEGRALS

1025


knots, the Jones and the HOMFLY polynomials 3 and their generalizations. The former depend on one, the latter on two variables, one of which occurs also with inverse powers, i.e., in this variable the polynomials are of the Laurent type. Other polynomials found in the literature, such as Conway, X-, or Kauffman’s bracket polynomials, are special cases of the HOMFLY polynomials. In addition, there exist a different type of Kauffman polynomials and of BLMHo polynomials F (a, z) and Q(x), respectively, which are capable of distinguishing some knots with accidentally degenerate HOMFLY polynomials. They will not be discussed here. For their definition see Appendix 16B. The X-polynomial X(a) is trivially related to the Jones polynomial J(t), to which it reduces after a variable change a = t1/4 . The X-polynomial is closely related to the Kauffman polynomial K(a) by X(a) = (−a)−3w K(a).

(16.110)

The number w is the cotorsion, also called twist number , Tait number , or writhe [6]. It is defined by giving the loop or link an orientation and attributing to each crossing a number 1 or −1 according to the following rule. At each crossing follows the overpass along the direction of orientation. If the underpass runs from right to left, the crossing carries the number 1, otherwise −1. The sum of these numbers is the cotorsion w. In the trefoil knot in Fig. 16.2 each crossing carries a −1 so that w = −3. The Kauffman polynomial is found by a very simple construction procedure. A set of n trivial loops is defined to have the Laurent polynomial Kn (a) = −(a2 + a−2 )n−1 .

(16.111)

Every knot or link can be reduced to such loops by changing the crossings recursively into two new configurations according to the graphical rule shown in Fig. 16.10.

= a

L+

+ a−1

L0

L∞

Figure 16.10 Graphical rule for removing crossing in generating Kauffman polynomial.

The first configuration is associated with a factor a, the second with a factor a−1 . The configuration receiving the factor a is most easily identified by approaching the crossing on the underpassing curve and taking a right turn. The two new 3

The word “HOMFLY” collects the initials of the authors (Hoste, Ocneanu, Millet, Freyd, Lickorish, Yetter). The papers are quoted in Notes and References.

1026


Figure 16.11 Kauffman decomposition of trefoil knot. The configuration 3 is the Hopf link. The calculation of the associated polynomials is shown in Table 16.4. Table 16.4 Kauffman polynomials in decomposition of trefoil knot.

link 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

bracket polynomial −a2 − a−2 1 1 −a2 − a−2 1 −a2 − a−2 −a2 − a−2 −a4 − 2 − a−4 −a−3 −a3 −a3 a5 + a −a4 − a−4 a6 a7 − a3 − a−5

rule Eq. 16.111 Eq. 16.111 Eq. 16.111 Eq. 16.111 Eq. 16.111 Eq. 16.111 Eq. 16.111 Eq. 16.111 ah14i + a−1 h15i, Fig. ah12i + a−1 h13i, Fig. ah10i + a−1 h11i, Fig. ah8i + a−1 h9i, Fig. ah6i + a−1 h7i, Fig. ah4i + a−1 h5i, Fig. ah2i + a−1 h3i, Fig.

16.10 16.10 16.10 16.10 16.10 16.10 16.10

configurations are processed further in the same way and so on until one arrives only at trivial loops. By applying these rules to a trefoil knot, we obtain a knot and a link known as the Hopf link . These are decomposed further as shown in Fig. 16.11. The Kauffman polynomials of each part are listed in Table 16.4. The polynomial of the trefoil knot is K(a) = a7 − a3 − a−5 .


1027


Since w = −3, we obtain with the factor (−a)−3w = −a9 the X-polynomial X(a) = −a16 + a12 + a4 .

(16.113)

This corresponds to a Jones polynomial J(t) = t + t3 − t4 . For the Jones polynomials, there exists a simple direct construction. According to J.H. Conway, any knot can be related to lower knots or links by performing the skein operations shown in Fig. 16.12 on any crossing in the projection plane. Either

Figure 16.12 Skein operations relating higher knots to lower ones.

a crossing L+ is transformed into L− and L+ , or L+ is transformed into L− and L0 . For knots related in this way one defines the Jones polynomial J(t) recursively by the skein relation √ 1 1 JL+ (t) − tJL− (t) = t − √ JL0 (t). t t !

(16.114)

The circular loop is defined to have the trivial polynomial J(t) ≡ 1. By applying the skein operations to two disjoint unknotted loops in Fig. 16.13, one finds the Jones polynomial √ √ J2 (t) = −( t + 1/ t). (16.115) Upon carrying this procedure to n such loops, we find √ √ Jn (t) = [−( t + 1/ t)]n−1 ,

(16.116)

in agreement with (16.111). For the lowest knots, the Jones polynomials are listed in Table 16.3. Up to nine crossings, the Jones polynomials distinguish mirrorsymmetric knots. Conway discovered the first skein relation in 1970 when trying to develop a computer program for calculating Alexander polynomials. He found the Alexander polynomials to obey modulo the normalization convention, the skein relation √ √ AL+ (t) − AL− (t) = ( t − 1/ t)AL0 (t), (16.117) which eventually reduces the polynomials of all knots to the trivial one A1 (t) = 1. The skein relation simplifies the procedure so much that Conway was able to work out by hand all polynomials known at that time. Because of the simplicity of

1028


Figure 16.13 Skein operations for calculating Jones polynomial of two disjoint unknotted loops.

Figure 16.14 Skein operation for calculating Jones polynomial of trefoil knot.

Figure 16.15 Skein operation for calculating Jones polynomial of Hopf link.

this procedure, the Alexander polynomials are now often referred to as AlexanderConway polynomials. Let us now calculate the Jones polynomial for the trefoil knot 31 of Fig. 16.7. First we apply the skein operation shown in Fig. 16.14. The loop L− is unknotted and has a unit polynomial. Thus we obtain the polynomial relation √ √ Jtrefoil (t)JL+ (t) = t2 · 1 + t( t − 1/ t)JL0 (t). (16.118) The configuration L0 is known as a Hopf link . It needs one more reduction4 via the operation shown in Fig. 16.15, resulting in the relation √ √ 1 JL 0 (t) = tJ2 (t) + ( t − 1/ t)J1 (t). t +

(16.119)

4

The Kauffman bracket polynomial of the Hopf link is KL0 (a) = −a4 − a−4 . Together with the cotorsion w√= −2, this amounts to an X-polynomial X(a) = −a10 − a2 and the Jones polynomial JL0 (t) = − t(1 + t2 ) as in (16.120). H. Kleinert, PATH INTEGRALS

1029


Using (16.111), we find √ JL0 (t) = − t(1 + t2 ).

(16.120)

Inserting this into (16.118) leads to the Jones polynomial of the trefoil knot Jtrefoil = t + t3 − t4 .

(16.121)

It differs from the result found above for the left-handed trefoil by the substitution t → t−1 . The HOMFLY polynomials HL (t, α) are obtained from a slight √ generalizations √ of the skein relation (16.114) of the Jones polynomials: The factor ( t − 1/ t) on the right-hand side is replaced by an arbitrary parameter α, leading to the skein relation 1 HL (t, α) − tHL− (t, α) = αHL0 (t, α). t +

(16.122)

The trivial knot is defined to have the trivial polynomial H1 (t, α) = 1. For two independent loops, the relation yields H2 (t, α) = (t−1 − t)α−1 .

(16.123)

The HOMFLY polynomials H(t, α) transform under a mirror reflection of the knot into H(−t−1 , α). Note that H2 (t, α) is mirror-symmetric [H1 (t, α) is trivially so].5 In general, the HOMFLY polynomials give reliable information on a possible mirror symmetry. There are, however, a few exceptions, i.e., mirror-related pairs of knots possessing the same HOMFLY polynomial.6 Examples for HOMFLY polynomials are Htrefoil(rh) (t, α) Htrefoil(lh) (t, α) HHopf(rh) (t, α) Hknot 41 (t, α)

= = = =

−t4 + 2t2 + t2 α2 , −t−4 + 2t−2 + t−2 α2 , (t − t3 )α−1 + tα, t−2 − 1 + t2 − α2 .

(16.124)

Setting α = t1/2 −t−1/2 produces the Jones polynomials, while t → 1, α → t1/2 −t−1/2 leads, with appropriate powers of t as normalization factors, back to the AlexanderConway polynomials. In Table 16.3, the HOMFLY polynomials are listed for knots up to 8 crossings. For mirror-unsymmetric knots, only one partner is recorded. The reflected polynomial is obtained by the substitution t → −t−1 . The meaning of the entries is best explained with an example: The knot 71 has an entry ([0]004 − 3)([0]00(10) − 4)([0]006 − 1)([0]001), which stands for the polynomial 5

For the Kauffman polynomials F (a, x) defined in Appendix 16B, mirror reflection implies F (a, x) → F (a−1 , x). 6 The first degeneracy of this type occurs for a link of 3 loops with 8 crossings.

1030


H(t, α) = (4t6 − 3t8 ) + (10t6 − 4t8 )α2 + (6t6 − t8 )α4 + t6 α6 . A bracket marks the position and coefficient of the zeroth power in t2 ; the numbers to the right and the left of it specify the coefficients of the adjacent higher and lower powers of t2 , respectively. Numbers with more than one digit are put in parentheses. The polynomial of the reflected knot is obtained by reflecting the numbers in parentheses on the associated bracket. The knots marked by an s are mirror-symmetric. The Alexander-Conway polynomials are special cases of the HOMFLY polynomials. A comparison with the skein relation (16.117) shows that they are obtained from them by setting t = 1 and replacing α by t1/2 − t−1/2 : AL (t) = HL (1, t1/2 − t−1/2 ).

(16.125)

The reducible granny and square knots in Fig. 16.5 are distinguished by the Jones and the HOMFLY polynomials; the latter are Hgranny (t, α) = (2t2 − t4 + t2 α2 )2 , Hsquare (t, α) = (2t2 − t4 + t2 α2 )(2t−2 − t−4 + t2 α2 );

(16.126)

the former are obtained by inserting α = t1/2 − t−1/2 . Up to now, there exists no complete algebraic classification scheme. For example, the Jones polynomials of the knots with 10 and 13 crossings shown in Fig. 16.16 are

Figure 16.16 Knots with 10 and 13 crossings, not distinguished by Jones polynomials.

the same.7 For further details, see the mathematical literature quoted at the end of the chapter. Even with the incomplete classification of knots, it has until now been impossible to calculate the probability distribution of the various equivalence classes of knots. Modern computers allow us to enumerate the different topological configurations for not too long polymers and to simulate their distributions by Monte Carlo methods. In Fig. 16.17 we show the result of a simulation by Michels and Wiegel, where they measure the fraction fN of unknotted polymers of N links. They fit their curve by a power law fN = CµN N α ,

(16.127)

7

The HOMFLY polynomials have their first degeneracy for prime knots with 9 crossings. It was checked that up to 13 crossings (amounting to 12 965 knots) no polynomial of a nontrivial knot is accidentally degenerate with the trivial polynomial of a circle. H. Kleinert, PATH INTEGRALS

1031

16.5 The Gauss Invariant of Two Curves

Figure 16.17 L = N a.

Fraction fN of unknotted closed polymers in ensemble of fixed length

with the parameters C ≈ 1.026, µ ≈ 0.99640, α ≈ 0.0088. Thus fN falls off exponentially in N like µN with µ < 1. The exponent α is extremely small. A more recent simulation by S. Windwer takes account of the fact that the line elements are self-avoiding. It yields the parameters8 C ≈ 1.2325, µ ≈ 0.9949, α ≈ 0.

(16.128)

For a polymer enclosed in a sphere of radius R, the distribution has the finite-size dependence fN (R) = e−A(N

β l/R)γ

,

(16.129)

where l is the length of a link. The “critical exponents” are β ≈ 0.76 and γ ≈ 3.

16.5

The Gauss Invariant of Two Curves

For any analytic calculation of topological properties one needs a functional of the polymer shape which is capable of distinguishing the different knot classes. Initially, 8

A first theoretical determination of these parameters has recently been given by mapping the problem onto a four-state Potts model. A presentation of this method which does not involve path integrals would go beyond the scope of this book. See the papers by A. Kholodenko quoted at the end of the chapter.

1032


a hopeful candidate was the one-loop version of the contour integral introduced almost two centuries ago by Gauss for a pair of closed curves C and C ′ : G(C, C ′) =

1 4π

I I

C′

C

[dx × dx′ ] ·

x − x′ . |x − x′ |3

(16.130)

Gauss proved this to be a topological invariant. In fact, we may rewrite (16.134) with the help of the δ-function (10A.8) as I

C

dx

I

C′

dx′ × (x − x′ ) =− |x − x′ |3

Z

3

d x (x; C) ·

"Z

R′ d x (x ; C ) × ′3 R 3 ′

′

′

#

(16.131)

The second integral is recognized as the gradient of the multivalued field Ω(x; C ′ ) defined by Eqs. (10A.18), which is the solid angle under which the contour C ′ is seen from the point x, so that G(C, C ′) = −

1 4π

Z

d3 x (x; C) · ∇Ω(x; C ′ ).

(16.132)

Inserting here Eq. (10A.27), where S ′ is any surface enclosed by the contour L′ , and using the fact that Z

d3 x (x; C) · ∇Ω(x; S ′ ) = −

Z

d3 x ∇ · (x; C)Ω(x; S ′ ) = 0,

(16.133)

due to (10A.9) and the fact that Ω(x; S) = 0 is single-valued, we obtain ′

G(C, C ) = −

Z

d3 x (x; C) · (x; S ′ ).

(16.134)

This is a purely topological integral. By rewriting it as ′

G(C, C ) = −

I

C

dxi δi (x; S ′ ),

(16.135)

we see that G(C, C ′) gives the linking number of C and C ′ . It is defined as the number of times by which one of the curves, say C ′ , perforates the surface S spanned by the other. Alternative expressions for the Gauss integral (16.134) are 1 I 1 I ′ ′ G(C, C ) = − dΩ (x ; C) = − dΩ(x, C ′ ), 4π C ′ 4π C ′

(16.136)

where where Ω′ (x′ ; S) is the solid angle under which the curve C is seen from the point x′ . The values of the Gauss integral for various pairs of linked curves up to 8 crossings are given in the third column of Table 16.5. All the intertwined pairs of curves labeled by 21 , 71 , 72 , 87 , for instance, have a Gauss integral G(C, C ′) = ±1. H. Kleinert, PATH INTEGRALS

1033

16.6 Bound States of Polymers and Ribbons

Let us end this section by another interpretation of the Gauss integral. According to Section 10A.1, the solid angle Ω is equal to the magnetic potential of a current 4π running through the curve C ′ . Its gradient is the magnetic field Bi = ∂i Ω.

(16.137)

Hence we can write G(C, C ′) = −

I

C

dxi Bi = −

I

C′

dx′i Bi′ .

(16.138)

According to this expression, G(C, C ′) gives the total work required to move a unit magnetic charge around the closed orbit C in the presence of the magnetic field due to a unit electric current along C ′ . Unfortunately, there exists no such topologically invariant integral for a single closed polymer. If we identify the curves C and C ′ , the Gauss integral ceases to be a topological invariant. It can, however, be used to classify self-entangled ribbons. These possess two separate edges identified with C and C ′ . Such ribbons play an important role in biophysics. The molecules of DNA, the carriers of genetic information on the structure of living organisms, can be considered as ribbons. They consist of two chains of molecules connected by weak hydrogen bridges. These can break up thermally or by means of enzymes decomposing the ribbon into two single chains.

16.6

Bound States of Polymers and Ribbons

Two or more polymers may line up parallel to each other and form a bound state. The most famous example is the molecule of DNA. It is a bound state of two long chains of molecules which may contain a few thousand up to several billion links. The distance d between the two chains is about 20rA. In equilibrium, the two chains are twisted up in the form of a double helix, rising by about 20rA (i.e., about 10 monomers) per turn (see Fig. 16.18). The DNA molecule may be idealized as an infinitesimally thin ribbon. The ribbon is always two-sided since the edges of the ribbon are made up of different phosphate groups whose chemical structure makes the binding unique. One-sided structures formed by a Möbius strip are excluded. Circular DNA molecules have interesting topological properties. In the double helix, one edge passes through the other an integer number of times. This is the linking number Lk of the double helix. Being a topological invariant, it does not change if the two closed edges become unbound and distorted into an arbitrary shape. If the total number of windings Nw in the DNA helix is different from the linking number Lk , a circular helix is always under mechanical stress. It can relax by forming a supercoil (see Fig. 16.19). The number of excess turns τ = Lk − Nw ,

(16.139)

1034


provides a measure for the supercoiling density which is defined by the ratio σ≡

τ . Nw

(16.140)

In natural DNA, the supercoiling density is usually −σ ∼ 0.03 − −0.1. The negative sign implies that the natural twist of the double helix is slightly decreased by the supercoiling. The negative sign seems to be essential in the main biological process, the replication. It may be varied by an enzyme, called DNA gyrase. A cell has a large arsenal of enzymes which can break one of the chains in the helix and unwind the linking number Lk by one or more units, changing the topology. Such enzymes run under the name of topoisomerases of type I. There is also one of a type II which breaks both chains and can tie or untie knots in the double helix of DNA as a whole. The biophysical importance of the supercoil derives from the fact that the stress carried by such a configuration can be relaxed by breaking a number of bonds between the two chains. In fact, a number θ = −σ of broken bonds leads to a complete relaxation. During a cell division, all bonds are broken. Note that this process would be energetically unfavorable if the supercoiling density were positive.

Figure 16.18 Small section and idealized view of circular DNA molecule. The link number Lk (defined as number of times one chain passes through arbitrary surface spanned by the other) is Lk ≈ 9.

Figure 16.19 A supercoiled DNA molecule. This is the natural shape when carefully extracted from a cell. The supercoiling is negative. H. Kleinert, PATH INTEGRALS


Figure 16.20 Simple links of two polymers up to 8 crossings.

1035

1036


Just as for knots, no complete topological invariants are known for such types of links of two (or more) closed polymers. Historically, generalized Alexander polynomials were used to achieve an approximate classification of links. They are polynomials of two variables. To construct them, we take one of the two polymers and label all underpasses in the same way as for knots. The same thing is done for the other polymer. For each underpass, a row of the Alexander matrix Aij is written down with two variables s and t. The Alexander polynomial A(s, t) is again given by any (n − 1) × (n − 1)-subdeterminant of the n × n-matrix Aij . The links up to 8 crossings are shown in Fig. 16.20. The Alexander polynomials associated with these are listed in Table 16.5. Note that the replacements s → 1/s, t → 1/t, or both, leave the Alexander polynomial invariant (due to the prescription of renormalizing the lowest coefficients to an integer). For unlinked polymers one has A(s, t) = 0. There is no need to go through the details of the procedure since the more recent and powerful Jones and HOMFLY polynomials can be constructed for arbitrary links without additional prescriptions. The latter are tabulated in the fourth column of Table 16.5. In many cases, a change in the orientation of the second loop gives rise to an inequivalent link. Then the table shows two entries underneath each other. For the knot 72 , the upper entry {−1}(1 ∗ −1)(1 ∗ 0 − 11)(∗ − 1 − 1) indicates the polynomial H(t, α) = α−1 (t−1 − t) + α(t−1 − t2 + t3 ) + α3 (−t − t3 ). The curly bracket shows the lowest power of α and the star marks the position of the zeroth power of t. The coefficients of t, t3 , . . . stand to the right of it, those of . . . , t−3 , t−1 to the left of it. For the special case of a circular ribbon such as a DNA molecule, the Gauss integral over the two edges, being a topological invariant, renders also a classification of the ribbon as a whole. As shown in Eq. (16.135), the Gauss integral yields precisely the linking number Lk . It is useful to calculate G(C, C ′) for a ribbon in the limit of a very small edgeto-edge distance d. This will also clarify why the Gauss integral G(C, C ′ ) in which both C and C ′ run over the same single loop is not a topological invariant. We start with the Gauss integral for the two edges C, C ′ of the ribbon 1 G(C, C ) = 4π ′

I I C

C′

[dx × dx′ ] ·

x − x′ |x − x′ |3

(16.141)

and shift the two neighboring integration contours C, C ′ both towards the ribbon ¯ Let ǫ measure the distance between the two edges, and let n(τ ) be axis called C. the unit vector orthogonal to the axis pointing from C to C ′ . Then we write x(τ ) − x(τ ′ ) − ǫn(τ ′ ) dτ . ¯ ¯ |x(τ ) − x(τ ′ ) − ǫn(τ ′ )|3 C C (16.142) ¯ C) ¯ [which then In the limit ǫ → 0, G(C, C ′ ) does not just become equal to G(C, ′ ′ would be the same as G(C, C) or G(C , C )]. A careful limiting procedure performed below shows that there is a remainder Tw , ¯ C) ¯ + Tw . Lk = G(C, (16.143) 1 G(C, C ) = 4π ′

I

I

′

˙ ) × (x(τ ˙ ′ ) + ǫn(τ ˙ ′ ))] · dτ [x(τ


1037


Table 16.5 Alexander polynomials A(s, t) and the coefficients of HOMFLY polynomials H(t, α) for simple links of two closed curves up to 8 minimal crossings, labeled as in Fig. 16.20. The value |A(1, 1)| is equal to the absolute value of the Gauss integral |G(C, C ′ )| for the two curves. The entries in the last column are explained in the text.

A(s,t) 21

1

|A|

H(t,α)

1 {−1}(∗1−1)(∗1) {−1}(∗01−1)(∗11) {−1}(∗01−1)(∗03−1)(∗01)

41 s+t

2

51 (s−1)(t−1)

0 {−1}(1∗−1)(−12∗−1)(1∗)

61

s2 +t2 +st

3

{−1}(∗001−1)(∗006−3)(∗005−1)(∗001) {−1}(∗001−1)(∗111)

62 st(s+t)−st+s+t

3 {−1}(∗001−1)(∗022−1)(∗011)

63 2st−(s+t)+2

2

71 s2 t2 −st(s+t)+st−(s+t)+1

1

72

st(s+t)−t2 −s2 −3st+s+t

1

{−1}(∗01−1)(∗021−1)(∗011) {−1}(1−10∗)(−21∗−1)(1∗) {−1}(1−1∗)(−24−3∗)(10∗) {−1}(∗1−1)(−11∗2−1)(1∗1) {−1}(∗1−1)(1∗0−11)(∗−1−1) {−1}(∗1−1)(−2∗5−2)(−1∗4−1)(∗1)

73 2(s−t)(t−1)

0 {−1}(1∗−1)(1∗−1−1)(∗−1−1)

74 (s−1)(t−1)(s2 +1)

0 {−1}(−13−2∗)(−25−3∗)(−14−1∗)(10∗)

75

2s3 t−t2 −s+2

2

{−1}(∗002−3)(∗014−3)(∗012) {−1}(−1∗3−2)(−2∗6−2)(−1∗4−1)(∗1)

76 (s+1)2 (s−1)(t−1)

0 {−1}(1∗−1)(1−2∗1)(1−3∗1)(−1∗)

77 s3 +t

2

78 (s−t)(t−1)

0 {−1}(−13−2∗)(−13−2∗)(10∗)

81

(s+t)(s2 +t2 )

82 st(s+t−1)(st+1)+s+t

4 4

83

2s2 t2 −st(s+t)+3st−(s+t)+2

3

84

s2 t2 (s+t)−2s2 t2 +2st(s+t)−2st+s+t

4

85 s2 t2 −2st(s+t)+3st−2(s+t)+1

3

86 2st−3(s+t)+2

2

87

s2 t2 −2st(s+t)+s2 +3st+t2 −2(s+t)+1

1

88

s2 t2 −2st(s+t)+s2 +st+t2 −2(s+t)+1

3

89 s3 +2s2 t−4s2 −4st+s+2t

2

{−1}(−1∗3−2)(−1∗4−1)(∗1) {−1}(∗002−31)(∗006−4)(∗005−1)(∗001)

{−1}(∗0001−1)(∗00010−6)(∗00015−5)(∗0007−1)(∗0001) {−1}(∗0001−1)(∗1111) {−1}(∗0001−1)(∗0034−3)(∗0044−1)(∗0011) {−1}(∗0001−1)(∗0212−1)(∗0111) {−1}(∗001−1)((∗0041−2)(∗0043−1)(∗0011) {−1}(1−100∗)(−200∗−1)(11∗) {−1}(∗0001−1)(∗0131−1)(∗0121) {−1}(∗0001−1)(∗0042−2)(∗0043−1)(∗0011) {−1}(∗001−1)(∗0130−1)(∗0121) {−1}(1−100∗)(1−42−2∗)(−23−1∗)(10∗) {−1}(∗01−1)(∗0201−1)(∗0111) {−1}(1−10∗)(−20∗1−1)(1∗1) {−1}(1−1∗)(1−4∗3−1)(−2∗3−1)(∗1) {−1}(∗1−1)(∗2−33−1)(∗1−32)(∗0−1) {−1}(∗1−1)(−1∗4−31)(−1∗3−2)(∗1) {−1}(∗1−22−1)(1∗1−33)(∗−1−2) {−1}(∗1−22−1)(1∗1−33)(∗−1−2) {−1}(∗1−22−1)(∗2−34−1)(∗1−32)(∗0−1)

810

(s2 −1)(t−1)

0 {−1}(1−2∗2−1)(1−4∗4−1)(1−3∗2)(−1∗)

811

s3 t−2s2 (s+t)+2s(s+t)−2(s+t)+1

2

812 (s2 −1)(t−1)

{−1}(∗002−31)(∗005−2−1)(∗0042−1)(∗0011) {−1}(2−3∗1)(1−5∗3−1)(−2∗3−1)(∗1)

0 {−1}(−13−2∗)(−14−4∗1)(2−3∗1)(−1∗)

813

(s2 +1)(s−1)(t−1)

0 {−1}(1∗−1)(∗1−21)(−1∗2−2)(∗1)

814

s3 t−4s2 t+4s2 +4st−4s+1

2

815 (s−1)(t−1) 816

s3 −2s(s+1)+1

{−1}(∗002−31)(∗005−2−1)(∗0131) {−1}(−1∗3−2)(1−3∗5−1)(−2∗3−1)(∗1)

0 {−1}(1−2∗2−1)(−2∗3−1)(∗1) 2

{−1}(∗1−22−1)(∗2−22)(∗0−1) {−1}(∗1−22−1)(∗3−43)(∗1−41)(∗0−1)

1038


The remainder is called the twist of the ribbon, defined by Tw ≡

1 I ˙ ) · [n(τ ) × n(τ ˙ )]/|x(τ ˙ )|. dτ x(τ 2π C¯

(16.144)

Incidentally, this integral makes sense also for a single curve if n(τ ) is taken to be the principal normal vector of the curve. Then Tw gives what is called the total integrated torsion of the single curve. The first term in (16.143), the Gauss integral ¯ is called in this context the writhing number of the curve for a single closed curve C, ¯ C) ¯ = Wr ≡ G(C,

1 4π

I

¯ C

dτ

I

¯ C

˙ ) × x(τ ˙ ′ )] · dτ ′ [x(τ

x(τ ) − x(τ ′ ) . |x(τ ) − x(τ ′ )|3

(16.145)

Thus one writes the relation (16.143) commonly in the form Lk = Wr + Tw .

(16.146)

Only the sum Wr + Tw is a topological invariant, with Tw containing the information ¯ This formula was found by Calagareau on the ribbon structure of the closed loop C. in 1959 and generalized by White in 1969. From what we have seen above in Eq. (16.136), the writhing number may also be written as an integral 1 Wr = − 4π

I

¯ C

dΩ(x),

(16.147)

where Ω(x) is the solid angle under which the axis of the ribbon is seen from another point on the axis. When rewritten in the form (16.138), it has the magnetic interpretation stated there. This interpretation is relevant for understanding the spacetime properties of the dionium atom which in turn may be viewed as a world ribbon whose two edges describe an electric and a magnetic charge. We have pointed out in Section 14.6 that for a half-integer charge parameter q, a dionium atom consisting of two bosons is a fermion. For this reason, a path integral over a fluctuating ribbon can be used to describe the quantum mechanics of a fermion in three spacetime dimensions [7]. Let us derive the relation (16.146). We split the integral (16.142) over τ into two parts: a small neighborhood of the point τ ′ , i.e., τ ∈ (τ ′ − δ, τ ′ + δ)

(16.148)

and the remainder, for which the integrand is regular. In the regular part, we can set the distance ǫ between the curves C and C ′ equal to zero, and obtain the Gauss ¯ C), ¯ i.e., the writhing number Wr . In the singular part, we approximate integral G(C, x(τ ) within the small neighborhood (16.148) by the straight line ˙ ′ )(τ − τ ′ ), x(τ ) ≈ x(τ ′ ) + x(τ ˙ ) ≈ x(τ ˙ ′ ). x(τ


1039


Figure 16.21 Illustration of Calagareau-White relation (16.146). The number of windp ings around the cylinder is Lk = 2, while Tw = Lk − p/ p2 + R2 , where p is the pitch of the helix and R the radius of the cylinder.

Then the τ -integral can be performed. For ǫ ≪ δ ≪ 1, we find

Tw

1 = − 4π

I

′

C′

′

′

′

2

˙ )] · n(τ ˙ )ǫ dτ [n(τ ) × x(τ

Z

τ ′ +δ

τ ′ −δ

1 I ˙ ′ ) × n(τ ′ )] · n(τ ˙ ′ )/|x(τ ˙ ′ )|. = dτ ′ [x(τ 2π C ′

1 dτ q ˙ ′ )|2 (τ − τ ′ )2 + ǫ2 |x(τ

(16.150)

It is worth emphasizing that in contrast to the Gauss integral for two curves C, C ′, the value of the Gauss integral for C¯ is not an integer but a continuous number. It depends on the shape of the ribbon, changing continuously when the ribbon is deformed isotopically. If one section of the ribbon passes through the other, however, it changes by 2 units. The defining equation shows that Wr vanishes if the ribbon axis has a center or a plane of symmetry. To give an example for a circular closed ribbon with an integer number Lk and an arbitrary writhing number Wr we follow Brook-Fuller and Crick. A cylinder with a closed ribbon is wound flat around the surface of a cylinder, returning along the cylinder axis (see Fig. 16.21). While the two edges of the ribbon perforate each other an integer number of times such that Lk = 2, the ribbon axis has a noninteger integral Wr depending on the ratio √ Gauss 2 2 between pitch and radius, Wr = 2 − p/ p + R .

1040

16.7


Chern-Simons Theory of Entanglements

The Gauss integral has a form very similar to the Biot-Savart law of magnetostatics found in 1820. That law supplies an action-at-a-distance formula for the interaction energy of two currents I, I ′ running along the curves C and C ′ : II ′ E=− 2 c

I I C

C′

dx · dx′

1 , |x − x′ |

(16.151)

where c is the light velocity. It was a decisive conceptual advance of Maxwell’s theory to explain formula (16.151) by means of a local field energy arising from a vector potential A(x). In view of the importance of the Gauss integral for the topological classification of entanglements, it is useful to derive a local field theory producing the Gauss integral as a topological action-at-a-distance. Imagine the two contours C and C ′ carrying stationary currents of some pseudo-charge which we normalize at first to unity. These currents are coupled to a vector potential A(x), which is now unrelated to magnetism and which will be called statisto-magnetic vector potential :

Ae,curr = −i

I

C

dxA(x) − i

I

C′

dxA(x).

(16.152)

The action is of the Euclidean type as indicated by the subscript e (recall the relation with the ordinary action A = iAe ). We now construct a field action for A(x) so that the field equations render an interaction between the two currents which is precisely of the form of the Gauss integral. This field action reads Ae,CS

i = 2

Z

d3 x A · (∇ × A)

(16.153)

and is called the Chern-Simons action. It shares with the ordinary Euclidean magnetic field action the quadratic dependence on the vector potential A(x), as well as the invariance under local gauge transformations A(x) → A(x) + ∇Λ(x),

(16.154)

which is obvious when transforming the second vector potential in (16.153). The gauge transformation of the first vector potential produces no change after a partial integration. Also the coupling in (16.152) to the contours C and C ′ is gauge-invariant after a partial integration, since the contours are closed, satisfying ∇·

I

dx = 0.

(16.155)

In contrast to the magnetic field energy, however, the action (16.153) is purely imaginary. The factor i is important for the applications in which the Chern-Simons ′ action will give rise to phase factors of the form ei2θG(C,C ) . H. Kleinert, PATH INTEGRALS

1041

16.7 Chern-Simons Theory of Entanglements

By extremizing the combined action, we obtain the field equation ∇ × A(x) =

I

C

+

I C′

dx.

(16.156)

Its solution is Ai (x) =

I

C

+

I C′

Gij (x, x′ )dx′j ,

(16.157)

where Gij (x, x′ ) is a suitable Green function solving the inhomogeneous field equation (16.157) with a δ-function source instead of the current. Due to the gauge invariance of the left-hand side, however, there is no unique solution. Given a solution Gij (x, x′ ), any gauge-transformed Green function Gij (x, x′ ) → Gij (x, x′ ) + ∇i Λj (x, x′ ) + ∇j Λi (x, x′ ) will give the same curl ∇ × A. Only the transverse part of the vector potential (16.157) is physical, and the Green function has to satisfy (3)

ǫijk ∇j Gkl (x, x′ ) = δij (x − x′ )T ,

(16.158)

where

(3)

δij (x − x′ )T = δij −

∇i ∇j (3) δ (x − x′ ) ∇2

(16.159)

is the transverse δ-function. The vector potential is then obtained from (16.157) in the transverse gauge with ∇ · A(x) = 0.

(16.160)

The solution of the differential equation (16.158) is easily found in the twodimensional transverse subspace. The Fourier transform of Eq. (16.158) reads iǫijk pj Gkl (p) = δil −

pj pl , p2

(16.161)

and this is obviously solved by Gij (p) = iǫikj pk

1 . p2

(16.162)

The transverse gauge (16.160) may be enforced in an action formalism by adding to the action (16.153) a gauge-fixing term AGF =

1 (∇ · A)2 , 2α

(16.163)

with an intermediate gauge parameter α which is taken to zero at the end. The field equation (16.158) is then changed to i ǫijk ∇j + ∇i ∇k Gkl (x, x′ ) = δik δ (3) (x − x′ ), α

(16.164)

1042


which reads in momentum space i iǫijk pj − pi pk Gkl (p) = δik . α

(16.165)

This has the unique solution pi pk Gik (p) = iǫijk pj + iα 2 p

!

1 , p2

(16.166)

whose α → 0 -limit is (16.162). Going back to configuration space, the Green function becomes ′

Gij (x, x ) =

Z

d3 p ip(x−x′ ) iǫikj pk 1 (x − x′ )k 1 1 e = ∇ . (16.167) ǫ = ǫ ikj k ijk (2π)3 p2 4π |x − x′ | 4π |x − x′ |3

Inserting this into Ae,curr + Ae,CS yields the interaction between the currents9 Ae,int = −i

I I C

C′

dxi dx′j Gij (x, x′ ).

(16.168)

Up to the prefactor −i, this is precisely the Gauss integral G(C, C ′) of the two curves C, C ′ . In addition there are the self-interactions of the two curves Ae,int = −

i 2

I I C

C

+

I

C′

I C′

dxi dx′j Gij (x, x′ ),

(16.169)

which are equal to −(i/2)[G(C, C) + G(C ′ , C ′ )]. Due to their nontopological nature discussed earlier, these have no quantized values and must be avoided. Such unquantized self-interactions can be avoided by considering systems whose orbits are subject to appropriate restrictions. They may, for instance, contain only lines which are not entangled with themselves and run in a preferred direction from −∞ to ∞. Ensembles of nonrelativistic particles in two space dimensions have precisely this type of worldlines in three-dimensional spacetime. They are therefore an ideal field for the Chern-Simons theory, as we shall see below in more detail. Another way of avoiding unquantized self-interactions is based on a suitable limiting procedure. If the lines C and C ′ coincide, the Gaussian integral G(C, C ′ ) over C = C ′ may be spread over a large number N of parallel running lines Ci (i = 1, . . . , N), each of which carries a topological charge 1/N. The sum P (1/N 2 ) ij G(Ci , Cj ) contains N-times the same self-interaction and N(N −1)-times the same integer-valued linking number Lk of pairs of lines. In the limit N → ∞, only the number Lk survives. The result coincides with the Gauss integral for the two frame lines C1 and CN of the ribbon. The number LK may therefore be called the frame linking number . This number depends obviously on the choice of the framing. There is a preferred choice for which Lk vanishes. This eliminates the self-interaction trivially. 9

Compare this with the derivation of Eq. (3.244). H. Kleinert, PATH INTEGRALS

1043

16.8 Entangled Pair of Polymers

Although the limiting procedure makes the self-interaction topological, the arbitrariness of the framing destroys all information on the knot classes. This information can be salvaged by means of a generalization of the above topological action leading to a nonabelian version of the Chern-Simons theory. That theory has the same arbitrariness in the choice of the framing. However, by choosing the framing to be the same as in the abelian case and calculating only ratios of observable quantities, it is possible to eliminate the framing freedom. This will enable us to distinguish the different knot classes after all.

16.8

Entangled Pair of Polymers

For a pair of polymers, the above problems with self-entanglement can be avoided by a slight modification of the Chern-Simons theory. This will allow us to study the entanglement of the pair. In particular, we shall be able to calculate the second topological moment of the entanglement, which is defined as the expectation value hm2 i of the square of the linking number m [8]. The self-entanglements will of the individual polymers will be ignored. The result will apply approximately to a polymer in an ensemble of many others, since these may be considered roughly as a single very long effective polymer. Consider two polymers running along the contours C1 and C2 which statistically can be linked with each other any number of times m = 0, 1, 2, . . . . The situation is illustrated in Fig. 16.22 for m = 2. The linking number (16.134) for these two polymers can be calculated with the help of

Figure 16.22 Closed polymers along the contours C1 , C2 respectively. a slight modification of the Chern-Simons action (16.153) and the couplings (16.152). We simply introduce two vector potentials and the Euclidean action Ae = Ae,CS12 + Ae,curr ,

(16.170)

where Ae,CS12 = i

Z

d3 x A1 · (∇ × A2 )

(16.171)

and Ae,curr = −i

I

C1

dxA1 (x) − i

I

dxA2 (x).

(16.172)

C2

If we choose the gauge fields to be transverse, as in (16.160), we obtain with the same technique as before the correlation functions of the gauge fields µν Dab (x, x′ ) ≡ hAµa (x)Aνb (x′ )i,

a, b = 1, 2

(16.173)

1044


are ij D11 (x, x′ ) = ij D12 (x, x′ ) =

=

ij D22 (x, x′ ) = 0, Z 1 1 d3 p ip(x−x′ ) iǫikj k k ij D21 (x, x′ ) = e = ǫikj ∇k (2π)3 p2 4π |x − x′ | 1 (x − x′ )k ǫijκ . 4π |x − x′ |3

0,

(16.174)

(16.175)

The transverse gauge is enforced by adding to the action (16.171) a gauge-fixing term AGF =

1 [(∇ · A1 )2 + (∇ · A2 )2 ], 2α

(16.176)

and taking the limit α → 0 at the end. Extremizing the extended action produces the Gaussian linking number −iG(C1 , C2 ) of Eq. (16.134). The calculation is completely analogous to that leading to Eq. (16.168). The partition function of the two polymers and the gauge fields is given by the path integral Z Z Z Z= Dx1 Dx2 DA1 DA2 e−Ae −AGF . (16.177) C1

C2

Performing the functional integral over the gauge fields, we obtain from the extremum Z Z Z = const × Dx1 Dx2 eiG(C1 ,C2 ) , C1

(16.178)

C2

where the constant is the trivial fluctuation factor of the vector potentials. The Gaussian integral G(C1 , C2 ) has the values m = 0, ±1, ±2, . . . of the linking numbers. In order to analyze the distribution of linking number in the two-polymer system we must be able to fix a certain linking number. This is possible by replacing phase factor eim in the path R the imλ −imλ integral (16.178) by e , and calculating Z(κ). An integral dκe Z(λ) will then select any specific linking number m. But a phase factor eimλ is simply produced in the partition function (16.178) by attaching to one of the current couplings in the interaction (16.172), say to that of C2 , a factor λ, thus changing (16.172) to I I Ae,curr,λ = −i dx A1 (x) − iλ dx A2 (x). (16.179) C1

C2

The λ-dependent partition function is then Z Z Z Z(λ) = Dx1 Dx2 DA1 DA2 e−Ae,CS12 −Ae,curr,λ −AGF C1 C Z 2 Z = const × Dx1 Dx2 eimλ . C1

(16.180)

C2

Ultimately, we want to find the probability distribution of the linking numbers m as a function of the lengths of C1 and C2 . The solution of this two-polymer problem may be considered as an approximation to a more interesting physical problem in which a particular polymer is linked to any number N of polymers, which are effectively replaced by a single long “effective” polymer [9]. Unfortunately, the full distribution of m is very hard to calculate. Only a calculation of the second topological moment is possible with limited effort. This quantity is given by the expectation value hm2 i of the square of the linking number m. Let PL1 ,L2 (x1 , x2 ; m) be the configurational probability to find the polymer C1 of length L1 with fixed coinciding endpoints at x1 and the polymer C2 of length L2 with fixed coinciding H. Kleinert, PATH INTEGRALS

1045


endpoints at x2 , entangled with a Gaussian linking number m. The second moment hm2 i is given by the ratio of integrals R 3 R +∞ d x1 d3 x2 −∞ dm m2 PL1 ,L2 (x1 , x2 ; m) 2 hm i = R , (16.181) R +∞ d3 x1 d3 x2 −∞ dm PL1 ,L2 (x1 , x2 ; m) performed for either of the two probabilities. The integrations in d3 x1 d3 x2 covers all positions of the endpoints. The denominator plays the role of a partition function of the system: Z Z +∞ Z ≡ d3 x1 d3 x2 dm PL1 ,L2 (x1 , x2 ; m). (16.182) −∞

Due to translational invariance of the system, the probabilities depend only on the differences between the endpoint coordinates: PL1 ,L2 (x1 , x2 ; m) = PL1 ,L2 (x1 − x2 ; m).

(16.183)

Thus, after the shift of variables, the spatial double integrals in (16.181) can be rewritten as Z Z d3 x1 d3 x2 PL1 ,L2 (x1 , x2 ; m) = V d3 x PL1 ,L2 (x; m), (16.184) where V denotes the total volume of the system.

16.8.1

Polymer Field Theory for Probabilities

The calculation of the path integral over all line configurations is conveniently done within the polymer field theory developed in Section 15.12. It permits us to rewrite the partition function (16.180) as a functional integral over two ψ1α1 (x1 ) and ψ2α2 (x2 ) with n1 and n2 replica (α1 = 1, . . . , n1 , α2 = 1, . . . , n2 ). At the end we shall take n1 , n2 → 0 to ensure that these fields describe only one polymer each, es explained in Section 15.12. For these fields we define an auxiliary probability P~z (~x1 , ~x2 ; λ) to find the polymer C1 with open ends at x1 , x′1 and the polymer C2 with open ends at x2 , x′2 . The double vectors ~x1 ≡ (x1 , x′1 ) and ~x2 ≡ (x2 , x′2 ) collect initial and final endpoints of the two polymers C1 and C2 . The auxiliary probability P~z (~x1 , ~x2 ; λ) is given by a functional integral Z P~z (~x1 , ~x2 ; λ) = lim D(fields) ψ1αi (x1 )ψ1∗α1 (x′1 )ψ2α2 (x2 )ψ2∗α2 (x′2 )e−A , (16.185) n1 ,n2 →0

where D(fields) indicates the measure of functional integration, and A the total action (16.180) governing the fluctuations. The expectation value is calculated for any fixed pair (α1 , α2 ) of replica labels, i.e., replica labels are not subject to Einstein’s summation convention of repeated indices. The action A consists of kinetic terms for the fields, a quartic interaction of the fields to account for the fact that two monomers of the polymers cannot occupy the same point, the so-called excludedvolume effect , and a Chern-Simons field to describe the linking number m. Neglecting at first the excluded-volume effect and focusing attention on the linking problem only, the action reads A = ACS12 + Ae,curr + Apol + AGF ,

(16.186)

with a polymer field action Apol =

2 Z X i=1

i 2 ¯ ψi | + m2 |Ψi |2 . d3 x |D i

(16.187)

They are coupled to the polymer fields by the covariant derivatives Di = ∇ + iγi Ai ,

(16.188)

1046


with the coupling constants γ1,2 given by γ1 = 1,

γ2 = λ.

(16.189)

The square masses of the polymer fields are given by m2i = 2M zi .

(16.190)

where M = 3/a, with a being the length of the polymer links [recall (15.79)], and zi the chemical potentials of the polymers, measured in units of the temperature. The chemical potentials are conjugate variables to the length parameters L1 and L2 , respectively. The symbols Ψi collect the replica of the two polymer fields Ψi = ψi1 , . . . , ψini , (16.191) and their absolute squares contain the sums over the replica ¯ i |2 = |Di Ψ

ni X

αi =1

|Di ψiαi |2 ,

ni X

|Ψi |2 =

αi =1

|ψiαi |2 .

(16.192)

Having specified the fields, we can now write down the measure of functional integration in Eq. (16.185): Z D(fields) = DAi1 DAj2 DΨ1 DΨ∗1 DΨ2 DΨ∗2 . (16.193) By Eq. (16.180), the parameter λ is conjugate to the linking number m. We can therefore calculate the probability PL1 ,L2 (~x1 , ~x2 ; m) in which the two polymers are open with different endpoints from the auxiliary one P~z (~x1 , ~x2 ; λ) by the following Laplace integral over ~z = (z1 , z1 ): PL1 ,L2 (~x1 , ~x2 ; m) = ′lim

x1 → x1 x′2 →x2

Z

c+i∞

c−i∞

M dz1 M dz2 z1 L1 +z2 L2 e 2πi 2πi

Z

∞

dke−imλ P~z (~x1 , ~x2 ; λ) .

−∞

(16.194)

16.8.2

Calculation of Partition Function

Let us use the polymer field theory to calculate the partition function (16.182). By Eq. (16.194), it is given by the integral over the auxiliary probabilities Z =

Z

3

3

d x1 d x2 ′lim

x1 →x1 x′2 →x2

Z

c+∞


c−i∞

Z

∞

dm

−∞

Z

+∞

dλe−imλ P~z (~x1 , ~x2 ; λ) .

−∞

(16.195) The integration over m is trivial and gives 2πδ(λ), enforcing λ = 0, so that Z =

Z

d3 x1 d3 x2 ′lim

x1 →x1 x′2 →x2

Z

c+i∞ c−i∞

M dz1 M dz2 z1 L1 +z2 L2 e P~z (~x1 , ~x2 ; 0) . 2πi

(16.196)

To calculate P~z (~x1 , ~x2 ; 0), we observe that the action A in Eq. (16.186) depends on λ only via the polymer part (16.187), and is quadratic in λ. Let us expand A as A = A0 + λA1 + λ2 A2 ,


1047

16.8 Entangled Pair of Polymers with the λ-independent part A0

≡

ACS12 + AGF +

Z

3

"

2

2

d x |D1 Ψ1 | + |∇Ψ2 | +

2 X i=1

2

|Ψi |

#

,

(16.198)

a linear coefficient A1 ≡

Z

d3 x j2 (x) · A2 (x)

(16.199)

containing a pseudo-current of the second polymer field j2 (x) = iΨ∗2 (x)∇Ψ2 (x),

(16.200)

and a quadratic coefficient A2 ≡

1 4

Z

d3 x A22 |Ψ2 (x)|2 .

With these definitions we write with the help of (16.198): Z P~z (~x1 , ~x2 ; 0) = D(fields)e−A0 ψ1α1 (x1 )ψ1∗α1 (x′1 )ψ2α2 (x2 )ψ2α2 (x′ ).

(16.201)

(16.202)

In the action (16.198), the fields Ψ2 , Ψ∗2 are obviously free, whereas the fields Ψ1 , Ψ∗1 are apparently not because of the couplings with the Chern-Simons fields in the covariant derivative D1 . This coupling is, however, without physical consequences. Indeed, by integrating out Ai2 in (16.202), we find from ACS12 the flatness condition: ∇ × A1 = 0.

(16.203)

On a flat space with vanishing boundary conditions at infinity this implies A1 = 0. As a consequence, the functional integral (16.202) factorizes as follows [compare (15.370)] P~z (~x1 , ~x2 ; 0) = G0 (x1 − x′1 ; z1 )G0 (x2 − x′2 ; z2 ),

(16.204)

where G0 (xi − x′i ; zi ) are the free correlation functions of the polymer fields: G0 (xi − x′i ; zi ) = hψiαi (xi )ψi∗αi (x′i )i.

(16.205)

In momentum space, the correlation functions are hψ˜iαi (ki )ψ˜i∗αi (k′i )i = δ (3) (ki − k′i )

1 , k2i + m2i

(16.206)

such that G0 (xi − x′i ; zi ) =

Z

d3 k ik·x 1 e , (2π)3 k2i + m2i

(16.207)

and G0 (xi − x′i ; Li )

= =

c+i∞

M dzi zi Li e G0 (xi − x′i ; zi ) 2πi c−i∞ 3/2 ′ 1 M e−M(xi −xi )/2Li . 2 4πLi

Z

(16.208)

1048


The partition function (16.196) is then given by the integral Z Z = 2π d3 x1 d3 x2 ′lim G0 (x1 − x′1 ; L1 )G0 (x2 − x′2 ; L2 ). x1 →x1 x′2 →x2

(16.209)

The integrals at coinciding endpoints can easily be performed, yielding Z=

2πM 3 V 2 (L1 L2 )−3/2 . (8π)3

(16.210)

It is important to realize that in Eq. (16.195) the limits of coinciding endpoints x′i → xi and the inverse Laplace transformations do not commute unless a proper renormalization scheme is chosen to eliminate the divergences caused by the insertion of the composite operators |ψ α (x)|2 . This can be seen for a single polymer. If we were to commuting the limit of coinciding endpoints with the Laplace transform, we would obtain Z c+i∞ Z c+i∞ dz zL dz zL ′ e lim G (x − x ; z) = e G0 (0, z), (16.211) 0 ′ →x x 2π 2πi c−i∞ c−i∞ where G0 (0; z) = h|ψ(x)|2 i.

(16.212)

This expectation value, however, is linearly divergent: Z d3 k h|ψ(x)|2 i = → ∞. 2 k + m2

16.8.3

(16.213)

Calculation of Numerator in Second Moment

Let us now turn to the numerator in Eq. (16.181): Z Z ∞ 3 3 N ≡ d x1 d x2 dm m2 PL1 ,L2 (x1 , x2 ; m).

(16.214)

−∞

We shall set up a functional integral for N in terms of the auxiliary probability P~z (~x1 , ~x2 ; 0) analogous to Eq. (16.195). First we observe that Z Z ∞ Z cτ i∞ M dzi M dz2 3 3 2 N = d x1 d r2 dm m ′lim x →x1 2πi 2πi −∞ c−i∞ 1 x′2 →x2

× ez1 L1 +z2 L2

Z

∞

dλe−imλ P~z (~x1 , ~x2 ; λ).

(16.215)

−∞

The integration in m is easily performed after noting that 2 Z ∞ Z ∞ ∂ −imλ 2 −imλ ~ dm m e P~z (~x1 , x2 ; λ) = − dm e P~z (~x1 , ~x2 ; λ). ∂λ2 −∞ −∞

(16.216)

After two integrations by parts in λ, and an integration in m, we obtain Z Z c+i∞ M dz1 M dz2 z1 L1 +z2 L2 N = d3 x1 d3 x2 ′lim (−1) e x1 →x1 2πi 2πi c−i∞ x′2 →x2

×

Z

∞

∂2 dλ δ(λ) P~z (~x1 , ~x2 ; λ) . ∂λ2 −∞

(16.217)


1049

16.8 Entangled Pair of Polymers Performing the now the trivial integration over λ yields Z Z c+i∞ M dz1 M dz2 z1 L1 +z2 L2 ∂ 2 ~ e P (~ x , x ; 0) . N = d3 x1 d3 x2 ′lim (−1) 1 2 ~ z x1 →x1 2πi 2πi ∂λ2 c−i∞

(16.218)

x′2 →x2

To compute the term in brackets, we use again (16.197) and Eqs. (16.198)–(16.223), to find N

=

Z

d x1 d x2 nlim →0

×

Z

×

3

3

1 n2 →0

Z

c+i∞

c−i∞


D(fields) exp(−A0 )|ψ1α1 (x1 )|2 |ψ2α2 (x2 )|2

"Z

3

d x A2 ·

Ψ∗2 ∇Ψ2

2

1 + 2

Z

3

d

x A22

2

|Ψ2 |

#

.

(16.219)

In this equation we have taken the limits of coinciding endpoint inside the Laplace integral over z1 , z2 . This will be justified later on the grounds that the potentially dangerous Feynman diagrams containing the insertions of operations like |Ψi |2 vanish in the limit n1 , n2 → 0. In order to calculate (16.219), we decompose the action into a free part # " Z 2 X 2 0 3 1 2 2 A0 ≡ ACS + d x |D Ψ1 | + |∇Ψ2 | + 2|Ψi | , (16.220) i=1

and interacting parts A01 ≡

Z

d3 x j1 (x) · A1 (x),

(16.221)

with a “current” of the first polymer field j1 (x) ≡ iΨ∗1 (x)∇Ψ1 (x),

(16.222)

and A20 ≡

1 4

Z

d3 x A21 |Ψ1 (x)|2 .

(16.223)

Expanding the exponential e

A0

=e

A00 +A10 +A20

=e

A0

1 2 1 (A0 ) 2 1−A0 + −A0 +. . . , 2

(16.224)

and keeping only the relevant terms, the functional integral (16.219) can be rewritten as a purely Gaussian expectation value Z Z c+i∞ M dz1 M dz2 z1 L1 +z2 L2 N = κ2 d3 x1 d3 x2 nlim e 1 →0 2πi 2πi c−i∞ n2 →0 Z × D(fields) exp(−A00 )|ψ1α1 (x1 )|2 |ψ2α2 (x2 )|2 "Z # 2 Z 1 × d3 x A1 · Ψ∗1 ∇Ψ1 + d3 x A21 |Ψ1 |2 2 # "Z 2 Z 1 3 ∗ 3 2 2 × d x A2 · Ψ2 ∇Ψ2 + d x A2 |Ψ2 | . (16.225) 2

1050


+

+

+

Figure 16.23 Four diagrams contributing in Eq. (16.225). The lines indicate correlation functions of Ψi -fields. The crossed circles with label i denote the insertion of |Ψi (xi )|2 . Note that the initially asymmetric treatment of polymers C1 and C2 in the action (16.187) has led to a completely symmetric expression for the second moment. Only four diagrams shown in Fig. 16.23 contribute in Eq. (16.225). The first diagram is divergent due to the divergence of the loop formed by two vector correlation functions. This infinity may be absorbed in the four-Ψ interaction accounting for the excluded volume effect which we do not consider at the moment. We now calculate the four diagrams separately.

16.8.4

First Diagram in Fig. 16.23

From Eq. (16.225) one has to evaluate the following integral N1 = ×

Z c+i∞ Z Z M dz1 M dz2 z1 L1 +z2 L2 κ2 3 3 lim e d x d x d3 x′1 d3 x′2 1 2 4 nn1 →0 2πi 2πi c−i∞ 2 →0 |ψ1α1 (x1 )|2 |ψ2α2 (x2 )|2 |Ψ1 |2 A21 x′ |Ψ2 |2 A22 x′ . 1

(16.226)

2

As mentioned before, there is an ultraviolet-divergent contribution which must be regularized. The system has, of course, a microscopic scale, which is the size of the monomers. This, however, is not the appropriate short-distance scale to be uses here. The model treats the polymers as random chains. However, the monomers of a polymer in the laboratory are usually not freely movable, so that polymers have a certain stiffness. This gives rise to a certain persistence length ξ0 over which a polymer is stiff. This length scale is increased to ξ > ξ0 by the excluded-volume effects. This is the length scale which should be used as a proper physical short-distance cutoff. We may impose this cutoff by imagining the model as being defined on a simple cubic lattice of spacing ξ. This would, of course, make analytical calculations quite difficult. Still, as we shall see, it is possible to estimate the dependence of the integral N1 and the others in the physically relevant limit in which the lengths of the polymers are much larger than the persistence length ξ. An alternative and simpler regularization is based on cutting off all ultraviolet-divergent continuum integrals at distances smaller than ξ. After such a regularization, the calculation of N1 is rather straightforward. Replacing the expectation values by the Wick contractions corresponding to the first diagram in Fig. 16.23, and performing the integrals as shown in Appendix 16A, we obtain N1

= ×

Z 1 Z 2 V M4 − 32 − 21 (L1 L2 ) ds [(1 − s)s] d3 xe−M x /2s(1−s) 6 4π (8π) 0 Z 1 Z Z 3 2 1 dt [(1 − t)t]− 2 d3 ye−M y /2t(1−t) d3 x′′1 ′′ 4 . |x1 | 0

(16.227)

The variables x and y have been rescaled with respect to the original ones in order to extract the behavior of √ N1 in L1 and √ L2 . As a consequence, the lattices where x and y are defined have now spacings ξ/ L1 and ξ/ L2 respectively. H. Kleinert, PATH INTEGRALS

1051


The x, y integrals may be explicitly computed in the physical limit L1 , L2 ≫ ξ, in which the above spacings become small. Moreover, it is possible to approximate the integral in x′′1 with an integral over a continuous variable l and a cutoff in the ultraviolet region: Z Z ∞ 1 dl . (16.228) d3 x′′1 ′′ 4 ∼ 4π 2 |x1 | l2 ξ After these approximations, we finally obtain N1 = V π 1/2

16.8.5

M (L1 L2 )−1/2 ξ −1 . (4π)3

(16.229)

Second and Third Diagrams in Fig. 16.23

Here we have to calculate Z Z Z c+i∞ M dz1 M dz2 z1 L1 +z2 L2 3 3 2 e d x1 d x2 d3 x′1 d3 x′′1 d3 x′2 N2 = κ nlim 1 →0 2πi 2πi c−i∞ n2 →0 α1 2 α2 2 ∗ ∗ 2 2 × |ψ1 (x1 )| |ψ2 (x2 )| ( A1 ·Ψ1 ∇Ψ1 )x′ ( A1 ·Ψ1 ∇Ψ1 )x′′ A2 |Ψ2 | x′ . 1

1

(16.230)

2

The above amplitude has no ultraviolet divergence, so that no regularization is required. The Wick contractions pictured in the second Feynman diagrams of Fig. 16.23 lead to the integral Z Z t √ M3 1 −1/2 N2 = −4 2V L2 L−1 dt dt′ C(t, t′ ), 1 π6 0 0

(16.231)

where C(t, t′ ) is a function independent of L1 and L2 : Z 2 ′ ′ ′ −3/2 C(t, t ) = [(1 − t)t (t − t )] d3 xd3 yd3 ze−M(y−x) /2(1−t) ×

[δ z · (z + x) − (z + x) z ] 2 ′ 2 ′ ij i j ∇jy e−M y /2t ∇ix e−M x /2(t−t ) . |z|3 |z + x|3

(16.232)

As in the previous section, the variables x, y, z have been rescaled with respect to the original ones in order to extract the behavior in L1 . If the polymer lengths are much larger than the persistence length one can ignore the fact that the monomers have a finite size and it is possible to compute C(t, t′ ) analytically, leading to −1/2

N2

=

−

V L2 L−1 1 M 3/2 4K, (2π)6

where K is the constant 1 3 1 1 5 1 7 1 1 9 1 19π K ≡ B , + B , −B , + B , = ≈ 0.154, 6 2 2 2 2 2 2 2 3 2 2 384

(16.233)

(16.234)

and B(a, b) = Γ(a)Γ(b)/Γ(a + b) is the Beta function. For large L1 → ∞, this diagram gives a negligible contribution with respect to N1 . The third diagram in Fig. 16.23 give the same as the second, except that L1 and L2 are interchanged. N3 = N2 |L1 ↔L2 .

(16.235)

1052


16.8.6

Fourth Diagram in Fig. 16.23

Here we have the integral N4

= −4κ2 ×

1 lim 2 nn1 →0 →0 2

Z

c+i∞

c−i∞


Z

d3 x1 d3 x2

Z

d3 x′1 d3 x′2 d3 x′′1 d3 x′′2

∗ 2 ∗ 2 |ψ1α1 (x1 )|2 |ψ2 (xα 2 )| (A1 ·Ψ1 ∇Ψ1 )x′ (A1 ·Ψ1 ∇Ψ1 )x′′ 1

1

× (A2 ·Ψ∗2 ∇Ψ2 )x′ (A2 · Ψ∗2 ∇Ψ2 )x′′ 2

2

,

(16.236)

which has no ultraviolet divergence. After some analytic effort we find N4 = −

1 M 5V (L1 L2 )−1/2 16 (2π)11

Z

0

1

ds

Z

s

ds′

0

Z

1

dt

0

Z

t

dt′ C(s, s′ , t, t′ ),

(16.237)

0

where −3/2

−3/2

C(s, s′ ; t, t′ ) = [(1 − s)s′ (s − s′ )] [(1 − t)t′ (t − t′ )] Z pα pβ pα pβ d3 p ǫ ǫ + ǫ ǫ (16.238) × ikα jlβ ilα jkβ (2π)3 p2 p2 p2 p2 Z i √ ′ ′ ′2 ′2 ′ 2 ′ × d3 x′ d3 y ′ e−i L1 p(x −y ) e−M x /2(1−s) ∇jy′ e−M y /2t ∇ix′ e−M(x−y) /2(s−s ) Z i √ ′ ′ ′2 ′2 ′ ′ ′ 2 ′ ∇kv′ e−M(u −v ) /2(t−t ) , × d3 u′ d3 v ′ e−i L2 p(u −v ) e−M v /2(1−t) ∇lu′ e−Mu /2t and x′ , y′ are scaled variables. To take √ into account the finite persistence length, they should be defined on√a lattice with spacing ξ/ L1 . Similarly, u′ , v′ should be considered on a lattice with spacing ξ/ L2 . Without performing the space integrations d3 x′ d3 y′ d3 u′ d3 v′ , the behavior of N4 as a function of the polymer lengths can be easily estimated in the following limits: 1. L1 ≫ 1; L1 ≫ L2 N4 ∝ L−1 1

(16.239)

N4 ∝ L−1 2

(16.240)

2. L2 ≫ 1; L2 ≫ L1

3. L1 , L2 ≫ 1, L2 /L1 = α = finite −3/2

N4 ∝ L 1

.

(16.241)

Moreover, if the lengths of the polymers are considerably larger than the persistence length, the function C(s, s′ , t, t′ ) can be computed in a closed form: N4

≈ −

128V M (L1 L2 )−1/2 π 5 π 3/2

Z

1

ds

0 −1/2

× [L1 t(1 − s) + L2 (1 − t)s]

Z

0

.

1

dt(1 − s)(1 − t)(st)1/2 (16.242)

It is simple to check that this expression has exactly the above behaviors. H. Kleinert, PATH INTEGRALS

16.9 Chern-Simons Theory of Statistical Interaction

16.8.7

1053

Second Topological Moment

Collecting all contributions we obtain the result for the second topological moment: hm2 i =

N1 + N2 + N3 + N4 , Z

(16.243)

with N1 , N2 , N3 , N4 , Z given by Eqs. (16.210), (16.229), (16.233), (16.235), and (16.231). In all formulas, we have assumed that the volume V of the system is much larger than the size of the volume occupied by a single polymer, i.e., V ≫ L31 To discuss the physical content of the result (16.232), we assume C2 to be a long effective polymer representing all polymers in a uniform solution. We introduce the polymer concentration l as the average mass density of the polymers per unit volume: l=

M , V

(16.244)

where M is the total mass of the polymers M=

Np X i=1

ma

Lk . a

(16.245)

Here ma is the mass of a single monomer of length a, Lk is the length of polymer Ck , and Np is the total number of polymers. Thus Lk /a is the number of monomers in the polymer Ck . The polymer C1 is singled out as any of the polymers Ck , say Ck¯ , of length L1 = Lk¯ . The remaining ones are replaced by a long effective polymer C2 of length L2 = σk6=k¯ Lk . From the above relations we may also write L2 ≈

aV l . ma

(16.246)

In this way, the length of the effective molecule C2 is expressed in terms of physical parameters, the concentration of polymers, the monomer length, and the mass and volume of the system. Inserting (16.246) into (16.232), with N1 , N2 , N3 , N4 , Z given by Eqs. (16.210), (16.229), (16.233), (16.235), and (16.231). and keeping only the leading terms for V ≫ 1, we find for the second topological moment hm2 i the approximation hm2 i ≈

N1 + N2 , Z

(16.247)

and this has the approximate form al hm2 i = ma

"

# 1/2 2KL1 ξ −1 Li − 4 3/2 , 2π 1/2 M 2 π M

(16.248)

with K of (16.234). Thus we have succeeded in setting up a topological field theory to describe two fluctuating polymers C1 and C2 , and calculated the second topological moment for the linking number m between C1 and C2 . The result is used as an approximation for the second moment for a single polymer with respect to all others in a solution of many polymers. An interesting remaining problem is to calculate the effect of the excluded volume.

16.9

Chern-Simons Theory of Statistical Interaction

The Chern-Simons theory (16.153) together with the coupling (16.152) generates the desired topological interaction corresponding to the Gaussian integral between

1054


pairs of curves C and C ′ . We now demonstrate that this topological interaction is the same as the statistical interaction introduced in Eq. (7.279) and encountered again in (16.26), where it governed the physics of a charged particle running around a magnetic flux tube. This observation will make the Gaussian integral and thus the Chern-Simons action relevant for the description of the statistical properties of nonrelativistic particle orbits. In contrast to the electromagnetic generation of fractional statistics for particle orbits in Section 16.2 via the Aharonov-Bohm effect, the field theory involving the statisto-magnetic vector potential has the advantage of removing the asymmetry between the particles, of which one had to carry a charge, the other one a magnetic flux. An arbitrary number of identical particle orbits can now be endowed with a fractional statistics, the same that was produced by topological interaction (7.279). To prove the equality between the two topological interactions in two space and one time dimension, consider an electron in a plane encircling an infinitely thin magnetic “flux tube” at the origin (the word flux tube stands between quotation marks since the “tube” is only a point in the two-dimensional space). In a Euclidean spacetime, the worldline of the electron C winds itself around the straight “flux tube” C ′ along the τ -axis. For this geometry, the integral over C ′ in (16.134) can easily √ R∞ 3 be done using the formula −∞ dt/ t2 + d2 = 2. The result is 1 G(C, C ) = 2π ′

Z

1 ˙ )∇ϕ(x(τ )) = dτ x(τ 2π

Z

dτ ϕ(x(τ ˙ )),

(16.249)

where ϕ(x(τ )) denotes the azimuthal angle of the electron with respect to the “flux tube” at the time τ Up to a factor 2π, the expression (16.249) agrees with the statistical interaction in (16.26) and (7.279). In two space and one time dimension, the behavior under particle exchange can be assigned to an amplitude at will by adding to the Euclidean action a Gaussian integral with a suitable prefactor. A phase factor eiθ is produced by the exchange when choosing the following Euclidean action-at-a-distance: Ae,int = i2¯ hθG(C, C ′).

(16.250)

This topological interaction is generated by the Chern-Simons action Ae,CS =

1 4θ¯ hi

Z

d3 x A · (∇ × A).

(16.251)

The phase angle θ is related to the former parameter µ0 of the statistical interactions (16.26) and (7.279) by θ = πµ0 .

(16.252)

For µ0 = ±1, ±3, ±5, . . . , the particle orbits have Fermi statistics; for µ0 = 0, ±2, ±4, . . . , they have Bose statistics. Fractional values of µ0 lead to fractional statistics. In contrast to the magnetic generation of fractional statistics, the ChernSimons mechanism applies to any number of particle orbits. By one of the methods H. Kleinert, PATH INTEGRALS

1055

16.9 Chern-Simons Theory of Statistical Interaction

discussed after Eq. (16.169) it must, however, be assured that the Gaussian “selfenergy” does not render any undesirable nontopological contributions. To maintain the analogy with the magnetic interactions as far as possible, we write the Chern-Simons action for a gas of electrons in the form Ae,CS

1 e2 = 4πi c2h ¯ µ0

Z

d3 x A × (∇ × A),

(16.253)

and the coupling of the statisto-magnetic vector potential to the particle orbits as Ae,curr = −i

e c

I

C

dx A(x) − i

e c

I

C′

dx A(x).

(16.254)

This looks precisely like the Euclidean coupling of an ordinary vector potential to electrons. For an arbitrary number of orbits we define the Euclidean two-dimensional current density j(x) ≡ ec

XI α

Cα

dxα δ (3) (x − xα )

(16.255)

and write the interaction (16.254) as

Ae,curr

1 = −i 2 c

Z

d3 x j(x)A(x).

(16.256)

The curl of the vector potential B ≡∇×A

(16.257)

is referred to as a statisto-magnetic field . By varying (16.253) plus (16.256) with respect to A(x), we obtain the field equation B(x) = µ0

2π¯ hc j(x). 2 e

(16.258)

With the help of the elementary flux quantum Φ0 , this can also be written as 1 B(x) = µ0 Φ0 j(x). e

(16.259)

To apply the above formulas, we must transform them from the three-dimensional Euclidean spacetime to the Minkowski space, where the curves Cα become particle orbits in two space dimensions whose coordinates x⊥ = (x, y) are functions of the time t. Specifically, we substitute the three coordinates (x1 , x2 , x3 ) as follows: (x1 , x2 ) → x⊥ ≡ (x, y), x3 → ix0 ≡ ict.

1056


The Euclidean field components A1,2,3 go over into the Minkowskian statisto-electric potential φ and two spatial components Ax,y . The three fields B1,2,3 turn into the Minkowskian statisto-electric fields Ey , Ex and a statisto-magnetic field Bz : A3 = iφ = iA0 , A1 = Ax , A2 = Ay , B3 = iBz , B1 = −iEy , B2 = iEx .

(16.260)

The Euclidean currents become, up to a factor i, the two-dimensional charge and current densities: j3 = ij0 = icρ(x⊥ ), j1 = ijx (x⊥ ) = e

ρ(x) ≡ e

X

(2)

X α

α

δ (2) (x⊥ − x⊥α ),

x˙ α δ (x⊥ − x⊥α ),

α

j2 = ijy (x⊥ ) = e

X

y˙α δ (2) (x⊥ − x⊥α ),

(16.261)

where ρ is the particle density per unit area. The motion of a particle in an external field φ, Ax , Ay is then governed by the interaction 1 dtd2 x ρφ − (jx Ax + jx Ay ) . (16.262) c Conversely, particles with fractional statistics in a 2+1-dimensional spacetime generate Minkowskian statisto-electromagnetic fields following the equations 1 1 Bz = µ0 Φ0 ρ, Ex = µ0 Φ0 jy , Ey = µ0 Φ0 jx . (16.263) c c The electromagnetic normalization in Eq. (16.262) has the advantage that a charged particle cannot distinguish a statisto-magnetic field from a true magnetic field. This property forms the basis for a simple interpretation of the fractional quantum Hall effect as will be seen in Section 18.9. Aint =

16.10

Z

Second-Quantized Anyon Fields

After the developments in Chapter 7, we should expect that the phase factor eiµ0 π , appearing in the path integral upon exchanging the endpoints of two anyonic orbits, can also be found in a second-quantized operator formulation. To verify this, we consider a free Bose field with the action (7.286) and couple it with a statistoelectromagnetic field subject to a Chern-Simons action. The resulting free anyon action reads Aanyon = ACS + Aboson ,

(16.264)

where Aboson =

Z

2

dx

Z

tb

ta

(

e dt ψ (x, t) i¯ h ∂t + i φ(x, t) + µ ψ(x, t) h ¯ ∗

2 h ¯ 2 e − ∇ − i A(x, t) ψ(x, t) . 2M h ¯c )

(16.265)


1057

16.10 Second-Quantized Anyon Fields

The latter corresponds to the action (7.286) in the continuum limit with the derivatives replaced by covariant derivatives according to the usual minimal substitution rule (2.636): e p0 → p0 − φ, c

e p → p − A. c

(16.266)

The first field equation in (16.263) now reads Bz (x, t) = µ0 Φ0 ψ † (x, t)ψ(x, t),

(16.267)

so that the vector potential satisfies the differential equation ∂x Ay (x, t) − ∂y Ax (x, t) = µ0 Φ0 ψ † (x, t)ψ(x, t).

(16.268)

It determines (Ax , Ay ) up to the gauge freedom (∂x Λ, ∂y Λ), where Λ(x, t) is an arbitrary single-valued function satisfying Schwarz’ integrability condition (∂x ∂y − ∂y ∂x )Λ(x, t) = 0.

(16.269)

In the present case, it is useful to allow for a violation of this condition by searching for a multivalued function α(x, t) whose gradient is equal to a given vector potential: (Ax , Ay ) = (∂x α, ∂y α).

(16.270)

This function must obey the differential equation (recall the discussion in Appendix 10A) (∂x ∂y − ∂y ∂x )α(x, t) = µ0

2π¯ hc † ψ (x, t)ψ(x, t). e

(16.271)

The Green function of this differential equation, which is the elementary building block for the construction of all multivalued functions in two dimensions, is the function used before in Eq. (16.249) [see also (10A.30)]: ϕ(x − x′ ) ≡ arctan[(y − y ′ )/(x − x′ )].

(16.272)

It gives the angle between the vectors x and x′ and violates the Schwarz integrability condition at the points where the vectors coincide [recall (10A.33)]: (∂x ∂y − ∂y ∂x )ϕ(x − x′ ) = 2πδ (2) (x − x′ ).

(16.273)

To satisfy this equation, the cut of the arctan in the complex plane must be avoided, which is always possible by deforming it appropriately. The function ϕ(x − x′ ) has the important property ϕ(x − x′ ) − ϕ(x′ − x) = π.

(16.274)

In the two terms on the left-hand side, the point x′ is moved around the point x in the anticlockwise sense.

1058


With the help of the multivalued Green function (16.272) we find immediately the solution of Eq. (16.271): α(x, t) = µ0

h ¯c e

Z

d2 xϕ(x − x′ )ψ † (x, t)ψ(x, t).

(16.275)

The relation (16.270) permits the elimination of the statisto-magnetic field from the action. Actually, this statement is true in general. One can always multiply the fields ψ(x, t) by a phase factor

exp −i

e h ¯c

Z

x

dx′ A(x′ , t) ,

where the contour integral is taken to x from any fixed point along some fixed path. In front of the transformed field −i(e/¯ hc)

Ψ(x, t) = e

Rx

dx′ A(x′ ,t)

ψ(x, t),

(16.276)

the covariant derivatives Di ψ(x, t) = (∂i − i

e Ai )ψ(x, t) h ¯c

(16.277)

become ordinary derivatives ∂i . Unfortunately, the new field Ψ(x, t) depends on the vector potential in a complicated nonlocal way so that this transformation is in general not worthwhile. In the present case, however, the equation of motion for the vector potential is so simple that the transformation can be done explicitly. In fact, the nonlocality has precisely the desired property of changing the statistics of the fields from Bose statistics to any statistics. ˆ t) which are canonically We show this by considering the field operators ψ(x, quantized according to Eq. (7.294). In the continuum limit they satisfy the commutation rules ˆ t), ψˆ† (x′ , t)] = δ (2) (x − x′ ), [ψ(x, [ψˆ† (x, t), ψˆ† (x′ , t)] = 0, ˆ t), ψ(x ˆ ′ , t)] = 0. [ψ(x,

(16.278)

The transformed field operators satisfy the corresponding commutation rules modified by a phase factor eiµ0 π : ˆ t)ψˆ† (x′ , t) − eiπµ0 ψˆ† (x′ , t)ψ(x, ˆ t) = δ (2) (x − x′ ), ψ(x, ψˆ† (x, t)ψˆ† (x′ , t) − eiπµ0 ψˆ† (x′ , t)ψˆ† (x, t) = 0, ˆ ′ , t)ψ(x, ˆ t)ψ(x ˆ ′ , t) − eiπµ0 ψ(x ˆ t), = 0. ψ(x,

(16.279)

As in (16.274), the vector x′ on the left-hand side has to be carried around x in the anticlockwise sense. Using the relation (16.270), the integral in the prefactor of H. Kleinert, PATH INTEGRALS

1059

16.11 Fractional Quantum Hall Effect

(16.276) can immediately be performed and the transformed fields are simply given by ˜ t) = e−i h¯ec α(x,t) ψ(x, t), ψ(x,

e ψ˜† (x, t) = ψ(x, t)ei h¯ c α(x,t) .

(16.280)

The same relations hold for the second-quantized field operators. This makes it quite simple to prove the commutation rules (16.279). We do this here only for the second rule which controls the behavior of the many-body wave functions under the exchange of any two-particle coordinates: ψˆ† (x, t)ψˆ† (x′ , t) − eiπµ0 ψˆ† (x′ , t)ψˆ† (x, t) = 0. This amounts to the relation e e e e ′ ˆ ˆ ′ ,t) ˆ† ˆ ψˆ† (x, t)ei h¯ c α(x,t) ψˆ† (x′ , t)ei h¯ c αˆ (x ,t) = eiπµ0 ψˆ† (x′ , t)ei h¯ c α(x ψ (x, t)ei h¯ c α(x,t) . (16.281)

The phase factors in the middle can be taken to the right-hand side by using the transformation formula ei

R

ˆ d2 x′ f (x′ ,t)ψˆ† (x,t)ψ(x,t)

ψˆ† (x, t)e−i

R

ˆ d2 x′ f (x′ ,t)ψˆ† (x,t)ψ(x,t)

= eif (x,t) ψˆ† (x, t), (16.282)

which follows from the Lie expansion [recall (1.432)] ˆ

ˆ

ˆ −iA = 1 + i[A, ˆ B] ˆ + eiA Be

i2 ˆ ˆ ˆ [A, [A, B]] + . . . . 2!

(16.283)

Setting f (x) equal to µ0 ϕ(x − x′ ), Eq. (16.281) goes over into e ′ ˆ α(x ˆ ′ ,t)] ψˆ† (x, t)ψˆ† (x′ , t)eiµ0 ϕ(x−x ) ei h¯ c [α(x,t)+ e ′ ˆ ′ ,t)+α(x,t)] ˆ = eiπµ0 ψˆ† (x′ , t)ψˆ† (x, t)eiµ0 ϕ(x −x) ei h¯ c [α(x .

(16.284)

The correctness of this equation follows directly from the property (16.274) of the ϕ(x) field and from the commutativity of the Bose fields ψ † (x, t) with each other. This proves the second of the anyon commutation rules (16.279). The others are obtained similarly. Note that we could just as well have constructed the anyon fields from Fermi fields by shifting the exchange phase by an angle π.

16.11

Fractional Quantum Hall Effect

If particles obeying fractional statistics move in an ordinary magnetic field, they are also subject to a statisto-magnetic field . As observed earlier, this acts upon each particle in the same way as an additional true magnetic field. This observation provides a key for the understanding of the fractional quantum Hall effect. The arguments will now be sketched. To measure the effect experimentally, a thin slab of conducting material (the original experiment used the compound Alx Ga1−x As) is placed at low temperatures

1060


(≈ 0.5 K) in the xy-plane traversed by a strong magnetic field Bz (between 10 and 200 kG) along the z-axis. An electric field Ex is applied in the x-direction and an electric Hall current jy per length unit is measured in the y-direction. Such a current is expected in a dissipative electron gas with a number density (per unit area) ρ, where the fields satisfy the relation Ex = −

1 jy Bz ρec

(16.285)

(see Appendix 16E). The transverse resistance defined by Bz ρec

Rxy ≡

(16.286)

rises linearly in Bz . Its dimension is sec/cm. In contrast with this naive expectation, the experimental data for Rxy rise stepwise with a number of plateaus whose resistance take the values h/e2 ν, where ν is a rational number with odd denominators: ν = 15 , 27 , 13 , 25 , 23 , . . . .

(16.287)

We have omitted the observed number ν = 52 since its theoretical explanation requires additional physical considerations (see the references at the end of the chapter). Similar plateaus had been observed at integer values of ν. Those are explained as follows. In an ideal Fermi liquid at zero temperature, the electron orbits have energies 2 p /2M. Their momenta fill a Fermi sphere of radius pF . The size of pF is determined from the particle number per unit area ρ via the phase space integral ρLx Ly = 2 ×

dpx dpy Lx Ly , (2π¯ h)2

Z

(16.288)

where Lx and Ly are the lengths of the rectangular layered material in the x- and y-directions. The factor 2 accounts for the two spin orientations. The rotationally invariant integration up to pF yields pF =

q

2πρ¯ h.

(16.289)

By switching on a magnetic field Bz , the rotational invariance is destroyed and the electrons circle with a velocity v = ωr on Landau orbits around the z-direction with the cyclotron frequency ω = eB/Mc. In quantum mechanics, the system corresponds to an ensemble of harmonic oscillators which in the gauge A = (0, Bx, 0) [see Eq. (9.93)] move back and forth in the x-direction and have a spectrum (n + 1/2)¯ hω R [see Eq. (9.100)]. The phase space integral in the x-direction dpx Lx /(2π¯ h) becomes therefore a sum over n. The center of oscillations is x0 = py /Mω [see H. Kleinert, PATH INTEGRALS

1061

16.11 Fractional Quantum Hall Effect R

Eq. (9.95)], so that the remaining phase space integral dpy Ly /(2π¯ h) can be integrated to MωLx /(2π¯ h). Thus (16.288) gives, for each spin orientation, nF 1 X . ρ = Mω 2π¯ h n=0

(16.290)

The number of filled levels is ν = nF + 1. In the vacuum, the levels of one orientation are degenerate with those of the opposite orientation at a neighboring n (up to radiative corrections of the order α ≈ 1/137). This is due to the anomalous magnetic moment of the spin-1/2 electron being equal to one Bohr magneton µB = e¯ h/2Mc ≈ 0.927 × 10−20 erg/gauss (i.e., twice as large as classically expected, the factor 2 being caused by the relativistic Thomas precession). Due to the factor 2, the energy levels for the two orientations are split by ω¯ h, which is precisely equal to the energy difference between levels of neighboring n. In a solid material, however, the anomalous magnetic moment is strongly renormalized and the degeneracy is removed. There, every level has a definite spin orientation. According to Eq. (16.290), the highest level is occupied completely if each level has taken up a particle number corresponding to its maximal filling density ρmax =

Mω . 2π¯ h

(16.291)

At smaller magnetic fields, this density is small and the electrons are spread over many levels whose number ν is given by Mω ν. 2π¯ h

(16.292)

Bz hc 1 = . ρ e ν

(16.293)

ρ= Expressing ω in terms of Bz leads to

Using the flux quantum Φ0 = hc/e, this equation states that the magnetic flux per electron Φ≡

Bz ρ

(16.294)

has the value Φ 1 = . Φ0 ν

(16.295)

If the magnetic field is increased, the Landau levels can accommodate more electrons which then reside in a decreasing number ν of levels. By inserting into Eq. (16.286) the values of Bz at which the highest level becomes depleted, one obtains precisely the experimentally observed quantized Hall resistances Rxy =

h1 e2 ν

(16.296)

1062

16 Polymers and Particle Orbits in Multiply Connected Spaces ν eff

1/7

4

3

8/15

8

8/17

7/13

7

7/15

6/11

6

6/13

5/19

5/9

5

5/11

5/21

4/15

4/7

4

4/9

4/17

3/17

3/11

3/5

3

3/7

3/13

3/19

2/11

2/7

2/3

2

2/5

2/9

2/13

1/5

1/3

1/1

1

1/3

1/5

1/7

2

1

1

m

2

3

1/9

4

Figure 16.24 Values of parameter ν, at which plateaus in fractional quantum Hall resistance h/e2 ν are expected theoretically. The right-hand side shows the values ν eff /(2mν eff + 1), the left-hand side ν eff /(2mν eff − 1). The full circles indicate the values found experimentally.

with integer values of ν. The assumption of a statisto-magnetic interaction makes it possible to explain the fractional quantum Hall effect by reducing it to the ordinary quantum Hall effect. In the fractional quantum Hall effect, the magnetic field is so strong that even the lowest Landau level is only partially filled. This is why one did not expect any plateaus at all. According to a simple idea due to Jain, however, it is possible to relate the fractional plateaus to the integer plateaus. For this one assumes that the electrons in the ground state of the fractional quantum Hall effect carry an even statisto-magnetic flux −2mΦ0 due to the presence of a Chern-Simons action. For the wave function, this amounts to a statistical phase factor ei2πm under the exchange of two particle coordinates; it leaves the Fermi statistics of the electrons unchanged. Now one takes advantage of the observation made in the last section that the electrons cannot distinguish a statisto-magnetic field from an external magnetic field. They move in Landau orbits enforced by the combined field Bzeff = Bz − Bzstat ,

Bzstat = 2mΦ0 ρ.

(16.297)

The cyclotron frequency of the electrons in their Landau orbits is ω eff = eBzeff /Mc.

(16.298)

Since the effective field is now much smaller than the external field, the Landau levels possess a greatly reduced capacity. Thus the electrons must be distributed H. Kleinert, PATH INTEGRALS

1063

16.12 Anyonic Superconductivity

over several levels in spite of the large magnetic field. The number decreases as the field grows further. The steps appear at those places where the effective magnetic field has its integer quantum Hall plateaus, i.e., at the effective magnetic fields Bzeff = ±ρΦ0 /ν eff ,

ν eff = 1, 2, 3, . . . .

(16.299)

The values of νeff are related to the ν-values of the external magnetic field as follows: ±

1 1 = − 2m. eff ν ν

(16.300)

ν eff . 2mν eff ± 1

(16.301)

From this one has ν=

The resulting values of ν on the integer-valued plane spanned by the numbers m and ν eff are shown in Fig. 16.24. Only odd denominators are allowed. The values of ν found by this simple hypothesis agree well with those of the lower experimental levels (16.287).

16.12

Anyonic Superconductivity

At the end of Section 16.3 we have mentioned that an ensemble of particles with fractional statistics in 2+1 spacetime dimensions exhibits Meissner screening. This has given rise to speculations that the presently poorly understood phenomenon of high-temperature superconductivity may be explained by anyons physics. The new kind of superconductivity is observed in materials which contain pronounced layer structures, and it is conceivable that the currents move in these two-dimensional subspaces without dissipation. With some effort it can indeed be shown that in 2+1 dimensions a Chern-Simons action may be generated in principle10 by integrating out Fermi fields. Accepting this, we can easily derive that an addition of this action to the usual electromagnetic field action gives the magnetic field a finite range, i.e., a finite penetration depth. The usual electromagnetic action reads 1 A= 8π

Z

dtd3 x[E2 − (∇ × A)2 ],

(16.302)

where E is the electric field E=−

1 ∂A − ∇A0 . c ∂t

(16.303)

In the Euclidean formulation with x4 = ict, the action becomes Ae = 10

1 Z 4 d x[E2 + (∇ × A)2 ]. 8πc

See Notes and References at the end of the chapter.

(16.304)

1064


To add the Chern-Simons action, we restrict the spacetime dimensionality to 3. The restriction is imposed by considering a system in 4 spacetime dimensions and assuming it to be translationally invariant along the fourth coordinate direction x4 . Then there are no electric fields, and the Euclidean action becomes Ae =

L 8πc

Z

d3 x(∇ × A)2 ,

(16.305)

where L denotes the length of the system in the x4 -direction. To this, we now add the Chern-Simons action (16.253) and the current coupling (16.256). By extremizing the total action, we obtain the field equation: L e2 1 ∇ × (∇ × A) + i ∇ × A = i j. 4πc 2πc2 h ¯ µ0 c2

(16.306)

For the magnetic field B = ∇ × A, the equation reads L 1 (∇ × B + iλ−1 B) = i 2 j, 4πc c

(16.307)

hc = fine-structure where the parameter λ denotes the following length (α = e2 /¯ constant ≈ 1/137): λ≡

µ0 c¯ hµ0 L = L. 2e2 2α

(16.308)

By multiplying (16.307) vectorially with ∇ and using the equation once more, we obtain 1 1 L (−∇2 + λ−2 )B = i 2 ∇ × j + 2 λ−1 j . (16.309) 4πc c c In the current-free case, the magnetic field is seen to have only a finite penetration depth λ into the material. In an ordinary superconductor, this phenomenon is known as the Meissner effect. There it can be understood as a consequence of the induction of supercurrents in an ideal (i.e., incompressible and frictionless) liquid of charged particles, which lowers the invading magnetic field according to Lenz’ rule. In the absence of friction, there is a complete extinction. Recall that a superconductor with time-dependent currents and fields is governed by the characteristic London equation (see Appendix 16D) ∇ × j ∝ B.

(16.310)

For a two-dimensional superconductor, this amounts to (∇ × j⊥ )z ∝ Bz .

(16.311)

The above anyonic system shows a similar induction phenomenon. In the absence of currents, Eq. (16.309) determines the magnetic field Bz from the particle density by Bz = µ0 Φ0 ρ.


1065

16.13 Non-Abelian Chern-Simons Theory

If there are currents in the xy-plane j⊥ = (jx , jy ), the magnetic field is increased by ∆Bz in accordance with the equation L 1 (−∇2 + λ−2 )∆Bz = 2 (∇ × j⊥ )z . 4πc c

(16.313)

This is the desired relation between the magnetic field and the curl of the current which indicates the superconducting character of the system expressed before in the London equation (16.311). The contact with the London equation is established by a restriction to smooth field configurations in which the first term in (16.313) can be ignored. Thus we conclude that the currents and magnetic fields in a two-dimensional system of anyons show Meissner screening. This is not sufficient to make the system superconductive since it does not automatically imply the absence of dissipation. In a usual superconductor, the existence of an energy gap makes the dissipative part of the current-current correlation function vanish for wave vectors smaller than some value kc . This value determines the critical current strength above which superconductivity returns to normal. In the anyonic system, the absence of dissipation was proved in an approximation. Recent studies of higher corrections, however, have shown the presence of dissipation after all, destroying the hope for an anyonic superconductor.

16.13

Non-Abelian Chern-Simons Theory

The topological field interaction (16.153) can be generalized to nonabelian gauge groups. For the local symmetry group SU(N) it reads Ae,CS

2 k Z 3 d xǫijk tr N Ai ∇j Ak + Ai Aj Ak , = 4πi 3

(16.314)

where Ai are Hermitian traceless N × N-matrices and tr N denotes the associated trace. In the nonabelian theory, the gauge transformations are Ai ↔ UAi U −1 + i(∂i U)U −1 .

(16.315)

It can be shown that they transform Ae,CS as follows [10]: Ae,CS → Ae,CS + 2πink¯ h,

n = integer.

(16.316)

Thus, the action is not completely gauge-invariant. For integer values of k, however, the additional 2πink¯ h does not have any effect upon the phase factor e−Ae,CS /¯h in the path integral associated with the orbital fluctuations. Thus there is gauge invariance for integer values of k (in contrast to the abelian case where k is arbitrary). In the nonabelian theory, gauge fixing is a nontrivial issue. It is no longer possible to simply add a gauge fixing functional of the type (16.163). The reason is that the volume in the field space of gauge transformations depends on the gauge field. For

1066


a consistent gauge fixing, this volume has to be divided out of the gauge-fixing functional as was first shown by Fadeev and Popov [11]. For an adequate discussion of this interesting topic which lies beyond the scope of this quantum-mechanical text the reader is referred to books on quantum field theory. As in the abelian case, the functional derivative of the Chern-Simons action with respect to the vector potential gives the field strength Bi ≡ ǫijk Fjk ,

(16.317)

where Fij is the nonabelian version of the curl: Fij ≡ ∂i Aj − ∂j Ai − i[Ai , Aj ].

(16.318)

In 1989, Witten found an important result: The expectation value of a gaugeinvariant integral defined for any loop L, ˆ [A] ≡ tr N Pˆ ei WL [A] ≡ tr N W

H

L

dxA

,

(16.319)

the so-called Wilson loop integral , possesses a close relationship with the Jones polynomials of knot theory. The loop L can consist of several components linked in an arbitrary way, in which case the integral in WL [A] runs successively over all components. The operator Pˆ in front of the exponential function denotes the path-ordering operator . It is defined in analogy with the time-ordering operator Tˆ in (1.241): If the exponential function in (16.319) is expanded into a Taylor series, it specifies the order in which the N × N-matricesAi (x), which do not commute for different x and i, appear in the products. If the path is labeled by a “time parameter”, the earlier matrices stand to the right of the later ones. The fluctuations of the vector potential are controlled by the Chern-Simons action (16.314). Expectation values of the loop integrals are defined by the functional integral hWL [A]i ≡

R

DAi e−Ae,CS /¯h WL [A] R . DAi e−Ae,CS /¯h

(16.320)

To calculate the self-interaction of a loop, we proceed as described in the abelian case after Eq. (16.169) by spreading the line out into an infinitely thin ribbon of parallel lines. The borders of the ribbon are positioned in such a way that their linking number Lk vanishes. If 0 denotes a circle, i.e., a trivial knot, one can show that hW0 [A]i =

q N/2 − q −N/2 , q 1/2 − q −1/2

(16.321)

with q ≡ e−2πi/(N +k) .

(16.322)

For an arbitrary link L one finds the skein relation (see Appendix 16C) q N/2 hWL+ [A]i − q −N/2 hWL− [A]i = (q 1/2 − q −1/2 )hWL0 [A]i.

(16.323)


Calculation of Feynman Diagrams in Polymer Entanglement 1067

Appendix 16A

If N = 2, this agrees up to the sign of the right-hand side with the relation (16.114) for the Jones polynomials JL (t). For general values of N 6= 2, we obviously obtain with t = q −N/2 and α = q 1/2 − q −1/2 = −(t1/N − t−1/N ) the skein relations (16.122) of the HOMFLY polynomials. The important relation is hWL [A]i = HL (t, −(t1/N − t−1/N )), hW0 [A]i

t = eπi/k .

(16.324)

Since the second variable in HL (t, α) appears only in even or odd powers, HL (t, −(t1/2 − t−1/2 )) is a Jones polynomial up to a sign (−1)s+1 , where s is the number of loops in L. A favored choice of framing is one in which the self-linking number Lk of each component is equal to the twist number or writhe w introduced in Eq. (16.110). Then the ribbon lies flat on the projection plane of the knot. This framing can easily be drawn on the blackboard by splitting the line L into two parallel running lines; it is therefore called the blackboard framing. Incidentally, each choice of framing can be drawn as a blackboard framing if one adds to the loop L an appropriate number of windings via a Reidemeister move of type I. These are trivial for lines and nontrivial only for ribbons (see Fig. 16.6). In the blackboard framing, each such winding changes the values Lk and w simultaneously by one unit. Thus Lk = w can be brought to any desired value. Take, for example, the trefoil knot in Fig. 16.2. In the blackboard framing it has the self-linking number Lk = w = −3. This can be brought to zero by adding three windings via a Reidemeister move of type I.11 In the framing Lk = w, the right-hand side of (16.324) carries an extra phase factor cw , where c = e−i2π(N

2 −1)/2N k

.

(16.325)

For comparison: In the abelian Chern-Simons theory, the phase factor is c = e , and the expectation hWL [A]i has the value cΣi6=j Lkij for a link of several loops labeled by i with vanishing individual self-linking numbers Lki . In the framing Lki = wi , the value is cΣi6=j Lkij +Σi wi . The investigation of the properties of loops with nonabelian topological interactions is an interesting task of present-day research. −2πi/k

Appendix 16A

Calculation of Feynman Diagrams in Polymer Entanglement

For the calculation of the amplitudes N1 , . . . , N4 in Eqs. (16.226), (16.230), (16.235), and (16.236), we need the following simple tensor formulas involving two completely antisymmetric tensors εijl : εijk εimn = δjm δkn − δjn δkm ,

εijk εijl = 2δkl .

(16A.1)

The Feynman diagrams shown in Fig. 16.23 corresponds to integrals over products of the polymer correlation functions G0 defined in Eq. (16.213), which have to be integrated over space and 11

Mathematicians usually prefer another framing in which the ribbons lie flat on the so-called Seifert surfaces.

1068


Laplace transformed. For the latter we make use of the convolution property of the integral over two Laplace transforms f˜(z) and g˜(z) of the functions f, g: Z c+i∞ Z L M dz zL ˜ e f (z)˜ g(z) = dsf (s)g(L − S). (16A.2) 2πi 0 c−i∞ All spatial integrals are Gaussian of the form Z 2 2 2 d3 xe−ax +2bx·y = (2π)3/2 a−3/2 eb y /a ,

a > 0.

(16A.3)

Contracting the fields in Eq. (16.226), and keeping only the contributions which do not vanish in the limit of zero replica indices, we find with the help of Eqs. (16A.1) and (16A.2): Z Z L1 Z L2 Z N1 = d3 x1 , d3 x2 ds dt d3 x′1 d3 x′2 G0 (x1 −x′1 ; s)G0 (x′1 − x1 ; L1 − s) 0

0

× G0 (x2 −

x′2 ; t)G0 (x′2

− x2 ; L2 − t)

Performing the changes of variables s′ =

s , L1

t′ =

t , L2

x=

|x′1

l . − x′2 |4

x1 − x′1 √ , L1

(16A.4)

y=

x2 − x′2 √ , L2

(16A.5)

and setting x′′1 ≡√x′1 − x′2 , we√easily derive (16.227). For small ξ/ L1 and ξ/ L2 , we use the approximation (16.228). The space integrals can be done using the formula (16A.3). After some work we obtain the result (16.240). For the amplitude N2 in Eq. (16.230) we obtain likewise the integral Z Z N2 = d3 x1 d3 x2 d3 x′1 d3 x′′1 d3 x′2 "Z Z S L1 i × ds ds′ G0 (x′1 − x1 ; L1 − s) ∇jx′′ G0 (x1 − x′′1 ; s′ )∇ix′1 G0 (x′′1 − x′1 ; s − s′ ) 0

×

Dik (x′1

1

0

−

x2 )Djk (x′′1

−

x′2 )

"Z

0

L2

dt G0 (x2 −

x′2 ; L2

−

t)G0 (x′2

#

− x2 ; t) ,

(16A.6)

where√ Dij (x, x′ ) are the correlation functions √ (16.174) and (16.175) of the vector potentials. Setting be easily evaluated x2 ≡ L2 u + x′2 and supposing that ξ/ L2 is small, the integral over u can √ ′′ with the help of the Gaussian integral (16A.3). After the substitutions x = L1 y + x1 x′1 = 1 √ √ ′ L1 (y − x) + x1 , x2 = L1 (y − x − z) + x1 and a rescaling of the variables s, s′ by a factor L−1 1 , we derive Eq. (16.231) with (16.232). √ √ For small ξ/ L1 , ξ/ L2 , the spatial integrals are easily evaluated leading to: r √ Z t −1/2 −1/2 Z 1 − 2V L2 L−1 t − t′ ′ ′ 1 M N2 = dt dt t (1 − t) . (16A.7) (4n)6 1 − t + t′ 0 0

After the change of variable t′ → t′′ = t − t′ , the double integral is reduced to a sum of integrals the type r Z 1 Z t t′ m ′ ′n c(n, m) = dtt dt t , m, n = integers . 1 − t′ 0 0

These can be simplified by replacing tm by dtm+1 /dt(m + 1), and doing the integrals by parts. In this way, we end up with a linear combination of integrals of the form: Z 1 1 tκ+ 2 3 1 dt √ =B κ+ , . (16A.8) 2 2 1−t 0 The calculations of N3 and N4 are very similar, and are therefore omitted.


1069

Appendix 16B

Kauffman and BLM/Ho polynomials

Appendix 16B

Kauffman and BLM/Ho Polynomials

The Kauffman polynomials are given by F (a, x) = a−w Λ(a, x), where w is the writhe and Λ(a, x) satisfies the skein relation ΛL+ (a, x) + ΛL− (a, x) = x[ΛL0 (a, x) + ΛL∞ (a, x)].

(16B.1)

The subscripts refer to the same loop configurations as in Figs. 16.10 and 16.12. The trivial loop has Λ(a, x) =

a + a−1 − 1. z

(16B.2)

While the Kauffman polynomial is a knot invariant, the Λ-polynomial is only a ribbon invariant.12 If a winding LT + or LT − is removed from a loop with the help of a Reidemeister move of type I in Fig. 16.6 (which for infinitely thin lines would be trivial while changing the writhe of a ribbon by one unit) then Λ(a, x) receives a factor a or a−1 , respectively (see Fig. 16.25).

= a× LT +

= a−1 × LT 0

LT −

LT 0

Figure 16.25 Trivial windings LT + and LT − . Their removal by means of Reidemeister move of type I decreases or increases writhe w by one unit. The Kauffman polynomials arise from Wilson loop integrals of a nonabelian Chern-Simons theory, if the action (16.314) is SO(N )- rather than SU(N )-symmetric. A list of these polynomials can be found in papers by Lickorish and Millet and by Doll and Hoste quoted at the end of the chapter. The BLMHo polynomials are special cases of the Kauffman polynomials. The relation between them is Q(x) ≡ F (1, x).

Appendix 16C

Skein Relation between Wilson Loop Integrals

Here we sketch the derivation of the skein relation (16.323) for the expectation values of Wilson’s loop integrals (16.319). Let us decompose Ai in terms of the N 2 − 1 generators Ta of the group SO(N ): X Ai = Aai Ta . (16C.1) a

They satisfy the commutation rules [Ta , Tb ] = ifabc Tc . 12

(16C.2)

More precisely, F (a, x) is invariant under the three Reidemeister moves which, in the projected picture of the knot in Fig. 16.6, define the ambient isotopy, whereas Λ changes under the first Reidemeister move, associated only with regular isotopy. whereas

1070


For simplicity, we assume k to be very large so that we can restrict the treatment to the lowest order in 1/k. To avoid inessential factors of the constants e, c, ¯h, we set these equal to 1. Under a small variation of the fields one has Z ˆ L [A] δW ˆ ˆ L [A], = i P dx′i δ (3) (x − x′ )Ta (x′ )W (16C.3) δAai (x) L where the path-ordering operator Pˆ arranges the expression to its right in such a way that Ta is ˆ L at the correct path-ordered place. To emphasize this, we have recorded the position situated in W of Ta by means of an x-argument. More precisely, if we discretize the loop integral and write ˆ L [A] = eiAi (¯x1 )∆x1i eiAi (¯x2 )∆x2i · · · eiAi (¯xn )∆xni · · · , W

(16C.4)

¯ n are the midpoints of the intervals ∆xn , a differentiation with respect to one of the where x n n n n Ai (¯ x )-fields replaces the associated factor eiAi (¯xn )∆xi by iTa eiAi (¯x )∆xi . With the δ-function on a line L defined in Eq. (10A.8), we write (16C.3) as ˆ L [A] δW ˆ L [A]. = iPˆ δi (x, L)Ta (x)W δAai (x)

(16C.5)

For simplicity, we assume x to be only once traversed by the loop L. ˆ L changes by If the shape of the loop is deformed infinitesimally by dSi = ǫijk dxi d′ xj , then W ˆ L = idxi d′ xj Pˆ Fija (x)Ta (x)W ˆ L, δW

(16C.6)

where Fija are the N 2 − 1 components of the nonabelian field strengths Fij = ∂i Aj − ∂j Ai − i[Ai , Aj ]

(16C.7)

and x the midpoints of the parallelograms spanned by dx and d′ x. The derivation of Eq. (16C.6) is based on the observation that a change of the path by a small parallelogram adds to the line ˆ L a factor W ˆ , which is a Wilson loop integral around the small parallelogram. The integral W latter is evaluated as follows: ˆ W

′

′

′

′

=

eiAi (x−d x/2)dxi eiAj (x+dx/2)d xj e−iAi (x+d x/2)dxi e−iAj (x−dx/2)d xj

= ×

ei[Ai (x)dxi −∂j Ai (x)dxi d xj +...] ei[Aj (x)d xj +∂i Aj (x)dxi d xj +...] ′ ′ ′ e−i[Ai (x)dxi +∂j Ai (x)dxi d xj +...] e−i[Aj (x)d xj −∂i Aj (x)dxi d xj +...]

=

′

′

′

′

eiFij (x)dxi d xj .

(16C.8)

The last line is found with the help of the Baker-Hausdorff formula eA eB = eA+B+[A,B]/2+... (recall Appendix 2A). ˆ . Their N × N -traces ˆ L [A] by W Let us denote the Chern-Simons functional integral over W L ¯ are WL [A] and WL . The latter differs from the expectation hWL [A]i in (16.320) by not containing the normalizing denominator, i.e., Z W L ≡ DAe−Ae,CS WL [A]. (16C.9) This changes under the loop deformation by Z δW L = DAδWL [A]e−Ae,CS Z = idxi d′ xj DAFija (x)Ta (x)WL [A]e−Ae,CS ,

(16C.10)


Appendix 16C

Skein Relation between Wilson Loop Integrals

1071

with the tacit agreement that a generator Ta (x) written in front of the trace has to be evaluated within the trace at the correct path-ordered position. Now we observe that Fija can also be obtained by applying a functional derivative to the Chern-Simons action (16.314): i

4π δAe,CS ǫijk a = Fija (x). k δAk (x)

(16C.11)

This allows us to rewrite (16C.10) as Z Z 4π δAe,CS − DA dSi a Ta (x)WL [A]e−Ae,CS k δAi (x) and further as

4π k

Z

DAdSi Ta (x)WL

δ e−Ae,CS . δAai (x)

A partial functional integration produces Z Z 4π δWL [A] −Ae,CS − DA dSi Ta (x) e , k δAai (x) which brings the variation to the form Z 4πi δW L = − DAdSi δi (x, L)Ta (x)Ta (x)WL [A]e−Ae,CS . k

(16C.12)

The expectation of Wilson’s loop integral W L changes under a deformation only if the loop crosses another line element. This property makes W L a ribbon invariant, i.e., an invariant of regular isotopy. For a finite deformation, the right-hand side has to be integrated over the area S across which the line has swept. Using the integral formula Z 1 pierces S dSi δi (x, L) = if the line L , (16C.13) 0 misses S S we obtain for each crossing W L+ − W L− ≡ ∆W L

4πi =− k

Z

DATa (x)Ta (x)WL [A]e−Ae,CS .

(16C.14)

The two generators Ta (x) lie path-ordered on the different line pieces of the crossing. To establish contact with the knot polynomials, the left-hand sides have been labeled by the loop subscripts L+ and L− appearing in the skein relations of Fig. 16.114. The product of the generators on the right-hand side is the Casimir operator of the N × N -representation of SO(N ): (Ta )αβ (Ta )γδ =

1 1 δαδ δβγ − δαβ δγδ . 2 2N

(16C.15)

When inserted into Eq. (16C.14), we obtain the graphical relation:

. The second graph on the right-hand side can be decomposed into

1072


2 . Taking these two terms to the left-hand side of (16C.14), we obtain the skein relation πi 2πi πi 1− W L+ − 1 + W L− = − W L0 . Nk Nk k

(16C.16)

We now apply this relation to the windings displayed in Fig. 16.25. They decompose into a line and a circle. Due to the trace operation in W L0 , the circle contributes a factor N . Thus we obtain the relation πi 2πi πi W LT + − 1 + W LT − = − (16C.17) 1− N W LT 0 . Nk Nk k Now we remove on the left-hand side the windings according to the graphical rules of Fig. 16.25. Under this operation, the Wilson loop integral is not invariant. Like BLMHo polynomials, it acquires a factor a or a−1 : W LT + = aW LT 0 ,

W LT − = a−1 W LT 0 .

(16C.18)

To be compatible with (16C.17), the parameter a must satisfy a=1−

πi (N 2 − 1), Nk

a−1 = 1 +

πi (N 2 − 1). Nk

(16C.19)

Due to (16C.18), the Wilson loop integral is only a ribbon invariant exhibiting regular isotopy. A proper knot invariant which distinguishes ambient isotopy classes arises when multiplying W L by a−w . The polynomials HL ≡ e−w W L satisfy the skein relation πi 2πi πi 1− a HL + − 1 + a−1 HL− = − HL 0 . (16C.20) Nk Nk k The prefactors on the left-hand side can be written for large k as 1 − 2πiN/k ≈ q N/2 and 1 + 2πiN/k ≈ q −N/2 with q = 1 − 2πiπ/k. The prefactor on the right-hand side is equal to q 1/2 − q −1/2 . To leading order in 1/k, we have thus derived the skein relation (16.323) for the HOMFLY polynomials HL .

Appendix 16D

London Equations

Consider an ideal fluid of charged particles. By definition, it is non-viscous and incompressible, satisfying ∇ · v = 0. If the charge of the particles is e (which we take to be negative for electrons), the electric current density is j = ρev,

(16D.1)

where ρ is the particle density. The current is obviously conserved. The equation of motion of the particles in an electric and magnetic field is governed by the Lorentz force and reads 1 M v˙ = e E + v × B . (16D.2) c H. Kleinert, PATH INTEGRALS

Appendix 16D

1073

London Equations

Using the kinematic identity dv ∂v ∂v = + (v · ∇)v = +∇ dt ∂t ∂t

1 2 v 2

− v × (∇ × v),

this leads to the partial differential equation for the velocity field v(x, t) ∂v M 2 e +∇ v = eE + M v · ∇ × v + B . M ∂t 2 Mc

(16D.3)

(16D.4)

Consider the time dependence of the vector field on the right-hand side X=∇×v+

e B. Mc

(16D.5)

Using Maxwell’s equation ∂ B = −c∇ × E, ∂t

(16D.6)

∂ X = ∇ × (∇ × X). ∂t

(16D.7)

we derive

Suppose now that there is initially no B-field in the ideal fluid at rest which therefore has X ≡ 0 everywhere. If a magnetic field is turned on, Eq. (16D.7) guarantees that X remains zero at all times. This implies that ∇×j=−

ρe2 B, Mc

(16D.8)

which is the first London equation. By inserting the first London equation into Eq. (16D.4), we find the second London equation ∂ M 2 (16D.9) v+∇ v = eE. ∂t 2 If the vector potential is taken in the transverse gauge ∇ · A = 0 (which in this context is also called London gauge), then the first London equation can be solved and yields j=−

ρe2 A. Mc

(16D.10)

By inserting this equation into the Maxwell equation with no electric field E, we obtain 4π ρ4πe2 j=− A. c M c2

(16D.11)

∇ × (∇ × A) + λ−2 A = 0,

(16D.12)

∇×B= When rewritten in the form

with λ−2 =

ρ4πe2 , M c2

(16D.13)

the equation exhibits directly the finite penetration depth λ of a magnetic field into the fluid, the celebrated Meissner effect. It is the ideal manifestation of the Lenz rule, according to which an incoming magnetic field induces currents reducing the magnetic field — in the present case to extinction.

1074

Appendix 16E


Hall Effect in Electron Gas

A gas of electrons with a density ρ carries an electric current j = ρev.

(16E.1)

In a magnetic field, the particle velocities change due to the Lorentz force by 1 M v˙ = e E + v × B . c

(16E.2)

If σ0 denotes the conductivity of the system without a magnetic field, the electric current is obviously given by 1 j = σ0 E + v × B c 1 j×B . (16E.3) = σ0 E + ρec The second term shows the classical Hall resistance (16.286).

Notes and References For the Aharonov-Bohm effect, see the original work by Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). For a review see S. Ruijsenaars, Ann. Phys. (N.Y.) 146, 1 (1983). See also the papers A. Inomata and V.A. Singh, J. Math. Phys. 19, 2318 (1978); E. Corinaldesi and F. Rafeli, Am. J. Phys. 46, 1185 (1978); M.V. Berry, Eur. J. Phys. 1, 240 (1980); S. Ruijsenaars, Ann. Phys. 146, 1 (1983); G. Morandi and E. Menossi, Eur. J. Phys. 5, 49 (1984); R. Jackiw, Ann. Phys. 201, 83 (1990); and in “M. A. B. Bég Memorial Volume” (A. Ali and P. Hoodbhoy, Eds.), World Scientific, Singapore, 1991; G. Amelino-Camelia, Phys. Lett. B 326, 282 (1994); Phys. Rev. D 51, 2000 (1995); C. Manuel and R. Tarrach, Phys. Lett. B 328, 113 (1994); S. Ouvry, Phys. Rev. D 50, 5296 (1994); C.R. Hagen, Phys. Rev. D 31, 848 (1985); D 52 2466 (1995); P. Giacconi, F. Maltoni, and R. Soldati, Phys. Rev. D 53, 952 (1996); R. Jackiw and S.-Y. Pi, Phys. Rev. D 42, 3500 (1990); O. Bergman and G. Lozano, Ann. Phys. (N.Y.) 229, 416 (1994); M. Boz, V. Fainberg, and N.K. Pak, Phys. Lett. A 207,1 (1995); Ann. Phys (N.Y.) 246, 347 (1996); M. Gomes, J.M.C. Malbouisson, and A.J. da Silva, Phys. Lett. A 236, 373 (1997); Int. J. Mod. Phys. A 13, 3157 (1998); (hep-th/0007080). Path integrals in multiply connected spaces and their history are discussed in the textbook L.S. Schulman, Techniques and Applications of Path Integration, Wiley, New York, 1981. Details on Lippmann-Schwinger equation is found in most standard textbooks, say S.S. Schweber, Relativistic Quantum Field Theory, Harper and Row, New York, 1961, Section 11b. In chemistry, the properties of self-entangled polymer rings, called catenanes, were first investigated by H. Kleinert, PATH INTEGRALS


1075

H.L. Frisch and E. Wasserman, J. Am. Chem. Soc. 83, 3789 (1961). Their existence was proved mass-spectroscopically by R. Wolovsky, J. Am. Chem. Soc. 92, 2132 (1961); D.A. Ben-Efraim, C. Batich, and E. Wasserman, J. Am. Chem. Soc. 92, 2133 (1970). In optics, the Kirchhoff diffraction formula can be rewritten as a path integral formula with linking terms: J.H. Hannay, Proc. Roy. Soc. Lond. A 450, 51 (1995), In biophysics, J.C. Wang, Accounts Chem. Res. D 10, 2455 (1974) showed that DNA molecules can get entangled and must be disentangled during replication. The path integral approach to the entanglement problem in polymer systems was pioneered by S.F. Edwards, Proc. Phys. Soc. 91, 513 (1967); S.F. Edwards, J. Phys. A 1, 15 (1968). See also M.G. Brereton and S. Shaw, J. Phys. A 13, 2751 (1980) and later works of these authors. Investigations via Monte Carlo simulations were made by A.V. Vologodskii, A.V. Lukashin, M.D. Frank-Kamenetskii, and V.V. Anshelevin, Sov. Phys. JETP 39, 1095 (1974); A.V. Vologodskii, A.V. Lukashin, and M.D. Frank-Kamenetskii, Sov. Phys.-JETP 40, 932 (1975). See also the review article by M.D. Frank-Kamenetskii and A.V. Vologodskii, Sov. Phys. Usp. 24, 679 (1982). This article also discussed ribbons. For further computer work on knot distributions see J.P.J. Michels and F.W. Wiegel, Phys. Lett. A 9, 381 (1982); Proc. Roy. Soc. A 403, 269 (1986), and references therein. The work is summarized in the textbook by F.W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer Science, World Scientific, Singapore, 1986. See also S. Windwer, J. Chem. Phys. 93, 765 (1990). The parameter C at the end of Section 6.4 was found by A. Kholodenko, Phys. Lett. A 159, 437 (1991), who mapped the problem onto a q-state Potts model with q = 4. This mapping gives α = 0 and C = 2e−π/6 ≈ 1.18477. For the Gauss integral as a topological invariant of links see the original paper by G.F. Gauss, Koenig. Ges. Wiss. Goettingen 5, 602 (1877). The writhing number Wr was introduced by F.B. Fuller, Proc. Nat. Acad. Sci. USA 68, 815 (1971), who applied the mathematical relation to DNA. See also F.H.C. Crick, Proc. Nat. Acad. Sci. USA 68, 2639 (1976). The relation Lk = Tw + Wr was first written down by G. Calagareau, Rev. Math. Pur. et Appl. 4, 58 (1959); Czech. Math. J. 4, 588 (1961), and extended by J.H. White, Am. J. Math. 90, 1321 (1968). In particle physics, ribbons are used to construct path integrals over fluctuating fermion orbits: A.M. Polyakov, Mod. Phys. Lett. A 3, 325 (1988). For more details see

1076


C.H. Tze, Int. J. Mod. Phys. A 3, 1959 (1988). The construction of the Alexander polynomial of links is described in A.V. Vologodskii, A.V. Lukashin, and M.D. Frank-Kamenetskii, JETP 40, 932 (1974). Their derivation from the skein relations is shown in J.H. Conway, An Enumeration of Knots and Links, Pergamon, London, 1970, pp. 329–358; L.H. Kauffman, Topology 20, 101 (1981). In the mathematical literature, the various knot polynomials are discussed by L.H. Kauffman, Topology 26, 395 (1987); Contemporary Mathematics AMS 78, 283 (1988); Trans. Amer. Math. Soc. 318, 417 (1990); On Knots, Princeton University Press, Princeton, 1987; Knots and Physics, World Scientific, Singapore, 1991; J. Math. Phys. 36, 2402 (1995). V. Jones, Bull. Am. Math. Soc. 12, 103 (1985); Ann. Math. 126, 335 (1987); P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K.C. Millet, and A. Ocneanu, Bull. Am. Math. Soc. 12, 239 (1985); W. B. R. Lickorish and K.C. Millet, Math. Magazine 61, 3 (1987). The lower HOMFLY polynomials are tabulated in the text. For the higher ones see the microfilm accompanying the article H. Doll and J. Hoste, Math. of Computation 57, 747 (1991) and the unpublished tables by M.B. Thistlethwaite, University of Knoxville, Tennessee. The author is grateful for a copy of them. A collection of many relevant articles is found in T. Kuhno (ed.), New Developments in the Theory of Knots, World Scientific, Singapore, 1990. A short introduction to the classification problem of knots is given in the popular articles W.F.R. Jones, Scientific American, November 1990, p. 52, I. Stewart, Spektrum der Wissenschaft, August 1990, p. 12. The Chern-Simons actions have in recent years received increasing attention due to their relevance for explaining the fractional quantum Hall effect and a possible presence in high-temperature superconductivity . Actions of this type were first observed in four-dimensional quantum field theories in the form of so-called anomalies by J. Wess and B. Zumino, Phys. Lett. B 36, 95 (1971). The action (16.253) in three spacetime dimensions was first analyzed by S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. 140, 372 (1982), who pointed out the connection with the Chern classes of differential geometry described by S. Chern, Complex Manifolds without Potential Theory, Springer, Berlin, 1979. In particular they found the mass of the electromagnetic field which was conjectured to be the origin of the Meissner effect in high-temperature superconductors. See A.L. Fetter, C. Hanna, and R.B. Laughlin, Phys. Rev. B 39, 9679 (1989); Y.-H. Chen and F. Wilczek, Int. J. Mod. Phys. B 3, 117 (1989); Y.-H. Chen, F. Wilczek, E. Witten, and B.I. Halperin, Int. J. Mod. Phys. B 3, 1001 (1989); A. Schakel, Phys. Rev. D 44, 1198 (1992). However, the recent finding of dissipation in anyonic systems by D.V. Khveshchenko and I.I. Kogan, Int. J. Mod. Phys. B 5, 2355 (1991) speaks against an anyon mechanism of this phenomenon. A Chern-Simons type of action appeared when integrating out fermions in H. Kleinert, Fortschr. Phys. 26, 565 (1978) (http://www.physik.fu-berlin.de/~kleinert/55). The relation with Chern classes was recognized by M.V. Berry, Proc. Roy. Soc. A 392, 45 (1984); B. Simon, Phys. Rev. Lett. 51, 2167 (1983). The Chern-Simons action in the text was derived for a degenerate electron liquid in two dimensions by H. Kleinert, PATH INTEGRALS

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T. Banks and J.D. Lykken, Nucl. Phys. B 336, 500 (1990); S. Randjbar-Daemi, A. Salam, and J. Strathdee, Nucl. Phys. B 340, 403 (1990), P.K. Panigrahi, R. Ray, and B. Sakita, Phys. Rev. B 42, 4036 (1990). There is related work by M. Stone, Phys. Rev. D 33, 1191 (1986); I.J.R. Aitchison, Acta Physica Polonica B 18, 207 (1987). See also the reprints of many papers on this subject in A. Shapere and F. Wilczek, Geometric Phases in Physics, World Scientific, Singapore, 1989. F. Wilczek, Fractional Statistics and Anyon Superconductivity, World Scientific, Singapore, 1990, which itself provides a clear introduction to the subject and contains many important reprints. A good review is also contained in the lectures J.J. Leinaas, Topological Charges in Gauge Theories, Nordita Preprint, 79/43, ISSN 0106-2646. Textbooks on this subject are A. Lerda, Anyons-Quantum Mechanics of Particles with Fractional Statistics, Lecture Notes in Physics, m14, Springer, Berlin 1992; A. Khare, Fractional Statistics and Quantum Theory, World Scientific, Singapore, 1997. The Lerda book contains many useful examples and explains the origin of difficulties in treating interacting anyons. The Khare book provides a well-motivated treatment and includes a brief introduction to the Braid group. Both include discussions of the Quantum Hall Effect and Anyon Superconductivity. For the relation between the Chern-Simons theory and knot polynomials see E. Witten, Comm. Math. Phys. 121, 351 (1989), Nucl. Phys. B 330, 225 (1990). See also P. Cotta-Ramusino, E. Guadagnini, M. Martellini, and M. Mintchev, Nucl. Phys. B 330, 557 (1990); G.V. Dunne, R. Jackiw, and C. Trugenburger, Ann. Phys. 194, 197 (1989); A. Polychronakos, Ann. Phys. 203, 231 (1990); E. Guadagnini, I. J. Mod. Phys. A 7, 877 (1992). The integer quantum Hall effect was found by K. vonKlitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980); the fractional one by D.C. Tsui, H.L. Stormer, and A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1980). Theoretical explanations are given in R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983), Phys. Rev. B 23, 3383 (1983); F.D.M. Haldane, Phys. Rev. Lett. 51, 605 (1983); B.I. Halperin, Phys. Rev. Lett. 52, 1583 (1984); D.P. Arovas, J.R. Schrieffer, F. Wilczek, Phys. Rev. Lett. 53, 722 (1984); D.P. Arovas, J.R. Schrieffer, F. Wilczek, and A. Zee, Nucl. Phys. B 251, 117 (1985); J.K. Jain, Phys. Rev. Lett. 63, 199 (1989). The exceptional filling factor ν = 52 is discussed in R. Willet et al., Phys. Rev. Lett. 59, 17765 (1987); S. Kivelson, C. Kallin, D.P. Arovas, and J.R. Schrieffer, Phys. Rev. Lett. 56, 873 (1986). The Chern-Simons path integral is treated semiclassically in D.H. Adams, Phys. Lett. B 417, 53 (1998) (hep-th/9709147). For a simple discussion of the change from Bose to Fermi statistics at the level of creation and annihilation operators via a topological interaction see E. Fradkin, Phys. Rev. Lett. 63, 322 (1989); Field Theories of Condensed Matter Physics, Addison-Wesley, 1991. The lattice form of the action

P

µ

x ǫµνλ A

(x)∇ν Aλ (x) used by that author is not correct since

1078


it violates gauge invariance. This can, however, easily be restored without destroying the results by replacing the first Aµ (x)-field by Aµ (x − eµ ), where eµ is the unit vector in the µ-direction. See the general discussion of lattice gauge transformations in H. Kleinert, Gauge Fields in Condensed Matter , Vol. I, World Scientific, Singapore, 1989, Chapter 8 (http://www.physik.fu-berlin.de/~kleinert/b1). Also D. Eliezer and G.W. Semenoff, Anyonization of Lattice Chern-Simons Theory, Ann. Phys. 217, 66 (1992). For the London equations see the original paper by F. London and H. London, Proc. Roy. Soc. A 147, 71 (1935) and the extension thereof: A.B. Pippard, ibid., A 216, 547 (1953). The individual citations refer to

[1] P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931); Phys. Rev. 74, 817 (1948), Phys. Rev. 74, 817 (1948). See also J. Schwinger, Particles, Sources and Fields, Vols. 1 and 2, Addison Wesley, Reading, Mass., 1970 and 1973.

[2] For a review, see G. Giacomelli, in Monopoles in Quantum Field Theory, World Scientific, Singapore, 1982, edited by N.S. Craigie, P. Goddard, and W. Nahm, p. 377. H. Kleinert, PATH INTEGRALS


1079

[3] B. Cabrera, Phys. Rev. Lett. 48, 1378 (1982). [4] For a detailed discussion of the physics of vortex lines in superconductors, see H. Kleinert, Gauge Fields in Condensed Matter , World Scientific, Singapore, 1989, Vol. I, p. 331 (http://www.physik.fu-berlin.de/~kleinert/b1). [5] See the reprint collection A. Shapere and F. Wilczek, Geometric Phases in Physics, World Scientific, Singapore, 1989. In partcular the paper by D.P. Arovas, Topics in Fractional Statistics, p.284. [6] The Tait number or writhe must not be confused with the writhing number Wr introduced in Section 16.6 which is in general noninteger. See P.G. Tait, On Knots I, II, and III , Scientific Papers, Vol. 1. Cambridge, England: University Press, pp. 273-347, 1898. [7] See Ref. [7] of Chapter 19. There exists, unfortunately, no obvious extension to four spacetime dimensions. [8] The development in this Section follows F. Ferrari, H. Kleinert, and I. Lazzizzera, Phys. Lett. A 276, 1 (2000) (cond-mat/0002049); Eur. Phys. J. B 18, 645 (2000) (cond-mat/0003355); nt. Jour. Mod. Phys. B 14, 3881 (2000) (cond-mat/0005300); Topological Polymers: An Application of Chern-Simons Field Theories, in K. Lederer and N. Aust (eds.), Chemical and Physical Aspects of Polymer Science and Engineering, 5-th Oesterreichische Polymertage, Leoben 2001, Macromolecular Symposia, 1st Edition, ISBN 3-527-30471-1, Wiley-VCH, Weinheim, 2002. [9] M.G. Brereton and S. Shah, J. Phys. A: Math. Gen. 15 , 989 (1982). [10] R. Jackiw, in Current Algebra and Anomalies, ed. by S.B. Treiman, R. Jackiw, B. Zumino, and E. Witten, World Scientific, Singapore, 1986, p. 211. [11] L.D. Faddeev and V.N. Popov, Phys. Lett. B 25 , 29 (1967).

H. Kleinert, PATH INTEGRALS June 16, 2007 ( /home/kleinert/kleinert/books/pathis/pthic17.tex) So many paths, that wind and wind, While just the art of being kind Is all the sad world needs. Ella Wilcox (1855-1919), The World’s Needs

17 Tunneling Tunneling processes govern the decay of metastable atomic and nuclear states, as well as the transition of overheated or undercooled thermodynamic phases to a stable equilibrium phase. Path integrals are an important tool for describing these processes theoretically. For high tunneling barriers, the decay proceeds slowly and its properties can usually be explained by a semiclassical expansion of a simple model path integral. By combining this expansion with the variational methods of Chapter 5, it is possible to extend the range of applications far into the regime of low barriers. In this chapter we present a novel theory of tunneling through high and low barriers and discuss several typical examples in detail. A useful fundamental application arises in the context of perturbation theory since the large-order behavior of perturbation expansions is governed by semiclassical tunneling processes. Here the new theory is used to calculate perturbation coefficients to any order with a high degree of accuracy.

17.1

Double-Well Potential

A simple model system for tunneling processes is the symmetric double-well potential of Eq. (5.78). It may be rewritten in the form V (x) =

ω2 (x − a)2 (x + a)2 , 8a2

(17.1)

which exhibits the two degenerate symmetric minima at x = ±a (see Fig. 17.1). The coupling strength is g = ω 2 /2a2 .

(17.2)

Near the minima, the potential looks approximately like a harmonic oscillator potential V± (x) = ω 2 (x ∓ a)2 /2: ω2 x∓a V (x) = (x ∓ a)2 1 ± + . . . ≡ V± (x) + ∆V± (x) + . . . . 2 a

1080

(17.3)

1081

17.1 Double-Well Potential V (x ) 0.4

0.3

0.2

0.1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

Figure 17.1 Plot of symmetric double-well potential V (x) = (x − a)2 (x + a)2 ω 2 /8a2 for ω = 1 and a = 1.

The height of the potential barrier at the center is Vmax

(ωa)2 = . 8

(17.4)

In the limit a → ∞ at a fixed frequency ω, the barrier height becomes infinite and the system decomposes into a sum of two independent harmonic-oscillator potentials widely separated from each other. Correspondingly, the wave functions of the system should tend to two separate sets of oscillator wave functions ψn (∆x± ) →

ω π¯ h

1/4

q 1 2 √ e−ω(∆x± ) /2¯h Hn (∆x± ω/¯ h), 2n/2 n!

(17.5)

where the quantities ∆x± ≡ x ± a

(17.6)

measure the distances of the point x from the respective minima. A similar separation occurs in the time evolution amplitude which decomposes into the sum of the amplitudes of the individual oscillators a→∞

(xb tb |xa ta ) − −−→ (xb tb |xa ta )− + (xb tb |xa ta )+ (17.7) Z t Z i b 1 2 ≡ Dx(t) exp dt [x˙ − ω 2(x + a)2 ] + (a → −a). h ¯ ta 2 For convenience, we have assumed a unit particle mass M = 1 in the Lagrangian of the system: L=

x˙ 2 − V (x). 2

(17.8)

1082

17 Tunneling

If a is no longer infinite, a particle in either of the two oscillator wells has a nonvanishing amplitude for tunneling through the barrier to the other well, and the wave functions of the right- and left-hand oscillators are mixed with each other. Since the action is symmetric under the mirror reflection x → −x, the solutions of the Schrödinger equation ˆ h∂t ψ(x, t), Hψ(x, t) = H(−i∂x , x)ψ(x, t) = i¯

(17.9)

with the Hamiltonian H(p, x) =

p2 + V (x), 2

(17.10)

can be separated into symmetric and antisymmetric wave functions. As usual, the symmetric states have a lower energy than the antisymmetric ones since a smaller number of nodes implies less kinetic energy for the particles. If the distance parameter a is very large, then, to leading order in a → ∞, the lowest two wave functions coincide approximately with the symmetric and antisymmetric combinations of the harmonic-oscillator wave functions 1 ψs,a ≈ √ [ψ0 (x − a) ± ψ0 (x + a)]. 2

(17.11)

Due to tunneling, the lowest two energies show some deviation from the harmonic ground state value 1 (0) (0) ¯ ω + ∆Es,a Es,a = h . 2

(17.12)

At a large distance parameter a, this deviation is very small. In quantum mechanics, the level shifts ∆Es,a can be calculated in lowest-order perturbation theory by inserting the approximate wave functions (17.11) into the formula Z

∆Es,a =

ˆ s,a . dxψs,a Hψ

(17.13)

Since the wave functions ψ0 (x ± a) of the individual potential wells fall off expo2 nentially like e−x /2 at large x, the level shifts ∆Es,a are exponentially small in the square distance a2 . In this chapter we derive the level shifts ∆Es , ∆Ea and the related tunneling amplitudes from the path integral of the system. For large a, this will be relatively simple since we can have recourse to the semiclassical approximation developed in Chapter 4 which becomes exact in the limit a → ∞. As long as we are interested only in the lowest two states, the problem can immediately be simplified. We take the spectral representation of the amplitude (xb tb |xa ta ) = =

Z

(i/¯ h)

Dx(t)e

X

R tb ta

dt[x˙ 2 /2−V (x)]

ψn (xb )ψn (xa )e−iEn (tb −ta )/¯h

(17.14)

n


1083

17.2 Classical Solutions — Kinks and Antikinks

to imaginary times ta,b → τa,b = ∓iL/2, where it becomes (xb L/2|xa − L/2) = =

Z

−(1/¯ h)

Dx(τ )e

X

R L/2

−L/2

dτ [x′2 /2+V (x)]

(17.15)

ψn (xb )ψn (xa )e−En L/¯h ,

n

with the notation x′ (τ ) ≡ dx(τ )/dτ . In the limit of large L, the spectral sum (17.15) is obviously most sensitive to the lowest energies, the contributions of the higher energies En being suppressed exponentially. Thus, to calculate the small level shifts of the two lowest states, ∆Es,a , we have only to find the leading and subleading exponential behaviors. Since the wave functions are largest close to the bottoms of the double well at x ∼ ±a, we may consider the amplitudes with the initial and final positions xa and xb lying precisely at the bottoms, once on the same side of the potential barrier, (a L/2|a − L/2) = (−a L/2| −a −L/2),

(17.16)

and once on the opposite sides (a L/2| −a −L/2) = (−a L/2|a −L/2).

(17.17)

For these amplitudes we now calculate the semiclassical approximation in the limit L → ∞. The results will lead to level shift formula in Section 17.7.

17.2

Classical Solutions — Kinks and Antikinks

According to Chapter 4, the leading exponential behavior of the semiclassical approximation is obtained from the classical solutions to the path integral. The fluctuation factor requires the calculation of the quadratic fluctuation correction. The result has the form X

class. solutions

exp{−Acl /¯ h} × F ,

(17.18)

where Acl denotes the action of each classical solution and F the fluctuation factor. The amplitude (17.16), which contains the bottom of the same well on either side, is dominated by a trivial classical solution which remains all the time at the same bottom: x(τ ) ≡ ±a.

(17.19)

Classical solutions exist also for the other amplitudes (17.17) which connect the different bottoms at −a and a. These solutions cross the barrier and read, in the limit L → ∞, x(τ ) = x± cl (τ ) ≡ ±a tanh[ω(τ − τ0 )/2],

(17.20)

1084

17 Tunneling τ 4 2

-2

2

x

-2 -4 10V (x)

Figure 17.2 Classical kink solution (solid curve) in double-well potential (short-dashed curve with units marked on the lower half of the vertical axis). The solution connects the two degenerate maxima in the reversed potential. The long-dashed curve shows a solution which starts out at a maximum and slides down into the adjacent abyss.

with an arbitrary parameter τ0 specifying the point on the imaginary time axis where the crossing takes place. The crossing takes place within a time of the order of 2/ω. For large positive and negative τ , the solution approaches ±a exponentially (see Fig. 17.2). Alluding to their shape, the solutions x± cl (τ ) are called kink and antikink 1 solutions, respectively. To derive these solutions, consider the equation of motion in real time, x¨(t) = −V ′ (x(t)),

(17.21)

where V ′ (x) ≡ dV (x)/dx. In the Euclidean version with τ = −it, this reads x′′ (τ ) = V ′ (x(τ )).

(17.22)

Since the differential equation is of second order, there is merely a sign change in front of the potential with respect to the real-time differential equation (17.21). The Euclidean equation of motion corresponds therefore to a usual equation of motion of a point particle in real time, whose potential is turned upside down with respect to Fig. 17.1. This is illustrated in Fig. 17.3. The reversed potential allows obviously for a classical solution which starts out at x = −a for τ → −∞ and arrives at x = a for τ → +∞. The particle needs an infinite time to leave the initial potential mountain and to climb up to the top of the final one. The movement through the central valley proceeds within the finite time ≈ 2/ω. If the particle does not start 1

In field-theoretic literature, such solutions are also referred to as instanton or anti-instanton solutions, since the valley is crossed within a short time interval. See the references quoted at the end of the chapter. H. Kleinert, PATH INTEGRALS

1085

17.2 Classical Solutions — Kinks and Antikinks

φ

−V (φ)

Figure 17.3 Reversed double-well potential governing motion of position x as function of imaginary time τ .

its movement exactly at the top but slightly displaced towards the valley, say at x = −a + ǫ, it will reach x = a − ǫ after a finite time, then return to x = −a + ǫ, and oscillate back and forth forever. In the limit ǫ → 0, the period of oscillation goes to infinity and only a single crossing of the valley remains. To calculate this movement, the differential equation (17.22) is integrated once after multiplying it by x′ = dx/dτ and rewriting it as d 1 d ′2 x = V (x(τ )). 2 dτ dτ

(17.23)

x′2 + [−V (x(τ ))] = const . 2

(17.24)

The integration gives

If τ is reinterpreted as the physical time, this is the law of energy conservation for the motion in the reversed potential −V (x). Thus we identify the integration constant in (17.24) as the total energy E in the reversed potential: const ≡ E.

(17.25)

Integrating (17.24) further gives 1 τ − τ0 = ± √ 2

Z

dx

x(τ )

x(τ0 )

q

E + V (x)

.

(17.26)

A look at the potential in Fig. 17.3 shows that an orbit starting out with the particle at rest for τ → −∞ must have E = 0. Inserting the explicit potential (17.1) into (17.26), we obtain for |x| < a 2a x dx′ 1 a+x = ± log ω 0 (a − x′ )(x′ + a) ω a−x 2 x = ± arctanh . ω a

τ − τ0 = ±

Z

(17.27)

1086

17 Tunneling

Thus we find the kink and antikink solutions crossing the barrier: xcl (τ ) = ±a tanh[(τ − τ0 )ω/2].

(17.28)

The Euclidean action of such a classical object can be calculated as follows [using (17.24) and (17.25)]: Acl =

x′ 2 dτ cl + V (xcl (τ )) = 2 −∞

Z

∞

"

= −EL +

#

Z

a

−a

Z

∞

dτ (x′2cl − E)

−∞

q

dx 2[E + V (x)].

(17.29)

The kink has E = 0, so that q

2[E + V (x)] =

ω 2 (a − x2 ), 2a

(17.30)

and the classical action becomes ω Za 2 2 ω3 2 2 Acl = . dx(a − x ) = a ω = 2a −a 3 3g

(17.31)

Note that for E = 0, the classical action is also given by the integral Acl =

Z

∞

−∞

dτ x′cl 2 .

(17.32)

There are also solutions starting out at the top of either mountain and sliding down into the adjacent exterior abyss, for instance (see again Fig. 17.3) 2a Z ∞ dx′ 1 x+a = ± log ′ ′ ω x (x − a)(x + a) ω x−a 2 x = ± arccoth . ω a

τ − τ0 = ∓

(17.33)

However, these solutions cannot connect the bottoms of the double well with each other and will not be considered further. Being in the possession of the classical solutions (17.19) and (17.28) with a finite action, we are now ready to write down the classical contributions to the amplitudes (17.16) and (17.17). According to the semiclassical formula (17.18), they are (a L/2|a − L/2) = 1 × Fω (L)

(17.34)

(a L/2| − a − L/2) = e−Acl /¯h × Fcl (L).

(17.35)

and

The factor 1 in (17.34) emphasizes the vanishing action of the trivial classical solution (17.19). The exponential e−Acl /¯h contains the action of the kink solutions (17.28). H. Kleinert, PATH INTEGRALS

1087

17.3 Quadratic Fluctuations

The degeneracy of the solutions in τ0 is accounted for by the fluctuation factor Fcl (L), as will be shown below. Actually, the classical kink and antikink solutions (17.28) do not occur exactly in (17.35) since they reach the well bottoms at x = ±a only at infinite Euclidean times τ → ±∞. For the amplitude to be calculated we need solutions for which x is equal to ±a at large but finite values τ = ±L/2. Fortunately, the error can be ignored since for large L the kink and antikink solutions approach ±a exponentially fast. As a consequence the action of a proper solution which would reach ±a at a finite L differs from the action Acl only by terms which tend to zero like e−ωL . Since we shall ultimately be interested only in the large-L limit we can neglect such exponentially small deviations. In the following section we determine the fluctuation factors Fcl .

17.3

Quadratic Fluctuations

The semiclassical limit includes the effects of the quadratic fluctuations. These are obtained after approximating the potential around each minimum by a harmonic potential and keeping only the lowest term in the expansion (17.3). The fluctuation factor of a pure harmonic oscillator of frequency ω and unit mass has been calculated in Section 2.3 with the result Fω (L) =

r

ω ∼ 2π¯ h sinh ωL

r

ω −ωL/2 + O(e−3ωL/2 ). e π¯ h

(17.36)

The leading exponential at large L displays the ground state energy ω/2, while the corrections contain all information on the exited states whose energy is (n + 1/2)ω with n = 1, 2, 3, . . . . Note q that according to the spectral representation of the amplitude (17.15), the factor ω/π¯ h in (17.36) must be equal to the square of the ground state wave function Ψ0 (∆x± ) at the potential minimum. This agrees with (17.5). Consider now the fluctuation factor of a single kink contribution. It is given by the path integral over the fluctuations y(τ ) ≡ δx(τ ) Fcl (L) =

Z

−(1/¯ h)

Dy(τ )e

R L/2

−L/2

′′

dτ (1/2)[y ′ 2 +V (xcl (τ ))y 2 ]

,

(17.37)

where xcl (τ ) is the kink solution, and y(τ ) vanishes at the endpoints: y(L/2) = y(−L/2) = 0.

(17.38)

Suppose for the moment that L = ∞. Then the kink solution is given by (17.28) and we obtain the fluctuation potential ′′

3 ω2 2 1 2 1 2 2 3 x (τ ) − ω = ω tanh [ω(τ − τ )/2] − 0 cl 2 a2 2 2 ! 2 3 1 = ω2 1 − . 2 cosh2 [ω(τ − τ0 )/2]

V (xcl (τ )) =

(17.39)

1088

17 Tunneling

Figure 17.4 Potential (17.41) for quadratic fluctuations around kink solution (17.28) in Schr¨ odinger equation. The dashed lines indicate the bound states at energies 0 and 3ω 2 /4.

Thus, the quadratic fluctuations are governed by the Euclidean action A0fl

"

!

#

1 3 1 = dτ y ′2 + ω 2 1 − y2 . 2 2 2 cosh [ω(τ − τ0 )/2] −L/2 Z

L/2

(17.40)

The rules for doing a functional integral with a quadratic exponent were explained in Chapter 2. The paths y(τ ) are expanded in terms of eigenfunctions of the differential equation "

d2 3 1 − 2 + ω2 1 − 2 dτ 2 cosh [ω(τ − τ0 )/2]

!#

yn (τ ) = λn yn (τ ),

(17.41)

with λn being the eigenvalues. This is a Schrödinger equation for a particle moving along the τ -axis in an attractive potential well of the Rosen-Morse type [compare (14.156) and see Fig. 17.4]:

V (τ ) = ω

2

!

3 1 1− . 2 2 cosh [ω(τ − τ0 )/2]

(17.42)

The eigenfunctions yn (τ ) satisfy the usual orthonormality condition Z

∞

−∞

dτ yn (τ )yn′ (τ ) = δnn′ .

(17.43)

Given a complete set of these solutions yn (τ ) with n = 0, 1, 2, . . . , we now perform the normal-mode expansion y ξ0 ,ξ1 ,... (τ ) =

∞ X

ξn yn (τ ).

(17.44)

n=0


1089


After inserting this into (17.40), we perform a partial integration in the kinetic term and the τ -integrals with the help of (17.43), and the Euclidean action of the quadratic fluctuations takes the simple form ∞ 1X A= λn ξn2 . 2 n=0

(17.45)

With this, the fluctuation factor (17.37) reduces to a product of Gaussian integrals over the normal modes Fcl (L) = N

" ∞ Z Y

∞

−∞

n=0

#

P∞ 2 dξ 1 √ n e− n=0 ξn λn /2¯h = N √ . Πn λn 2π¯ h

(17.46)

The normalization constant N , to be calculated below, accounts for the Jacobian which relates the time-sliced measure to the normal mode measure. First we shall calculate the eigenvalues λn . For this we use the amplitude of the Rosen-Morse potential obtained via the Duru-Kleinert transformation (14.159) in Eq. (14.160) [1]. If the potential is written in the form V (τ ) = ω 2 −

V0 , cosh [m(τ − τ0 )] 2

(17.47)

there are bound states for n = 0, 1, 2, . . . , nmax < s, where s is defined by 

1 s ≡ −1 + 2

s



V0 1 + 4 2. m

(17.48)

Their wave functions are, according to (14.162), and (14.164), yn (τ ) =

2n−s mq Γ(s − n)Γ(1 + 2s − n) coshn−s [m(τ − τ0 )] n! Γ(1 + s − n) × F (−n, 1 + 2s − n; s − n + 1; 12 (1 − tanh[m(τ − τ0 )])), (17.49)

r

where F (a, b; c; z) are the hypergeometric functions (1.450). In terms of s, the parameter V0 becomes V0 = m2 s(s + 1).

(17.50)

λ2n = ω 2 − m2 (s − n)2 .

(17.51)

The bound-state energies are

In the Schrödinger equation (17.41), we have m = ω/2, V0 = 3ω 2 /2, so that s = 2 and there are exactly two bound-state solutions. These are s

y0 (τ ) = −

3ω 1 2 8 cosh [ω(τ − τ0 )/2]

(17.52)

1090

17 Tunneling

and y1 (τ ) =

s

=

s

1 3ω F (−1, 4; 2; 12 (1 − tanh[ω(τ − τ0 )/2])) 4 cosh[ω(τ − τ0 )/2] 3ω sinh[ω(τ − τ0 )/2] . 4 cosh2 [ω(τ − τ0 )/2]

(17.53)

The negative sign in (17.52) is a matter of convention, to give y1 (τ ) the same sign as xcl (τ − τ0 ) in Eq. (17.89) below. The normalization factors can be checked using the formula Z ∞ 1 µ+1 ν −µ sinhµ x = B( , ), (17.54) ν cosh x 2 2 2 0 with B(x, y) ≡ Γ(x)Γ(y)/Γ(x + y) being the Beta function. The corresponding eigenvalues are λ0 = 0,

λ1 = 3ω 2/4.

(17.55)

The existence of a zero-eigenvalue mode is a general property of fluctuations around localized classical solutions in a system which is translationally invariant along the τ -axis. It prevents an immediate application of the quadratic approximation, since a zero-eigenvalue mode is not controlled by a Gaussian integral as the others are in Eq. (17.46). This difficulty and its solution will be discussed in Subsection 17.3.1. In addition to the two bound states, there are continuum wave functions with λn ≥ ω 2 . For an energy λk = ω 2 + k 2 ,

(17.56)

they are given by a linear combination of yk (τ ) ∝ Aeikτ F (s + 1, −s; 1 − ik/m; 12 (1 − tanh[m(τ − τ0 )]))

(17.57)

and its complex-conjugate (see the end of Subsection 14.4.4). Using the identity for the hypergeometric function2 Γ(c)Γ(c − a − b) F (a, b; a + b − c + 1; 1 − z) Γ(c − a)Γ(c − b) Γ(c)Γ(−c + a + b) + (1 − z)c−a−b F (c − a, c − b; c − a − b + 1; 1 − z), (17.58) Γ(a)Γ(b)

F (a, b, c; z) =

and F (a, b; c; 0) = 1, we find the asymptotic behavior τ →∞

F − −−→ 1, τ →−∞

−−→ F − 2

(17.59)

Γ(−ik/m)Γ(1 − ik/m) Γ(ik/m)Γ(1 − ik/m) + e−2ikτ . Γ(−s − ik/m)Γ(s + 1 − ik/m) Γ(−s)Γ(1 + s) (17.60)

M. Abramowitz and I. Stegun, op. cit., formula 15.3.6. H. Kleinert, PATH INTEGRALS

1091


These limits determine the asymptotic behavior of the wave functions (17.57). With an appropriate choice of the normalization factor in (17.57) we fulfill the standard scattering boundary conditions ψ(τ ) →

(

eikτ + Rk e−ikτ , Tk eikτ ,

τ → −∞, τ → ∞.

(17.61)

These define the transmission and reflection amplitudes. From (17.59) and (17.60) we calculate directly Γ(−s − ik/m)Γ(s + 1 − ik/m) , (17.62) Γ(−ik/m)Γ(1 − ik/m) Γ(−s − ik/m)Γ(s + 1 − ik/m) Γ(ik/m) Γ(ik/m)Γ(1 − ik/m) Rk = = Tk . (17.63) Γ(−s)Γ(1 + s) Γ(−ik/m) Γ(−s)Γ(1 + s) Tk =

Using the relation Γ(z) = π/ sin(πz)Γ(1 − z), this can be written as Γ(s + 1 − ik/m) Γ(1 + ik/m) sin(ik/m) , Γ(s + 1 + ik/m) Γ(1 − ik/m) sin(s + ik/m) sin(πs) = Tk . sin(ik/m)

Tk =

(17.64)

Rk

(17.65)

The scattering matrix Tk Rk Rk Tk

Sk =

!

(17.66)

is unitary since Rk Tk∗ + Rk∗ Tk = 0,

|Tk |2 + |Rk |2 = 1.

(17.67)

It is diagonal on the state vectors 1 ψ =√ 2

1 1

e

!

,

1 ψ =√ 2 ø

1 −1

!

,

(17.68)

which, as we shall prove below, correspond to odd and even partial waves. The 2iδke,ø respective eigenvalues λe,ø define the phase shifts δke,ø , in terms of which k = e ø 1 2iδe (e k + e2iδk ), 2 ø 1 2iδe = (e k − e2iδk ). 2

Tk =

(17.69)

Rk

(17.70)

Let us verify the association of the eigenvectors (17.68) with the even and odd partial waves. For this we add to the wave function (17.61) the mirror-reflected solution r

ψ (τ ) →

(

Tk e−ikτ , e−ikτ + Rk eikτ ,

τ → −∞, τ → ∞,

(17.71)

1092

17 Tunneling

and obtain e

r

ψ (τ ) = ψ(τ ) + ψ (τ ) →

(

eikτ + (Rk + Tk )e−ikτ , e−ikτ + (Rk + Tk )eikτ ,

τ → −∞, τ → ∞.

(17.72)

Inserting (17.69), this can be rewritten as e

ψ (τ ) →

(

e

e

e

e

eiδk [ei(kτ −δk ) + e−i(kτ −δk ) ] = 2eiδk cos(k|τ | + δke ), e e e e eiδk [e−i(kτ +δk ) + ei(kτ +δk ) ] = 2eiδk cos(k|τ | + δke ),

τ → −∞, (17.73) τ → ∞.

The odd combination, on the other hand, gives o

r

ψ (τ ) = ψ(τ ) − ψ (τ ) →

(

eikτ + (Rk − Tk )e−ikτ , −e−ikτ − (Rk − Tk )eikτ ,

τ → −∞, τ → ∞,

(17.74)

and becomes with (17.69): o

ψ (τ ) →

(

ø

e

ø

e

eiδk [ei(kτ −δk ) − e−i(kτ −δk ) ] = 2ieiδk sin(k|τ | + δkø ), ø e e e −eiδk [e−i(kτ +δk ) − ei(kτ +δk ) ] = −2ieiδk sin(k|τ | + δkø ),

τ → −∞, (17.75) τ → ∞.

From Eqs. (17.69), (17.70) we see that |Tk |2 = cos2 (δke − δkø ),

|Rk |2 = sin2 (δke − δkø ),

(17.76)

and further e

ø

e2i(δk +δk ) = (Tk + Rk )(Tk − Rk ) = Tk2 +

Tk Tk Rk Rk∗ Tk = Tk2 + (1 − Tk Tk∗ ) ∗ = ∗ . (17.77) ∗ Tk Tk Tk

From this we find the explicit equation for the sum of even and odd phase shifts 1 Tk δke + δkø = arg ∗ = arg Tk . 2 Tk

(17.78)

Similarly we derive the equation for the difference of the phase shifts: −i sin[2(δke − δkø )] = Tk Rk∗ − Tk∗ Rk = 2Tk Rk∗ = −2

Rk∗ |Tk |2 . ∗ Tk

(17.79)

Dividing this by the first equation in (17.76), we obtain Rk∗ sinh(ik/m) = − , Tk∗ sin(πs)

(17.80)

sin(πs) sin(πs). sinh(k/m)

(17.81)

−i tan(δke − δkø ) = − and thus δke − δkø = arctan


1093


For s = integer, the even and odd phase shifts become equal, δke = δkø ≡ δk ,

(17.82)

so that the reflection amplitude vanishes, and the transmission amplitude reduces to a pure phase factor Tk = e2iδk , with the phase shift determined from (17.78) to 2δk = −i log Tk .

(17.83)

Now the wave functions have the simple asymptotic behavior yk (τ ) → ei(kτ ±δk ) ,

τ → ±∞,

(17.84)

and (17.64) reduces for both even and odd phase shifts to e2iδk = (−1)s

Γ(s + 1 − ik/m) Γ(1 + ik/m) . Γ(1 − ik/m) Γ(s + 1 + ik/m)

(17.85)

For the Schrödinger equation (17.41) with s = 2, this becomes simply e2iδk =

2 − ik/m 1 − ik/m , 2 + ik/m 1 + ik/m

(17.86)

and hence δk = arctan[k/m] + arctan[k/2m].

(17.87)

If we now try to evaluate the product of eigenvalues in (17.46), two difficulties arise: 1) The zero eigenvalue causes the result to be infinite. Q √ 2) The continuum states make the evaluation of the product of eigenvalues n λn nontrivial. These two difficulties will be removed in the following two subsections.

17.3.1

Zero-Eigenvalue Mode

The physical origin of the infinity caused by the zero-eigenvalue solution in the fluctuation integral over ξ0 in (17.46) lies in the time-translational invariance of the system. This fact supplies the key to removing the infinity. A kink at an imaginary time τ0 contributes as much to the path integral as a kink at any other time τ0′ . The difference between two adjacent solutions at an infinitesimal temporal distance can be viewed as a small kink fluctuation which does not change the Euclidean action. There is a zero eigenvalue associated with this difference. If there is only a single zero-eigenvalue solution as in the Schrödinger equation (17.41) its wave function must be proportional to the derivative of the kink solution: y0

"

xcl (τ − τ0 ) − xcl (τ − τ1 ) = α τ0 − τ1 aω/2 = −α . 2 cosh [ω(τ − τ0 )/2]

#

τ0 →τ1

= αx′cl (τ − τ0 ) (17.88)

1094

17 Tunneling

This coincides indeed with the zero-eigenvalue solution (17.52). The normalization factor α is fixed by the integral α=

Z

∞

−∞

dτ x′2 cl

−1/2

.

(17.89)

With the help of (17.32) the right-hand side can also be expressed in terms of the kink action: α=√

1 . Acl

(17.90)

Inserting (17.2) and (17.31), this becomes α=

q

q

3/2a2 ω =

3g/ω 3

(17.91)

[the positive square root corresponding to the negative sign in (17.52)]. The zeroeigenvalue mode is associated with a shift of the position of the kink solution from τ0 to any other place τ0′ . For an infinite length L of the τ -axis, this is the source of the infinity of the integral over ξ0 in the product (17.46). At a finite L, on the other hand, only √ a τ -interval of length L is available for displacements. Therefore, the infinite 1/ λ0 should become proportional to L for large L: 1 √ = const · L. λ0

(17.92)

What is the proportionality constant? We find it by transforming the integration measure (17.46), N

Z

∞ dξ Y √ 0 2π¯ h n=1

∞

−∞

"Z

∞

−∞

#

dξ √ n , 2π¯ h

(17.93)

to a form in which the translational degree of freedom appears explicitly 1 N√ 2π¯ h

Z

∞

−∞

YZ

dτ0

∞

n6=0 −∞

"

dξ √ n 2π¯ h

# ∂ξ 0 (ξ 1 , ξ2 , . . .) . ∂τ0

(17.94)

The Jacobian appearing in the integrand satisfies the identity ∂ξ 0 dτ0 δ(ξ0 ) ∂τ0

Z

= 1.

(17.95)

Let us calculate this Jacobian using the method developed by Faddeev and Popov [2]. Given an arbitrary fluctuation y ξ0,ξ1 ,ξ2 ,..., the parameter ξ0 may be recovered by forming the scalar product with the zero-eigenvalue wave function y0 (τ ): ξ0 =

Z

∞

−∞

dτ y ξ0 ,ξ1 ,ξ2 ,...(τ )y0 (τ ).


1095


Moreover, it is easy to see that the fluctuation y ξ0,ξ1 ,ξ2 ,... can be replaced by the full path xξ0 ,ξ1 ,ξ2 ,...(τ ) = xcl (τ ) + y ξ0 ,ξ1 ,ξ2 ,... (τ ),

(17.97)

so that we can also write ξ0 =

Z

∞

−∞

dτ xξ0 ,ξ1 ,ξ2 ,...(τ )y0 (τ ).

(17.98)

This follows from the fact that the additional kink solution xcl (τ ) does not have any overlap with its derivative. Indeed, Z

∞ −∞

dτ xcl (τ − τ0 )y0 (τ − τ0 ) ∝ ∝

Z

∞

−∞

dτ xcl (τ − τ0 )x′cl (τ − τ0 )

1 2 ∞ x = 0. 2 cl −∞

(17.99)

With (17.98), the delta function δ(ξ0 ) appearing in (17.95) can be rewritten in the form δ(ξ0 ) = δ

Z

∞

ξ0 ,ξ1 ,ξ2 ,...

dτ x

−∞

(τ )y0 (τ ) .

(17.100)

To establish the relation between the normal coordinate ξ0 and the kink position τ0 , we now replace xξ0 ,ξ1 ,ξ2 ,... by an alternative parametrization of the paths in which the role of the variable ξ0 is traded against the position of the kink solution, i.e., we rewrite the fluctuating path xξ0 ,ξ1 ,ξ2 ,... (τ ) = xcl (τ ) +

∞ X

ξn yn (τ )

(17.101)

∞ X

ξn yn (τ − τ0 ).

(17.102)

n=0

in the form τ0 ,ξ1 ,ξ2 ,...

x

(τ ) ≡ xcl (τ − τ0 ) +

n=1

By definition the point τ0 = 0 coincides with ξ0 = 0. Thus, if we insert (17.102) into (17.100) and use this δ-function in the identity (17.95), we find the condition for the Jacobian ∂ξ0 /∂τ0 : Z

∞ −∞

dτ0 δ

Z

∞ −∞

τ0 ,ξ1 ,ξ2 ,...

dτ x

∂ξ0 (τ )y0 (τ ) ∂τ0

= 1.

(17.103)

Since the δ-function has a vanishing argument at τ0 = 0, we expand the argument in powers of τ0 , keeping only the lowest order. Writing y0 (τ ) = αx′cl (τ ), we obtain Z

∞

dτ xτ0 ,ξ1 ,ξ2 ,... (τ )y0 (τ ) = −ατ0

−∞

"Z

∞

−∞

∞ X

dτ x′2 cl +

n=1

ξn

Z

∞

#

dτ x′cl yn′ + O(τ02 ).

−∞

(17.104)

1096

17 Tunneling

Using (17.32) and abbreviating the scalar products in the brackets as rn ≡

∞

Z

−∞

dτ x′cl yn′ ,

(17.105)

we may express the right-hand side of (17.104) more succinctly as "

−ατ0 Acl +

∞ X

n=1

#

ξn rn + O(τ02 ).

(17.106)

Inserting this into (17.103), and using (17.90), we arrive at the Jacobian ∂ξ 0 ∂τ0

= Acl

1/2

1 + Acl

−1

∞ X

!

ξ n rn .

n=1

(17.107)

As a consequence, the zero-eigenvalue modes contribute to the fluctuation factor (17.46) as follows: Fcl (L) = N =N

" ∞ Z Y

∞

−∞ n=1 " Z ∞ ∞ Y −∞

n=1

= N qQ

1 ∞ n=1

#

P∞ 2 ∞ dξ dξ 2 √ n e−(1/2¯h) n=1 ξn λn √ 0 e−(1/2¯h)ξ0 λ0 −∞ 2π¯ h 2π¯ h #Z ! ∞ ∞ X dξn −(1/2¯h) P∞ ξn2 λn dτ0 n=1 √ √ e Acl 1/2 1+Acl−1 ξ n rn −∞ 2π¯ h 2π¯ h n=1

λn

s

Acl 2π¯ h

Z

Z

∞

−∞

dτ0 .

(17.108)

The linear terms in the large parentheses disappear at the level of quadratic fluctuations since they are odd in ξ. Thus we may use for quadratic fluctuations the simple mnemonic rule −

q ∂x(τ )/∂τ0 x˙ cl (τ ) + . . . ∂ξ0 1 = = = + . . . = Acl + . . . , ∂τ0 ∂x(τ )/∂ξ0 αx˙ cl (τ ) α

(17.109)

where . . . stands for the irrelevant linear terms in ξn inside the integral (17.108). For higher-order fluctuations, the linear terms in the large parentheses cannot be ignored. They contribute an effective Euclidean action Aeff e

"

= −¯ h log 1 + Acl

−1

∞ X

n=1

#

ξn rn = −¯ h log 1 + A−1 cl

Z

dτ x′cl (τ )y ′ (τ ) , (17.110)

which will be needed for calculations in Section 17.8. The integral over τ0 is now an appropriate place to impose the finiteness of the time interval τ ∈ (−L/2, L/2), namely Z

L/2

−L/2

dτ0 = L.

(17.111)

A comparison of (17.108) with (17.46) shows that √ the correct evaluation of the formally diverging zero-eigenvalue contribution 1/ λ0 is equivalent to the following replacement: 1 √ ≡ λ0

Z

dξ 2 √ 0 e−(1/2¯h)λ0 ξ0 − − −→ 2π¯ h

s

Acl 2π¯ h

Z

L/2 −L/2

dτ0 =

s

Acl L. 2π¯ h

(17.112)


1097


17.3.2

Continuum Part of Fluctuation Factor

We now turn to the second problem, the calculation of the product of the continuum eigenvalues in (17.46). To avoid carrying around the overall normalization factor N it is convenient to factor out the quadratic fluctuations (17.36) in the absence of a kink solution. Let λ0n be the associated eigenvalues. Their fluctuation potential V ′′ (xcl (τ )) = ω 2 x2 is harmonic [compare (17.39)] with a fluctuation factor known from (2.164). Comparing this with the expression (17.46) without a kink, we obtain for the normalization factor the equation 1

N qQ

n

λ0n

= Fω (L) =

r

ω ∼ 2π¯ h sinh ωL

r

ω −ωL/2 e . π¯ h

(17.113)

Pulling this factor out of the product on the right-hand side of (17.46), we are left with a ratio of eigenvalue products: 1

Fcl (L) = N √Q

n

1

λn

= N qQ

n

λ0n

sQ

sQ

0 λ0 n λn = Fω (L) Qn n . Q n λn n λn

(17.114)

As long as L is finite, the continuum wave functions are all discrete. Let ∂n/∂k denote their density of states per momentum interval. Then the ratio of the continuum eigenvalues can be written for large L as sQ

"

1 λ0n = exp − Q 2 n λn cont n

Z

0

∞

!

#

∂n ∂n dk log λn . − ∂k ∂k 0

(17.115)

The density of states, in turn, may be extracted from the phase shifts (17.85). For this we observe that for a very large L where boundary conditions are a matter of choice, we may impose the periodic boundary condition y(τ + L) = y(τ ).

(17.116)

Together with the asymptotic forms (17.84), this implies ei(kL/2+δk ) = e−i(kL/2+δk ) ,

(17.117)

which quantizes the wave vectors k to discrete values satisfying kL + 2δk = 2πn.

(17.118)

The derivative with respect to k yields the density of states ∂n L 1 dδk = + . ∂k 2π π dk

(17.119)

Since the phase shifts vanish in the absence of a kink solution, this implies ∂n L , = ∂k 0 2π

(17.120)

1098

17 Tunneling

and the general formula (17.115) becomes simply sQ

"

#

λ0n 1 Z ∞ dδk = exp − dk log(ω 2 + k 2 ) . Q 2π 0 dk n λn cont

n

(17.121)

To calculate the integral for our specific fluctuation problem (17.41), we use the expression (17.85) to find the derivative of the phase shift for any integer value of s: "

#

1 2 dδk s =− + ...+ 2 . 2 dk m 4 + (k/m) s + (k/m)2

(17.122)

For s = 2, the exponent in (17.121) becomes h i 1 Z∞ 1 2 2 2 2 2 dx + log ω (1 + x m /ω ) . 2π −∞ 1 + x2 4 + x2

(17.123)

The ω 2 -term in the logarithmR can be separated from this using Levinson’s theorem. It states that the integral 0∞ dk(∂n/∂k − ∂n/∂k|0 ) is equal to the number of bound states: Z

∞ 0

!

∂n ∂n 1 dk − = ∂k ∂k 0 π

Z

∞

dk

0

dδk = s. dk

(17.124)

This relation is obviously fulfilled by (17.122). It is a consequence of the fact that a potential with s bound states has s states less in the continuum than a free system. Using this property, the integral (17.123) can be rewritten as log ω 2 +

Z

∞ −∞

dx 1 2 + log(1 + x2 m2 /ω 2). 2 2π 1 + x 4 + x2

(17.125)

The rescaled integral is calculated using the formula 1 r dx log(1 + p2 x2 ) = log 1 + p . 2 2 2 2π r + s x rs s

(17.126)

m m log ω + log 1 + 1 + log 1 + 2 . ω ω

(17.127)

m 2 . ω

(17.128)

Z

∞ −∞

The result is

2

When inserted into the exponent of (17.121), it yields sQ

λ0n m = ω2 1 + Q ω n λn cont n

1+

In our case with m = ω/2, this reduces to 3ω 2 . Including the bound-state eigenvalue λ1 of Eq. (17.55) in the denominator, this amounts to v uQ 0 u n λn tQ ′

√ 1 =q 3ω 2 = 12ω ≡ K ′ . n λn 3ω 2/4

(17.129)


1099

17.4 General Formula for Eigenvalue Ratios

Multiplying this with the zero-eigenvalue contribution as evaluated in Eq. (17.108), we arrive at the final result for the fluctuation factor in the presence of a kink or an antikink solution: Fcl (L) = Fω (L)KL,

(17.130)

with 1 K=√ K′ = λ0 L

17.4

s

Acl √ 12ω. 2π¯ h

(17.131)

General Formula for Eigenvalue Ratios

The above-calculated ratio of eigenvalue products sQ

λ0n Q n λn cont n

(17.132)

of the Rosen-Morse Schrödinger equation appears in many applications with different potential strength parameters s. It is therefore useful to derive a formula for this ratio which is valid for any s. The eigenvalue equation reads d2 m2 s(s + 1) 2 − 2 +ω − yn (τ ) = λn yn (τ ). dτ cosh2 m(τ − τ0 )

"

#

(17.133)

First we consider the case of an arbitrary integer value of s. Following the previous discussion, the ratio of eigenvalue products is found to be sQ

"

#

1 ∞ dδk λ0n = exp − dk (17.134) log(ω 2 + k 2 ) Q λ 2π dk 0 cont n n ( ) Z s X 1 ∞ n 2 2 2 = exp − d(k/m) 2 log[ω + (k/m) m ] . 2 2π 0 n=1 n + (k/m) n

Z

The ω 2 -term in the logarithm is eliminated by the generalization of (17.124) to any integer s: 1 π

Z

∞

−∞

d(k/m)

s X

n 2 = s. 2 n=1 n + (k/m)

(17.135)

Hence sQ

s X λ0n 1 Z∞ n s = ω exp dx log(1 + x2 m2 /ω 2) . Q 2 2 2π −∞ n=1 n + x n λn cont n

"

#

(17.136)

The integrals can be done using formula (17.126), and we obtain sQ

s Y λ0n m s = ω 1+ n . Q ω n λn cont n=1 n

(17.137)

1100

17 Tunneling

For s = 2, and m = ω/2, this reduces to the previous result (17.128). Only a little more work is required to find the ratio of all discrete and continuous eigenvalue products for a noninteger value of s. Introduce a new parameter z, let s be a parameter smaller than 1 so that there are no bound states, and consider the fluctuation equation d2 s(s + 1) − 2 + m2 z − dτ cosh2 m(τ − τ0 )

"

!#

yn (τ ) = λn yn (τ ).

(17.138)

The general Schrödinger operator under consideration (17.133) corresponds to z = ω 2 /m2 . Since there are no bound states by assumption, the first line in formula (17.134) now gives the ratio of all eigenvalues: sQ

"

#

λ0n 1 Z∞ dδk = exp − dk log(m2 z + k 2 ) . Q 2π −∞ dk n λn n

(17.139)

Here δk is equal to the average of even and odd phase shifts (δke + δkø )/2. For the same reason, we can replace log(m2 z + k 2 ) by log(z + k 2 /m2 ) without error [using the generalization of (17.124)]. After substituting k 2 → ω 2 ǫ, we find s

" # √ Πn λ0n 1 Z dδω ǫ log(z + ǫ) , = exp − dǫ Πn λn 2π C dǫ

(17.140)

where the contour of integration C encircles the right-hand cut clockwise in the ǫ-plane. A partial integration brings this to exp

1 2π

Z

C

dǫ δm√ǫ

1 . z+ǫ

(17.141)

For z < 0, the contour of integration can be deformed to encircle the only pole at ǫ = −z counterclockwise, yielding sQ Q

λ0n = exp[iδm√−z ]. λ n n n

(17.142)

Inserting for δk the average of even and odd phase shifts from (17.78), we obtain " √ #1/2 √ λ0n Γ( z − s)Γ( z + s + 1) √ √ = . Q Γ( z)Γ( z + 1) n λn

sQ

n

(17.143)

In the fluctuation equation (17.41), the parameters m2 and ω 2 = zm2 are such as to create a zero eigenvalue at n = 0 according to formula (17.51). Then z = s2 . In the neighborhood of this z-value, the eigenvalue λ0 = m2 [z − s2 ]


1101

17.5 Fluctuation Determinant from Classical Solution

is a would-be zero eigenvalue. Dividing it out of the product (17.143), we obtain an equation which remains valid in the limit z → s2 : #1/2 √ √ Γ( z − s + 1)Γ( z + s + 1) √ √ = m ( z + s) . Γ( z)Γ( z + 1) n λn

v uQ 0 u n λn tQ ′

"

√

(17.145)

This can be continued analytically to arbitrary z and s, as long as z remains sufficiently close to s = 2. For s = 2 and z = 4 (corresponding to m = ω/2) we recover the earlier result (17.129): v uQ 0 u n λn tQ ′ n

17.5

λn

=

√

12ω.

(17.146)

Fluctuation Determinant from Classical Solution

The above evaluations of the fluctuation determinant require the complete knowledge of the bound and continuum spectrum of the fluctuation equation. Fortunately, there exists a way to find the determinant which needs much less information, requiring only the knowledge of the large-τ behavior of the classical solution and the value of its action. The basis for this derivation is the Gelfand-Yaglom formula derived in Section 2.4. According to it, the fluctuation determinant of a differential operator 2 m2 s(s + 1) ˆ = − d + ω2 − O dτ 2 cosh2 [m(τ − τ0 )]

(17.147)

is given by the value of the zero-eigenvalue solution D(τ ) at the final τ value τ = L/2 ˆ = N D(L/2), det O

(17.148)

provided that it was chosen to satisfy the initial conditions at τ = −L/2: D(−L/2) = 0,

˙ D(−L/2) = 1.

(17.149)

The normalization factor N is irrelevant when considering ratios of fluctuation determinants, as we do in the problem at hand.3 To satisfy the boundary conditions (17.149), we need two linearly independent solutions of zero eigenvalue. One is known from the invariance under time translations. It is proportional to the time derivative of the classical solution [see (17.88)]: y0 (τ ) = αx′cl (τ ).

(17.150)

ˆ with d/dτ replaced by the If the determinant is calculated for the time-sliced operator O difference operator ∇τ , the normalization is N = 1/ǫ where ǫ is the thickness of the time slices. See Chapter 2. 3

1102

17 Tunneling

In the above fluctuation problem (17.41) with z = 2 and m = ω/2, the classical solution is xcl (τ ) = arctanh[ω(τ − τ0 )/2]. It has the asymptotic behavior y0 (τ ) →

ω −ω|τ | e 2

for τ → ±∞,

(17.151)

with a symmetric exponential falloff in both directions of the τ -axis. In the sequel it will be convenient to work with zero-eigenvalue solutions without the prefactor ω/2, which behave asymptotically like a pure exponential. These will be denoted by ξ(τ ) and η(τ ), the solution ξ(τ ) being proportional to y0 , i.e., ξ(τ ) → e−ω|τ |

for τ → ±∞.

(17.152)

The second independent solution can be found from d’Alembert’s formula (2.229). Its explicit form is not required; only its asymptotic behavior is relevant. Assuming the Lagrangian to be invariant under time reversal, which is usually the (2) case, this asymptotic behavior is found via the following argument: Since φ0 (τ ) is (1) linearly independent of φ0 (τ ), we can be sure that it has asymptotically the opposite exponential behavior (i.e., it grows with τ ) and the opposite symmetry under time reversal (i.e., it is antisymmetric). Thus η must behave as follows: η(τ ) → ±eω|τ |

for τ → ±∞.

(17.153)

We now form the linear combination which satisfies the boundary conditions (17.149) for large negative τ = −L/2: D(τ ) =

1 [ξ(−L/2)η(τ ) − η(−L/2)ξ(τ )] , W

(17.154)

where ˙ ) W ≡ W [ξ(τ )η(τ )] = ξ(τ )η(τ ˙ ) − η(τ )ξ(τ

(17.155)

is the Wronskian of the two solutions. It is independent of τ and can be evaluated from the asymptotic behavior as W = 2ω.

(17.156)

Inserting (17.152) and (17.153) into (17.154), we find the solution D(τ ) =

i 1 h −ωL/2 e η(τ ) + eωL/2 ξ(τ ) . W (1)

(17.157)

(2)

Even without knowing the solutions φ0 (τ ), φ0 (τ ) at a finite τ , the fluctuation determinant at large τ = L/2 can be written down: D(L/2) =

2 1 = . W ω


1103

17.5 Fluctuation Determinant from Classical Solution

For fluctuations around the constant classical solution, the zero-eigenvalue solution with the boundary conditions (17.149) is 1 sinh[ω(τ + L/2)]. ω

D (0) (τ ) =

(17.159)

It behaves for large τ = L like D (0) (L/2) →

1 ωL e 2ω

for large L.

(17.160)

for large L.

(17.161)

The ratio is therefore D (0) (L/2) 1 → eωL D(L/2) 2

This exponentially large number is a signal for the presence of a would-be zero eigenvalue in D(L/2). Since the τ -interval (−L/2, L/2) is finite, there exists no exactly vanishing eigenvalue. In the finite interval, the derivative (17.150) of the kink solution does not quite satisfy the Dirichlet boundary condition. If the vanishing at the endpoints was properly enforced, the particle distribution would have to be compressed somewhat, and this would shift the energy slightly upwards. The shift is exponentially small for large L, so that the would-be zero eigenvalue has an exponentially small eigenvalue λ0 ∝ e−ωL . A finite result for L → ∞ is obtained by removing this mode from the ratio (17.161) and considering the limit Q

λn 0 D (0) (L/2) λ0 . = lim L→∞ D(L/2) n λn

n

(17.162)

Q′

The leading e−ωL -behavior of the would-be zero eigenvalue can be found perturbatively using as before only the asymptotic behavior of the two independent solutions. To lowest order in perturbation theory, an eigenfunction satisfying the Dirichlet boundary condition at finite L is obtained from an eigenfunction φ0 which vanishes at τ = −L/2 by the formula φL0 (τ )

λ0 = φ0 (τ ) + W

Z

τ

−L/2

dτ ′ [ξ(τ )η(τ ′ ) − η(τ )ξ(τ ′ )] φ0 (τ ′ ).

(17.163)

The limits of integration ensure that φL0 (τ ) vanishes at τ = −L/2. The eigenvalue λ0 is determined by enforcing the vanishing also at τ = L/2. Taking for φ0 (τ ) the zero-eigenvalue solution D(τ ) "

λ0 = −D(L/2)W ξ(L/2)

Z

L/2

−L/2

dτ η(τ )D(τ ) − η(L/2)

Z

L/2

−L/2

#−1

dτ ξ(τ )D(τ ) . (17.164)

Inserting (17.154) and using the orthogonality of ξ(τ ) and η(τ ) (following from the fact that the first is symmetric and the second antisymmetric), this becomes "

2

Z

λ0 = −D(L/2)W ξ(−L/2)ξ(L/2)

L/2

−L/2

2

Z

dτ η (τ ) + η(−L/2)η(L/2)

L/2

−L/2

2

dτ ξ (τ )

#−1

.

(17.165)

1104

17 Tunneling

Invoking once more the symmetry of ξ(τ ) and η(τ ) and the asymptotic behavior (17.152) and (17.153), we obtain λ0 = −D(L/2)W

2

"

L/2

Z

−ωL

e

−L/2

2

ωL

dτ η (τ ) − e

Z

L/2

−L/2

2

dτ ξ (τ )

#−1

.

(17.166)

The first integral diverges like eωL ; the second is finite. The prefactor makes the second integral much larger than the first, so that we find for large L the would-be zero eigenvalue W2 . 2 −∞ dτ ξ (τ )

λ0 = D(L/2)e−ωL R ∞

(17.167)

This eigenvalue is exponentially small and positive, as expected. Inserting it into (17.162) and using (17.156) and (17.160), we find the eigenvalue ratio Q

1 λ0n . = lim 2ω R ∞ 2 L→∞ n λn −∞ dτ ξ (τ ) n

(17.168)

Q′

The determinant D(L/2) has disappeared and the only nontrivial quantity to be evaluated is the normalization integral over the translational eigenfunction ξ(τ ). The normalization integral requires the knowledge of the full τ -behavior of the (1) zero-eigenvalue solution φ0 (τ ); the asymptotic behavior used up to this point is insufficient. Fortunately, the classical solution xcl (τ ) also supplies this information. The normalized solution is y0 = αx′cl (τ ) behaving asymptotically like 2aαωe−ωτ . (1) Imposing the normalization convention (17.152) for φ0 (τ ), we identify ξ(τ ) =

1 αx′ (τ ). 2aωα cl

(17.169)

Using the relation (17.32), the normalization integral is simply Z

∞

dτ ξ(τ )2 =

−∞

Acl . 4a2 ω 2

(17.170)

With it the eigenvalue ratio (17.168) becomes Q

λ0n 4a2 ω = 2ω . Acl n λn n

Q′

(17.171)

By inserting the value of the classical action Acl = 2a2 ω/3 from (17.31), we obtain λ0n = 12ω 2, λ n n

Q

n

Q′

(17.172)

just as in (17.146) and (17.129). It is remarkable that the calculation of the ratio of the fluctuation determinants with this method requires only the knowledge of the classical solution xcl (τ ). H. Kleinert, PATH INTEGRALS

1105

17.6 Wave Functions of Double-Well

17.6

Wave Functions of Double-Well

The semiclassical result for the amplitudes (a L/2|a −L/2),

(a L/2| −a −L/2),

(17.173)

with the endpoints situated at the bottoms of the potential wells can easily be extended to variable endpoints xb 6= a, xa 6= ±a, as long as these are situated near the bottoms. The extended amplitudes lead to approximate particle wave functions for the lowest two states. The extension is trivial for the formula (17.34) without a kink solution. We simply multiply the fluctuation factor by the exponential exp(−Acl /¯ h) containing the classical action of the path from xa to xb . If xa and xb are both near one of the bottoms of the well, the entire classical orbit remains √ near this bottom. If the distance of the orbit from the bottom is less than 1/a ω, the potential can be approximated by the harmonic potential ω 2 x2 . Thus near the bottom at x = a, we have the simple approximation to the action Acl ≈

n o ω [(xa − a)2 + (xb − a)2 ] cosh ωL − 2(xb − a)(xa − a) . 2¯ h sinh ωL (17.174)

For a very long Euclidean time L, this tends to Acl ≈

ω [(xb − a)2 + (xa − a)2 ]. 2¯ h

(17.175)

The amplitude (17.34) can therefore be generalized to (xb L/2|xa L/2) ≈

r

ω −(ω/2¯h)[(xb −a)2 +(xa −a)2 ] −ωL/2¯h e . e π¯ h

This can also be written in terms of the bound-state wave functions (17.5) for n = 0 as (xb L/2|xa L/2) ≈ ψ0 (xb − a)ψ0 (xa − a)e−ωL/2¯h .

(17.176)

For the amplitude (17.35) with the path running from one potential valley to the other, the construction is more subtle. The approximate solution is obtained by combining a harmonic classical path running from (xa , −L/2) to (a, −L/4), a kink solution running from (a, −L/4) to (a, L/4), and a third harmonic classical path running from (a, L/4) to (xb , L/2). This yields the amplitude r

2 ω −(ω/2¯h)(xb −a)2 e KLe−Acl /¯h e−(ω/2¯h)(xa −a) . π¯ h

(17.177)

Note that by patching the three pieces together, it is impossible to obtain a true classical solution. For this we would have to solve the equations of motion containing a kink with the modified boundary conditions x(−L/2) = xa , x(L/2) = xb . From the exponential convergence of x(τ ) → ±a (like e−ωL ) it is, however, obvious that the

1106

17 Tunneling

true classical action differs from the action of the patched path only by exponentially small terms. As before, the prefactor in (17.177) can be attributed to the ground state wave functions ψ0 (x), and we find the amplitude for xb close to −a and xa close to a: (xb L/2|xa − L/2) ≈ ψ0 (xb + a)ψ0 (xa − a)KLe−Acl /¯h e−ωL/2¯h .

17.7

(17.178)

Gas of Kinks and Antikinks and Level Splitting Formula

The above semiclassical treatment is correct to leading order in e−ωL . This accuracy is not sufficient to calculate the degree of level splitting between the two lowest states of the double well caused by tunneling. Further semiclassical contributions to the path integral must be included. These can be found without further effort. For very large L, it is quite easy to accommodate many kinks and antikinks along the τ -axis without a significant deviation of the path from the equation of motion. Due to the fast approach to the potential bottoms x = ±a near each kink or antikink solution, an approximate solution can be constructed by smoothly combining a number of individual solutions as long as they are widely separated from each other. The deviations from a true classical solution are all exponentially small if the separation distance ∆τ on the τ -axis is much larger than the size of an individual kink (i.e., ∆τ ≫ 1/ω). The combined solution may be thought of as a very dilute gas of kinks and antikinks on the τ -axis. This situation is referred to as the dilute-gas limit. Consider such an “almost-classical solution” consisting of N kink-antikink solutions xcl (τ ) = ±atanh[ω(τ −τi )/2] in alternating order positioned at, say, τ1 ≫ τ2 ≫ τ3 ≫ . . . ≫ τN and smoothly connected at some intermediate points τ¯1 , . . . , τ¯N −1 . In the dilute-gas approximation, the combined action is given by the sum of the individual actions. For the amplitude (17.34) in which the paths connect the same potential valleys, the number of kinks must be equal to the number of antikinks. The action combined is then an even multiple of the single kink action: A2n ≈ 2nAcl .

(17.179)

For the amplitude (17.35), where the total number is odd, the combined action is A2n+1 ≈ (2n + 1)Acl .

(17.180)

As the kinks and antikinks are localized objects of size 2/ω, it does not matter how they are distributed on the large-τ interval [−L/2, L/2], as long as their distances are large compared with their size. In the dilute-gas limit, we can neglect the sizes. In the path integral, the translational degree of freedom of widely spaced N kinks and antikinks leads, via the zero-eigenvalue modes, to the multiple integral Z

L/2

−L/2

dτN

LN dτN −1 . . . dτ1 = . N! −L/2 −L/2

Z

τN

Z

τ1


1107

17.7 Gas of Kinks and Antikinks and Level Splitting Formula

The Jacobian associated with these N integrals is [see (17.112)] s

N

Acl . 2π¯ h

(17.182)

The fluctuations around the combined solution yield a product of the individual fluctuation factors. For a given set of connection points we have 1

1 qQ ′

λ n n

LN

qQ ′

λ n n

× . . . × qQ′

1

,

(17.183)

n λn

LN−1

L1

¯ i ≡ τ¯i − τ¯i−1 are the patches on the τ -axis in which the individual solutions where L are exact. Their total sum is L=

N X

Li .

(17.184)

i=1

We now include the effect of the fluctuations at the intermediate times τ¯i where the individual solutions are connected. Remembering the amplitudes (17.176), we see that the fluctuation factor for arbitrary endpoints xi , xi−1 near the bottom of the potential valley must be multiplied at each end with a wave function ratio ψ0 (x ± a)/ψ0 (0). Thus we have to replace 1 √Q

n

λn

ψ0 (xi ± a) ψ0† (xi−1 ± a) 1 √Q . ψ0 (0) ψ0† (0) n λn

→

(17.185)

The adjacent xi -values of all fluctuation factors are set equal and integrated out, giving 1 √Q

n

λn

= L

Z

ψ0 (xN −1 − a)ψ0† (xN −1 − a) 1 √Q 2 |ψ0 (0)| n λn LN n λn LN−1

1 dxN · · · dx1 √Q

×...×

ψ0 (x1 − a)ψ0† (x1 − a) 1 . (17.186) √Q 2 |ψ0 (0)| n λn L1

Due to the unit normalization of the ground state wave functions, the integrals are trivial. Only the |ψ0 (0)|2 -denominators survive. They yield a factor 1 |ψ0 (0)|2(N −1)

=

r

ω −(N −1) . π¯ h

(17.187)

It is convenient to multiply and divide the result by the square root of the product of eigenvalues of the harmonic kink-free fluctuations, whose total fluctuation factor is known to be 1 q

Πn λ0n |L

=

r

ω −ωL/2¯h e . π¯ h

(17.188)

1108

17 Tunneling

Then we obtain the total corrected fluctuation factor r

s ω −(N −2) −ωL/2¯h Y 0 1 e λn qQ ′ π¯ h L n λn n

1

L1

qQ ′

λ n n

L2

× . . . × qQ

1

′ n λn L

. (17.189)

N

We q that the harmonic fluctuation factor (17.188) for the entire interval qQ now observe 0 ω/π¯ h exp(−ωL/2¯ h) can be factorized into a product of such factors n λn | L = for each interval τ¯i , τ¯i−1 as follows: s Y n

λ0n

= L

r

ω π¯ h

−(N −1) sY n

λ0n

···

L1

s Y

λ0n . LN n

(17.190)

The total corrected fluctuation factor can therefore be rewritten as r

v uQ

v uQ

0 0 u ω −ωL/2¯h u t Qn λn × . . . × t Qn λn . e ′ ′ π¯ h n λn L1 n λn LN

(17.191)

Each eigenvalue ratio gives the Li -independent result K′ =

v u Q 0 u n λn tQ ′ λn n

,

(17.192)

Li

with K ′ of Eq. (17.131). Expressing K ′ in terms of K via ′

K = q

s

Acl 2π¯ h

−1

K,

(17.193)

−1

the factors Acl /2π¯ h remove the Jacobian factors (17.182) arising from the positional integrals (17.181). Altogether, the total fluctuation factor of N kink-antikink solutions with all possible distributions on the τ -axis is r

ω −ωL/2¯h LN N −N Acl /¯h . e K e π¯ h N!

(17.194)

Summing over all even and odd kink-antikink configurations, we thus obtain (a L/2| ± a − L/2) =

r

ω −ωL/2¯h X 1 N e (KLe−Acl /¯h ) . π¯ h even N!

(17.195)

odd

This can be summed up to (a L/2| ± a − L/2) = ×

r

ω −ωL/2¯h e π¯ h

(17.196)

i 1h exp KLe−Acl /¯h ± exp −KLe−Acl /¯h . 2 H. Kleinert, PATH INTEGRALS

1109

17.7 Gas of Kinks and Antikinks and Level Splitting Formula

As in the previous section, we generalize this result toqpositions xb , xa near the ¯ /ω). Using the classical potential minima (with a maximal distance of the order of h action (17.175) and expressing it in terms of ground state wave functions, we can now add the contribution of the amplitudes for all possible configurations, arriving at (xb L/2|xa − L/2) = e−ωL/2¯h (17.197) n h i 1 × ψ0 (xb − a)ψ0 (xa − a) exp(Ke−Acl /¯h L) + exp(−Ke−Acl /¯h L) 2 i 1h +ψ0 (xb − a)ψ0 (xa − a) exp(Ke−Acl /¯h L) − exp(−Ke−Acl /¯h L) 2 o +(xb → −xb ) + (xa → −xa ) + (xb → −xb , xa → −xa ) .

The right-hand side is recombined to

1 1 √ [ψ0 (xb − a) + ψ0 (xb + a)] × √ [ψ0 (xa − a) + ψ0 (xa + a)] 2 2 ω × exp − − Ke−Acl /¯h L 2 1 1 + √ [ψ0 (xb − a) − ψ0 (xb + a)] × √ [ψ0 (xa − a) − ψ0 (xa + a)] 2 2 ω −Acl /¯ h + Ke L . × exp − 2

(17.198)

Here we identify the ground state wave function as the symmetric combination of the ground state wave functions of the individual wells 1 Ψ0 (x) = √ [ψ0 (x − a) + ψ0 (x + a)]. 2

(17.199)

Its energy is E

(0)

=E

(0)

∆E (0) = ω/2 − Ke− − 2

Acl /¯ h

h ¯.

(17.200)

The first excited state has the antisymmetric wave function 1 Ψ1 (x) = √ [ψ0 (x − a) − ψ0 (x + a)] 2

(17.201)

and the slightly higher energy E

(1)

=E

(0)

∆E (0) + = ω/2 + Ke−Acl /¯h h ¯. 2

(17.202)

The level splitting is therefore ∆E = 2K¯ he−Acl /¯h .

(17.203)

1110

17 Tunneling

Inserting K from (17.131), we obtain the formula √

s

∆E = 4 3

Acl h ¯ ωe−Acl /¯h , 2π¯ h

(17.204)

with Acl = (2/3)a2 ω. When expressing the action in terms of the height of the potential barrier Vmax = a2 ω 2 /8 = 3ωAcl/16, the formula reads s

√ 8Vmax ∆E = 4 3 h ¯ ωe−16Vmax /3¯hω . 3πω¯ h

(17.205)

The level splitting decreases exponentially with increasing barrier height. Note that Vmax is related to the coupling constant of the x4 -interaction by Vmax = ω 4 /16g. To ensure the consistency of the approximation we have to check that the assumption of a low density gas of kinks and antikinks is self-consistent. When looking at the series (17.195) for the exponential (17.196), we see that the average number of contributing terms is given by ¯ ≈ KLe−Acl /¯h = ∆E L. N 2¯ h

(17.206)

The associated average separation between kinks and antikinks is ∆L ≡ 2¯ h/∆E.

(17.207)

If we compare this with their size 2/ω, we find the ratio distance h ¯ω ≈ . size ∆E

(17.208)

For increasing barrier height, the level splitting decreases and the dilution increases exponentially. Thus the dilute-gas approximation becomes exact in the limit of infinite barrier height.

17.8

Fluctuation Correction to Level Splitting

Let us calculate the first fluctuation correction to the level splitting formula (17.204). For this we write the potential (17.1) as in (5.78): V (x) = −

ω2 2 g 4 1 x + x + , 4 4 4g

(17.209)

with the interaction strength g≡

ω2 . 2a2

(17.210)

Expanding the action around the classical solution, we obtain the action of the fluctuations y(τ ) = x(τ ) − xcl (τ ). Its quadratic part was given in Eq. (17.40) which we write as Z 1 A0fl = dτ dτ ′ y(τ )Oω (τ, τ ′ )y(τ ′ ), (17.211) 2 H. Kleinert, PATH INTEGRALS

1111

17.8 Fluctuation Correction to Level Splitting with the functional matrix ′ d2 3 1 2 Oω (τ, τ ) ≡ − 2 + ω 1 − δ(τ − τ ′ ) dτ 2 cosh2 [ω(τ − τ0 )/2] ′

(17.212)

associated with the Schrödinger operator for a particle in a Rosen-Morse potential (14.157). The prime indicates the absence of the zero eigenvalue in the spectral decomposition of Oω (τ, τ ′ ). Since the associated mode does not perform Gaussian fluctuations, it must be removed from y(τ ) and treated separately. At the semiclassical level, this was done in Subsection 17.3.1, and the zero eigenvalue appeared in the level splitting formula (17.204) as a factor (17.112). The removal gave rise to an additional effective interaction (17.110): Z −1 ′ ′ Aeff = −¯ h log 1 + A dτ x (τ )y (τ ) . (17.213) e cl cl With (17.88)–(17.91), this can be rewritten after a partial integration as # " r Z 3g eff ′ dτ y0 (τ )y(τ ) . Ae = −¯h log 1 − ω3 The interaction between the fluctuations is Z g int Afl = dτ y 4 (τ ) + 4xcl (τ )y 3 (τ ) . 4

(17.214)

(17.215) int

eff

In the path integral, we now perform a Taylor series expansion of the exponential e−(Afl +Ae )/¯h in powers of the coupling strength g. A perturbative evaluation of the correlation functions of the fluctuations y(τ ) according to the rules of Section 3.20 produces a correction factor to the path integral g¯h C = 1 − (I1 + I2 + I3 ) 3 + O(g 2 ) , (17.216) ω where I1 , I2 , and I3 are the dimensionless integrals running over the entire τ -axis: Z ω3 dτ hy 4 (τ )iOω , I1 = 4¯h2 Z ω3g I2 = − 3 dτ dτ ′ xcl (τ )hy 3 (τ )y 3 (τ ′ )iOω xcl (τ ′ ), 2¯h r Z ω 3 3g I3 = − 2 dτ dτ ′ y0′ (τ )hy(τ )y 3 (τ ′ )iOω xcl (τ ′ ). ω3 ¯h

(17.217)

In order to check the dimensions we observe that the classical solution (17.28) can be written p p with (17.210) as xcl (τ ) = ω 2 /2g tanh[ω(τ − τ0 )/2], while y(τ ) and τ have the dimensions ¯h/ω and 1/ω, respectively. The Dirac brackets h. . .iOω denote the expectation with respect to the quadratic fluctuations controlled by the action (17.211). Due to the absence of a zero eigenvalue, the fluctuations are harmonic. The expectation values of the various powers of y(τ ) can therefore be expanded according to the Wick rule of Section 3.17 into a sum of pair contractions involving products of Green functions G′Oω (τ, τ ′ ) =

hy(τ )y(τ ′ )iOω = h ¯ Oω−1 (τ, τ ′ ),

where Oω−1 (τ, τ ′ ) denotes the inverse of the functional matrix (17.212). The first term in (17.217) gives rise to three Wick contractions and becomes Z 3ω 3 I1 = dτ G′O2ω(τ, τ ). 4¯h2

(17.218)

(17.219)

1112

17 Tunneling

The integrand contains an asymptotically constant term which produces a linear divergence for large L. This divergence is subtracted out as follows: Z 3ω 3ω 3 ¯h2 I1 = L + 2 dτ G′O2ω(τ, τ ) − . (17.220) 16 4ω 2 4¯h The first term is part of the first-order fluctuation correction without the classical solution, i.e., it contributes to the constant background energy of the classical solution. It is obtained by replacing G′O2ω(τ, τ ′ ) → ¯hGω (τ − τ ′ ) =

¯h −ω|τ −τ ′| e 2ω

(17.221)

[recall (3.301) and (3.246)]. In the amplitudes (17.195), the background energy changes only the 3 exponential prefactor e−ωL/2¯h to e−(1+3g¯h/16ω )ωL/2¯h and does not contribute to the level splitting. The level splitting formula receives a correction factor g¯h g¯h C ′ = 1 − c1 3 + . . . = 1 − (I1′ + I2′ + I3′ ) 3 + O(g 2 ) , (17.222) ω ω in which all contributions proportional to L are removed. Thus I1 is replaced by its subtracted part I1′ ≡ I1 − L3ω/16. The integral I2 has 15 Wick contractions which decompose into two classes: Z gω 3 dτ dτ ′ I2 ≡ I21 + I22 = − 3 2¯ h × xcl (τ ) 6G′O3ω(τ, τ ′ ) + 9G′Oω (τ, τ )G′Oω (τ, τ ′ )G′Oω (τ ′ , τ ′ ) xcl (τ ′ ). (17.223)

Each of the two subintegrals I21 and I22 contains a divergence with L which can again be found ′ via the replacement (17.221). The subtracted integrals in (17.222) are I21 = I21 + ωL/8 and ′ −ωL/2¯ h I22 = I22 +3ωL/16. Thus, altogether, the exponential prefactor e in the amplitudes (17.195) 3 h/2ω 3 )ωL/2¯ h is changed to e−[1/2+(3/16−1/8−9/16)g¯h/ω )ωL/2¯h = e−(1/2−g¯ , in agreement with (5.258). √ To compare the two expressions, we have to set ω = 2 since the present ω is the frequency at the bottom of the potential wells whereas the ω in Chapter 5 [which is set equal to 1 in (5.258)] parametrized the negative curvature at x = 0. The Wick contractions of the third term lead to the finite integral r Z ω 3 3g I3 = I3′ = −3 2 dτ dτ ′ y0′ (τ )G′Oω (τ, τ ′ )G′Oω (τ ′ , τ ′ )xcl (τ ′ ). (17.224) ω3 ¯h The correction factor (17.216) can be pictured by means of Feynman diagrams as C =1−3 + 3

1 + 6 2! + O(g 2 ),

+9

! (17.225)

where the vertices and lines represent the analytic expressions shown in Fig. 17.5. For the evaluation of the integrals we need an explicit expression for G′Oω (τ, τ ′ ). This is easily found from the results of Section 14.4.4. In Eq. (14.160), we gave the fixed-energy amplitude (xb |xa )ERM ,EPT solving the Schrödinger equation ¯h2 d2 ¯h2 EPT − − ERM + − (xb |xa )ERM ,EPT = −i¯hδ(xb − xa ). 2µ dx2 2µ cosh2 x (17.226) H. Kleinert, PATH INTEGRALS

1113

17.8 Fluctuation Correction to Level Splitting Inserting EPT = (¯ h2 /2µ)s(s + 1), the amplitude reads for xb > xa (xb |xa )ERM ,EPT =

−iµ Γ(m(ERM ) − s)Γ(s + m(ERM ) + 1) ¯h × Ps−m(ERM ) (tanh xb )Ps−m(ERM ) (− tanh xa ),

(17.227)

with q 1 − 2µERM /¯h2 .

(17.228)

¯ h ωτ ωτ ′ Γ(m − 2)Γ(m + 3) P2−m (tanh ) P2−m (− tanh ), ω 2 2

(17.229)

m(ERM ) =

After a variable change x = ωτ /2 and h ¯ 2 /µ = ω 2 /2, we set s = 2 and insert the energy ERM = 2 −3ω /4. Then the operator in Eq. (17.226) coincides with Oω (τ, τ ′ ) of Eq. (17.212), and we obtain the desired Green function for τ > τ ′ GOω (τ, τ ′ ) =

with m = 2. Due to translational invariance along the τ -axis, this Green function has a pole at ERM = −3ω 2 /4 which must be removed before going to this energy. The result is the subtracted Green function G′Oω (τ, τ ′ ) which we need for the perturbation expansion. The subtraction procedure is most easily performed using the formula G′Oω = (d/dERM )ERM GOω |ERM =−3ω2 /4 . In terms of the parameter m, this amounts to 1 d G′Oω (τ, τ ′ ) = . (17.230) (m2 − 4)GOω (τ, τ ′ ) 2m dm m=2

Inserting into (17.227) the Legendre polynomials from (14.164), P2−m (z)

1 = Γ(1 + m)

1+z 1−z

−m/2 1−

3 3 2 (1 − z) + (1 − z) , 1+m (1 + m)(2 + m)

(17.231)

the Green function (17.230) can be written as G′Oω (τ, τ ′ ) = h ¯ [Y0 (τ> )y0 (τ< ) + y0 (−τ> )Y0 (−τ< )],

(17.232)

where τ> and τ< are the greater and the smaller of the two times τ and τ ′ , respectively, and y0 (τ ), Y0 (τ ) are the wave functions r √ ωτ 3ω 1 −2 y0 (τ ) = −2 6ωP2 − tanh =− , (17.233) 2 8 cosh2 ωτ 2 g xcl (τ ) ¯h g 4¯ h r 3g ′ y (τ ) ω3 0 G′Oω (τ, τ ′ )

Figure 17.5 Eq. (17.225).

Vertices and lines of Feynman diagrams for correction factor C in

1114

17 Tunneling

1 1 Y0 (τ ) = √ 2 6ω 2ωm

1 d ωτ (m2 − 4)Γ(m − 2)Γ(m + 3) P2−m tanh 2 dm 2 2 d −m ωτ P2 tanh . + (m − 4)Γ(m − 2)Γ(m + 3) dm 2 m=2

(17.234)

From (17.231) we see that

√ d −m 6 ωτ P tanh = y0 (τ )[6(3 − 2γ + ωτ ) − e−ωτ (8 + e−ωτ )], dm 2 2 m=2 144

(17.235)

where γ ≈ 0.5773156649 is the Euler-Mascheroni constant (2.461). Hence Y0 (τ )

=

1 y0 (τ )[e−ωτ (e−ωτ + 8) − 2(2 + 3ωτ )]. 12ω 2

(17.236)

For τ = τ ′ , the Green function is G′Oω (τ, τ )

=

¯ h 1 2ω cosh4

ωτ 2

11 4 ωτ 2 ωτ cosh + cosh − . 2 2 8

(17.237)

Note that an application of the Schr¨ odinger operator (17.212) to the wave functions Y0 (τ ) and y0 (τ ) produces −y0 (τ ) and 0, respectively. These properties can be used to construct the Green function G′Oω (τ, τ ′ ) by a slight modification of the Wronski method of Chapter 3. Instead of the differential equation Oω G(τ, τ ′ ) = h ¯ δ(τ − τ ′ ), we must solve the projected equation Oω′ G′Oω (τ, τ ′ ) = h ¯ [δ(τ − τ ′ ) − y0 (τ )y0 (τ ′ )],

(17.238)

where the right-hand side is the completeness relation without the zero-eigenvalue solution: X yn (τ )yn (τ ′ ) = δ(τ − τ ′ ) − y0 (τ )y0 (τ ′ ). (17.239) n6=0

The solution of the projected equation (17.238) is precisely given by the combination (17.232) of the solutions Y0 (τ ) and y0 (τ ) with the above-stated properties. The evaluation of the Feynman integrals I1 , I21 , I22 , I3 is somewhat tedious and is therefore described in Appendix 17A. The result is I1′ =

97 , 560

′ I21 =

53 , 420

′ I22 =

117 , 560

I3 =

49 . 20

(17.240)

These constants yield for the correction factor (17.222) 71 g¯h 2 + O(g ) , C′ = 1 − 24 ω 3

(17.241)

modifying the level splitting formula (17.204) for the ground state energy to r √ 3 3 ω 3 /3g (0) ∆E = 4 3 ¯hωe−ω /3g¯h−71g¯h/24ω +... . 2π¯h

(17.242)

This expression can be compared with the known energy eigenvalues of the lowest two double-well states. In Section 5.15, we have calculated the variational approximation W3 (x0 ) to the effective classical potential of the double well and obtained for small g an energy (see Fig. 5.24) which did not yet incorporate the effects of tunneling. We now add to this the level shifts ±∆E (0) /2 from Eq. (17.242) and obtain the curves also shown in Fig. 5.24. They agree reasonably well with the Schrödinger energies. H. Kleinert, PATH INTEGRALS

1115

17.9 Tunneling and Decay

Figure 17.6 Positions of extrema xex in asymmetric double-well potential, plotted as function of asymmetry parameter ǫ. If rotated by 900 , the plot shows the typical cubic shape. Between ǫ> and ǫ< , there are two minima and one central maximum. The branches denoted by “min” are absolute minima; those denoted by “rel min” are relative minima.

17.9

Tunneling and Decay

The previous discussion of level splitting leads us naturally to another important tunneling phenomenon of quantum theory: the decay of metastable states. Suppose that the potential is not completely symmetric. For definiteness, let us add to V (x) of Eq. (17.1) a linear term which breaks the symmetry x → −x: ∆V = −ǫ

x−a . 2a

(17.243)

For small ǫ > 0, this slightly depresses the left minimum at x = −a. The positions of the extrema are found from the cubic equation ω2a V (xex ) = 2 ′

"

xex a

3

#

ǫ xex − − 2 2 = 0. a ω a

(17.244)

They are shown in Fig. 17.6. For large ǫ, there is only one extremum, and this is always a minimum. In the region where xex has three solutions, say x− , x0 , x+ , the branches denoted by “rel min.” in Fig. 17.6 correspond to relative minima which lie higher than the absolute minimum. The central branch corresponds to a maximum. As ǫ decreases from large positive to large negative values, a classical particle at rest at the minimum follows the upper branch of the curve and drops to the lower branch as ǫ becomes smaller than ǫ< . Quantum-mechanically, however, there is tunneling

1116

17 Tunneling

to the lower state before ǫ< . Tunneling sets in as soon as ǫ becomes negative, i.e., as soon as the initial minimum at x+ comes to lie higher than the other minimum at x− . The state whose wave packet is localized initially around x+ decays into the lower minimum around x− . After some finite time, the wave packet is concentrated around x− . A state with a finite lifetime is described analytically by an energy which lies in the lower half of the complex energy plane, i.e., which carries a negative imaginary part E im . The imaginary part gives half the decay rate Γ/2¯ h. This follows directly from the temporal behavior of a wave function with an energy E = E re + iE im which is given by ψ(x)e−iEt/¯h = ψ(x)e−iE

re t/¯ h

eE

im t/¯ h

= ψ(x)e−iE

re t/¯ h

e−Γt/2¯h .

(17.245)

The last factor leads to an exponential decay of the norm of the state Z

d3 x|ψ(x)|2 = e−Γt/¯h ,

(17.246)

which shows that h ¯ /Γ is the lifetime of the state. A positive sign of the imaginary part of the energy is ruled out since it would imply the state to have an exponentially growing norm. We are now going to calculate Γ for the lowest state.4 If ǫ has a small negative value, the initial probability is concentrated in the potential valley around the righthand minimum x = x+ ≈ a. We assume the potential barrier to be high compared to the ground state energy. Then a semiclassical treatment is adequate. In this approximation we evaluate the amplitude (x+ tb |x+ ta ). It contains the desired information on the lifetime of the lowest state by behaving, for large tb − ta , as (x+ tb |x+ ta ) ∼ ψ0 (0)ψ0 (0)e−iE

re (t −t )/¯ a h b

e−Γ(tb −ta )/2¯h .

As before, it is convenient to work with the Euclidean amplitude with τa = −L/2 and τb = L/2, (x+ L/2|x+ − L/2),

(17.247)

which behaves for large L as (x+ L/2|x+ − L/2) ∼ ψ0 (0)ψ0 (0)e−E

re L/¯ h

eiΓL/2¯h .

(17.248)

The classical approximation to this amplitude is dominated by the path solving the imaginary-time equation of motion which corresponds to a real-time motion in the reversed potential −V (x) (see Fig. 17.7). The particle starts out at x = x+ for 4

Due to the finite lifetime this state is not stationary. For sufficiently long lifetimes, however, it is approximately stationary for a finite time. H. Kleinert, PATH INTEGRALS

1117


Figure 17.7 Classical bubble solution in reversed asymmetric quartic potential for ǫ < 0, starting out at the potential maximum at x+ , crossing the valley, and returning to the maximum.

τ = −L/2, traverses the minimum of −V (x) at some finite value τ = τ0 , and comes back to x+ at τ = L/2. This solution is sometimes called a bounce solution, because of its returning to the initial point. There exists an important application of the tunneling theory to the vaporization process of overheated water, to be discussed in Section 17.11. There the same type of solutions plays the role of critical bubbles triggering the phase transition. Since bounce solutions were first discussed in this context [3], we shall call them bubble solutions or critical bubbles. We now proceed as in the previous section, i.e., we calculate a) the classical action of a bubble solution, b) the quadratic fluctuations around a bubble solution, c) the sum over infinitely many bubble solutions. By following these three steps naively, we obtain the amplitude (x+ L/2|x+ − L/2) =

r

q ω −ωL/2¯h e exp Acl /2π¯ hK ′ Le−Acl /¯h . π¯ h

(17.249)

Here Acl is the action of the bubble solution and K ′ collects the fluctuations of all nonzero-eigenvalue modes in the presence of the bubble solution as in (17.192): K′ =

v u Q0 u λ n t Qn ′ λn n

.

(17.250)

L

The translational invariance makes the imaginary part in the exponent proportional to the total length L of the τ -axis. From the large-L behavior of the amplitude (17.249) we obtain the ground state energy E (0)



ω = − 2

s



Acl ′ −Acl /¯h  Ke . 2π¯ h

(17.251)

In order to deduce the finite lifetime of the state from this formula we note that, just like the kink solution, the bubble solution has a zero-eigenvalue fluctuation

1118

17 Tunneling

associated with the time translation invariance of the system. As before, its wave function is given by the time derivative of the bubble solution y0 (τ ) = √

1 ′ x (τ ). Acl cl

(17.252)

In contrast to the kink solution, however, the bubble solution returns to the initial position, implying that xcl (τ ) has a maximum. Thus, the zero-eigenvalue mode ∝ x′cl (τ ) contains a sign change (see Fig. 17.7). In wave mechanics, such a place is called a node of the wave function. A wave function with a node cannot be the ground state of the Schrödinger equation governing the fluctuations "

d2 − 2 + V ′′ (xcl (τ )) yn (τ ) = λn yn (τ ). dτ #

(17.253)

A symmetric wave function without a node must exist, which will have a lower energy than the zero-eigenvalue mode, i.e., it will have a negative eigenvalue λ−1 < 0. The associated wave function is denoted by y−1 (τ ). It corresponds to a size fluctuation of the bubble solution. The nodeless wave function y−1 (τ ) is the ground state. There can be no further negative-eigenvalue solution [4]. It is instructive to trace the origin of the negative sign within the efficient calculation method of the fluctuation determinant in Section 17.5. In contrast to the instanton treated there, the bubble solution has opposite symmetry, with an antisymmetric translational mode x′cl (τ ). From this we may construct again two linearly independent solutions to find the determinant D(τ ) to be used in Eq. (17.154). The negative eigenvalue λ−1 enters in the calculation of the functional integral (17.46) via a fluctuation integral Z

dξ 2 √ 1 e−(1/2¯h)ξ1 λ−1 . 2π¯ h

(17.254)

This integral diverges. The harmonic fluctuations of the integration variable take place around a maximum; they are unstable. At first sight one might hope to obtain a correct result by a naive analytic continuation doing first the integral for λ−1 > 0, where it gives Z

dξ 1 2 √ −1 e−(1/2¯h)ξ−1 λ−1 = √ , λ−1 2π¯ h

(17.255)

and then continuing the right-hand side analytically to negative λ−1 . The result would be Z dξ i 2 √ −1 e−(1/2¯h)ξ−1 λ−1 = ± q . (17.256) 2π¯ h |λ−1 |

From (17.250) and (17.251) we then might expect the formula for the decay rate to be s

1 Acl ′ −Acl /¯h Γ = −2i Ke h ¯ 2π¯ h

(wrong),


1119


Figure 17.8 Action of deformed bubble solution as function of deformation parameter ξ. The maximum at ξ = 1 represents the critical bubble.

with ′

′

K = i|K | =

v u Q0 u t Q n λn

q n6=0,−1 λn L

i

|λ−1 |

.

(17.258)

However, this naive manipulation does not quite give the correct result. As we shall see immediately, the error consists in a missing factor 1/2 which has a simple physical explanation. A more careful analytic continuation is necessary to find this factor [3]. As a function of ξ, it behaves as shown in Fig. 17.8. For a proper analytic continuation, consider a continuous sequence of paths in the functional space and parametrize it by some variable ξ. Let the trivial path x(τ ) ≡ x+

(17.259)

correspond to ξ = 0, and the bubble solution x(τ ) = xcl (τ )

(17.260)

to ξ = 1. The action of the trivial path is zero, that of the bubble solution is A = Acl . As the parameter ξ increases to values > 1, the bubble solution is deformed with a growing portion of the curve moving down towards the bottom of the lower potential valley (see Fig. 17.9). This lowers the action more and more. There is a maximum at the bubble solution ξ = 1. The negative eigenvalue λ−1 < 0 of the fluctuation equation (17.285) is proportional to the negative curvature at the maximum. Since there exists only a single negative eigenvalue, the fluctuation determinant of the remaining modes is positive. It does not influence the process of analytic continuation. Thus we may study the analytic continuation within a simple model integral designed to have the qualitative behavior described above: Z=

Z

0

∞

dξ 2 3 √ eλ(ξ +αξ ) . 2π

(17.261)

The parameter λ stands for the negative eigenvalue λ−1 , whereas α is an auxiliary parameter to help perform the analytic continuation. For α > 0, the integral is

1120

17 Tunneling

Figure 17.9 Sequence of paths as function of parameter ξ, starting out at ξ = 0, with a constant solution in the metastable valley x(τ ) ≡ x ≡ x+ , reaching the extremal bubble solution x(τ ) ≡ xcl (τ ) for ξ = 1, and sliding more and more down towards the stable minimum for 1 < ξ → ∞.

stable and well defined. For α < 0, the “Euclidean action” in the exponent A = −λ(ξ 2 + αξ 3 ) has a maximum at ξm = −

2 . 3α

(17.262)

Near the maximum, it has the expansion 4 A = −λ − (ξ − ξm )2 + . . . . 2 27α

(17.263)

The second term possesses a negative curvature λ which represents the negative eigenvalue λ−1 . The parameter α2 plays the role of h ¯ /(−λAcl ) in the bubble discussion, and the semiclassical expansion of the path integral corresponds to an expansion of the model integral in powers of α2 . We want to show that the lowest two orders of this expansion yield an imaginary part 2

Im Z ∼ eλ4/27α

1 1 q , 2 |λ|

(17.264)

where the exponential is the classical contribution and the factor contains the fluctuation correction. To derive (17.264), we continue the integral (17.261) analytically from α > 0, where it is well defined, into the complex α-plane. It is convenient to introduce a new variable t = αξ. Then Z becomes 1 Z= α

Z

0

∞

"

#

dt λ √ exp 2 (t2 + t3 ) . α 2π

(17.265)

Since λ < 0 this integral converges for α > 0. To continue it to negative real values of α, we set α ≡ |α|eiϕ and increase the angle ϕ from zero to π/2. While doing so, H. Kleinert, PATH INTEGRALS

1121


Figure 17.10 Lines of constant Re (t2 + t3 ) in complex t-plane and integration contours Ci for various phase angles of α (shown in the insert) which maintain convergence of the integral (17.265).

we deform the contour in the t-plane in order to maintain convergence. Thus we introduce an auxiliary real variable t′ and set t = ei2ϕ/3 t′ ,

t′ ∈ (0, ∞).

(17.266)

The continued integral is then performed as dt = ei2ϕ/3 0∞ dt′ . From the geometric viewpoint, the convergence is maintained for the following reason: For α > 0, the real part of the “action” −(λ/α2 )(t2 + t3 ) has asymptotically three mountains at azimuthal angles ϕ = 0, 2π/3, 4π/3, and three valleys at ϕ = π/3, π, 5π/3 (see Fig. 17.10). As α is rotated by the phase eiϕ , these mountains rotate with 2/3 of the angle ϕ anticlockwise in the t-plane. Since the contour keeps running up the same mountain, the integral continues to converge, rendering an analytic function of α. After α has been rotated to eiπ α = −α, the exponent in (17.265) takes back the original form, but the contour C runs up the mountain at ϕ = 2π/3. It does not matter which particular shape is chosen for the contour in the finite regime. We may deform the contour to the shape C2 shown in Fig. 17.10. Next we observe that the point −α can also be reached by rotating α in the clockwise sense with −ϕ increasing to π. In this case the final contour will run like C3 in Fig. 17.10. The difference between the two analytic continuations is R

1 ∆Z ≡ Z(|α|e−iπ ) − Z(|α|eiπ ) = |α|

Z

C4

R

(

)

dt λ 2 √ exp (t + t3 ) , α2 2π

(17.267)

where the contour C4 = C2 − C3 connects the mountain at ϕ = 4π/3 with that at ϕ = 2π/3. The convergence of the combined integral is most rapid if the contour

1122

17 Tunneling

C4 is chosen to run along the line of steepest slope. This traverses the minimum at t = −1 vertically in the complex t-plane. The fact that ∆Z is nonzero implies that the partition function has a cut in the complex α-plane along the negative real axis. Since Z is real for α > 0, it is a real analytic function in the complex α-plane and the difference ∆Z gives a purely imaginary discontinuity across the cut: ∆Z ≡ disc Z = Z(−|α| − iη) − Z(−|α| + iη).

(17.268)

Let us calculate the discontinuity in the limit of small α where the dominant contribution comes from the neighborhood of the point t = −1. While the action at this point has a local maximum along the real t-axis, it has a local minimum along the vertical contour in the complex t-plane. For small α2 , the integral can be found via the saddle point approximation calculating the local minimum in the quadratic approximation: λ4/27α2

disc Z ≈ e

Z

i∞

−i∞

2

= eλ4/27α √

2 dξ √ eλ(ξ−ξ0 ) 2π

(17.269)

i . −λ

Due to the real analyticity of Z, the imaginary part of Z is equal to one half of this: 2

Im Z(−|α| ∓ iη) = ±eλ4/27α

1 √ . 2 −λ

(17.270)

The contour leading up to the extremal point adds only a real part to Z. The result (17.270) is therefore the exact leading contributing to the imaginary part in the ¯ → 0. limit α2 → 0, corresponding to the semiclassical limit h The exponent in (17.270) is the action of the model integral at the saddle point. The second factor produces the desired imaginary part. For a sequence of paths in functional space whose action depends on ξ as in Fig. 17.9, the result can be phrased as follows: Z ∞ Z 1 Z 1−i∞ dξ −A(ξ)/¯h dξ −A(ξ)/¯h dξ −A′′ (1)(ξ−1)2 /2¯h −A(1)/¯ h √ √ √ e = e +e e 0 0 1 2π¯ h 2π¯ h 2π¯ h Z 1 dξ −A(ξ)/¯h i −A(1)/¯h 1 q √ ≈ e + e . (17.271) 2 0 2π¯ h −A′′ (1)

After translating this result to the form (17.254), we conclude that the integration over the negative-eigenvalue mode Z

dξ 2 √ −1 e−ξ−1 λ−1 /2¯h 2π¯ h

(17.272)

becomes, for λ−1 < 0 and after a proper analytic continuation, Z

dξ i 1 2 √ −1 e−ξ−1 λ−1 /2¯h = q . 2 |λ−1 | 2π¯ h

(17.273)


17.10 Large-Order Behavior of Perturbation Expansions

1123

It is easy to give a physical interpretation to the factor 1/2 appearing in this formula, in contrast to the naively continued formula (17.287). At the extremum, the classical solution, which plays the role of a critical bubble, can equally well contract or expand in size. In the first case, the path x(τ ) returns towards to the original valley and the bubble disappears. In the second case, the path moves more and more towards the lower valley at x = −x− , thereby transforming the system into the stable ground state. The factor 1/2 accounts for the fact that only the expansion of the bubble solution produces a stable ground state, not the contraction. The factor 1/2 multiplies the naively calculated imaginary part of the partition function which becomes 1q Acl /2π¯ h|K ′ |Le−Acl /¯h . 2

Im Z(−|g| − iη) ≈

(17.274)

The summation over an infinite number of bubble solutions moves the imaginary contribution to Z into the exponent as follows: Re Z + Im Z = Re Z(1 + Im Z/Re Z) − −−→ Re ZeIm Z/Re Z infinite sum

(17.275)

as in Eqs. (17.195), (17.196). By comparison with (17.248), we obtain the correct semiclassical tunneling rate formula [rather than (17.257)]: 1 Γ= h ¯

s

Acl ′ −Acl /¯h , |K |e 2π¯ h

(17.276)

where K ′ is the square root of the eigenvalue ratios, with the zero-eigenvalue mode removed. The prefactor has the dimension of a frequency. It defines the bubble decay frequency ωatt =

s

Acl |K ′ |. 2π¯ h

(17.277)

The exponential in (17.276) is a “quantum Boltzmann factor” which suppresses the formation of a bubble triggering the tunneling process via its expansion. The subscript indicates that the frequency plays the role of an attempt frequency by which the metastable state attempts to tunnel through the barrier into the stable ground state.

17.10

Large-Order Behavior of Perturbation Expansions

The above semiclassical approach of the decay rate of a metastable state has an important fundamental application. At the end of Chapter 3 we have remarked that the perturbation expansion of the anharmonic oscillator has a zero radius of convergence. This property is typical for many quantum systems. The precise form of the divergence is controlled by the tunneling rate formula (17.276), as we shall see now.

1124

17.10.1

17 Tunneling

Growth Properties of Expansion Coefficients

As a specific, but typical, example we consider the anharmonic oscillator with the action A=

Z

L/2 −L/2

dτ

"

x′2 ω 2 2 g 4 − x − x , 2 2 4 #

(17.278)

and study the partition function as an analytic function of g. It is given by the path integral at large L (which now represents the imaginary time β = 1/kB T , setting h ¯ = 1) Z(g) =

Z

Dx(τ )eA .

(17.279)

The L-dependence of the partition function follows from the spectral representation X

Z(g) =

e−E

(n) (g)L

,

(17.280)

n

where E (n) (g) are the energy eigenvalues of the system. In the limit L → ∞, this becomes an expansion for the ground state energy E (0) (g). In the limit L → ∞, Z(g) behaves like Z(g) → e−E

(0) (g)L

,

(17.281)

exhibiting directly the ground state energy. Since the path integral can be done exactly at the point g = 0, it is suggestive to expand the exponential in powers of g and to calculate the perturbation series Z(g) =

X

Zk

k=0

g ω3

k

.

(17.282)

As shown in Section 3.20, the expansion coefficients are given by the path integrals Zk

L/2 (−g)k = Dx(τ ) dτ x4 (τ ) k! −L/2 * + k Z L/2 (−g) = Z −1 dτ x4 (τ ) . k! −L/2 ω

Z

"Z

#k

"

exp −

Z

L/2

−L/2

dτ

1 2 ω2 2 x˙ + x 2 2

!#

(17.283)

By selecting the connected Feynman diagrams in Fig. 3.7 contributing to this path integral, we obtain the perturbation expansion in powers of g for the free energy F . In the limit L → ∞, this becomes an expansion for the ground state energy E (0) (g), in accordance with (17.281). By following the method in Section 3.18, we find similar expansions for all excited energies E (n) (g) in powers of g. For g = 0, (n) the energies are, of course, those of a harmonic oscillator, E0 = ω(n + 1/2). In general, we find the series E

(n)

(g) =

∞ X

k=0

(n) Ek

k

g 4

.

(17.284)


1125


Most perturbation expansions have the grave deficiency observed in Eq. (3C.27). Their coefficients grow for large order k like a factorial k! causing a vanishing radius of convergence. They can yield approximate results only for very small values of g. (n) Then the expansion terms Ek (g/4)k decrease at least for an initial sequence of kvalues, say for k = 0, . . . , N. For large k-values, the factorial growth prevails. Such series are called asymptotic. Their optimal evaluation requires a truncation after the smallest correction term. In general, the large-order behavior of perturbation expansions may be parametrized as Ek = γp

γ1 γ2 + 2 + ... , k (−4a) (pk)! 1 + k k

β+1 β

k

(17.285)

where the leading term (pk)! grows like p

√

p k (1−p)/2

(pk)! = (k!) (p ) k

p

(2π)(p−1)/2

[1 + O(1/k)] .

(17.286)

This behavior is found by approximating n! via Stirling’s formula (5.204). It is easy to see that the kth term of the series (17.284) is minimal at 1 . p(a|g|)1/p

k ≈ kmin ≡

(17.287)

This is found by applying Stirling’s formula once more to (k!)p and by minimizing ′ γ(k!)p k β (pp a|g|)k with β ′ = β + (1 − p)/2, which yields the equation p log k + log(pp a|g|) + (β + p/2)/k + . . . = 0.

(17.288)

An equivalent way of writing (17.285) is "

#

c1 c2 Ek = γp(−4a)k Γ(pk + β + 1) 1 + + + . . . .(17.289) pk + β (pk + β)(pk + β − 1) The simplest example for a function with such strongly growing expansion coefficients can be constructed with the help of the exponential integral E1 (g) =

Z

∞

g

dt −t e . t

(17.290)

Defining 1 E(g) ≡ e1/g E1 (1/g) = g

Z

0

∞

dt

1 e−t , 1 + gt

(17.291)

this has the diverging expansion E(g) = 1 − g + 2!g 2 − 3!g 3 + . . . + (−1)N N!g N + . . . .

(17.292)

At a small value of g, such as g = 0.05, the series can nevertheless be evaluated quite accurately if truncated at an appropriate value of N. The minimal correction

1126

17 Tunneling

is reached at N = 1/g = 20 where the relative error with respect to the true value E ≈ 0.9543709099 is equal to ∆E/E ≈ 1.14 · 10−8. At a somewhat larger value g = 0.2, on the other hand, the optimal evaluation up to N = 5 yields the much larger relative error ≈ 1.8%, the true value being E ≈ 0.852110880. The integrand on the right-hand side of (17.291), the function 1 , 1+t

B(t) =

(17.293)

is the so-called Borel transform of the function E(g). It has a power series expansion which can be obtained from the divergent series (17.292) for E(g) by removing in each term the catastrophically growing factor k!. This produces the convergent series B(t) = 1 − t + t2 − t3 + . . . ,

(17.294)

which sums up to (17.293). The integral F (g) =

Z

0

∞

dt −t/g e B(t) g

(17.295)

restores the original function by reinstalling, in each term tk , the removed k!-factor. Functions F (g) of this type are called Borel-resummable. They possess a convergent Borel transform B(t) from which F (g) can be recovered with the help of the integral (17.295). The resummability is ensured by the fact that B(t) has no singularities on the integration path t ∈ [0, ∞), including a wedge-like neighborhood around it. In the above example, B(t) contains only a pole at t = −1, and the function E(g) is Borel-resummable. Alternating signs of the expansion coefficients of F (g) are a typical signal for the resummability. The best-known quantum field theory, quantum electrodynamics, has divergent perturbation expansions, as was first pointed out by Dyson [5]. The expansion parameter g in that theory is the fine-structure constant α = 1/137.035963(15) ≈ 0.0073.

(17.296)

Fortunately, this is so small that an evaluation of observable quantities, such as the anomalous magnetic moment of the electron 2

∆µ 1α α ae = = − 0.328 478 965 7 µ 2π π

+ 1.1765(13)

3

α π

+ ... ,

(17.297)

gives an extremely accurate result: atheor = (1 159 652 478 ± 140) · 10−12 . e

(17.298)

The experimental value differs from this only in the last three digits, which are 200 ± 40. The divergence of the series sets in only after the 137th order. H. Kleinert, PATH INTEGRALS


1127

A function E(g) with factorially growing expansion coefficients cannot be analytic at the origin. We shall demonstrate below that it has a left-hand cut in the complex g-plane. Thus it satisfies a dispersion relation 1 E(g) = 2πi

Z

∞

0

dg ′

disc E(−g ′ ) , g′ + g

(17.299)

where disc E(g ′ ) denotes the discontinuity across the left-hand cut disc E(g) ≡ E(g − iη) − E(g + iη).

(17.300)

It is then easy to see that the above large-order behavior (17.289) is in one-to-one correspondence with a discontinuity which has an expansion, around the tip of the cut, p

disc E(−|g|)=2πiγ(a|g|)−(β+1)/p e−1/(a|g|) [1 + c1 (a|g|)1/p + c2 (a|g|)2/p + . . .] . (17.301) The parameters are the same as in (17.289). The one-to-one correspondence is proved by expanding the dispersion relation (17.299) in powers of g/4, giving k

Ek = (−4)

Z

0

∞

dg ′ 1 disc E(−g ′ ). ′k+1 2πi g

(17.302)

The expansion coefficients are given by moment integrals of the discontinuity with respect to the inverse coupling constant 1/g. Inserting (17.301) and using the integral formula5 Z

0

∞

dg

1 (1/p) e−1/(a|g|) = aα pΓ(pα), α+1 |g|

(17.303)

we indeed recover (17.289). From the strong-coupling limit of the ground state energy of the anharmonic oscillator Eq. (5.168) we see that the discontinuity grow for large g like g 1/3 . In this case, the dispersion relation (17.304) needs a subtraction and reads g Z ∞ dg ′ disc E(−g ′ ) E(g) = E(0) + . 2πi 0 g ′ g′ + g

(17.304)

This does not influence the moment formula (17.302) for the expansion coefficients, except that the lowest coefficient is no longer calculable from the discontinuity. Since the lowest coefficient is known, there is no essential restriction.

17.10.2

Semiclassical Large-Order Behavior

The large-order behavior of many divergent perturbation expansions can be determined with the help of the tunneling theory developed above. Consider the potential 5

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 3.478.

1128

17 Tunneling

of the anharmonic oscillator at a small negative coupling constant g (see Fig. 17.11). The minimum at the origin is obviously metastable so that the ground state has only a finite lifetime. There are barriers to the right and left of the metastable minimum, which are very high for very small negative coupling constants. In this limit, the lifetime can be calculated accurately with the semiclassical methods of the last section. The fluctuation determinant yields an imaginary part of Z(g) of the form (17.270), which determines the imaginary part of the ground state energy via (17.276), which is accurate near the tip of the left-hand cut in the complex g-plane. From this imaginary part, the dispersion relation (17.302) determines the large-order behavior of the perturbation coefficients. The classical equation of motion as a function of τ is x′′ (τ ) − V ′ (x(τ )) = 0.

(17.305)

The differential equation is integrated as in (17.26), using the first integral of motion 1 ′2 1 2 2 g 4 x − ω x − x = E = const , (17.306) 2 2 4 from which we find the solutions for E = 0 1 τ − τ0 = ± ω

1

Z

dx q = x 1 − (|g|/2ω 2)x2

or

x(τ ) = xcl (τ ) ≡

v u 2 u 2ω 1  ∓ arcosh t



1 , |g| x

ω

v u 2 u 2ω ±t

1 . |g| cosh[ω(τ − τ0 )]

(17.307)

(17.308)

They represent excursions towards the abysses outside the barriers and correspond precisely to the bubble solutions of the tunneling discussion in the last section. The excursion towards the abyss on the right-hand side is illustrated in Fig. 17.11. The associated action is calculated as in (17.29): Acl =

1 ′2 x (τ ) + V (xcl (τ )) = 2 dτ 2 cl −L/2

Z

= 2

L/2

Z

xm

0

Z

L/2

0

dτ [x′2 cl (τ ) − E]

q

dx 2(E + V ) − EL,

(17.309)

where xm is the maximum of the solution. The bubble solution has E = 0, so that Z xm √ 4ω 3 Acl = 2 dx 2V = . (17.310) 3|g| 0 Inserting the fluctuating path x(τ ) = xcl (τ ) + y(τ ) into the action (17.278) and expanding it in powers of y(τ ), we find an action for the quadratic fluctuations of the same form as in Eq. (17.211), but with a functional matrix #′

Oω (τ, τ ) =

"

d2 − 2 + ω 2 + 3gx2cl (τ ) δ(τ − τ ′ ) dτ

=

"

d2 6 − 2 + ω2 1 − 2 dτ cosh [ω(τ − τ0 )]

′

!#′

δ(τ − τ ′ ).

(17.311)



1129

Figure 17.11 Potential of anharmonic oscillator (17.278) for small negative coupling g. The ground state centered at the origin is metastable. It decays via a classical solution which makes an excursion towards the abyss as indicated by the oriented curve.

This is once more the operator of the Rosen-Morse type encountered in Eq. (17.138) with m = ω, z = 1, and s = 2. The subscript ω on the operator symbol indicates the asymptotic harmonic form of the potential. The potential accommodates again two bound states with the normalized wave functions6 and energies (see Fig. 17.12) s

y0 (τ ) = − y−1 (τ ) =

s

3ω sinh[ω(τ − τ0 )] 2 cosh2 [ω(τ − τ0 )]

3ω 1 2 4 cosh [ω(τ − τ0 )]

with λ0 = 0,

(17.312)

with λ−1 = −3ω 2 .

(17.313)

These are the same functions as in (17.52), (17.53), apart from the fact that m is now ω rather than ω/2. However, the energies are shifted with respect to the earlier case. Now the first excited state has a zero eigenvalue so that the ground state has a negative eigenvalue. This is responsible for the finite lifetime of the ground state. The fluctuation determinant is obtained by any of the above procedures, for instance from the general formula (17.143), √ √ λ0n Γ( z − s)Γ( z + s + 1) √ √ = , Q Γ( z)Γ( z + 1) n λn Q

n

6

The sign of y0 is chosen to agree with that of x′cl (τ ) in accordance with (17.88).

(17.314)

1130

17 Tunneling

Figure 17.12 Rosen-Morse Potential for fluctuations around the classical bubble solution.

by inserting the parameters z = 1 and s = 2. The zero eigenvalue is removed by multiplying this with (z − 1)ω 2 , resulting in the eigenvalue ratio " √ # √ Q 0 √ √ n λn 2 Γ( z − 2)Γ( z + 3) √ √ = lim( z − 1)( z + 1)ω = −12ω 2 . (17.315) Q′ z→1 λ Γ( z)Γ( z + 1) n n

The negative sign due to the negative-eigenvalue solution in (17.313) accounts for the instability of the fluctuations. Using formula (17.274), we find the imaginary part of the partition function Im Z(−|g| − iη) ≈

s v u 3 6u t 4ω ωLe−4ω3 /3|g| e−ωL/2 .

π

3|g|

(17.316)

After summing over all bubble solutions, as in (17.275), we obtain the imaginary part of the ground state energy s v u 3 6u (0) t 4ω e−4ω3 /3|g| . Im E (−|g| − iη)= − ω

π

3|g|

(17.317)

A comparison of this with (17.301) fixes the growth parameters of the large-order perturbation coefficients to 1 ω a = 3/4ω 3, β = − , γ = − 2 π

s

6 , p = 1. π

(17.318)

Recalling the one-to-one correspondence between (17.301) and (17.289), we see that the large-order behavior of the perturbation coefficients of the ground state energy E (0) (g) is (0) Ek

ω =− π

s

6 (−3/ω 3 )k Γ(k + 1/2). π


1131


It is just as easy to find the large-k behavior of the excited states. Their decay is triggered by a periodic classical solution with a very long but finite Euclidean period q 2 L, which moves back and forth between positions x< 6= 0 and x> < 2ω /|g|. Its action is approximately given by 4ω 3 (1 − 12e−ωL ). 3|g|

A′cl ≈

(17.320)

In comparison with the limit L → ∞, the Boltzmann-like factor e−Acl of this solution is replaced by −A′cl

e

−Acl

=e

∞ X

n=0

Ancl

12n −nωL e . n!

(17.321)

The exponentials in the sum raise the reference energies in the imaginary part of Z(−|g|−iη) in (17.316) from ω/2 to ω(n+1/2). The imaginary parts for the energies to the nth excited states become s v u

3 12n 6u t 4ω Im E (n) (−|g| − iη)= − ω n! π 3|g|

1+2n

e−4ω

3 /3|g|

,

(17.322)

implying an asymptotic behavior of the perturbation coefficients: (n) Ek

ω =− π

s

6 12n (−3/ω 3 )k Γ(k + n + 1/2). π n!

(17.323)

It is worth mentioning that within the semiclassical approximation, the dispersion integrals for the energies can be derived directly from the path integral (17.279). This can obviously be rewritten as Z(g) =

Z

×

Z

i∞

−i∞

dλ 2πi

Z

da −(ga+λa)/4 e 4 ( "

∞ 0

Dx(τ ) exp −

Z

L/2

L/2

dτ

1 ′2 ω 2 2 λ 4 x + x − x 2 2 4

#)

.

(17.324)

The integration over λ generates a δ-function 4δ( dτ x4 (τ ) − a) which eliminates the additionally introduced a-integration. The integral over a is easily performed. It yields a factor 1/(λ + g), so that we obtain the integral formula R

Z(g) =

Z

i∞

−i∞

dλ 1 Z(−λ). 2πi λ + g

(17.325)

The integrand has a pole at λ = −g and a cut on the positive real λ-axis. We now deform the contour of integration in λ until it encloses the cut tightly in the clockwise sense. In the semiclassical approximation, the discontinuity across the cut is given by Eq. (17.316), i.e., with the present variable λ: s s

Im Z(−|λ| − iη)=

6 π

4ω 3 −4ω3 /3λ −ωL/2 e e . 3λ

(17.326)

1132

17 Tunneling

√ On the upper branch of the cut, λ is positive, on the lower negative. Thus we arrive at a simple dispersion integral from λ = 0 to λ = ∞: Z(g)=2ω

Z

0

∞

dλ 1 2π λ + g

s s

6 π

4ω 3 −4ω3 /3λ −ωL/2 e e . 3λ

(17.327)

s s

(17.328)

For the ground state energy, this implies E

(0)

(g)= − 2ω

Z

∞

0

dλ 1 2π λ + g

6 π

4ω 3 −4ω3 /3λ . e 3λ

Of course, this expression is just an approximation, since the integrand is valid only at small λ. In fact, the integral converges only in this approximation. If the full imaginary part is inserted, the integral diverges. We shall see below that for large λ, the imaginary part grows like λ1/3 . Thus a subtraction is necessary. A convergent integral representation exists for E (0) (g) − E (0) (0). With E (0) (0) being equal to ω/2, we find the convergent dispersion integral Z ∞ ω dλ 1 (0) E (g)= + 2ωg 2 2π λ(λ + g) 0

s s

6 π

4ω 3 −4ω3 /3λ e . 3λ

(17.329)

The subtraction is advantageous also if the initial integral converges since it suppresses the influence of the large-λ regime on which the semiclassical tunneling calculation contains no information. After substituting λ → 4g/3tω 3, the integral (17.329) is seen to become a Borel integral of the form (17.295). By expanding 1/(λ + g) in a power series in g ∞ X 1 = (−1)k g k λ−k−1 , λ + g k=0

(17.330)

we obtain the expansion coefficients as the moment integrals of the imaginary part as a function of 1/g: (0)

Ek = −2ω (−4)k

Z

0

∞

dλ 1 2π λk+1

s s

6 π

4ω 3 −4ω3 /3λ e . 3λ

(17.331)

This leads again to the large-k behavior (17.319). The direct treatment of the path integral has the virtue that it can be generalized also to systems which do not possess a Borel-resummable perturbation series. As an example, one may derive and study the integral representation for the level splitting formula in Section 17.7.

17.10.3

Fluctuation Correction to the Imaginary Part and Large-Order Behavior

It is instructive to calculate the first nonleading term c1 a|g| in the imaginary part (17.301), which gives rise to a correction factor 1 + c1 /k in the large-order behavior H. Kleinert, PATH INTEGRALS

1133


(17.289). As in Section 17.8, we expand the action around the classical solution. The interaction between the fluctuations y(τ ) is the same as before in (17.215). The quadratic fluctuations are now governed by the differential operator d2 6 Oω (τ, τ ′ ) = − 2 + ω 2 1 − 2 dτ cosh ω(τ − τ0 ) "

!#′

δ(τ − τ ′ ),

(17.332)

the prime indicating the absence of the zero eigenvalue. Its removal gives rise to the factor 

h log 1 − Aeff e = −¯

s

3|g| 4ω 3

Z



dτ y0′ (τ )y(τ ) .

(17.333) int

eff

After expanding, in the path integral, the exponential e−(Afl +Ae )/¯h in powers of the interaction up to the second order, a perturbative evaluation of the correlation functions of the fluctuations y(τ ) according to the rules of Section 3.20 yields a correction factor "

#

|g|¯ h C = 1 + (I1 + I2 + I3 ) 3 + O(g 2) , ω

(17.334)

with the same τ -integrals as in Eqs. (17.217), (17.219), (17.223), and (17.224), after replacing g by |g|. The correction parameter C has again a diagrammatic expansion (17.225), where the vertices stand for the same analytic expressions as in Fig. 17.5, except for the third vertex, which is now s

3|g| ′ y (τ ). 4ω 3 0

(17.335)

The lines represent the subtracted Green function G′Oω (τ, τ ′ ) = hy(τ )y(τ ′)iOω = h ¯ Oω−1 (τ, τ ′ ),

(17.336)

where Oω−1 (τ, τ ′ ) is the inverse of the functional matrix (17.332). In contrast to the level splitting calculation in Section 17.8, only the integral I1 requires a subtraction, 3ω 3 I1 = 2 4¯ h

Z

dτ

G′O2ω(τ, τ )

3ω 3ω 3 =L + 2 16 4¯ h

Z

dτ

"

G′O2ω(τ, τ )

h ¯2 − 2 , 4ω #

(17.337)

and Eq. (17.334) assumes that I1 is subtracted, i.e., I1 should be replaced by I1′ ≡ I1 − L3ω/16. The correction factor for the tunneling rate reads, therefore, ′

"

C = 1+

(I1′

#

|g|¯ h + I2 + I3 ) 3 + O(g 2 ) . ω

(17.338)

The subtracted integral contributes only to the real part of the ground state energy which we know to be (1/2 + 3g¯ h/16ω 3)¯ hω.

1134

17 Tunneling

As in Section 17.8, the explicit Green function G′Oω (τ, τ ′ ) is found from the amplitude (17.227). By a change of variables x = ωτ and h ¯ 2 /2µ = ω 2 , setting s = 2, the Schrödinger operator in (17.226) coincides with that in (17.212), provided we set ERM = 0. The amplitude (17.227) then yields the Green functions for τ > τ ′ h ¯ Γ(m − 2)Γ(m + 3) × P2−m (tanh ωτ )P2−m(− tanh ωτ ′ ), (17.339) GOω (τ, τ ′ ) = 2ω with m = 1. Due to translational invariance along the τ -axis, this Green function has a pole at ERM = 0 [just like the Green function (17.229)]. The pole must be removed before going to this energy, and the result is the subtracted Green function G′Oω (τ, τ ′ ), given by G′Oω (τ, τ ′ )

1 d = . (m2 − 1)GOω (τ, τ ′ ) 2m dm m=1

(17.340)

Using (17.231), we find the subtracted Green function

G′Oω (τ, τ ′ ) = h ¯ [Y0 (τ> )y0 (τ< ) + y0 (−τ> )Y0 (−τ< )],

(17.341)

with s

s

3ω −1 3ω sinh ωτ y0 (τ ) = 2 P2 (− tanh ωτ ) = − , 2 2 cosh2 ωτ Y0 (τ ) =

s

2 1 3ω 8ωm

s

(

"

(17.342)

#

1 d (m2 − 1)Γ(m − 2)Γ(m + 3) P2−m (tanh ωτ ) 2 dm ) h i d 2 −m + (m − 1)Γ(m − 2)Γ(m + 3) P2 (tanh ωτ ) dm m=1 "

#

3 2 3 1 1 sinh ωτ 1 = − + − ωτ − − e−ωτ . 2 3 3ω 4 cosh ωτ 4 8 cosh ωτ 4

(17.343)

For τ = τ ′ h ¯ 1 (cosh2 ωτ − 1)(cosh2 ωτ − 1/2). (17.344) 2ω cosh2 ωτ The evaluation of the integrals I1′ , I21 , I22 , I3 proceeds as in Section 17.8 (performed in Appendix 17A), yielding [6] 11 · 29 71 3 · 13 53 I1′ = − 4 , I21 = − 5 , I22 = 4 , I3 = − 4 . (17.345) 2 ·5·7 2 ·3·7 2 ·7 2 ·5 The correction factor (17.338) is therefore G′Oω (τ, τ ) =

"

#

95 3|g|¯ h C = 1− + O(g 2) . 3 72 4ω ′

(17.346)

Using the one-to-one correspondence between (17.289) and (17.301), this yields the large-k behavior of the expansion coefficients of the ground state energy: (0)

Ek

ω =− π

s

6 (−3/ω 3 )k Γ(k + 1/2)[1 − 95/72k + . . .]. π

(17.347)


1135


17.10.4

Variational Approach to Tunneling. Perturbation Coefficients to All Orders

The semiclassical calculations of tunneling amplitudes are valid only for very high barriers. It is possible to remove this limitation with the help of a variational approach [7] similar to the one described in Chapter 5. For simplicity, we discuss here only the case of an anharmonic oscillator at zero temperature. For the lowest energy levels we shall derive highly accurate imaginary parts over the entire left-hand cut in the coupling constant plane. The accuracy can be tested by inserting these imaginary parts into the dispersion relation (17.329) to recover the perturbation coefficients of the energies. These turn out to be in good agreement with the exact ones to all orders. For the path integral of the anharmonic oscillator Z(g) =

Z

(

Dx(τ ) exp −

Z

L/2

−L/2

dτ

"

1 ′2 ω 2 2 g 4 x + x + x 2 2 4

#)

,

(17.348)

the variational energy (5.32) at zero temperature is given by W1 =

Ω ω 2 − Ω2 2 3g 4 + a + a, 2 2 4

(17.349)

with a2 = 1/2Ω. We have omitted the path average argument x0 since, by symmetry of the potential, the minimum lies at x0 = 0. The energy has to be extremized in Ω2 . This yields the cubic equation Ω3 − ω 2 Ω − 3g/2 = 0. The physically relevant solution starts out with ω at g = 0 and has two branches: For g ∈ (−g (0) , 0) with √ g (0) = 4ω 3/9 3 [compare (5.163)], it is given by 2ω π 1 Ω = √ cos − arccos (−g/g (0) ) . 3 3 3

(17.350)

For large negative coupling constants g < −g (0) , the solution is ω Ωre = √ cosh(γ/3), Ωim = ω sinh(γ/3); 3

γ = arcosh (−g/g (0) ).

(17.351)

In this regime, the ground state energy acquires an imaginary part 1 3g Im W1 = Ωi (1 − 1/|Ω|2 ) − Ωre Ωim /2|Ω|4 . 4 4

(17.352)

This imaginary part describes the instability of the system to slide down into the two abysses situated at large positive and negative x. In this regime of coupling constants, the barriers to the right and left of the origin are no obstacle to the decay since they are smaller than the zero-point energy. In the first regime of small negative coupling constants g ∈ (−g (0) , 0), the barriers are high enough to prevent at least one long-lived ground state from sliding down. Its energy is approximately given by the minimum of (17.349). It can decay towards

1136

17 Tunneling

the abysses via an extremal excursion across the trial potential Ω2 x2 /2 + gx4/4. The associated bubble solution reads, according to (17.308), q

x(τ ) = xcl (τ ) ≡ ± 2Ω2 /|g|

1 . cosh[Ω(τ − τ0 )]

(17.353)

It has the action Acl = 4Ω3 /3|g|. Its fluctuation determinant is given by (17.315), if q ω is replaced by the trial frequency Ω. Translations contribute a factor βΩ Acl /2π. Thus, the partition function has an imaginary part s v u 3 6u t 4Ω e−βΩ/2−4Ω3 /3|g| . Im Z(−|g| − iη)=βΩ

π

3|g|

(17.354)

In the variational approach, this replaces the semiclassical expression (17.316), which im will henceforth be denoted by Zsc (g). The expression (17.354) receives fluctuation corrections. To lowest order, they int produce a factor exp(−hAint fl,tot iOΩ ), where the action Afl,tot contains the interaction terms (17.215) and (17.213) of the fluctuations, with ω replaced by Ω, plus additional terms arising from the variational ansatz. They compensate for the fact that we are using the trial potential Ω2 x2 /2 rather than the proper ω 2 x2 /2 as the zeroth-order potential for the perturbation expansion. These compensation terms have the action Aint fl,var

ω 2 − Ω2 2 = dτ x (τ ) 2 −∞ Z ∞ ω 2 − Ω2 2 = [xcl (τ ) + 2xcl (τ )y(τ ) + y 2 (τ )]. dτ 2 −∞ Z

∞

(17.355)

The expectations h. . .iOΩ in the perturbation correction are calculated with respect to fluctuations governed by the operator (17.332), in which ω is replaced by Ω. As before, all correlation functions are expanded by Wick’s rule into sums of products of the simple correlation functions G′OΩ (τ, τ ′ ) of (17.336). Using the integral formula (17.54), we have Z

∞

−∞

The expectation of Z

∞

−∞

R∞

−∞ 2

dτ x2cl (τ ) = 4Ω/|g|.

(17.356)

dτ y(τ )2 is found with the help of (17.344) as

dτ hy (τ )iOΩ

h i 1 1Z∞ = L + dτ G′OΩ (τ, τ ) − 1/2 2Ω Ω −∞ 1 7 = L . − 2Ω 6Ω2

(17.357)

The second term can be obtained quite simply by differentiating the logarithm of (17.314) with respect to Ω2 z. The linearly divergent term L/2Ω contributes to the earlier-calculated term proportional to L in the integral (17.337) (with ω replaced by Ω); together they yield H. Kleinert, PATH INTEGRALS

1137


L-times W1 of (17.349). Thus we can remove a factor e−LW1 from Im Z, write Z as ≈ Re ZeImZ/ReZ = e−LW1 +ImZ/ReZ [as in (17.275)], and deduce the imaginary part of the energy from the exponent. We now go over to the cumulants in accordance with the rules of perturbation theory in Eqs. (3.480)–(3.484) involving the integrals (17.217) (with g and ω replaced by |g| and Ω, respectively). Using (17.345) we find the correction factor e−A0 −A1 with ! 95 |g| 1 2 7 4Ω 2 , A1 = (ω − Ω ) . (17.358) A0 = − 96 Ω3 2 |g| 6Ω2 If we want to find all terms contributing to the imaginary part up to the order g, we must continue the perturbation expansion to the next order. This yields a further factor 1 int 2 int 2 exp [hAfl,tot iOΩ − hAfl,tot iOΩ ] = exp(−A2 − A3 − A4 ), (17.359) 2 with the integrals Z 1 2 2 2 dτ dτ ′ xcl (τ )hy(τ )y(τ ′)iOΩ xcl (τ ′ ), A2 = − (ω − Ω ) 2 s Z 3|g| A3 = −(ω 2 − Ω2 ) dτ dτ ′ y0′ (τ )hy(τ )y(τ ′)iOΩ xcl (τ ′ ), 3 4Ω Z A4 = (ω 2 − Ω2 )|g|

dτ dτ ′ xcl (τ )hy(τ )y 3(τ ′ )iOΩ xcl (τ ′ ).

(17.360)

Performing the Wick contractions in the correlation functions, the integrals are conveniently rewritten as 1 1 A2 = − (ω 2 − Ω2 )2 a2 , 2 Ω|g| 1 A3 = −(ω 2 − Ω2 ) 2 a3 , Ω 1 A4 = (ω 2 − Ω2 ) 2 a4 , (17.361) Ω where a2 , a3 , a4 are given by a2 = |g|Ω 2

a3 = Ω

s

Z

dτ dτ ′ xcl (τ )G′OΩ (τ, τ ′ )xcl (τ ′ ),

3|g| Z dτ dτ ′ y0′ (τ )G′OΩ (τ, τ ′ )xcl (τ ′ ), 4ΩZ3

a4 = 3|g|Ω2

dτ dτ ′ xcl (τ )G′OΩ (τ, τ ′ )G′OΩ (τ ′ , τ ′ )xcl (τ ′ ).

(17.362)

In terms of these, the imaginary part of the energy reads Im E(−|g| − iη) =

s v u 3 6u t 4Ω e−4Ω3 /3|g|−c1 3|g|/4Ω3 −Ω

ω 2 − Ω2 × exp − 2 "

π

3|g|

4Ω 7 a3 − a4 − −2 2 |g| 6Ω Ω2

!

(ω 2 − Ω2 )2 + a2 , (17.363) 2Ω|g| #

1138

17 Tunneling

evaluated at the Ω-value (17.350). To best visualize the higher-order effect of fluctuations, we factorize (17.363) into the semiclassical part (17.317) and a correction factor εi (g), Im E(−|g| − iη) =

s v u 3 6u t 4ω e−4ω3 /3|g| ε (g), −ω

π

3|g|

i

(17.364)

where Ω 5/2 Ω3 − ω 3 3|g| εi (g) = exp − 4 − c1 3 ω 3|g| 4Ω ! 2 2 ω − Ω 4Ω 7 a3 − a4 (ω 2 − Ω2 )2 − − 2 + − a 2 . 2 |g| 6Ω2 Ω2 2Ω|g|

(17.365)

The calculation of the integrals (17.362) proceeds as in Appendix 17A, yielding

Figure 17.13 Reduced imaginary part of lowest three energy levels of anharmonic oscillator for negative couplings plotted against g′ ≡ g/ω 3 . The semiclassical limit corresponds to εi ≡ 1. The small-|g′ | branch is due to tunneling, the large-|g′ | branch to direct decay (sliding). Solid and dotted curves show the imaginary parts of the variational approximations W1 and W3 , respectively; dashed straight lines indicate the exactly known slopes.

a2 = −1, a3 = 3/4, a4 = 1/12. The result is shown in Fig. 17.13. The slope of εi (g) at g = 0 maintains the firstorder value c1 3/4 = 95/96, i.e., the additional terms in the exponent of (17.365) cancel each other to first order in g. H. Kleinert, PATH INTEGRALS


1139

There exists a short derivation of this result using the same method as in the derivation of Eq. (5.190). We take the fluctuation-corrected semiclassical approximation at the frequency ω Im E(−|g| − iη) =

s v u 3 6u t 4ω e−4ω3 /3|g|−3c1 |g|/4ω3 , −ω

π

3|g|

(17.366)

move the ω-dependent prefactor into the exponent with the help of the logarithm, q q 2 2 2 2 replace everywhere ω by Ω − (Ω − ω ) = Ω − gr 2 /2 with r 2 = 2(Ω2 − ω 2 )/g, and expand the exponent in powers of g including all orders of g to which the exponent of (17.366) is known (treating r as a quantity of order unity). This leads again to (17.364) with (17.365). The imaginary part is inserted into the dispersion relation (17.329) and yields for positive g the energy E

(0)

Z ∞ ω dλ 1 (g)= + 2ωg 2 2π λ(λ + g) 0

s s

6 π

4ω 3 −4ω3 /3λ e εi (λ). 3λ

(17.367)

Expanding the integrand in powers of g gives an integral formula for the perturbation coefficients analogous to (17.331). Its evaluation yields the numbers shown in Table 17.1. They are compared with exact previous larger-order values (17.319) which follow from εi ≡ 1. The improvement of our knowledge on the imaginary part of the energy makes it possible to extend the previous large-order results to low orders. Even the lowest coefficient with k = 1 is reproduced very well [8]. The high degree of accuracy of the low-order coefficients is improved further by going to the higher variational approximation W3 of Eq. (5.192) and extracting from it the imaginary part Im W3 (0) at zero temperature [9]. When continuing the coupling constant g to the sliding regime, we obtain the dotted curve in Fig. 17.13. It merges rather smoothly into the tunneling branch at g ≈ −0.24. Plotting the merging regime with more resolution, we find two closely lying intersections at g ′ = −0.229 and g ′ = −0.254. We choose the first of these to cross over from one branch to the other. After inserting the imaginary part into the integral (17.331), we obtain the fifth column in Table 17.1. For k = 1, the accuracy is now better than 0.05%. To make the approximation completely consistent, the tunneling amplitude should also be calculated to the corresponding order. This would yield a further improvement in the low-order coefficients. It is instructive to test the accuracy of our low-order results by evaluating the dispersion relation (17.367) for the g-dependent ground state energy E (0) (g). The results shown in Fig. 17.14 compare well with the exact curves. They are only slightly worse than the original Feynman-Kleinert approximation W1 evaluated at positive values of g. We do not show the approximation W3 since it is indistinguishable from the exact energy on this plot. The approximation obtained from the dispersion relation has the advantage of possessing the properly diverging power series expansion and a reliable information

1140

17 Tunneling

Table 17.1 Comparison between exact perturbation coefficients, semiclassical ones, and those obtained from moment integrals over the imaginary parts consisting of (17.363) in the tunneling regime and the analytic continuation of the variational approximations W1 and W3 in the sliding regime. An alternating sign (−1)k−1 is omitted and ω is set equal to 1. k 1 2 3 4 5 6 7 8 9 10

Ek 0.75 2.625 20.8125 241.289063 3580.98047 63982.8135 1329733.73 31448214.7 833541603 24478940700

Eksc 1.16954520 5.26295341 39.4721506 414.457581 5595.17734 92320.4261 1800248.43 40505587.0 1032892468 29437435332

Ekvar1+disp 0.76306206 2.49885978 18.3870038 205.886443 3093.38043 57436.2852 1244339.99 30397396.0 822446267 24420208763

Ekvar3+disp 0.74932168 2.61462012 20.7186128 240.857317 3590.69587 64432.5387 1342857.03 31791078.0 842273537 24703889150

on the analytic cut structure in the complex g-plane. Also here, the third-order (0) result Evar3+disp based on the imaginary part of W3 for g < 0 is so accurate that it cannot be distinguished from the exact ones on the plot. The strong-coupling behavior is well reproduced by our curves. Recall the limiting expression for the middle curve given in Eq. (5.77) and the exact one (5.226) with the coefficients of Table 5.9. The calculation of the imaginary part in the sliding regime can be accelerated by removing from the perturbation coefficients the portion which is due to the imaginary part of the tunneling amplitude. By adding the energy associated with this portion in the form of a dispersion relation it is possible to find variational approximations which for positive coupling constants are not only numerically accurate but which also have power series expansions with the correct large-order behavior [which was not the case for the earlier approximations WN (g)]. The entire treatment can be generalized to excited states. The variational energies are then replaced by the minima of the expressions derived in Section 5.19, (n)

W1

= Ωn2 +

ω 2 − Ω2 n2 g n4 , + 2 Ω 4 Ω2

(17.368)

with n2 = n + 1/2 and n4 = (3/2)(n2 + n + 1/2). The optimal Ω-values√are given by the solutions (17.350), (17.351), with g (0) replaced by g (n) = 2n2 /3 3n4 . For g ∈ (−g (n) , 0), the energies are real; for g < −g (n) they possess the imaginary part (n) Im W1

1 ω2 g re im n4 = Ωi 1 − n − Ω Ω . 2 2 |Ω|2 2 |Ω|4 !

(17.369)

For g ∈ (−g (n) , 0), the imaginary part arises from the bubble solution. In the semiclassical limit it produces a factor 12n Ancl /n! for n > 0 as in Eq. (17.322) (with H. Kleinert, PATH INTEGRALS

1141


Figure 17.14 Energies of anharmonic oscillator as function of g′ ≡ g/ω 3 , obtained from variational imaginary part, and the dispersion relation (17.328) as a function of the coupling constant g. Comparison is made with the exact curve and the Feynman-Kleinert variational energy for g > 0.

ω replaced by Ω). Also here, the variational approach can easily be continued higher order approximations W2 , W3 , . . . . To first order in g, the imaginary part is known from a WKB calculation [10]. It reads s v u

3 12n 6u t 4ω Im E (n) (−|g| − iη)= − ω n! π 3|g|

1+2n

e−4ω

3 /3|g|−c(n) 3|g|/4ω 3 1

,

(17.370)

with the slope parameter

(n)

(n) c1

95 29 17 d εi = + n + n2 ω −3 . = 3 d(g/ω ) 96 16 16

(17.371)

Following the procedure described after Eq. (17.366), we obtain from this a variational expression for the imaginary part which generalizes Eq. (17.364) to any n: Im E (n) (−|g| − iη)= − with a correction factor (n)

εi (g) =

Ω ω

3n+5/2

exp − 4

n

12 n!

s v 1+2n u 3 6u 4ω 3 (n) t ω e−4ω /3|g| ε (g),

π

3|g|

Ω3 − ω 3 (n) 3|g| − c1 3|g| 4Ω3

i

(17.372)

1142

17 Tunneling

Im εi (g)

Figure 17.15 Reduced imaginary part of ground state energy of anharmonic oscillator from variational perturbation theory plotted for small negative g against log(−g/4). The fat curve is the analytic continuation of the strong-coupling expansion (5.226) with the expansion coefficients up to the 22nd order listed in Table 5.9. The thin curve is the divergent semiclassical expansion of the contribution of the classical solution in Eq. (17.374).

ω 2 − Ω2 − 2

4Ω 3n + 5/2 (ω 2 − Ω2 )2 − − . (17.373) |g| Ω2 2Ω|g| !

Inserting for Ω the optimal value Ω(n) , we obtain the solid curves shown in Fig. 17.13. Their slopes have the exact values (17.371). The sliding regime for the excited states can be obtained from an analytic continuation of the variational energies. For n = 1, 2 the resulting imaginary parts are shown as dotted curves in Fig. 17.13. They merge smoothly with the corresponding tunneling branches obtained from W1 . If we extend the variational evaluation of the perturbation expansions to high orders in g, we find the imaginary part over the left-hand cut extending deeper and deeper into the regime dominated by the classical solution and the fluctuations around it [9]. This is shown in the double-logarithmic plot of Fig. 17.15. The result may be compared with the the divergent semiclassical expansion around the classical solution [12] 2

4 g g log ε (g) = − + k1 + k2 3g 4 4 i

+ ... .

(17.374)

also plotted in Fig. 17.15. The coefficients are listed in Table 17.2. The inclusion of finite temperatures is possible by summing over the imaginary parts of the energies weighted by a Boltzmann factor with these energies. This opens the road to applications in many branches of physics where tunneling phenomena are relevant. H. Kleinert, PATH INTEGRALS

1143


Table 17.2 Coefficients kn of semiclassical expansion (17.374) around classical solution. 1 3.95833

2 19.3437500

3 174.2092014

6 632817.0536

7 1.357206 × 107

8 3.2924 × 108

4 2177.286133

5 34045.58329

9 8.92 × 109

10 2.65 × 1011

It will be interesting to generalize this procedure to quantum field theories, where it can give rise to the development of much more efficient resummation techniques for perturbation series. One will be able to set up system-dependent basis functions in terms of which these series possess a convergent re-expansion. The critical exponents of the O(N)-symmetric ϕ4 -theory should then be calculable from the presently known five-loop results [13] with a much greater accuracy than before.

17.10.5

Convergence of Variational Perturbation Expansion

The knowledge of the discontinuity across the left-hand makes it possible to understand roughly the convergence properties of the variational perturbation expansion developed in Section 5.14. The ground state energy satisfies the subtracted dispersion relation [compare (17.299)] ω g − 2 2πi

E (0) (g) =

Z

−∞

0

dg ′ disc E (0) (g ′) , g′ g′ − g

(17.375)

where disc E (0) (g ′) denotes the discontinuity across the left-hand cut in the complex g-plane. An expansion of the integrand in powers of g yields the perturbation series E

(0)

(g) = ω

N X

(0) Ek

k=0

g 4ω 3

k

.

(17.376)

The associated variational energy has the form [compare (5.206)] WNΩ (g) = Ω

N X

k=0

(0)

εk

g 4Ω3

k

.

(17.377)

It is obtained from (17.376) by the replacement (5.188) and a re-expansion in powers of g. In the present context, we write this replacement as ω− − −→ Ω(1 − σˆ g )1/2 ,

(17.378)

where gˆ is the dimensionless coupling constant g/Ω3 , and σ = Ω(Ω2 − 1)/g [recall Eqs. (5.213) and (5.208)].

(17.379)

1144

17 Tunneling

There is a simple way of obtaining the same re-expansion from the dispersion relation (17.375). Introducing the dimensionless coupling constant g¯ ≡ g/ω 3, the replacement (17.378) amounts to g¯ − − −→ g˜(ˆ g) ≡

gˆ . (1 − σˆ g )3/2

(17.380)

Since Eq. (17.375) represents an energy, it can be written as ω times a dimensionless function of g¯. Apart from the replacement (17.380) in the argument, it receives an overall factor Ω/ω = (1 − σˆ g )1/2 . We introduce the reduced energies ˆ g ) ≡ E(g)/Ω, E(ˆ

(17.381)

which depends only on the reduced coupling constant gˆ, the dispersion relation (17.375) for E (0) (g) implies a dispersion relation for Eˆ (0) (ˆ g ): Eˆ (0) (g) = (1 − σˆ g )1/2

"

¯ (0) (¯ g ) Z −∞ d¯ 1 g˜(ˆ g ′ disc E g ′) + . 2 2πi 0 g¯′ g¯′ − g˜(ˆ g) #

(17.382)

The resummed perturbation series is obtained from this by an expansion in powers of gˆ/4 up to order N. It should be emphasized that only the truncation of the expansion causes a difference between the two expressions (17.375) and (17.382), since g¯ and g˜ are the same numbers, as can be verified by inserting (17.379) into the right-hand side of (17.380). To find the re-expansion coefficients we observe that the expression (17.382) satisfies a dispersion relation in the complex gˆ-plane. If C denotes the cuts in this g ) is the discontinuity across these cuts, the dispersion relation plane and discC E(ˆ reads 1 gˆ Eˆ (0) (ˆ g) = + 2 2πi

Z

C

ˆ (0) (ˆ dˆ g ′ discC E g ′) . gˆ′ gˆ′ − gˆ

(17.383)

We have changed the argument of the energy from g¯ to gˆ since this will be the relevant variable in the sequel. When expanding the denominator in the integrand in powers of gˆ/4, the expan(0) sion coefficients εl are found to be moment integrals with respect to the inverse coupling constant 1/ˆ g [compare (17.302)]: (0)

εk = −

4k Z dˆ g ˆ (0) (ˆ discC E g ). k+1 2πi C gˆ

(17.384)

In the complex gˆ-plane, the integral (17.382) has in principle cuts along the contours C1 , C¯1 , C2 , C¯2 , and C3 , as shown in Fig. 17.16. The first four cuts are the images of the left-hand cut in the complex g-plane; the curve C3 is due to the square root of 1 − σˆ g in the mapping (17.380) and the prefactor of (17.382). H. Kleinert, PATH INTEGRALS

1145


Figure 17.16 Cuts in complex gˆ-plane whose moments with respect to inverse coupling constant determine re-expansion coefficients. The cuts inside the shaded circle happen to be absent due to the convergence of the strong-coupling expansion for g > gs .

¯ g ) abbreviate the reduced discontinuity in the original dispersion relation Let D(¯ (17.375): ¯ g ) ≡ disc E¯ (0) (¯ D(¯ g ) = 2iIm E¯ (0) (¯ g − iη),

g¯ ≤ 0.

(17.385)

Then the discontinuities across the various cuts are disc

C1,¯1,2,¯2

¯ g (1 − σˆ Eˆ (0) (ˆ g ) = (1 − σˆ g )1/2 D(ˆ g )−3/2 ),

(17.386)

disc Eˆ (0) (ˆ g ) = −2i(σˆ g − 1)1/2 C3

1 × − 2

Z

∞

0

d¯ g ′ gˆ(σˆ g − 1)−3/2 ¯ D(−¯ g′) . ′2 2 −3 2π g¯ + gˆ (σˆ g − 1)

(17.387)

For small negative g¯, the discontinuity is given by the semiclassical limit (17.317): s s

¯ g ) ≈ −2i 6 D¯ π

4 4/3¯g e . −3¯ g

(17.388)

(0)

We denote by εk (Ci ) the contributions of the different cuts to the integral (17.384) for the coefficients. After inserting (17.388) into Eq. (17.386), we obtain from the cut along C1 the semiclassical approximation (0)

εk (C1 ) ≈ −2 4k

Z

C1

dˆ g 1 2π gˆk+1

s v u 6u g )5/2 4(1−σˆg )3/2 /3ˆg t− 4(1 − σˆ e .

π

3ˆ g

(17.389)

For the kth term Sk of the series this yields an estimate Sk ∝

"Z

Cγ

#

dγ fk (γ) e (σˆ g )k , 2π

(17.390)

1146

17 Tunneling

where fk (γ) is the function of γ ≡ σˆ g

fk (γ) = − k +

3 4σ log(−γ) + (1 − γ)3/2 . 2 3γ

(17.391)

For large k, the integral may be evaluated via the saddle point approximation of Subsection 4.2.1. The extremum of fk (γ) satisfies the equation −k +

4σ 3 = (1 − γ)1/2 (1 + 12 γ) , 2 3γ

(17.392)

which is solved by γ− −−→ γk = −4σ/3k. k→∞

(17.393)

At the extremum, fk (γ) has the value fk − −−→ k log(3k/4eσ) − 2σ. k→∞

(17.394)

The constant −2σ in this limiting expression arises when expanding the second term of Eq. (17.391) into a Taylor series, (4σ/3γ)(1 − γ)3/2 = 4σ/3γk − 2σ + . . . . Only the first two terms survive the large-k limit. Thus, to leading order in k, the kth term of the re-expanded series becomes −2σ

Sk ∝ e

−3k e

!k

gˆ 4

!k

.

(17.395)

The corresponding re-expansion coefficients are (0)

(0)

εk ∝ e−2σ Ek .

(17.396)

They have the remarkable property of growing in precisely the same manner with (0) k as the initial expansion coefficients Ek , except for an overall suppression factor e−2σ . This property was found empirically in Fig. 5.20b. In order to estimate the convergence of the variational perturbation expansion, we note that with Ω(Ω2 − 1) σ= g

(17.397)

1 . Ω2

(17.398)

and gˆ from (5.213), we have σˆ g =1−

For large Ω, this expression is smaller than unity. Hence the powers (σˆ g )k alone yield a convergent series. An optimal re-expansion of the energy can be achieved H. Kleinert, PATH INTEGRALS

1147


by choosing, for a given large maximal order N of the expansion, a parameter σ proportional to N: σ ≈ σN ≡ cN.

(17.399)

Inserting this into (17.391), we obtain for large k = N "

#

4c fN (γ) ≈ N − log(−γ) + (1 − γ)3/2 . 3γ

(17.400)

The extremum of this function lies at 1+

4c (1 − γ)1/2 (1 + 12 γ) = 0. 3γ

(17.401)

The constant c is now chosen in such a way that the large exponent proportional to N in the exponential function efN (γ) due to the first term in (17.400) is canceled by an equally large contribution from the second term, i.e., we require at the extremum fN (γ) = 0.

(17.402)

The two equations (17.401) and (17.402) are solved by γ = −0.242 964 029 973 520 . . . ,

c = 0.186 047 272 987 975 . . . .

(17.403)

In contrast to the extremal γ in Eq. (17.393) which dominates the large-k limit, the extremal γ of the present limit, in which k is also large but of the order of N, remains finite (the previous estimate holds for k ≫ N). Accordingly, the second term (4c/3γ)(1 − γ)3/2 in fN (γ) contributes in full, not merely via the first two Taylor expansion terms of (1 − γ)3/2 , as it did in (17.394). Since fN (γ) vanishes at the extremum, the Nth term in the re-expansion has the order of magnitude 1 SN ∝ (σN gˆN )N = 1 − 2 ΩN

!N

.

(17.404)

According to (17.397) and (17.399), the frequency ΩN grows for large N like 1/3

ΩN ∼ σN g 1/3 ∼ (cNg)1/3 .

(17.405)

As a consequence, the last term of the series decreases for large N like "

1 SN (C1 ) ∝ 1 − (σN g)2/3

#N

2/3

≈ e−N/(σg)

≈ e−N

1/3 /(cg)2/3

.

(17.406)

This estimate does not yet explain the convergence of the variational perturbation expansion in the strong-coupling limit observed in Figs. 5.21 and 5.22. For the contribution of the cut C1 to SN , the derivation of such a behavior requires

1148

17 Tunneling

including a little more information into the estimate. This information is supplied by the empirically observed property, that the best ΩN -values lie for finite N on a curve [recall Eq. (5.211)]:

σN ∼ cN 1 +

6.85 . N 2/3

(17.407)

Thus the asymptotic behavior (17.399) receives, at a finite N, a rather large correction. By inserting this σN into fN (γ) of (17.400), we find an extra exponential factor ∆fN

e

4c (1 − γ)3/2 6.85 ≈ exp N 3 γ N 2/3 6.85 1/3 = exp −N log(−γ) 2/3 ≈ e−9.7N . N "

#

(17.408)

This reduces the size of the last term due to the cut C1 in (17.406) to −2/3 ]N 1/3

SN (C1 ) ∝ e−[9.7+(cg)

,

(17.409)

which agrees with the convergence seen in Figs. 5.21 and 5.22. There is no need to evaluate the effect of the shift in the extremal value of γ caused by the correction term in (17.407), since this would be of second order in 1/N 2/3 . How about the contributions of the other cuts? For C¯1 , the integrals in (17.384) run from gˆ = −2/σ to −∞ and decrease like (−2/σ)−k . The associated last term SN (C¯1 ) is of the negligible order e−N log N . For the cuts C2,¯2,3 , the integrals (17.384) start at gˆ = 1/σ and have therefore the leading behavior (0)

εk (C2,¯2,3 ) ∼ σ k .

(17.410)

This implies a contribution to the Nth term in the re-expansion of the order of SN (C2,¯2,3 ) ∼ (σˆ g )N ,

(17.411)

which decreases merely like (17.406) and does not explain the empirically observed convergence in the strong-coupling limit. As before, an additional information produces a better estimate. The cuts in Fig. 17.16 do not really reach the point σˆ g = 1. (0) ˆ There exists a small circle of radius ∆ˆ g > 0 in which E (ˆ g ) has no singularities at all. This is a consequence of the fact unused up to this point that the strong-coupling expansion (5.231) converges for g > gs . For the reduced energy, this expansion reads: Eˆ (0) (ˆ g) =

gˆ 4

!1/3   

α0 + α1

"

gˆ 1 3 4ω (1 − σˆ g )3/2

#−2/3

+ α2

"

gˆ 1 3 4ω (1 − σˆ g )3/2

#−4/3

 

+ ... . 

(17.412)

The convergence of (5.231) for g > gs implies that (17.412) converges for all σˆ g in H. Kleinert, PATH INTEGRALS

1149

17.10 Large-Order Behavior of Perturbation Expansions exp(6.41 − 9.42N 1/3 ) 2

3

4

5

-10

|SN | -20

-30

N 1/3 -40

Figure 17.17 Theoretically obtained convergence behavior of N th approximants for α0 , to be compared with the empirically found behavior in Fig. 5.21.

e9.23N

1/3

−5.14

SN N 1/3

Figure 17.18 Theoretically obtained oscillatory behavior around exponentially fast asymptotic approach of α0 to its exact value as a function of the order N of the approximant, to be compared with the empirically found behavior in Fig. 5.22, averaged between even and odd orders.

a neighborhood of the point σˆ g = 1 with a radius ∆(σˆ g) ∼

gˆ −¯ gs

!2/3

=

(

)2/3

1 [1 + ∆(σˆ g )] −σ¯ gs

,

(17.413)

where g¯s ≡ gs /ω 3 . For large N, ∆(σˆ g ) goes to zero like 1/(N|¯ g s|c)2/3 . Thus the (0) integration contours of the moment integrals (17.384) for the contributions εk (Ci ) of the other cuts do not begin at the point σˆ g = 1, but a little distance ∆(σˆ g ) away from it. This generates an additional suppression factor (σˆ g )−N ∼ [1 + ∆(σˆ g )]−N .

(17.414)

1150

17 Tunneling

Let us set −¯ gs = |¯ gs | exp(iϕs ) and xs ≡ (−ˆ g /¯ gs )2/3 = −|xs | exp(iθ), and introduce the parameter a ≡ 1/[|¯ gs |c]2/3 . Since there are two complex conjugate contributions we obtain, for large N a last term of the re-expanded series the order of SN (C2,¯2,3 ) ≈ e−N

1/3 a cos θ

cos(N 1/3 a sin θ).

(17.415)

By choosing |¯ gs | ∼ 0.160,

θ ∼ −0.467,

(17.416)

we obtain the curves shown in Figs. 17.17 and 17.18 which agree very well with the 1/3 observed Figs. 5.21 and 5.22. Their envelope has the asymptotic falloff e−9.23N . Rn

n

Figure 17.19 Comparison of ratios Rn between successive expansion coefficients of strong-coupling expansion (dots) with ratios Rnas of expansion of superposition of two singularities at g = 0.156 × exp(±0.69) (crosses).

Let us see how the positions of the leading Bender-Wu singularities determined by (17.416) compare with what we can extract directly from the strong-coupling series (5.231) up to order 22. For a pair of square root singularities at xs = −|xs | exp(±iθ), P the coefficients of a power series αn xn have the asymptotic ratios Rn ≡ αn+1 /αn ∼ Rnas ≡ − cos[(n + 1)θ + δ]/|xs | cos(nθ + δ). In Fig. 17.19 we have plotted these ratios against the ratios Rn obtained from the coefficients αn of Table 5.9 . For large n, the agreement is good if we choose |xs | = 1/0.117,

θ = −0.467,

(17.417)

with an irrelevant phase angle δ = −0.15. The angle θ is in excellent agreement with the value found in (17.416). From |xs | we obtain |¯ gs | = 4|1/xs |3/2 = 0.160, again in excellent agreement with (17.416). This convergence radius is compatible with the heuristic convergence of the strong-coupling series up to order 22, as can be seen in Fig. 17.19 by comparing the curves resulting from the series with the exact curve. H. Kleinert, PATH INTEGRALS

1151

17.11 Decay of Supercurrent in Thin Closed Wire E (0) (g)

3rd order PT strong coupling 2nd order PT exact

g

Figure 17.20 Strong-coupling expansion of ground state energy in comparison with exact values and perturbative results of 2nd and 3rd order. The convergence radius in 1/g is larger than 1/0.2.

It is possible to extend the convergence proof to the more general divergent power series discussed in Section 5.17, whose strong-coupling expansions have the more general growth parameters p and q [14]. The convergence is assured for 1/2 < R 2/q < 1 [15]. If the interaction of the anharmonic oscillator is dτ xn (τ ) with n 6= 4, the dimensionless expansion parameter for the energies is g/ω n/2+1 rather than g/ω 3. Then q = n/2 + 1, such that for n ≥ 6 the convergence is lost. This can be verified by trying to resum the expansions for the ground state energies of n = 6 and n = 8, for example. For n = 6, the cut in Fig. 17.16 becomes circular such that there is no more shaded circle C3 in which the strong-coupling series converges.

17.11

Decay of Supercurrent in Thin Closed Wire

An important physical application of the above tunneling theory explains the temperature behavior of the resistance of a thin7 superconducting wire. The superconducting state is described by a complex order parameter ψ(z) depending on the spatial variable z along the wire. We then speak of an order field . The variable z plays the role of the Euclidean time τ in the previous sections. We shall consider a closed wire where ψ(z) satisfies the periodic boundary condition ψ(z) = ψ(z + L).

(17.418)

The energy density of the system is described approximately by a Ginzburg-Landau expansion in powers of ψ and its gradients containing only the terms g ε(z) = |∂z ψ(z)|2 + m2 |ψ(z)|2 + |ψ(z)|4 . 4 7

(17.419)

A superconducting wire is called thin if it is much smaller than the coherence length to be defined in Eq. (17.425).

1152

17 Tunneling

The total fluctuating energy is given by the functional ∗

E[ψ , ψ] =

Z

L/2

−L/2

dz ε(z),

(17.420)

and the probability of each fluctuation is determined by the Boltzmann factor exp{−E[ψ ∗ , ψ]/kB T }. The parameter m2 in front of |ψ(z)|2 is called the mass term of the field. It vanishes at the critical temperature Tc and behaves near Tc like m2 ≈ m20

T −1 . Tc

(17.421)

Below Tc , the square mass is negative and the wire becomes superconducting. One can easily estimate, that each term in the Landau expansion is of the order of |1 − T /Tc |2 and any higher expansion term in (17.419) would be smaller than that by at least a power |1 − T /Tc |1/2 . The partition function of the system is given by the path integral Z=

Z

Dψ ∗ (z)Dψ(z)e−E[ψ

∗ ,ψ]/k

BT

.

(17.422)

If T does not lie too close to Tc [although close enough to justify the Landau expansion, i.e., the neglect of higher expansion terms in (17.419) suppressed by a factor |1 − T /Tc |1/2 ], this path integral can be treated semiclassically in the way described earlier in this chapter [16]. The basic microscopic mechanism responsible for the phenomenon of superconductivity will be irrelevant for the subsequent discussion. Let us only recall the following facts: A superconductor is a metal at low temperatures whose electrons near the surface of the Fermi sea overcome their Coulomb repulsion due to phonon exchange. This enables them to form bound states between two electrons of opposite spin orientations in a relative s-wave, the celebrated Cooper pairs.8 The attraction which binds the Cooper pairs is extremely weak. This is why the temperature has to be very small to keep the pairs from being destroyed by thermal fluctuations. The critical temperature Tc , where the pairs break up, is related to the binding energy of the Cooper pairs by Epair = kB Tc . The field-theoretic process called phonon exchange is a way of describing the accumulation of positive ions along the path of an electron which acts as an attractive potential wake upon another electron while screening the Coulomb repulsion. The attraction is very weak and leads to a bound state only in the s-wave (the centrifugal barrier ∝ l(l + 1)/r2 preventing the formation of a bound state in higher partial waves). The potential between the electrons may well be approximated by a δ-function potential V (x) ≈ −gδ(r). The critical 8

We consider here only with old-fashioned superconductivity which sets in below a very small critical temperature of a few-degree Kelvin. The physics of the recently discovered hightemperature superconductors is at present not sufficiently understood to be discussed along the same lines. H. Kleinert, PATH INTEGRALS

17.11 Decay of Supercurrent in Thin Closed Wire

1153

Figure 17.21 Renormalization group trajectories in the g, µ plane of superconducting electrons (g=attractive coupling constant, µ=Debye temperature, kB = 1). Curves with same Tc imply identical superconducting properties. The renormalization group determines the reparametrizations of a fixed superconductive system along any of these curves.

temperature Tc , usually a few degrees Kelvin, is found to satisfy the characteristic exponential relation Tc kB = µe−1/g .

(17.423)

The parameter µ denotes the upper energy cutoff of the phonon spectrum TD kB , where TD is the Debye temperature of the lattice vibration. An important result of the theory, confirmed by experiment, is that all T dependent characteristic equilibrium properties of the superconductor near Tc depend only on the single parameter Tc . Thus, many quite different systems with different microscopic parameters µ ≡ TD and g will have the same superconducting properties (see Fig. 17.21). The critical temperature is an important prototype for the understanding of the so-called dimensionally transmuted coupling constant in quantum field theories, which plays a completely analogous role in specifying the system. In quantum field theory, an arbitrary mass parameter µ is needed to define the coupling strength of a renormalized theory and physical quantities depend only on the combination9 Mc = µe−1/g(µ) .

(17.424)

The set of all changes µ which are accompanied by a simultaneous change of g(µ) such as to stay on a fixed curve with Mc from the renormalization group. The curve µ, g(µ) is called the renormalization group trajectory [15]. 9

In quantum chromodynamics, this dimensionally transmuted coupling constant is of the order of the pion mass and usually denoted by Λ.

1154

17 Tunneling

If one works in natural units with h ¯ = kB = M = 1, the critical temperature corresponds to a length of ≈ 1000rA. This length sets the scale for the spatial correlations of the Cooper pairs near the critical point via the relation

ξ(T ) =

const. T 1− Tc Tc

−1/2

≈ 1000rA 1 −

T Tc

−1/2

.

(17.425)

The Cooper pairs are much larger than the lattice spacing, which is of the order of 1rA. Their size is determined by the ratio h ¯ 2 kF /me πkB Tc , where kF is the wave number of electrons of mass me on the surface of the Fermi sphere. The temperature Tc in conventional superconductors of the order of 1 K corresponds to 1/11604.447 eV. Thus the thermal energy kB Tc is smaller than the atomic energy EH = 27.210 eV (recall the atomic units defined on p. 13.7) by a factor 2.345 × 10−3 , and we find that h ¯ 2 /me πkB Tc is of the order of 102 a2H . Since aH is of the order of 1/kF , we estimate the size of the Cooper pairs as being roughly 100 times larger than the lattice spacing. This justifies a posteriori the δ-function approximation for the attractive potential, whose range is just a few lattice spacings, i.e., much smaller than ξ(T ). The presence of such large bound states causes the superconductor to be coherent over the large distance ξ(T ). For this reason, ξ(T ) is called the coherence length. Similar Cooper pairs exist in other low-temperature fermion systems such as 3 He, where they give rise to the phenomenon of superfluidity. There, the interatomic potential contains a hard repulsive core for r < 2.7rA. This prevents the formation of an s-wave bound state. In addition it produces a strong spin-spin correlation in the almost fully degenerate Fermi liquid, with a preference of parallel spin configurations. Because of the necessary antisymmetry of the pair wave function of the electrons, this amounts to a repulsion in any even partial wave. For this reason, Cooper pairs can only exist in the p-wave spin triplet state. The binding energy is much weaker than in a superconductor, suppressing the critical temperature by roughly a factor thousand. Experimentally, one finds Tc = 27mK at a pressure of p = 35 bar. Since the masses of the 3 He atoms are larger than those of the electrons by about the same factor thousand, the coherence length ξ has the same order of magnitude in both systems, i.e., 1/Tc has the same length when measured in units of rA. The theoretical description of the behavior of the condensate is greatly simplified by re-expressing the fundamental Euclidean action in terms of a Cooper pair field which is the composite field ψpair (x) = ψe (x)ψe (x).

(17.426)

Such a change of field variables can easily be performed in a path integral formulation of the field theory. The method is very similar to the introduction of the auxiliary field ϕ(x) in the polymer field theory of Section 15.12. Since this subject has been H. Kleinert, PATH INTEGRALS

1155


treated extensively elsewhere10 we shall not go into details. The partition function of the system reads Z=

Z

∗

Dψe∗ (x)Dψe (x)e−A[ψe ,ψe ] .

(17.427)

By going from integration variables ψe to ψpair , we can derive the alternative pair partition function Z=

Z

∗

∗ Dψpair (x)Dψpair (x)e−A[ψpair ,ψpair ] ,

(17.428)

where ψpair is the Cooper pair field (17.426). In general, the new action is very complicated. For temperatures close to Tc , however, it can be expanded in powers of the field ψpair and its derivatives, leading to a Landau expansion of the type (17.419). For static fields the Euclidean field action is Z 1 ∗ d3 x ε(x) (17.429) A[ψpair , ψpair ] = E/kB T = kB T " ! # Z µ 1 1 1 1 3 2 4 2 d x − log + 2 |ψpair | + |ψpair | + 2 |∇ψpair | + . . . , = kB T T g 2Tc2 Tc where the dots denote the omitted each accompanied by an additional Let us discuss the path integral that with the critical temperature written as

higher powers of ψpair and of their derivatives, factor 1/Tc . (17.429) first in the classical limit. We observe (17.423), the mass term in the energy can be

T Tc |ψpair |2 ∼ − 1 − |ψpair |2 . − log T Tc

(17.430)

It has the “wrong sign” for T < Tc , so that the field has no stable minimum at ψpair = 0. It fluctuates around one of the infinitely many nonzero values with the fixed absolute value s

|ψpair,0 | = Tc 1 −

T . Tc

(17.431)

It is then useful to take a factor Tc (1 − T /Tc )1/2 out of the field ψpair , define ψ(x) ≡ ψpair (x)

1 , Tc (1 − T /Tc )1/2

(17.432)

and write the renormalized energy density as 1 ε(x) = |∇ψ|2 − |ψ|2 + |ψ|4 . 2 10

(17.433)

The way to describe the pair formation by means of path integrals is explained in H. Kleinert, Collective Quantum Fields, Fortschr. Phys. 26 , 565 (1978) (http://www.physik.fu-berlin.de/~kleinert/55).

1156

17 Tunneling

Here we have made use of the coherence length (17.425) to introduce a dimensionless space variable x, replacing x → x ξ. We also have dropped an overall energy density factor proportional to (1 − T /Tc )2 Tc2 . In the rescaled form (17.433), the minimum of the energy lies at |ψ0 | = 1, where it has the density ε = εc = −1/2.

(17.434)

The negative energy accounts for the binding of the Cooper pairs in the condensate (in the present natural units) and is therefore called the condensation energy. In terms of (17.433), the partition function in equilibrium can be written as Z=

Z

Dψ ∗ (x)Dψ(x)e−(1/T )

R

d3 x ε(x)

.

(17.435)

We are now prepared to discuss the flow properties of an electric current of the system carried by the Cooper pairs. It is carried by the divergenceless pair current [compare (1.102)] j(x) =

↔ 1 ∗ ψ (x) ∇ ψ(x) 2i

(17.436)

associated with the transport of the number of pairs, apart from a charge factor of the pairs, which is equal to twice the electron charge. The important question to be understood by the theory is: How can this current become “super”, and stay alive for a very long time (in practice ranging from hours to years, as far as the patience of the experimentalist may last) [17]. To see this let us set up a current in a long circular wire and assume that the wire thickness is much smaller than the coherence length ξ(T ). Then transverse variations of the pair field ψ(x) are strongly suppressed with respect to longitudinal ones (by the gradient terms |∇ψ(x)|2 in the Boltzmann factor) and the system depends mainly on the coordinate z along the wire, so that the above formalism can be applied. If the cross section of the wire is absorbed into the inverse temperature prefactor in the Boltzmann factor in (17.435), we may simply study the partition function (17.435) for a one-dimensional problem along the z-axis. The energy density (17.433) is precisely of the form announced in the beginning in Eq. (17.419). It is convenient to decompose the complex field ψ(z) into polar coordinates ψ(z) = ρ(z)eiγ(z) ,

(17.437)

in terms of which the energy density reads 1 ε(z) = −ρ2 + ρ4 + ρ2z + ρ2 γz2 , 2

(17.438)

where the subscript z indicates a derivative with respect to z. The field equations are j(z) = ρ2 (z)γz (z) = const



1157

Figure 17.22 Potential V (ρ) = −ρ2 + ρ4 /2 − j 2 /ρ2 showing barrier in superconducting wire to the left of ρ0 to be penetrated if the supercurrent is to relax.

and ρzz = −ρ + ρ3 +

j2 . ρ3

(17.440)

If z is reinterpreted as an imaginary “time”, the latter equation can be interpreted as describing the mechanical motion of a mass point at the position ρ(z) moving as a function of the “time” z in the potential 1 j2 −V (ρ) ≡ ρ2 − ρ4 + 2 , 2 ρ

(17.441)

which is the potential shown in Fig. 17.22 turned upside down. Certainly, the time-sliced path integral in ρ would suffer from the phenomenon of path collapse described in Chapter 8. At the level of the semiclassical approximation to be performed here, however, this does not happen. There are two types of extremal solutions. The trivial solutions are √ γ(z) = kz, ρ(z) ≡ ρ0 = 1 − k 2 . (17.442) Since the wire is closed, the phase γ(z) has to be periodic over the total length L of the wire. This implies the quantization of the wave number k, kn =

2π n, L

n = 0, ±1, ±2, . . . .

(17.443)

1158

17 Tunneling

Figure 17.23 The condensation energy as function of velocity parameter kn = 2πn/L.

The current associated with these solutions is j = ρ20 k = (1 − k 2 )k.

(17.444)

As a function of k, this has an absolute maximum at the so-called critical current jc , i.e., 2 |j| < jc ≡ √ . 3 3

(17.445)

No solution of the field equations can carry a larger current than this. The critical wave number is 1 kc ≡ √ , 3

(17.446)

1 ec (k) = V (ρ0 ) = − (1 − k 2 )2 . 2

(17.447)

and the energy density:

It is plotted in Fig. 17.23. Note that the k-values (17.446) for which a supercurrent can exist between the turning points. The energy ec (k) represents the negative condensation energy of the state in the presence of the current. For k → 0 it goes against the current-free value (17.434). We can now understand why all states of current jn smaller than jc are, in fact, “super” in the sense of having an extremely long lifetime. At each value of kn , the wire carries a metastable current which can only decay by a slow tunneling. To see this, we picture the field configuration as a spiral of radius ρ wound around the wire with the azimuthal angle representing the phase γ(z) = kn z (see Fig. 17.24). At H. Kleinert, PATH INTEGRALS

1159


z−axis

Figure 17.24 Order parameter ∆(z) = ρ(z)eiγ(z) of superconducting thin circular wire neglecting fluctuations. The order parameter is pictured as a spiral of radius ρ0 and pitch ∂γ(z)/∂z = 2πn/L winding around the wire. At T = 0, the supercurrent is absolutely stable since the winding number n is fixed topologically.

zero temperature, the size ρ of the order parameter is frozen at ρ0 and the winding number is absolutely stable on topological grounds. Then, each metastable state with wave number kn has an infinite lifetime. If the current is to relax by one unit of n it is necessary that at some place z, thermal fluctuations carry ρ(z) to zero. There the phase becomes undefined and may slip by 2π. At the typical low temperatures of these systems, such phase slips are extremely rare. To have a local excursion of ρ(z) to ρ ≈ 0 at one place z, with an appreciable measure in the functional integral (17.435), it must start from a nontrivial solution of the equations of motion which carries ρ(z) as closely as possible to zero. From our experience with the mechanical motion of a mass point in a potential such as −V (ρ) of Eq. (17.441), √ it is easily realized that there exists such a solution. It carries ρ(z) from ρ = 1 − k 2 at √ 0 z = −∞ across the potential barrier to the small value ρ1 = 2k and back once more across the barrier to ρ0 at z = ∞ (see Fig. 17.25). Using the first integral of motion of the differential equation (17.440), the law of energy conservation 1 2 1 1 1 ρz − V (ρ) = E = − V (ρ0 ) = ρ0 (ρ0 + 2ρ1 ) (17.448) 2 2 2 4 leads to the equation ρz = This is solved by the integral z − z1 =

q

2E + V (ρ).

√ Z 2

1 = √ 2

ρ

ρ1

Z

ρ21

yielding 2

ρdρ − + 4Eρ2 − 2j 2 dρ2 q , (ρ2 − ρ21 )(ρ2 − ρ20 )

√

ρ2

(17.449)

ρ6

z − z1 = − q 2(ρ20 − ρ21 )

2ρ4

v u 2 uρ arctanht

− ρ21 . ρ20 − ρ21

(17.450)

(17.451)

1160

17 Tunneling

Inverting this, we find the bubble solution ρ2cl (z) = 1 − k 2 −

ω 2/2 , cosh2 [ω(z − z1 )/2]

(17.452)

where ω=

q

2(ρ20 − ρ21 )

(17.453)

is the curvature of V (ρ) close to ρ0 , i.e., V (ρ) ≈ ω 2(ρ − ρ0 )2 + . . . .

(17.454)

The extra energy of the bubble solution is Ecl =

Z

0

L

4q 4 dz [e(ρcl ) − ec (k)] = ω = 2(1 − 3k 2 ). 3 3

(17.455)

The explicit solution (17.452) reaches the point of smallest ρ at z1 , where its value is √ ρ1 ≡ ρ(z1 ) = 2k. (17.456) This value is still nonzero and does not yet permit a phase slip. However, we shall now demonstrate that quadratic fluctuations around the solution (17.456) do, in fact, to reduce the current. For this, we insert the fluctuating order field ρ(z) = ρcl (z) + δρ(z)

(17.457)

Figure 17.25 Extremal excursion of order parameter in superconducting wire. It corresponds to a mass point starting out at √ ρ0 , rolling under the influence of “negative gravity” up the mountain unto the point ρ1 = 2k, and returning back to ρ0 , with the variable z playing the role of a time variable. H. Kleinert, PATH INTEGRALS

1161


Figure 17.26 Infinitesimal translation of critical bubble yields antisymmetric wave function of zero energy ρ′cl solving differential equation (17.509). Since this wave function has a node, there must be a negative-energy bound state.

into the free energy. With ρcl being extremal, the lowest variation of E is of second order in δρ(z) δ2E =

Z

L

0

dz δρ(z)[−∂z2 + V ′′ (ρ)]δρ(z).

(17.458)

This expression is not positive definite as can be verified by studying the eigenvalue problem [−∂z2

′′

+ V (ρcl )]ψn (z) =

"

−∂z2

−1+

3ρ2cl

j4 − 3 4 ψn (z) = λn ψ(z). ρcl #

(17.459)

The potential V ′′ (ρcl (z)) has asymptotically the value ω 2 . When approaching z = z1 from the right, it develops a minimum at a negative value (see Fig. 17.26). After that it goes again against ω 2 . The energy eigenvalues λ0 and λ−1 lie as indicated in the figure. The fact that there is precisely one negative eigenvalue λ−1 can be proved without an explicit solution by the same physical argument that was used to show the instability of the fluctuation problem (17.253): A small temporal translation of the classical solution corresponds to a wave function which has no energy and a zero implying the existence of precisely one lower wave function with λ−1 < 0 and no zero. The negative eigenvalue makes the critical bubble solution unstable against contraction or expansion. The former makes the fluctuation return to the spiral classical solution (17.452) of Fig. 17.24, the second removes one unit from the winding number of the spiral and reduces the supercurrent. For the precise calculation of the

1162

17 Tunneling

decay rate, the reader is referred to the references quoted at the end of the chapter. Here we only give the final result which is [18] rate = const × L ω(k)e−Ecl/kB T ,

(17.460)

with the k-dependent prefactor

Figure 17.27 Logarithmic plot of resistance of thin superconducting wire as function of temperature at current 0.2µA in comparison with experimental data (vertical axis is normalized by the Ohmic resistance Rn = 0.5Ω measured at T > Tc , see papers quoted at end of chapter).

√ √ " !# 2 7/4 2 (1 − 3k ) 3 2k 1 − 3k √ ω(k) = 2|λ′−1 | exp − √ arctan , (17.461) (1 − k 2 )1/2 1 − 3k 2 2k where i1/2 1 h λ′−1 ≡ − (1 + k 2 )2 + 3(1 − 3k 2 )2 − (1 + k 2 ) < 0 (17.462) 2 is the negative eigenvalue of the fluctuations in the complex field ψ(z) [which is not directly related to λ−1 of Eq. (17.459) and requires a separate discussion of the initial path integral (17.435)]. This complicated-looking expression has a simple quite accurate approximation which had previously been deduced from a numerical evaluation of the fluctuation determinant [19]: √ ω(k) ≈ (1 − 3k)15/4 (1 + k 2 /4). (17.463) √ Both expressions vanish at the critical value k = kc = 1/ 3. The resistance of a thin superconducting wire following from this calculation is compared with experimental data in Fig. 17.27. H. Kleinert, PATH INTEGRALS

17.12 Decay of Metastable Thermodynamic Phases

17.12

1163

Decay of Metastable Thermodynamic Phases

A generalization of this decay mechanism can be found in the first-order phase transitions of many-particle systems. These possess some order parameter with an effective potential which has two minima corresponding to two different thermodynamic phases. Take, for instance, water near the boiling point. At the boiling temperature, the liquid and gas phases have the same energy. This situation corresponds to the symmetric potential. At a slightly higher temperature, the liquid phase is overheated and becomes metastable. The potential is now slightly asymmetric. The decay of the overheated phase proceeds by the formation of critical bubbles [3]. Their outside consists of the metastable water phase, their inside is filled with vapor lying close to the stable minimum of the potential. The radius of the critical bubble is determined by the equilibrium between the gain in volume energy and the cost in surface energy. If σ is the surface tension and ǫ the difference in energy density, the energy of bubble solution depends on the radius as follows: E ∝ σ 4πR2 − ǫ

4π 3 R . 3

(17.464)

A plot of this energy in Fig. 17.28 looks just like that of the action A(ξ) in Fig. 17.8. Thus the role of the deformation parameter ξ is played here by the bubble radius R.

Figure 17.28 Bubble energy as function of its radius R.

At the critical bubble, the energy has a maximum. The fluctuations of the critical bubble must therefore have a negative eigenvalue. This negative eigenvalue mode accounts for the fact that the critical bubble is unstable against expansion and contraction. When expanding, the bubble transforms the entire liquid into the stable gas phase. When contracting, the bubble disappears and the liquid remains in the overheated phase. Only the first half of the fluctuations have to be counted when calculating the lifetime of the overheated phase. It is instructive to take a comparative look at the instability of a critical bubble to see how the different spatial dimensions modify the properties of the solution. We shall discuss first the case of three space dimensions. As in the case of superconductivity, the description of the liquid-vapor phase transition makes use of a space-dependent order parameter, the real order field ϕ(x). The two minima of the

1164

17 Tunneling

potential V (ϕ) describe the two phases of the system. The kinetic term x′2 in the path integral is now a field gradient term [(∂x ϕ(x))2 ] which ensures finite correlations between neighboring field configurations. The Euclidean action controlling the fluctuations is therefore of the form A[ϕ] =

Z

1 d x [∇ϕ(x)]2 + V (ϕ) , 2 3

(17.465)

where V (ϕ) is the same potential as in Eq. (17.1), but it is extended by the asymmetric energy (17.243). Within classical statistics, the thermal fluctuations are controlled by the path integral for the partition function Z=

Z

Dϕ(x)e−A[ϕ]/T .

(17.466)

Here T is the temperature measured in multiples of the Boltzmann constant kB . R The path integral Dϕ(x) is defined by cutting the three-dimensional space into small cubes of size ǫ and performing one field integration at each point. The critical bubble extremizes the action. Assuming spherical symmetry, the bubble satisfies in D dimensions the classical Euler-Lagrange field equation d2 D−1 d − 2− ϕcl + V ′ (ϕcl (r)) = 0. dr r dr !

(17.467)

This differs from the equation (17.305) for the one-dimensional bubble solution by the extra gradient term −[(D − 1)/r]∂r ϕcl (r). Such a term is an obstacle to an exact solution of the equation via the energy conservation law (17.306). The relevant qualitative properties of the solution can nevertheless be seen in a similar way as for the bubble solution. As in Fig. 17.11 we plot the reversed potential and imagine the solution ϕ(r) to describe the motion of a mass point in this potential with −r playing the role of a “time”. Setting ϕcl (r) = x(−t), the field equation (17.467) takes the form x¨(t) −

D−1 x(t) ˙ − V ′ (x(t)) = 0. t

(17.468)

In this notation, the second term, i.e., the term −[(D − 1)/r]∂r ϕcl (r) in (17.467) plays the role of a negative “friction” accelerating motion of the particle along x(t). This effect decreases with time like 1/t. With our everyday experience of mechanical systems, the qualitative behavior of the solution can immediately be plotted qualitatively as shown in Fig. 17.29. For D = 1, the energy conservation makes the particle reach the right-hand zero of the potential. For D > 1, the “antifriction” makes the trajectory overshoot. At r = 0, the solution is closest to the stable minimum (the maximum of the reversed potential) on the left-hand side. In the superheated water system, this corresponds to the inside of the bubble being filled with vapor. As r moves outward in the bubble, the state moves closer to the metastable state, i.e., it becomes more and more liquid. The antifriction term has the effect that the H. Kleinert, PATH INTEGRALS

1165

17.12 Decay of Metastable Thermodynamic Phases r

φcl (r)

r=0

−

m2 2 g φ − φ4 2 4! r=∞

r=0

φcl (r)

Figure 17.29 Qualitative behavior of critical bubble solution as function of its radius.

point of departure on the left-hand side lies energetically below the final value of the metastable state. Consider now the fluctuations of such a critical bubble in D = 3 dimensions. Suppose that the field deviates from the solution of the field equation (17.467) by δϕ(x). The deviations satisfy the differential equation ˆ2 d2 2 d L − 2− + 2 + V ′′ (ϕcl (r)) δϕ(x) = λ δϕ(x), dr r dr r

"

#

(17.469)

ˆ 2 is the differential operator of orbital angular momentum (in units h where L ¯ = 1). Taking advantage of rotational invariance, we expand δϕ(x) into eigenfunctions of angular momentum ϕnlm , the spherical harmonics Ylm (ˆ x): φ(x) =

X

ϕnlm (r)Ylm (ˆ x).

(17.470)

nlm

The coefficients ϕnlm satisfy the radial differential equation "

d2 2 d l(l + 1) − 2− + + V ′′ (ϕcl (r)) ϕnlm (r) = λnl ϕnlm (r), dr r dr r2 #

(17.471)

1166

17 Tunneling

with ω2 3 ω2 2 + ϕ (r). (17.472) 2 2 a2 cl One set of solutions is easily found, namely those associated with the translational motion of the classical solution. Indeed, if we take the bubble at the origin, V ′′ (ϕcl (r)) = −

ϕcl (x) = ϕcl (r),

(17.473)

to another place x + a, we find, to lowest order in a, ϕcl (x + a) = ϕcl (x) + a∂x ϕcl (x) = ϕcl (r) + aˆ x∂r ϕcl (r).

(17.474)

ˆ is just the Cartesian way of writing the three components of the spherical But x harmonics Y1m (ˆ x). If we introduce the spherical components of a vector as follows x3 √ x0      x1  ≡  (x1 + ix2 )/ √2  , x−1 −(x1 − ix2 )/ 2 

we see that





xˆm =

s



4π Y1m (ˆ x). 3

(17.475)

(17.476)

Thus, δϕ(x) = aˆ x∂r ϕ(x) must be a solution of Eq. (17.469) with zero eigenvalue ˆ 2 to have the eigenˆ causes L λ. This can easily be verified directly: The factor x value 2, and the accompanying radial derivative δϕ(x) = ∂r ϕcl (r) is a solution of Eq. (17.471) for l = 1 and λnl = 0, as is seen by differentiating the Euler-Lagrange equation (17.467) with respect to r. Choosing the principal quantum number of ˆ ∂r ϕcl (r) these translational modes to be n = 1, we assign the three components of x to represent the eigenmodes ϕ1,1,m . As long as the bubble radius is large compared to the thickness of the wall, which is of the order 1/ω, the 1/r2 -terms will be very small. There exists then an entire family of solutions ϕ1lm (x) with all possible values of l which all have approximately the same radial wave function ∂r ϕcl (r). Their eigenvalues are found by a perturbation expansion. The perturbation consists in the centrifugal barrier but with the l = 1 barrier subtracted since it is already contained in the derivative ∂r ϕcl (r), i.e., Vpert = [l(l + 1) − 2]/2r 2 .

(17.477)

The bound-state wave functions ϕ1lm are normalizable and differ appreciably from zero only in the neighborhood of the bubble wall. To lowest approximation, the perturbation expansion produces therefore an energy λnl ≈

l(l + 1) − 2 , rc2


1167


where rc is the radius of the critical bubble. As a consequence, the lowest l = 0 eigenstate has a negative energy λ00 ≈ −

1 . rc2

(17.479)

Physically, this single l = 0 -mode corresponds to an infinitesimal radial vibration of the bubble. As already explained above it is not astonishing that a radial vibration has a negative eigenvalue. The critical bubble lies at a maximum of the action. Expansion or contraction is energetically favorable. Since Y00 (x) is a constant, the wave function is proportional to (d/dr)ϕcl (r) itself without an angular factor. This is seen directly by performing an infinitesimal radial contraction ϕcl ((1 − ǫ)r) = ϕcl (r) − ǫr∂r ϕcl (r).

(17.480)

The variation r∂r ϕcl (r) is almost zero except in the vicinity of the critical radius rc , so that r∂r ϕcl (r) ≈ rc ∂r ϕcl (r) which is the above wave function. Being the ground state of the Schrödinger equation (17.469), it should be denoted by ϕ000 (r). Since it solves approximately the Schrödinger equation (17.471) with l = 1, it also solves this equation approximately with l = 0 and the energy (17.479). Finally let us point out that in D > 1 dimensions, the value of the negative eigenvalue can be calculated very simply from a phenomenological consideration of the bubble action. Since the inside of the bubble is very close to the true ground state of the system whose energy density lies lower than that of the metastable one by ǫ, the volume energy of a bubble of an arbitrary radius R is RD ǫ, EV = −SD D

(17.481)

where SD RD−1 is the surface of the bubble and SD RD /D its volume. The surface energy can be parametrized as ES = SD RD−1 σ,

(17.482)

where σ is a constant proportional to the surface tension. Adding the two terms and differentiating with respect to R, we obtain a critical bubble radius at R = rc = (D − 1)σ/ǫ,

(17.483)

with a critical bubble energy D SD D−1 SD SD D D−1 σ Ec = R σ= R ǫ= (D − 1) . D c D(D − 1) c D ǫD−1

(17.484)

The second derivative with respect to the radius R is, at the critical radius, d2 E D−1 = −DE . c dR2 R=rc rc2

(17.485)

1168

17 Tunneling

Identifying the critical bubble energy Ec with the classical Euclidean action Acl we find the variation of the bubble action as D−1 1 δ 2 Acl ≈ − (δR)2 D Acl 2 . 2 rc

(17.486)

We now express the dilational variation of the bubble radius in terms of the normal coordinate of (17.470). The normalized wave function is obviously ϕ000 (r) = qR

∂r ϕcl (r) dD x(∂r ϕcl )2

.

(17.487)

But the expression under the square root is exactly D times the action of the critical bubble Z

dD x(∂r ϕcl )2 = DAcl .

(17.488)

To prove this we introduce a scale factor s into the solution of the bubble and evaluate the action 1 d x ([∂r ϕcl (sr)]2 + V (ϕcl (sr)) 2 2 Z 1 s 2 D = D d x [∂r ϕcl (r)] + V (ϕcl (r)) . s 2

A˜cl =

Z

D

(17.489)

Since A˜cl is extremal at s = 1, it has to satisfy ∂ A˜cl = 0, ∂s s=1

or

(17.490)

Z

1 d x (D − 2) [∂r ϕcl ]2 + DV (ϕcl (r)) = 0. 2

(17.491)

Z

D−2 d xV (ϕcl (r)) = − D

(17.492)

D

Hence D

Z

1 dD x [∂r ϕcl (r)]2 , 2

implying that 1 D−2 − dD x[∂r ϕcl (r)]2 2 2D Z 1 = dD x[∂r ϕcl (r)]2 . D

Acl =

Z

(17.493)

With (17.487), the ϕ000 contribution to δϕ(x) reads ∂r ϕ δϕ(x) = ξ000 ϕ000 (r) = ξ000 √ , DAcl


1169


and we arrive at ξ000 δR = √ . DAcl

(17.495)

Inserting this into (17.486) shows that the second variation of the Euclidean action δ 2 Acl can be written in terms of the normal coordinates associated with the normalized fluctuation wave function ϕ000 as 2 δ 2 Acl = −ξ000

D−1 . 2rc2

(17.496)

From this relation, we read off the negative eigenvalue λ00 = −

D−1 . 2rc2

(17.497)

For D = 3, this is in agreement with the D = 3 value (17.479). For general D, the eigenvalue corresponding to (17.479) would have been derived with the arguments employed there from the derivative term −[(D − 1)/r]d/dr in the Lagrangian and would also have resulted in (17.497). All other multipole modes ϕnlm have a positive energy. Close to the bubble wall (as compared with the radius), the classical solutions (1/r)ϕnlm (r) can be taken approximately from the solvable one-dimensional equation 1 d2 ω2 3 1 − + 1 − 2 2 dr 2 2 2 cosh [ω(r − rc )/2]

"

!#

1 ˜ n 1 ϕnlm . (17.498) ϕnlm ≈ λ r r

The wave functions with n = 0 are ϕ0lm ≈

s

3ω 1 , 2 8 cosh [ω(r − rc )/2]

(17.499)

and have the eigenvalues λ0l ≈

l(l + 1) − 2 . 2rc2

(17.500)

The n = 1 -bound states are ϕ1lm ≈

s

3ω sinh[ω(r − rc )/2] , 4 cosh2 [ω(r − rc )/2]

(17.501)

with eigenvalues 3 l(l + 2) − 2 λ1l ≈ ω 2 + . 8 2rc2

(17.502)

1170

17 Tunneling

Figure 17.30 Decay of metastable false vacuum in Minkowski space. It proceeds as a shock wave which after some time traverses the world almost with light velocity, converting the false into the true vacuum.

17.13

Decay of Metastable Vacuum State in Quantum Field Theory

The theory of decay presented in the last section has an interesting quantum fieldtheoretic application. Consider a metastable scalar field system in a D-dimensional Euclidean spacetime at temperature zero. At a fixed time, there will be a certain average number of bubbles, regulated by the “quantum Boltzmann factor” exp(−Acl /¯ h). If the bubble gas is sufficiently dilute (i.e., if the distances between bubbles is much larger than the radii), each bubble is described quite√accurately by the classical solution. In Minkowski space, a Euclidean radius r = x2 + c2 τ 2 √ corresponds to r = x2 − c2 t2 , where c is the light velocity. The critical bubble has therefore the spacetime behavior √ ϕcl (x, t) = ϕcl (r = x2 − c2 t2 ). (17.503) From the above discussion in Euclidean space we know that ϕ will be equal to the metastable false vacuum in the outer region r > rc , i.e., for x2 − c2 t2 > rc2 .

(17.504)

x2 − c2 t2 < rc2

(17.505)

The inside region

contains the true vacuum state with the lower energy. Thus a critical bubble in spacetime has the hyperbolic structure drawn in Fig. 17.30. Therefore, the Euclidean critical bubble describes in Minkowski space the growth of a bubble as a function of time. The bubble starts life at some time t = rc /c and expands almost instantly to a radius of order rc . The position of the shock wave is described by x2 − c2 t2 = rc2 .


17.14 Crossover from Quantum Tunneling to Thermally Driven Decay

1171

This implies that a shock wave that runs through space with a velocity v=

c |x| =q t 1 − rc2 /c2 t2

(17.507)

and converts the metastable into the stable vacuum — a global catastrophe. A Euclidean bubble centered at another place xb , τb would correspond to the same process starting at xb and a time tb = rc /c + τb .

(17.508)

A finite time after the creation of a bubble, of the order rc /c, the velocity of the shock wave approaches the speed of light (in many-body systems the speed of sound). Thus, we would hardly be able to see precursors of such a catastrophe warning us ahead of time. We would be annihilated with the present universe before we could even notice.

17.14

Crossover from Quantum Tunneling to Thermally Driven Decay

For completeness, we discuss here the difference between a decay caused by a quantum-mechanical tunneling process at T = 0, and a pure thermally driven decay at large temperatures. Consider a one-dimensional system possessing, at some place x∗ , a high potential barrier, much higher than the thermal energy kB T , with a shape similar to Fig. 17.10. Let the well to the left of the barrier be filled with a grand-canonical ensemble of noninteracting particles of mass M in a nearly perfect equilibrium. Their distribution of momenta and positions in phase space is 2 governed by the Boltzmann factor e−β[p /2M +V (x)] . The rate, at which the particles escape across the barrier, is given by the classical statistical integral Γcl =

Zcl−1

Z

dx

dp −β[p2 /2M +V (x)] p e δ(x − x∗ ) Θ(p), 2π¯ h M

Z

(17.509)

where Zcl is the classical partition function Zcl =

Z

dx

Z

dp −β[p2 /2M +V (x)] e . 2π¯ h

(17.510)

The step function Θ(p) selects the particles running to the right across the top of the potential barrier. Performing the phase space path integral in (17.509) yields Γcl =

Zcl−1 −V (x∗ ) e . 2π¯ hβ

(17.511)

If the metastable minimum of the potential is smooth, V (x) can be replaced approximately in the neighborhood of x0 by the harmonic expression V (x) ≈

M 2 ω (x − x0 )2 . 2 0

(17.512)

1172

17 Tunneling

The classical partition function is then given approximately by Zcl ≈

1 , h ¯ βω0

(17.513)

and the decay rate follows the simple formula Γcl ≈

ω0 −βV (x∗ ) . e 2π

(17.514)

Let us compare this result with the decay rate due to pure quantum tunneling. In the limit of small temperatures, the decay proceeds from the ground state, and the partition function is approximately equal to Z ≈ e−β(E

(0) −i¯ hΓ/2)

.

(17.515)

The decay rate is given by the small imaginary part of the partition function: Γ− −−→ T →0

2 Im Z . h ¯ β Re Z

(17.516)

In contrast to this, the thermal rate formula (17.511) implies for the hightemperature regime, where Γ becomes equal to Γcl , the relation: Γ− −−→ T →∞

ω∗ Im Zcl . π Re Zcl

(17.517)

The frequency ω∗ is determined by the curvature of the potential at the top of the barrier, where it behaves like V (x) ≈ −

M 2 ω (x − x∗ )2 . 2 ∗

(17.518)

The relation (17.517) follows immediately by calculating in the integral (17.510) the contribution of the neighborhood of the top of the barrier in the saddle point approximation. As in the integral, this is done (17.261) by rotating the contour of integration which starts at x = x∗ into the upper complex half-plane. Writing x = x∗ + iy this leads to the following integral: Im Zcl ≈

Z

0

dy

∞

q

2π¯ h2 β/M

M 2 2 e−β [V (x∗ )+ 2 ω∗ y ] ≈

1 e−βV (x∗ ) . 2¯ hβω∗

(17.519)

Since the real part is given by (17.513) we find the ratio Im Zcl ω0 −βV (x∗ ) ≈ e , Re Zcl 2ω∗

(17.520)

so that (17.511) is equivalent to (17.517). The two formulas (17.516) and (17.517) are derived for the two extreme regimes T ≫ T0 and T ≪ T0 , respectively, where T0 denotes the characteristic temperature H. Kleinert, PATH INTEGRALS

Appendix 17A

1173

Feynman Integrals for Fluctuation Correction

associated with the curvature of the potential at the metastable minimum T0 = h ¯ ω0 /kB . Numerical studies have shown that the applicability extends into the close neighborhood of T0 on each side of the temperature axis. The crossover regime is quite small, of the order O(¯ h3/2 ). Note that given the knowledge of the imaginary parts of all excited states, which can be obtained as in Section 17.9, it will be possible to calculate the average lifetime of a metastable state at all temperatures without the restrictions of the semiclassical approximation. This remains to be done.

Appendix 17A

Feynman Integrals for Fluctuation Correction

For the integral (17.219) we obtain with (17.237) immediately the result stated in Eq. (17.240) I1′ =

97 . 560

(17A.1)

To calculate the remaining three double integrals I21 , I22 , I3 in (17.223) and (17.224) we observe that because of the symmetry of the Green function GOω (τ, τ ′ ) in τ, τ ′ the measure of integration R∞ Rt can be rewritten as 2 −∞ dτ −∞ dτ ′ . We further introduce the dimensionless classical functions r g x˜cl (τ ) ≡ xcl (τ ) = tanh(τ /2), 2ω 2 r 8 1 , (17A.2) y˜0 (τ ) ≡ y0 (τ ) = 3ω cosh2 (ωτ /2) use natural units with ω = 1, g = 1, since these quantities cancel in all integrals, and define xG (τ )

≡

xKG (τ )

≡

xK3G (τ )

≡

x ˜cl (τ )GOω (τ, τ ) Z τ GOω (τ, τ ′ )xG (τ ′ )dτ ′ −∞ Z τ A 3 ′ ′ ′ GOω (τ, τ )xG (τ )dt , −∞

(17A.3)

A

where the subscript A denotes the antisymmetric part in τ . Because of the antisymmetry of xcl , the symmetric part gives no contribution to the integrals which can be written as Z ∞ I21 = 6 dτ x ˜cl (τ )xK3G (τ ), (17A.4) 0 Z ∞ I22 = 9 dτ xG (τ )xKG (τ ), (17A.5) 0 Z ∞ I3 = 24 dτ y0′ (τ )xKG (τ ). (17A.6) 0

When evaluating the integrals in (17A.3) the antisymmetry of y0 (τ ) is useful. We easily find xKG (τ ) =

1 12τ + tanh(τ /2) . 4 12cosh2 (τ /2)

Inserting this into I22 and I3 we encounter integrals of two types Z ∞ Z ∞ dτ sinhm (τ /2)/ coshn (τ /2) and τ sinhm (τ /2)/ coshn (τ /2). 0

0

(17A.7)

1174

17 Tunneling

The former can be performed with the help of formula (17.54), the latter require integrations by parts of the type Z ∞ Z ∞ τ τ f ′ (τ /2) ln(2 cosh ) dτ, (17A.8) f (τ /2) tanh dτ = − 2 2 0 0 which lead to a finite sum of integrals of the first type plus integrals of the type11 Z ∞ log cosh(τ /2) sinhm (τ /2)/ coshn (τ /2). 0

R∞ These, in turn, are equal to −∂ν 0 dτ sinhm (τ /2)/ coshν+1 (τ /2) so that it is evaluated again via (17.54). After performing the subtraction in I22 we obtain the values given in Eq. (17.240). The evaluation of the integral I21 is more tedious since we must integrate over the third power of the Green function, which is itself a lengthy function of τ . It is once more useful to exploit the symmetry properties of the integrand. We introduce the abbreviations fnS/A (τ ) FnS/A (τ ) Nn

1 n [HR (τ ) ± HRn (−τ )]y03−n (τ ), 2 Z τ ≡ fnS/A (τ ′ )˜ xcl (τ ′ )dτ ′ , 0 Z ∞ ≡ HRn (τ )y03−n (τ ′ )˜ xcl (τ ′ )dτ ′ , ≡

(17A.9)

0

and find xK3G in the form xK3G (τ )

=

f3A (τ )[N0 − F0S (τ )] + 3f2A (τ )[N1 − F1S (τ )] + 3f1A (τ )[N2 − F2S (τ )] +3f2S(τ )F1A (τ ) + 3f1S (τ )F2A (τ ) + f0S (τ )F3A (τ ),

(17A.10)

with N0 =

3 , 32

N1 = −

7 , 128

N2 =

203 log2 − . 512 2

(17A.11)

Explicitly: " sech7 τ2 τ 3τ 5τ xK3G (τ ) = 3 τ 58 cosh − 27 cosh − 3 cosh 3 · 29 2 2 2 τ 3τ 5τ 7τ − 753 sinh + 48 sinh + 22 sinh + sinh 2 2 2 2 τ τ + 36 cosh ln(2 cosh ) (6 τ + 8 sinh τ + sinh 2τ ) 2 2 Z τ # τ τ′ ′ ′ − 108 cosh dτ τ tanh . 2 2 0

(17A.12)

The integrals to be performed are of the same types as before, except for the one involving the last term which requires one further partial integration: Z ∞ Z Z Z τ τ ′ ′ τ′ 1 ∞ d h τi τ ′ ′ τ′ − dτ x˜cl (τ )sech6 dτ τ tanh = dτ sech6 dτ τ tanh 2 0 2 3 0 dτ 2 0 2 0 Z 1 ∞ τ τ 16 = − dτ sech6 τ tanh = − . (17A.13) 3 0 2 2 135 11

I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 2.417. H. Kleinert, PATH INTEGRALS

1175


After the necessary subtraction of the divergent term we find the value of I21 given in Eq. (17.240). The final result for the first coefficient of the Taylor expansion of the subtracted fluctuation factor C ′ in Eq. (17.222) is therefore c1 =

71 ≈ 2.958. 24

(17A.14)

This number was calculated in Ref. [12] by solving the Schrödinger equation [20]. With the help of the WKB approximation he derived a recursion relation for the higher coefficients ck of the expansion of C ′ : " # 2 3 g¯h g¯h g¯h ′ − c3 + ... . (17A.15) C = 1 − c1 3 − c2 ω ω3 ω3 From this he calculated the next nine coefficients c2 =

315 65953 ≈ 9.84376, c3 = ≈ 57.2509. 32 1152

(17A.16)

Their large behavior is ck ∼

9 π

k 3 k![ln(6k) + γ]. 2

(17A.17)

Notes and References The path integral theory of tunneling was first discussed by A.I. Vainshtein, Decaying Systems and the Divergence of Perturbation Series, Novosibirsk Report (1964), in Russian (unpublished), and in the context of the nucleation of first-order phase transitions by J.S. Langer, Ann. Phys. 41, 108 (1967). The subject was studied further in the field-theoretic literature: M.B. Voloshin, I.Y. Kobzarev, L.B. Okun, Yad. Fiz. 20. 1229 (1974); Sov. J. Nucl. Phys. 20, 644 (1975); R. Rajaraman, Phys. Rep. 21, 227 (1975), R. Rajaraman, Phys. Rep. 21, 227 (1975), P. Frampton, Phys. Rev. Lett. 37, 1378 (1976) and Phys. Rev. D 15, 2922 (1977), S. Coleman, Phys. Rev. D 15, 2929 (1977); also in The Whys of Subnuclear Physics, Erice Lectures 1977, Plenum Press, 1979, ed. by A. Zichichi, I. Affleck, Phys. Rev. Lett. 46, 388 (1981). For a finite-temperature discussion see L. Dolan and J. Kiskies, Phys. Rev. D 20, 505-513 (1979). For multidimensional tunneling processes see H. Kleinert and R. Kaul, J. Low Temp. Phys. 38, 539 (1979) (http://www.physik.fu-berlin. de/~kleinert/66), A. Auerbach and S. Kivelson, Nucl. Phys. B 257, 799 (1985). The fact that tunneling calculations can be used to derive the growth behavior of large-order perturbation coefficients was first noticed by A.I. Vainshtein, Novosibirsk Preprint 1964, unpublished. A different but closely related way of deriving this behavior was proposed by L.N. Lipatov, JETP Lett. 24, 157 (1976); 25, 104 (1977); 44, 216 (1977); 45, 216 (1977).

1176

17 Tunneling

A review on this subject is given in J. Zinn-Justin, Phys. Rep. 49, 205 (1979), and in Recent Advances in Field Theory and Statistical Mechanics, Les Houches Lectures 1982, Elsevier Science 1984, ed. by J.-B. Zuber and R. Stora. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon, Oxford, 1990. The important applications to the ǫ-expansion of critical exponents in O(N )-symmetric ϕ4 field theories were made by E. Brezin, J.C. Le Guillou, and J. Zinn-Justin, Phys. Rev. D 15, 1544, 1558 (1977). E. Brezin and G. Parisi, J. Stat. Phys. 19, 269 (1978). The perturbation expansion of the ϕ4 -theory up to fifth-order was calculated in Ref. [13]. For two quartic interactions of different symmetries (one O(N )-symmetric, the other of cubic symmetry) see: H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995) (cond-mat/9503038) A detailed discussion and a comprehensive list of the references to the original papers are contained in the textbook [15]. The calculation of the anomalous magnetic moments in quantum electrodynamics is described in T. Kinoshita and W.B. Lindquist, Phys. Rev. D 27, 853 (1983). M.J. Levine and R. Roskies, in Proceedings of the Second International Conference on Precision Measurements and Fundamental Constants, ed. by B.N. Taylor and W.D. Phillips, Natl. Bur. Std. US, Spec. Publ. 617 (1981). The analytic result for the semiclassical decay rate of a supercurrent in a thin wire was found by ¨ I.H. Duru, H. Kleinert, and N. Unal, J. Low Temp. Phys. 42, 137 (1981) (ibid.http/74), H. Kleinert and T. Sauer, J. Low Temp. Physics 81, 123 (1990) (ibid.http/204). The experimental situation is explained in M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York, 1975. See, in particular, Chapter 7, Sections 7.1–7.3. For quantum corrections to the decay rate see N. Giordano, Phys. Rev. Lett. 61, 2137 (1988); N. Giordano and E.R. Schuler, Phys. Rev. Lett. 63, 2417 (1989). For thermally driven tunneling processes see the review article P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990). Tunneling processes with dissipation were first discussed at T = 0 by A.O. Caldeira and A.J. Leggett, Ann. Phys. 149, 374 (1983), 153, 445 (1973) (Erratum) and for T 6= 0 by H. Grabert, U. Weiss and P. Hänggi, Phys. Rev. Lett. 52, 2193 (1984), A.I. Larkin and Y.N. Ovchinnikov, Sov. Phys. JETP 59, 420 (1984). Papers on coherent tunneling: A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987), U. Weiss, H. Grabert, P. Hänggi, and P. Riseborough, Phys. Rev. B 35, 9535 (1987), H. Grabert, P. Olschowski, and U. Weiss, Phys. Rev. B 36, 1931 (1987), On the use of periodic orbits see: P. Hänggi and W. Hontscha, Ber. Bunsen-Ges. Phys. Chemie 95, 379 (1991). The individual citations refer to [1] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965, §25, Problem 3. H. Kleinert, PATH INTEGRALS


1177

[2] L.D. Faddeev and V.N. Popov, Phys. Lett. B 25, 29 (1967). [3] J.S. Langer, Ann. Phys. 41, 108 (1967). [4] For a general proof see S. Coleman, Nucl. Phys. B 298, 178 (1988). [5] F.J. Dyson, Phys. Rev. 85, 631 (1952). [6] Compare the result in Eq. (17.345) with J.C. Collins and D.E. Soper, Ann. Phys. 112, 209 (1978). [7] The variational approach to tunneling was initiated in H. Kleinert, Phys. Lett. B 300, 261 (1993). It has led to very precise tunneling rates in subsequent work by R. Karrlein and H. Kleinert, Phys. Lett. A 187, 133 (1994); H. Kleinert and I. Mustapic, Int. J. Mod. Phys. A 11, 4383 (1995), most precisely in Ref. [11]. [8] H. Kleinert, Phys. Lett. B 300, 261 (1993) (ibid.http/214). [9] R. Karrlein and H. Kleinert, Phys. Lett. A 187, 133 (1994) (hep-th/9504048). [10] C.M. Bender and T.T. Wu, Phys. Rev. D 7 , 1620 (1973), Eq. (5.22). [11] B. Hamprecht and H. Kleinert, Tunneling Amplitudes by Perturbation Theory, Phys. Lett. B 564, 111 (2003) (hep-th/0302124). [12] J. Zinn-Justin, J. Math. Phys. 22, 511 (1981); Table III. [13] H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin, and S.A. Larin, Phys. Lett. B 272, 39 (1991) (hep-th/9503230). [14] H. Kleinert, Phys. Rev. D 57 , 2264 (1998); Addendum: Phys. Rev. D 58 , 107702 (1998) (cond-mat/9803268). [15] For details and applications to see the textbook H. Kleinert and V. Schulte-Frohlinde, Critical Phenomena in φ4 -Theory, World Scientific, Singapore, 2001 (ibid.http/b8). [16] The path integral treatment of the decay rate of a supercurrent in a thin wire was initiated by J.S. Langer and V. Ambegaokar, Phys. Rev. 164, 498 (1967), D.E. McCumber and B.I. Halperin, Phys. Rev. B 1, 1054 (1970). [17] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York, 1975. [18] H. Kleinert and T. Sauer, J. Low Temp. Physics 81, 123 (1990) (ibid.http/204). [19] D.E. McCumber and B.I. Halperin Phys. Rev. B 1, 1054 (1970). [20] Divide the numbers in the Table IV of Ref. [12] by 6n4n to get our cn .


Path, motive, guide, original, and end. Samuel Johnson (1709-1784), The Rambler

18 Nonequilibrium Quantum Statistics Quantum statistics described by the theoretical tools of the previous chapters is quite limited. The physical system under consideration must be in thermodynamic equilibrium, with a constant temperature enforced by a thermal reservoir. In this situation, partition function and the density matrix can be calculated from an analytic continuation of quantum-mechanical time evolution amplitudes to an imaginary h/kB T . In this chapter we want to go beyond such equilibrium time tb − ta = −i¯ physics and extend the path integral formalism to nonequilibrium time-dependent phenomena. The tunneling processes discussed in Chapter 17 belong really to this class of phenomena, and their full understanding requires the theoretical framework of this chapter. In the earlier treatment this was circumvented by addressing only certain quasi-equilibrium questions. These were answered by applying the equilibrium formalism to the quantum system at positive coupling constant, which guaranteed perfect equilibrium, and extending the results to the quasi-equilibrium situation analytic continuation to small negative coupling constants. Before we can set up a path integral formulation capable of dealing with true nonequilibrium phenomena, some preparatory work is useful based on the traditional tools of operator quantum mechanics.

18.1

Linear Response and Time-Dependent Green Functions for T / =0

If the deviations of a quantum system from thermal equilibrium are small, the easiest description of nonequilibrium phenomena proceeds via the theory of linear response. In operator quantum mechanics, this theory is introduced as follows. ˆ First, the system is assumed to have a time-independent Hamiltonian operator H. The ground state is determined by the Schrödinger equation, evolving as a function of time according to the equation ˆ

|ΨS (t)i = e−iHt |ΨS (0)i

(18.1)

(in natural units with h ¯ = 1, kB = 1). The subscript S denotes the Schrödinger picture. 1178

18.1 Linear Response and Time-Dependent Green Functions for T 6= 0

1179

ˆ a time-dependent external Next, the system is slightly disturbed by adding to H interaction, ˆ →H ˆ +H ˆ ext (t), H

(18.2)

−iHt ˆ |Ψdist UH (t)|ΨS (0)i, S (t)i = e

ˆ

(18.3)

˙ ˆ ext (t)UˆH (t), iUˆH (t) = H H

(18.4)

ˆ ˆ ext ˆ ˆ ext (t) ≡ eiHt H H (t)e−iHt . H

(18.5)

ˆ ext (t) is assumed to set in at some time t0 , i.e., H ˆ ext (t) vanishes identically where H for t < t0 . The disturbed Schrödinger ground state has the time dependence where UˆH (t) is the time translation operator in the Heisenberg picture. It satisfies the equation of motion

with1 To lowest-order perturbation theory, the operator UˆH (t) is given by UˆH (t) = 1 − i

Z

t

t0

ˆ ext (t′ ) + · · · . dt′ H H

(18.6)

In the sequel, we shall assume the onset of the disturbance to lie at t0 = −∞. ˆ whose Heisenberg Consider an arbitrary time-independent Schrödinger observable O representation has the time dependence ˆ ˆ −iHt ˆ ˆ H (t) = eiHt O Oe .

(18.7)

Its time-dependent expectation value in the disturbed state |Ψdist S (t)i is given by ˆ

ˆ

iHt ˆ −iHt ˆ ˆ dist ˆ† hΨdist Oe UH (t)|ΨS (0)i S (t)|O|ΨS (t)i = hΨS (0)|UH (t)e

≈ hΨS (0)| 1 + i

× 1−i

Z

t

−∞ t

Z

−∞

ˆ ext (t′ ) + . . . O ˆ H (t) dt′ H H

ext ′ ˆH dt′ H (t ) + . . . |ΨS (0)i

ˆ H (t)|ΨH i − ihΨH | = hΨH |O

Z

t

−∞

h

(18.8) i

ˆ H (t), H ˆ ext (t′ ) |ΨH i + . . . . dt′ O H

We have identified the time-independent Heisenberg state with the time-dependent Schrödinger state at zero time in the usual manner, i.e., |ΨH i ≡ |ΨS (0)i. Thus the ˆ deviates from equilibrium by expectation value of O dist ˆ S (t)i ≡ hΨdist(t)|O(t)|Ψ ˆ ˆ δhΨS (t)|O|Ψ S S (t)i − hΨS (t)|O(t)|ΨS (t)i

= −i 1

Z

t

−∞

h

i

ˆ H (t), H ˆ ext (t′ ) |ΨH i. dt′ hΨH | O H

(18.9)

ext Note that after the replacements H → H0 , HH → HIint , Eq. (18.4) coincides with the equation for the time evolution operator in the interaction picture to appear in Section 18.7. In contrast to that section, however, the present interaction is a nonpermanent artifact to be set equal to zero at the end, and H is the complicated total Hamiltonian, not a simple free one. This is why we do not speak of an interaction picture here.

1180

18 Nonequilibrium Quantum Statistics

If the left-hand side is transformed into the Heisenberg picture, it becomes ˆ S (t)i = δhΨH |O ˆ H (t)|ΨH i = hΨH |δ O ˆ H (t)|ΨH i, δhΨS (t)|O|Ψ so that Eq. (18.9) takes the form ˆ H (t)|ΨH i = −i hΨH |δ O

Z

t −∞

h

i

ˆ H (t), H ˆ ext (t′ ) |ΨH i. dt′ hΨH | O H

(18.10)

ˆ H (t′ ) in ˆ H (t) and H It is useful to use the retarded Green function of the operators O the state |ΨH i [compare (3.40)]: h

i

′ ′ ˆ ˆ ′ GR OH (t, t ) ≡ Θ(t − t )hΨH | OH (t), HH (t ) |ΨH i.

(18.11)

Then the deviation from equilibrium is given by the integral ˆ H (t)|ΨH i = −i hΨH |δ O

Z

∞ −∞

′ dt′ GR OH (t, t ).

(18.12)

ˆ H (t) is capable of undergoing oscillations. Suppose now that the observable O ˆ H (t) will in general excite these oscillaThen an external disturbance coupled to O tions. The simplest coupling is a linear one, with an interaction energy ˆ ext (t) = −O ˆ H (t)δj(t), H

(18.13)

where j(t) is some external source. Inserting (18.13) into (18.12) yields the linearresponse formula ˆ H (t)|ΨH i = i hΨH |δ O

Z

∞

−∞

′ ′ dt′ GR OO (t, t )δj(t ),

(18.14)

ˆ where GR OO is the retarded Green function of two operators O: h

i

′ ′ ˆ ˆ ′ GR OO (t, t ) = Θ(t − t )hΨH | OH (t), OH (t ) |ΨH i.

(18.15)

At frequencies where the Fourier transform of GOO (t, t′ ) is singular, the slightest disturbance causes a large response. This is the well-known resonance phenomenon found in any oscillating system. Whenever the external frequency ω hits an eigenfrequency, the Fourier transform of the Green function diverges. Usually, the eigenfrequencies of a complicated N-body system are determined by calculating (18.15) and by finding the singularities in ω. It is easy to generalize this description to a thermal ensemble at a nonzero temperature. The principal modification consists in the replacement of the ground state expectation by the thermal average ˆ ˆ Tr (e−H/T O) ˆ hOiT ≡ . ˆ Tr (e−H/T )


18.2 Spectral Representations of Green Functions for T 6= 0

1181

Using the free energy ˆ

F = −T log Tr (e−H/T ), this can also be written as ˆ ˆ ˆ T = eF/T Tr (e−H/T O). hOi

(18.16)

ˆ must be replaced by H ˆ − µNˆ and F by its grandIn a grand-canonical ensemble, H canonical version FG (see Section 1.17). At finite temperatures, the linear-response formula (18.14) becomes ˆ δhO(t)i T = i

Z

∞

−∞

′ ′ dt′ GR OO (t, t )δj(t ),

(18.17)

′ where GR OO (t, t ) is the retarded Green function at nonzero temperature defined by [recall (1.302)] ˆ ′ R ′ ′ F/T ˆ H (t), O ˆ H (t′ ) GR Tr e−H/T O OO (t, t ) ≡ GOO (t − t ) ≡ Θ(t − t ) e

n

h

io

. (18.18)

ˆ i (t) for In a realistic physical system, there are usually many observables, say O H i = 1, 2, . . . , l, which perform coupled oscillations. Then the relevant retarded Green function is some l × l matrix ˆ ′ R ′ ′ F/T ˆ i (t), O ˆ j (t′ ) GR Tr e−H/T O ij (t, t ) ≡ Gij (t − t ) ≡ Θ(t − t ) e H H

n

h

io

.

(18.19)

After a Fourier transformation and diagonalization, the singularities of this matrix render the important physical information on the resonance properties of the system. The retarded Green function at T 6= 0 occupies an intermediate place between the real-time Green function of field theories at T = 0, and the imaginary-time Green function used before to describe thermal equilibria at T 6= 0 (see Subsection 3.8.2). The Green function (18.19) depends both on the real time and on the temperature via an imaginary time.

18.2

Spectral Representations of Green Functions for T / =0

The retarded Green functions are related to the imaginary-time Green functions of 1 ˆH equilibrium physics by an analytic continuation. For two arbitrary operators O , 2 ˆ OH , the latter is defined by the thermal average ˆ 1 2 ˆH ˆH G12 (τ, 0) ≡ G12 (τ ) ≡ eF/T Tr e−H/T Tˆτ O (τ )O (0) ,

h

i

(18.20)

ˆ H (τ ) is the imaginary-time Heisenberg operator where O ˆ ˆ −Hτ ˆ ˆ H (τ ) ≡ eHτ O Oe .

(18.21)

1182


To see the relation between G12 (τ ) and the retarded Green function GR 12 (t), we take 1 2 ˆ ˆ a complete set of states |ni, insert them between the operators O , O , and expand G12 (τ ) for τ ≥ 0 into the spectral representation G12 (τ ) = eF/T

ˆ 1 |n′ ihn′ |O ˆ 2|ni. e−En /T e(En −En′ )τ hn|O

X

n,n′

(18.22)

Since G12 (τ ) is periodic under τ → τ + 1/T , its Fourier representation contains only the discrete Matsubara frequencies ωm = 2πmT : Z

G12 (ωm ) =

1/T

0 F/T

= e

dτ eiωm τ G12 (τ ) X

n,n′

ˆ 1 |n′ ihn′ |O ˆ 2|ni e−En /T 1 − e(En −En′ )/T hn|O ×

−1 . iωm − En′ + En

(18.23)

The retarded Green function satisfies no periodic (or antiperiodic) boundary condition. It possesses Fourier components with all real frequencies ω: GR 12 (ω)

=

Z

∞

−∞

= eF/T

iωt

dt e Z

0

∞

F/T

Θ(t)e

dt eiωt

Xh

n,n′

ˆ −H/T

Tr e

h

i

ˆ 2 (0) ˆ 1 (t), O O H H

∓

ˆ 1 |n′ ihn′ |O ˆ 2|ni e−En /T ei(En −En′ )t hn|O i

ˆ 2|n′ ihn′ |O ˆ 1 |ni . (18.24) ∓e−En /T e−i(En −En′ )t hn|O

In the second sum we exchange n and n′ and perform the integral, after having attached to ω an infinitesimal positive-imaginary part iη to ensure convergence [recall the discussion after Eq. (3.84)]. The result is F/T GR 12 (ω) = e

X

n,n′

h

i

ˆ 1 |n′ ihn′ |O ˆ 2|ni e−En /T 1 − e(En −En′ )/T hn|O ×

i . ω − En′ + En + iη

(18.25)

By comparing this with (18.23) we see that the thermal Green functions are obtained from the retarded ones by replacing [1] i −1 . → ω − En′ + En + iη iωm − En′ + En

(18.26)

ˆ i (which are not observable). A similar procedure holds for fermion operators O There are only two changes with respect to the boson case. First, in the Fourier expansion of the imaginary-time Green functions, the bosonic Matsubara frequencies ωm in (18.23) become fermionic. Second, in the definition of the retarded Green H. Kleinert, PATH INTEGRALS

18.2 Spectral Representations of Green Functions for T 6= 0

1183

functions (18.19), the commutator is replaced by an anticommutator, i.e., the reˆ i is defined by tarded Green function of fermion operators O H ′ GR ij (t, t )

≡

GR ij (t

′

′

F/T

− t ) ≡ Θ(t − t )e

ˆ −H/T

Tr e

h

j ˆ i (t), O ˆH O (t′ ) H

i +

. (18.27)

These changes produce an opposite sign in front of the e(En −En′ )/T -term in both of the formulas (18.23) and (18.25). Apart from that, the relation between the two Green functions is again given by the replacement rule (18.26). At this point it is customary to introduce the spectral function ρ12 (ω ′ ) =

′

1 ∓ e−ω /T eF/T

×

P

n,n′

ˆ 1|n′ ihn′ |O ˆ 2 |ni, e−En /T 2πδ(ω − En′ + En )hn|O

(18.28)

where the upper and the lower sign hold for bosons and fermions, respectively. Under an interchange of the two operators it behaves like ρ12 (ω ′ ) = ∓ρ12 (−ω ′ ).

(18.29)

Using this spectral function, we may rewrite the Fourier-transformed retarded and thermal Green functions as the following spectral integrals: GR 12 (ω) =

Z

∞

G12 (ωm ) =

Z

∞

−∞

−∞

i dω ′ ρ12 (ω ′ ) , 2π ω − ω ′ + iη dω ′ −1 . ρ12 (ω ′ ) 2π iωm − ω ′

(18.30) (18.31)

These equations show how the imaginary-time Green functions arise from the retarded Green functions by a simple analytic continuation in the complex frequency plane to the discrete Matsubara frequencies, ω → iωm . The inverse problem of reconstructing the retarded Green functions in the entire upper half-plane of ω from the imaginary-time Green functions defined only at the Matsubara frequencies ωm is not solvable in general but only if other information is available [2]. For instance, the sum rules for canonical fields to be derived later in Eq. (18.66) with the ensuing asymptotic condition (18.67) are sufficient to make the continuation unique [3]. Going back to the time variables t and τ , the Green functions are GR 12 (t) = Θ(t) G12 (τ ) =

Z

∞

−∞

Z

∞

−∞

dω ′ ′ ρ12 (ω ′ )e−iω t , 2π

X dω ′ −1 ρ12 (ω ′)T e−iωm τ . 2π iωm − ω ′ ωm

(18.32) (18.33)

The sum over even or odd Matsubara frequencies on the right-hand side of G12 (τ ) was evaluated in Section 3.3 for bosons and fermions as X −1 1 T e−iωm τ = Gpω,e (τ ) = e−ω(τ −1/2T ) iωm − ω 2 sin(ω/2T ) n −ωτ = e (1 + nω ) (18.34)

1184


and T

X n

e−iωm τ

−1 1 = Gaω,e (τ ) = e−ω(τ −1/2T ) iωm − ω 2 cos(ω/2T ) −ωτ = e (1 − nω ),

(18.35)

with the Bose and Fermi distribution functions [see (3.93), (7.529), (7.531)] nω =

1 eω/T

∓1

,

(18.36)

respectively.

18.3

Other Important Green Functions

In studying the dynamics of systems at finite temperature, several other Green functions are useful whose spectral functions we shall now derive. In complete analogy with the retarded Green functions for bosonic and fermionic operators, we may introduce their counterparts, the so-called advanced Green functions (compare page 38)

ˆ ′ A ′ ′ F/T ˆ 1 (t), O ˆ 2 (t′ ) Tr e−H/T O GA 12 (t, t ) ≡ G12 (t − t ) = −Θ(t − t)e H H

h

i ∓

. (18.37)

Their Fourier transforms have the spectral representation GA 12 (ω)

=

Z

∞

−∞

i dω ′ ρ12 (ω ′) , 2π ω − ω ′ − iη

(18.38)

differing from the retarded case (18.30) only by the sign of the iη-term. This makes the Fourier transforms vanish for t > 0, so that the time-dependent Green function has the spectral representation [compare (18.32)] GA 12 (t) = −Θ(−t)

Z

∞ −∞

dω ρ12 (ω)e−iωt . 2π

(18.39)

By subtracting retarded and advanced Green functions, we obtain the thermal expectation value of commutator or anticommutator:

ˆ ˆ 1 (t), O ˆ 2 (t′ ) C12 (t, t′ ) = eF/T Tr e−H/T O H H

h

i ∓

′ A ′ = GR 12 (t, t ) − G12 (t, t ). (18.40)

Note the simple relations: ′ ′ ′ GR 12 (t, t ) = Θ(t − t )C12 (t, t ), ′ ′ ′ GA 12 (t, t ) = −Θ(t − t)C12 (t, t ).

(18.41) (18.42) H. Kleinert, PATH INTEGRALS

1185

18.3 Other Important Green Functions

When inserting the spectral representations (18.30) and (18.39) of GR 12 (t) and A G12 (t) into (18.40), and using the identity (1.324), i i η = 2πδ(ω − ω ′), − = 2 ′ ′ ω − ω + iη ω − ω − iη (ω − ω ′)2 + η 2

(18.43)

we obtain the spectral integral representation for the commutator function:2 C12 (t) =

Z

∞

−∞

dω ρ12 (ω)e−iωt . 2π

(18.44)

Thus a knowledge of the commutator function C12 (t) determines directly the spectral function ρ12 (ω) by its Fourier components C12 (ω) = ρ12 (ω).

(18.45)

An important role in studying the dynamics of a system in a thermal environment is played by the time-ordered Green functions. They are defined by ˆ ˆ 1 (t)O ˆ 2 (t′ ) . G12 (t, t′ ) ≡ G12 (t − t′ ) = eF/T Tr e−H/T TÔ H H

h

i

(18.46)

Inserting intermediate states as in (18.23) we find the spectral representation G12 (ω) = +

Z

∞

−∞ ∞

Z

−∞

ˆ ˆ 1 (t)O ˆ 2 (0) dt eiωt Θ(t) eF/T Tr e−H/T O H H

n

o

ˆ 2 1 ˆH ˆH dt eiωt Θ(−t)eF/T Tr e−H/T O (t)O (0)

F/T

= e

Z

± eF/T

Z

∞ 0

dt eiωt

−∞

dt eiωt

o

X

ˆ 1 |n′ ihn′ |O ˆ 2|ni e−En /T ei(En −En′ )t hn|O

X

ˆ 2 |n′ ihn′ |O ˆ 1|ni . e−En /T e−i(En −En′ )t hn|O

n,n′ 0

n

n,n′

(18.47)

Interchanging again n and n′ , this can be written in terms of the spectral function (18.28) as G12 (ω) =

Z

∞

−∞

dω ′ 1 i 1 i ρ12 (ω ′ ) + . (18.48) ′ /T ′ /T −ω ′ ω 2π 1∓e ω − ω + iη 1 ∓ e ω − ω ′ − iη "

#

Let us also write down the spectral decomposition of a further operator expression complementary to C12 (t) of (18.40), in which boson or fermion fields appear with the “wrong” commutator: ′

F/T

A12 (t − t ) ≡ e 2

ˆ −H/T

Tr e

h

ˆ 1 (t), O ˆ 2 (t′ ) O H H

i ±

.

(18.49)

Due to the relation (18.41), the same representation is found by dropping the factor Θ(t) in (18.32).

1186


1 2 This function characterizes the size of fluctuations of the operators OH and OH . Inserting intermediate states, we find

A12 (ω) =

Z

∞

iωt F/T

−∞

= eF/T

dt e e Z

∞

−∞

ˆ −H/T

Tr e

dt eiωt

Xh

h

i

1 2 ˆH ˆH O (t), O (0)

±

ˆ 1 |n′ ihn′ |O ˆ 2|ni e−En /T ei(En −En′ )t hn|O

n,n′

i

ˆ 2 |n′ ihn′ |O ˆ 1|ni . (18.50) ±e−En /T e−i(En −En′ )t hn|O

In the second sum we exchange n and n′ and perform the integral, which runs now over the entire time interval and gives therefore a δ-function: A12 (ω) = eF/T

X

n,n′

h

i

ˆ 1 |n′ ihn′ |O ˆ 2|ni e−En /T 1 ± e(En −En′ )/T hn|O × 2πδ(ω − En′ + En ).

(18.51)

In terms of the spectral function (18.28), this has the simple form A12 (ω) =

∞

Z

−∞

ω′ ω dω ′ tanh∓1 ρ12 (ω ′ ) 2πδ(ω − ω ′) = tanh∓1 ρ12 (ω). (18.52) 2π 2T 2T

Thus the expectation value (18.49) of the “wrong” commutator has the time dependence ′

′

A12 (t, t ) ≡ A12 (t − t ) =

Z

∞

−∞

dω ω −iω(t−t′ ) ρ12 (ω) tanh∓1 e . 2π 2T

(18.53)

There exists another way of writing the spectral representation of the various A Green functions. For retarded and advanced Green functions GR 12 , G12 , we decompose in the spectral representations (18.30) and (18.38) according to the rule (1.325): i P = i ∓ iπδ(ω − ω ′ ) , ω − ω ′ ± iη ω − ω′

(18.54)

where P indicates principal value integration across the singularity, and write GR,A 12 (ω) = i

Z

∞

−∞

dω ′ P ρ12 (ω ′ ) ∓ iπδ(ω − ω ′) . 2π ω − ω′

(18.55)

Inserting (18.54) into (18.48) we find the alternative representation of the timeordered Green function G12 (ω) = i

Z

∞

−∞

dω ′ P ω ρ12 (ω ′) − iπ tanh∓1 δ(ω − ω ′ ) . (18.56) ′ 2π ω−ω 2T

The term proportional to δ(ω − ω ′ ) in the spectral representation is commonly referred to as the absorptive or dissipative part of the Green function. The first term proportional to the principal value is called the dispersive or fluctuation part. H. Kleinert, PATH INTEGRALS

1187

18.4 Hermitian Adjoint Operators

The relevance of the spectral function ρ12 (ω ′) in determining both the fluctuation part as well as the dissipative part of the time-ordered Green function is the content of the important fluctuation-dissipation theorem. In more detail, this may be restated as follows: The common spectral function ρ12 (ω ′ ) of the commutator function in (18.44), the retarded Green function in (18.30), and the fluctuation part of the time-ordered Green function in (18.56) determines, after being multiplied by a factor tanh∓1 (ω ′ /2T ), the dissipative part of the time-ordered Green function in Eq. (18.56). A The three Green functions −iG12 (ω), −iGR 12 (ω), and −iG12 (ω) have the same real parts. By comparing Eqs. (18.30) and (18.31) we found that retarded and advanced Green functions are simply related to the imaginary-time Green function via an analytic continuation. The spectral decomposition (18.56) shows this is not true for the time-ordered Green function, due to the extra factor tanh∓1 (ω/2T ) in the absorptive term. Another representation of the time-ordered Green is useful. It is obtained by expressing tan∓1 in terms of the Bose and Fermi distribution functions (18.36) as tan∓1 = 1 ± 2nω . Then we can decompose G12 (ω) =

18.4

Z

∞

−∞

i dω ′ ρ12 (ω ′ ) ± 2πnω δ(ω − ω ′ ) . 2π ω − ω ′ + iη "

#

(18.57)

Hermitian Adjoint Operators

ˆ 1 (t), O ˆ 2 (t) are Hermitian adjoint to each other, If the two operators O H H ˆ 2 (t) = [O ˆ 1 (t)]† , O H H

(18.58)

the spectral function (18.28) can be rewritten as ′

ρ12 (ω ′ ) = (1 ∓ e−ω /T )eF/T X ˆ 1 (t)|n′ ik2 . × e−En /T 2πδ(ω ′ − En′ + En )|hn|O H

(18.59)

n,n′

This shows that ρ12 (ω ′ )ω ′ ≥ 0 ′

ρ12 (ω )

for bosons, (18.60)

≥0

for fermions.

This property permits us to derive several useful inequalities between various diagonal Green functions in Appendix 18A. Under the condition (18A.9), the expectation values of anticommutators and commutators satisfy the time-reversal relations ′ GA 12 (t, t ) A12 (t, t′ ) C12 (t, t′ ) G12 (t, t′ )

= = = =

′ ∗ ∓GR 21 (t , t) , ±A21 (t′ , t)∗ , ∓C21 (t′ , t)∗ . ±G21 (t′ , t)∗ .

(18.61) (18.62) (18.63) (18.64)

1188


Examples are the corresponding functions for creation and annihilation operators which will be treated in detail below. More generally, this properties hold for any ˆ 1 (t) = ψˆp (t), O ˆ 2 (t) = ψˆ† (t) of a specific interacting nonrelativistic particle fields O H H p momentum p. Such operators satisfy, in addition, the canonical equal-time commutation rules at each momentum ψˆp (t), ψˆp† (t) = 1

h

i

(18.65)

(see Sections 7.6, 7.9). Using (18.40), (18.44) we derive from this spectral function sum rule: Z

∞

−∞

dω ′ ρ12 (ω ′ ) = 1. 2π

(18.66)

For a canonical free field with ρ12 (ω ′) = 2πδ(ω ′ −ω), this sum rule is of course trivially fulfilled. In general, the sum rule ensures the large-ω behavior of imaginary-time, retarded, and advanced Green functions of canonically conjugate field operators to be the same as for a free particle, i.e., −−→ G12 (ωm ) −

ωm →∞

18.5

i , ωm

GA,R −−→ 12 (ω) − ω→∞

1 . ω

(18.67)

Harmonic Oscillator Green Functions for T / =0

As an example, consider a single harmonic oscillator of frequency Ω or, equivalently, a free particle at a point in the second-quantized field formalism (see Chapter 7). We shall start with the second representation.

18.5.1

Creation Annihilation Operators

ˆ 1 (t) and O ˆ 2 (t) are the creation and annihilation operators in the The operators O H H Heisenberg picture a ˆ†H (t) = aˆ† eiΩt ,

a ˆH (t) = a ê−iΩt .

(18.68)

The eigenstates of the Hamiltonian operator 1 ˆ = 1 pˆ2 + Ω2 xˆ2 = ω a H ˆ† a ˆ+a âˆ† = ω a ˆ† a ˆ± 2 2 2

are

1 |ni = √ (ˆ a† )n |0i, n!

(18.69)

(18.70)

with the eigenvalues En = (n ± 1/2)Ω for n = 0, 1, 2, 3, . . . or n = 0, 1, if a ˆ† and a ˆ commute or anticommute, respectively [compare Eq. (7.551)]. In the secondquantized field interpretation the energies are En = nΩ and the final Green functions H. Kleinert, PATH INTEGRALS

18.5 Harmonic Oscillator Green Functions for T 6= 0

1189

are the same. The spectral function ρ12 (ω ′ ) is trivial to calculate. The Schrödinger ˆ 2 = aˆ† can connect the state |ni only to hn + 1|, with the matrix element operator O √ ˆ1 = a n + 1. The operator O ˆ does the opposite. Hence we have ′

′

−Ω/T

ρ12 (ω ) = 2πδ(ω − Ω)(1 ∓ e

F/T

)e

∞,1 X

e−(n±1/2)Ω/T (n + 1).

(18.71)

n

Now we make use of the explicit partition functions of the oscillator whose paths satisfy periodic and antiperiodic boundary conditions: ∞,1 X

−(n±1/2)Ω/T

e

−F/T

=e

n

=

(

[2 sinh(Ω/2T )]−1 2 cosh(Ω/2T )

bosons fermions

for

)

.

(18.72)

These allow us to calculate the sum in (18.71) as follows ∞,1 X

−(n±1/2)Ω/T

e

n

1 = 2

(

!

∂ 1 −F/T (n + 1) = −T e + ∂Ω 2 coth(Ω/2T ) tanh(Ω/2T )

)

!

(18.73)

+ 1 e−F/T = 1 ∓ e−Ω/T

−1

e−F/T .

The spectral function ρ12 (ω ′) of the a single oscillator quantum of frequency Ω is therefore given by ρ12 (ω ′ ) = 2πδ(ω ′ − Ω).

(18.74)

With it, the retarded and imaginary-time Green functions become ′

′ ′ −Ω(t−t ) GR , Ω (t, t ) = Θ(t − t )e ′

GΩ (τ, τ ) = −T

∞ X

e−iωm (τ −τ

m=−∞

= e−Ω(τ −τ

′)

  

1 ± nΩ ±nΩ

(18.75) ′)

1 iΩm − Ω for τ

≥ ′ τ,
0, this coincides precisely with the periodic Green function Gpe (τ, τ ′ ) = Gpe (τ − τ ′ ) at imaginary-times τ > τ ′ [see (3.248)], if τ and τ ′ are continued analytically to it and it′ , respectively. Decomposing (18.101) into real and imaginary parts we see by comparison with (18.100) that anticommutator and commutator functions are the doubled real and imaginary parts of the time-ordered Green function: A(t, t′ ) = 2 Re G(t, t′ ),

C(t, t′ ) = 2i Im G(t, t′ ).

(18.104)

In the fermionic case, the hyperbolic functions cosh and sinh in numerator and denominator are simply interchanged, and the result coincides with the analytically continued antiperiodic imaginary-time Green function (3.263). H. Kleinert, PATH INTEGRALS

1193

18.6 Nonequilibrium Green Functions

The time-reversal properties (18.61)–(18.64) of the Green functions become for real fields ϕ(t): ˆ GA (t, t′ ) A(t, t′ ) C(t, t′ ) G(t, t′ )

18.6

= = = =

∓GR (t′ , t), ±A(t′ , t), ∓C(t′ , t), ±G(t′ , t).

(18.105) (18.106) (18.107) (18.108)

Nonequilibrium Green Functions

Up to this point we have assumed the system to be in intimate contact with a heat reservoir which ensures a constant temperature throughout the volume. The disturbance in (18.3) was taken to be small, so that only a small fraction of the particles could be excited. If the disturbance grows larger, large clouds of excitations can be formed in a local region. Such a system leaves thermal equilibrium, and the response is necessarily nonlinear. The system must be studied in its full quantummechanical time evolution. In order to describe such a process theoretically, we shall assume an initial equilibrium characterized by some density operator [compare (2.359)] ρˆ =

X n

ρn |nihn|,

(18.109)

with eigenvalues ρn = e−En /T .

(18.110)

The disturbance sets in at some time t0 . If the initial state is out of equilibrium, the formalism to be described remains applicable, with only a few adaptations, if the initial state at t0 is still characterized by a density operator of type (18.109), but has probabilities ρn different from (18.110). Of course, in the limit of very small deviations from thermal equilibrium, the formalism to be described reduces to the previously treated linear-response theory. We first develop a perturbation theory for the time evolution of operators in a nonequilibrium situation. This serves to set up a path integral formalism for the description of the dynamical behavior of a single particle in contact with a thermal reservoir. This description can, in principle, be extended to ensembles of many particles by considering a similar path integral for a fluctuating field. After the discussion in Chapter 7, the necessary second quantization is straightforward and requires no detailed presentation. The perturbation theory for nonequilibrium quantum-statistical mechanics to be developed now is known under the name of closed-time path Green function formalism (CTPGF). This formalism was developed by Schwinger [4] and Keldysh [5], and has been applied successfully to many nonequilibrium problems in statistical physics, in particular to superconductivity and plasma physics.

1194


The fundamental problem of nonequilibrium statistical mechanics is finding the time evolution of thermodynamic averages of products of Heisenberg operators ϕˆH (t). For interesting applications it is useful to keep the formulation general and deal with relativistic fields of operators ϕˆH (x, t). As in Section 7.6, an extra spatial argument x allows for a different time-dependent operator ϕ(t) ˆ at each point x in space. In order to prepare ourselves for the most interesting study of electromagnetic fields, we consider the simplest relativistically invariant classical action describing an observable field in D dimensions which has the form A0 =

Z

dtdD x L0 (x, t) ≡

1 2

Z

n

o

dtdD x [ϕ(x, ˙ t)]2 − [∇ϕ(x, t)]2 − m2 ϕ2 (x, t) . (18.111)

As in Section 7.6, we go over to a countable set of infinite points x assuming that space is a fine lattice of spacing ǫ, with the continuum limit ǫ → 0 taken at the end. The associated Euler-Lagrange equation extremizing the action is the Klein-Gordon equation ϕ(x, ¨ t) + (−∂x2 + m2 )ϕ(x, t) = 0.

(18.112)

This is solved by plane waves 1 e−iωp t+ipx , fp (x, t) = q 2ωp V

1 f¯p (x, t) = q eiωp t+ipx ωp V

(18.113)

of positive and negative energy. As in Section 7.6, we imagine the system to be confined to a finite cubic volume V . Then the momenta p are discrete. The solutions (18.113) behave like an infinite set of harmonic oscillator solution, one for each momentum vector p, with the p-dependent frequencies ωp ≡

q

p2 + m2 .

(18.114)

The general solution of (18.112) may be expanded as ϕ(x, t) =

X p

1 −iωp +ipx ap e + a∗p eiωp t+ipx . 2ωp V

(18.115)

The canonical momenta of the field variables ϕ(x, t) are the field velocities ˙ t). π(x, t) ≡ px (t) ≡ ϕ(x,

(18.116)

The fields are quantized by the canonical commutation rules [ˆ π (x, t), ϕ(x, t)] = −iδxx′ .

(18.117)

The quantum field is now expanded as in (18.115), but in terms of operators a ˆp and their Hermitian adjoint operators aˆ†p . These satisfy the usual canonical commutation rules of creation and annihilation operators of Eq. (7.294): [ˆ ap (t), a ˆ†p′ (t)] = δpp′ ,

[ˆ a†p (t), a ˆ†p′ (t)] = 0,

[ˆ ap (t), a ˆp′ (t)] = 0.

(18.118)


1195


The simplest nonequilibrium quantities to be studied are the thermal averages of one or two such field operators. More generally, we may investigate the averages of one or two fields with respect to an arbitrary initial density operator ρˆ, the so-called ρ-averages: hϕˆH (x)iρ = Tr [(ˆ ρ ϕˆH (x)] ,

hϕˆH (x)ϕˆH (y)iρ = Tr [ˆ ρ ϕˆH (x)ϕˆH (y)] .

(18.119)

For brevity, we have gone over to a four-vector notation and use spacetime coordinates x ≡ (x, t) to write ϕˆH (x, t) as ϕˆH (x). In general, the fields ϕ(x, t) will interact with each other and with further fields, adding to (18.111) some interaction Aint . The behavior of an interacting field system can then be studied in perturbation theory. This is done by techniques related to those in Section 1.7. First we identify a time-independent part of the Hamiltonian for which we can solve the Schrödinger equation exactly. This is called the free part ˆ 0 . For the field ϕ(x, t) at hand this follows from the action of the Hamiltonian H (18.111) via the usual Legendre transformation (1.13). Its operator version is ˆ0 = H

Z

o 1Z D n ˙ ˆ d x H0 (x, t) ≡ d x [ϕ(x, ˆ t)]2 + [∇ϕ(x, ˆ t)]2 + m2 ϕˆ2 (x, t) . (18.120) 2 D

ˆ int (t). Then we The interaction Aint gives rise to an interaction Hamiltonian H introduce the field operators in Dirac’s interaction picture ϕ(x). ˆ These are related ˆ to the Heisenberg operators via the free Hamiltonian H0 , by ˆ

ˆ

ϕ(x) ˆ ≡ eiH0 (t−t0 ) ϕˆH (x, t0 )e−iH0 (t−t0 ) .

(18.121)

The operators in the two pictures are equal to each other at a time t0 at which the density operator ρˆ is known. We also introduce the interaction picture for the interaction Hamiltonian3 ˆ ˆ int ˆ ˆ int (t) ≡ eiHt H H (t)e−iHt . I

(18.122)

This operator is used to set up the time evolution operator in the interaction picture Z

ˆ t0 ) ≡ Tˆ exp i U(t,

t

t0

ˆ Iint (t′ ) . dt′ H

(18.123)

It allows us to express the time dependence of the field operators ϕ(x) ˆ as follows: ϕˆH (x) = Uˆ (t0 , t)ϕ(x) ˆ Uˆ (t, t0 ). 3

(18.124)

For consistency, the field operator ϕ(x) ˆ should carry the same subscript I which is, however, omitted to shorten the notation.

1196


The ρ-averages of the Heisenberg fields in the interaction representation are therefore h

i

ˆ t0 ) , hϕˆH (x)iρ = Tr ρˆ Uˆ (t0 , t)ϕ(x) ˆ U(t, ′

hϕˆH (x)ϕˆH (x )iρ =

    

h

(18.125) i

ˆ 0 , t)ϕ(x) ˆ ′ , t0 ) , Tr ρˆ U(t ˆ Uˆ (t, t′ )ϕ(x ˆ ′ )U(t h

i

ˆ 0 , t′ )ϕ(x ˆ ′ , t)ϕ(x) Tr ρˆ U(t ˆ ′ )U(t ˆ Uˆ (t, t0 ) ,

t > t′ , t′ > t.

(18.126)

Now, suppose that the interaction has been active for a very long time, i.e., we let t0 → −∞. In this limit, (18.125) can be rewritten in terms of the scattering operator ˆ Sˆ ≡ U(∞, −∞) of the system.4 Using the time-ordering operator Tˆ of Eq. (1.241), we may write h

i

hϕˆH (x)iρ = Tr ρˆ Sˆ† TˆSˆϕ(x) ˆ , h

(18.127) i

hϕˆH (x)ϕˆH (y)iρ = Tr ρˆ Sˆ† TˆSˆϕ(x) ˆ ϕ(y) ˆ .

(18.128)

These expressions are indeed the same as those in (18.125) and (18.125); for instance

Sˆ† Tˆ Sˆϕ(x) ˆ

ˆ = U(−∞, t)Uˆ (t, ∞)Tˆ Uˆ (∞, t)ϕ(x) ˆ Uˆ (t, −∞) ˆ = U(−∞, t)ϕ(x) ˆ Uˆ (t, −∞).

(18.129)

For further development it is useful to realize that the operators in the expectations (18.127) and (18.128) can be reinterpreted time-ordered products of a new type, ordered along a closed-time contour which extends from t = −∞ to t = ∞ and back . This contour is imagined to encircle the time axis in the complex t-plane as shown in Figure 18.1. The contour runs from t = −∞ to t = ∞ above the real

Figure 18.1 Closed-time contour in forward–backward path integrals.

time axis and returns below it. Accordingly, we distinguish t values from the upper branch and the lower branch by writing them as t+ and t− , respectively. Similarly we define x(t+ ) ≡ x+ and x(t− ) ≡ x− . When viewed as a function of the closed-time contour, the operator

Sˆ† Tˆ Sˆϕ(x) ˆ 4

(18.130)

The matrix elements of Sˆ between momentum eigenstates form the so-called S-matrix. H. Kleinert, PATH INTEGRALS

1197


can be rewritten as

TˆP Sˆ† Sˆϕ(x ˆ +) ,

(18.131)

where TˆP performs a time ordering along the closed-time contour. The coordinate x lies on the positive branch of the contour, where it is denoted by x+ . The operator TˆP is called path-ordering operator . We can then write down immediately a generating functional for an arbitrary product of field operators ordered along the closed-time path:

Z

TˆP Sˆ† Sˆ exp i

dx j(x+ )ϕ(x ˆ +)

,

(18.132)

where dx is short for d3 xdt. Functional differentiation with respect to j(x+ ) produces ϕ(x ˆ + ) ≡ ϕ(x). ˆ For symmetry reasons it is also useful to introduce the source j(x− ) coupled to the field on the lower time branch ϕ(x ˆ − ). Thus we shall work with the symmetric generating functional

Z

Z[jP ] = Tr ρˆ TˆP Sˆ† Sˆ exp i

dx j(x+ )ϕ(x ˆ +) +

Z

dx j(x− )ϕ(x ˆ −)

.

It can be written as

Z

Z[jP ] = Tr ρˆ TˆP Sˆ† Sˆ exp i

P

dx jP (x)ϕˆP (x)

,

(18.133)

with the subscript p distinguishing the time branches. The path-ordering symbol serves to write down a useful formal expression for ˆ5 the interaction representation of the operator Sˆ† S:

Sˆ† Sˆ = TˆP exp −i

Z

P

ˆ int (t) . dt H I

(18.134)

In terms of this, Z[jP ] takes the suggestive form

Z[jP ] = Tr ρˆ TˆP exp −i

Z

P

ˆ int (t) + i dtH I

Z

P

dx jP (x)ϕˆP (x)

.

(18.135)

To calculate the integrals along the closed-time contour p, it is advantageous to traverse the lower time branch in the same direction as the upper from t = −∞ to ∞ (since we are used to integrating in this direction), and rewrite the closed-contour integral in the source term, Z

P

dx jP (x)ϕˆP (x) =

Z

3

dx

Z

∞

−∞

dt j(x, t+ )ϕ(x ˆ +) +

Z

−∞

∞

dt j(x, t− )ϕ(x ˆ −) , (18.136)

Note that the left-hand side is equal to 1 due to Sˆ being unitary. However, this identity cannot be inserted into the path-ordered expressions (18.131)–(18.133), since the current terms require a factorization of Sˆ or Sˆ† at specific times and an insertion of field operators between the factors. 5

1198


as Z

P


Z

3

dx

Z

∞

−∞

dt [j(x, t+ )ϕ(x ˆ + ) − j(x, t− )ϕ(x ˆ − )] . (18.137)

Obviously, the functional derivative with respect to −j(x, t− ) produces a factor ϕ(x ˆ − ). Correspondingly, we shall imagine the two fields ϕ(x ˆ + ), ϕ(x ˆ − ) as two components of a vector 

ˆ~ (x) =  ϕ

with the associated current



ϕ(x ˆ +) ϕ(x ˆ −)

j(x+ )

~(x) = 

−j(x− )

In this vector notation, the source term reads Z



(18.138)



(18.139)

,

.

ˆ~ (x), dx ~(x)ϕ

(18.140)

and all closed-time path formulas go directly over into vector or matrix formulas whose integrals run only once along the positive time axis, for example Z

′

P

′

dx jP (x)GP (x, x )jP (x ) =

Z

dx ~(x)G(x, x′ )~(x′ ),

(18.141)

where G(x, x′ ) on the right-hand side denotes the 2 × 2 matrix 

G(x, y) = 

G++ (x, y) G+− (x, y) G−+ (x, y) G−− (x, y)

 



≡



G(x+ , y+ ) G(x+ , y− )

.

G(x− , y+ ) G(x− , y− )

(18.142)

~ˆ , we shall identify the closedSince all formulas for jP and ϕˆP hold also for ~ and ϕ time path objects with the corresponding vectors and matrices. Differentiating the generating functional with respect to jP produces all Green functions of the theory. Forming two derivatives gives the two-point Green function GP (x, y) =

δ

δ

iδjP (x) iδjP (y)

Z[jP ]

jP =0

h

i

= Tr ρˆ TˆP Sˆ† Sˆ ϕˆP (x)ϕˆP (y) ,

(18.143)

which we decompose according to the branches of the closed-time contour in the same way as the matrix (18.142): 

GP (x, y) = 

G++ (x, y) G+− (x, y) G−+ (x, y) G−− (x, y)



.

(18.144)

The four matrix elements collect precisely the four physically relevant timedependent Green functions discussed in the last section for the case of ρˆ being an H. Kleinert, PATH INTEGRALS


1199

equilibrium density operator. Here they may be out of thermal equilibrium, formed with an arbitrary ρ-average rather than the thermal average at a given temperature. Going back from the interaction picture to the Heisenberg picture, the matrix GP (x, y) is the expectation GP (x, y) = hTˆP ϕˆH (xP )ϕˆH (yP)iρ ,

(18.145)

where xP can be x+ or x− . Considering the different components we observe that the path order is trivial as soon as x and y lie on different branches of the time axis. Since y+ lies always before x− , the path-ordering operator can be omitted so that G−+ (x, y) = hϕˆH (x)ϕˆH (y)iρ.

(18.146)

In the opposite configuration, the path order is opposite. When reestablishing the original order, a negative sign arises for fermion fields. Hence, G+− (x, y) = hϕˆH (y)ϕˆH (x)iρ = ±hϕˆH (x)ϕˆH (y)iρ.

(18.147)

In either case, a distinction of the upper and lower time branches is superfluous after an explicit path ordering. If both x and y lie on the upper branch, the path order coincides with the usual time order so that G++ (x, y) is equal to the expectation G++ (x, y) = hTˆ ϕˆH (x)ϕˆH (y)iρ ≡ G(x, y),

(18.148)

i.e., the ρ-average of the usual time-ordered Green function. Similarly, if x and y both lie on the lower branch, the path order coincides with the usual anti-time order and ¯ y). G−− (x, y) = hTˆ ϕˆH (x)ϕˆH (y)iρ ≡ G(x,

(18.149)

From these relations it is easy to see that only three of the four matrix elements of GP (x, y) are linearly independent, since there exists the relation G++ + G−− = G+− + G−+ .

(18.150)

This can be verified by writing out explicitly the time order and antiorder on the left-hand side. In the linear-response theory of Sections 18.1 and 18.2, the most convenient independent Green functions are the retarded and the advanced ones, together with the expectation of the anticommutator (the commutator for fermions). By analogy, we also define here, in the nonequilibrium case, GR (x, y) =

Θ(x − y)h[ϕˆH (x), ϕˆH (y)]∓ iρ ,

(18.151)

GA (x, y) = −Θ(y − x)h[ϕˆH (x), ϕˆH (y)]∓ iρ ,

(18.152)

A(x, y) = h[ϕˆH (x), ϕˆH (y)]± iρ .

(18.153)

1200


As in (18.53), the last expression coincides with the absorptive or dissipative part of the Green function. The expectation of the commutator (the anticommutator for fermions), C(x, y) = h[ϕˆH (x), ϕˆH (y)]∓ iρ ,

(18.154)

is not an independent quantity. It is related to the others by C(x, y) = GR (x, y) − GA (x, y).

(18.155)

A comparison of the Fourier decomposition of the field (18.115) with (18.92) shows that the Green functions are simple plane-wave superpositions of harmonic oscillator of all momenta p and frequency Ω = ωp . The normalization factor h ¯ /M becomes 1/V . For instance GR (x, x′ ) =

X p

M ip(x−x′ ) R G (t, t′ )|Ω=ωp . e h ¯V

(18.156)

In the continuum limit, where the sum over momenta goes over into an integral with the rule (7.558), this becomes, from (18.95), R

′

′

G (x, x ) = −Θ(x − x )

Z

dD p ′ eik(x−x ) 2i sin ωp (t − t′ ). D 2ωp (2π)

(18.157)

Similarly we find from (18.102) ′

A(x, x ) =

Z

dD p ′ eik(x−x ) 2 cos ωp (t − t′ ). D 2ωp (2π)

(18.158)

These and the other Green functions satisfy identities analogous to those formed from the position operator ϕ(t) ˆ of a simple harmonic oscillator in (18.105)–(18.105): GA (x, x′ ) A(x, x′ ) C(x, x′ ) G(x, x′ )

= = = =

∓GR (x′ , x), ±A(x′ , x), ∓C(x′ , x). ±G(x′ , x)∗ .

(18.159) (18.160) (18.161) (18.162)

It is now easy to express the matrix elements of the 2×2 Green function GP (x, y) in (18.144) in terms of the three independent quantities (18.153). Since GR = G−+ − G−− = G++ − G+− , GA = G+− − G−− = G++ − G−+ , A = G−+ + G+− = G++ + G−− ,

(18.163)

C = G−+ − G+− = GR − GA , H. Kleinert, PATH INTEGRALS

1201


we find 1 (A + C) = 12 (A + GR − GA ), 2 1 = (A − C) = 12 (A − GR + GA ), 2

G−+ = G+−

(18.164)

and G++ = GR + G+− = 12 (A + GR + GA ), G−− = G+− + G−+ − G++

(18.165)

= A − G++ = 12 (A − GR − GA ).

The matrix GP (x, y) can therefore be written as follows: 



R A R A 1 A+G +G A−G +G  . GP =  2 A + GR − GA A − GR − GA

(18.166)

For actual calculations it is somewhat more convenient to use a transformation introduced by Keldysh [5]. It arises from the similarity transformation ˜ = QGP Q−1 , G



1 −1

1 with Q = √  2 1

1

 

= (QT )−1 ,

(18.167)

producing the simpler triangular Green function matrix 







R A R A 1  1 −1  1  A + G + G A − G + G  ˜ G(x, y) = √ 2 1 1 2 A + GR − GA A − GR − GA









A 1  1 1   0 G  ×√ = . 2 −1 1 GR A

(18.168)

Due to the calculational advantages it is worth re-expressing all quantities in the new basis. The linear source term, for example, becomes R

P


R

=

R

with the source vectors ˜ˆ ϕ(x) ≡Q

ϕ(x ˆ +) ϕ(x ˆ −)

!



dx (j(x+ ), −j(x− )) 

ϕ(x ˆ +) ϕ(x ˆ −)

ˆ˜ dx ˜(x)ϕ(x),

1 =√ 2

ϕ(x ˆ + ) − ϕ(x ˆ −) ϕ(x ˆ + ) + ϕ(x ˆ −)

!

 

(18.169)

,

(18.170)

and the field vectors 

j1 (x)





j(x+ )







j(x+ ) + j(x− ) 1  = Q = √  . ˜(x) ≡  2 j(x+ ) − j(x− ) j2 (x) −j(x− )

(18.171)

1202


The quadratic source term Z

dx dx′ jP (x)GP (x, x′ )jP (x′ ) =

Z

′

dx dx

becomes

(18.172) 

(j(x+ ), −j(x− ))  Z

G++ G+− G−+ G−−





 (x, x′ ) 

j(x′+ ) −j(x′− )

 

˜ x′ )˜ dx dx′ ˜T (x)G(x, (x′ ).

(18.173)

˜ (1) and G ˜ (2) has the same triangular form as The product6 of two Green functions G each factor. The three nonzero entries are composed as follows: (2) −1 −1 ˜ 12 = G ˜ (1) G ˜ (2) = QG(1) G P Q QGP Q ! A 0 GA 1 G2 = . R A GR GR 1 G2 1 A2 + A1 G2

(18.174)

More details on these Green functions can be found in the literature [6].

18.7

Perturbation Theory for Nonequilibrium Green Functions

The interaction picture can be used to develop a perturbation expansion for nonequilibrium Green functions. For this we go back to the generating functional (18.135) and assume that the interaction depends only on the field operators. Usually it will be a local interaction, i.e., a spacetime integral over an interaction density:

exp −i

Z

P

Z

ˆ int (t) = exp i dt H I

P

dt

Z

3

int

d x L (ϕˆP (x, t)) .

(18.175)

The subsequent formal development applies also to the case of a more general nonlocal interaction n

o

exp iAint ˆP ] . P [ϕ

(18.176)

To account for the interaction, we use the fact used in Section 3.18, that within the expectation (18.135) the field ϕˆP can be written as a differential operator δ/iδjP (x) applied to the source term. In this form, the interaction term can be moved outside the thermal expectation. The result is the generating functional in the interaction picture n

o

Z[jP ] = exp iAint P [δ/iδjP ] Z0 [jP ], 6

The product isRmeant in the functional sense, i.e., G(2) )(x, y) = dz G(1) (x, z)G(2) (z, y).

(18.177)

(1)

(G


1203

18.7 Perturbation Theory for Nonequilibrium Green Functions

where

Z

Z0 [jP ] = Tr ρˆ TˆP exp i

P

dx ϕˆP(x) jP (x)

(18.178)

is the free partition function. To apply this formula, we have to find Z0 [jP ] explicitly. By expanding the exponential in powers of iAint P [δ/iδjP ] and performing the functional derivatives δ/iδjP , we obtain the desired perturbation expansion for Z[jP ]. For a general density operator ρˆ, the free partition function Z0 [jP ] cannot be written down in closed form. Here we give Z0 [jP ] explicitly only for a harmonic system in thermal equilibrium, where the ρ-averages h. . .iρ are the thermal averages h. . .iT calculated in Sections 18.1 and 18.2. Since the fluctuation terms in the field ϕ(t) are quadratic, Z0 [jP ] must have an exponent quadratic in the sources jP . To satisfy (18.143), the functional is necessarily given by 1 Z0 [jP ] = exp − 2

Z

dxdy jP (x)GP (x, y)jP(y) .

(18.179)

Inserting the 4 × 4-matrix (18.166), this becomes Z Z 0 GA j+ 1 ′ −1 Z0 [j+ , j− ] = exp − dx dx (j+ , −j− )Q Q R 2 G A −j− Z Z h n 1 = exp − dx dx′ (j+ + j− )(x)GA (x, x′ )(j+ − j− )(x′ ) 4 + (j+ − j− )(x)GR (x, x′ )(j+ + j− )(x′ ) "

!

o

+ (j+ − j− )(x)A(x, x′ )(j+ − j− )(x′ )] ,

!#

(18.180)

where j+ (x) ≡ j(x+ ),

j− (x) ≡ j(x− ).

(18.181)

The advanced Green functions are different from zero only for t < t′ . Using relation (18.159), the second term is seen to be the same as the first. For the real field at hand, these terms are purely imaginary [see (18.156)]. The anticommutation function A(x, x′ ) is symmetric by (18.160). We therefore rewrite (18.180) as Z0 [j+ , j− ] = exp h

1 − 2

Z

dx

Z

dx′ Θ(x′ − x)

(18.182)

× (j+ − j− )(x)GR (x, x′ )(j+ + j− )(x′ ) + (j+ − j− )(x)A(x, x′ )(j+ − j− )(x′ )

i

.

For any given spectral function, the exponent can easily be written down explicitly using the spectral representations (18.44) and (18.53). As an important example consider the simple case of a single harmonic oscillator of frequency Ω. Then the field ϕ(x) ˆ depends only on the time t, and the commutator

1204


and “wrong” commutator functions are given by (18.93) and (18.102). Reintroducing all factors h ¯ and kB , we have Z 1 Z Z0 [j+ , j− ] = exp − 2 dt dt′ Θ(t − t′ ) (18.183) 2¯ h h

× (j+ − j− )(t)C(t, t′ )(j+ + j− )(t′ ) + (j+ − j− )(t)A(t, t′ )(j+ − j− )(t′ )

or, more explicitly, 1 dt dt′ Θ(t′ − t) 2MΩ¯ h h × − (j+ − j− )(t) i sin Ω(t − t′ ) (j+ + j− )(t′ )

Z0 [j+ , j− ] = exp

Z

−

i

Z

(18.184)

h ¯Ω cos Ω(t − t′ ) (j+ − j− )(t′ ) . 2kB T We have taken advantage of the presence of the Heaviside function to express the retarded Green function for t > t′ as a commutator function C(t, t′ ) [recall (18.151), (18.154)]. Together with the anticommutator function A(t, t′ ), we obtain for t > t′ i

+ (j+ − j− )(t) coth

Ω hβ − i(t − t′ )] cosh [¯ 1 h ¯ ′ ′ ′ 2 G(t, t ) = [A(t, t ) + C(t, t )] = , (18.185) h ¯ Ωβ 2 2MΩ sinh 2 which coincides with the time-ordered Green function (18.101) for t > t′ , and thus with the analytically continued periodic imaginary-time Green function (3.248). The exponent in this generating functional is thus quite similar to the equilibrium source term (3.218). The generating functional (18.180) can, of course, be derived without the previous operator discussion completely in terms of path integrals for the harmonic oscilˆ lator in thermal equilibrium. Using the notation X(t) for a purely time-dependent oscillator field ϕ(x), ˆ we can take the generating functional directly from Eq. (3.168): i tb M ˙2 dt (X − Ω2 X 2 ) + jX h ¯ ta 2 = e(i/¯h)Acl,j FΩ,j (tb , ta ).

(Xb tb |Xa ta )jΩ =

Z

DX(t) exp

Z

with a total classical action h i 1 MΩ Acl,j = (Xb2 + Xa2 ) cos Ω(tb − ta )−2XbXa 2 sin Ω(tb − ta ) Z tb 1 + dt[Xa sin Ω(tb − t) + Xb sin Ω(t − ta )]j(t), sin Ω(tb − ta ) ta

(18.186)

(18.187)

and the fluctuation factor (3.170), and express (18.187) as in (3.171) in terms of the two independent solutions Da (t) and Db (t) of the homogenous differential equations (3.48) introduced in Eqs. (2.221) and (2.222): Acl,j =

i M h 2˙ Xb Da (tb )−Xa2 D˙ b (ta )−2XbXa 2Da (tb ) H. Kleinert, PATH INTEGRALS

1205

18.8 Path Integral Coupled to Thermal Reservoir

1 Z tb dt [Xb Da (t)+Xa Db (t)]j(t). Da (tb ) ta

+

(18.188)

The fluctuation factor is taken as in (3.172). Then we calculate the thermal average of the forward–backward path integral of the oscillator X(t) via the Gaussian integral Z0 [j+ , j− ] =

Z

j ∗

− dXb dXa (Xb h ¯ β|Xa 0)Ω (Xb tb |Xa ta )j+ Ω (Xb tb |Xa ta )Ω . (18.189)

Here (Xb h ¯ β|Xa 0)Ω is the imaginary-time amplitude (2.403): 1

(Xb h ¯ β|Xa 0) = q 2π¯ h/M

s

Ω sinh h ¯β

(

)

1 MΩ × exp − ¯ βΩ − 2Xb Xa ] . [(Xb2 + Xa2 ) cosh h 2¯ h sinh h ¯ βΩ

(18.190)

We have preferred deriving Z0 [j+ , j− ] in the operator language since this illuminates better the physical meaning of the different terms in the result (18.185).

18.8

Path Integral Coupled to Thermal Reservoir

After these preparations, we can embark on a study of a simple but typical problem of nonequilibrium thermodynamics. We would like to understand the quantummechanical behavior of a particle coupled to a thermal reservoir of temperature T and moving in an arbitrary potential V (x) [7]. Without the reservoir, the probability of going from xa , ta to xb , tb would be given by7 2

|(xb tb |xa ta )| =

Z

iZ M 2 Dx(t) exp dt x˙ − V (x) h ¯ 2

2 .

(18.191)

This may be written as a path integral over two independent orbits, to be called x+ (t) and x− (t): (xb tb |xa ta )(xb tb |xa ta )∗ = × exp

Z

i h ¯

Dx+ (t) Dx− (t) Z

tb

ta

dt

(18.192)

M 2 (x˙ − x˙ 2− ) − (V (x+ ) − V (x− )) 2 +

.

In accordance with the development in Section 18.7, the two orbits are reinterpreted as two branches of a single closed-time orbit xP (t). The time coordinate tP of the path runs from ta to tb slightly above the real time axis and returns slightly below it, just as in Fig. 18.1. The probability distribution (18.191) can then be written as a path integral over the closed-time contour encircling the interval (ta , tb ): |(xb tb |xa ta )|2 = 7

Z

DxP exp

i h ¯

Z

P

dt

M 2 x˙ − V (xP ) 2 P

In the sequel, we display the constants h ¯ and kB explicitly.

.

(18.193)

1206


We now introduce a coupling to a thermal reservoir for which we use, as in the equilibrium discussion in Section 3.13, a bath of independent harmonic oscillators ϕî (t) of masses Mi and frequencies Ωi in thermal equilibrium at temperature T . For simplicity, the coupling is assumed to be linear in ϕî (t) and the position of the particle x(t). The bath contributes to (18.193) a factor involving the thermal expectation of the linear interaction 2

|(xb tb |xa ta )| =

Z

i DxP exp h ¯ (

× Tr

M 2 dt x˙ − V (xP ) 2" P P #) X Z i i ρˆ TˆP exp ci dt ϕˆP(t) xP (t) . h ¯ i P

Z

(18.194)

Here, ϕîP (t) for i = 1, 2, 3, . . . are the position operators of the auxiliary harmonic oscillators. Since the oscillators are independent, the trace of the exponentials factorizes into a product of single-oscillator expressions Tr

(

"

iX ρˆ TˆP exp ci h ¯ i

Z

P

dt ϕîP (t)xP (t)

#)

=

Y i

i Tr ρˆ TˆP exp ci h ¯

Z

P

dt ϕîP (t)xP (t)

.

(18.195) The density operator ρˆ has the eigenvalues (18.110). Each factor on the right-hand side is of the form (18.178) with ϕ(t) ˆ = ci ϕîP (t)/¯ h and j+,− = x+,− (t), so that (18.195) leads to the partition function (18.183), which reads here Z0b [x+ , x− ]

= exp

h

1 − 2 2¯ h

Z

dt

Z

dt′ Θ(t − t′ )

′

(18.196)

′

′

′

× (x+ − x− )(t)Cb (t, t )(x+ + x− )(t ) + (x+ − x− )(t)Ab (t, t )(x+ − x− )(t )

i

,

where Cb (t, t′ ) and Ab (t, t′ ) collect the commutator and anticommutator functions of the bath. They are sums of correlation functions (18.93) and (18.102) of the individual oscillators of mass Mi and frequency Ωi , each contributing with a weight c2i . Thus we may write Cb (t, t′ ) =

c2i h[ϕî (t), ϕî (t′ )]iT

Z

∞ dω ′

ρb (ω ′ )i sin ω ′(t−t′ ), (18.197) 2π −∞ i Z ∞ X dω ′ h ¯ ω′ ′ 2 ′ Ab (t, t ) = ci h{ϕî (t), ϕî (t )}iT = h ¯ ρb (ω ′ ) coth cos ω ′(t−t′ ), (18.198) 2π 2k T −∞ B i X

= −¯ h

where the ensemble averages at a fixed temperature T are now denoted by a subscript T , and ρb (ω ′ ) ≡ 2π

X i

c2i [δ(ω ′ − Ωi ) − δ(ω ′ + Ωi )] 2Mi Ωi

(18.199)

is the spectral function of the bath. It is the antisymmetric continuation of the spectral function (3.405) to negative ω ′ . Since the spectral function of the bath ρb (ω ′ ) H. Kleinert, PATH INTEGRALS

1207


of (18.199) is odd in ω ′ , we can replace both trigonometric functions −i sin ω ′(t − t′ ) ′ ′ and cos ω ′(t − t′ ) in (18.199) by the exponentials e−iω (t−t ) . The expression in the exponent of (18.196) may be considered as an effective action in the path integral, caused by the thermal bath. We shall therefore write i FV i FV AD [x+ , x− ] + AFV [x , x ] ,(18.200) Z0 [x+ , x− ] = exp A [x+ , x− ] = exp + − F h ¯ h ¯

where the effective action AFV [x+ , x− ] consists of a dissipative part AFV D [x+ , x− ] and FV a fluctuation part AF [x+ , x− ]. The expression Z0 [x+ , x− ] is the famous influence functional first introduced by Feynman and Vernon. Inserting (18.200) into (18.194) and displaying explicitly the two branches of the path xP (t) with the proper limits of time integrations, we obtain from (18.194) the probability for the particle to move from xa ta to xb tb as the path integral |(xb tb |xa ta )|2 = × exp

Z

Dx+ (t)

Z

Dx− (t) ×

i M 2 i Z tb dt (x˙ + − x˙ 2− ) − (V (x+ ) − V (x− )) + AFV [x+ , x− ] . (18.201) h ¯ ta 2 h ¯

For a better understanding of the influence functional, we introduce an auxiliary retarded function γ(t − t′ ) ≡ Θ(t − t′ )

1 M

Z

∞ −∞

dω σb (ω) −iω(t−t′ ) e . 2π ω

(18.202)

Then we can write Θ(t − t′ )Cb (t, t′ ) = i¯ hM γ(t ˙ − t′ ) + i¯ hM∆ω 2 δ(t − t′ ),

(18.203)

where the quantity ∆ω 2 ≡ −

1 Z ∞ dω ′ σb (ω ′) 1 X c2i = − M −∞ 2π ω ′ M i Mi Ω2i

(18.204)

was introduced before in Eq. (3.417). Inserting the first term of the decomposition (18.203) into (18.196), the dissipative part of the influence functional can be integrated by parts in t′ and becomes M Z tb Z tb ′ dt dt (x+ − x− )(t)γ(t − t′ )(x˙ + + x˙ − )(t′ ) 2 ta ta Z M tb + dt(x+ − x− )(t)γ(t − tb )(x+ + x− )(ta ). (18.205) 2 ta

AFV D [x+ , x− ] = −

The δ-function in (18.203) contributes to AFV D [x+ , x− ] a term analogous to (3.418) M ∆Aloc [x+ , x− ] = 2

Z

tb

ta

dt∆ω 2 (x2+ − x2− )(t),

(18.206)

1208


which may simply be absorbed into the potential terms of the path integral (18.201), renormalizing them to i − h ¯

Z

tb

ta

dt [Vren (x+ ) − Vren (x− )] .

(18.207)

This renormalization is completely analogous to that in the imaginary-time formula (3.420). The odd bath function ρb (ω ′ ) can be expanded in a power series with only odd powers of ω ′ . The lowest approximation ρb (ω ′ ) ≈ 2Mγω ′ ,

(18.208)

describes Ohmic dissipation with some friction constant γ [recall (3.424)]. For frequencies much larger than the atomic relaxation rates, the friction goes to zero. This behavior is modeled by the Drude form (3.425) of the spectral function ρb (ω ′ ) ≈ 2Mγω ′

2 ωD . 2 ωD + ω ′2

(18.209)

Inserting this into Eq. (18.202), we obtain the Drude form of the function γ(t): R γD (t) ≡ Θ(t) γωD e−ωD t .

(18.210)

The superscript emphasizes the retarded nature. This can also be written as a Fourier integral R (t) γD

=

Z

∞

−∞

dω ′ R ′ −iω′ t , γ (ω )e 2π D

(18.211)

with the Fourier components R γD (ω ′ ) = γ

iωD . ω ′ + iωD

(18.212)

The position of the pole in the lower half-plane ensures the retarded nature of the friction term by producing the Heaviside function in (18.210) [recall (1.308)]. The imaginary-time expansion coefficients γm of Eq. (3.428) are related to these by γm = γ(ω ′ )|ω′ =i|ωm | ,

(18.213)

by close analogy with the relation between the retarded and imaginary-time Green functions (18.30) and (18.31). In the Ohmic limit (18.208), the dissipative part of the influence functional simR plifies. Then γD (t) becomes narrowly peaked at positive t, and may be approximated by a right-sided retarded δ-function as R γD (t) → γ δ R (t),



1209

whose superscript R indicates the retarded asymmetry of the δ-function. this, (18.205) becomes a local action

With

AFV D [x+ , x− ]

M Z tb M =− γ dt(x+ − x− )(x˙ + + x˙ − )R − γ(x2+ − x2− )(ta ). 2 2 ta

(18.215)

The right-sided nature of the function δ R (t) causes an infinitesimal negative shift in the time argument of the velocities (x˙ + + x˙ − )(t) with respect to the factor (x+ − x− )(t), indicated by the superscript R. It expresses the causality of the friction forces and will be seen to be crucial in producing a probability conserving time evolution of the probability distribution. The second term changes only the curvature of the effective potential at the initial time, and can be ignored. It is useful to incorporate the slope information (18.208) also into the bath correlation function Ab (t, t′ ) in (18.198), and factorize it as Ab (t, t′ ) = 2MγkB T K(t, t′ ),

(18.216)

where X 1 c2 h{ϕî (t), ϕî (t′ )}iT . 2MγkB T i i

K(t, t′ ) = K(t − t′ ) ≡

(18.217)

The prefactor in (18.216) is conveniently abbreviated by the constant w ≡ 2MγkB T,

(18.218)

which is related to the so-called diffusion constant D ≡ kB T /Mγ

(18.219)

w = 2γ 2 M 2 D.

(18.220)

by

The Fourier decomposition of (18.217) is K(t, t′ ) =

Z

∞

−∞

dω ′ ′ ′ K(ω ′ )e−iω (t−t ) , 2π

(18.221)

with K(ω ′ ) ≡

1 ρb (ω ′) h ¯ ω′ h ¯ ω′ coth . 2Mγ ω ′ 2kB T 2kB T

(18.222)

In the limit of a purely Ohmic dissipation this simplifies to K(ω ′ ) → K Ohm (ω ′ ) ≡

h ¯ ω′ h ¯ ω′ coth . 2kB T 2kB T

(18.223)

1210


The function K(ω ′) has the normalization K(0) = 1, giving K(t−t′ ) a unit temporal area: Z

∞

−∞

dt K(t − t′ ) = 1.

(18.224)

In the classical limit h ¯ → 0, the Drude spectral function (18.209) leads to cl KD (ω ′)

2 ωD = ′2 , 2 ω + ωD

(18.225)

with the Fourier transform cl (t − t′ ) = KD

1 −ωD (t−t′ ) e . 2ωD

(18.226)

In the limit of Ohmic dissipation, this becomes a δ-function. Thus K(t − t′ ) may be viewed as a δ-function broadened by quantum fluctuations and relaxation effects. With the function K(t, t′ ), the fluctuation part of the influence functional in (18.196), (18.200), (18.201) becomes AFV F [x+ , x− ]

w Z tb Z tb ′ =i dt dt (x+ − x− )(t) K(t, t′ ) (x+ − x− )(t′ ). (18.227) 2¯ h ta ta

Here we have used the symmetry of the function K(t, t′ ) to remove the Heaviside function Θ(t − t′ ) from the integrand, and to extend the range of t′ -integration to the entire interval (ta , tb ). In the Ohmic limit, the probability of the particle to move from xa ta to xb tb is given by the path integral 2

|(xb tb |xa ta )| = i × exp h ¯

Z

Dx+ (t)

Z

Dx− (t)

M 2 dt (x˙ − x˙ 2− ) − (V (x+ ) − V (x− )) 2 + ta Z tb Mγ (x+ − x− )(t)(x˙ + + x˙ − )R (t) × exp −i dt 2¯ h ta w Z tb Z tb ′ Ohm ′ ′ − 2 dt dt (x+ − x− )(t) K (t, t ) (x+ − x− )(t ) . (18.228) ta 2¯ h ta

Z

tb

This is the closed-time path integral of a particle in contact with a thermal reservoir. The paths x+ (t), x− (t) may also be associated with a forward and a backward movement of the particle in time. For this reason, (18.228) is also called a forward– backward path integral. The hyphen is pronounced as minus, to emphasize the opposite signs in the partial actions. It is now convenient to change integration variables and go over to average and relative coordinates of the two paths x+ , x− : x ≡ (x+ + x− )/2, y ≡ x+ − x− .


1211

18.9 Fokker-Planck Equation

Then (18.228) becomes |(xb tb |xa ta )|

2

=

Z

Dx(t)

× exp

18.9

Z

i − h ¯

Dy(t) Z

tb

ta

w − 2¯ h2

R

dt M −y˙ x˙ + γy x˙

Z

tb

ta

dt

Z

tb

ta

′

dt y(t)K

+V

Ohm

′

y y x+ −V x− 2 2 ′

(t, t )y(t ) .

(18.230)

Fokker-Planck Equation

At high-temperatures, the Fourier transform of the Kernel K(t, t′ ) in Eq. (18.223) tends to unity such that K(t, t′ ) becomes a δ-function, so that the path integral (18.230) for the probability distribution of a particle coupled to a thermal bath simplifies to 2

P (xb tb |xa ta ) ≡ |(xb tb |xa ta )| =

Z

Dx(t)

Z

Dy(t)

w Z tb i tb R ′ × exp − dt y[M x¨ + Mγ x˙ + V (x)] − 2 dt y 2 . (18.231) h ¯ ta 2¯ h ta The superscript R records the infinitesimal backward shift of the time argument as in Eq. (18.215). The y-variable can be integrated out, and we obtain

Z

1 tb P (xb tb |xa ta ) = N Dx(t) exp − dt [M x¨ + Mγ x˙ R + V ′ (x)]2 . (18.232) 2w ta The proportionality constant N can be fixed by the normalization integral

Z

Z

Z

dxb P (xb tb |xa ta ) = 1.

(18.233)

Since the particle is initially concentrated around xa , the normalization may also be fixed by the initial condition lim P (xb tb |xa ta ) = δ(xb − xa ).

tb →ta

(18.234)

The right-hand side of (18.232) looks like a Euclidean path integral associated with the Lagrangian [8] 1 [M x¨ + Mγ x˙ + V ′ (x)]2 . (18.235) 2w The result will, however, be different, due to time-ordering of the x˙ R -term. Apart from this, the Lagrangian is not of the conventional type since it involves a second time derivative. The action principle δA = 0 now yields the Euler-Lagrange equation Le =

∂L d ∂L d2 ∂L − + 2 = 0. (18.236) ∂x dt ∂ x˙ dt ∂ x¨ This equation can also be derived via the usual Lagrange formalism by considering x and x˙ as independent generalized coordinates x, v.

1212


18.9.1

Canonical Path Integral for Probability Distribution

In Section 2.1 we have constructed path integrals for time evolution amplitudes to solve the Schrödinger equation. By analogy, we expect the path integral (18.232) for the probability distribution to satisfy a differential equation of the Schrödinger type. This equation is known as a Fokker-Planck equation. As in Section 2.1, the relation is established by rewriting the path integral in canonical form. Treating v = x˙ as an independent dynamical variable, the canonical momenta of x and v are [9] Mγ Mγ ∂L =i [M x¨ + Mγ x˙ + V ′ (x)] = i [M v˙ + Mγv + V ′ (x)], ∂ x˙ w w 1 ∂L = p. (18.237) = i ∂ x¨ γ

p = i pv

The Hamiltonian is given by the Legendre transform 2 X

∂Le x˙ i = Le (v, v) ˙ + ipv + ipv v, ˙ ˙i i=1 ∂ x

H(p, pv , x, v) = Le (x, ˙ x¨) −

(18.238)

where v˙ has to be eliminated in favor of pv using (18.237). This leads to H(p, pv , x, v) =

w 2 1 pv − ipv [γv + V ′ (x)] + ipv. 2 2M M

(18.239)

The canonical path integral representation for the probability distribution reads therefore P (xb tb |xa ta ) =

Z

Z Dp Z Dpv Dx Dv 2π 2π Z Z

tb

× exp

dt [i(px˙ + pv v) ˙ − H(p, pv , x, v)] .

ta

(18.240)

It is easy to verify that the path integral over p enforces v ≡ x, ˙ after which the path integral over pv leads back to the initial expression (18.232). We may keep the auxiliary variable v(t) as an independent fluctuating quantity in all formulas and decompose the probability distribution P (xb tb |xa ta ) with respect to the content of v as an integral P (xb tb |xa ta ) =

Z

∞

dvb

−∞

Z

∞

−∞

dva P (xb vb tb |xa va ta ).

(18.241)

The more detailed probability distribution on the right-hand side has the path integral representation 2

P (xb vb tb |xa va ta ) = |(xb vb tb |xa va ta )| = × exp

Z

tb

ta

Z

Dx

Z

Z Dp Z Dpv Dv 2π 2π

dt [i(px˙ + pv v) ˙ − H(p, pv , x, v)] ,

(18.242)


1213


where the endpoints of v are now kept fixed at vb = v(tb ), va = v(ta ). We now use the relation between a canonical path integral and the Schrödinger equation discussed in Section 2.1 to conclude that the probability distribution (18.242) satisfies the Schrödinger-like differential equation [10]: H(ˆ p, pˆv , x, v)P (x v tb |xa va ta ) = −∂t P (x v t|xa va ta ).

(18.243)

This is the Fokker-Planck equation in the presence of inertial forces. At this place we note that when going over from the classical Hamiltonian (18.239) to the Hamiltonian operator in the differential equation (18.243), there is an operator ordering problem. Such a problem was encountered in Section 10.5 and discussed further at the end of Section 11.3. In this respect the analogy with the simple path integrals in Section 2.1 is not perfect. When writing down Eq. (18.243) we do not know in which order the momentum operator pˆv must stand with respect to v. If we were dealing with an ordinary functional integral in (18.232) we would know the order. It would be found as in the case of the electromagnetic interaction in Eq. (11.89) to have the symmetric order −(ˆ pv v + v pˆv )/2. On physical grounds, it is easy to guess the correct order. The differential equation (18.243) has to conserve the total probability Z

dx dvP (x v tb |xa va ta ) = 1

(18.244)

for all times t. This is guaranteed if all momentum operators stand to the left of all coordinates in the Hamiltonian operator. Indeed, integrating the Fokker-Planck equation (18.243) over x and v, only a left-hand position of the momentum operators leads to a vanishing integral, and thus to a time independent total probability. We suspect that this order must be derivable from the retarded nature of the velocity in the term y x˙ R in (18.231). This will now be shown.

18.9.2

Solving the Operator Ordering Problem

The ordering problem in the Hamiltonian operator associated with (18.239) does not involve the potential V (x). We may therefore study this problem most simply by considering the free Hamiltonian H0 (p, pv , x, v) =

w 2 p − iγpv v + ipv, 2M 2 v

(18.245)

associated with the Lagrangian path integral P0 (xb tb |xa ta ) = N

Z

1 Dx(t) exp − 2w

Z

tb

ta

R 2

dt [M x¨ + Mγ x˙ ]

.

(18.246)

We furthermore may concentrate on the probability distribution with xb = xa = 0, and assume tb − ta to be very large. Then the frequencies of all Fourier decompositions are continuous.

1214


In spite of the restrictions to large tb − ta , the result to be derived will be valid for any time interval. The reason is that operator order is a property of extremely short time intervals, so that it does not matter, how long the time interval is on which we solve the problem. Forgetting for a moment the retarded nature of the velocity x, ˙ the Gaussian path integral can immediately be done and yields P0 (0 tb |0 ta ) ∝ Det −1 (−∂t2 − γ∂t ) "

∝ exp −(tb − ta )

Z

∞

−∞

dω ′ log(ω ′2 − iγω ′ ) , 2π #

(18.247)

where γ is positive. The integral on the right-hand side diverges. This is a consequence of the fact that we have not used Feynman’s time slicing procedure for defining the path integral. As for an ordinary harmonic oscillator discussed in detail in Sections 2.3 and 2.14), this would lead to a finite integral in which ω ′ is replaced by ω ˜ ′ ≡ (2 − 2 cos aω ′ )/a2 :

γ dω ′ log[˜ ω ′2 + γ 2 ] = 0 + . 2 −∞ −∞ −∞ 2π (18.248) For a derivation see Section 2.14, in particular the first term in Eq. (2.477). The same result can equally well be obtained without time slicing by regularizing the divergent integral in (18.247) analytically, as shown in (2.496). Recall the discussion in Section 10.6 where analytic regularization was seen to be the only method that allows to define path integrals without time slicing in such a way that they are invariant under coordinate transformations [11]. It is therefore suggestive to apply the same procedure also to the present path integrals with dissipation and to use the dimensionally regularized formula (2.533): 1 2

Z

∞

1 dω ′ ˜ ′2 ] = log[˜ ω ′4 + γ 2 ω 2π 2

Z

∞

−∞

Z

∞

1 dω ′ log ω ˜ ′2 + 2π 2

dω ′ γ log(ω ′ ± iγ) = , 2π 2

Z

∞

γ > 0.

(18.249)

Applying this to the functional determinant in (18.247) yields Det(−∂t2 − γ∂t ) = Det(i∂t )Det(i∂t + iγ) = exp [Tr log(i∂t ) + Tr log(i∂t + iγ)] γ = exp (tb − ta ) , (18.250) 2 and thus γ P0 (0 tb |0 ta ) ∝ exp −(tb − ta ) . (18.251) 2 This corresponds to an energy γ/2, and an ordering −iγ(ˆ pv v + v pˆv )/2 in the Hamiltonian operator. We now take the retardation of the time argument of x˙ R into account. Specifically, we replace the term γy x˙ R in (18.246) by the Drude form on the left-hand side (18.214) before going to the limit ωD → ∞: γy x˙ R (t) →

Z

R dt′ y(t) γD (t − t′ ) x(t′ ),


1215


containing now explicitly the retarded Drude function (18.210) of the friction. Then the frequency integral in (18.247) becomes dω ′ ω ′ ωD log ω ′2 − γ ′ = 2π ω +iωD !

i dω h −log(ω ′ +iωD ) + log ω ′2 +iω ′ωD −γωD , −∞ −∞ 2π (18.253) ′ where we have omitted a vanishing integral over log ω on account of (18.249). We now decompose

Z

∞

Z

∞

log ω ′2 +iω ′ωD −γωD = log(ω ′ +iω1 ) + log(ω ′ +iω2 ), with

ω1,2

ωD = 1± 2

s

!

4γ 1− , ωD

(18.254)

(18.255)

and use formula (2.533) to find i ωD ω1 ω2 dω h −log(ω ′ +iωD )+log ω ′2 +iω ′ωD −γωD = − + + = 0. (18.256) 2 2 2 −∞ 2π

Z

∞

The vanishing frequency integral implies that the retarded functional determinant is trivial: h

i

Det(−∂t2 − γ∂tR ) = exp Tr log(−∂t2 − γ∂tR ) = 1,

(18.257)

instead of (18.250) obtained from the frequency integral without the Drude modification. With the determinant (18.257), the probability becomes a constant P0 (0 tb |0 ta ) = const.

(18.258)

This shows that the retarded nature of the friction force has subtracted an energy γ/2 from the energy in (18.251). Since the ordinary path integral corresponds to a Hamiltonian operator with a symmetrized term −i(ˆ pv v + v pˆv )/2, the subtraction of γ/2 changes this term to −iγ pˆv v. Note that the opposite case of an advanced velocity term x˙ A in (18.246) would A R be approximated by a Drude function γD (t) which looks just like γD (t) in (18.212), but with negative ωD . The right-hand side of (18.256) would then become 2γ rather than zero. The corresponding formula for the functional determinant is h

i

Det(−∂t2 − γ∂tA ) = exp Tr log(−∂t2 − γ∂tA ) = exp [(tb − ta )γ] ,

(18.259)

where γ∂tA stands for the advanced version of the functional matrix (18.252) in which ωD is replaced by −ωD . Thus we would find P0 (0 tb |0 ta ) ∝ exp [−(tb − ta )γ] ,

(18.260)

1216


with an additional energy γ/2 with respect to the ordinary formula (18.251). This ˆ 0 , which would corresponds to the opposite (unphysical) operator order −iγv pˆv in H violate the probability conservation of time evolution twice as much as the symmetric order. The above formulas for the functional determinants can easily be extended to the slightly more general case where V (x) is the potential of a harmonic oscillator V (x) = Mω02 x2 /2. Then the path integral (18.232) for the probability distribution becomes P0 (xb tb |xa ta ) = N

Z

1 Dx(t) exp − 2w

Z

tb

ta

R

dt [M x¨ + Mγ x˙ +

ω02 x]2

, (18.261)

which we evaluate at xb = xa = 0, where it is given by the properly retarded expression P0 (0 tb |0 ta ) ∝ Det −1 (−∂t2 − γ∂t + ω02 ) " # Z ∞ dω ′ ′2 ′ 2 ∝ exp −(tb − ta ) log(ω − iγω − ω0 ) . −∞ 2π

(18.262)

The logarithm can be decomposed into a sum log(ω ′ + iω1 ) + log(ω ′ + iω2 ) with 

v u



u γ 4ω 2 = 1 ± t1 − 20  . 2 γ

ω1,2

(18.263)

We now apply the analytically regularized formula (2.533) to obtain Z

∞

−∞

dω ′ ω1 ω2 [log(ω ′ + iω1 )+log(ω ′ + iω2 )] = + = γ. 2π 2 2

(18.264)

Both under- and overdamped motion yield the same result. This is one of the situations where our remarks after Eqs. (2.536) and (2.535) concerning the cancellation of oscillatory parts apply. For the functional determinant (18.262), the result is h

i

Det(−∂t2 − γ∂t − ω02 ) = exp Tr log(−∂t2 − γ∂t − ω02 ) = exp (tb − ta )

γ . (18.265) 2

Note that the result is independent of ω0 . This can simply be understood by forming the derivative of the logarithm of the functional determinant in (18.250) with respect to ω02 . Since log DetM = Tr log M, this yields the trace of the associated Green function: ∂ Tr log(−∂t2 − γ∂t − ω02) = − 2 ∂ω0

Z

dt (−∂t2 − γ∂t − ω02 )−1 (t, t).

(18.266)

In Fourier space, the right-hand side turns into the frequency integral −

Z

dω ′ 1 . 2π (ω ′ + iω1 )(ω ′ + iω1 )


1217


Since the two poles lie below the contour of integration, we may close it in the upper half-plane and obtain zero. Closing it in the lower half plane would initially lead to two nonzero contributions from the residues of the two poles which, however, cancel each other. The Green function (18.266) is causal, in contrast to the oscillator Green function in Section 3.3 whose left-hand pole lies in the upper half-plane (recall Fig. 3.3). Thus it carries a Heaviside function as a prefactor [recall Eq. (1.301) and the discussion of causality there]. The vanishing of the integral (18.266) may be interpreted as being caused by the Heaviside function (1.300). The γ-dependence of (18.265) can be calculated likewise: ∂ log Det∂t (−∂t2 − γ∂t − ω02 ) = − ∂γ

Z

dt [∂t (−∂t2 − γ∂t − ω02 )−1 ](t, t). (18.268)

We perform the trace in frequency space: i

Z

ω′ dω ′ . 2π (ω ′ + iω1 )(ω ′ + iω1 )

(18.269)

If we now close the contour of integration with an infinite semicircle in the upper half plane to obtain a vanishing integral from the residue theorem, we must subtract the R ′ integral over the semicircle i dω /2πω ′ and obtain 1/2, in agreement with (18.265). Formula (18.265) can be generalized further to time-dependent coefficients Det

h

−∂t2 −γ(t)∂t

i

2

n

− Ω (t) = exp Tr log

h

−∂t2 −γ(t)∂t

2

− Ω (t)

io

= exp

"Z

tb

ta

#

γ(t) dt . 2 (18.270)

This follows from the factorization h

i

Det −∂t2 − γ(t)∂t − Ω2 (t) = Det[∂t + Ω1 (t)] Det[∂t + Ω2 (t)] , with

∂t Ω2 (t) + Ω1 (t)Ω2 (t) = Ω2 (t),

Ω1 (t) + Ω2 (t) = γ(t),

(18.271)

(18.272)

and applying formula (3.134). The probability obtained from the general path integral (18.232) without retardation of the velocity term is therefore

P0 (0 tb |0 ta ) ∝ exp −(tb − ta )

γ , 2

(18.273)

as in (18.251). Let us now introduce retardation of the velocity term by using the ω ′ -dependent Drude expression (18.212) for the friction coefficient. First we consider again the harmonic path integral (18.261), for which (18.262) becomes (

P0 (0 tb |0 ta ) ∝ exp −(tb − ta )

Z

∞

−∞

h i dω ′ R log ω ′2 − iγD (ω ′ )ω ′ − ω02 . 2π )

(18.274)

1218


Rewriting the logarithm as − log(ω ′ + iωD ) + Σ3k=1 log(ω ′ + iωk ) with ω1,2

γ = 2

v u u 1 ± t1 − 



4ω02  , γ2

ω3 = ωD − γ

(18.275)

[recall Eq. (3.451) in the equilibrium discussion of Section 3.15], we use again formula (2.533) to find Z

∞

−∞

3 3 X X dω ′ ωk − log(ω ′ + iωD ) + log(ω ′ + iωk ) = −ωD + = 0. (18.276) 2π k=1 k=1 2

"

#

Thus γ and ω0 disappear from the functional determinant, and we remain with P0 (0 tb |0 ta ) = const .

(18.277)

This implies a unit functional determinant [12] Det(∂t2 + iγ∂tR + ω02) = 1,

(18.278)

in contrast to the unretarded determinant (18.265). The γ-independence of this can also be seen heuristically as in (18.266) by forming the derivative with respect to γ: ∂ Det(−∂t2 − γ∂tR − ω02) = − ∂γ

Z

dt [∂tR (∂t2 − γ∂t − ω02)−1 (t, t).

(18.279)

Since the retarded derivative carries a Heaviside factor Θ(t − t′ ) of (1.300), we find zero for t = t′ . The result 1/2 of the unretarded derivative in (18.268) can similarly be understood as a consequence of the average Heaviside function (1.309) at t = t′ . An advanced time derivative in the determinant (18.278) would, of course, have produced the result Det(∂t2 + iγ∂tA + ω02) = γ.

(18.280)

By analogy with (18.271), the general retarded determinant is also independent of γ(t) and Ω(t). h

i

Det −∂t2 − γ(t)∂tR − Ω2 (t) = 1. In the advanced case, we would find similarly Det

h

−∂t2

−

γ(t)∂tA

2

i

− Ω (t) = exp

Z

(18.281)

dt γ(t) .

(18.282)

By comparing the functional determinants (18.265) and (18.278) we see that the retardation prescription can be avoided by a trivial additive change of the Lagrangian (18.235) to 1 γ 2 [¨ x + Mγ x˙ + V ′ (x)] − . (18.283) 2w 2 From this the path integral can be calculated with the usual time slicing described in Section 10.5 [13]. Le (x, x) ˙ =


1219


18.9.3

Strong Damping

For γ ≫ V ′′ (x)/M, the dynamics is dominated by dissipation, and the Lagrangian (18.235) takes a more conventional form in which only x and x˙ appear: "

i2 1 h 1 1 ′ Le (x, x) ˙ = Mγ x˙ R + V ′ (x) = x˙ R + V (x) 2w 4D Mγ

#2

,

(18.284)

where x˙ R lies slightly earlier V ′ (x(t)). The probability distribution P (xb tb |xa ta ) = N

Z

Dx exp −

Z

tb

ta

R

dt Le (x, x˙ )

(18.285)

looks like an ordinary Euclidean path integral for the density matrix of a particle of mass M = 1/2D. As such it obeys a differential equation of the Schrödinger type. Forgetting for a moment the subtleties of the retardation, we introduce an auxiliary momentum integration and go over to the canonical representation of (18.285): P (xb tb |xa ta ) =

Z

Dx

Z

Dp exp 2π

(Z

tb

ta

p2 1 ′ dt ipx˙ − 2D + ip V (x) 2 Mγ "

#)

. (18.286)

This probability distribution satisfies therefore the Schrödinger type of equation H(ˆ pb, xb )P (xb tb |xa ta ) = −∂tb P (xb tb |xa ta ),

(18.287)

with the Hamiltonian operator pˆ2 1 ′ 1 H(ˆ p, x) ≡ 2D − iˆ p V (x) = −D ∂x ∂x + V ′ (x) . 2 Mγ DMγ "

#

(18.288)

In order to conserve probability, the momentum operator has to stand to the left of the potential term. Only then does the integral over xb of Eq. (18.287) vanish. Equation (18.287) is the overdamped or ordinary Fokker-Planck equation. Without the retardation on x˙ in (18.285), the path integral would give a symˆ This follows from the fact that metrized operator −i[ˆ pV ′ (x) + V ′ (x)ˆ p]/2 in H. ′ the coupling (1/2DMγ)xV ˙ (x) looks precisely like the coupling of a particle to a magnetic field with a “vector potential” A(x) = (1/2DMγ)V ′ (x) [see (10.170)]. Realizing this it is not difficult to account quite explicitly for the effect of retardation of the velocity in the path integral (18.284) Let us assume, for a moment, that w is very small. Then the path integral (18.285) without the retardation, P0 (xb tb |xa ta ) = N

Z

1 Z tb 2 Dx exp − dt [Mγ x˙ + V ′ (x)] , 2w ta

(18.289)

can be performed in the Gaussian approximation resulting for xb = xa = 0 in the inverse functional determinant P0 (0 tb |0 ta ) = Det −1 [∂t + V ′′ (x)/Mγ] ,

(18.290)

1220


whose value is according to formula (3.134) Det [∂t + V ′′ (x)/Mγ] = exp

Z

dt V ′′ (x)/2Mγ .

(18.291)

The retarded version of this determinant is trivial: h

i

Det ∂tR + V ′′ (x)/Mγ = 1,

(18.292)

as we learned from Eq. (18.281. The advanced version would be [compare (18.282)] Det

h

∂tA

i

′′

+ V (x)/Mγ = exp

Z

′′

dt V (x)/Mγ .

(18.293)

Although the determinants (18.291), (18.292), and (18.293) were discussed here only for a large time interval tb − ta , the formulas remain true for all time intervals, due to the trivial first-order nature of the differential operator. In a short time interval, however, the second derivative is approximately time-independent. For this reason the difference between ordinary and retarded path integrals (18.285) is given by the difference between the functional determinants (18.291) and (18.292) not only if w is very small but for all w. Thus we can avoid the retardation of the velocity as in Eq. (18.283) by adding to the Lagrangian (18.284) a term containing the second derivative of the potential: "

1 1 ′ ˙ = x˙ + V (x) Le (x, x) 4D Mγ

#2

−

1 V ′′ (x). 2Mγ

(18.294)

From this, the path integral can be calculated with the same slicing as for the gauge-invariant coupling in Section 10.5: P0 (xb tb |xa ta ) = N

Z



Dx(t) exp −

Z

tb

ta

dt

 

1

 4D

"

x˙ +

′

V (x) Mγ

#2

 V (x)  . 2Mγ  ′′

−

(18.295)

As an example consider a harmonic potential V (x) = Mω02 x2 /2 where the Lagrangian (18.294) becomes Le (x, x) ˙ =

1 κ (x˙ + κx)2 − , 4D 2

(18.296)

where we have introduced the frequency κ ≡ ω02 /γ. The equation of motion reads −¨ x + κ2 x = 0,

(18.297)

and its solution connecting xa , ta with xb , tb is x(t) =

h i 1 κ(t+ta ) κ(−t+ta+2κtb ) κ(t+tb ) κ(−t+2 ta +tb ) e x −e x −e x +e x . (18.298) a a b b e2κta −e2κtb H. Kleinert, PATH INTEGRALS

1221


This has the total Euclidean action 2

κ κ(eκta xa − eκtb xb ) Ae = − . 2 D (e2κta − e2κtb ) 2

(18.299)

The fluctuation determinant is from Eq. (2.164), after an appropriate substitution of variables, 1 Fκ (tb − ta ) = q . 2π sinh κ(tb − ta )

(18.300)

The probability distribution is then given by −Ae

P (xb tb |xa ta ) = Fκ (tb − ta )e

[xb − x¯(tb − ta )]2 =q exp − ,(18.301) 2σ 2 (tb − ta ) 2πσ 2 (t) (

1

)

where x¯(t), σ 2 (t) are the averages x¯(t) ≡ hx(t)i = xa e−κt ,

σ 2 (t) ≡ h[x(t) − x¯(t)]2 i =

D 1 − e−κt , κ

(18.302)

obtained from the integrals x¯(tb − ta ) ≡ hx(tb − ta )i ≡ h[x(tb − ta ) − x¯(tb − ta )]2 i ≡

Z

∞

−∞ ∞

Z

−∞

xb P (xb tb |xa ta ),

(18.303)

[xb − x¯(tb − ta )]2 P (xb tb |xa ta ). (18.304)

It is easy to verify that (18.301) satisfies the Fokker-Planck equation (18.287):

−D∂x2b − κ∂xb xb P (xb tb |xa ta ) = −∂tb P (xb tb |xa ta ).

(18.305)

For tb → ta , the probability distribution (18.301) starts out as a δ-function around the initial position xa . In the limit of large tb − ta , it converges against the limiting distribution lim P (xb tb |xa ta ) =

tb →∞

r

κ x2 exp −κ b . 2πD 2D (

)

(18.306)

Replacing κ again by ω02 /γ, and D from (18.219), this becomes lim P (xb tb |xa ta ) =

tb →∞

s

V ′′ (0) 1 exp − V (xb ) . 2πkB T kB T

(18.307)

Thus, the limiting distribution of (18.285) depends only on xb . It is given by the Boltzmann factor associated with the potential V (x), in which the particle moves. This result can be generalized to a large class of potentials.

1222


An interesting related result can be derived by introducing an external source term jb x(tb ) into the Lagrangian (18.294). By repeated functional differentiation with respect to jb we find that the expectation values n

n

hx i = lim hx (tb )i = lim tb →∞

tb →∞

R

have the large-time limit

−

n

Dx x (tb )e −

Dx e

R

R tb ta

R tb ta

dt Le (x,x˙ R )

(18.308)

dt Le (x,x˙ R )

dx xn e−V (x)/kB T lim hx (tb )i = hx i = R . tb →∞ dx e−V (x)/kB T n

R

n

(18.309)

The generalization of this relation to quantum field theory forms the basis of stochastic quantization in Section 18.11.

18.10

Langevin Equations

Consider the forward–backward path integral (18.230) for high γT . Then the second exponent limits the fluctuations of y to satisfy |y| ≪ |x|, and K(t, t′ ) will be assumed to take the Drude form (18.226), which becomes a δ-function for ωD → ∞. Then we can expand

V x+

y −V 2

y 2

x−

∼ yV ′ (x) +

y 3 ′′′ V (x) + . . . , 24

(18.310)

keeping only the first term. We further introduce an auxiliary quantity η(t) by η(t) ≡ M x¨(t) + Mγ x˙ R (t) + V ′ (x(t)).

(18.311)

With this, the exponential function in (18.230) becomes after a partial integration of the first term using the endpoint properties y(tb) = y(ta ) = 0: i exp − h ¯

Z

tb

ta

w dt yη − 2 2¯ h

Z

tb ta

dt

Z

tb

ta

′

′

′

dt y(t)K(t, t )y(t ) .

(18.312)

The variable y can obviously be integrated out and we find a probability distribution

P [η] ∝ exp −

1 2w

Z

tb ta

dt

Z

tb

ta

dt′ η(t)K −1 (t, t′ )η(t′ ) ,

(18.313)

where the fluctuation width w was given in (18.218), and K −1 (t, t′ ) denotes the inverse functional matrix of K(t, t′ ). The defining equation (18.311) for η(t) may be viewed as a stochastic differential equation to be solved for arbitrary initial positions x(ta ) = xa and velocities x(t ˙ a) = va . The differential equation is driven by a Gaussian random noise variable η(t) with a correlation function hη(t)iη = 0,

hη(t)η(t′ )iη = w K Ohm (t − t′ ),

(18.314)


1223

18.10 Langevin Equations

where the expectation value of an arbitrary functional of F [x] is defined by the path integral hF [x]iη ≡ N

Z

x(ta )=xa

Dx P [η]F [x].

(18.315)

The normalization factor N is fixed by the condition N Dη P [η] = 1, so that h 1 iη = 1. In the sequel, this factor will always be absorbed in the measure Dη. For each noise function η(t), the solution of the differential equation yields a path xη (xa , xb , ta ) with a final position xb = xη (xa , xb , tb ) and velocity vb = x˙ η (xa , xb , tb ), all being functionals of η(t). From this we can calculate the distribution P (xb vb tb |xa va ta ) of the final xb and vb by summing over all paths resulting from the noise functions η(t) with the probability distribution (18.313). The result is of course the same as the distribution (18.242) obtained previously from the canonical path integral. It is useful to exhibit clearly the dependence on initial and final velocities by separating the stochastic differential equation (18.311) into two first-order equations R

M v(t) ˙ + Mγv R (t) + V ′ (x(t)) = η(t), x(t) ˙ = v(t),

(18.316) (18.317)

to be solved for initial values x(ta ) = xa and x(t ˙ a ) = va . For a given noise function η(t), the final positions and velocities have the probability distribution Pη (xb vb tb |xa va ta ) = δ(xη (t) − xb )δ(x˙ η (t) − vb ).

(18.318)

Given these distributions for all possible noise functions η(t), we find the final probability distribution P (xb vb tb |xa va ta ) from the path integral over all η(t) calculated with the noise distribution (18.313). We shall write this in the form P (xb vb tb |xa va ta ) = hPη (xb vb tb |xa va ta )iη .

(18.319)

Let us change of integration variable from x(t) to η(t). This produces a Jacobian J[x] ≡ Det[δη(t)/δx(t′ )] = det [M∂t2 + Mγ∂tR + V ′′ (x(t))].

(18.320)

In Eq. (18.281) we have seen that due to the retardation of ∂tR , this Jacobian is unity. Hence we can rewrite the expectation value (18.315) as a functional integral hF [x]iη ≡

Z

Dη P [η] F [x]

x(ta )=xa

.

(18.321)

From the probability distribution P (xb vb tb |xa va ta ) we find the pure position probability P (xb tb |xa ta ) by integrating over all initial and final velocities as in Eq. (18.241). Thus we have shown that a solution of the forward–backward path integral at high temperature (18.232) can be obtained from a solution of the stochastic differential equations (18.311), or more specifically, from the pair of stochastic differential equations (18.316) and (18.317).

1224


The stochastic differential equation (18.311) together with the correlation function (18.314) is called semiclassical Langevin equation. The fluctuation width w in (18.314) was given in (18.218). The attribute semiclassical emphasizes the truncation of the expansion (18.310) after the first term, which can be justified only for nearly harmonic potentials. For a discussion of the range of applicability of the truncation see the literature [14]. The untruncated path integral is equivalent to an operator form of the Langevin equation, the so-called quantum Langevin equation [15]. This equivalence will be discussed further in Subsection 18.16. The physical interpretation of Eq. (18.314) goes as follows. For T → 0 and h ¯ → 0 at h ¯ /T = const, the random variable η(t) does not fluctuate at all and (18.311) reduces to the classical equation of motion of a particle in a potential V (x), with an additional friction term proportional to γ. For T and h ¯ both finite, the particle is shaken around its classical path by thermal and quantum fluctuations. At high temperatures (at fixed h ¯ ), K(ω ′ ) reduces to lim K(ω ′ ) ≡ 1.

(18.322)

T →∞

Then, η(t) is an instantaneous random variable with the correlation functions hη(t)η(t′)iη = w δ(t − t′ ),

hη(t)iη = 0,

(18.323)

and (18.311) with (18.314) reduces to the classical Langevin equation with inertia [16]. In the opposite limit of small temperatures, K(ω ′ ) diverges like K(ω ′ ) − −−→ T →0

h ¯ |ω ′| , 2kB T

(18.324)

so that w K(ω ′ ) has the finite limit lim wK(ω ′) = Mγ¯ h|ω ′|.

(18.325)

T →0

To find the Fourier transformation of this, we use the Fourier decomposition of the Heaviside function (1.302) Θ(ω ′) =

1 2π

Z

∞

′

−∞

dt e−iω t

i t + iη

(18.326)

to form the antisymmetric combination 1 Z∞ i i ′ Θ(ω ) − Θ(−ω ) = dt e−iω t + 2π −∞ t + iη t − iη Z ∞ i ′ P ≡ dt e−iω t . π −∞ t ′

′

!

(18.327)

A multiplication by ω ′ yields 1 ∞ 1 ′ P dt ∂t e−iω t = π −∞ t π Z ∞ 1 ′ P = − dt e−iω t 2 . π −∞ t

|ω ′ | = ω ′ [Θ(ω ′ ) − Θ(−ω ′ )] = −

Z

Z

∞

−∞

′

dt e−iω t ∂t

P t


1225

18.10 Langevin Equations

By comparison with (18.325) we see that the pure quantum limit of K(t − t′ ) can be written as wK(t − t′ ) = − T =0

Mγ¯ h P . π (t − t′ )2

(18.329)

Hence the quantum-mechanical motion in contact with a thermal reservoir looks just like a classical motion, but disturbed by a random source with temporally long-range correlations hη(t)η(t′ )iη = −

Mγ¯ h P . π (t − t′ )2

(18.330)

The temporal range is found from the temporal average ∂2 1 h(∆t) it ≡ d∆t (∆t) K(∆t) = − ′2 K(ω ′ ) =− ∂ω 6 −∞ ω ′ =0 2

Z

∞

2

h ¯ kB T

!2

. (18.331)

Apart from the negative sign (which would be positive for Euclidean times), the random variable acquires more and more memory as the temperature decreases and the system moves deeper into the quantum regime. Note that no extra normalization factor is required to form the temporal average (18.331), due to the unit normalization of K(t − t′ ) in (18.224). In the overdamped limit, the classical Langevin equation with inertia (18.311) reduced to the overdamped Langevin equation: η(t) ≡ Mγ x(t) ˙ + V ′ (x(t)).

(18.332)

A stochastic movement of this type is called a Wiener process. The probability distribution is calculated as in Eqs. (18.319), (18.315) from the path integral P (xb ta |xa ta ) =

Z

Dη P [η] δ(xη (t) − xb ),

(18.333)

and Dη is normalized so that Dη P [η] = 1. A path integral representation closely related to this is obtained by using the identity R

Z

x(tb )=xb

x(ta )=xa

Dx δ[x˙ − η] = δ(xη (tb ) − xb ),

(18.334)

which can easily be proved by time-slicing the Fourier representation of the δfunctional δ[x˙ − η] =

Z

Dp ei

R

dt p (x−η) ˙

(18.335)

and performing all the momentum integrals. This brings the path integral (18.333) to the form P (xb tb |xa ta ) =

Z

x(tb )=xb

x(ta )=xa

Dx

Z

Dη P [η] δ[x˙ − η].

(18.336)

1226


18.11

Stochastic Quantization

In Eq. (18.308) we observed that the expectation value of powers of a classical variable x in a potential V (x) can be recovered as a result of a path integral associated with the Lagrangian (18.294). From Eq. (18.333) we know that the path integral (18.308) can be replaced by the stochastic path integral: n

n

hx i = lim hx (s)i = lim s→∞

s→∞

Z

Dη xnη (s)P [η] ,

(18.337)

where P [η] ≡

Z

−(1/4kB T )

Dηe

Rs

sa

ds′ η2 (s′ )

,

(18.338)

To simplify the equations, we have replaced the physical time by a rescaled parameter s = t/Mγ. Equivalently we may say that we obtain the expectation values (18.337) by solving the stochastic differential equation x′ (s) = −V ′ (x) + η(s),

(18.339)

where η(s) is a white noise with the pair correlation functions hη(s)iT = 0,

hη(s)η(s′)iT = 2kB T δ(s − s′ ),

(18.340)

and going to the large-s limit of the expectation values hxn (s)i. This can easily be generalized to Euclidean quantum mechanics. Suppose we want to calculate the correlation functions (3.295) hx(τ1 )x(τ2 ) · · · x(τn )i ≡ Z −1

Z

1 Dx x(τ1 )x(τ2 ) · · · x(τn ) exp − Ae . (18.341) h ¯

We introduce an additional auxiliary time variable s and set up a stochastic differential equation ∂s x(τ ; s) = −

δAe + η(τ ; s), δx(τ ; s)

(18.342)

where η(τ ; s) has correlation functions hη(τ ; s)i = 0,

hη(τ ; s)η(τ ′ ; s′ )i = 2¯ hδ(τ − τ ′ )δ(s − s′ ).

(18.343)

The role of the thermal fluctuation width 2kB T in (18.340) is now played by 2¯ h. The correlation functions (18.341) can now be calculated from the auxiliary correlation functions of x(τ, s) in the large-s limit: hx(τ1 )x(τ2 ) · · · x(τn )i = s→∞ lim hx(τ1 , s)x(τ2 , s) · · · x(τn , s)i.

(18.344)


1227

18.11 Stochastic Quantization

Due to the extra time variable of stochastic variable x(τ ; s) with respect to (18.339), the probability distribution associated with the stochastic differential equation (18.361) is a functional P [xb (τ), sb ; xa (τ), sa ] given by the functional generalization of the path integral (18.295): Z

P [xb (τb), s; xa (τ), sa ] = N −

× e

Dx(τ ; s)

R sb sa

ds

n

1 4¯ h

R∞

δ dτ [∂s x(τ ;s)+ δx(τ A − 1h ;s) e ] 2¯ −∞

δ2 Ae δx(τ ;s)2

o

. (18.345)

This satisfies the functional generalization of the Fokker-Planck equation (18.287): H[ˆ p(τ ), x(τ )]P (x(τ )s|xa (τ ); sa ) = −∂s P (x(τ )s|xa (τ ); sa ),

(18.346)

with the Hamiltonian "

#

δ H[ˆ p(τ ), x(τ )] = Ae , dτ h ¯ pˆ (τ ) − iˆ p(τ ) δx(τ ) −∞ Z

∞

2

(18.347)

where pˆ(τ ) ≡ δ/δx(τ ). We have dropped the subscript b of the final state, for brevity. Explicitly, the Fokker-Planck equation (18.346) reads "

#

h ¯δ δAe h ¯δ dτ + P [x(τ), s; xa (τ), sa ] = −¯ h∂s P [x(τ), s; xa (τ), sa ].(18.348) − δx(τ) δx(τ) δx(τ) −∞ Z

∞

For s → ∞, the distribution becomes independent of the initial path xa (τ), and has the limit [compare (18.308)] lim P [x(τ), s; xa (τ), sa ] = Z

s→∞

e−Ae [x]/¯h

,

(18.349)

Dx(τ) e−Ae [x]/¯h

and the correlation functions (18.360) are given by the usual path integral, apart from the normalization which is here such that h1i = 1. As an example, consider a harmonic oscillator where Eq. (18.342) reads ∂s x(τ ; s) = −M(−∂τ2 + ω 2)x(τ ; s) + η(τ ; s).

(18.350)

This is solved by s

Z

x(τ ; s) =

0

2

ds′ e−M (−∂τ +ω

2 )(s′ −s)

η(τ ; s′ ).

(18.351)

The correlation function reads, therefore, hx(τ1 ; s1 )x(τ2 ; s2 )i =

Z

s1

ds′1

0

Z

s2

2

ds′2 eM (−∂τ +ω

0

2 )(s′ +s′ −s −s ) 1 2 1 2

hη(τ1 ; s′1 )η(τ2 ; s′2 )i. (18.352)

Inserting (18.343), this becomes hx(τ1 ; s1 )x(τ2 ; s2 )i = h ¯

Z

∞ 0

h

2

ds e−M (−∂τ +ω

2 )(s+|s

1 −s2 |)

2

− e−M (−∂τ +ω

2 )(s+s

1 +s2 )

i

,(18.353)

1228


or hx(τ1 ; s1 )x(τ2 ; s2 )i =

i h 1 h ¯ −M (−∂τ2 +ω 2 )|s1 −s2 | −M (−∂τ2 +ω 2 )(s1 +s2 ) e − e . (18.354) M −∂τ2 + ω 2

For Dirichlet boundary conditions (xb = xa = 0) where operator (−∂τ2 + ω 2 ) has the sinusoidal eigenfunctions of the form (3.63) with eigenfrequencies (3.64), this has the spectral representation ∞ 2 X h ¯ 1 hx(τ1 ; s1 )x(τ2 ; s2 )i = sin νn (τ1 − τa ) sin νn (τ2 − τa ) 2 M tb − ta n=1 νn + ω 2

×

h

2

e−M (νn +ω

2 )|s

1 −s2 |

2

− e−M (−νn +ω

2 )(s

1 +s2 )

i

.

(18.355)

For large s1 , s2 , the second term can be omitted. If, in addition, s1 = s2 , we obtain the imaginary-time correlation function [compare (3.69), (3.301), and (3.36)]: s1

1 h ¯ (τ1 , τ2 ) 2 M −∂τ + ω 2 h ¯ sinh ω(τb − τ> ) sinh ω(τ< − τa ) . = M ω sinh ω(τb − τa )

lim hx(τ1 ; s)x(τ2 ; s)i = hx(τ1 )x(τ2 )i = =s →∞ 2

(18.356)

We can use these results to calculate the time evolution amplitude according to an imaginary-time version of Eq. (3.315): −Ae (xb ,xa ;τb −τa )/¯ h −

(xb τb |xa τa ) = C(xb , xa )e

e

R τb τa

M 2

dτb′ hLe,fl (xb ,x˙ b )i/¯ h

,

(18.357)

where Ae (xb , xa ; τb − τa ) is the Euclidean version of the classical action (4.80). If the Lagrangian has the standard form, then hLe,fl (xb , x˙ b )i =

M hδ x˙ 2b i, 2

(18.358)

and we obtain the imaginary-time evolution amplitude in an expression like (3.315). The constant of integration is determined by solving the differential equation (3.316), and a similar equation for xa . From this we find as before that C(xb , xa ) is independent of xb and xa . For the harmonic oscillator with Dirichlet boundary conditions we calculate from this M h ¯ω hδ x˙ 2b i = D coth ω(τb − τa ). (18.359) 2 2 Integrating this over τb yields h ¯ (D/2) log[2 sinh ω(τb − τa )], so that the second exponential in (18.357) reduces to the correct fluctuation factor in the D-dimensional imaginary-time amplitude [compare (2.403)]. The formalism can easily be carried over to real-time quantum mechanics. We replace t → −iτ and Ae → −iA, and find that the real-time correlation functions are obtained from the large-s limit hx(t1 )x(t2 ) · · · x(τn )i = s→∞ lim hx(t1 , s)x(t2 , s) · · · x(tn , s)i,

(18.360)


1229

18.12 Stochastic Calculus

where x(t; s) satisfies the stochastic differential equation h ¯ ∂s x(t; s) = i

δA + η(t; s), ∂x(t; s)

(18.361)

where the noise η(t; s) has the same correlation functions as in (18.343), if we replace τ by t. This procedure of calculating quantum-mechanical correlation functions is called stochastic quantization [17].

18.12

Stochastic Calculus

The relation between Langevin and Fokker-Planck equations is a major subject of the so-called stochastic calculus. Given a Langevin equation, the time order of the potential V (x) with respect to x˙ and x¨ is a matter of choice. Different choices form the basis of the Itô or the Stratonovich calculus. The retarded position which appears naturally in the derivation from the forward–backward path integral favors the use of the Itô calculus. A midpoint ordering as in the gauge-invariant path integrals in Section 10.5 corresponds to the Stratonovich calculus.

18.12.1

Kubo’s stochastic Liouville equation

It is worthwhile to trace how the retarded operator order of the friction term enters the framework of stochastic calculus. Thus we assume that the stochastic differential equations (18.316) and (18.317) have been solved for a specific noise function η(t) such that we know the probability distribution Pη (x v t|xa va ta ) in (18.318). Now we observe that the time dependence of this distribution is governed by a simple differential equation known as Kubo’s stochastic Liouville equation [18], which is derived as follows [19]. A time derivative of (18.318) yields ∂t Pη (x v t|xa va ta ) = x˙ η (t)δ ′ (xη (t) − x)δ(x˙ η (t) − v) + x¨η (t)δ(xη (t) − x)δ ′ (x˙ η (t) − v). (18.362) The derivatives of the δ-functions are initially with respect to the arguments xη (t) and x˙ η (t). These can, however, be expressed in terms of derivatives with respect to −x and −v. However, since x¨η (t) depends on x˙ η (t) we have to be careful where to put the derivative −∂v . The general formula for such an operation may be expressed as follows: Given an arbitrary dynamical variable z(t) which may be any local function (local in the temporal sense) of x(t) and x(t), ˙ and whose derivative is some function of z(t), i.e., z(t) ˙ = F (z(t)), then d ∂ ∂ ∂ δ(z(t)−z) = z(t) ˙ δ(z(t)−z) = − [z(t)δ(z(t)−z)] ˙ = − [F (z)δ(z(t)−z)]. dt ∂z(t) ∂z ∂z (18.363) To prove this formula, we multiply each expression by an arbitrary smooth test function g(z) and integrate over z. Each integral yields indeed the same result

1230


′ g(z(t)) ˙ = z(t)g ˙ (z(t)) = F (z)g ′ (z(t)). Applying the identity (18.363) to (18.362), we obtain an equation for Pη (x v t|xa va ta ):

∂t Pη (x v t|xa va ta ) = −[∂x x˙ η (t) + ∂v x¨η (t)]Pη (x v t|xa va ta ).

(18.364)

We now express x¨η (t) with the help of the Langevin equation (18.311) in terms of the friction force −Mγ x˙ η (t), the force −V ′ (xη (t)), and the noise η(t). In the presence of the δ-function δ(x˙ η (t) − v), the velocity x˙ η (t) can everywhere be replaced by v, and Eq. (18.364) becomes

∂t Pη (x v t|xa va ta ) = − v∂x +

1 [η(t) + f (x, v)] Pη (x v t|xa va ta ), M

(18.365)

where f (x, v) ≡ −Mγv − V ′ (x)

(18.366)

is the sum of potential and friction forces. This is Kubo’s stochastic Liouville equation which, together with the correlation function (18.218) of the noise variable and the prescription (18.319) of forming expectation values, determines the temporal behavior of the probability distribution P (x v t|xa va ta ).

18.12.2

From Kubo’s to Fokker-Planck Equations

Let us calculate the expectation value of Pη (x v t|xa va ta ) with respect to noise fluctuations and show that P (x v t|xa va ta ) of Eq. (18.319) satisfies the Fokker-Planck equation with inertia (18.243). First we observe that in a Gaussian expectation value (18.315), the multiplication of a functional F [η] by η produces the same result as the functional differentiation with respect to η with a subsequent functional multiplication by the correlation function hη(t)η(t′ )i: hη(t)F [η]iη =

Z

′

′

dt hη(t)η(t )iη

*

+

δη(t) F [η] δη(t′ )

.

(18.367)

η

This follows from the fact that η(t) can be obtained from a functional derivative of the Gaussian distribution in (18.315) as: 1

η(t)e− 2w

R

dtdt′ η(t)K −1 (t,t′ )η(t′ )

Z

= −w dt′ K(t, t′ )

R 1 δ − 2w dtdt′ η(t)K −1 (t,t′ )η(t′ ) e . (18.368) ′ δη(t )

Inside the functional integral (18.315) over η(t), an integration by parts moves the functional derivative −δ/δη(t′ ) in front of F [η] with a sign change. The surface terms can be discarded since the integrand decrease exponentially fast for large noises η(t). Thus we obtain indeed the useful formula (18.367). With the goal of a Gaussian average (18.315) in mind, we can therefore replace Eq. (18.365) by (

" Z

1 ∂t Pη (x v t|xa va ta ) = − v∂x + ∂v w M

#)

δ dt K(t, t ) + f (x, v) Pη (x v t|xa va ta ). δη(t′ ) (18.369) ′

′


1231


After this, we observe that "

#

δ δxη (t) δ x˙ η (t) δ(xη (t) −x)δ(x˙ η (t) −v) = − ∂x + ∂v δ(xη (t) −x)δ(x˙ η (t) −v). ′ ′ δη(t ) δη(t ) δη(t′ ) (18.370) From the stochastic differential equation (18.311) we deduce the following behavior of the functional derivatives: δ¨ xη (t) 1 = δ(t − t′ ) − γΘ(t − t′ ) + smooth function of t − t′ , ′ δη(t ) M δ x˙ η (t) 1 = Θ(t − t′ ) + O(t − t′ ), ′ δη(t ) M δxη (t) = O((t − t′ )2 ). δη(t′ )

(18.371) (18.372) (18.373)

Inserting (18.362) with (18.372) and (18.373) into (18.369), the functional derivatives (18.372) and (18.373) are multiplied by K(t, t′ ) and integrated over t′ . Consider now the regime of large temperatures. There the function K(t, t′ ) is narrowly peaked around t = t′ , forming almost a δ-function [recall the unit normalization (18.322)]. We shall emphasize this by writing K(t, t′ ) ≡ δǫ (t − t′ ), with the subscript indicating the width ǫ of K(t, t′ ) which goes to zero like h ¯ /kB T for large T [recall (18.331)]. In this limit, the contribution of the derivative (18.373) vanishes, whereas (18.372) contributes to (18.369) a term δ (18.374) δ(xη (t)−x)δ(x˙ η (t)−v) δη(t′ ) Z δ x˙ η (t) 1 = − dt′ δǫ (t−t′ ) ∂v δ(xη (t)−x)δ(x˙ η (t)−v) = − ∂v δ(xη (t)−x)δ(x˙ η (t)−v). ′ δη(t ) 2M Z

dt′ K(t, t′ )

The factor 1/2 on the right-hand side arises from the fact that the would-be δfunction δǫ (t−t′ ) is symmetric in t − t′ , so that its convolution with the Heaviside function Θ(t − t′ ) is nonzero only over half the peak. Taking the noise average (18.319), we obtain from (18.369) the Fokker-Planck equation with inertia (18.243):

∂t P (x v t|xa va ta ) = −v∂x +

1 w ∂v ∂v − f (x, v) M 2M

P (x v t|xa va ta ). (18.375)

Note that the differential operators have precisely the same order as in Eq. (18.239) as a consequence of formula, here (18.363). In the overdamped limit, the derivation of the Fokker-Planck equation becomes simple. Then we have to consider only the pure x-space distribution Pη (x t|xa ta ) =

Z

dv Pη (x v t|xa va ta ) = δ(xη (t) − x),

(18.376)

1232


whose time derivative is given by ∂t Pη (x t|xa va ta ) = −∂x x˙ η (t)Pη (x t|xa va ta ) 1 = − ∂x [η(t) − V ′ (x)]Pη (x t|xa va ta ). Mγ

(18.377)

After treating the noise term η(t) according to the rule (18.367), η(t) → w

Z

dt′ δǫ (t − t′ )

δ , δη(t′ )

(18.378)

we use δ δxη (t) δ(xη (t) − x) = − δ(xη (t) − x) ′ δη(t ) δη(t′ )

(18.379)

and δ x˙ η (t) 1 = δ(t − t′ ) + smooth function of t − t′ , ′ δη(t ) Mγ δxη (t) 1 = Θ(t − t′ ) + O(t − t′ ), ′ δη(t ) Mγ

(18.380)

to find the overdamped Fokker-Planck equation (18.287): "

#

1 ′ V (x) P (x t|xa ta ). ∂t P (x t|xa ta ) = D∂ + Mγ 2

(18.381)

The distributions P (x t|xa ta ) and P (x v t|xa va ta ) develop from initial δ-function distributions P (x ta |xa ta ) = δ(x − xa ) and P (x v ta |xa ta ) = δ(x − xa )δ(v − xa ). Let us multiply these δ-functions with arbitrary initial probabilities P (x, ta ) and P (x v, ta ) and integrate over x and v. Then we obtain the stochastic path integrals P (x , t) = P (x v, t) =

Z Z

Dη e−(1/2w) Dη e−(1/2w)

R R

dtdt′ η(t)K −1 (t,t′ )η(t′ )

P (xaη (t), ta ),

(18.382)

dtdt′ η(t)K −1 (t,t′ )η(t′ )

P (xaη (t), vaη , t),

(18.383)

where xaη and vaη are initial positions and velocities of paths which arrive at the final x and v following the equation of motion with a fixed noise η(t): xaη (t) = x −

Z

t

ta

′

′

dt x(t ˙ ),

vaη (t) = x −

Z

t

ta

dt′ v(t ˙ ′ ).

(18.384)

At high temperatures, the overdamped equation can be written with (18.332) as P (x , t) =

Z

−(1/2w)

Dη e

R

dt η2 (t)

1 P x− Mγ

Z

t

ta

′

′

′

′

!

dt [η(t ) − V (x(t ))] , t . (18.385) H. Kleinert, PATH INTEGRALS

1233


The time evolution equation (18.381) follows from this by calculating for a short time increment ǫ: P (x , t + ǫ) =

Z

−(1/2w)

Dη e

R

dt η2 (t)

(

−

ǫ Mγ

Z

t+ǫ

t

dt′ [η(t′ ) − V ′ (x(t′ ))] ∂x

) 1 Z t+ǫ ′ Z t+ǫ ′′ ′ ′ ′ ′′ ′ ′′ 2 + dt dt [η(t ) − V (x(t ))] [η(t ) − V (x(t ))] ∂x + . . . 2M 2 γ 2 t t ! Z t 1 ′ ′ ′ ′ ×P x− dt [η(t ) − V (x(t ))] , t . (18.386) Mγ ta

We now use the correlation functions (18.323), ignore all powers higher than linear in ǫ, and find in the limit ǫ → 0 directly the equation (18.381).

18.12.3

Itˆ o’s Lemma

The procedure applied to Eq. (18.385) to derive the overdamped Fokker-Planck equation (18.381) is a special case of a more general method of stochastic calculus. Consider an arbitrary function f (x(t)) of the fluctuating variable x(t) at a time t. At a slightly later time t + ǫ, x has the value x(t + ǫ) = x(t) + ∆x(t), where ∆x(t) ≡

Z

t

t+ǫ

dt′ x(t ˙ ′ ),

(18.387)

and the function f (x(t + ǫ)) has the Taylor expansion f (x(t + ǫ)) = f (x(t)) + f ′ (x(t))∆x(t) 1 1 + f ′′ (x(t))[∆x(t)]2 + f (3) [∆x(t)]3 + . . . . 2 3!

(18.388)

We now assume that x(t) ˙ fluctuates with a white noise around its average hx(t)i ˙ according to a stochastic differential equation x(t) ˙ = hx(t)i ˙ + η(t),

(18.389)

where the noise is harmonic, has a zero average hη(t)i = 0, and the pair correlation function hη(t)η(t′ )i = σ 2 δ(t − t′ ).

(18.390)

This makes (18.389) a Wiener process [compare (18.332)], The average hx(t)i, ˙ is called the drift of the process. Then the linear term in ∆x(t) on the right-hand side of (18.388) has the averages h∆x(t)i =

Z

t

t+ǫ

dt′ hx(t ˙ ′ ) + η(t′ )i =

Z

t

t+ǫ

dt′ hx(t ˙ ′ )i,

(18.391)

1234


and the quadratic term 2

h[∆x(t)] i = =

t+ǫ

Z

t

dt1

t+ǫ

Z

t

dt1

t+ǫ

Z

t

Z

dt2 h[hx(t ˙ 1 )i + η(t1 )] [hx(t ˙ 2 )i + η(t2 )]i

t+ǫ

t

dt2 [hx(t ˙ 1 )ihx(t ˙ 2 )i + hη(t1 )η(t2 )i] .

Since hx(t)i ˙ is a smooth function of t, the first term is of the order ǫ2 . The second term, however, is of the order ǫ due to the δ-function in the correlation function (18.390). Thus we find h[∆x(t)]2 i = ǫσ 2 + O(ǫ2 ).

(18.392)

The average of the cubic term h[∆x(t)]3 i is given by the integral Z

t+ǫ t

=

=

dt1

Z

t

Z

t+ǫ

Z

t+ǫ

t

t

t+ǫ

dt1

dt1

dt2 t+ǫ

Z

t+ǫ

t

t+ǫ

dt3 h[hx(t ˙ 1 )i + η(t1 )] [hx(t ˙ 2 )i + η(t2 )] [hx(t ˙ 3 )i + η(t2 )]i

t

Z

t

Z

dt2

dt2

Z

t+ǫ

Z

t+ǫ

t

t

h

dt3 hx(t ˙ 1 )ihx(t ˙ 2 )ihx(t ˙ 3 )i + hx(t ˙ 1 )ihη(t2)η(t3 )i ˙ 3 )ihη(t1 )η(t2 )i + hx(t ˙ 2 )ihη(t1 )η(t3 )i + hx(t

dt3 hx(t ˙ 1 )ihx(t ˙ 2 )ihx(t ˙ 3 )i + 3ǫσ 2

Z

t+ǫ

t

i

dthx(t)i ˙ = O(ǫ2 ).(18.393)

The averages of the higher powers [∆x(t)]n are obviously at least of order ǫn/2 . For small ǫ we derive from this the simple formula ˙ hf(x(t))i = hf ′ (x(t))ihx(t)i ˙ +

σ 2 ′′ hf (x(t))i. 2

(18.394)

Note that in a time-sliced formulation, f (x(t))x(t) ˙ has the form f (xn )(xn+1 − xn )/ǫ, with independently fluctuating xn and xn+1 , so that we may treat xn and (xn+1 − xn )/ǫ as independent fluctuating variables. In the continuum limit x(t) and x(t) ˙ become independent. The important point noted by Itô is now that this result is not only true for the averages but also for the derivative f˙(x(t)) itself, i.e., f (x(t)) obeys the stochastic differential equation σ2 f˙(x(t)) = f ′ (x(t)) x(t) ˙ + f ′′ (x(t)), 2

(18.395)

which is known as Itô’s Lemma. To prove this we must show that the omitted fluctuations in the higher powers [∆x(t)]n for n ≥ 2 are of higher order in ǫ than the leading fluctuation of ∆x(t) which is of order ǫ. Indeed, let us denote the fluctuation part of [∆x(t)]n by zn (t). The lowest ones are The lowest ones are z1 (t) =

Z

t

t+ǫ

dt η(t),


1235


and z2 (t) ≡ [z2,1 (t) + z2,2 (t)] ,

where z2,1 (t) = 2

t+ǫ

Z

t

2 z2,2 (t) = [z1 (t)] (18.397) .

dt1 hx(t ˙ 1 )i z1 (t),

The fluctuations of z2,1 (t) are smaller than the leading ones of z1 (t) by a factor ǫ, so that they can be ignored in the limit ǫ → 0. The size of the fluctuations of z2,2 (t) are estimated by calculating its variance h[z2,2 (t)]2 i − hz2,2 (t)i2 . The first term is D

2

[z2,2 (t)]

E

=

Z

t+ǫ

t

dt1

Z

t+ǫ

t

dt2

t+ǫ

Z

dt3

t

Z

t+ǫ t

dt4 hη(t1 )η(t2 )η(t3 )η(t4 )i. (18.398)

According to Wick’s rule (3.302) for harmonic fluctuations, the expectation value on the right-hand sid is equal to the sum of three pair contractions hη(t1 )η(t2 )ihη(t3 )η(t4 )i + hη(t1 )η(t3 )ihη(t2 )η(t4 )ihη(t1 )η(t4 )ihη(t2 )η(t3 )i. (18.399) Inserting (18.390) and performing the integrals yields D

E

[z2,2 (t)]2 = 3σ 4 ǫ2

(18.400)

so that we obtain for the variance of z2,2 (t): h[z2,2 (t)]2 i − hz2,2 (t)i2 = 2σ 4 ǫ2 .

(18.401)

This must be compared with the variance of the leading fluctuations z1 (t) in (18.395): h[z1 (t)]2 i − hz1 (t)i2 =

Z

t+ǫ

dt1

t

Z

t+ǫ

t

dt2 hη(t1 )η(t2 )i = ǫσ 2 .

(18.402)

√ Thus the fluctuating part of z2,2 (t) is by a factor ǫ smaller than that of z1 (t), so that it can be ignored in the continuum limit ǫ → 0. Similar estimates can be derived for all higher fluctuations zn (t) in the Taylor expansion (18.388), thus proving Itô’s Lemma (18.395). For an exponential function, Itô’s Lemma yields d Px σ2 P 2 P x e = P x˙ + e . dt 2 !

(18.403)

This can be integrated to Rt

eP x = e

0

dt′ P x˙ P 2 σ2 t/2

e

.

(18.404)

The expectation value of this can also be formulated as a rule for calculating the expectation value of an exponential of an integral over a Gaussian noise variable with zero average:

P

e

Rt 0

dt′ η(t′ )

= eP

2

Rt 0

dt′

Rt 0

dt′′ hη(t′ )η(t′′ )i

= eP

2 σ 2 t/2

.

(18.405)

1236


This rule can also be derived directly from Wick’s rule (3.307). The right-hand side corresponds to the Debye-Waller factor introduced in solid-state physics to describe the reduction of the intensities of Bragg peaks by thermal fluctuations of the atomic positions [see Eq. (3.308)]. There is a simple mnemonic way of formalizing this derivation of Eq. (18.395) in a sloppy differential notation. We expand 1 f (x(t + dt)) = f (x(t) + xdt) ˙ = f (x(t)) + f ′ (x(t))x(t)dt ˙ + f ′′ (x(t))x˙ 2 (t)dt2 + . . . , 2 (18.406) and insert in the higher-order expansion terms x˙ = hxi ˙ + η(t) where hη(t)i = 0 and the expectation hη 2 (t)i dt = σ 2 ,

(18.407)

which is a sloppy infinitesimal form of the correct equation Z

t+ǫ t

′

′

dt hη(t )η(t)i =

Z

t+ǫ

t

dt′ σ 2 δ(t′ − t) = σ 2 .

(18.408)

The variable x˙ 2 (t)dt2 has an expectation value σ 2 dt and a variance h[x˙ 2 (t)dt2 ]2 − hx˙ 2 (t)dti2 i = 2σ 2 dt2 , so that x˙ 2 (t)dt2 in (18.406) can be replaced according to the rule (18.409) x˙ 2 (t)dt2 → σ 2 dt/2. All higher powers x˙ n (t)dtn can be estimated as follows x˙ n (t)dtn ≈ O(dtn/2 ).

(18.410)

and can be dropped from (18.406), so that we reobtain the properly derived relation (18.394). It must be realized that Itô’s Lemma is valid only in the limit ǫ → 0. For a discrete time axis with small but finite time intervals ∆t = ǫ, the fluctuations of √ zn (t) cannot strictly be ignored but are only suppressed by a small factor σ ǫ which is typically of the order of a few percent. The discrete version of Itô’s lemma expands the fluctuating difference ∆f (x(tn )) ≡ f (x(tn+1 )) as follows: ∆f (x(tn )) = f ′ (x(tn ))∆x(tn ) +

18.13

√ σ 2 ′′ f (x(tn )) + O(σ ∆t), 2

(18.411)

Supersymmetry

Recalling the origin (18.291) of the extra last term in the exponent of the path integral (18.295), this can be rewritten in a slightly more implicit but useful way as P0 (xb tb |xa ta ) ∝

Z



 V ′′ (x) Dx(t) Det ∂t + exp −  Mγ "

#

Z

tb

ta



" #2 1 V ′ (x)  dt x˙ + . 4D Mγ 

(18.412)


1237

18.13 Supersymmetry ′

In this expression, the time ordering of the velocity with respect to VM(x) is arbitrary. γ It may be quantum-mechanical (Stratonovich-like), but equally well retarded (Itôlike) or advanced, as long as it is used consistently in both the Lagrangian and the determinant. An interesting mathematical structure arises if one generates the determinant with the help of an auxiliary fermion field c(t) from a path integral over c(t): ′′

det [∂t + V (x(t))/Mγ] ∝

Z

DcD¯ c e−

R

dt¯ c(t)[M γ∂t +V ′′ (x(t))]c(t)

.

(18.413)

In quantum field theory, such auxiliary fermionic fields are referred to as ghost fields. With these we can rewrite the path integral (18.285) for the probability distribution as an ordinary path integral P (xb tb |xa ta ) =

Z

Dx

Z

DcD¯ c exp {−APS [x, c, c¯]} ,

(18.414)

where APS is the Euclidean action APS

1 = 2DM 2 γ 2

Z

tb

ta

1 2 dt [Mγ x˙ + V ′ (x)] + c¯(t) [Mγ∂t + V ′′ (x(t))] c(t) , (18.415) 2

first written down by Parisi and Sourlas [20] and by McKane [21]. This action has a particular property. If we denote the expression in the first brackets by Ux ≡ Mγ∂t x + V ′ (x),

(18.416)

the operator between the Grassmann variables in (18.415) is simply the functional derivative of Ux : Uxy ≡

δUx = Mγ∂t + V ′′ (x). δy

(18.417)

Thus we may write APS =

1 2D

Z

tb

ta

dt

1 2 U + c¯(t) Uxy c(t) , 2 x

(18.418)

where Uxy c(t) is the usual short notation for the functional matrix multiplication dt′ Uxy (t, t′ )c(t′ ). The relation between the two terms makes this action supersymmetric. It is invariant under transformations which mix the Fermi and Bose degrees of freedom. Denoting by ε and ε¯ a small anticommuting Grassmann variable and its conjugate (see Section 7.10), the action is invariant under the field transformations

R

δx(t) = ε¯c(t) + c¯(t)ε, δ¯ c(t) = −¯ εUx , δc(t) = Ux ε.

(18.419) (18.420) (18.421)

The invariance follows immediately after observing that δUx = ε¯Uxy c(t) + c¯(t)Uxy ε.

(18.422)

1238


Formally, a similar construction is also possible for a particle with inertia in the path integral (18.232), which is an ordinary path integral involving the Lagrangian (18.283). Here we can write P (xb tb |xa ta ) = N

Z

1 Dx J[x] exp − 2w

Z

tb

ta

′

2

dt [M x¨ + Mγ x˙ + V (x)]

, (18.423)

where J[x] abbreviates the determinant J[x] = det [M∂t2 + Mγ∂t + V ′′ (x(t))],

(18.424)

which is known from formula (18.270). The path integral (18.423) is valid for any ordering of the velocity term, as long as it is the same in the exponent and the functional determinant. We may now express the functional determinant as a path integral over fermionic ghost fields J[x] =

det [M∂t2

′′

+ Mγ∂t + V (x(t))] ∝

Z

DcD¯ c e−

R

dt c¯(t)[M ∂t2 +M γ∂t +V ′′ (x(t))]c(t)

,

(18.425) and rewrite the probability distribution P (xb tb |xa ta ) as an ordinary path integral P (xb tb |xa ta ) ∝

Z

Dx

Z

DcD¯ c exp{−AKS [x, c, c¯]},

(18.426)

where A[x, c, c¯] is the Euclidean action KS

A [x, c, c¯] ≡

Z

tb

ta

h i 1 ′ (x)]2 + c¯(t) M∂t2 +Mγ∂t +V ′′ (x(t)) c(t) . [M x¨ +Mγ x+V ˙ dt 2w (18.427)

This formal expression contains subtleties arising from the boundary conditions when calculating the Jacobian (18.425) from the functional integral on the righthand side. It is necessary to factorize the second-order operator in the functional determinant and express the determinant of each first-order factor as a functional integral over Grassmann variables as in (18.413). At the end, the action is again supersymmetric, but there are twice as many auxiliary Fermi fields [22]. As a check of this formula, we may let the coupling to the thermal reservoir go to zero, γ → 0. Then the first factor in (18.426), Z

exp −

tb

ta

1 ′ dt [M x¨ +Mγ x+V ˙ (x)]2 2w

becomes proportional to a δ-functional δ[M x¨ +V ′ (x)]. The argument is simply the functional derivative of the original action of the quantum system in (18.191), so that we obtain in the limit δ[δA/δx]. The functional matrix between the Grassmann H. Kleinert, PATH INTEGRALS

1239

18.13 Supersymmetry

fields in (18.426), on the other hand, reduces to δ 2 A/δx(t)δx(t′ ), and we arrive at the path integral P (xb tb |xa ta )

∝

γ→0

Z Z

×

Dx δ[δA/δx]

DcD¯ c exp −

Z

tb

ta

dt

Z

tb

ta

dt′ c¯(t)δ 2 A/∂x(t)∂x(t′ )c(t′ ) .(18.428)

Performing the integral over the Grassmann variables yields P (xb tb |xa ta )

∝

γ→0

Z

h

i

Dx δ[δA/δx] Det δ 2 A/∂x(t)∂x(t′ ) .

(18.429)

The δ-functional selects from all paths only those which obey the Euler-Lagrange equations of motion. With the help of the functional identity δ[M x¨ + V ′ (x)] = δ[x − xcl ] × Det −1 [M x¨ + V ′′ (x)],

(18.430)

which generalizes identity δ(f (x)) = δ(x)/f ′ (x) if f (0) = 0, the above path integral becomes simply P (xb tb |xa ta )

∝

γ→0

Z

Dx δ[x − xcl ],

(18.431)

which is the correct probability distribution of classical physics. Note that by a Fourier decomposition of the δ-functional (18.428) we obtain the alternative path integral representation of classical physics P (xb tb |xa ta ) ∝

γ→0

Z

−

DxDλDcD¯ ce

R tb ta

dtδA/δx(t)λ(t)−

R tb R tb ta

dt

ta

dt′ c¯(t)δ2A/δx(t)δx(t′ )c(t′ )

.(18.432)

This is supersymmetric under the transformations δx = ε¯c ,

δc = 0 ,

δ¯ c = −¯ ελ ,

δλ = 0 ,

(18.433)

as observed by Gozzi [23]. There exists a compact way of rewriting the action using superfields. We define a three-dimensional superspace consisting of time and two auxiliary Grassmann ¯ Then we define a superfield variables θ and θ. ¯ ¯ ¯ X(t) ≡ x(t) + iθc(t) − iθc(t) − θλ(t).

(18.434)

We now consider the superaction Asuper ≡

Z

¯ dθdθA[X] ≡

Z

¯ ¯ − iθ¯ ¯ dθdθA[x + iθc c − θθλ]

(18.435)

and expand the action into a functional Taylor series: Z

δA ¯ 1 ¯ ∂2A ¯ ¯ ¯ ¯ ¯ dθdθ A[x] + (iθc − iθ¯ c − θθλ) + (iθc − iθ¯ c − θθλ) (iθc − iθ¯ cc − θθλ) . δx 2 δxδx (

)

1240


Due to the nilpotency (7.375) of the Grassmann variables, the expansion stops after the second term. Recalling now the integration rules (7.378) and (7.379), this becomes 1 ∂2A δA λ + c¯ c, δx 2 δxδx which is precisely the short-hand functional notation for the negative exponent in the path integral (18.432).

18.14

Stochastic Quantum Liouville Equation

At lower temperatures, where quantum fluctuations become important, the forward– backward path integral (18.230) does not allow us to derive a Schrödinger-like differential equation for the probability distribution P (x v t|xa va tt ). To see the obstacle, we go over to the canonical representation of (18.230): 2

|(xb tb |xa ta )| =

Z

Dx Dy

Z

Dp Dpy i exp 2π 2π h ¯

Z

tb

ta

dt [px˙ + py y˙ − HT ] , (18.436)

where HT =

1 w ˆ Ohm py px + γpy y + V (x + y/2) − V (x − y/2) − i y K y M 2¯ h

(18.437)

plays the role of a temperature-dependent quasi-Hamiltonian for an Ohmic system associated with the Lagrangian of the forward–backward path integral (18.230). The ˆ Ohm y(t) abbreviates the product of the functional matrix K Ohm (t, t′ ) with notation K ˆ Ohm y(t) ≡ R dt′ K Ohm (t, t′ )y(t′). Hence is HT the functional vector y(t′ ) defined by K a nonlocal object (in the temporal sense), and this is the reason for calling it quasiHamiltonian. It is useful to omit y-integrations at the endpoints in the path integral (18.436), and set up a path integral representation for the product of amplitudes U(xb yb tb |xa ya ta ) ≡ (xb + yb /2 tb |xa + ya /2 ta )(xb − yb /2 tb |xa − ya /2 ta )∗ . (18.438) Given some initial density matrix ρ(x+ , x− ; t) = ρ(x + y/2, x − y/2; t) at time t = ta , which may actually be in equilibrium and time-independent, as in Eq. (2.359), the functional matrix U(xb yb tb |xa ya ta ) allows us to calculate ρ(x+ , x− ; t) at any time by the time evolution equation Z

ρ(x + y/2, x − y/2; t) = dxa dya U(x y t|xa ya ta ) ρ(xa + ya /2, xa − ya /2; ta ). (18.439) Recall that the Fourier transform of ρ(x + y/2, x − y/2; t) with respect to y is the Wigner function (1.224). When considering the change of U(x y t|xa ya ta ) over a small time interval ǫ, the momentum variables p and py have the same effect as differential operators −i∂xb H. Kleinert, PATH INTEGRALS

18.14 Stochastic Quantum Liouville Equation

1241

and −i∂yb , respectively. The last term in HT , however, is nonlocal in time, thus preventing a derivation of a Schrödinger-like differential equation. The locality problem can be removed by introducing a noise variable η(t) with the correlation function determined by (18.315): hη(t)η(t′ )iT =

w Ohm −1 [K ] (t, t′ ). 2

(18.440)

Then we can define a temporally local η-dependent Hamiltonian operator ˆ η ≡ 1 (ˆ px + γy) pˆy + V (x + y/2) − V (x − y/2) − yη, H M

(18.441)

which governs the evolution of η-dependent versions of the amplitude products (18.438) via the stochastic Schrödinger equation ˆ η Uη (x y t|xa ya ta ). i¯ h∂t Uη (x y t|xa ya ta ) = H

(18.442)

The same equation is obeyed by the noise-dependent density matrix ρη (x, y; t). Averaging these equation over η with the distribution (18.315) yields for ya = yb = 0 the same probability distribution as the forward–backward path integral (18.230): |(xb tb |xa ta )|2 = U(xb 0 tb |xa 0 ta ) ≡ hU(xb 0 tb |xa ya ta )iη .

(18.443)

At high temperatures, the noise averaged stochastic Schrödinger equation (18.442) takes the form ˆ¯ U(x y t|x y t ), i¯ h∂t U(x y t|xa ya ta ) = H T a a a

(18.444)

ˆ¯ is now a local (in the temporal sense) where H ˆ¯ ≡ 1 pˆ pˆ + γy pˆ + V (x + y/2) − V (x − y/2) − i w y 2, H T y x y M 2¯ h

(18.445)

arising from the Hamiltonian (18.437) in the high-temperature limit K Ohm → 1 [recall (18.223)]. In terms of the separate path positions x± = x ± y/2 where px = ∂+ + ∂− and py = (∂+ − ∂− )/2, this takes the more familiar form [24] w ˆ¯ ≡ 1 pˆ2 − pˆ2 + V (x ) − V (x ) + γ (x − x )(ˆ H ˆ− ) − i (x+ − x− )2 . T + − + − p+ − p + − 2M 2 2¯ h (18.446) The last term is often written as −i¯ hΛ(x+ −x− )2 , where Λ is the so-called decoherence rate per square distance

Λ≡

w MγkB T . 2 = 2¯ h h ¯2

(18.447)

It is composed of the damping rate γ and the squared thermal length (2.345): Λ=

2πγ , le2 (¯ hβ)

(18.448)

1242


and controls the decay of interference peaks [25]. Note that the order of the operators in the mixed term of the form y pˆy in Eq. (18.445) is opposite to the mixed term −iˆ pv v in the differential operator (18.239) of the Fokker-Planck equation. This order is necessary to guarantee the conservation of probability. Indeed, multiplying the time evolution equation (18.444) by δ(y), and integrating both sides over x and y, the left-hand side vanishes. The correctness of this order can be verified by calculating the fluctuation determinant of the path integral for the product of amplitudes (18.438) in the Lagrangian form, which looks just like (18.230), except that the difference between forward and backward trajectories y(t) = x+ (t) − x− (t) is nonzero at the endpoints. For the fluctuation which vanish at the endpoints, this is irrelevant. As explained before, the order is a short-time issue, and we can take tb − ta → ∞. Moreover, since the order is independent of the potential, we may consider only the free case V (x ± y/2) ≡ 0. The relevant fluctuation determinant was calculated in formula (18.250). In the Hamiltonian operator (18.445), this implies an additional energy −iγ/2 with respect to the symmetrically ordered term γ{y, pˆy }/2, which brings it to γy pˆy , and thus the order in (18.446).

18.15

Master Equation for Time Evolution

In the high-temperature limit, the Hamiltonian (18.446) becomes local. Then the evolution equation (18.439) for the density matrix ρ(x+a , x−a ; ta ) can be converted into an operator equation ˆ¯ ρ(x , x ; t ), i¯ h∂t ρ(x+ , x− ; ta ) = H T + − a

(18.449)

ˆ¯ is the operator version of the temperature-dependent Hamiltonian where H T (18.446). Such an equation does not exist at low temperatures, due to the nonlocality of the last term in (18.437). Then one cannot avoid solving the stochastic Schrödinger equation (18.442) with the subsequent averaging (18.443). For moderately high temperatures, however, a Hamiltonian formalism can still be set up, although it requires solving a recursion relation. For this purpose we write down the quasi-Hamiltonian in D dimensions ˆ¯ H T ≡

Mγ 1 2 ˆ˙ + + x ˆ˙ − )R ˆ+ − p ˆ 2− + V (x+ ) − V (x− ) + ˆ − )(x p (ˆ x+ − x 2M 2 w Ohm ˆ ˆ − )K ˆ − ), − i (ˆ x+ − x (ˆ x+ − x (18.450) 2¯ h

where the Fourier transform of K Ohm (t, t′ ) is expanded in powers of ω ′ [recall (18.223)] K

Ohm

1 (ω ) = 1 + 3 ′

h ¯ ω′ 2kB T

!2

+ ... .

(18.451)


1243

18.15 Master Equation for Time Evolution

In this way we find for the last term the locally looking high-temperature expansion ˆ Ohm (ˆ ˆ − )K ˆ − ) = −i(ˆ ˆ − )2 + i −i(ˆ x+ − x x+ − x x+ − x

w¯ h ˆ˙ + − x ˆ˙ − )2 + . . . . (18.452) (x 2 24(kB T )

ˆ˙ is defined implicitly as an The expression is not really local, since the operator x abbreviations for the commutator ˆ¯ , x ˆ˙ ≡ i [H x (18.453) T ˆ ]. h ¯ If the expansion (18.452) is carried further, higher derivatives of x arise, which are all defined recursively: ˆ¯ , x], ˆ ˆ¨ ≡ i [H x T ˙ h ¯

ˆ¯ , x ˆ ≡ i [H ˆ ˙¨ x T ¨ ], . . . . h ¯

(18.454)

Thus Eq. (18.450) with the expansion (18.452) is a recursive equation for the Hamilˆ¯ . For small γ (and thus w = 2Mγk T ), the recursion can be tonian operator H T B ˆ ˆ /M into Eq. (18.455). solved iteratively, in the first step by inserting x˙ ≈ p It is useful to re-express (18.449) in the Dirac operator form where the density P ˆ 2 /2M + Vˆ matrix has a bra–ket representation ρˆ(t) = mn ρmn (t)|mihn|. Denoting p ˆ we obtain with the expansion (18.452) the local master equation: in (18.450) by H, ˆ¯ ρˆ ≡ [H, ˆ ρˆ] + Mγ x ˆ˙ ρ − ρˆxˆ ˆ˙ x + x ˆ˙ − x ˆ˙ ρˆ x ˆ xˆ ˆ ρˆ x ˆ i¯ h∂t ρˆ = H T 2 iw iw¯ h2 ˆ ˆ ˙ [x, ˙ ρˆ]] + . . . . [ˆ x, [ˆ x, ρˆ]] − − [x, 2¯ h 24(kB T )2

(18.455)

The validity of the above iterative procedure is most easily proved in the timesliced path integral. The final slice of infinitesimal width ǫ reads U(x+b , x−b , tb |x+a , x−a , tb − ǫ) Z Z dp+ (tb ) dp− (tb ) i {p+ (tb )[x+ (tb )−x+ (tb −ǫ)]−p− x˙ − −H¯ T (tb )} = e h¯ . (18.456) (2π)3 (2π)3 ¯ T (t). When Consider now a term of the generic form F˙+ (x+ (t))F− (x− (t)) in H differentiating U(x+b , x−b , tb |x+a , x−a , tb − ǫ) with respect to the final time tb , the ¯ T (tb ). At tb , the term F˙+ (x+ (t))F− (x− (t)) in H ¯ T (t) integrand receives a factor −H has the explicit form ǫ−1 [F+ (x+ (tb )) − F+ (x+ (tb − ǫ))] F− (x− (tb )). It can be taken out of the integral, yielding ǫ−1 [F+ (x+ (tb ))U − UF+ (x+ (tb − ǫ))] F− (x− (tb )).

(18.457)

ˆ¯ /¯ In operator language, the amplitude U is associated with Uˆ ≈ 1 − iǫH T h, such the ˆ ˙ ¯ term F+ (x+ (t))F− (x− (t)) in H T yields a Schrödinger operator i i h ˆ¯ ˆ H T , F+ (x+ ) F− (x− ) h ¯

(18.458)

1244


in the time evolution equation (18.455). ¨ we have to split off the last two time For functions of the second derivative x slices in (18.456) and convert the two intermediate integrals over x into operator ˆ¯ with x ˆ , and expressions, which obviously leads to the repeated commutator of H T so on. The operator order in the terms in the parentheses of Eq. (18.455) is fixed by the retardation of x˙ ± with respect to x± in (18.446). This implies that the associated ˆ˙ ˆ ± , thus acting operator x(t) has a time argument which lies slightly before that of x ˆ ˆ . This puts x(t) ˙ ˆ , i.e., next to ρˆ. On the right-hand upon ρˆ before x to the right of x ˆ˙ must lie to the left side of ρˆ, the time runs in the opposite direction such that x ˆ , again next to ρˆ. In this way we obtain an operator order which ensures that of x Eq. (18.455) conserves the total probability. This property and the positivity of ρˆ are actually guaranteed by the observation, that the master equation (18.455) can be written in the Lindblad form [26] 2 X 1 ˆ ˆ† 1 ˆ ˆ† i ˆ ˆ † ˆL ˆn , ρˆ] − Ln Ln ρˆ + ρˆL ∂t ρˆ = − [H, n Ln − Ln ρ h ¯ 2 2 n=1

with the two Lindblad operators [27] √ ! √ w 3w h ¯ ˆ1 ≡ ˆ2 ≡ ˆ˙ . ˆ−i ˆ, L x L x x 2¯ h 2¯ h 3kB T

(18.459)

(18.460)

ˆ˙ ρ from being a ˆ xˆ Note that the operator order in Eq. (18.455) prevents the term x pure divergence. If we rewrite it as a sum of a commutator and an anticommutator, ˆ˙ ˆ˙ [ˆ x, x]/2 + {ˆ x, x}/2, then the latter term is a pure divergence, and we can think of the first two γ-terms in (18.455) as being due to an additional anti-Hermitian term ˆ the dissipation operator in the Hamiltonian operator H, ˆ γ = γM 1 [ˆ ˆ˙ H x, x]. 4

18.16

(18.461)

Relation to Quantum Langevin Equation

The stochastic Liouville equation (18.442) can also be derived from an operator version of the Langevin equation (18.311), the so-called Quantum Langevin equation ¨ˆ(t) + Mγ xˆ˙ (t) + V ′ (ˆ Mx x(t)) = ηˆ(t),

(18.462)

where ηˆ(t) is an operator noise variable with the commutation rule [ˆ ηt , ηˆt′ ] = w

i¯ h ∂t δ(t − t′ ), kB T

(18.463)

and the correlation function [28] 1 h[ˆ ηt , ηˆt′ ]+ iηˆ = w K(t, t′ ). 2


18.17 Electromagnetic Dissipation and Decoherence

1245

The commutator (18.463) and the correlation function (18.464) are related to each other as required by the fluctuation-dissipation theorem: By omitting the factor coth(¯ hω/2kB T ) in Eq. (18.223), the Fourier integral (18.221) for K(t, t′ ) reduces to (¯ h/2kB T )∂t δ(t − t′ ). A comparison with the general spectral representation (18.53) shows that the expectation value (18.464) has the spectral function ρb (ω ′ ) = 2Mγ¯ hω ′ .

(18.465)

By inserting this into the spectral representation (18.53) we obtain the right-hand side of the commutator equation (18.463). A noise variable with the properties (18.463) and (18.464) can be constructed explicitly by superimposing quantized oscillator velocities of frequencies ω as follows: s

M¯ hγ ηˆ(t) = −i π

Z

0

∞

√ ′ ′ dΩ′ Ω′ [aΩ′ e−iω t − a†ω′ eiω t ].

(18.466)

It is worth pointing out that there exists a direct derivation of the quantum Langevin equation (18.462), whose noise operator ηˆ(t) satisfies the commutator and fluctuation properties (18.463) and (18.464), from Kubo’s stochastic Liouville equation, and thus from the forward–backward path integral (18.230) [29].

18.17

Electromagnetic Dissipation and Decoherence

There exists a thermal bath of particular importance: atoms are usually observed at a finite temperature where they interact with a grand-canonical ensemble of photons in thermal equilibrium. This interaction will broaden the natural line width of atomic levels even if all major mechanisms for the broadening are removed. To study this situation, let us set up a forward–backward path integral description for a bath of photons, and derive from it a master equation for the density matrix which describes electromagnetic dissipation and decoherence. As an application, we shall calculate the Wigner-Weisskopf formula for the natural line width of an atomic state at zero temperature, find the finite-temperature effects, and calculate the Lamb shift between atomic s- and p-wave states of principal quantum number n = 2 with the term notation 2S1/2 and 2P1/2 . The master equation may eventually have applications to dilute interstellar gases or to few-particle systems in cavities.

18.17.1

Forward–Backward Path Integral

With the application to atomic physics in mind, we shall consider a threedimensional quantum system described by a time-dependent quantum-mechanical density matrix ρ(x+ , x− ; t). In contrast to Eq. (18.439), we use here the forward and backward variables as arguments, and write the time evolution equation as ρ(x+b , x−a ; tb ) =

Z

dx+a dx−a U(x+b , x−b , tb |x+a , x−a , ta )ρ(x+a , x−a ; ta ). (18.467)

1246


In an external electromagnetic vector potential A(x, t), the time-evolution kernel is determined by a forward–backward path integral of the type (18.192), in which the forward and backward paths start at different initial and final points x+a , x−a and x+b , x−b , respectively: ∗

U(x+b , x−b , tb |x+a , x−a , ta ) ≡ (x+b , tb |x+a , ta )(x−b , tb |x−a , ta ) = i × exp h ¯

Z

tb

ta

Z

Dx+ Dx−

M 2 e e 2 x+ − x− − V (x+ ) + V (x− ) − x˙ + A(x+ , t) + x˙ − A(x− , t) . 2 c c (18.468)

The vector potential A(x, t) is a superposition of oscillators Xk (t) of frequency Ωk = c|k| in a volume V : A(x, t) =

X

ck (x)Xk (t),

k

ck = √

eikx , 2Ωk V

X k

=

Z

d3 kV . (2π)3

(18.469)

At a finite temperature T , these oscillators are assumed to be in equilibrium, where we shall write their time-ordered correlation functions as ij tr ′ ′ ′ ⊥ ij ′ ˆ î ˆj ′ Gij kk′ (t, t ) = hT Xk (t), X−k′ (t )i = δkk′ GΩk (t, t ) ≡ δkk Pk GΩk (t, t ). (18.470)

The transverse projection matrix is the result of the sum over the transverse polarization vectors of the photons: Pk⊥ ij =

X

h=±

ǫi (k, h)ǫj ∗ (k, h) = (δ ij − k i k j /k2 ).

(18.471)

The function GΩk (t, t′ ) on the right-hand side of (18.470) is the Green function (18.185) of a single oscillator of frequency Ωk . It is decomposed into real and imaginary parts, defining AΩk (t, t′ ) and CΩk (t, t′ ) as in (18.185), which are commutator and anticommutator functions of the oscillator at temperature T : ˆ ˆ ′ )]iT and AΩ (t, t′ ) ≡ h[X(t), ˆ ˆ ′ )]iT , respectively. CΩk (t, t′ ) ≡ h[X(t), X(t X(t k The thermal average of the evolution kernel (18.468) is then given by the forward– backward path integral U(x+b , x−b , tb |x+a , x−a , ta ) = × exp

i h ¯

Z

tb

ta

dt

Z

Dx+ (t)

Z

Dx− (t)

M 2 i (x˙ + − x˙ 2− ) − (V (x+ ) − V (x− )) + AFV [x+ , x− ] , (18.472) 2 h ¯

FV

where exp{iA [x+ , x− ]/¯ h} is the Feynman-Vernon influence functional defined in Eq. (18.200). The influence action AFV [x+ , x− ] is the sum of a dissipative and a FV fluctuating part AFV D [x+ , x− ] and AF [x+ , x− ], whose explicit forms are now ie2 dt dt′ Θ(t − t′ ) 2 2¯ h c h × x˙ + (t)Cb (x+ t, x′+ t′ )x˙ + (t′ ) − x˙ + (t)Cb (x+ t, x′− t′ )x˙ − (t′ )

AFV D [x+ , x− ] =

Z

Z

i

− x˙ − (t)Cb (x− t, x′+ t′ )x˙ + (t′ ) + x˙ − (t)Cb (x− t, x′− t′ )x˙ − (t′ ) , (18.473) H. Kleinert, PATH INTEGRALS

1247


and Z ie2 Z dt dt′ Θ(t − t′ ) 2 2¯ h c h × x˙ + (t)Ab (x+ t, x′+ t′ )x˙ + (t′ ) + x˙ + (t)Ab (x+ t, x′− t′ )x˙ − (t′ )

AFV F [x+ , x− ] =

i

+ x˙ − (t)Ab (x− t, x′+ t′ )x˙ + (t′ ) + x˙ − (t)Ab (x− t, x′− t′ )x˙ − (t′ ) ,(18.474)

with Cb (x− t, x′− t′ ) and Ab (x− t, x′− t′ ) collecting the 3 × 3 commutator and anticommutator functions of the bath of photons. They are sums of correlation functions over the bath of the oscillators of frequency Ωk , each contributing with a ′ weight ck (x)c− k(x′ ) = eik(x−x ) /2Ωk V . Thus we may write, generalizing (18.197) and (18.198), Cbij (x t, x′ t′ ) =

X

D

E

ˆ i (t), X ˆ j (t′ )] c−k (x)ck (x′ ) [X −k k

k

T

dω ′d3 k ′ = −i¯ h ρk (ω ′)Pk⊥ ij eik(x−x ) sin ω ′(t − t′ ), 4 (2π) Dn oE X ij ′ ′ ˆ j (t′ ) ˆ i (t), X c−k (x)ck (x′ ) X Ab (x t, x t ) = −k k Z

T

k

= h ¯

Z

(18.475)

dω ′d3 k h ¯ ω ′ ik(x−x′ ) ′ ⊥ ij e ρ (ω )P coth cos ω ′(t − t′ ),(18.476) k k (2π)4 2kB T

where ρk (ω ′ ) is the spectral density contributed by the oscillator of momentum k: ρk (ω ′ ) ≡

2π [δ(ω ′ − Ωk ) − δ(ω ′ + Ωk )]. 2Ωk

(18.477)

At zero temperature, we recognize in (18.475) and (18.476) twice the imaginary and real parts of the Feynman propagator of a massless particle for t > t′ , which in four-vector notation with k = (ω/c, k) and x = (ct, x) reads 1 d4 k ik(x−x′ ) i¯ h G(x, x ) = [A(x, x′ ) + C(x, x′ )] = e 4 2 2 (2π) k + iη Z 3 dω d k ic¯ h ′ ′ = e−i[ω(t−t )−k(x−x )] , 2 4 2 (2π) ω − Ωk + iη Z

′

(18.478)

where η is an infinitesimally small number > 0. We shall now focus attention upon systems which are so small that the effects of retardation can be neglected. Then we can ignore the x-dependence in (18.476) and (18.477) and find h ¯ 2 ij δ ∂t δ(t − t′ ). (18.479) 2πc 3 Inserting this into (18.473) and integrating by parts, we obtain two contributions. The first is a diverging term Cbij (x t, x′ t′ ) ≈ Cbij (t, t′ ) = i

∆M ∆Aloc [x+ , x− ] = 2

Z

tb ta

dt (x˙ 2+ − x˙ 2− )(t),

(18.480)

1248


where ∆M ≡ −

e2 c2

Z

dω ′d3 k σk (ω ′) ij tr e2 δ = − (2π)4 ω ′ kk 3π 2 c3

Z

0

∞

dk

(18.481)

diverges linearly. This simply renormalizes the kinetic terms in the path integral (18.472), renormalizing them to i h ¯

Z

tb

ta

dt

Mren 2 x˙ + − x˙ 2− . 2

(18.482)

By identifying M with Mren this renormalization may be ignored. The second term has the form [compare (18.205)] AFV D [x+ , x− ]

M Z tb ¨ − )R (t), = −γ dt (x˙ + − x˙ − )(t)(¨ x+ + x 2 ta

(18.483)

with the friction constant of the photon bath encountered before in Eq. (3.441): γ≡

e2 2 α = , 3 6πc M 3 ωM

(18.484)

hc ≈ 1/137 is the fine-structure constant (1.502) and ωM ≡ Mc2 /¯ h the where α ≡ e2 /¯ Compton frequency associated with the mass M. Note once more that in contrast to the usual friction constant γ in Section 3.13, this has the dimension 1/frequency. As discussed in Section 18.8, the retardation enforced by the Heaviside function in the exponent of (18.473) removes the left-hand half of the δ-function [see (18.214)]. It ensures the causality of the dissipation forces, which has been shown in Section 18.9.2 to be crucial for producing a probability conserving time evolution of the probability distribution [13]. The superscript R in (18.483) shifts the accelera¨ − )(t) slightly towards an earlier time with respect to the velocity factor tion (¨ x+ + x (x˙ + − x˙ − )(t). We now turn to the anticommutator function. Inserting (18.477) and the friction constant γ from (18.484), it becomes e2 Ab (x t, x′ t′ ) ≈ 2γkB T K Ohm (t, t′ ), c2

(18.485)

as in Eq. (18.216), with the same function K Ohm (t, t′ ) as in Eq. (18.223), whose high-temperature expansion starts out as in Eq. (18.451). In terms of the function K Ohm (t, t′ ), the fluctuation part of the influence functional in (18.474), (18.473), (18.472) becomes [compare (18.227)] AFV F [x+ , x− ] = i

w Z tb Z tb ′ dt dt (x˙ + − x˙ − )(t) K Ohm (t, t′ ) (x˙ + − x˙ − )(t′ ). (18.486) 2¯ h ta ta

Here we have used the symmetry of the function K Ohm (t, t′ ) to remove the Heaviside function Θ(t − t′ ) from the integrand, extending the range of t′ -integration to the entire interval (ta , tb ). We also have introduced the constant w ≡ 2MkB T γ,


1249


for brevity. In the high-temperature limit, the time evolution amplitude for the density matrix is given by the path integral U(x+b , x−b , tb |x+a , x−a , ta ) =

Dx+ (t)

Z

Dx− (t)

M 2 (x˙ − x˙ 2− ) − (V (x+ ) − V (x− )) (18.488) 2 + ta Z tb Z i w tb R 2 ¨−) − 2 × exp − Mγ dt (x˙ + − x˙ − )(¨ x+ + x dt (x˙ + − x˙ − ) , 2¯ h ta 2¯ h ta

× exp

i h ¯

Z

Z

tb

dt

where the last term is now local since K Ohm (t, t′ ) → δ(t − t′ ). In this limit (as in the classical limit h ¯ → 0), this term squeezes the forward and backward paths together. The density matrix (18.488) becomes diagonal. The γ-term, however, remains and describes classical radiation damping. At moderately high temperature, we should include also the first correction term in (18.451) which adds to the exponent an additional term Z tb w ¨ − )2 . dt (¨ x+ − x − 24(kB T )2 ta

(18.489)

The extended expression is the desired closed-time path integral of a particle in contact with a thermal reservoir.

18.17.2

Master Equation for Time Evolution in Photon Bath

It is possible to derive a master equation for the evolution of the density matrix ρ(x+a , x−a ; ta ) analogous to Eq. (18.455) for a quantum particle in a photon bath. Since the dissipative and fluctuating parts of the influence functional in Eq. (18.480) and (18.486) coincide with the corresponding terms in (18.230), except for an extra dot on top of the coordinates, the associated temperature-dependent Hamiltonian operator is directly obtained from (18.450) with the expansion (18.452) by adding the extra dots: In the high-temperature limit we obtain Mγ ˆ ˆ≡ 1 p ˆ˙ − )(x ˆ¨ + + x ˆ¨ − )R − i w (x ˆ˙ + − x ˆ˙ − )2 , ˆ 2− + V (x+ ) − V (x− ) + ˆ 2+ − p H (x˙ + − x 2M 2 2¯ h (18.490) extended at moderately high temperatures by the Hamiltonian corresponding to (18.489):

ˆ¯ ≡ i ∆H T

w¯ h ˆ¨ + − x ˆ¨ − )2 . (x 24(kB T )2

(18.491)

The master equation corresponding to the Ohmic equation (18.455) reads now ˆ¯ ρˆ ≡ [H, ˆ ρˆ] + Mγ x ˆ˙ x ˆ¨ ρˆ − ρˆx ˆ¨ x ˆ˙ + x ˆ˙ ρˆ x ˆ¨ − x ˆ¨ ρˆ x ˆ˙ i¯ h∂t ρˆ = H T 2 iw ˆ ˆ iw¯ h2 ˆ ˆ ˙ [x, ˙ ρˆ]] − ¨ , [x ¨ , ρˆ]]. − [x, [x 2¯ h 24(kB T )2

(18.492)

1250


The conservation of total probability and the positivity of ρˆ are ensured by the observation, that Eq. (18.492) can be written in the Lindblad form 2 X i ˆ 1 ˆ ˆ† 1 ˆ ˆ† ˆ † ˆL ˆn , ∂t ρˆ = − [H, ρˆ] − Ln Ln ρˆ + ρˆL n Ln − Ln ρ h ¯ 2 2 n=1

with the two Lindblad operators √ ! √ w 3w h ¯ ˆ1 ≡ ˆ˙ − i ˆ˙ ˆ2 ≡ ˆ¨ . L x x, L x 2¯ h 2¯ h 3kB T

(18.493)

(18.494)

As noted in the discussion of Eq. (18.455), the operator order in (18.492) prevents ˆ˙ x ˆ¨ ρˆ from being a pure divergence. By rewriting it as a sum of a commutator the term x ˆ˙ x ˆ¨ ]/2 + {x, ˆ˙ x ˆ¨ }/2, the latter term is a pure divergence, and an anticommutator, [x, and we can think of the first two γ-terms in (18.492) as being due to an additional anti-Hermitian dissipation operator ˆ γ = γM 1 [x, ˆ˙ x ˆ¨ ]. H 4

(18.495)

ˆ p ˆ˙ ± = p ˆ ] = 0, one has x ˆ ± /M to all For a free particle with V (x) ≡ 0 and [H, orders in γ, such that the time evolution equation (18.492) becomes ˆ ρˆ] − i¯ h∂t ρˆ = [H,

iw [ˆ p, [ˆ p, ρˆ]]. 2M 2 h ¯

(18.496)

In the momentum representation of the density matrix ρˆ = pp′ ρpp′ |pihp′ |, the last term simplifies to −iΓ ≡ −iw(p − p′ )2 /2M 2 h ¯ 2 multiplying ρˆ, which shows that a free particle does not dissipate energy by radiation, and that the off-diagonal matrix elements decay with the rate Γ. For small e2 , the implicit equation Eq. (18.490) with the expansion term (18.491) ˆ˙ ≈ p ˆ /M and can be solved approximately in a single iteration step, inserting x ˆ ¨ ≈ −∇V /M. x P

18.17.3

Line Width

Let us apply the master equation (18.492) to atoms, where V (x) is the Coulomb potential, assuming it to be initially in an eigenstate |ii of H, with a density matrix ρˆ(0) = |iihi|. Since atoms decay rather slowly, we may treat the γ-term in (18.492) perturbatively. It leads to a time derivative of the density matrix ∂t hi|ˆ ρ(t)|ii = −

γ γ X ˆ p ˆ] p ˆ ρˆ(0)|ii = hi|[H, ωif hi|p|f ihf |p|ii h ¯M M f 6=i

= −Mγ

X f

3 ωif |xf i |2 ,

(18.497)

where h ¯ ωif ≡ Ei − Ef , and xf i ≡ hf |x|ii are the matrix elements of the dipole operator. H. Kleinert, PATH INTEGRALS


1251

An extra width comes from the last two terms in (18.492): w w 2 hi|p˙ 2 |ii 2 hi|p |ii − 2 2 2 12M (kB T ) M h ¯ " # 2 X h ¯ 2 ωif 2 ωif 1 + |xf i |2 . = −w 2 12(k T ) B n

∂t hi|ˆ ρ(t)|ii = −

(18.498)

This time dependence is caused by spontaneous emission and induced emission and absorption. To identify the different contributions, we rewrite the spectral decompositions (18.475) and (18.476) in the x-independent approximation as Cb (t, t′ ) + Ab (t, t′ ) (18.499) ( ) Z ′ 3 ′ 4π dω d k π h ¯ω ′ ′ −iω ′ (t−t′ ) 1 + coth [δ(ω − Ω ) − δ(ω + Ω )] e , = h ¯ k k 3 (2π)4 2MΩk 2kB T or Cb (t, t′ ) + Ab (t, t′ ) (18.500) Z ′ 3 4π dω d k π 2 ′ ′ ′ −iω ′ (t−t′ ) = h ¯ 2δ(ω − Ω )+ [δ(ω −Ω ) + δ(ω +Ω )] e . k k k h ¯ Ω /k T 3 (2π)4 2MΩk e k B −1 Following Einstein’s intuitive interpretation, the first term in curly brackets is due to spontaneous emission, the other two terms accompanied by the Bose occupation function account for induced emission and absorption. For high and intermediate temperatures, (18.500) has the expansion 4π h ¯ 3

Z

dω ′d3 k π 2δ(ω ′ − Ωk ) 4 (2π) 2MΩk ! ) 1h 2kB T ¯ Ωk ′ ′ ′ ′ + −1+ [δ(ω − Ωk ) + δ(ω + Ωk )] e−iω (t−t ) . (18.501) h ¯ Ωk 6 kB T (

The first term in curly brackets corresponds to the spontaneous emission. It conP 3 tributes to the rate of change ∂t hi|ˆ ρ(t)|ii a term −2Mγ f 0, since the δ-function allows only for decays. Second, there is an extra factor 2. Indeed, by comparing (18.499) with (18.501) we see that the spontaneous emission receives equal contributions from the 1 and the coth(¯ hω ′/2kB T ) in the curly brackets of (18.499), i.e., from dissipation and fluctuation terms Cb (t, t′ ) and Ab (t, t′ ). Thus our master equation yields for the natural line width of atomic levels the equation Γ = 2Mγ

X

f

3 ωif |xf i |2 ,

in agreement with the historic Wigner-Weisskopf formula.

(18.502)

1252


In terms of Γ, the rate (18.497) can therefore be written as ρ(t)|ii = −Γ + Mγ ∂t hi|ˆ

X

f

3 ωif |xf i |2 + Mγ

X

f >i

|ωif |3 |xf i |2 .

(18.503)

The second and third terms do not contribute to the total rate of change of hi|ˆ ρ(t)|ii since they are canceled by the induced emission and absorption terms associated with the −1 in the big parentheses of the fluctuation part of (18.501). The finite lifetime changes the time dependence of the state |i, ti from |i, ti = |i, 0ie−iEt to |i, 0ie−iEt−Γt/2 . Note that due to the restriction to f < i in (18.502), there is no operator local in time whose expectation value is Γ. Only the combination of spontaneous and induced emissions and absorptions in (18.503) can be obtained from a local operator, which is in fact the dissipation operator (18.495). For all temperatures, the spontaneous and induced transitions together lead to the rate of change of hi|ˆ ρ(t)|ii: ∂t hi|ˆ ρ(t)|ii =

 X −2Mγ  ω 3

if

f

+

X f

3 ωif

1 e¯hωif /kB T − 1

 

|xf i |2 .

(18.504)

For a state with principal quantum number n the temperature effects become detectable only if T becomes larger than −1/(n+ 1)2 + 1/n2 ≈ 2/n3 times the Rydberg > 20 to have observable temperature TRy = 157886.601K. Thus we have to go to n ∼ effects at room temperature.

18.17.4

Lamb shift

For atoms, the Feynman influence functional (18.472) allows us to calculate the celebrated Lamb shift. Being interested in the time behavior of the pure-state density matrix ρ = |iihi|, we may calculate the effect of the actions (18.473) and (18.474) perturbatively. For this, consider the dissipative part of the influence action (18.473), and in it the first term involving x+ (t) and x+ (t′ ), and integrate the external positions in the path integral (18.472) over the initial wave functions, forming Uii,tb ;ii,ta =

Z

dx+b dx−b

Z

dx+a dx−a hi|x+b ihi|x−bi × U(x+b , x−b , tb |x+a , x−a , ta )hx+b |iihx−b |ii.

(18.505)

To lowest order in γ, the effect of the Cb -term in (18.473) can be evaluated in the local approximation (18.479) as follows. We take the linear approximation to the R R exponential exp[ dtdt′ O(t, t′ )] ≈ 1 + dtdt′ O(t, t′ ) and propagate the initial state with the help of the amplitude Uii,t′ ;ii,ta to the first time t′ , then with Uf i,t;f i,t′ to the later time t, and finally with Uii,ta ;ii,t to the final time tb . The intermediate state between the times t and t′ are arbitrary and must be summed. Details how to do such a perturbation expansion are given in Section 3.17. Thus we find ∆C Uii,tb ;ii,ta = i

e2 2¯ h2 c2

Z

tb

ta

dtdt′

XZ f

dx+

Z

dx′+ Uii,ta ;ii,t hi|x+ ix+ hx+ |f i H. Kleinert, PATH INTEGRALS

1253


×[∂t ∂t′ Cb (t, t′ )]Uf i,t;f i,t′ hf |x′+ ix′+ hx′+ |iiUii,t′ ;ii,ta .

(18.506)

Inserting Uii,ta ;ii,t = e−iEi (ta −t)/¯h etc., this becomes ∆C Uii,tb ;ii,ta

tb e2 ˆ (t′ )|ii = − 2 2 dtdt′ hi|ˆ x(t) [∂t ∂t′ Cb (t, t′ )] x 2¯ h c ta Z e2 X tb ′ ˆ˙ iCb(t, t′ )hf |x|ii. ˆ˙ = − 2 2 dtdt′ eiωif (t−t ) hi|x|f (18.507) 2¯ h c f ta

Z

Expressing Cbij (t, t′ ) of Eq. (18.479) in the form Cbij (t, t′ )

h ¯ 2 ij δ = 2πc 3

Z

dω ′ ω e−iω(t−t ) , 2π

dt

Z

(18.508)

the integration over t and t′ yields ∆C Uii,tb ;ii,ta

e2 2 = −i 4π¯ hc3 3

Z

tb ta

dω X ω ˆ˙ f i |2 . |x 2π f ω − ωif − iη

(18.509)

The same treatment is applied to the Ab in the action (18.474), where the first term involving x+ (t) and x+ (t′ ) changes (18.510) to e2 2 ∆Uii,tb ;ii,ta= −i 4π¯ hc3 3

Z

tb

ta

dω X ω h ¯ω 1+coth 2π f ω − ωif + iη 2kB T

Z

dt

!

ˆ˙ f i |2 .(18.510) |x

The ω-integral is conveniently split into a zero-temperature part I(ωif , 0) ≡

Z

∞

0

dω X ω , π f ω − ωif + iη

(18.511)

and a finite-temperature correction ∆IT (ωif , T ) ≡ 2

Z

0

∞

dω X ω 1 . h ¯ ω/k T −1 B π f ω − ωif + iη e

(18.512)

Decomposing 1/(ωif − ω + iη) = P/(ω − ωif ) − iπδ(ωif − ω), the imaginary part of the ω-integral yields half of the natural line width in (18.497). The other half comes from the part of the integral (18.473) involving x− (t) and x− (t′ ). The principalvalue part of the zero-temperature integral diverges linearly, the divergence yielding again the mass renormalization (18.481). Subtracting this divergence from I(ωif , 0), the remaining integral has the same form as I(ωif , 0), but with ω in the numerator replaced by ωif = 0. This integral diverges logarithmically like (ωif /π) log[(Λ − ωif )/|ωif |], where Λ is Bethe’s cutoff [30]. For Λ ≫ |ωif |, the result (18.510) implies an energy shift of the atomic level |ii: ∆Ei =

e2 2 X 3 Λ 2 ω |ˆ x | log , f i if 4πc3 3π f |ωif |

(18.513)

1254


which is the Lamb shift. Usually, the weakly varying logarithm is approximated by a weighted average L = log[Λ/h|ωif |i] over energy levels and taken out of the integral. Then contribution of the term (18.510) can be attributed to an extra term ˆ LS ≈ −i L γM 1 [x, ˆ˙ x ˆ¨ ] H π 4

(18.514)

in the Hamiltonian (18.492). In this form, the Lamb shift appears as a Hermitian logarithmically divergent correction to the operator (18.495) governing the spontaneous emission of photons. To lowest order in γ, the commutator is for a Coulomb potential V (x) = −e2 /r equal to i h ¯ h ¯ 2c α 2 ˆ ˙ = 2 ∇ V (x) = − 2 [ˆ p, p] 4πδ (3) (x), 2 M M M

(18.515)

¯3 4α2h hi|δ (3) (x)|ii. ∆Ei = 2 3M c

(18.516)

leading to

For an atomic state of principal quantum number n with a wave function ψn (x), this becomes ∆En =

4α¯ h3 αL |ψn (0)|2 . 3M 2 c2

(18.517)

Only atomic s-states can contribute, since the wave functions of all other angular momenta vanish at the origin. Explicitly, the s-states of the hydrogen atom (13.213) have the value at the origin ψn (0) = √

1 n3 π

1 aH

3/2

,

(18.518)

where aH = h ¯ /Mcα is the Bohr radius (4.339). If the nuclear charge is Z, then aB , is diminished by this factor. Thus we obtain the energy shift 4α2h ¯3 ∆En = 3M 2 c

Mcα h ¯

3

L . n3 π

(18.519)

For a hydrogen atom with n = 2, this becomes ∆E2 =

α3 2 α Mc2 L. 6π

(18.520)

The quantity Mc2 α2 is the unit energy of atomic physics determining the hydrogen spectrum to be En = −Mc2 α2 /2n2 . Thus Mα2 = 4.36 × 10−11 erg = 27.21eV = 2 Ry = 2 · 3.288 × 1015 Hz.

(18.521)


1255


Inserting this together with α ≈ 1/137.036 into (18.520) yields8 ∆E2 ≈ 135.6MHz × L.

(18.522)

The constant L can be calculated approximately as L ≈ 9.3,

(18.523)

∆E2 ≈ 1261MHz.

(18.524)

∆ELamb shift ≈ 1057 MHz

(18.525)

leading to the estimate

The experimental Lamb shift

is indeed contained in this range. In this calculation, two effects have been ignored: the vacuum polarization of the photon and the form factor of the electron caused by radiative corrections. They reduce the frequency (18.524) by (27.3 + 51)MHz bringing the theoretical number closer to experiment. The vacuum polarization will be discussed in detail in Section 19.5. At finite temperature, (18.513) changes to 

e2 2 X 3 Λ kB T ωif |ˆ xf i |2 log ∆Ei = + 3 4πc 3π f |ωif | h ¯ ωij

!2

where J(z) denotes the integral

J(z) ≡ z

Z

0

∞

J

!

h ¯ ωif  , kB T

P z′ dz ′ , z − z ez ′ − 1

(18.526)

(18.527)

which has the low-temperature (large-z) expansion J(z) = −π 2 /6 − 2ζ(3)/z + . . . , and goes to zero for high temperature (small z) like −z log z, as shown in Fig. 18.2. z=h ¯ ωij /kB T 10

20

30

40

50

-0.25 -0.5 -0.75

6J(z)/π 2

-1 -1.25 -1.5

Figure 18.2 Behavior of function 6J(z)/π 2 in finite-temperature Lamb shift.

The above equations may have applications to dilute interstellar gases or, after a reformulation in a finite volume, to few-particle systems contained in cavities. So far, a master equation has been set up only for a finite number of modes [31]. 8

The precise value of the Lamb constant α4 M/6π is 135.641± 0.004 MHz.

1256

18.17.5


Langevin Equations

For high γT , the last term in the forward–backward path integral (18.488) makes the size of the fluctuations in the difference between the paths y(t) ≡ x+ (t) − x− (t) very small. It is then convenient to introduce the average of the two paths as x(t) ≡ [x+ (t) + x− (t)] /2, and expand

V x+

y y −V x− 2 2

∼ y · ∇V (x) + O(y3 ) . . . ,

(18.528)

keeping only the first term. We further introduce an auxiliary quantity (t) by ¨ (t) − Mγ ˙¨ ˙ (t) ≡ M x x(t) + ∇V (x(t)).

(18.529)

With this, the exponential function in (18.488) becomes

exp −

i h ¯

Z

tb

ta

dt y˙ −

w 2¯ h2

Z

tb

ta

dt y˙ 2 (t) ,

(18.530)

where w is the constant (18.487). Consider now the diagonal part of the amplitude (18.528) with x+b = x−b ≡ xb and x+a = x−a ≡ xa , implying that yb = ya = 0. It represents a probability distribution P (xb tb |xa ta ) ≡ |(xb , tb |xa , ta )|2 ≡ U(xb , xb , tb |xa , xa , ta ).

(18.531)

Now the variable y can simply be integrated out in (18.530), and we find the probability distribution 1 P [ ] ∝ exp − 2w

Z

tb

ta

dt

2

(t) .

(18.532)

The expectation value of an arbitrary functional of F [x] can be calculated from the path integral hF [x]iη ≡ N

Z

Dx P [ ]F [x],

(18.533)

where the normalization factor N is fixed by the condition h 1 i = 1. By a change of integration variables from x(t) to η(t), the expectation value (18.533) can be rewritten as a functional integral hF [x]iη ≡ N

Z

D P [ ] F [x].

(18.534)

Note that the probability distribution (18.532) is h ¯ -independent. Hence in the approximation (18.528) we obtain the classical Langevin equation. In principle, the integrand contains a factor J −1 [x], where J[x] is the functional Jacobian

J[x] ≡ Det[δη i (t)/δxj (t′ )] = det [ M∂t2 − Mγ∂t3R δij + ∇i ∇j V (x(t))]. (18.535) H. Kleinert, PATH INTEGRALS

1257

18.18 Fokker-Planck Equation in Spaces with Curvature and Torsion

By the same procedure as in Section 18.9.2 it can be shown that the determinant is unity, due to the retardation of the friction term, thus justifying its omission in (18.534). The path integral (18.534) may be interpreted as an expectation value with respect to the solutions of a stochastic differential equation (18.529) driven by a Gaussian random noise variable η(t) with a correlation function hη i (t)η j (t′ )iT = δ ij w δ(t − t′ ).

(18.536)

Since the dissipation carries a third time derivative, the treatment of the initial conditions is nontrivial and will be discussed elsewhere. In most physical applications γ leads to slow decay rates. In this case the simplest procedure to solve (18.529) is to write the stochastic equation as ¨ (t) + ∇V (x(t)) = ˙ (t) + Mγ ˙¨ x(t), Mx

(18.537)

and solve it iteratively, first without the γ-term, inserting the solution on the righthand side, and such a procedure is equivalent to a perturbative expansion in γ in Eq. (18.488). Note that the lowest iteration of Eq. (18.537) with ≡ 0 can be multiplied by x˙ and leads to the equation for the energy change of the particle d M 2 ˙ x = −Mγ x ¨ 2. x˙ + V (x) − Mγ x¨ dt 2

(18.538)

The right-hand side is the classical electromagnetic power radiated by an accelerated particle. The extra term in the brackets is known as Schott term [32].

18.18

Fokker-Planck Equation in Spaces with Curvature and Torsion

According to the new equivalence principle found in Chapter 10, equations of motion can be transformed by a nonholonomic transformation dxi = ei µ (q)dq µ into spaces with curvature and torsion, where they are applicable to the diffusion of atoms in crystals with defects [33]. If we denote gµν q˙ν by vµ , the Langevin equation (18.311) goes over into v˙ µ = Fµ (q, v) + ei µ (q)ηi

(18.539)

where Fµ (q, v) is the sum of all forces after the nonholonomic transformation: h

i

Fµ (q, v) ≡ M Γνλµ (q)v ν v λ − γvµ − ∂µ V (q).

(18.540)

In addition to the transformed force (18.366), Fµ (q, v) contains the apparent forces resulting from the coordinate transformation. For a distribution Pη (qvt|qa va ta ) = δ(qη (t) − q)δ(q˙η − vµ )

(18.541)

1258


one obtains, instead of (18.365), the Kubo equation

∂t Pη (q v t|qa va ta ) = −∂µ g µν (q)vν −

i 1 vh i ∂µ e µ (q)ηi + Fµ (q, v) Pη (q v t|qa va ta ), M (18.542)

and from this the generalization of the Fokker-Planck equation (18.375) to spaces with curvature and torsion: w v 1 ∂t P (x v t|xa va ta ) = −∂µ g vν + ∂µv ∂ − Fµ (q, v) M 2M µ

µν

P (x v t|xa va ta ). (18.543)

In the overdamped limit, the integrated probability distributions P (q t|qa ta ) ≡

Z

dD vP (q v t|qa va ta )

(18.544)

satisfies the equation [generalizing (18.381)] ∂t P (q t|qa t) =

"

#

1 ∂ν g µν Vν (q) P (q t|qa t), D∂µ ei µ ∂ν ei ν + Mγ

(18.545)

where Vν (q) ≡ ∂ν V (q). In the Fokker-Planck equations (18.542) and (18.545), the probability distributions P (q v t|qa va ta ) and P (qt|qa t) have the unit normalizations Z

D

D

d qd v P (q v t|qa va ta ) = 1,

Z

dD q P (q t|qa ta ) = 1,

(18.546)

as can be seen from the definitions (18.541) and (18.544). For distributions normalR √ ized with the invariant volume integral dD q g, to be denoted by P inv (qt|qa ta ) ≡ √ −1 g P (qt|qa t), we obtain from (18.545) the following invariant Fokker-Planck equation: ∂t P

inv

(q t|qa ta ) =

(

D 1 √ Vν (x) √ ∂µ g µν g ∂ν + 2Sν + g kB T

)

P inv (q t|qa ta ). (18.547)

The first term on the right-hand side contains the Laplace-Beltrami operator (11.13). With the help of the covariant derivative Dµ∗ defined in Eq. (11.96), which arises from Dµ by a partial integration, this equation can also be written as ∂t P

inv

"

(q t|qa ta ) = D g

µν

Dµ∗ Dν∗

#

1 ¯ µ + Dµ V (x) P inv (q t|qa ta ), Mγ

(18.548)

¯ µ is the covariant derivative (10.37) associated with the Christoffel symbol. where D H. Kleinert, PATH INTEGRALS

1259

18.19 Stochastic Interpretation of Quantum-Mechanical Amplitudes

18.19

Stochastic Interpretation of Quantum-Mechanical Amplitudes

In the last section we have seen that the probability distribution |(xb , tb |xa , ta )|2 is the result of a stochastic differential equation describing a classical path disturbed by a noise term η(t) with the correlation function (18.314). It is interesting to observe that the quantum-mechanical amplitude (xb , tb |xa , ta ) possesses quite a similar stochastic interpretation, albeit with some imaginary factors i and an unsatisfactory aspect as we shall see. Recall the path integral representation of the time evolution amplitude in Eq. (2.704). It involves the action A(x, t; xa , ta ) from the initial point xa to the actual particle position x. Recalling the definition of the fluctuation factor F (xb , xa ; tb − ta ) in Eq. (4.90), we see that this factor is given by the path integral i F (xb , xa ; tb − ta ) = Dx exp h ¯ (xa ,ta );(xb ,tb )

Z

Z

tb

ta

M dt (x˙ − v)2 , 2

(18.549)

where v(x, t) = (1/M)∂x A(x, xa ; tb − ta ) is the classical particle velocity. Up to a factor i and the absence of the retardation symbol, this path integral has the same form as the one for the probability (18.285) at large damping. As in (2.705) we introduce the momentum variable p(t) and obtain the canonical path integral F (xb , xa ; tb −ta ) =

Z

(xa ,ta );(xb ,tb )

D′x

2 (t)/2M Dp (i/¯h) Rttb dt{p(t)[x(t)−v(x(t),t)]−p ˙ } , (18.550) a e 2π¯ h

which looks similar to the stochastic path integral (18.285). The role of the diffusion constant D = kB T /Mγ is now played by h ¯ /2. By analogy with the path integral of a particle in a magnetic field in (2.646), the fluctuation factor satisfies a Schrödingerlike equation pˆ2b 1 + {ˆ pb , vb } F (xb , xa ; tb − ta ) = i¯ hF (xb , xa ; tb − ta ). 2M 2 !

(18.551)

This can easily be verified for the free particle, where vb =q(xb − xa )/(tb − ta ), and the fluctuation factor is from (2.120) F (xb , xa ; tb − ta ) = 2πi¯ h(tb − ta ). Note the symmetric operator order of the product pˆv in accordance with the time slicing in Section 10.5, and the ensuing operator order observed in Eq. (11.88). By reordering the Hamiltonian operator on the left-hand side of (18.551) to position pˆ to the left of the velocity, 2 ˆ → pˆ + pˆv + i ∇v. H 2M 2

(18.552)

Without the last term, the path integral (18.550) would describe fluctuating paths obeying the stochastic differential equation analogous to the classical Langevin equation (18.332) [34]: x(t) ˙ − v(x(t), t) = p(t)/M.

(18.553)

1260


The momentum variable p(t) plays the role of the noise variable η(t). Up to a factor i, this quantum noise has the same correlation functions as a white-noise variable: hδ(t − t′ ). hp(t)p(t′ )i = −iM¯

(18.554)

The “Fokker-Planck equation” associated with this “process” would be the ordinary Schrödinger equation for the amplitude (xb tb |xa ta ). For a free particle, the ordering problem can be solved by noting that in the path integral (18.549), the constant v can be removed from the path integral leaving −iA(xb ,xa ;tb −ta )/¯ h

F (xb , xa ; tb − ta ) = e

Z

(xa ,ta );(xb ,tb )

Dx exp

i h ¯

Z

tb

ta

M dt x˙ 2 2

, (18.555)

the right-hand factor being the path integral for the amplitude (xb tb |xa ta ) itself. This is identical with the path integral for the probability distribution of Brownian motion, and quantum-mechanical fluctuations are determined by the process x(t) ˙ = p(t)/M,

(18.556)

with the quantum noise (18.554). But also in the presence of a potential, it is possible to specify a process which properly represents quantum-mechanical fluctuations, although the situation is more involved [35]. To find it we rewrite the action in (18.549) as A=

Z

tb

ta

"

#

Z tb M M i¯ h 2 dt (x˙ − v) = dt (x˙ − s)2 − s′2 , 2 2 2 ta

(18.557)

with some as yet unknown function s(x). The associated Hamiltonian is now 2 pˆ2 ˆ → pˆ + 1 {ˆ H p, s} + i¯ hs′2 = + pˆs, 2M 2 2M

(18.558)

with the proper operator order. In order for (18.557) to hold, the function s(x) must satisfy the equations v = s′ ,

− i¯ hs′2 + s2 = v 2 .

(18.559)

Recalling Eqs. (4.12) and (4.5) we see that the equations in (18.559) can be satisfied with the help of the full eikonal S(x): s(x) = S(x)/M.

(18.560)

The process which describes the quantum-mechanical fluctuations is therefore x(t) ˙ − S(x)/M = p(t)/M.

(18.561)

The analogy is, however, not really satisfactory, since the full eikonal contains information on all fluctuations. Indeed, by the definition (4.4), it is given by the logarithm H. Kleinert, PATH INTEGRALS

1261

18.20 Stochastic Equation for Schr¨ odinger Wave Function

of the amplitude (x t|xa ta ), or any superposition ψ(x, t) = it:

R

dxa (x t|xa ta )ψ(xa ta ) of

S(x) = −i¯ h log(x t|xa ta ).

(18.562)

For the fluctuation factor this implies s(x) − v(x) = δv(x) ≡ −

i h ¯ F (x t|xa ta ). M

(18.563)

The path integral representation (18.549) for the fluctuation factor which has maximal analogy with the stochastic path integral (18.295) is therefore (

i F (xb tb , xa ta ) = Dx exp h ¯ (xa ,ta );(xb ,tb ) Z

Z

tb

ta

2 M R i dt x˙ − v + h ¯ log F (xb tb , xa ta ) . 2 M (18.564)

)

Since we have to know F (xb tb , xa ta ) to describe quantum-mechanical fluctuations as a process, this representation is of little practical use. The initial representation (18.549) which does not correspond to a proper process can, however, be used to solve quantum-mechanical problems.

18.20

Stochastic Equation for Schr¨ odinger Wave Function

It is possible to write a stochastic type of path integral for the Schrödinger wave function ψ(x, t) in D dimensions. By close analogy with Eq. (18.385), it reads ψ(xb , tb ) =

Z

(i/¯ h)

Dv e

R tb ta

dt [(M/2)v 2 (t)−V (xv (tb ,t))]

ψ (xv (tb , ta ), ta ) ,

(18.565)

where xv (tb , t) ≡ x(tb ) −

Z

t

tb

dt′ v(t′ )

(18.566)

is a functional of v(t′ ) parameterizing all possible fluctuating paths arriving at the fixed final point x(tb ) after having started from an arbitrary initial point x(t). They are Brownian bridges between the two points. The variables v(t) are the independently fluctuating velocities of the particle. The natural appearance of the velocities in the measure of the stochastic path integral (18.565) is in agreement with our observation in Eq. (10.141) that the time-sliced measure should contain the coordinate differences ∆xn as the integration variables rather than the coordinates themselves, which was the starting point for the nonholonomic coordinate transformations to spaces with curvature and torsion. We easily verify that (18.565) satisfies the Schrödinger equation by calculating the wave function at a slightly later time tb + ǫ, and expanding the right-hand side in powers of ǫ. Using the correlation functions hv i(t)i = 0,

hv i (t)v j (t′ )i = i¯ hδ ij δ(t − t′ ),

(18.567)

1262


we find, via a similar intermediate step as in (18.386), the desired result: h ¯2 2 i∂t ψ(x, t) = − ∂ + V (x) ψ(x, t). 2M x "

#

(18.568)

One may also write down a corresponding path integral for the time evolution amplitude (xb tb |xa ta ) =

Z

(i/¯ h)

Dv e

R tb ta

dt [(M/2)v 2 (t)−V (xv (tb ,t))] (D)

δ

(xa − xb +

Z

tb

ta

dt v(t)), (18.569)

which returns the Schrödinger amplitude (18.565) after convolution with ψ(xa , ta ) (and reduces to δ (D) (xa − xb ) for tb → ta , as it should). The addition of an interaction with a vector potential is nontrivial. The electromagnetic interaction in (10.167) Aem =

Z

t

ta

dt A(x(t)) · x˙

(18.570)

cannot be simply inserted into the exponent of the path integral (18.565) since in the evaluation via the correlation functions (18.567) assumes the independence of the noise variables v(t). This, however, is not true in the interaction (18.570). Recall the discussion in Section 10.6 which showed that the time-sliced version of the interaction (18.570) must contain the midpoint ordering of the vector potential with respect to the intervals ∆x to be compatible with the classical field equation. In Section 11.3 we have furthermore seen that this guaranteed gauge invariance. For the time-sliced short-time action, this implies that (18.570) has the form [see (10.178)] Aǫem = A(¯ x) · ∆x.

(18.571)

¯ , implying that in the sum over In this expression, a variation of ∆x changes also x all sliced actions, the ∆x are not independent. This is only achieved by the reexpanded postpoint interaction [see 10.177]. In the continuum, we shall indicate the postpoint product as before in (18.231) by a retardation symbol R, and rewrite (18.570) as Aem

"

#

h ¯ = dt A(x(t))x˙ R (t) − iǫ ∇·A(x(t)) . 2M ta Z

t

(18.572)

In the theory of stochastic differential equations, this postpoint expression is called an Itô integral . The ordinary midpoint integral (18.570) is referred to as Stratonovich integral . ˙ The Itô integral can now be added to the action in (18.569) with x(t) replaced by v(t), and we obtain i Z tb M 2 dt v (t) + A(xv (tb , t))vR (t) − V (xv (tb , t)) h ¯ 2 ta ( Z " #) Z tb i tb h ¯ × exp dt −iǫ ∇·A(xv (tb , t)) δ (D) (xa − xb + dt v(t)). (18.573) h ¯ ta 2M ta

(xb tb |xa ta ) =

Z

Dv exp


18.21 Real Stochastic and Deterministic Equation for Schr¨ odinger Wave Function 1263

Expanding the functional integrand in powers of v(t) as in (18.385) and using the correlation functions (18.567) we obtain the Schrödinger equation h ¯2 i∂t (x t|xa ta ) = − [∇ − iA(x)]2 + V (x) (x t|xa ta ). 2M "

#

(18.574)

The advantage of the Itô integral is that such a calculation becomes quite simple using the correlation functions (18.567). The integral itself, however, is awkward to handle since it cannot be modified by partial integration. This is only possible for the ordinary, Stratonovich integral.

18.21

Real Stochastic and Deterministic Equation for Schr¨ odinger Wave Function

The noise variable in the previous stochastic differential equation had an imaginary correlation function (18.554). It is possible to set up a completely real stochastic differential equation and modify this into a simple deterministic model which possesses the quantum properties of a particle in an arbitrary potential. In particular, the model has a discrete energy spectrum with a definite ground state energy, in this respect going beyond an earlier model by ’t Hooft [36], whose spectrum was unbounded from below. Let u(x) = (u1 (x), u2 (x)) be a time-independent field in two dimensions to be called mother field . The reparametrization freedom of the spatial coordinates is fixed by choosing harmonic coordinates in which ∇2 u(x) = 0,

(18.575)

where ∇2 is the Laplace operator. Equivalently, the components u1 (x) and u2 (x) may be assumed to satisfy the Cauchy-Riemann equations ∂µ uν = ǫµ ρ ǫν σ ∂ρ uσ ,

(µ, ν, . . . = 1, 2),

(18.576)

where ǫµν is the antisymmetric Levi-Civita pseudotensor. The metric is δµν , so that indices can be sub- or superscripts.

18.21.1

Stochastic Differential Equation

Consider now a point particle in contact with a heat bath of “temperature” h ¯ . Its classical orbit x(t) is assumed to follow a stochastic differential equation consisting of a fixed rotation and a random translation in the diagonal direction n ≡ (1, 1): ˙ x(t) =

× x(t) + n η(t),

(18.577)

where is the rotation vector of length ω pointing orthogonal to the plane, and η(t) a white-noise variable with zero expectation and the correlation function hη(t)η(t′ )i = h ¯ δ(t − t′ ).

(18.578)

1264


For a particle starting at x(0) = x, the position x(t) at a later time t is a function of x and a functional of the noise variable η(t′ ) for 0 < t′ < t: x(t) = Xη (x, t).

(18.579)

As earlier in Section 18.12, a subscript η is used to indicate the functional dependence on the noise variable ν. We now use the orbits ending at all possible final points x = x(t) to define a time-dependent field u(x; t) which is equal to u(x) at t = 0, and evolves with time as follows: u(x; t) = ut [x; η] ≡ u (X0 [t, x; η]) ,

(18.580)

where the notation ut [x; η] indicates the variables as in (18.579). As a consequence of the dynamic equation (18.577), the change of the field u(x, t) in a small time interval from t = 0 to t = ∆t has the expansion ∆uη (x, 0) = ∆t [ × x] · ∇ uη (x, 0) + +

1 2

Z

0

∆t

dt′

Z

∆t

0

3/2

The omitted terms are of order ∆t

18.21.2

Z

0

∆t

dt′ η(t′ ) (n · ∇) uη (x, 0)

dt′′ η(t′ )η(t′′ ) (n · ∇)2 uη (x, 0) + . . . . (18.581)

.

Equation for Noise Average

We now perform the noise average of Eq. (??), defining the average field u(x, t) ≡ huη (x, t)i.

(18.582)

Using the vanishing average of η(t) and the correlation function (18.578), we obtain in the limit ∆t → 0 the time derivative ˆ u(x, t), at t = 0, ∂t u(x, t) = H (18.583) with the time evolution operator ¯ ˆ ≡ {[ × x] · ∇} + h H (n · ∇)2 . (18.584) 2 ˆ time-independent. For this reason, the The average over η has made the operator H average field u(x, t) at an arbitrary time t is obtained by the operation u(x, t) = Uˆ (t)u(x, 0), (18.585) ˆ is a simple exponential where U(t)

ˆ ˆ ≡ eHt U(t) ,

(18.586) ˆ U(t) ˆ = U(t) ˆ H. ˆ as follows immediately from (18.583) and the trivial property H 2 ˆ commutes with the Laplace operator ∇ , thus ensuring Note that the operator H that the harmonic property (18.575) of u(x) remains true for all times, i.e., ∇2 u(x, t) ≡ 0.


18.21 Real Stochastic and Deterministic Equation for Schr¨ odinger Wave Function 1265

18.21.3

Harmonic Oscillator

We now show that Eq. (18.583) describes the quantum mechanics of a harmonic oscillator. Let us restrict our attention to the line with arbitrary x1 ≡ x and x2 = 0. Applying the Cauchy-Riemann equations (18.576), we can rewrite Eq. (18.583) in the pure x-form h ¯ ω x ∂x u2 (x, t) − ∂x2 u2 (x, t), 2 h ¯ ∂t u2 (x, t) = −ω x ∂x u1 (x, t) + ∂x2 u1 (x, t), 2 ∂t u1 (x, t) =

(18.588) (18.589)

where we have omitted the second spatial coordinates x2 = 0. Now we introduce a complex field ψ(x, t) ≡ e−ωx

2 /2¯ h

This satisfies the differential equation

h

i

u1 (x, t) + iu2 (x, t) .

h ¯2 ω2 h ¯ω i¯ h∂t ψ(x, t) = − ∂x2 + x2 − 2 2 2

!

ψ(x, t),

(18.590)

(18.591)

which is the Schrödinger equation of a harmonic oscillator with the discrete energy spectrum En = (n + 1/2)¯ hω, n = 0, 1, 2, . . . .

18.21.4

General Potential

The method can easily be generalized to an arbitrary potential. We simply replace (18.577) by x˙ 1 (t) = −∂2 S 1 (x(t)) + n1 η(t), x˙ 2 (t) = −∂1 S 1 (x(t)) + n2 η(t),

(18.592)

where S(x) shares with u(x) the harmonic property (18.575): ∇2 S(x) = 0,

(18.593)

i.e., the functions S µ (x) with µ = 1, 2 fulfill Cauchy-Riemann equations like uµ (x) in (18.576). Repeating the above steps we find, instead of the operator (18.584), ¯ ˆ ≡ −(∂2 S 1 )∂1 − (∂1 S 1 )∂2 + h H (n · ∇)2 , 2

(18.594)

and Eqs. (18.588) and (18.589) become: h ¯ (∂x S 1 )∂x u2 (x, t) − ∂x2 u2 (x, t), 2 h ¯ ∂t u2 (x, t) = −(∂x S 1 ) ∂x u1 (x, t) + ∂x2 u1 (x, t). 2 ∂t u1 (x, t) =

(18.595) (18.596)

1266


This time evolution preserves the harmonic nature of u(x). Indeed, using the harmonic property ∇2 S(x) = 0 we can easily derive the following time dependence of the Cauchy-Riemann combinations in Eq. (18.576): ˆ 1 u1 − ∂2 u2 ) − ∂2 ∂1 S 1 (∂1 u1 − ∂2 u2 ) + ∂ 2 S 1 (∂2 u1 + ∂1 u2 ), ∂t (∂1 u1 − ∂2 u2 ) = H(∂ 2 1 2 1 2 1 1 2 ˆ ∂t (∂2 u + ∂1 u ) = H(∂2 u + ∂1 u ) − ∂2 ∂1 S (∂2 u + ∂1 u ) − ∂22 S 1 (∂1 u1 − ∂2 u2). Thus ∂1 u1 − ∂2 u2 and ∂2 u1 + ∂1 u2 which are zero at any time remain zero at all times. On account of Eqs. (18.596), the combination ψ(x, t) ≡ e−S

1 (x)/¯ h

satisfies the Schrödinger equation

h

u1 (x, t) + iu2 (x, t)

i

(18.597)

h ¯2 i¯ h∂t ψ(x, t) = − ∂x2 + V (x) ψ(x, t), 2 "

#

(18.598)

where the potential is related to S 1 (x) by the Riccati differential equation 1 h ¯ V (x) = [∂x S 1 (x)]2 − ∂x2 S 1 (x). 2 2

(18.599)

The harmonic oscillator is recovered for the pair of functions S 1 (x) + iS 2 (x) = ω(x1 + ix2 )2 /2.

18.21.5

(18.600)

Deterministic Equation

The noise η(t) in the stochastic differential equation Eq. (18.592) can also be replaced by a source composed of deterministic classical oscillators qk (t), k = 1, 2, . . . with the equations of motion p˙k = −ωk2 qk ,

q˙k = pk ,

(18.601)

as η(t) ≡

X

q˙k (t).

(18.602)

k

The initial positions qk (0) and momenta pk (0) are assumed to be randomly distributed with a Boltzmann factor e−βHosc /¯h , such that hqk (0)qk (0)i = h ¯ /ωk2 ,

hpk (0)pk (0)i = h ¯.

(18.603)

Using the equation of motion q˙k (t) = ωk qk (0) sin ωk t + pk (0) sin ωk t,


18.22 Heisenberg Picture for Probability Evolution

1267

we find the correlation function hq˙k (t)q˙k (t′ )i = ωk2 cos ωk t cos ωk t′ hqk (0)qk (0)i + sin ωk t sin ωk thpk (0)pk (0)i = cos ωk (t − t′ ). (18.605) We may now assume that the oscillators qk (t) are the Fourier components of a massless field, for instance the gravitational field whose frequencies are ωk = k, and whose random initial conditions are caused by the big bang. If the sum over k is R∞ simply a momentum integral −∞ dk, then (18.605) yields a white-noise correlation function (18.578) for η(t). Thus it is indeed possible to simulate the quantum-mechanical wave functions ψ(x, t) and the energy spectrum of an arbitrary potential problem by deterministic equations with random initial conditions at the beginning of the universe. It remains to solve the open problem of finding a classical origin of the second important ingredient of quantum theory: the theory of quantum measurement to be extracted from the wave function ψ(x, t). Only then shall we understand how God throws dice [37].

18.22

Heisenberg Picture for Probability Evolution

The parallels between the evolution of probabilities and the Schrödinger theory of probability amplitudes can be exploited further. For example, it is possible to develop a Heisenberg operator description of the time dependence of thermal expectations. This goes by complete analogy with the development in Section 2.23 for the quantum-mechanical time evolution amplitude. To see this consider the thermal expectations of x and x2 for a particle which sits at the initial time t = ta at xa . They are given by the integrals Z ∞ hxi ≡ dxb xb P (xb tb |xa ta ), (18.606) −∞ Z ∞ (18.607) hx2 i ≡ dxb x2b P (xb tb |xa ta ). −∞

For simplicity, let us first look at the case of a dominant friction term. As in quantum mechanics, it is useful to introduce a bra-ket notation, but for the probabilities rather than the amplitudes, hxb tb |xa ta i ≡ |(xb tb |xa ta )|2 .

(18.608)

The fact that this probability satisfies the Fokker-Planck equation implies that we can write it as hxb tb |xa ta i = e−(tb −ta )H(pˆb ,xb ) δ(xb − xa ).

(18.609)

Thus we may introduce time-independent basis vectors |xa i satisfying hxb |xa i = δ(xb − xa ).

(18.610)

On this basis, the operators pˆ, x ˆ are defined in the usual way. They satisfy hxb |ˆ x = hxb |ˆ p =

xb hxb |, ∂ hxb |. −i ∂xb

(18.611)

1268


Then we may rewrite (18.609) in bra-ket notation as ˆ x)(tb −ta ) hxb tb |xa ta i = hxb |e−H(p,ˆ |xa i.

(18.612)

The expectation value of any function f (x) is calculated as follows Z ∞ ˆ x) hf (x)i = dxb f (xb )hxb |e−(tb −ta )H(p,ˆ |xa i −∞ Z ∞ ˆ x) = dxb hxb |f (ˆ x)e−(tb −ta )H(p,ˆ |xa i −∞ Z ∞ Z ∞ ˆ x) = dxb dxP hxb |e−(tb −ta )H(p,ˆ |xP ihxP |f (ˆ x(tb − ta ))|xa i. −∞

(18.613)

−∞

In the last term we have introduced the time-dependent Heisenberg type of operator ˆ x) ˆ x) x ˆ(t) ≡ etH(p,ˆ x ê−tH(p,ˆ .

(18.614)

The probability P (xb tb |xa ta ) satisfies the normalization condition Z ∞ Z ∞ dxb P (xb tb |xa ta ) = dxb hxb tb |xa ta i −∞ −∞ Z ∞ ˆ x) = dxb hxb |e−(tb −ta )H(p,ˆ |xa i = 1.

(18.615)

−∞

Using this in the last line of (18.614), we arrive at the simple formula Z ∞ hf (x)i = dxb hxb |f (ˆ x(tb − ta ))|xa i.

(18.616)

−∞

For the Brownian motion of a point particle where Le

=

H

=

x˙ 2 , 4D Dp2 ,

(18.617)

the Heisenberg operators are pˆ(t) =

pˆ,

xˆ(t) =

eHt x ê−Ht = xˆ − i2Dpˆt,

ˆ

ˆ

(18.618)

and x ˆ2 (t)

= =

x ˆ2 − i2D · (ˆ px ˆ+x ˆpˆ)t − 4D2 pˆ2 t2 , x ˆ2 + 2Dt − i2D · 2ˆ xpˆ − 4D2 pˆ2 t.

It is easy to calculate the following matrix Z ∞ dxb hxb |ˆ x|xa i = −∞ Z ∞ dxb hxb |ˆ p|xa i = −∞ Z ∞ dxb hxb |ˆ x2 |xa i = −∞ Z ∞

Z

−∞ ∞

−∞

dxb hxb |ˆ p2 |xa i dxb hxb |ˆ pxˆ|xa i

(18.619)

elements: xa , Z ∞ ∂ −i dxb δ(xb − xa ) = 0, ∂x b −∞ Z ∞ dxb xb 2 δ(xb − xa ) = xa 2 , −∞ Z ∞

(18.620)

∂2 δ(xb − xa ) = 0, ∂xb 2 −∞ Z ∞ ∂ = −i dxb δ(xb − xa )xa = 0. ∂x b −∞ = −

dxb


1269

18.22 Heisenberg Picture for Probability Evolution The vanishing integrals reflect the translational invariance of the integrated bra state which is therefore annihilated by a translational operator pˆb on its right: Z ∞ dxb hxb |ˆ p = 0.

R∞

−∞

dxb hxb |, (18.621)

−∞

With the help of Eqs. (18.620), we obtain hxi = hx2 i =

xa , xa 2 + 2D(tb − ta ),

(18.622)

and h(x − xa )2 i = 2D(tb − ta ).

(18.623)

Clearly, a similar formalism can be developed for the general case with the Lagrangian containing x¨-terms. All we have to do is define time-dependent Heisenberg operators for both sets of canonical coordinates x, p, v, pv . For instance, consider the case of a free particle, where V (x) = 0 and the Hamiltonian (18.239) reduces to w 2 p − iγpv v + ipv. 2M 2 v

H=

(18.624)

If we want to calculate expectations such as hˆ x2 i for a particle initially at xa , we have to evaluate integrals of the form Z ∞ 2 hϕˆ i = dxb x2b P (xb tb |xa ta ) −∞ Z ∞ Z ∞ = dxb dx2b xb 2 P (xb x2b tb |xa x2a ta ). (18.625) −∞

−∞

Here we introduce basis vectors |xvi which diagonalize the operators x ˆ, vˆ. The momentum operators satisfy hxv|ˆ p = hxv|ˆ pv

=

∂ hxv|, ∂x ∂ − hxv|. ∂v

−

(18.626)

Then we can write hx2 i =

Z

∞

dxb

−∞

Z

∞ −∞

dx2b hxb x2b |ˆ x2 (tb − ta )|xa x2a i,

(18.627)

where x ˆ(t) is the Heisenberg operator defined by ˆ pˆv ,ˆ x,ˆ v) ˆ pˆv ,ˆ x,ˆ v) x ˆ(t) = etH(p, x ê−tH(p, .

(18.628)

The Heisenberg equations of motions are pˆ˙ (t) ˙ pˆv (t) x ˆ˙ (t) vˆ˙ (t)

ˆ pˆ(t)] = 0, = [H, ˆ pˆv (t)] = γ pˆv (t) − pˆ(t), = [H, ˆ x = [H, ˆ(t)] = vˆ(t), ˆ vˆ(t)] = −i w pˆv (t) − γˆ = [H, v (t). M2

According to the first equation, pˆ(t) is a constant operator: pˆ(t) ≡ pˆ = const.

(18.629)

1270


The second equation is solved by pˆv (t) = pˆv eγt −

1 pˆ(eγt − 1), γ

(18.630)

where pˆv is the initial value of pˆv (t) at t = 0. With this, the fourth equation can be integrated to give Z t ′ w vˆ(t) = vê−γt − i 2 dt′ e−γ(t−t ) pˆv (t′ ) (18.631) M 0 w 1 = vê−γt − i pˆv sinh γt − pˆ(cosh γt − 1) , 2 γM γ so that 1 w 1 x ˆ(t) = x ˆ + vˆ (1 − e−γt ) − i p cosh γt − p(sinh γt − γt) . v γ γM 2 γ

(18.632)

Using again the relations Z

∞

−∞

dxb

Z

∞

−∞

dx2b hxb x2b |

pˆ pˆv

=0

(18.633)

to express the translational invariance of the integrated bra state, as in Eq. (18.621), we find directly 1 hˆ xi = xa + x˙ a (1 − e−γt ), γ

(18.634)

and h(x − xb )2 i =

Appendix 18A

1 (1 − e−γt )2 γ2 kB T + 2 [2γt − 3 + 4e−γt − e−2γt ]. γ M x˙ 2a

(18.635)

Inequalities for Diagonal Green Functions

Let us introduce several diagonal Green functions consisting of thermal averages of equal-time commutators and anticommutators of bosonic and fermionic field operators, elementary or composite. For brevity, we write h i h i ˆ ˆ ) = h. . .iT h. . .iT = Tr exp(−H/T ) . . . Tr exp(−H/T (18A.1) and define the averages with obvious spectral representations Z ∞ dω ˆ ψˆ† ]∓ iT = ρ12 (ω), c ≡ h[ψ, −∞ 2π Z ∞ dω ω ˆ ψˆ† ]± iT = a ≡ h[ψ, ρ12 (ω) tanh∓1 . 2T −∞ 2π

(18A.2)

We shall also introduce a quantity obtained by integrating the imaginary-time Green function over a period τ ∈ [0, 1/T ). This gives for boson and fermion fields the nonnegative expression [see (18.23)] Z 1/T ˆ )ψˆ† (0)iT g ≡ G(ωm = 0) = dτ hψ(τ 0 ( ) Z ∞ 1 dω 1 = ρ12 (ω) ≥ 0. (18A.3) ω tanh(ω/2T ) −∞ 2π H. Kleinert, PATH INTEGRALS

Appendix 18A


1271

Note that for fermion fields, the spectral weight in this integral is accompanied by an extra factor tanh(ω/2T ). This is due to the fact that ωm = 0 is no fermionic Matsubara frequency, but a “wrong” bosonic one. In fact, the sum (18.23) for G(ωm ) contains a factor 1 − e(En −En′ )/T for both bosons and fermions, while ρ12 (ω) in the spectral representation (18A.3) introduces, via (18.59), a factor 1 − e−ω/T for bosons and a factor 1 + eω/T for fermions, thus explaining the relative factor tanh(ω/2T ) in (18A.3). The integration over τ leads to the factor 1/ω in (18A.3). This factor is also found by integrating the retarded Green function G12 (t) and the commutator function C12 (t) over all real times, resulting in the spectral representations Z ∞ Z dω 1 ˆ dt Θ(t)h[ψ(t), ψˆ† (0)]∓ iT = ρ12 (ω) , (18A.4) i 2π ω −∞ Z ∞ Z dω 1 ω ˆ i dt Θ(t)h[ψ(t), ψˆ† (0)]± iT = ρ12 (ω) tanh∓1 . (18A.5) 2π ω 2T −∞ Another set of thermal expectation values involves products of field operators with time derivatives rather than integrals. Their spectral representations contain an extra factor ω. For example, the τ -derivative of the expectation value in (18A.3) leads to Z ∞ dω ˙ˆ † ˆ ρ12 (ω)ω(1 ± nω ). (18A.6) −hψ(0), ψ (0)iT = −∞ 2π The real-time derivatives of the expectation values in (18A.4) and (18A.5) have the spectral integrals Z ∞ dω ˙ˆ d ≡ ih[ψ(0), ψˆ† (0)]∓ iT = ρ12 (ω)ω, (18A.7) −∞ 2π Z ∞ ω dω ˙ˆ ρ12 (ω)ω tanh∓1 . (18A.8) e ≡ ih[ψ(0), ψˆ† (0)]± iT = 2T −∞ 2π The expectation values c, a, g, d, e satisfy several rigorous inequalities. To derive these, we observe that ( ) 1 1 1 1 µ(ω) = ρ12 (ω) (18A.9) g 2π ω tanh(ω/2T ) is a positive function. This follows directly from (18.59), according to which ρ12 (ω) at negative ω is negative for bosons and positive for fermions. Having divided out the total integral g defined in (18A.3), the integral over µ(ω) is normalized to unity, Z ∞ dω µ(ω) = 1, (18A.10) −∞

for both bosons and fermions. Using µ(ω), we form the following ratios: ( ) Z ∞ 1 c = dω µ(ω)ω , g coth(ω/2T ) −∞ a g d g e g

=

Z

∞

Z

∞

Z

∞

dω µ(ω)ω

(

dω µ(ω)ω

2

(

1 coth(ω/2T )

)

,

(18A.13)

dω µ(ω)ω

2

(

coth(ω/2T ) 1

)

.

(18A.14)

−∞

=

−∞

=

−∞

coth(ω/2T ) 1

)

(18A.11)

,

(18A.12)

1272


The inequalities to be derived are based on the Jensen-Peierls inequality for convex functions derived in Chapter 5. Recall that a convex function f (ω) satisfies ω1 + ω2 f (ω1 ) + f (ω2 ) f ≤ , (18A.15) 2 2 which is generalized to X

f

µi ωi

i

!

X

≤

µi f (ωi ),

(18A.16)

i

P where µi is an arbitrary set of positive numbers with i µi = 1. In the continuum limit, this becomes Z ∞ Z ∞ dω µ(ω)f (ω). (18A.17) f dω µ(ω)ω ≤ −∞

−∞

It is obvious that a similar Jensen-Peierls inequality holds also for concave functions with inequality sign in the opposite direction. The Jensen-Peierls inequality (18A.17) is now applied to the function f (ω) = ω coth

ω , 2T

(18A.18)

which looks like a slightly distorted hyperbola coming in from infinity along the diagonal lines |ω| and crossing the f -axis at ω = 0, f (0) = 2T . The second derivative of f (ω) is positive everywhere, ensuring the convexity. The function (18A.18) appears in the integrand of the boson part of Eq. (18A.12). The right-hand side of (18A.17) can therefore be written as a/g. The left-hand side is obviously equal to (c/g) coth(c/2T g). Hence we arrive at the inequality c coth

c ≤ a. 2T g

(18A.19)

In terms of the original field operators, this amounts to ˆ ψˆ† ]+ iT ≥ h[ψ, ˆ ψˆ† ]− iT coth h[ψ,

ˆ ψˆ† ]− iT h[ψ, . R 1/T ˆ )ψˆ† (0)iT 2T 0 dτ hψ(τ

(18A.20)

In the special case that ψˆ is a canonical interacting boson field of momentum p, the commutator ˆ ψˆ† ]− = 1, and the inequality becomes is simply [ψ, † ˆ 1 + 2hψˆp ψp iT

≥ coth(1/2T g) = 1 +

i.e., † ˆ hψˆp ψp iT ≥

2 = 1 + 2ng−1 , e1/gT − 1

1 ≡ ng−1 , e1/gT − 1

(18A.21)

(18A.22)

where ng−1 is the free-boson distribution function (18.36) for an energy g −1 . This is quite an interesting relation. The quantity g is the Euclidean equilibrium Green function G(ωm , p) at ωm = 0. For free particles in contact with a reservoir, it is given by g −1 = G(0, p)−1 =

p2 − µ ≡ ξ(p), 2M

(18A.23)

i.e., it is equal to the particle energy measured with respect to the chemical potential µ. Moreover, we know that for free particles † ˆ hψˆp ψp iT = nξ(p) ,


Appendix 18A


1273

so that the inequality (18A.22) becomes an equality. The content of the inequality (18A.22) may therefore be phrased as follows: For any interaction, the occupation of a state with momentum p is never smaller than for a free boson level of energy g −1 = G(0, p). Another inequality can be derived from the concave function √ y √ ¯ , (18A.25) f (y) = y coth 2T using y = ω 2 and the measure Z

∞

dω µ(ω) =

−∞

Z

∞

0

dy √ √ µ( y) = 1. y

(18A.26)

As argued before, concave functions satisfy the inequality opposite to (18A.17), from which we derive the inequality Z ∞ Z ∞ dy √ dy √ ¯ f (18A.27) √ µ( y)y ≥ √ µ( y)f¯(y), y y 0 0 which can be rewritten as f¯

Z

∞

dω µ(ω)ω 2

−∞

≥

Z

∞

dω µ(ω)f¯(ω 2 ).

(18A.28)

−∞

Again, the right-hand side is a/g, but now it is bounded from above by s s ! d 1 d a ≤ coth . g g 2T g

(18A.29)

The combined inequality p c c coth ≤ a ≤ dg coth 2T g

1 2T

s ! d g

(18A.30)

may be used to derive further inequalities: c2

≤

c coth(d/2T c) ≤

dg, a,

g

≤

coth(c/2T a),

c

≤

d tanh(c/2T a),

c

≤

a tanh(d/2T c).

(18A.31)

For fermion fields we see that an inequality like (18A.19) holds with c and a interchanged, i.e., a coth

a ≤ c, 2T g

(18A.32)

which leads to ˆ ψˆ† ]− iT ≤ h[ψ, ˆ ψˆ† ]+ iT tanh h[ψ,

ˆ ψˆ† ]+ iT h[ψ, . R 1/T ˆ )ψˆ† (0)iT 2T 0 dτ hψ(τ

(18A.33)

ˆ ψˆ† ]+ = 1, this becomes For canonical fermion fields with [ψ, ˆT 1 − 2hψˆ† ψi

≤

tanh(1/2gT ) = 1 −

2 , e1/gT + 1

(18A.34)

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i.e., the fermionic counterpart of (18A.22): † ˆ hψˆp ψp iT ≤

1 = ng−1 , e1/gT + 1

(18A.35)

where ng−1 is the free-fermion distribution function (18.36) at an energy g −1 . As in the Bose case, free particles fulfill † ˆ hψˆp ψp iT = nξ(p) ,

(18A.36)

with g −1 = ξ(p), so that the inequality (18A.35) becomes an equality. The inequality implies that an interacting Fermi level is never occupied more than a free fermion level of energy g −1 = G(0, p)−1 . Also the second inequality in (18A.31) can be taken over to fermions which amounts to (18A.32), but with a and d replaced by c and e.

Appendix 18B

General Generating Functional

For a field operator a(t) of frequency Ω and its Hermitian conjugate a† (t), the retarded and advanced Green functions and the expectation values of commutators and anticommutators were derived in Eqs. (18.68)–(18.77): ′ GR Ω (t, t ) = ′ GA Ω (t, t ) =

CΩ (t, t′ ) = AΩ (t, t′ ) =

′

Θ(t − t′ )e−iΩ(t−t ) ,

′

−Θ(t′ − t)e−iΩ(t−t ) , ′

e−iΩ(t−t ) , ∓1 ′ Ω tanh e−iΩ(t−t ) . 2T

(18B.1)

Introducing complex sources η(t) and η † (t) associated with these operators, the generating functional for these functions is Z tb † dt(ˆ a† η + η † a ˆ) . (18B.2) Z0 [ηP , ηP ] = Tr ρˆ TˆP exp −i ta

The complex sources are distinguished according to the closed-time contour branches by a subscript P. The generating functional can then be written down immediately as Z Z † ′ † ′ ′ Z0 [ηP , ηP ] = exp − dt dt ηP (t)GP (t, t )ηP (t ) , (18B.3) generalizing (18.179), where the matrix Gp =

1 2

A AΩ + GR Ω + GΩ R A AΩ + GΩ − GΩ

A AΩ − GR Ω + GΩ R AΩ − GΩ − GA Ω

(18B.4)

contains the following operator expectations on the two time branches: Gp (t, t′ )

= =

hTˆP a ˆH (t+ )ˆ a†H (t′+ )iT hTˆP a ˆH (t− )ˆ a†H (t′+ )iT hTâ ˆH (t+ )ˆ a†H (t′+ )iT hˆ aH (t− )ˆ a†H (t′+ )iT

hTˆP a ˆH (t+ )ˆ a†H (t′− )iT hTˆP a ˆH (t− )ˆ a†H (t′− )iT ! ±hˆ a†H (t′− )ˆ aH (t+ )iT . hTˆ a ˆ (t )ˆ a† (t′ )i H

−

H

−

(18B.5)

T


Appendix 18B


1275

Note that a ˆH (t+ ), a ˆ†H (t+ ) and a ˆH (t− ), a ˆ†H (t− ) obey the Heisenberg equations of motion with the Hamiltonians h i ˆ+ ≡ Ω a ˆ†H (t+ )ˆ aH (t+ ) ± a ˆH (t+ )ˆ a†H (t+ ) , H 2 i Ωh † ˆ H− ≡ − a ˆH (t− )ˆ aH (t− ) ± a ˆH (t− )ˆ a†H (t− ) . (18B.6) 2 In the second-quantized field interpretation, they read ˆ+ H ˆ− H

Ω † a ˆ (t+ )ˆ aH (t+ ), 2 H Ω † ≡ − a ˆ (t− )ˆ aH (t− ). 2 H ≡

The explicit time dependence of the matrix elements of GP (t, t′ ) in Eq. (18B.5) is ′ Θ(t − t′ ) ± nΩ ±nΩ GP (t, t′ ) = e−iΩ(t−t ) , 1 ± nΩ Θ(t′ − t) ± nΩ

(18B.7)

with nΩ = (eΩ/T ± 1)−1 . This Green function can, incidentally, be decomposed as ′ G0P (t, t′ ) + GN P (t, t ),

(18B.8)

where G0P (t, t′ ) is the Green function at zero temperature, i.e., the expression (18B.7) for nΩ ≡ 0. The matrix GN (t, t′ ) contains the expectations of the normal products: â â hN ˆH (t+ )ˆ a†H (t′+ )iT hN ˆH (t+ )ˆ a†H (t′− )iT ′ (t, t ) ≡ GN P â â hN ˆH (t− )ˆ a†H (t′+ )iT hN ˆH (t− )ˆ a†H (t′− )iT † ′ hˆ aH (t+ )ˆ aH (t+ )iT hˆ a†H (t′− )ˆ aH (t+ )iT ≡ ± . (18B.9) hˆ a†H (t′+ )ˆ aH (t− )iT hˆ a†H (t′− )ˆ aH (t− )iT ˆ (. . .) is defined by reordering the For an arbitrary product of operators, the normal product N operators so that all annihilation operators come to act first upon the state on the right-hand side. At the end, the product receives the phase factor (−)F , where F is the number of fermion permutations to arrive at the normal order. A similar decomposition exists at the operator level ˆ ˆ ′ ) which are linear combinations before taking expectation values. For any pair of operators A(t), B(t of creation and annihilation operators, the time-ordered product can be decomposed as ˆ B(t ˆ ′ ) = hTÂ(t) ˆ B(t ˆ ′ )i0 + N ˆ A(t) ˆ B(t ˆ ′ ), TÂ(t)

(18B.10)

where h. . .i0 ≡ Tr (|0ih0| . . .) denotes the zero-temperature expectation. This decomposition is proved in Appendix 18C, where it is also generalized to products of more than two operators. Let us also go to the Keldysh basis here: 1 1 −1 η˜P = QηP = √ ηˆP . (18B.11) 1 1 2 Then the generating functional becomes [instead of (18.180)]: !# " Z ! Z 0 GA η+ Ω ∗ ′ ∗ ∗ −1 Z0 [ηP , ηP ] = exp − dt dt (η+ , −η− )Q Q GR A −η− Ω Z n 1Z ∗ ∗ ′ ′ = exp − dt dt′ (η+ − η− )(t)GR Ω (t − t )(η+ + η− )(t ) 2 ∗ ∗ ′ ′ + (η+ + η− )(t)GA Ω (t − t )(η+ − η− )(t ) o ∗ ∗ + (η− − η− )(t)AΩ (t − t′ )(η+ − η− )(t′ ) ,

(18B.12)

1276


where we have used the notation η+ (t) ≡ η(t+ ),

η− (t) ≡ η(t− ).

(18B.13)

Expression (18B.12) can be simplified [as before (18.180)] using the time reversal relation (18.62) in the form AΩ (t, t′ ) = Θ(t, t′ )AΩ (t, t′ ) ± Θ(t′ − t)AΩ (t′ , t),

(18B.14)

leading to ∗ Z0 [ηP , ηP ]

= exp

Z Z t 1 ∞ dt dt′ 2 −∞ −∞ h ′ ′ × (η+ − η− )∗ (t)GR Ω (t, t )(η+ + η− )(t )

−

′ ∗ ∗ ′ − (η+ − η− )(t)GR Ω (t, t ) (η+ + η− ) (t )

+ (η+ − η− )∗ (t)AΩ (t, t′ )(η+ − η− )(t′ )

i + (η+ − η− )(t)AΩ (t, t′ )∗ (η+ − η− )∗ (t′ ) .

(18B.15)

For the case of a second-quantized field, this is the most useful generating functional. The expression (18B.15) can be used to derive the generating functional for correlation functions between one or more ϕ(t) and the associated canonically-conjugate momenta. As an example, consider immediately a harmonic oscillator with ϕ(t) = x(t) and the momentum p(t). We would like to find the generating functional Z Z[jP , kP ] = Tr ρˆ TˆP exp i dx [jP (t)xP (t) + kP (t)pP (t)] . (18B.16) P

The position variable x(t) is decomposed as in (18.92) into a sum of creation and an annihilation operators: r ¯h −iΩt x ˆ(t) = a ê +a ˆ† eiΩt . (18B.17) 2M Ω

The inverse of this decomposition is √ a ˆ = (M Ω ϕˆ ± iˆ p)/ 2M Ω¯h, † a ˆ

and there is an analogous relation of the complex sources: √ η = (j ± iM Ω k) / 2M Ω¯h. † η

(18B.18)

(18B.19)

Inserting these sources into (18B.15), we obtain the generating functional Z ∞ Z t 1 Z0 [jP , kP ] = exp − dt dt′ (j+ − j− )(t) 2M Ω −∞ −∞ ′ ′ × [Re AΩ (t, t′ ) + iIm GR Ω (t, t )]j+ (t ) ′ ′ − [Re AΩ (t, t′ ) − iIm GR (18B.20) Ω (t, t )]j− (t ) Z ∞ Z t 1 ′ ′ − dt dt′ (k+ − k− )(t) {[Im AΩ (t, t′ ) − iRe GR Ω (t, t )]j+ (t ) 2 −∞ −∞ ′ R ′ ′ − [Im AΩ (t, t ) + iRe GΩ (t, t )]j− (t ) + (j ↔ kM Ω) . H. Kleinert, PATH INTEGRALS

Appendix 18B

1277


Here it is useful to introduce the quantities 1 ′ Re AΩ (t, t′ ) + iIm GR Ω (t, t ) , 2M Ω 1 ′ Im AΩ (t, t′ ) − iRe GR Ω (t, t ) . 2M Ω

α(t, t′ ) = β(t, t′ ) =

(18B.21)

Then the generating functional reads Z ∞ Z t Z0 [j+ , j− , k+ , k− ] = exp − dt dt′ (j+ −j− )(t) [α(t, t′ )j+ (t′ )−α∗ (t, t′ )j− (t′ )] + (j ↔ kM Ω) −M Ω

Z

−∞ −∞ Z t ∞ ′

dt

−∞

dt (k+ −k− )(t) [β(t, t )j+ (t )−β (t, t )j− (t )] + (j ↔ kM Ω) . ′

′

∗

′

′

−∞

(18B.22)

If the oscillator is coupled only to the real source j, i.e., if its generating functional reads Z ˆ Z[jP ] = Tr ρˆ TP exp i dx jP (t)ˆ xP (t) , (18B.23) P (18B.22) and have we can drop all but the first line in the exponent of Z ∞ Z t Z0 [j+ , j− ] = exp − dt dt′ (j+ − j− )(t) [α(t, t′ )j+ (t′ ) − α∗ (t, t′ )j− (t′ )] . −∞

−∞

(18B.24) Since (18B.22) and (18B.24) contain only the causal temporal order t > t′ , the retarded Green ′ function GR Ω (t, t ) in (18B.21) can be replaced by the expectation value of the commutator [see (18.40), (18.41), and (18.42)]. Thus, for t > t′ , the functions α(t, t′ ) and β(t, t′ ) are equal to9 α(t, t′ ) = β(t, t′ ) =

1 [Re AΩ (t, t′ ) + iIm CΩ (t, t′ )] , 2M Ω 1 [Im AΩ (t, t′ ) − iRe CΩ (t, t′ )] , 2M Ω

t > t′ , t > t′ .

(18B.25)

For a single oscillator of frequency Ω, we use the spectral function (18.74) properties (18.44) and (18.53) of AΩ (t, t′ ) and CΩ (t, t′ ), and find the simple expressions: " ( ) # Ω coth 2T 1 ′ −iΩ(t−t′ ) −iΩ(t−t′ ) + iIm e α(t, t ) = Re e Ω 2M Ω tanh 2T " ( ) # Ω coth 2T 1 ′ ′ = cos Ω(t − t ) − i sin Ω(t − t ) , (18B.26) Ω 2M Ω tanh 2T " ( ) # Ω coth ′ ′ 1 2T β(t, t′ ) = Im e−iΩ(t−t ) − iRe e−iΩ(t−t ) Ω 2M Ω tanh 2T " ( ) # Ω coth 2T 1 ′ ′ =− sin Ω(t − t ) + i cos Ω(t − t ) . (18B.27) Ω 2M Ω tanh 2T ′ Note that real and imaginary parts of the functions α(t−t ¯ ) can be combined into a single expression (β = 1/T )  ′   cosh[Ω(β/2 − i(t − t )] for bosons,  1 sinh(Ωβ/2) α(t ¯ − t′ ) = (18B.28) sinh[Ω(β/2 − i(t − t′ )] 2M Ω   for fermions.  cosh(Ωβ/2) 9

Note that α(t, t′ ) = hx(t)x(t′ )iT .

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The bosonic function agrees with the time-ordered Green function (18.101) for t > t′ and continues it analytically to t < t′ . In Fourier space, the functions (18B.26) and (18B.27) correspond to π ω′ coth±1 + 1 [δ(ω ′ − Ω) − δ(ω ′ + Ω)] , α(ω ′ ) = 2M Ω 2T iπ ω′ coth±1 + 1 [δ(ω ′ − Ω) + δ(ω ′ + Ω)] . β(ω ′ ) = − 2M Ω 2T Let us split these functions into a zero-temperature contribution plus a remainder π 1 ′ ′ ′ ′ α(ω ) = δ(ω − Ω) ± Ω/T [δ(ω − Ω) + δ(ω + Ω)] , MΩ e ∓1 π 1 ′ ′ ′ ′ [δ(ω − Ω) − δ(ω + Ω)] . β(ω ) = δ(ω − Ω) ± Ω/T MΩ e ∓1 On the basis of this formula, Einstein first explained the induced emission and absorption of light by atoms which he considered as harmonically oscillating dipoles in contact with a thermal reservoir. He imagined them to be harmonically oscillating dipole moments coupled to a thermal bath consisting of the Fourier components of the electromagnetic field in thermal equilibrium. Such a thermal bath is called a black body. The first purely dissipative and temperature-independent term in α(ω ′ ) was attributed by Einstein to the spontaneous emission of photons. The second term is caused by the bath fluctuations, making energy go in and out via induced emission and absorption of photons. It is proportional to the occupation number of the oscillator state nΩ = (e−Ω/T ∓ 1)−1 . The equality of the prefactors in front of the two terms is the important manifestation of the fluctuation-dissipation theorem found earlier [see (18.53)].

Appendix 18C

Wick Decomposition of Operator Products

ˆ and B(t) ˆ Consider two operators A(t) which are linear combinations of creation and annihilation operators ˆ A(t) = ˆ B(t) =

α1 a ˆ(t) + α2 a ˆ† (t), β1 a ˆ(t) + β2 a ˆ† (t).

(18C.1)

We want to show that the time-ordered product of two operators has the decomposition quoted in Eq. (18B.10): ˆ B(t) ˆ = hTÂ(t) ˆ B(t)i ˆ ˆˆ ˆ TÂ(t) 0 + N A(t)B(t).

(18C.2)

The first term on the right-hand side is the thermal expectation of the time-ordered product at zero temperature; the second term is the normal product of the two operators. ˆ are both creation or annihilation operators, the statement is trivial with hTÂˆBi ˆ 0= If Aˆ and B ˆ ˆ 0. If one of the two, say A(t), is a creation operator and the other, B(t), an annihilation operator, then Tâ ˆ(t)ˆ a† (t′ )

= Θ(t − t′ )ˆ a(t)ˆ a† (t′ ) ± Θ(t′ − t)ˆ a† (t′ )ˆ a(t)

= Θ(t − t′ )[ˆ a(t)ˆ a† (t′ )]∓ ± a ˆ† (t′ )ˆ a(t).

(18C.3)

Due to the commutator (anticommutator) the first term is a c-number. As such it is equal to the expectation value of the time-ordered product at zero temperature. The second term is a normal product, so that we can write â Tâ ˆ(t)ˆ a† (t′ ) = hTâ ˆ(t)ˆ a† (t′ )i0 + N ˆ(t)ˆ a† (t′ ).

(18C.4) H. Kleinert, PATH INTEGRALS

1279


The same thing is true if a and a† are interchanged (such an interchange produces merely a sign ˆ B(t ˆ ′ ) follows from the change on both sides of the equation). The general statement for A(t) bilinearity of the product. The decomposition (18C.2) of the time-ordered product of two operators can be extended to a product of n operators, where it reads ˆ 1 ) . . . A(t ˆ n) = TÂ(t

n X

˙ˆ ˙ˆ ˆ A(t ˆ N 1 ) . . . A(ti ) . . . A(tn ).

(18C.5)

i=2

A common pair of dots on top of a pair of operators denotes a Wick contraction of Section 3.10. It ˆ 1 )A(t ˆ i )i0 , multiplied indicates that the pair of operators has been replaced by the expectation hTÂ(t F by a factor (−) , if F = fermion permutations were necessary to bring the contracted operator to the adjacent positions. The remaining factors are contracted further in the same way. In this way, any time-ordered product ˆ 1 ) · · · A(t ˆ n) TÂ(t

(18C.6)

can be expanded into a sum of normal products of these operators containing successively one, two, three, etc. pairs of contracted operators. The expansion rule can be phrased most compactly by means of a generating functional R∞ R∞ R∞ ˆ A(t ˆ ′ )i0 j(t′ ) ˆ ˆ dtdt′ j(t)hTˆ A(t) i dtA(t)j(t) −1 ˆ ei −∞ dtA(t)j(t) . N (18C.7) Tê −∞ = e 2 −∞ Differentiations with respect to the source j(t) on both sides produce precisely the above decompositions. By going to thermal expectation values of (18C.7) at a temperature T , we find R∞ R∞ ˆ i dtA(t)j(t) − 2i dtdt′ j(t)G(t,t′ )j(t′ ) ˆ −∞ −∞ Te =e , (18C.8) T

with ˆ A(t ˆ ′ )i0 + hN ˆ A(t) ˆ A(t ˆ ′ )iT . G(t, t′ ) = hTÂ(t)

(18C.9)

The first term on the right-hand side is calculated at zero temperature. All finite temperature effects reside in the second term.

Notes and References The fluctuation-dissipation theorem was first formulated by H.B. Callen and T.A. Welton, Phys. Rev. 83, 34 (1951). It generalizes the relation between the diffusion constant and the viscosity discovered by A. Einstein, Ann. Phys. (Leipzig) 17, 549 (1905), and an analogous relation for induced light emission in A. Einstein, ”Strahlungs-Emission und -Absorption nach der Quantentheorie”, Verhandlungen der Deutschen Physikalischen Gesellschaft 18, 318 (1916), where he derived Planck’s black-body formula. See also the functioning of this theorem in the thermal noise in a resistor: H. Nyquist, Phys. Rev. 32, 110 (1928). K.V. Keldysh, Z. Eksp. Teor. Fiz. 47, 1515 (1964); Sov. Phys. JETP 20, 1018 (1965). See also V. Korenman, Ann. Phys. (N. Y.) 39, 72 (1966); D. Dubois, in Lectures in Theoretical Physics, Vol. IX C, ed. by W.E. Brittin, Gordon and Breach,

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New York, 1967; D. Langreth, in Linear and Nonlinear Electronic Transport in Solids, ed. by J.T. Devreese and V. Van Doren, Plenum, New York, 1976; A.M. Tremblay, B. Patton, P.C. Martin, and P. Maldague, Phys. Rev. A 19, 1721 (1979). For the derivation of the Langevin equation from the forward–backward path integral see S.A. Adelman, Chem. Phys. Lett. 40, 495 (1976); and especially A. Schmid, J. Low Temp. Phys. 49, 609 (1982). To solve the operator ordering problem, Schmid assumes that a time-sliced derivation of the forward–backward path integral would yield a sliced version of the stochastic differential equation (18.311) ηn ≡ (M/ǫ)(xn − 2xn−1 + xn−2 ) + +(M γ/2)(xn − xn−2 ) + ǫV ′ (xn−1 . The matrix ∂η/∂x has a constant determinant (M/ǫ)N (1 + ǫγ/2)N . His argument [cited also in the textbook by U. Weiss, Quantum Dissipative Systems, World Scientific, 1993, in the discussion following Eq. (5.93)] is unacceptable for two reasons: First, his slicing is not R derived. Second, the resulting determinant has the wrong continuum limit proportional to exp dt γ/2 for ǫ → 0, N = (tb − ta )/ǫ → ∞, corresponding to the unretarded functional determinant (18.270), whereas the correct limit should be γ-independent, by Eq. (18.278). The above textbook by U. Weiss contains many applications of nonequilibrium path integrals. More on Langevin and Fokker-Planck equations can be found in S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943); N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981; P. Hänggi and H. Thomas, Phys. Rep. 88, 207 (1982); C.W. Gardiner, Handbook of Stochastic Methods, Springer Series in Synergetics, 1983, Vol. 13; H. Risken, The Fokker-Planck Equation, ibid., 1983, Vol. 18; R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II , Springer, Berlin, 1985; H. Grabert, P. Schramm, and G.-L. Ingold, Phys. Rep. 168, 116 (1988). The stochastic Schrödinger equation with the Hamiltonian operator (18.446) was derived by A.O. Caldeira and A.J. Leggett, Physica A 121, 587 (1983); A 130 374(E) (1985). See also A.O. Caldeira and A.J. Leggett, Phys. Rev. A 31, 1059 (1985). A recent discussion of the relation between time slicing and Itô versus Stratonovich calculus can be found in H. Nakazato, K. Okano, L. Sch¨ ulke, and Y. Yamanaka, Nucl. Phys. B 346, 611 (1990). For the Heisenberg operator approach to stochastic calculus see N. Saito and M. Namiki, Progr. Theor. Phys. 16, 71 (1956). Recent applications of the Langevin equation to decay problems and quantum fluctuations are discussed in U. Eckern, W. Lehr, A. Menzel-Dorwarth, F. Pelzer. See also their references and those quoted at the end of Chapter 3. The quantum Langevin equation is discussed in G.W. Ford, J.T. Lewis und R.F. O’Connell, Phys. Rev. Lett. 55, 42273 (1985); Phys. Rev. A 37, 4419 (1988); Ann. of Phys. 185, 270 (1988). Deterministic models for Schrödinger wave functions are discussed in G. ’t Hooft, Class. Quant. Grav. 16, 3263 (1999) (gr-gc/9903084); hep-th/0003005; Int. J. Theor. Phys. 42, 355 (2003) (hep-th/0104080); hep-th/0105105; Found. Phys. Lett. 10, 105 (1997) (quant-ph/9612018). See also the Lecture H. Kleinert, PATH INTEGRALS


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G. ’t Hooft, How Does God Throw Dice? in Fluctuating Paths and Fields - Dedicated to Hagen Kleinert on the Occasion of his 60th Birthday, Eds. W. Janke, A. Pelster, H.-J. Schmidt, and M. Bachmann, World Scientific, Singapore, 2001 (http://www.physik.fu-berlin.de/~kleinert/fest.html). The representation in Section 18.21 is due to Z. Haba and H. Kleinert, Phys. Lett. A 294, 139 (2002) (quant-ph/0106095). See also F. Haas, Stochastic Quantization of the Time-Dependent Harmonic Oscillator , Int. J. Theor. Phys. 44, 1 (2005) (quant/ph-0406062). M. Blasone, P. Jizba, and H. Kleinert, Phys. Rev. A 71, 2005; Braz. J. Phys. 35, 479 (2005) (quant/ph-0504047); Annals Phys. 320, 468 (2005) (quant/ph-0504200). Another improvement is due to M. Blasone, P. Jizba, G. Vitiello, Phys. Lett. A 287, 205 (2001) (hep-th/0007138); M. Blasone, E. Celeghini, P. Jizba, G. Vitiello, Quantization, Group Contraction and Zero-Point Energy, Phys. Lett. A 310, 393 (2003) (quant-ph/0208012). The individual citations refer to [1] Some authors define G12 (τ ) as having an extra minus sign and the retarded Green function with a factor −i, so that the relation is more direct: GR 12 (ω) = G12 (ωm = −iω + η). See A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Sov. Phys. JETP 9, 636 (1959); or Methods of Quantum Field Theory in Statistical Physics, Dover, New York, 1975; also A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. Our definition (18.20) without a minus sign conforms with the definition of the fixed-energy amplitude in Chapter 9, Eq. (1.321), which is also a retarded Green function. [2] E.S. Fradkin, The Green’s Function Method in Quantum Statistics, Sov. Phys. JETP 9, 912 (1959). [3] G. Baym and D. Mermin, J. Math. Phys. 2, 232 (1961). One extrapolation uses Padé approximations: H.J. Vidberg and J.W. Serene, J. Low Temp. Phys. 29, 179 (1977); W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in Fortran, Cambridge Univ. Press (1992), Chapter 12.5. Since the thermal Green function are usually known only approximately, the continuation is not unique. A maximal-entropy method by R.N. Silver, D.S. Sivia, and J.E. Gubernatis, Phys. Rev. B 41, 2380 (1990) selects the most reliable result. [4] J. Schwinger, J. Math. Phys. 2, 407 (1961). [5] K.V. Keldysh, Z. Eksp. Teor. Fiz. 47, 1515 (1964); Sov. Phys. JETP 20, 1018 (1965). [6] K.-C. Chou, Z.-B. Su, B.-L. Hao, and L. Yu, Phys. Rep. 118, 1 (1985); also K.-C. Chou et al., Phys. Rev. B 22, 3385 (1980). [7] R.P. Feynman and F.L. Vernon, Ann. Phys. 24, 118 (1963); R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965, Sections 12.8 and 12.9. [8] The solution of path integrals with second time derivatives in the Lagrangian is given in H. Kleinert, J. Math. Phys. 27, 3003 (1986) (http://www.physik.fu-berlin.de/~kleinert/144). [9] H. Kleinert, Gauge Fields in Condensed Matter , op. cit., Vol. II, Section 17.3 (ibid.http/b2), and references therein.

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[10] See the review paper by S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). [11] H. Kleinert, A. Chervyakov, Phys. Lett. B 464, 257 (1999) (hep-th/9906156); Phys. Lett. B 477, 373 (2000) (quant-ph/9912056); Eur. Phys. J. C 19, 743-747 (2001) (quantph/0002067); Phys. Lett. A 273, 1 (2000) (quant-ph/0003095); Int. J. Mod. Phys. A 17, 2019 (2002) (quant-ph/0208067); Phys. Lett. A 308, 85 (2003) (quant-ph/0204067); Int. J. Mod. Phys. A 18, 5521 (2003) (quant-ph/0301081) . [12] In a first attempt to show that this functional the determinant is unity, A. Schmid, J. Low Temp. Phys. 49, 609 (1982). contrived a suitable time slicing of the differential operator in (18.320) to achieve this goal. However, since this was not derived from a time slicing of the initial forward–backwards path integral (18.230), this procedure cannot be considered as a proof. [13] The operator-ordering problem was first solved by H. Kleinert, Ann. of Phys. 291, 14 (2001) (quant-ph/0008109). [14] R. Benguria and M. Kac, Phys. Rev. Lett. 46, 1 (1981); Y.C. Chen, M.P.A. Fisher and A.J. Leggett, J. Appl. Phys. 64, 3119 (1988). [15] G.W. Ford, M. Kac, and P. Mazur, J. Math. Phys. 6, 504 (1965). G.W. Ford and M. Kac, J. Stat. Phys. 46, 803 (1987). [16] P. Langevin, Comptes Rendues 146, 530 (1908). [17] G. Parisi and Y.S. Wu, Scientia Sinica 24, 483 (1981). [18] R. Kubo, J.Math.Phys. 4, 174 (1963); R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II (Nonequilibrium Statistical Mechanics), Springer-Verlag, Berlin, 1985 (Chap. 2). [19] J. Zinn-Justin, Critical Phenomena, Clarendon, Oxford, 1989. [20] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 744 (1979); J. de Phys. 41, L403 (1981); Nucl. Phys. B 206 , 321 (1982); [21] A.J. McKane, Phys. Lett. A 76 , 22 (1980). [22] H. Kleinert and S. Shabanov, Phys. Lett. A 235, 105 (1997) (quant-ph/9705042). [23] E. Gozzi, Phys. Lett. B 201, 525 (1988). [24] A.O. Caldeira and A.J. Leggett, Physica A 121, 587 (1983); A 130 374(E) (1985). [25] More on this subject is found in the collection of articles D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.O. Stamatescu, H.D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin, 1996. [26] G. Lindblad, Comm. Math. Phys. 48, 119 (1976). This paper shows that the form (18.459) of the master equation (18.449) guarantees the positivity of the probabilities derived from the solutions. The right-hand side can be more generally P 1 ˆ ˆ 1ˆ ˆ ˆ n ρˆL ˆ m + h.c. . L L ρ ˆ + ρ ˆ L L − L h m n m n mn mn 2 2

[27] L. Diosi, Europhys. Lett. 22, 1 (1993).

[28] C.W. Gardiner, IBM J. Res. Develop. 32, 127 (1988). [29] H. Kleinert and S. Shabanov, Phys. Lett. A 200, 224 (1995) (quant-ph/9503004); K. Tsusaka, Phys. Rev. E 59, 4931 (1999). [30] H.A. Bethe, Phys. Rev. 72, 339 (1947). H. Kleinert, PATH INTEGRALS


1283

[31] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics, Wiley, New York, 1992. [32] R. Rohrlich, Am. J. Phys. 68, 1109 (2000). [33] H. Kleinert and S. Shabanov, J. Phys. A: Math. Gen. 31, 7005 (1998) (cond-mat/9504121); R. Bausch, R. Schmitz, and L.A. Turski, Phys. Rev. Lett. 73, 2382 (1994); Z. Phys. B 97, 171 (1995). [34] M. Roncadelli, Europhys. Lett. 16, 609 (1991); J. Phys. A 25, L997 (1992); A. Defendi and M. Roncadelli, Europhys. Lett. 21, 127 (1993). [35] Z. Haba, Lett. Math. Phys. 37, 223 (1996). [36] G. ’t Hooft, Found. Phys. Lett. 10, 105 (1997) (quant-ph/9612018). [37] G. ’t Hooft, How Does God Throw Dice? in Fluctuating Paths and Fields - Dedicated to Hagen Kleinert on the Occasion of his 60th Birthday, Eds. W. Janke, A. Pelster, H.-J. Schmidt, and M. Bachmann, World Scientific, Singapore, 2001 (http://www.physik.fu-berlin.de/~kleinert/fest.html).


Agri non omnes frugiferi sunt. Not all fields are fruitful. Cicero, Tusc. Quaest., 2, 5, 13

19 Relativistic Particle Orbits Particles moving at large velocities near the speed of light are called relativistic particles. If such particles interact with each other or with an external potential, they exhibit quantum effects which cannot be described by fluctuations of a single particle orbit. Within short time intervals, additional particles or pairs of particles and antiparticles are created or annihilated, and the total number of particle orbits is no longer invariant. Ordinary quantum mechanics which always assumes a fixed number of particles cannot describe such processes. The associated path integral has the same problem since it is a sum over a given set of particle orbits. Thus, even if relativistic kinematics is properly incorporated, a path integral cannot yield an accurate description of relativistic particles. An extension becomes necessary which includes an arbitrary number of mutually linked and branching fluctuating orbits. Fortunately, there exists a more efficient way of dealing with relativistic particles. It is provided by quantum field theory. We have demonstrated in Section 7.14 that a grand-canonical ensemble of particle orbits can be described by a functional integral of a single fluctuating field. Branch points of newly created particle lines are accounted for by anharmonic terms in the field action. The calculation of their effects proceeds by perturbation theory which is systematically performed in terms of Feynman diagrams with calculation rules very similar to those in Section 3.18. There are again lines and interaction vertices, and the main difference lies in the lines which are correlation functions of fields rather than position variables x(t). The lines and vertices represent direct pictures of the topology of the worldlines of the particles and their possible collisions and creations. Quantum field theory has been so successful that it is generally advantageous to describe the statistical mechanics of many completely different types of line-like objects in terms of fluctuating fields. One important example is the polymer field theory in Section 15.12. Another important domain where field theory has been extremely successful is in the theory of line-like defects in crystals, superfluids, and superconductors. In the latter two systems, the defects occur in the form of quantized vortex lines or quantized magnetic flux lines, respectively. The entropy of their classical shape fluctuations determines the temperature where the phase transitions take place. Instead of the usual way of describing these systems as ensembles of 1284

1285 particles with their interactions, a field theory has been developed whose Feynman diagrams are the direct pictures of the line-like defects, called disorder field theory [1]. The most important advantage of field theory is that it can describe most easily phase transitions, in which particles form a condensate. The disorder theory is therefore particularly suited to understand phase transitions in which defect-, vortex-, or flux-lines proliferate, which happens in the processes of crystal melting, superfluid to normal, or superconductor to normal transitions, respectively. In fact, the disorder theory is so far the only theory in which the critical behavior of the superconductor near the transition is properly understood [2]. A particular quantum field theory, called quantum electrodynamics describes with great success the electromagnetic interactions of electrons, muons, quarks, and photons. It has been extended successfully to include the weak interactions among these particles and, in addition, neutrinos, using only a few quantized Dirac fields and a quantized electromagnetic vector potential. The inclusion of a nonabelian gauge field, the gluon field, is a good candidate for explaining all known features of strong interactions. It is certainly unnecessary to reproduce in an orbital formulation the great amount of results obtained in the past from the existing field theory of weak, electromagnetic, and strong interactions. The orbital formulation was, in fact, proposed by Feynman back in 1950 [3], but never pursued very far due to the success of quantum field theory. Recently, however, this program was revived in a number of publications [4, 5]. The main motivation for this lies in another field of fundamental research: the string theory of fundamental particles. In this theory, all elementary particles are supposed to be excitations of a single line-like object with tension, and various difficulties in obtaining a consistent theory in the physical spacetime have led to an extension by fermionic degrees of freedom, the result being the so-called superstring. Strings moving in spacetime form worldsurfaces rather than worldlines. They do not possess a second-quantized field theoretic formulation. Elaborate rules have been developed for the functional integrals describing the splitting and merging of strings. If one cancels one degree of freedom in such a superstring, one has a theory of splitting and merging particle worldlines. As an application of the calculation rules for strings, processes which have been known from calculations within the quantum field theory have been recalculated using these reduced superstring rules. In this textbook, we shall give a small taste of such calculations by evaluating the change in the vacuum energy of electromagnetic fields caused by fluctuating relativistic spinless and spin-1/2 particles. It should be noted that since up to now, no physical result has emerged from superstring theory,1 there is at present no urgency to dwell deeper into the subject. By giving a short introduction into this subject we shall be able to pay tribute to some historic developments in quantum mechanics, where the relativistic generaliza1

This theory really deserves a price for having the highest popularity-per-physicality ratio in the history of science, enjoying a great amount of financial support. The situation is very similar to the geocentric medieval picture of the world.

1286

19 Relativistic Particle Orbits

tion of the Schrödinger equation was an important step towards the development of quantum field theory [6]. For this reason, many textbooks on quantum field theory begin with a discussion of relativistic quantum mechanics. By analogy, we shall incorporate relativistic kinematics into path integrals. It should be noted that an esthetic possibility to give a path Fermi statistics is based on the Chern-Simons theory of entanglement of Chapter 16. However, this approach is still restricted to 2 + 1 spacetime dimensions [7], and an extension to the physical 3 + 1 spacetime dimensions is not yet in sight.

19.1

Special Features of Relativistic Path Integrals

Consider a free point particle of mass M moving through the 3 + 1 spacetime dimensions of Minkowski space at relativistic velocity. Its path integral description is conveniently formulated in four-dimensional Euclidean spacetime where the fluctuating worldlines look very similar to the fluctuating polymers discussed in Chapter 15. Thus, time is taken to be imaginary, i.e., t = −iτ = −ix4 /c,

(19.1)

and the length of a four-vector x = (x, x4 ) is given by x2 = x2 + (x4 )2 = x2 + c2 τ 2 .

(19.2)

If xµ (λ) is an arbitrarily parametrized orbit, the well-known classical Euclidean action is proportional to the invariant length of the orbit in spacetime: L=

Z

λb

λa

q

dλ x′2 (λ),

(19.3)

and reads Acl,e = McL,

(19.4)

or, explicitly, Acl,e = Mc

Z

λb λa

ds(λ),

(19.5)

with ds(λ) ≡ dλ

q

q

x′2 (λ) = dλ x′2 (λ) + c2 τ ′2 (λ).

(19.6)

The prime denotes the derivative with respect to the parameter λ. The action is independent of the choice of the parametrization. If λ is replaced by a new parameter ¯ = f (λ), then λ λ → f (λ), 1 ′2 x′2 → x , f ′2 dλ → dλ f ′,

(19.7)


19.1 Special Features of Relativistic Path Integrals

1287

so that ds and the action remain invariant. We now calculate the Euclidean amplitude for the worldline of the particle to run from the spacetime point xa = (xa , cτa ) to xb = (xb , cτb ). For the sake of generality, we treat the case of D Euclidean spacetime dimensions. First we assume the total length L of the orbit to be fixed. The sum over all different lengths L will be performed later, since it does not depend on the shape of the path — the probability for the presence of the length L being determined by the Boltzmann-like factor e−A/¯h = e−LM c/¯h . The final result after the sum over all lengths L is well known: It has to coincide with the Green function of a Klein-Gordon field which in Euclidean time obeys the field equation (−∂b2 + M 2 c2 /¯ h2 )(xb |xa ) = δ (D) (xb − xa ),

(19.8)

solved by the Fourier integral (xb |xa ) =

Z

1 dD k eik(xb −xa ) . D 2 (2π) k + M 2 c2 /¯ h2

The sum over all path configurations of a fixed L is done with the techniques developed for polymers in Section 15.3. Each path is decomposed into N + 1 small sections of a fixed length a. For R2 ≪ aL, we obtain the limiting distribution of Eq. (15.49): (0) PL (R)

=

s

D

D 2 e−DR /2aL , 2πaL

(19.9)

where R denotes the distance between the endpoints in the D-dimensional Euclidean spacetime: R=

q

(xb − xa )2 .

(19.10)

Together with the probability factor e−A/¯h , the paths of a fixed length L yield the amplitude −M cL/¯ h

(xb L|xa 0) ≈ e

s

D

D 2 e−DR /2aL . 2πaL

(19.11)

It possesses the Fourier decomposition (xb L|xa 0) ≈

Z

dD k −M cL/¯h−ak2 L/2D ik(xb −xa ) e e . (2π)D

(19.12)

After an integration over L, the continuum limit a → 0 should yield the amplitude (19.9), up to some normalization factor. This, however, is impossible since the limit

1288


removes the k 2 -term in the exponent of (19.12). This term must survive if we want to obtain the correct amplitude, which satisfies (xb L|xa 0) ≈

Z

dD k M 2 c2 h ¯ L exp − + k2 2 D (2π) Mc h ¯ "

!#

eik(xb −xa ) .

(19.13)

The polymer expression (19.12) takes this form if the fluctuating relativistic orbit of a scalar particle is assumed to possess a finite effective length parameter aeff = 2D

h ¯ , Mc

(19.14)

which survives the continuum limit a → 0. In the polymer language, the orbits are stiff over a finite persistence length ξ ≈ aeff [see (15.158)]. In the present context, we call this property of relativistic particle orbits quantum stiffnessThe length scale contained in aeff , λC M ≡

h ¯ , Mc

(19.15)

is the well-known Compton wavelength of a particle of mass M [recall Eq. (4.340)]. It is a measure for the persistence length of quantum stiffness. To account for quantum stiffness in the action, we have to extend (19.4) by a curvature energy to Ae = Mc

Z

L 0



h ¯2 ds 1 + 2 2 M c

∂u ∂s

!2 

,

(19.16)

where uµ is the tangent unit vector pointing in the direction of the orbit. In the general parametrization of the orbit by λ, the tangent vector is given by x′µ uµ = √ , x′2

(19.17)

∂ µ 1 x′′µ (x′ x′′ )x′µ u = √ ∂λ uµ = ′2 − . ∂s x (x′2 )2 x′2

(19.18)

so that

Using this, the action can also be written as Ae = Mc

Z

λb λa

√ h ¯ 2 x′′ 2 − (x′ x′′ )2 /x′2 dλ x′2 1 + . Mc2 x′4 "

#

(19.19)

This expression is invariant under reparametrization λ → f (λ),


19.2 Proper Action for Fluctuating Relativistic Particle Orbits

1289

with arbitrary functions f (λ) > 0. In the classical limit h ¯ → 0, the quantum stiffness term disappears and the particle orbit is governed only by the first term proportional to the length L. Since the spatial distribution of a stiff polymer is not of the purely Gaussian type, the stiffness term alone is insufficient to find the correct correlation function of the Klein-Gordon field. These would contain undesirable fluctuation effects which must be eliminated by including higher derivative terms (∂s2 u)2 , . . . into the action, to every order in h ¯. Obviously, a polymer-like path integral description of a relativistic particle becomes extremely complicated and must therefore be rejected.

19.2

Proper Action for Fluctuating Relativistic Particle Orbits

To find a more satisfactory path integral description of a relativistic particle propagator we set up an improved action whose fluctuations differ from those in (19.5) without changing the classical orbits.

19.2.1

Gauge-Invariant Formulation

The new Euclidean action has the form = A¯(34) e

Z

λb

λa

#

"

Mc Mc ′2 x (λ) + h(λ) . dλ 2h(λ) 2

(19.21)

It has the advantage of containing the particle orbit quadratically as in the free nonrelativistic action, but at the expense of an extra dimensionless fluctuating variable h(λ). The Euclidean metric in (19.21) is δµν . Let us check that the new action coincides with the initial one (19.5) at the classical level. Extremizing A¯e in h(λ) gives the relation q

x′2 (λ).

h(λ) =

(19.22)

Inserting this back into A¯e renders the classical action Acl,e = Mc

Z

λb

λa

q

dλ x′2 (λ),

(19.23)

which is the same as (19.5). The new action shares with the old action (19.5) the reparametrization invariance (19.20) for arbitrary fluctuating path configurations. We only have to assign an appropriate transformation behavior to the extra field h(λ). If λ is replaced by a ¯ = f (λ), then x′2 and dλ transform as in (19.7) and the action new parameter λ remains invariant, if h(λ) is simultaneously changed as h → h/f ′ .

(19.24)

1290


We now set up a path integral starting from the action (19.21). First we sum over the orbital fluctuations at a fixed h(λ). To find the correct measure of integration, we use the canonical formulation in which the Euclidean action reads A¯e [p, x] =

Z

λb λa

"

#

h(λ) 2 dλ −ipx + p + M 2 c2 . 2Mc ′

(19.25)

This must be sliced in the length parameter λ. We form N + 1 slices as usual, choosing arbitrary small parameter differences ǫn = λn − λn−1 depending on n, and write the sliced action as A¯N e [p, x]

=

N +1 X n=1

"

p2 Mc −ipn (xn − xn−1 ) + hn ǫn n + ǫn hn . 2Mc 2 #

(19.26)

The path integral has then a universal measure and reads Z

D

D x

Z

" # NY N Z +1 Z D D p −A¯e [p,x]/¯h Y dD pn −A¯Ne [p,x]/¯h D e ≈ d xn e . (19.27) (2π¯ h)D (2π¯ h)D n=1 n=1

The momenta are integrated out to give (setting λN +1 ≡ λb , hN +1 ≡ hb ) 1 q

2π¯ hǫb hb /Mc

D

N Y

n=1

 Z   q

D

d xn

2π¯ hǫn hn /Mc



 D  exp

1 − A¯N [x] , h ¯ e

(19.28)

with the time-sliced action in configuration space A¯N e [x] =

N +1 X n=1

Mc Mc (∆xn )2 + ǫn hn . 2hn ǫn 2

(19.29)

The Gaussian integrals over xn in (19.28) can now be done successively as in Chapter 2 and we find Mc (xb − xa )2 Mc exp − − L , q D 2¯ h L 2¯ h 2π¯ hL/Mc 1

"

#

(19.30)

where L is the total sliced length of the orbit N +1 X

L≡

ǫn hn ,

(19.31)

dλ h(λ).

(19.32)

n=1

whose continuum limit is L=

Z

λb

λa

Remarkably, the result (19.30) does not depend on the function h(λ) but only on L. This is a reflection of the reparametrization invariance of the path integral. While H. Kleinert, PATH INTEGRALS


1291

the total λ-interval changes under the transformation, the total length L of (19.32) is invariant under the joint transformations (19.20) and (19.24). This invariance permits only the invariant length L to appear in the integrated expression (19.30), and the path integral over h(λ) can be reduced to a simple integral over L. The appropriate path integral for the time evolution amplitude reads (xb |xa ) = N

Z

∞

0

dL

Z

Dh Φ[h]

Z

¯

D D x e−Ae /¯h ,

(19.33)

where N is some normalization factor and Φ[h] an appropriate gauge-fixing functional .

19.2.2

Simplest Gauge Fixing

The simplest choice for the latter is a δ-functional, Φ[h] = δ[h − c],

(19.34)

which fixes h(λ) to be equal to the light velocity everywhere, and relates L = c(λb − λa ).

(19.35)

This relation makes the dimension of the parameter λ timelike. This Lorentzinvariant time parameter is the so-called proper time of special relativity, and should not be confused with the parameter time τ contained in the Dth component xD = cτ [see Eq. (19.1) for D = 4]. By analogy with the discussion of thermodynamics in Chapter 2 we shall then denote λb − λa by h ¯ β and write (19.35) as ¯ β. L = c(λb − λa ) ≡ c h

(19.36)

If we further use translational invariance to set λa = 0, we arrive at the gauge-fixed path integral (xb |xa ) = N c h ¯

Z

∞ 0

−βM c2 /2

dβ e

Z

D D x e−A0,e /¯h ,

(19.37)

with A0,e =

Z

0

hβ ¯

dλ

M 2 x˙ . 2

(19.38)

Since λ is now timelike, we use a dot to denote the derivative: x(λ) ˙ ≡ dx(λ)/dλ. Remarkably, the gauge-fixed action coincides with the action of a free nonrelativistic particle in D Euclidean spacetime dimensions. Having taken the trivial term R ¯hβ h out of the action (19.25), the expression (19.37) contains a Boltz0 dλ Mc/2¯ 2 mann weight e−βM c /2 multiplying each particle orbit of mass M.

1292


The solution of the path integral is then given by ¯ (xb |xa ) = N c h

Z

∞

0

1

dβ q D 2π¯ h2 β/M

M (xb − xa )2 Mc2 exp − . −β 2¯ h h ¯β 2 "

#

(19.39)

By Fourier-transforming the x-dependence, this amplitude can also be written as (xb |xa ) = N c h ¯

∞

Z

0

−βM c2 /2

Z

dD k h ¯ 2k2 exp ik(x − x ) − β , b a (2π)D 2M

(19.40)

2Mc h ¯

Z

dD k 1 eik(xb −xa ) . D 2 (2π) k + M 2 c2 /¯ h2

(19.41)

dβ e

"

#

and evaluated to (xb |xa ) = N Upon setting λC h ¯ M = , 2 2Mc

N =

(19.42)

this becomes the standard Green function of the Klein-Gordon field (19.9). In the Fourier representation (19.41), the integral over k [or the integral over β in (19.39)] can be performed with the explicit result for the Green function 1 (xb |xa ) = (2π)D/2

Mc √ h ¯ x2

!D/2−1

√ KD/2−1 Mc x2 /¯ h ,

(19.43)

where Kν (z) denotes the modified Bessel function and xq≡ xb − xa . In the nonrelativistic limit c → ∞, the asymptotic behavior Kν (z) → π/2ze−z leads to c→∞

(xb |xa ) = (xb τa |xa τa ) − −−→

h ¯ −M c2 (τb −τa )/¯h e (xb τb |xa τa )Schr , 2Mc

(19.44)

with the usual Euclidean time evolution amplitude of the free Schrödinger equation 1

(xb τb |xa τa )Schr = q D−1 2π¯ h(τb − τa )/M

M (xb − xa )2 exp − . 2¯ h τb − τa (

)

(19.45)

The exponential prefactor in (19.44) contains the effect of the rest energy Mc2 which is ignored in the nonrelativistic Schrödinger theory. Note that the same limit may be calculated conveniently in the saddle point approximation to the β-integral (19.39). For c → ∞, the exponent has a sharp extremum at β=

q

(xb −xa )2 c¯ h

=

q

(xb −xa )2 +c2 (τb −τa )2 c¯ h

→ c→∞

τb −τa (xb −xa )2 + 2 + . . . ,(19.46) h ¯ 2c h ¯ (τb − τa )

and the β-integral can be evaluated in a quadratic approximation around this value. This yields once again (19.44). H. Kleinert, PATH INTEGRALS

1293


19.2.3

Partition Function of Ensemble of Closed Particle Loops

The diagonal amplitude (19.37) with xb = xa contains the sum over all lengths and shapes of a closed particle loop in spacetime. This sum can be made a partition function of a closed loop if we remove a degeneracy factor proportional to 1/L from the integral over L. Then all cyclic permutations of the points of the loop are counted only once. Apart from an arbitrary normalization factor to be fixed later, the partition function of a single closed loop reads Z1 =

Z

∞

0

dβ −βM c2 /2 e β

Z

D D x e−A0,e /¯h .

(19.47)

Inserting the right-hand integral in (19.40) for the path integral (with xb = xa ), this becomes Z 1 = VD

Z

∞ 0

dβ −βM c2 /2 Z dD k h ¯ 2 k2 e exp −β , β (2π)D 2M !

(19.48)

immediately. The where VD is the total volume of spacetime. This can be evaluated q Gaussian integral gives for each of the D dimensions a factor 1/ 2π¯ h2 β/M, after which formula (2.490) leads to Z 1 = VD

Z

0

∞

VD dβ −βM c2 /2 1 1 e Γ(1 − D/2), q D = C D β (4π)D/2 2 λ M 2π¯ h β/M

(19.49)

where λC M is the Compton wavelength (19.15). With the help of the sloppy formula (2.498) of analytic regularization which implies the minimal subtraction explained in Subsection 2.15.1, the right-hand side of (19.48) can also be written as Z1 = −VD

Z

dD k 2 2 2 2 log k + M c /¯ h . (2π)D

(19.50)

The right hand side can be expressed in functional form as

Z1 = −Tr log −∂ 2 + M 2 c2 /¯ h2 = −Tr log −¯h2 ∂ 2 + M 2 c2 ,

(19.51)

the two expressions being equal in the analytic regularization of Section 2.15, since a constant inside the logarithm gives no contribution by Veltman’s rule (2.500). The partition function of a grand-canonical ensemble is obtained by exponentiating this: Z = eZ1 = e−Tr log(−¯h

2 2 ∂ +M 2 c2

).

(19.52)

In order to interprete this expression physically, we separate the integral dD k/(2π)D into an integral over the temporal component k D and a spatial remainder, and write

R

k 2 + M 2 c2 /¯ h2 = k D

2

+ ωk2 /c2 ,

(19.53)

1294


with the frequencies q

ωk ≡ c k2 + M 2 c2 /¯ h2 .

(19.54)

Recalling the result (2.497) of the integral (2.483) we obtain Z1 = −2VD

Z

dD−1 k h ¯ ωk . D−1 (2π) 2c

(19.55)

The exponent is the sum of two ground state energies of oscillators of energy h ¯ ωk /2, which are the vacuum energies associated with two relativistic particles. In quantum field theory one learns that these are particles and antiparticles. Many neutral particles are identical to their antiparticles, for example photons, gravitons, and the pion with zero charge. For these, the factor 2 is absent. Then the integral (19.48) contains a factor 1/2 accounting for the fact that paths running along the same curve in spacetime but in the opposite sense are identified. Comparing (19.52) with (3.556) and (3.619) for j = 0 we identify −Z1 h ¯ with −W [0] and the Euclidean effective action Γ of the ensemble of loops: h = Γe /¯ h. −Z1 = −W [0]/¯

19.2.4

(19.56)

Fixed-Energy Amplitude

The fixed-energy amplitude is related to (19.33) by a Laplace transformation: (xb |xa )E ≡ −i

Z

∞

τa

dτb eE(τb −τa )/¯h (xb |xa ) ,

(19.57)

where τb , τa are once more the time components in xb , xa . As explained in Chapter 9, the poles and the cut along the energy axis in this amplitude contain all information on the bound and continuous eigenstates of the system. The fixed-energy amplitude has the reparametrization-invariant path integral representation [here with the conventions of Eq. (19.21)] h ¯ (xb |xa )E = 2Mc with the Euclidean action A¯e,E =

Z

λb

λa

Z

0

∞

dL

Z

Dh Φ[h]

Z

¯

D D x e−Ae,E /¯h ,

Mc ′2 E2 Mc dλ x (λ) − h(λ) + h(λ) . 3 2h(λ) 2Mc 2 "

#

(19.58)

(19.59)

To prove this, we write the temporal xD -part of the sliced D-dimensional action (19.29) in the canonical form (19.26). In the associated path integral (19.27), we integrate out all xD n -variables, producing N δ-functions. These remove the integrals D over N momentum variables pD n , leaving only a single integral over a common p . D The Laplace transform (19.57), finally, eliminates also this integral making p equal to −iE/c. In the continuum limit, we thus obtain the action (19.59). The path integral (19.58) forms the basis for studying relativistic potential problems. Only the physically most relevant example will be treated here. H. Kleinert, PATH INTEGRALS

1295

19.3 Tunneling in Relativistic Physics

19.3

Tunneling in Relativistic Physics

Relativistic harbors several new tunneling phenomena, of which we want to discuss two especially interesting ones.

19.3.1

Decay Rate of Vacuum in Electric Field

In relativistic physics, an empty Minkowski space with a constant electric field E is unstable. There is a finite probability that a particle-antiparticle pair can be created. For particles of mass M, this requires the energy Epair = 2Mc2 .

(19.60)

This energy can be supplied by the external electric field. If the pair of charge ±e is separated by a distance which is roughly twice the Compton wavelength λC M = C h ¯ /Mc of Eq. (19.15), it gains an energy 2|E|λM e. The decay will therefore become significant when |E| > Ec =

M 2 c3 . e¯h

(19.61)

Euclidean Action In Chapter 17 we have shown that in the semiclassical limit where the decay-rate is small, it is proportional to a Boltzmann-like factor e−Acl,e /¯h , where Acl,e is the action of a Euclidean classical solution mediating the decay. Such a solution is easily found. We use the classical action in the form (19.23) and choose the parameter λ to measure the imaginary time λ = τ = it = x4 /c. Then the action takes the form Acl,e =

Z

τb

τa

q

dτ Mc2 1 + x˙ 2 (τ )/c2 − e E · x(τ ) .

(19.62)

This is extremized by the classical equation of motion M

˙ ) x(τ d q = −eE, dτ 1 + x˙ 2 (τ )/c2

(19.63)

whose solutions are circles in spacetime of a fixed E-dependent radius lE : 2

2

2

(x − x0 ) + c (τ − τ0 ) =

lE2

≡

Mc2 eE

!2

,

E ≡ |E|.

(19.64)

ˆ − τ -plane, where E ˆ is To calculate their action we parametrize the circles in the E the unit vector in the direction of E, by an angle θ as ˆ cos θ + x0 , x(θ) = lE E

τ (θ) =

lE sin θ + τ0 . c

(19.65)

1296


A closed circle has action Acl,e = Mc2

lE c

Z

2π 0

dθ cos θ

1 Ec − cos θ = Mc lE π = h ¯ π. cos θ E

(19.66)

The decay rate of the vacuum is therefore proportional to Γ ∝ e−πEc /E .

(19.67)

The circles (19.64) are, of course, the space-time pictures of the creation and the annihilation of particle-antiparticle pairs at times τ0 − lE /c and τ0 + lE /c and positions x0 , respectively (see Fig. 19.1). A particle can also run around the circle (x0 , cτ0 + lE ) r

?

6 (x0 cτ0 )

r (x0 , cτ0 − lE )

Figure 19.1 Spacetime picture of pair creation at the point x0 and time τ0 − lE /c and annihilation at the later time τ0 + lE /c.

repeatedly. This leads to the formula Γ∝

∞ X

Fn e−nπEc /E ,

(19.68)

n=1

with fluctuation factors Fn which we are now going to determine. Fluctuations As explained above, fluctuations must be calculated with the relativistic Euclidean action (19.21), in which we have to include the electric field by minimal coupling: A¯e =

Z

λb λa

"

#

M ′2 Mc e dλ x (λ) + h(λ) − i A(x(λ)) x′ (λ) . 2h(λ) 2 c

(19.69)

The vacuum will decay by the creation of an ensemble of pairs which in Euclidean spacetime corresponds to an ensemble of particle loops. For free particles, the partition function Z was given in (19.52) as an exponential of the one-loop partition function Z1 in (19.47). The corresponding Z1 in the presence of an electromagnetic field has the one-loop partition function Z1 =

Z

0

∞


Z

¯

D 4 x e−Ae /¯h ,


1297


with the Euclidean action A¯e =

Z

hβ ¯

dλ

0

M ′2 e x (λ) − i A(x(λ)) x′ (λ) . 2 c

(19.71)

The equations of motion are now e (x′4 E − i x′ × B) , Mc

x′′ =

x′′4 = −

e ′ x · E, Mc

(19.72)

If both a constant electric and a constant magnetic field are present, the vector potential is Aµ = −Fµν xν /2 and the action (19.71) takes the simple quadratic form A¯e =

Z

hβ ¯

0

M ′2 e dλ x − i Fµν xµ x′ν , 2 2c

(19.73)

and the equations of motion (19.72) are e x′′µ = −i F µν x′ν , c

with Fij = −ǫijk B k , F i4 = iF i0 = iE i .

(19.74)

In a purely electric field, the solutions are circular orbits: ˆ A cos ω E (λ − λ0 ) + c2 , x(λ) = E L

x4 (λ) = A sin ωLE (λ − λ0 ) + c4 ,

(19.75)

with the electric version of the Landau or cyclotron frequency (2.640) ωLE ≡

eE . Mc

(19.76)

The circular orbits are the same as those in the previous formulation (19.65). If E points in the z-direction, the action (19.73) decomposes into two decoupled quadratic actions A¯(12) + A¯(34) for the motion in the x1 − x2 and x3 − x4 -planes, respectively, e e and the one-loop partition function (19.70) factorizes as follows: Z1 =

Z

≡

Z

∞

0 ∞

0

dβ −βM c2 /2 ¯(12) e D 2 x(12) e−Ae /¯h β dβ −βM c2 /2 (12) e Z (0)Z (34) (E). β Z

Z

¯(34) /¯ h

D 2 x(34) e−Ae

(19.77)

The path integral for Z (12) (0) collecting the fluctuations in the x1 − x2 -plane have the trivial action A¯(12) = e

Z

0

hβ ¯

dλ

M ′2 (x + x′2 2 ), 2 1

(19.78)

with the trivial fluctuation determinant Det (−∂λ2 ) = 1, so that we obtain for Z (12) (0) the free-particle partition function in two dimensions Z

(12)

(0) = ∆x1 ∆x2

s

2

M . 2π¯ h2 β

(19.79)

1298


The factor ∆x1 ∆x2 is the total area of the system in the x1 − x2 -plane. Note that upper and lower indices are the same in the present Euclidean metric. For the motion in the x3 − x4 -plane, the quadratic fluctuations with periodic boundary conditions have a functional determinant −∂λ2 −iωLE ∂λ Det iωLE ∂λ −∂λ2

!

= Det −∂λ2 ×Det −∂λ2 + ωLE

2

=1×

sinh ωLE h ¯β , (19.80) E ωL h ¯β

leading to the partition function becomes Z

(34)

(E) = ∆x3 ∆x4

s

2

M sinh ωLE h ¯β . 2 E ωL h ¯β 2π¯ hβ

(19.81)

This result can, of course, also be obtained without calculation from the observation that the Euclidean electric path integral is completely analogous to the real-time magnetic path integral solved in Section 2.18. Indeed, with the E-field pointing in the z-direction, the action (19.73) becomes for the motion in the x3 − x4 -plane A¯(34) = e

Z

hβ ¯

0

dλ

M ′2 e x3 + x′4 2 + E(x3 x′4 − x4 x′3 ) . 2 c

(19.82)

This coincides with the magnetic action (2.627) in real time, if we insert there the magnetic vector potential (2.628) and replace B by E. The equations of motion (19.72) reduces to x′′3 = ωLE x′4 ,

x′′4 = −ωLE x′3 ,

(19.83)

in agreement with the magnetic equations of motion (2.664) in real time. Thus the motion in the x3 − x4 -plane as a function of the pseudotime λ is the same as the real-time motion in the x − y -plane, if the magnetic field B in the z-direction is exchanged by an E-field of the same size pointing in the x3 -axis. The amplitude can therefore be taken directly from (2.660) — we must merely replace the real time difference tb − ta by h ¯ β. Inserting (19.79) and (19.81) into (19.77), we obtain the partition function of a single closed particle orbit in four Euclidean spacetime dimensions: Z1 = ∆x4 V

Z

0

∞

dβ β

s

4

M ωLE h ¯ β/2 −βM c2 /2 e , 2 ¯ β/2 2π¯ h β sin ωLE h

(19.84)

where V ≡ ∆x1 ∆x2 ∆x3 is the total spatial volume. We now go over to real times by setting ∆x4 = ic∆t. By exponentiating the subtracted expression (19.84) as in Eq. (19.52), we go to a grand-canonical ensemble, and may identify Z1 with i times the effective electromagnetic action caused by fluctuating ensemble of particle loops Z1 = i∆Aeff /¯ h = i∆tV ∆Leff /¯ h.

(19.85)

The integral over β in (19.84) is divergent. In order to make it convergent, we perform two subtractions. In the subtracted expression we change the integration H. Kleinert, PATH INTEGRALS

1299


variable to the dimensionless ζ = βMc2 /2, and obtain the effective Lagrangian density ∆L

eff

Mc =h ¯c h ¯

4

Eζ/Ec E 2 ζ 2 −ζ −1− e . sin Eζ/Ec 6Ec3 !

1 Z ∞ dζ 4(2π)2 0 ζ 3

(19.86)

The first subtraction has removed the divergence coming from the 1/ζ 3-singularity in the integrand. This produces a real infinite field-independent contribution to the effective action which can be omitted since it is unobservable by electromagnetic experiments.2 After subtracting this divergence, the integral still contains a logarithmic divergence. It can be interpreted as a contribution proportional to E 2 to the Lagrangian density ∆Leff div

Mc =h ¯c h ¯

4

1 4(2π)2

! E 2 Z ∞ dζ α Z ∞ dζ cos ζ = cos ζ , 6Ec2 0 ζ 24π 0 ζ

(19.87)

which changes the original Maxwell term E 2 /2 to ZA E 2 /2 with α ZA = 1 + 12π

Z

0

∞

dζ cos ζ. ζ

(19.88)

According to the rules of renormalization theory, the prefactor is removed by renor1/2 malizing the field strength, replacing E → E/ZA , and identifying the replaced field 1/2 with the physical, renormalized field E/ZA ≡ ER . Due to the presence of the affective action, the vacuum is no longer time independent, put depends on time like e−i(H−iΓ¯h/2)∆t/¯h . Thus the decay rate of the vacuum per unit volume is given by the imaginary part of the effective Lagrangian density Γ = 2 Im ∆Leff . V

(19.89)

In order to calculate this we replace Eζ/Ec by z in (19.86), and expand in the integrand ∞ ∞ X X z z2 ζ2 n = 1 + 2 (−1)n 2 = 2 (−1) , sin z z − n2 π 2 ζ 2 − ζn2 n=1 n=1

ζn ≡ nπ

Ec . (19.90) E

Adding to the poles the usual infinitesimal iη-shifts in the complex plane (see p. 114), we replace with the help of the decomposition (1.325) ζ2 2

ζ ζ π π P → 2 = i δ(ζ + ζn ) − i δ(ζ − ζn ) + ζ 2 . 2 2 − ζn ζ − ζn + iη 2 2 ζ − ζn2

(19.91)

This energy would, however, be observable in the cosmological evolution to be discussed in Subsection 19.3.3.

1300


The δ-functions yield imaginary parts and thus directly the decay rate Γ Mc = 2 Im Leff = c V h ¯

4

E Ec

2

∞ 1 1X 1 (−1)n−1 2 e−nπEc /E . 3 4π 2 n=1 n

(19.92)

The principal values produces a real effective Lagrangian density ∆Leff P

Mc = h ¯c h ¯

4

1 P 4(2π)2

Z

0

∞

dζ ζ3

Eζ/Ec E 2 ζ 2 −ζ −1+ e . (19.93) sin Eζ/Ec 6Ec2 !

If we expand z z2 7 4 31 6 =1+ + z + z + ... sin z 6 360 15120

(19.94)

and perform the integrals over ζ, we find the expansion terms L

7α2 31πα3 6 4 = E + E + ... . 360M 4 630M 6

eff

(19.95)

The subscript P of L has been omitted since the imaginary part of the effective Lagrangian density (19.101) has an identically vanishing Taylor expansion. Each coefficient in (19.95) is exact to leading order in α. This expansion shows that due to the virtual creation and annihilation of particle-antiparticle pairs, the physical vacuum has a nontrivial E-dependent dielectric constant, whose lowest expansion term is is ǫ=

7α2 2 ∂ 2 Leff = E + ... . ∂E 2 30M 2

(19.96)

Such a term gives rise to a small amplitude for photon-photon scattering in the vacuum, a process which has been observed in the laboratory. Including Constant Magnetic Field parallel to Electric Field Let us see how the decay rate (19.92) and the effective Lagrangian (19.86) are influenced by the presence of an additional constant B-field. This will at first be assumed to be parallel to the E-field, with both fields pointing in the z-direction. Then the action (19.78) for the motion in the x1 − x2 -plane becomes A¯(12) = e

Z

0

hβ ¯

M ′2 e dλ x1 + x′2 2 + iB(x1 x′2 − x2 x′1 ) . 2 c

(19.97)

The partition function in the x1 − x2 -plane will therefore have the same form as (19.81), except with ωLE replaced by iωLB : Z (12) (B) = Z (34) (iB).


1301


Thus the B-field changes the partition function (19.84) of a single closed orbit to Z

Z1 = ∆x4 V

∞

0

dβ β

s

4

ωLB h M ωLE h ¯ β/2 ¯ β/2 −βM c2 /2 e , 2 E ¯ β/2 sinh ωLB h ¯ β/2 2π¯ h β sin ωL h

(19.99)

and the effective Lagrangian (19.86) becomes Mc ∆L = h ¯c h ¯

eff

4

1 4(2π)2

Z

∞

0

dζ Eζ/Ec Bζ/Ec (E 2 − B 2 )ζ 2 −ζ − 1 − e . ζ 3 sin Eζ/Ec sinh Bζ/Ec 6Ec3 (19.100) "

#

In the subtracted free-field term (19.87), the term E 2 is changed to the Lorentzinvariant combination E 2 − B 2 . The decay rate (19.92) is modified to Mc Γ = 2Im ∆Leff = c V h ¯

4

E Ec

2

∞ 1 1X 1 nπB/E (−1)n−1 2 e−nπEc /E. (19.101) 3 4π 2 n=1 n sinh nπB/E

Including Constant Magnetic Field in any Direction The case of a constant magnetic field pointing in any direction can be reduced to the parallel case by a simple Lorentz transformation. It is always possible to find a Lorentz frame in which the fields become parallel. This special frame will be called center-of-fields frame, and the transformed fields in this frame will be denoted by BCF and ECF . The transformation has the form v γ2 v v ECF = γ E + × B − ·E , c γ+1 c c v γ2 v v BCF = γ B − × E − ·B , c γ+1c c

(19.102) (19.103)

with a velocity of the transformation determined by E×B v/c = , 2 1 + (|v|/c) |E|2 + |B|2

(19.104)

and γ ≡ [1 − (|v|/c)2]−1/2 . The fields |ECF | ≡ ε and |BCF | ≡ β are, of course, Lorentz-invariant quantities which can be expressed in terms of the two quadratic Lorentz invariants of the electromagnetic field: the scalar S and the pseudoscalar P defined by 1 1 2 1 2 S ≡ − Fµν F µν = E − B2 = ε − β2 , 4 2 2

1 P ≡ − Fµν F˜ µν = E B = εβ. 4 (19.105)

Solving these equation yields ( )

ε β

≡

q√

S2

+

P2

r 1 q 2 ±S = √ (E − B2 )2 + 4(E B)2 ± (E2 − B2 ). (19.106) 2

1302


As a result we find that the formulas (19.100) and (19.101) are valid for arbitrary constant fields E and B if we replace E and B by the Lorentz invariants ε and β. After this we may expand the integrand in (19.100) in powers of ε and β using Eq. (19.94), eετ 1 e2 2 1 eβτ 2 4 τ 4 2 2 4 − ε − β + e 7ε − 10ε β + 7β = τ 3 sin eβτ sinh eετ τ 3 6τ 360 3 τ − e6 31ε6 − 49ε4 β 2 + 49ε2 β 4 − 31β 6 + . . . . (19.107) 1520

and obtain the effective Lagrangian the fields, whose lowest two terms yield the following direct generalization of (19.95): ∆L

eff

7α2 = 360M 4 31πα3 + 630M 8

n

(E2 − B2 )2 + 4(EB)2

n

o

o

(E2 − B2 )[2(E2 − B2 )2 − 4(EB)2] + . . . .

(19.108)

Spin-1/2 Particles We anticipate the small modification which is necessary to obtain the analogous result for spin-1/2 fermions: The tools for this will be developed in Subsections 19.6.1– 19.6.3. The relevant formula has actually been derived before in Eq. (7.512). The bosonic result receives for fermions a factor −1 and a fluctuation factor which is the square root of the functional determinant e Fµν 4 Det −δµν ∂λ − i Mc

e Fµν h = 4 det cosh ¯β . Mc

(19.109)

The normalization factor follows from Eqs. (7.415) and (7.418), where we found that the path integral of single complex fermion field carries a normalization factor 2. For a purely electric field, the square-root of the right-hand side of (19.109) is 2 det 1/2 [cos(e/Mc)Fµν ] = 2 cos(eE/Mc). Multiplying the cos-factor into the expansion (19.90), this is modified to z

∞ ∞ X X cos z z2 ζ2 =1+2 = 2 , 2 2 2 2 2 sin z n=1 z − n π n=1 ζ − ζn

ζn ≡ nπ

Ec . E

(19.110)

Performing now the singular integral over ζ in (19.86), we obtain for the decay rate the same formula as in (19.92), except that the alternating signs are absent. The factor 4 of the fermionic determinant reduces to a factor 2 for the effective action. The resulting effective Lagrangian density for spin-1/2 fermions is therefore ∆Leff spin 12

Mc = −¯ hc h ¯

4

1 2(2π)2

Z

0

∞

dζ ζ3

Eζ/Ec E 2 ζ 2 −ζ −1+ e , (19.111) tan Eζ/Ec 3Ec3 !


1303


a result first derived by Heisenberg and Euler in 1935 [8]. Its imaginary part yields the decay rate of the vacuum due to pair creation Γspin 1

2

V

=

2 Im ∆Leff spin 12

Mc =c h ¯

4

E Ec

2

∞ 1 X 1 −nπEc /E e . 3 4π n=1 n2

(19.112)

The reason for the reduction from 4 to 2 is that the sum over bosonic paths has to be first divided by a factor 2 to remove their orientation, before the fermionic factor 4 is applied. This procedure is not so obvious at this point but will be understood later in Subsection 19.6.2. The remaining factor 2 accounts for the two spin orientations of the charged particles. The Taylor series z2 1 2 6 z =1− − z4 − z −... tan z 3 45 945

(19.113)

in the integrand of (19.111) leads to the expansion 2α2 4 32πα3 6 E + E + ... . 45M 2 315M 4

∆Leff =

(19.114)

As in the boson result (19.95), each coefficient is exact to leading order in α, and the virtual creation and annihilation of fermion-antifermion pairs gives the physical vacuum a nontrivial E-dependent dielectric constant ǫ=

∂ 2 Leff 24α2 2 = E + ... . ∂E 2 45M 2

(19.115)

The resulting small amplitude for photon-photon scattering in the vacuum has also been observed in the laboratory. If we admit also a constant B-field parallel to E, the formulas (19.112) and (19.111) for the spin- 12 -particles are modified in the same way as the bosonic formulas (19.92) and (19.93), except that the determinant (19.109) in spinor space introduces a further factor cosh(eB/Mc). Thus we obtain ∆Leff hc spin 12 = −¯

Mc 4 1 Z ∞ dζ Eζ/Ec Bζ/Ec (E 2 −B 2 )ζ 2 −ζ − 1 + e . h ¯ 2(2π)2 0 ζ 3 tan Eζ/Ec tanh Bζ/Ec 3Ec3 (19.116)

"

#

For a general combination of constant electric and magnetic fields, we simply exchange E and B by the Lorentz invariants ε and β. From the imaginary part we obtain the decay rate Γspin 1

2

V

=

2 Im ∆Leff spin 12

Mc =c h ¯

4

ε Ec

2

∞ 1 X 1 nπβ/ε e−nπEc /ε . (19.117) 3 2 4π n=1 n tanh nπβ/ε

1304

19.3.2


Birth of Universe

A similar tunneling phenomenon could explain the birth of the expanding universe [9]. As an idealization of the observed density of matter, the universe is usually assumed to be isotropic and homogeneous. Then it is convenient to describe it by a coordinate frame in which the metric is rotationally invariant. To account for the expansion, we have to allow for an explicit time dependence of the spatial part of the metric. In the spatial part, we choose coordinates which participate in the expansion. They can be imagined as being attached to the gas particles in a homogenized universe. Then the time passing at each coordinate point is the proper time. In this context it is the so-called cosmic standard time to be denoted by t. We imagine being an observer at a coordinate point with dxi /dt = 0, and measure t by counting the number of orbits of an electron around a proton in a hydrogen atom, starting from the moment of the big bang (forgetting the fact that in the early time of the universe, the atom does not yet exist). Geometry With this time calibration, the component g00 of the metric tensor is identically equal to unity g00 (x) ≡ 1,

(19.118)

such that at a fixed coordinate point, the proper time coincides with the coordinate time, dτ = dt. Moreover, since all clocks in space follow the same prescription, there is no mixing between time and space coordinates, a property called time orthogonality, so that g0i (x) ≡ 0.

(19.119)

¯ 00 µ [recall (10.7)] vanishes identically: As a consequence, the Christoffel symbol Γ ¯ 00 µ ≡ 1 g µν (∂0 g0ν + ∂0 g0ν − g00 ) ≡ 0. Γ 2

(19.120)

This is the mathematical way of expressing the fact that a particle sitting at a coordinate point, which has dxi /dt = 0, and thus dxµ /dt = uµ = (c, 0, 0, 0), experiences no acceleration duµ ¯ 00 µ c2 = 0. = −Γ dτ

(19.121)

The coordinates themselves are trivially comoving. Under the above condition, the invariant distance has the general form ds2 = c2 dt2 − (3) gij (x)dxi dxj .


1305


We now impose the spatial isotropy upon the spatial metric gij . Denote the spatial length element by dl, so that dl2 = (3) gij (x)dxi dxj .

(19.123)

The isotropy and homogeneity of space is most easily expressed by considering the spatial curvature (3) Rijk l calculated from the spatial metric (3) gij (x). The space corresponds to a spherical surface. If its radius is a, the curvature tensor is, according to Eq. (10.160), i 1 h(3) (3) (3) (3) g (x) g (x) − g (x) g (x) . (19.124) il jk ik jl a2 The derivation of this expression in Section 10.4 was based on the assumption of a spherical space whose curvature K ≡ 1/a2 is positive. If we allow also for hyperbolic and parabolic spaces with negative and vanishing curvature, and characterize these by a constant (3)

Rijkl (x) =

  



1 0 k=   −1

 spherical  parabolic universe,   hyperbolic

(19.125)

then the prefactor 1/a2 in (19.124) is replaced by K ≡ k/a2 . For k = −1 and 0, the space has an open topology and an infinite total volume. The Ricci tensor and curvature scalar are in these three cases [compare (10.162) and (10.155)] 2 6 gil (x), (3) R = k 2 . (19.126) 2 a a By construction it is obvious that for k = 1, the three-dimensional space has a closed topology and a finite spatial volume which is equal to the surface of a sphere of radius a in four dimensions (3)

Ril = k

4

S a = 2π 2 a3 .

(19.127)

A circle in this space has maximal radius a and a maximal circumference 2πa. A sphere with radius r0 < a has a volume (3)

Vra0

=

Z

0

2π

dϕ



Z

0

π

dθ sin θ

Z

0

r

r2 dr q 1 − r 2 /a2 s

(19.128)



a3 r0 a2 r0 r0 2 = 4π  arcsin − 1 − 2 . 2 a 2 a

For small r0 , the curvature is irrelevant and the volume depends on r0 like the usual volume of a sphere in three dimensions: 4π 2 (3) a Vr 0 ≈ Vr 0 = (19.129) r0 . 3 For r0 → a, however, (3) Vra0 approaches a saturation volume 2πa3 . The analogous expressions for negative and zero curvature are obvious.

1306


Robertson-Walker Metric In spherical coordinates, the four-dimensional invariant distance (19.122) defines the Robertson-Walker metric. ds2 = c2 dt2 − dl2 dr 2 dl2 = + r 2 (dθ2 + sin2 θdϕ2 ). 1 − kr 2 /a2

(19.130) (19.131)

It will be convenient to introduce, instead of r, the angle α on the surface of the four-sphere, such that r = a sin α.

(19.132)

Then the metric has the four-dimensional angular form ds2 = c2 dt2 − a2 (t)[dα2 + f 2 (α)(dθ2 + sin2 θdϕ2 )],

(19.133)

where for spherical, parabolic, and hyperbolic spaces, f (α) is equal to   

sin α f (α) = α   sinh α

k = 1, k = 0, k = −1.

(19.134)

In order to have maximal symmetry, it is useful to absorb a(t) into the time and define a new timelike variable η by c dt = a(η) dη,

(19.135)

so that the invariant distance is measured by ds2 = a2 (η)[dη 2 − dα2 − f 2 (α)(dθ2 + sin2 θdϕ2 )].

(19.136)

Then the metric is simply 

1

  

gµν = a2 (η) 

−1

−f 2 (α)



−f 2 (α) sin2 θ,

   

(19.137)

and the Christoffel symbols become Γ00 0 =

aη aη aη , Γ00 i = 0, Γ0i 0 = 0, Γ0i j = δi j , Γij 0 = − 3 gij , Γij k = 0, (19.138) a a a

where the subscripts denote derivatives with respect to the corresponding variables: aη ≡

da a da a = ≡ at . dη c dt c


1307


We now calculate the 00-component of the Ricci tensor: R00 = ∂µ Γ00 µ − ∂0 Γµ0 µ − Γµ0 ν Γ0ν µ + Γ00 µ Γνµ ν .

(19.140)

Inserting the Christoffel symbols (19.138) we find 1 d aη = −3 2 aηη a − a2η , (19.141) dη a a 2 2 aη aη = Γ00 0 Γ00 0 + Γ00 i Γ0i 0 + Γi0 0 Γ00 i + Γi0 j Γ0j i = +3 , (19.142) a a 2 2 aη aη 0 0 0 i i 0 i k = Γ00 Γ00 + Γ00 Γi0 + Γ00 Γ0i + Γ00 Γki = +3 , (19.143) a a

∂µ Γ00 µ − ∂0 Γµ0 µ = −∂0 Γi0 i = −3 Γµ0 ν Γ0ν µ Γ00 µ Γνµ ν so that R00 = −

3 (aaηη − a2η ), a2

R0 0 = g 00 R00 = −

3 (aaηη − a2η ). a4

(19.144)

The other components can be determined by relating them to the three-dimensional curvature tensor (3) Rij which has the simple form (19.124). So we calculate Rij = Rµij µ = Rkij k + R0ij 0 = (3) Rij − Γkj 0 Γi0 k + Γij 0 Γk0k + R0ij 0 .

(19.145)

Inserting R0ij 0 = ∂0 Γij 0 − ∂i Γ0j 0 − Γ0j l Γil 0 − Γ0j 0 Γi0 0 + Γij l Γ0l 0 + Γij 0 Γ00 0 , (3)

Rij = k

2 gij a2

(19.146)

(19.147)

and the above Christoffel symbols (19.138) gives Rij = −

1 (2ka2 + a2η + aaηη )gij a4

(19.148)

and thus a curvature scalar 1 3 3 R = g R00 + g Rij = − 2 2 (aaηη − a2η ) − 4 (2ka2 + a2η + aaηη ) a a a 6 = − 3 (aηη + ka). (19.149) a 00

ij

Action and Field Equation In the absence of matter, the Einstein-Hilbert action of the gravitational field is f

A=

Z

Z f √ √ 1 d4 x −g L= − d4 x −g(R + 2λ), 2κ

(19.150)

1308


were κ is related to Newton’s gravitational constant GN ≈ 6.673 · 10−8 cm3 g−1 s−2

(19.151)

1 c3 . = κ 8πGN

(19.152)

by

A natural length scale of gravitational physics is the Planck length, which can be formed from combinations of Newton’s gravitational constant (19.151), the light velocity c ≈ 3 × 1010 cm/s, and Planck’s constant h ¯ ≈ 1.05459 × 10−27 : c3 GN h ¯

lP =

!−1/2

≈ 1.615 × 10−33 cm.

(19.153)

It is the Compton wavelength lP ≡ h ¯ /mP c associated with the Planck mass mP =

c¯ h GN

!1/2

≈ 2.177 × 10−5 g = 1.22 × 1022 MeV/c2 .

(19.154)

The constant 1/κ in the action (19.150) can be expressed in terms of the Planck length as 1 h ¯ = . κ 8πlP2

(19.155)

If we add to (19.150) a matter action and vary to combined action with respect to the metric gµν , we obtain the Einstein equation 1 1 Rµν − gµν R − λgµν = Tµν , κ 2

(19.156)

where Tµν is the energy-momentum tensor of matter. The constant λ is called the cosmological constant. It is believed to arise from the zero-point oscillations of all quantum fields in the universe. f

A single field contributes to the Lagrangian density L in (19.150) a term −Λ ≡ −λ/κ which is typically of the order of h ¯ /lP4 . For bosons, the sign is positive, for fermions negative, reflecting the filling of all negative-energy in the vacuum. A constant of this size is much larger than the present experimental estimate. In the literature one usually finds estimates for the dimensionless quantity Ωλ0 ≡

λ c2 . 3H02

(19.157)

where H0 is the Hubble constant, whose inverse is roughly the lifetime of the universe H0−1 ≈ 14 × 109 years.


1309


Present fits to distant supernovae and other cosmological data yield the estimate [10] Ωλ0 ≈ 0.68 ± 0.10.

(19.159)

The associated cosmological constant λ has the value λ = Ωλ0

3H02 Ωλ0 Ωλ0 Ωλ0 ≈ ≈ ≈ . (19.160) 2 27 2 9 2 c (6.55 × 10 cm) (6.93 × 10 ly) (2.14 Runiverse )2

Note that in the presence of λ, the Schwarzschild solution around a mass M has the metric ds2 = B(r)c2 dt2 − B −1 dr 2 − r 2 dθ2 − r 2 sin2 θdφ2 ,

(19.161)

with B(r) = 1 −

2MGN M lP 2 2 2 r2 = 1 − . − λ r − Ω λ0 c2 r 3 mP r 3 (2.14 Runiverse )2

(19.162)

The λ-term adds a small repulsion to Newton’s force between mass points. if the distances are the order of the radius of the universe. The value of the constant Λ associated with (19.159) is Λ=

λ 3H 2 l2 h ¯ = Ωλ0 2 0 P ≈ 10−122 4 . κ c 8π lP

(19.163)

Such a small prefactor can only arise from an almost perfect cancellation of the contributions of boson and fermion fields. This cancellation is the main reason for postulating a broken supersymmetry in the universe, in which every boson has a fermionic counterpart. So far, the known particle spectra show no trace of such a symmetry. Thus there is need to explain it by some other not yet understood mechanism. The simplest model of the universe governed by the action (19.150) is called Friedmann model or Friedmann universe.

19.3.3

Friedmann Model

Inserting (19.144) and (19.149) into the 00-component of the Einstein equation (19.156), we obtain the equation for the energy 3 2 2 a + ka − λ = κT0 0 . η 4 a

(19.164)

In terms of the cosmic standard time t, the general equation reads 3

"

at a

2

c2 + k 2 − λc2 = c2 κT0 0 . a #

(19.165)

1310


The simplest Friedmann model is based on an energy-momentum tensor T0 0 of an ideal pressure-less gas of mass density ρ: Tµ ν = cρuµ uν ,

(19.166) q

where uµ are the four-vectors uµ = (γ, γv/c) of the particles with γ ≡ 1/ 1 − v 2 /c2 . The gas is assumed to be at rest in our comoving coordinates. so that only the T0 0 component is nonzero: T0 0 = cρ.

(19.167)

This component is invariant under the time transformation (19.135). As a fortunate accident, this component of the Einstein equation has no aηη a term. Thus we may simply study the first-order differential equation 3 2 2 a + ka − λ = cκρ. η a4

(19.168)

Since the total volume of the universe is 2πa3 , we can express ρ in terms of the total mass M as follows ρ=

M . 2π 2 a3

(19.169)

In this way we arrive at the differential equation 3 2 κMc 4GN M (aη + ka2 ) − λ = 2 3 = . 4 a 2π a πc2 a3

(19.170)

This equation of motion can also be obtained in another way. We express the action (19.150) in terms of a(η) using the equation (19.184) for R. We use the volume (19.127) and the relation (19.135) to rewrite the integration measure as Z

4

√

d x −g =

Z

dt

(4)

a

S = 2π

2

Z

dη a4 (η),

(19.171)

so that f

A =

2π 2 2κ

Z

i

h

dη 6a(aηη + ka) − 2λa4 = ˆ

2π 2 κ

Z

i

h

dη −3a2η + 3ka2 − λa4 .(19.172)

The second expression arises from the first by a partial integration and ignoring the boundary terms which do not influence the equation of motion. The above matter is described by the action m

A =−

Z

4

√

d x −gcρ = −2π

2

Z

dη

Mc a(η). 2π 2

(19.173)

Variation with respect to a yields the Euler-Lagrange equation i 1h Mc 6(aηη + ka) − 4λa3 − 2 = 0. κ 2π


1311


Note that in terms of the Robertson-Walker time t [recall (19.135)], the equation of motion reads a¨ =

1 κMc λ a− . 3 6 2π 2 a2

(19.175)

As one should expect, the cosmological expansion is slowed down by matter, due to the gravitational attraction. A positive cosmological constant, on the other hand, accelerates the expansion. At the special value λ = λEinstein ≡

κMc 4GN M 4πGN ρ , = = 2 3 2 3 2π a πc a c

(19.176)

the two effects cancel each other and there exist a time-independent solution at a radius a and a density ρ. This is the cosmological constant which Einstein chose before Hubble’s discovery of the expanding universe to agree with Hoyle’s steadystate model of the universe (a choice which he later called the biggest blunder of his life). Multiplying (19.174) by aη and integration over η yields the pseudo-energy conservation law 3(a2η + ka2 ) − λa4 −

κMc a = const, 2π 2

(19.177)

in agreement with (19.170) for a vanishing pseudo-energy. This equation may also be written as λ a2η + ka2 − amax a − a4 = 0, 3

(19.178)

where amax ≡

κMc 4GN M = . 6π 2 3πc2

(19.179)

This looks like the energy conservation law for a point particle of mass 2 in an effective potential of the universe λ V univ (a) = ka2 − amax a − a4 , 3

(19.180)

at zero total energy. The potential is plotted for the spherical case k = 1 in Fig. 19.2. Friedmann neglects the cosmological constant and considers the equation a2η + ka2 − amax a = 0.

(19.181)

The solution of the differential equation for this trajectory is found by direct integration. Assuming k = 1 we obtain η=

Z

da q

−V univ (a)

=

Z

da q

−(a − amax /2)2 + a2max /4

= − arccos

2a . (19.182) amax

1312


V univ (a)/a2max

4 2

1

2

6

3

4

5

6

a amax

-2

a0

-4

amax

-6

Figure 19.2 Potential of closed Friedman universe as a function of the reduced radius a/amax for λa2max = 0.1. Note the metastable minimum which leads to a possible solution a ≡ a0 . A tunneling process to the right leads to an expanding universe.

With the initial condition a(0) = 0, this implies a(η) =

amax (1 − cos η). 2

(19.183)

Integrating Eq. (19.135), we find the relation between η and the physical (=proper) time t=

1 c

Z

dη a(η) =

amax 2c

Z

dη (1 − cos η) =

amax (η − sin η). 2c

(19.184)

The solution a(t) is the cycloid pictured in Fig. 19.3. The radius of the universe bounces periodically with period t0 = πamax /c from zero to amax . Thus it emerges from a big bang, expands with a decreasing expansion velocity due to the gravitational attraction, and recontracts to a point.

2.5 2

a amax

1.5 1 0.5 0.2

0.4

0.6

0.8

1

1.2

t/t0

Figure 19.3 Radius of universe as a function of time in Friedman universe, measured in terms of the period t0 ≡ πamax /c. (solid curve=closed, dashed curve=hyperbolic, dotted curve =parabolic). The curve for the closed universe is a cycloid.

Certainly, for high densities the solution is inapplicable since the pressure-less ideal gas approximation (19.167) breaks down. H. Kleinert, PATH INTEGRALS

1313


Consider now the case of negative curvature with k = −1. Then the differential equation (19.181) reads λ a2η − a2 − amax a − a4 = 0. 3

(19.185)

In order to compare the curves we shall again introduce a mass parameter M and rewrite the density as in (19.169), although M has no longer the meaning of the total mass of the universe (which is now infinite). The solution for λ = 0 is now amax (cosh η − 1), 2 amax t = (sinh η − η). 2c

a(η) =

(19.186) (19.187)

The solution is again depicted in Fig. 19.3. After a big bang, the universe expands forever, although with decreasing speed, due to the gravitational pull. The quantity amax is no longer the maximal radius nor is t0 the period. Consider finally the parabolic case k = 0, where the equation of motion (19.181) reads λ a2η − amax a − a4 = 0, 3

(19.188)

where M is a mass parameter defined as before in the negative curvature case. Now the solution for λ = 0 is simply η=2

s

a amax

,

(19.189)

η2 . 4

(19.190)

which is inverted to a(η) = amax Now we solve (19.135) with t=

amax 3 η , 12c

(19.191)

so that a(t) =

9 amax 4

1/3

(ct)2/3 .

(19.192)

This solution is simply the continuation of leading term in the previous two solutions to large t.

1314


19.3.4

Tunneling of Expanding Universe

It is now interesting to observe that the potential for the spherical universe in Fig. 19.2 allows for a time-independent solution in which the radius lies at the metastable minimum which we may call a0 . The solution a ≡ a0 is a timeindependent universe. We may now imagine that the expanding universe arises from this by a tunneling process towards the abyss on the right of the potential [9]. Its rate can be calculated from the Euclidean action of the associated classical solution in imaginary time corresponding to the motion from a0 towards the right in the inverted potential −V univ (a). Observe that this birth can only lead to universe of positive curvature. For a negative curvature, where the a2 -term in (19.180) has the opposite sign, the metastable minimum is absent.

19.4

Relativistic Coulomb System

An external time-independent potential V (x) is introduced into the path integral (19.58) by substituting the energy E by E − V (x). In the case of an attractive Coulomb potential, the second term in the action (19.59) becomes Aint = −

Z

λb

λa

dλ h(λ)

(E + e2 /r)2 , 2Mc2

(19.193)

where r = |x|. The associated path integral is calculated via a Duru-Kleinert transformation as follows [11]. Consider the three-dimensional Coulomb system where the spacetime dimension is D = 4. Then we increase the three-dimensional space in a trivial way by a dummy fourth component x4 , just as in the nonrelativistic treatment in Section 13.4. The R R 4 4 additional variable x is eliminated at the end by an integral dxa /ra = dγa , as in (13.115) and (13.122). Then we perform a Kustaanheimo-Stiefel transformation (13.101) dxµ = 2A(u)µ ν duν . This changes x′µ2 into 4~u2~u′ 2 , with the vector symbol indicating the four-vector nature. The transformed action reads: A˜e,E =

Z

λb λa

4M~u2 ′ 2 h(λ) e4 2 4 2 2 2 dλ ~u (λ) + (M c − E )~ u −2Ee − 2h(λ) 2Mc2 ~u2 ~u2

"

#

. (19.194)

We now choose the gauge h(λ) = 1, and go from λ to a new parameter s via the path-dependent time transformation dλ = f ds with f = ~u2 . Result is the DKtransformed action A¯DK e,E =

Z

sb

sa

" # 4M ′ 2 1 e4 2 4 2 2 2 ds ~u (s) + M c − E ~u − 2Ee − 2 . 2 2Mc2 ~u

(19.195)

It describes a particle of mass µ = 4M moving as a function of the “pseudotime” s in a harmonic oscillator potential of frequency 1 √ 2 4 ω= M c − E2. (19.196) 2Mc H. Kleinert, PATH INTEGRALS

1315

19.4 Relativistic Coulomb System

The oscillator possesses an additional attractive potential −e4 /2Mc2 ~u2 , which is conveniently parametrized in the form of a centrifugal barrier ¯2 Vextra = h

2 lextra , 2µ~u2

(19.197)

whose squared angular momentum has the negative value 2 lextra ≡ −4α2 .

(19.198)

Here α denotes the fine-structure constant α ≡ e2 /¯ hc ≈ 1/137. In addition, there is also a trivial constant potential Vconst = −

E 2 e. Mc2

(19.199)

If we ignore, for the moment, the centrifugal barrier Vextra , the solution of the path integral can immediately be written down [see (13.122)]: (xb |xa )E = −i

h ¯ 1 2Mc 16

Z

0

∞

dL ee

2 EL/M c2 ¯ h

Z

4π 0

dγa (~ub L|~ua 0) ,

(19.200)

where (~ub L|~ua 0) is the time evolution amplitude of the four-dimensional harmonic oscillator. There are no time slicing corrections for the same reason as in the threedimensional case. This is ensured by the affine connection of the KustaanheimoStiefel transformation satisfying Γµ µλ = g µν ei λ ∂µ ei ν = 0

(19.201)

(see the discussion in Section 13.6). Performing the integral over γa in (19.200), we obtain ! √ Z 2 ̺q h ¯ Mκ 1 ̺−ν (xb |xa )E = −i d̺ I0 2κ (rb ra + xb xa )/2 2Mc π¯ h 0 (1 − ̺)2 1−̺ " # 1+̺ × exp −κ (rb + ra ) , (19.202) 1−̺ with the variable h ≡ e−2ωL ,

(19.203)

and the parameters e2 E α =q , 2 2ω¯ h Mc M 2 c4 /E 2 − 1 µω 1√ 2 4 Eα κ = = M c − E2 = . 2¯ h h ¯c h ¯c ν ν =

(19.204) (19.205)

1316


As in the further treatment of (13.198), the use of formula (13.202) I0 (z cos(θ/2)) =

∞ 2X (2l + 1)Pl (cos θ)I2l+1 (z) z l=0

(19.206)

provides us with a partial wave decomposition ∞ 1 X 2l + 1 (rb |ra )E,l Pl (cos θ) rb ra l=0 4π

(xb |xa )E =

∞ l X 1 X ∗ (rb |ra )E,l Ylm (ˆ xb )Ylm (ˆ xa ). rb ra l=0 m=−l

=

(19.207)

The radial amplitude is normalized slightly differently from (13.205): (rb |ra )E,l

h ¯ √ 2M = −i rb ra 2Mc h ¯

Z

0

∞

dy

1 e2νy sinh y

(19.208)

√ × exp [−κ coth y(rb + ra )] I2l+1 2κ rb ra

!

1 . sinh y

At this place, we incorporate the additional centrifugal barrier via the replacement 2l + 1 → 2˜l + 1 ≡

q

2 (2l + 1)2 + lextra ,

(19.209)

as in Eqs. (8.145) and (14.238). The integration over y according to (9.29) yields (rb |ra )E,l = −i

h ¯ M Γ(−ν + ˜l + 1) Wν,˜l+1/2 (2κrb ) Mν,˜l+1/2 (2κra ) . (19.210) 2Mc h ¯ κ (2˜l + 1)!

This expression possesses poles in the Gamma function whose positions satisfy the equations ν − ˜l − 1 = 0, 1, 2, . . . . These determine the bound states of the Coulomb system. To simplify subsequent expressions, we introduce the small positive l-dependent parameter δl ≡ l − ˜l = l + 1/2 −

q

(l + 1/2)2 − α2 ≈

α2 + O(α4 ). 2l + 1

(19.211)

Then the pole positions satisfy ν = n ˜ l ≡ n − δl , with n = l + 1, l + 2, l + 3, . . . . Using the relation (19.204), we obtain the bound-state energies: Enl

#−1/2

α2 = ±Mc 1 + (n − δl )2 " # α2 α4 1 3 2 6 ≈ ±Mc 1 − 2 − 3 − + O(α ) . 2n n 2l + 1 8n 2

"

(19.212)

Note the appearance of the plus-minus sign as a characteristic property of energies in relativistic quantum mechanics. A correct interpretation of the negative energies as H. Kleinert, PATH INTEGRALS

1317

19.4 Relativistic Coulomb System

positive energies of antiparticles is straightforward only within quantum field theory, and will not be discussed here. Even if we ignore the negative energies, there is poor agreement with the experimental spectrum of the hydrogen atom. The spin of the electron must be included to get more satisfactory results. To find the wave functions, we approximate near the poles ν ≈ n ˜l : (−)nr 1 Γ(−ν + ˜l + 1) ≈ − , nr ! ν − n ˜l ¯ 2 κ2 E 2 2Mc2 1 2h ≈ , 2 ν−n ˜l n ˜ l 2M Mc2 E 2 − Enl E 1 1 κ ≈ , Mc2 aH n ˜l

(19.213)

with the radial quantum number nr = n − l − 1. By analogy with the nonrelativistic equation (13.207), the last equation can be rewritten as κ=

1 1 , a ˜H ν

(19.214)

where a ˜H ≡ aH

Mc2 E

(19.215)

denotes a modified energy-dependent Bohr radius [compare (4.339)]. It sets the length scale of relativistic bound states in terms of the energy E. Instead of being 1/α ≈ 137 times the Compton wavelength of the electron h ¯ /Mc, the modified Bohr radius is equal to 1/α times h ¯ c/E. Near the positive-energy poles, we now approximate M (−)nr 1 −iΓ(−ν + ˜l + 1) ≈ 2 h ¯κ n ˜ l nr ! a ˜H

E Mc2

2

2Mc2 i¯ h . 2 2 E − Enl

(19.216)

Using this behavior and formula (9.48) for the Whittaker functions [together with (9.50)] we write the contribution of the bound states to the spectral representation of the fixed-energy amplitude as (rb |ra )E,l =

∞ h ¯ X E Mc n=l+1 Mc2

2

2Mc2 i¯ h Rnl (rb )Rnl (ra ) + . . . . 2 2 E − Enl

(19.217)

A comparison between the pole terms in (19.210) and (19.217) renders the radial wave functions Rnl (r) =

1

1 1/2 ˜ a ˜H n ˜ l (2l + 1)! ˜

v u u t

(˜ nl + ˜l)! (n − l − 1)!

×(2r/˜ nl a ˜H )l+1 e−r/ñl a˜H M(−n + l + 1, 2˜l + 2, 2r/˜ nl a ˜H ) =

1 1/2 a ˜H n ˜l

v u u (n − l − 1)! −r/ña˜ H t e (2r/˜ n

(˜ n + ˜l)!

(19.218)

˜ ˜ ˜H )l+1 L2n˜l+1 (2r/˜ nl a ˜H ). la l −l−1

1318


The properly normalized total wave functions are 1 ψnlm (x) = Rnl (r)Ylm (ˆ x). r

(19.219)

The continuous wave functions are obtained in the same way as from the nonrelativistic amplitude in formulas (13.215)–(13.223).

19.5

Relativistic Particle in Electromagnetic Field

Consider now the relativistic particle in a general spacetime-dependent electromagnetic vector field Aµ (x).

19.5.1

Action and Partition Function

An electromagnetic field Aµ (x) is included into the canonical action (19.25) in the usual way by the minimal substitution (2.636): A¯e [p, x] =

Z

λb

λa

(

h(λ) dλ −ipx + 2Mc ′

"

2

e p− A c

2 2

+M c

#)

,

(19.220)

and the amplitude (19.33): h ¯ (xb |xa ) = 2Mc

Z

∞

0

dL

Z

Dh Φ[h]

Z

¯

D D x e−Ae /¯h ,

(19.221)

with the minimally coupled action [compare (2.698)] A¯e =

Z

λb

λa

"

#

e Mc Mc ′2 dλ x (λ) + i x′ (λ)A(x(λ)) + h(λ) , 2h(λ) c 2

(19.222)

which reduces in the simplest gauge (19.34) to the obvious extension of (19.37): h ¯2 2M

(xb |xa ) =

Z

∞

0

dβ e−βM c

2 /2

Z

D 4 x e−Ae ,

(19.223)

with the action Ae = Ae,0 + Ae,int ≡

Z

0

hβ ¯

M 2 e dτ x˙ (τ ) + i x(τ ˙ )A(x(τ )) . 2 c

(19.224)

The partition function of a single closed particle loop of all shapes and lengths in an external electromagnetic field is from (19.47) Z1 =

Z

0

∞

dβ −βM c2 /2 Z D −Ae /¯h e D xe . β

(19.225)

As in (19.55) and (19.56) this yields, up to a factor 1/¯ h the effective action of an ensemble of closed particle loops in an external electromagnetic field. H. Kleinert, PATH INTEGRALS

1319

19.5 Relativistic Particle in Electromagnetic Field

19.5.2


Since the electromagnetic coupling is rather small, we can split the exponent e−A/¯h into e−A0 /¯h e−Aint /¯h and expand the second factor in powers of Aint : −Aint /¯ h

e

n Z ¯ hβ (−ie/¯ hc)n Y = dτi x(τ ˙ i )A(x(τi )) . n! 0 n=0 i=1 ∞ X

(19.226)

If the noninteracting effective action implied by Eqs. (19.49), (19.47), and (19.56) is denoted by Z ∞ dβ −βM c2 /2 VD 1 Γe,0 VD ≡ −Z1 = − e Γ(1−D/2), (19.227) q D = − C D h ¯ β 0 λM (4π)D/2 2π¯ h2 β/M

with the Compton wavelength λC M of Eq. (19.15), we obtain the perturbation expansion Γe Γe,0 = − h ¯ h ¯

Z

∞ dβ

β

0

VD

−βM c2 /2

e

q

2π¯ h2 β/M

D

∞ X (−ie/c)n

n!

n=1

*

n Z Y

i=1

0

hβ ¯

dτi x(τ ˙ i )A(x(τi ))

+

,

0

(19.228)

where h . . . i0 denotes the free-particle expectation values [compare (3.480)–(3.483)] taken in the free path integral with periodic paths with a fixed β [compare (19.48) and (19.51)]: hO[x]i0 ≡

R

D D x O[x] e−Ae,0 /¯h R . D D x e−Ae,0 /¯h

(19.229)

D

q

The denominator is equal to VD / 2π¯ h2 β/M . The free effective action in the expansion (19.228) can be omitted by letting the sum start with n = 0. The evaluation of the cumulants proceeds by Fourier decomposing the vector fields as A(x) =

Z

dD k ikx e A(k), (2π)D

(19.230)

and rewriting (19.228) as Γe Γe,0 = − h ¯ h ¯

Z

0

∞

dβ −βM c2 /2 VD e q D β 2π¯ h2 β/M

"Z #* n Z + n hβ ¯ Y (−ie/¯ hc)n Y dD ki µi µi iki x(τi ) × A (ki ) dτi x˙ (τi )e .(19.231) n! (2π)D 0 n=1 i=1 i=1 0 ∞ X

First we evaluate the expectation values D

x(τ ˙ 1 )eik1 x(τ1 ) · · · x(τ ˙ n )eikn x(τn )

E

0

.

(19.232)

1320


Due to the periodic boundary conditions, we separate, as in Section 3.25, the path average x0 = x¯(τ ) [recall (3.804)], writing x(τ ) = x0 + δx(τ ),

(19.233)

and factorize (19.232) as D

ei(k1 +...+kn )x0

E D

δ x(τ ˙ 1 )eik1 δx(τ1 ) · · · δ x(τ ˙ n )eikn δx(τn )

0

E

0

.

(19.234)

The first average can be found as an average with respect to the x0 -part of the path integral whose measure was given in Eq. (3.808). It yields a δ function ensuring the conservation of the total energy and momenta of the n photons involved: D

ei(k1 +...+kn )x0

E

0

=

1 (2π)D δ (D) (k1 + . . . + kn ). VD

(19.235)

The denominator comes from the normalization of the expectation value which has R an integral dD x0 in the denominator. The second average is obtained using Wick’s theorem. The correlation function hδxµ (τ1 )δxν (τ2 )i0 , is obtained from Eq. (3.839) in the limit Ω → 0 for τ1 , τ2 ∈ (0, β). It is the periodic propagator with subtracted zero mode: hδxµ (τ1 )δxν (τ2 )i0 = δ µν G(τ1 , τ2 ) = δ µν

h ¯ ¯ ∆(τ1 , τ2 ), M

(19.236)

¯ 1 − τ2 ) is the subtracted periodic Green function Ga (τ − τ ′ ) of the where ∆(τ ω,e differential operator −∂τ2 calculated in Eq. (3.251) in the short notation of Subsection 10.12.1 [see Eq. (10.564)]. In the presently used physical units it reads: (τ − τ ′ )2 τ − τ ′ h ¯β ′ ′ ¯ ¯ ∆(τ, τ ) ≡ ∆(τ − τ ) = − + , 2¯ hβ 2 12

τ ∈ [0, h ¯ β].

(19.237)

The time derivatives of (19.237) are from (10.565): τ − τ ′ ǫ(τ − τ ′ ) ¯ τ ′ ) = −∆˙(τ, ¯ − , ˙∆(τ, τ ′) ≡ h ¯β 2

τ, τ ′ ∈ [0, h ¯ β].

(19.238)

With these functions it is straightforward to calculate the expectation value using P the Wick rule (3.307) for j(τ ) = ni ki δ(τ − τi ): D

eik1 δx(τ1 ) · · · eikn δx(τn )

By rewriting the right-hand side as − 12

e

Pn

i,j=1

ki kj G(τi ,τj )

− 12

=e

Pn

i,j=1

E

0

− 12

=e

eik1 δx(τ1 ) · · · eikn δx(τn )

E

0

i,j=1

ki kj G(τi ,τj )

ki kj [G(τi ,τj )−G(τi ,τi )]− 12 (

we see that if the momenta ki add up to zero, by D

Pn

 

= exp − 

n X i<j

Pn

i=1

Pn

.

(19.239)

2

i=1

ki ) G(τi ,τi )

, (19.240)

ki = 0, we can replace (19.239)  

ki kj [G(τi , τj ) − G(τi , τi )] .

(19.241)




1321


It is therefore useful to introduce the subtracted Green function G′ (τi , τj ) ≡ G(τi , τj ) − G(τi , τi ).

(19.242)

Recall the similar situation in the evaluation (5.377). An obvious extension of (19.239) is D

˙ 1 )] ˙ n )] ei[k1 δx(τ1 )+q1 x(τ · · · ei[kn δx(τn )+qn x(τ

Pn 1

−2

=e

i,j=1

E

0 1 q k i j ˙G (τi ,τj )− 2 i,j=1

Pn 1

ki kj G(τi ,τj )− 2

(19.243) Pn

i,j=1

Pn 1

ki qj G˙(τi ,τj )− 2

where the dots have the same meaning as in (10.394).

19.5.3

q q ˙G˙(τi ,τj ) i,j=1 i j

,

Lowest-Order Vacuum Polarization

Consider the lowest nontrivial case n = 2. By differentiating (19.241) with respect to iq1 and iq2 , and setting qi = 0, we obtain for k2 = −k1 = −k: D

E

h

i

x(τ ˙ 1 )eikδx(τ1 ) x(τ ˙ 2 )e−ikδx(τ2 ) = ˙G˙(τ1 , τ2 )+k 2 ˙G (τ1 , τ2 )G˙(τ1 , τ2 ) ek 0

2 [G(τ ,τ )−G(τ ,τ )] 1 2 1 1

.

(19.244)

Inserting this into (19.231) after factorization according to (19.234), we obtain the lowest correction to the effective action e2 ∆Γe=− 2¯ hc2

Z

×

Z

β

0

hβ ¯ 0

∞ dβ

dτ1

Z

e

q

2π¯ h2 β/M

hβ ¯

0

1

−βM c2 /2

D

" 2 Z Y

i=1

dD ki (2π)Dδ (D)(k1+k2 )Aµ (k1 )Aν (k2 ) (2π)D #

2

dτ2 [˙G˙(τ1 , τ2 )δ µν + k1µ k1ν ˙G (τ1 , τ2 ) G˙(τ1 , τ2 )] ek1 G (τ1 ,τ2 ) , ′

(19.245)

where we have displayed the proper vector indices. A partial integration over τ1 brings the second line to the form −

Z

0

hβ ¯

dτ1

Z

hβ ¯ 0

2

dτ2 k12 δ µν − k1µ k1ν ˙G (τ1 , τ2 ) G˙(τ1 , τ2 ) ek1 G (τ1 ,τ2 ) . h

′

(19.246)

i

¯ 1 − τ2 ) − ∆(0) ¯ Expressing G′ (τ1 , τ2 ) as (¯ h/M) ∆(τ and using the periodicity in τ2 , we calculate the integral Z ¯hβ h ¯ 2 2 µν µ ν ˙ ′ 2 (τ )e¯hk12 M [∆′p (τ )−∆′p (0)] . k δ − k k h ¯ β dτ ∆ 1 1 1 p 2 M 0

(19.247)

We now introduce reduced times u ≡ τ /¯ hβ and rewrite the Green functions for τ ∈ (0, h ¯ β), u ∈ (0, 1) as ¯β 1 ¯ 1 − τ2 ) = − h ∆(τ u(1 − u) − , 2 6 ¯˙ 1 − τ2 ) = u − 1 , ∆(τ 2

(19.248) (19.249)

1322


such that the integral in (19.247) becomes 1 4

Z

1

0

du (2u − 1)2 e−β

2 h ¯ 2 k1 u(1−u) 2M

.

(19.250)

Inserting this into (19.245) and dropping the irrelevant subscript of k1 we arrive at Z e2 Z ∞ dβ −βM c2 /2 1 dD k µ ∆Γe = e A (k)Aν (−k) q D 2¯ hc2 0 β (2π)D 2 2π¯ h β/M

× k 2 δ µν − k µ k ν

h ¯ 2 k2 h ¯ 4β 2 Z 1 du (2u − 1)2 e−β 2M u(1−u) . 2 4M 0

(19.251)

After replacing h ¯ 2 β/2M → β, the integral over β can easily be performed using the formula (2.490) with the result ∆Γe =

¯ 1 1 Z dD k µ e2h ν 2 µν µ ν A (k)A (−k) k δ − k k c2 (4π)D/2 2c (2π)D

× Γ (2 − D/2)

Z

0

1

h

du (2u − 1)2 u(1 − u)k 2 + M 2 c2 /¯ h2

iD/2−2

.

(19.252)

In the prefactor we recognize the fine structure constant α = e2 /¯ hc [recall (1.502)]. The momentum integral can be rewritten as 1 2

Z

dD k µ 1 ν 2 µν µ ν A (k)A (−k) k δ − k k = (2π)D 4

Z

dD k Fµν (−k)Fµν (k), (2π)D

(19.253)

where Fµν (x) = ∂µ Aν (x) − ∂ν Aµ (x)

(19.254)

is the tensor of electromagnetic field strengths. We now abbreviate the integral over u as follows: Π(k 2 ) ≡ α

4π Γ (2−D/2) (4π)D/2

Z

1

0

h

du (2u − 1)2 u(1 − u)k 2 + M 2 c2 /¯ h2

iD/2−2

.(19.255)

This allows us to re-express (19.252) in configuration space: ∆Γe =

1 16πc

Z

d4 xFµν (x)Π(−∂ 2 )Fµν (x),

(19.256)

where Fµν (x) is the Euclidean version of the gauge-invariant 4-dimensional curl of the vector potential: Fµν (x) = ∂µ Aν (x) − ∂ν Aµ (x).

(19.257)

In Minkowski space, the components of Fµν are the electric and magnetic fields: F0i = −F 0i = −∂ 0 Ai + ∂ i A0 = −∂0 Ai − ∂i A0 = −E i , Fij = F ij = ∂ i Aj + ∂ j Ai = −∂i Aj + ∂j Ai = −ǫijk B k .

(19.258) (19.259)


1323


This is in accordance with the electrodynamic definitions 1˙ E≡− A − ∇φ, c

B ≡ ∇ × A,

(19.260)

where A0 (x) is identified with the electric potential φ(x). In terms of Fµν (x), the Maxwell action in the presence of a charge density ρ(x) and a electric current density j(x) Aem =

Z

dt d3x

i 1 1 h 2 E (x) − B2 (x) − ρ(x)φ(x) − j(x) · A(x) 4π c

,

(19.261)

can be written covariantly as Aem = −

Z

d4 x

1 2 1 Fµν (x) + 2 j µ (x)Aµ (x) , 8πc c

(19.262)

where jµ (x) = (cρ(x), j(x))

(19.263)

is the four-vector formed by charge density and electric current. By extremizing the action (19.262) in the vector field Aµ (x) we find the Maxwell equations in the covariant form 1 ∂ν F νµ (x) = j µ (x), c

(19.264)

whose zeroth and spatial components reduce to the time-honored laws of Gauss and Ampère: ∇ · E = 4πρ (Gauss′ s law), 4π ∇×B = j (Ampère’s law). c

(19.265) (19.266)

Expressing E(x) in terms of the potential using Eq. (19.260) and inserting this into Gauss’s law, we obtain for a static point charge e at the origin the Poisson equation −∇2 φ(x) = 4πeδ (3) (x).

(19.267)

An electron of charge −e experiences an attractive mechanical potential V (x) = −eφ(x). In momentum space this satisfies the equation reads k2 V (k) = −4πe2 .

(19.268)

From this we find directly the Coulomb potential of a hydrogen atom V (x) = (∇2 )−1 4πeδ (3) (x) = −

Z

d3 k 4πe2 e2 = − , (2π)3 k2 r

r ≡ |x|,

(19.269)

1324


where e2 can be expressed in terms of the fine-structure constant α as e2 = h ¯ c α. (1.502). The Euclidean result (19.256) implies that a fluctuating closed particle orbit changes the first term in the Maxwell action (19.262) to Aeff em = −

Z

dD x

h i 1 Fµν (x) 1 + Π(−∂ 2 ) Fµν (x). 16πc

(19.270)

The quantity Π(−∂ 2 ) is the self-energy of the electromagnetic field caused by the fluctuating closed particle orbit. The self-energy changes the Maxwell equations (19.265) and (19.266) into h

1 + Π(−∂ 2 )

i

∇ · E = 4πρ, h i 4π 1 + Π(−∂ 2 ) ∇ × B = j. c

(19.271)

The static equation (19.268) for the atomic potential changes therefore into h

i

1 + Π(k2 ) k2 V (k) = −4πe2 .

(19.272)

Since Π(k2 ) is of order α ≈ 1/137, this can be solved approximately by h

V (k) ≡ −4πe2 1 − Π(k2 )

i

1 . k2

(19.273)

In real space, the attractive atomic potential is changed to lowest order in α as −

h i α α → − 1 − Π(∇2 ) . r r

(19.274)

Let us calculate this change explicitly. For small ǫ and k 2 , we expand the selfenergy (19.255) in D = 4 − ǫ dimensions as α 2 M 2 c2 eγ α¯ h2 k 2 k2 Π(k ) = − + log − + O ǫ, . (19.275) 24π ǫ 160πM 2 c2 4π¯ h2 M 2 c2 /¯ h2 2

"

#

!

Inserting this into (19.273), or into (19.273) and using the Poisson equation −∇2 × 1/r = 4πδ (3) (x), we see that the self energy changes the Coulomb potential as follows: α α2 h ¯ 2 (3) − δ (x). r 40M 2 c2 (19.276) The first term amounts to a small renormalization of the electromagnetic coupling by the factor in curly brackets, which is close to unity for finite ǫ since α is small. We are, however, interested in the result in D = 4 spacetime dimensions where ǫ → 0 and (19.276) diverges. The physical resolution of this divergence problem is to assume h iα α α 2 M 2 c2 eγ − → − 1−Π(∇2 ) ≈− 1− − +log r r 24π 2 ǫ 4π¯ h2 (

"

#)


1325

19.6 Path Integral for Spin-1/2 Particle

the initial point charge e0 in the electromagnetic interaction to be different from the experimentally observed e to precisely compensate the renormalization factor, i.e., e20

2

=e

(

M 2 c2 eγ α 2 +log 1+ − 24π 2 ǫ 4π¯ h2 "

#)

.

(19.277)

Thus, the result Eq. (19.276) is really obtained in terms of e0 , i.e., with α replaced by α0 . Then, using (19.277), we find that up to order α2 , the atomic potential is V eff (x) = −

α2 h α ¯ 2 (3) − δ (x). r 40M 2 c2

(19.278)

The second is an additional attractive contact interaction. It shifts the energies of the s-wave bound states in Eq. (19.212) slightly downwards.

19.6

Path Integral for Spin-1/2 Particle

For particles of spin 1/2 the path integral formulation becomes algebraically more involved. Let us first recall a few facts from Dirac’s theory of the electron.

19.6.1

Dirac Theory

In the Dirac theory, electrons are described by a four-component field ψα (x) in spacetime parametrized by xµ = (ct, x) with µ = 0, 1, 2, 3. The field satisfies the wave equation (i¯ h∂/ − Mc) ψ(x) = 0,

(19.279)

where ∂/ is a short notation for γ µ ∂µ and γ µ are 4 × 4 Dirac matrices satisfying the anticommutation rules {γ µ , γ ν } = 2g µν ,

(19.280)

where gµν is now the Minkowski metric  

gµν =   

1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1



  . 

(19.281)

An explicit representation of these rules is most easily written in terms of the Pauli matrices (1.445): 0

γ =

1 0 0 −1

!

,

i

γ =

0 σi − σi 0

!

,

(19.282)

1326


where 1 is a 2 × 2 unit matrix. The anticommutation rules (19.280) follow directly from the multiplication rules for the Pauli matrices: σ i σ j = δ ij + iǫijk σ k .

(19.283)

The action of the Dirac field is A=

Z

¯ (i¯ d4 x ψ(x) h∂/ − Mc) ψ(x),

(19.284)

¯ where the conjugate field ψ(x) is defined as ¯ ψ(x) ≡ ψ † (x)γ 0 .

(19.285)

¯ It can be shown that this makes ψ(x)ψ(x) a scalar field under Lorentz transformaµ ¯ tions, ψ(x)γ ψ(x) a vector field, and A an invariant. If we decompose ψ(x) into its Fourier components ψ(x) =

X k

1 √ eikx ψk (t), V

(19.286)

where V is the spatial volume, the action reads A=

Z

tb ta

dt

X k

ψk† (t) [i¯ h∂t − H(¯ hk)] ψk (t),

(19.287)

with the 4 × 4 Hamiltonian matrix H(p) ≡ γ 0 p c + γ 0 Mc2 .

(19.288)

This can be rewritten in terms of 2 × 2-submatrices as Mc p − p −Mc

H(p) =

!

c.

(19.289)

Since the matrix is Hermitian, it can be diagonalized by a unitary transformation to H d(p) =

εk 0 0 −εk

!

,

(19.290)

where q

εk ≡ c p2 + M 2 c2

(19.291)

are energies of the relativistic particles of mass M and momentum p. Each entry in (19.290) is a 2 × 2-submatrix. This is achieved by the Foldy-Wouthuysen transformation H d = eiS He−iS ,


1327


where S = −i · /2,

≡ arctan (v/c),

v ≡ p/M = velocity.

(19.293)

The vector is called rapidity. It points in the direction of the velocity v and has the length ζ = arctan (v/c), such that cos ζ = √

p2

Mc , + M 2 c2

sin ζ = √

p2

|p| . + M 2 c2

(19.294)

A function of a vector v is defined by its Taylor series where even powers of v are ˆ denotes as usual scalars v2n = v 2n and odd powers are vectors v2n+1 = v 2n v. If v ˆ ≡ v/|v|, the matrix · v ˆ has the property that all even powers the direction vector v ˆ )2n = (−1)n . of it are equal to a 4 × 4 unity matrix up to an alternating sign: ( · v ˆ ζ and the Taylor series of eiS reads explictly Thus S = −i · v (−1)n e = n=0,2,4,... n! iS

X

ζ 2

!n

(−1)n−1 ˆ) +( ·v n! n=1,3,5,... X

ζ 2

!n

= cos

ζ + 2

ζ ˆ sin . (19.295) ·v 2

Now, S commutes trivially with · p = · ˆ |p|, while anticommuting with γ 0 due to the anticommutation rules (19.280). Hence we can move the right-hand transformation in (19.292) simply to the left-hand side with a sign change of S, and obtain H d = e2iS H.

(19.296)

It is easy to calculate e2iS : we merely have to double the rapidity in (19.295) and obtain e2iS = cos ζ +

ˆ sin ζ = √ ·v

p2

Mc (1 + + M 2 c2

· p/Mc) .

(19.297)

Hence we obtain H d = e2iS H = √

Mc (1 + p2 + M 2 c2

· p/Mc) Mc2 γ 0 (1 +

· p/Mc) . (19.298)

Taking the right-hand parentheses to the left of γ 0 changes the sign of product (1 + · p/Mc) (1 − · p/Mc) is simply 1 + p2 /M 2 c2 , such that q

H d = c p2 + M 2 c2 γ 0 = εk γ 0 = h ¯ ωk γ 0 .

. The

(19.299)

Remembering γ 0 from Eq. (19.282) shows that H d has indeed the diagonal form (19.290). Going to the diagonal fields ψkd (t) = eiS ψk (t), the action becomes A=

Z

tb

ta

dt

X k

h

i

ψkd† (t) i¯ h∂t − H d (¯ hk) ψkd (t).

(19.300)

1328


Thus a Dirac field is equivalent to a sum of infinitely many momentum states, each being associated with four harmonic oscillators of the Fermi type. The path integral is a product of independent harmonic path integrals of frequencies ±ωk . It is then easy to calculate the quantum-mechanical partition function using the result (7.419) for each oscillator, continued to real times: ZQM =

o

Yn

2 cosh4 [ωk (tb − ta )/2] .

k

(19.301)

This can also be written as ZQM = exp 4

X k

!

log {2 cosh[ωk (tb − ta )/2]} ,

(19.302)

or as "

ZQM = exp 4

X k

#

Tr log (i¯ h∂t − h ¯ ωk ) = exp

(

X k

h

d

Tr log i¯ h∂t − H (¯ hk)

) i

. (19.303)

Since the trace is invariant under unitary transformations, we can rewrite this as ZQM = exp

(

X k

)

Tr log [i¯ h∂t − H(¯ hk)] ,

(19.304)

or, since the determinant of γ 0 is unity, as ZQM = exp

( X k

h

0

0

d

Tr log i¯ hγ ∂t −γ H (¯ hk)

) i

= exp

( X k

h

0

Tr log i¯ hγ ∂t −¯ hc k−Mc

2

) i

.

If we include the spatial coordinates into the functional trace, this can also be written as h

ZQM = exp Tr log i¯ hγ 0 ∂t − i¯ hc ∇−Mc2

i

= exp {Tr log [c (i¯ h∂/ − Mc)]} .

In analytic regularization of Section 2.15, the factor c in the tracelog can be dropped. Moreover, there exists a simple algebraic identity (i¯ h ∂/ + Mc)(i¯ h ∂/ − Mc) = −¯h2 ∂ 2 − M 2 c2 .

(19.305)

The factors on the left-hand side have the same functional determinant since [compare (7.338) and (7.420)] V

Det(i¯ h ∂/ − Mc) = eTr log(i¯h ∂/ −M c) = e 4 = Det(−i¯ h ∂/ − Mc).

R

d4 p Tr (2π)4

log(¯ h p/ −M c)

V4

=e

R

d4 p Tr (2π)4

log(−¯ h p/ −M c)

(19.306)

This allows us, as a generalization of (7.420), to write Det(i¯ h ∂/ − Mc) = Det(i¯ h ∂/ + Mc) =

q

Det(−¯ h2 ∂ 2 − M 2 c2 )14×4 ,

(19.307)


1329


where 14×4 is a 4 × 4 unit matrix. In this way we arrive at the quantum-mechanical partition function 1 f Tr log(−¯ h2 ∂ 2 − M 2 c2 ) ≡ eiΓ0 /¯h . 2

ZQM = exp 4 ×

(19.308)

The factor 4 comes from the trace in the 4 × 4 matrix space, whose indices have disappeared in the formula. The exponent determines the effective action Γ of the quantum system by analogy with the Euclidean relation Eq. (19.56). The Green function of the Dirac equation (19.279) is a 4 × 4 -matrix defined by (i¯ h∂/ − Mc)αβ (x|xa )βγ = i¯ hδ (D) (x − xa )δαγ .

(19.309)

Suppressing the Dirac indices, it has the spectral representation (xb |xa ) =

Z

d4 p i¯ h ′ e−ip(x−x )/¯h . 4 (2π¯ h) p/ − Mc + iη

(19.310)

This can be written more formally as a functional matrix (xb |xa ) = hxb |

i¯ h |xa i, i¯ h∂/ − Mc

(19.311)

which obviously satisfies the differential equation (19.309).

19.6.2

Path Integral

It is straightforward to write down a path integral representation for the amplitude (19.311): (xb |xa ) =

Z

∞

0

xb =x(τb )

Z

dS

xa =x(τa )

D4x

Z

D 4 p iA/¯h e , (2π¯ h)4

(19.312)

with the action A[x, p] =

Z

S

0

dτ [−px˙ + (/ p − Mc)] ,

(19.313)

where x˙ ≡ dx(τ )/dτ . We are using the proper time τ to parametrize the orbits.3 Thus S is the total time, in contrast to the length parameter L in the Euclidean discussion in the previous section [see Eq. (19.3)]. The minus sign in front of px˙ is necessary to have the positive sign for the spatial part px˙ in the Minkowski metric (19.281). The action (19.313) can immediately be generalized to ¯ p] = A[x, 3

Z

0

S

dτ [−px˙ + h(τ ) (/ p − Mc)] ,

(19.314)

It corresponds to the parameter λ in Subsection 19.3.1. From now on we prefer the letter τ , since there will be no danger of confusing the proper time with the coordinate time in Subsection 19.3.1.

1330


with any function h(τ ) > 0. This makes it invariant under the reparametrization τ → f (τ ),

h(τ ) → h(τ )/f (τ ).

(19.315)

The path integral (19.312) contains then an extra functional integration over h(τ ) with some gauge-fixing functional Φ[h], as in (19.33), which has been chosen in (19.313) as Φ[h] = δ[h − 1]. The path integral alone yields an amplitude hx|eiS(i¯h ∂/ −M c)/¯h |xa i,

(19.316)

and the integral over S in (19.312) produces indeed the propagator (19.311). In evaluating this we must assume, as usual, that the mass carries an infinitesimal negative imaginary part iη. This is also necessary to guarantee the convergence of the path integral (19.312). Electromagnetism is introduced as usual by the minimal substitution (2.636). In the operator version, we have to substitute e ∂µ − − −→ ∂µ + i Aµ . (19.317) h ¯c Thus we obtain the gauge-invariant action ¯ p] = A[x,

Z

S

0

"

e¯ h / − Mc dτ −px˙ + h(τ ) p/ − A c

!#

.

(19.318)

Another path integral representation which is closer to the spinless case is obtained by rewriting (19.311) as (x|xa ) = (i¯ h ∂/ + Mc) hx|

−¯ h2 ∂ 2

i¯ h |xa i, − M 2 c2

(19.319)

where we have omitted the negative infinitesimal imaginary part −i¯ h of the mass, for brevity, and used the fact that (i¯ h ∂/ + Mc)(i¯ h ∂/ − Mc) = −¯ h2 ∂ 2 − M 2 c2 ,

(19.320)

on account of the anticommutation relation (19.280). By rewriting (19.319) as a proper-time integral (x|xa ) =

1 (i¯ h ∂/ + Mc) 2M

Z

∞ 0

dShx|eiS (−¯h

)/2M ¯h |x i, a

(19.321)

D 4 p iA/¯h e , (2π¯ h)4

(19.322)

2 2 ∂ −M 2 c2

we find immediately the canonical path integral 1 (x|xa ) = (i¯ h∂/ + Mc) 2Mc

Z

0

∞

dS

Z

x=x(τb )

xa =x(τa )

4

D x

Z

with the action A[x, p] =

Z

0

S

1 2 2 2 dτ −px˙ + p −M c . 2M


1331


The suppressed Dirac indices of the 4 × 4 -amplitude on the left-hand side, (x|xa )αβ , are entirely due to the prefactor (i¯ h∂/ + Mc)αβ on the right-hand side. As in the generalization of (19.313) to (19.313), this action can be generalized to ¯ p] = A[x,

Z

0

S

"

#

h(τ ) 2 dτ −px˙ + p − M 2 c2 , 2Mc

(19.324)

with any function h(τ ) > 0, thus becoming invariant under the reparametrization (19.315), and the path integral (19.312) contains then an extra functional integration R Dh(τ ) Φ[h]. The action (19.324), is precisely the Minkowski version of the path integral of a spinless particle of the previous section [see Eq. (19.25)]. Introducing here electromagnetism by the minimal substitution (19.317) in the prefactor of (19.319) and on the left-hand side of (19.320), the latter becomes then e / + Mc i¯ h∂/ − A c

e i¯ h∂/ − A ¯2 / − Mc = h c

"

2

e i∂ − A h ¯c

#

e − Σµν Fµν − M 2 c2 , h ¯c (19.325)

where i Σµν ≡ [γ µ , γ ν ] = −Σνµ 4

(19.326)

are the generators of Lorentz transformations in the space of Dirac spinors. For any fixed index µ, they satisfy the commutation rules: [Σµν , Σµκ ] = ig µµ Σνκ .

(19.327)

Due to the antisymmetry in the two indices, this determines all nonzero commutators of the Lorentz group. Using Eqs. (19.258), we can write the last interaction term in (19.325) as Σµν Fµν = −2Σi B i + 2Σ0i E i ,

(19.328)

where Σi are the generators of rotation σi 0 0 σi

1 1 Σ ≡ ǫijk Σjk = 2 2 i

!

,

(19.329)

and 0i

i

0 i

Σ ≡ iα ≡ iγ γ = i

−σ i 0 0 σi

!

(19.330)

are the generators of rotation-free Lorentz transformations. Thus µν

Σ Fµν = −

(B + iE)

0 0 (B − iE)

!

.

(19.331)

1332


19.6.3

Amplitude with Electromagnetic Interaction

The obvious generalization of the path integral (19.322) which includes minimal electromagnetic interactions is then (x|xa ) =

1 2M

e i¯ h∂/ − A / +Mc c

Z

∞

Z

dS Dh(τ ) Φ[h]

0

Z

x=x(τb )

xa =x(τa )

D4x

Z

D 4 p ˆ iA/¯h Te , (2π¯ h)4 (19.332)

with the action ¯ p] = A[x,

Z

S

0

(

h(τ ) dτ −px˙ + 2Mc

"

2

e p− A c

h ¯e − Σµν Fµν − M 2 c2 c

#)

.

(19.333)

The symbol Tˆ is the time-ordering operator defined in (1.241), now with respect to the proper time τ , which has to be present to account for the possible noncommutativity of Fµν Σµν /2 at different τ . Integrating out the momentum variables yields the configuration-space path integral 1 e (x|xa ) = i¯ h ∂/ − A / +Mc 2M c with the action

¯ = A[x]

Z

0

S

Z

∞

0

dS

Z

Dh(τ ) Φ[h]

Z

x=x(τb )

xa =x(τa )

D 4 x Tˆ eiA/¯h ,(19.334) #

"

h ¯ e µν Mc Mc 2 e x˙ − xA ˙ − h(τ ) Σ F − h(τ ) . dτ − µν 2h(τ ) c 2Mc2 2

(19.335)

The coupling to the magnetic field adds to the rest energy Mc2 an interaction energy Hint = −

h ¯e · B. Mc

(19.336)

From this we extract the magnetic moment of the electron. We compare (19.336) with the general interaction energy (8.315), and identify the magnetic moment as =

h ¯e . Mc

(19.337)

Recall that in 1926, Uhlenbeck and Goudsmit explained the observed Zeeman splitting of atomic levels by attributing to an electron a half-integer spin. However, the magnetic moment of the electron turned out to be roughly twice as large as what one would expect from a charged rotating sphere of angular momentum L, whose magnetic moment is = µB

L , h ¯

(19.338)

where µB ≡ h ¯ e/Mc is the Bohr magneton (2.641). On account of this relation, it is customary to parametrize the magnetic moment of an elementary particle of spin S as follows: S = gµB . h ¯


1333


The dimensionless ratio g with respect to (19.338) is called the gyromagnetic ratio or Landé factor . For a spin-1/2 particle, S is equal to /2, and comparison with (19.337) yields the gyromagnetic ratio g = 2,

(19.340)

the famous result found first by Dirac, predicting the intrinsic magnetic moment µ of an electron to be equal to the Bohr magneton µB , thus being twice as large as expected from the relation (19.338), if we insert there the spin 1/2 for the orbital angular momentum. In quantum electrodynamics one can calculate further corrections to this Dirac result as a perturbation expansion in powers of the fine-structure constant α [recall (1.502)]. The first correction to g due to one-loop Feynman diagrams was found by Schwinger: α g =2× 1+ 2π

≈ 2 × 1.001161,

(19.341)

where α is the fine-structure constant (1.502). Experimentally, the gyromagnetic ratio has been measured to an incredible accuracy: g = 2 × 1.001 159 652 193(10),

(19.342)

in excellent agreement with (19.341). If the perturbation expansion is carried to higher orders, one is able to reach agreement up to the last experimentally known digits [13]. In the literature, there exist other representations of path integrals for Dirac particles involving Grassmann variables. For this we recall the discussion in Subsection 7.11.3 that a path integral over four real Grassmann fields θµ , µ = 0, 1, 2, 3 Z

D 4 θ exp

i h ¯

Z

i dτ θµ θ˙µ , 2

(19.343)

generates a matrix space corresponding to operators θˆµ with the anticommutation rules θˆµ , θˆν = g µν ,

n

and the matrix elements

o

hβ|θˆµ |αi = (γ5 γ µ )βα ,

β, α = 1, 2, 3, 4.

(19.344)

(19.345)

It is then possible to replace path integral (19.334) by (x|xa ) = ×

1 2M Z

Z

0

∞

dS

DχΦ[χ]

Z

Z

Dh Φ[h]

D4θ

Z

Dx

Z

Dp i(A[x]+A µ ¯ h G [θ ,A])/¯ e , (2π¯ h)4

(19.346)

1334


with the action of a relativistic spinless particle [the action (19.335) without the spin coupling] ¯ p] = A[x,

Z

S

0

(

h(τ ) dτ −px˙ + 2Mc

"

2

e p− A c

− M 2 c2

#)

,

(19.347)

and an action involving the Grassmann fields θµ : µ

AG [θ , A] =

Z

S

0

dτ

(

)

i¯ h i¯ he Fµν (x(τ ))θµ (τ )θν (τ ) . (19.348) θµ (τ )θ˙µ (τ ) − h(τ ) 4 4Mc2

This follows directly from Eq. (7.508). The function h(τ ) is the same as in the bosonic actions (19.25) and the path integral (19.33) guaranteeing the reparametrization invariance (19.24). After integrating out the momentum variables in the path integral (19.346), the canonical action is of course replaced by the configuration space action (19.222). In the simplest gauge (19.34), the total action reads h ¯ Mc2 i¯ h ¯ θµ ] = xA ˙ +i Fµν θµ θν − + θµ (τ )θ˙µ (τ ) . A[x, 4M 2 4 0 (19.349) Note that the Grassmann variables can always be integrated out, yielding the functional determinant [compare with the real-time formula (7.512)] Z

Z

i

Dx e− 4¯h

R

S

!

"

M e dτ − x˙ 2 − 2 c

dτ [θµ (τ )θ˙µ (τ )+(e/4M c)Fµν θ µ θ ν ]

= 4Det1/2 δµν ∂τ − i

#

e Fµν (x(τ )) . (19.350) Mc

For a constant field tensor, and with the usual antiperiodic boundary conditions, the result has been given before in Eq. (19.109).

19.6.4

Effective Action in Electromagnetic Field

In the absence of electromagnetism, the effective action of the fermion orbits is given by (19.308). Its Euclidean version differs from the Klein-Gordon expression in (19.48) only by a factor −2: h i Γfe,0 = −2 Tr log −¯ h2 ∂ 2 + M 2 c2 . h ¯

(19.351)

Explicitly we have from (19.49), (19.51), and (19.56): Z ∞ Γfe,0 dβ −βM c2 /2 1 VD 1 = 2VD e Γ(1 − D/2). (19.352) q D = 2 C D D/2 h ¯ β (4π) 0 2 λ M 2π¯ h β/M

The factor 2 may be thought of as 4 × 1/2 where the factor 4 comes from the free path integral over the Grassmann field, Z

D D θ e−Ae,0 [θ]/¯h = 4.


1335


counts the four components of the Dirac field. Recall that by (19.290), the Dirac field carries four modes, one of energy h ¯ ωk , with two spin degrees of freedom, the other of energy −¯ hωk with two spin degrees. The latter are shown in quantum field theory to correspond to an antiparticle with spin 1/2. The path integral over x (τ ) which counts paths in opposite directions with the ground state energy (19.49) describes particles and antiparticles [recall the remarks after Eq. 19.55]. This explains why only the spin factor 2 remains in (19.352). By including the vector potential via the minimal substitution pˆ → pˆ−(e/c)A, we obtain the Euclidean effective action from Eq. (19.225), and thus obtain immediately the path integral representation Z ∞ Z ˜f dβ −βM c2 /2 Γ e D D x e−Ae /¯h , =2 e h ¯ β 0

(19.354)

with the Euclidean action (19.224). This is not yet the true partition function Γe of the spin-1/2 particle, since the proper path integral contains the additional Grassmann terms of the action (19.349). In the Euclidean version, the full interaction is e i i x˙ µ (τ )Aµ (x(τ )) − Fµν (x(τ ))θµ (τ )θν (τ ) . (19.355) c 4M 0 Thus we obtain the path integral representation Z

Ae,int [x, θ] = Γfe =2 h ¯

Z

hβ ¯

dτ


∞

0

Z

D

D x

Z

D D θ e−{Ae,0 [x,θ]+Ae,int [x,θ]}/¯h,

(19.356)

where the free part of the Euclidean action is Ae,0 [x, θ] = Ae,0 [x] + Ae,0 [θ] ≡

19.6.5

Z

hβ ¯

0

M dτ x˙ 2 (τ ) + 2

Z

hβ ¯

0

dτ

h ¯ µ θ (τ )θ˙µ (τ ). 4

(19.357)


The perturbation expansion is a straightforward generalization of the expansion (19.228): ∞ X Γfe,0 Z ∞ dβ −βM c2 /2 Γfe 2VD (−ie/c)n = + e q D h ¯ h ¯ β n! 0 2π¯ h2 β/M n=1

×

*

n Y

i=1

(Z

0

hβ ¯

dτi

"

h ¯ x˙ µ (τi )Aµ (x(τi )) − Fµν (x(τi ))θµ (τi )θν (τi ) 4M

(19.358) #) +

. 0

The leading free effective action coincides, of course, with the n = 0-term of the sum [compare (19.352)]. The expectation values are now defined by the Grassmann extension of the Gaussian path integral (19.229): D D x D D θ O[x, θ] e−Ae,0 [x,θ]/¯h hO[x, θ]i0 ≡ R D −Ae,0 [x]/¯h R D −Ae,0 [θ]/¯h . D xe D θe R

R

(19.359)

1336

19 Relativistic Particle Orbits q

D

where the denominator is equal to (1/2)VD / 2π¯ h2 β/M × 4. There exists also an expansion analogous to (19.231), where the vector potentials have been Fourier decomposed according to (19.230). Then we obtain an expansion just like (19.231), except for a factor −2 and with the expectation values replaced as follows: *

n Z Y

i=1

0

hβ ¯

µi

iki x(τi )

dτi x˙ (τi )e →

*

n Z Y

i=1

0

hβ ¯

+

dτi

"0

#

i¯ h νi νi k θ (τi )θµi (τi ) eiki x(τi ) x˙ (τi ) + 2M i µi

+

. (19.360) 0

The evaluation of these expectation values proceeds as in Eqs. (19.232)–(19.243), except that we also have to form Wick contractions of Grassmann variables which have the free correlation functions hθµ (τ )θν (τ ′ )i = 2δ µν Gaω,e (τ − τ ′ ),

(19.361)

where 1 Gaω,e (τ − τ ′ ) = ǫ(τ ), 2

τ ∈ [−¯ hβ, h ¯ β)

(19.362)

is the Euclidean version of the antiperiodic Green function (3.109) solving the inhomogeneous equation ∂τ Gaω,e (τ ) = δ(τ ).

(19.363)

Outside the basic interval [−¯ hβ, h ¯ β) the function is to be continued antiperiodically. in accordance with the fermionic nature of the Grassmann variables. In operator language, the correlation function (19.361) is the time-ordered exˆ )θ(τ ˆ ′ )i0 [recall (3.296)]. By letting τ → τ ′ once from above pectation value hTˆ θ(τ and once from below, the correlation function shows agreement with the anticommutation rule (19.344). In verifying this we must use the fact that the time ordered product of fermion operators is defined by the following modification of the bosonic definition in Eq. (1.241): ˆ n (tn ) · · · O ˆ 1 (t1 )) ≡ ǫP O ˆ in (tin ) · · · O ˆ i1 (ti1 ), Tˆ (O

(19.364)

where tin , . . . , ti1 are the times tn , . . . , t1 relabeled in the causal order, so that tin > tin−1 > . . . > ti1 .

(19.365)

The difference lies in the sign factor ǫP which is equal to 1 for an even and −1 for an odd number of permutations of fermion variables. H. Kleinert, PATH INTEGRALS

1337


19.6.6

Vacuum Polarization

Let us see how the fluctuations of an electron loop change the electromagnetic field action. To lowest order, we must form the expectation value (19.360) for n = 0 and k1 = −k2 ≡ k: *"

"

#

#

+

i¯ h ν1 ν1 i¯ h ν2 ν2 x˙ (τ1 )+ k θ (τ1 )θµ1 (τ1 ) eikδx(τ1 ) x˙ µ2 (τ2 )− k θ (τ2 )θµ2 (τ2 ) e−ikδx(τ2 ) . 2M 2M 0 (19.366) µ1

From the contraction of the velocities x˙ µ1 (τ1 ) and x˙ µ2 (τ2 ) we obtain again the spinless result (19.244) leading in (19.246) to the integrand

k12 δ µ1 ν2

−

k1µ1 k1µ2

2

˙G (τ1 , τ2 ) =

k12 δ µ1 ν2

−

k1µ1 k1µ2

h ¯2 (u − 1/2)2. 2 M

(19.367)

In addition, there are the Wick contractions of the Grassmann variables: *"

"

#

#

h ¯ ν1 ν1 i¯ h ν2 ν2 k θ (τ1 )θµ1 (τ1 ) eikδx(τ1 ) k θ (τ2 )θµ2 (τ2 ) e−ikδx(τ2 ) 2M 2M

= − k12 δ µ1 ν2 − k1µ1 k1µ2

+

0

h ¯2 1 2 ǫ (τ1 − τ2 ). M2 4

(19.368)

Since ǫ2 (τ1 − τ2 ) = 1, this changes the spinless result (19.367) to

k12 δ µ1 ν2 − k1µ1 k1µ2 ˙G 2 (τ1 , τ2 ) = k12 δ µ1 ν2 − k1µ1 k1µ2

h ¯2 [(u − 1/2)2 − 1/4]. (19.369) M2

Remembering the factor −2 in the expansion (19.359) with respect to the spinless one, we find that the vacuum polarization due to fluctuating spin-1/2 orbits is obtained from the spinless result (19.255) by changing the factor 4(u − 1/2)2 = (2u − 1)2 in the integrand to −2 × 4u(u − 1) = 8u(1 − u). The resulting function Π(k 2 ) has the expansion 1 2 M 2 c2 eγ h ¯ 2k2 k2 Π(k ) = − log − + O ǫ, . 3π ǫ 15πM 2 c2 4π¯ h2 M 2 c2 /¯ h2 2

"

#

!

(19.370)

The first term produces a renormalization of the charge which is treated as in the bosonic case [recall (19.275)–(19.278)], which causes an additional contact interaction α α 4α2h ¯ 2 (3) − →− − δ (x). r r 15M 2 c2

(19.371)

There, the vacuum polarization has the effect of lowering the state 2S1/2 , which is the s-state of principal quantum number n = 2, against the p-state 2P1/2 by 27.3 MHz. The experimental frequency shift is positive ≈ 1057 MHz [recall Eq. (18.525)],

1338


and is mainly due to the effect of the electron moving through a bath of photons as calculated in Eq. (18.524). The effect of vacuum polarization was first calculated by Uehling [12], who assumed it to be the main cause for the Lamb shift. He was disappointed to find only 3% of the experimental result, and a wrong sign. The situation in muonic atoms is different. There the vacuum polarization does produce the dominant contribution to the Lamb shift for a simple reason: The other effects contain in a factor M/Mµ2 , where Mµ2 is the mass of the muon, whereas the vacuum polarization still involves an electron loop containing only the electron mass M, thus being enhanced by a factor (Mµ /M)2 ≈ 2102 over the others. The calculations for the electron in an atom have been performed to quite high orders [13] within quantum electrodynamics. We have gone through the above calculation only to show that it is possible to re-obtain quantum field-theoretic result within the path integral formalism. More details are given in the review article [5], As mentioned in the beginning, the above calculations are greatly simplified version of analogous calculations within superstring theory, which so far have not produced any physical results. If this ever happens, one should expect that also in this field a second-quantized field theory would be extremely useful to extract efficiently observable consequences. Such a theory still need development [14].

19.7

Supersymmetry

It is noteworthy that the various actions for a spin-1/2 particle is invariant under certain supersymmetry transformations.

19.7.1

Global Invariance

Consider first the fixed-gauge action (19.349). Its appearance can be made somewhat q ¯ /2M into the Grassmann variables θµ (τ ), more symmetric by absorbing a factor h so that it reads ¯ θµ ] = A[x,

Z

0

S

"

M e dτ − x˙ 2 − 2 c

i Mc2 M xA ˙ + Fµν θµ θν − + iθµ (τ )θ˙µ (τ ) . (19.372) 2 2 2

#

The correlation functions (19.361) of the θ-variables are now hθµ (τ )θν (τ ′ )i = δ µν Gf (τ, τ ′ ) ≡ δ µν

h ¯ f ∆0 (τ − τ ′ ), M

(19.373)

with ∆f0 (τ − τ ′ ) = ǫ(τ − τ ′ )/2. In this normalization, Gf (τ, τ ′ ) coincides, up to a sign, with the first term in the derivative ˙G(τ, τ ′ ) of the bosonic correlation function [recall (19.236) and the first term in (19.238)]. Let us apply to the variables the infinitesimal transformations δxµ (τ ) = iαθµ (τ ),

δθµ (τ ) = αx˙ µ (τ ).


1339

19.7 Supersymmetry

where α is an arbitrary Grassmann parameter. For the free terms this is obvious. The interacting terms change by Z e −iα d4 x θ˙µ Aµ + Fµν x˙ µ θν . (19.375) c Inserting Fµν x˙ µ (τ ) = dAν (x(τ ))/dτ − ∂ν [Aµ (x(τ ))x˙ µ (τ )], the first term cancels and the second is a pure surface term, such that the action is indeed invariant. Supersymmetric theories have a compact representation in an extended space called superspace. This space is formed by pairs (τ, ζ), where ζ is a Grassmann variable playing the role of a supersymmetric partner of the time parameter τ . The coordinates xµ (τ ) are extended likewise by defining X µ (τ ) ≡ xµ (τ ) + iζθµ (τ ).

(19.376)

A supersymmetric derivative is defined by ∂ ∂ + iζ ∂ζ ∂τ

µ

DX (τ ) ≡

!

X µ (τ ) = iθµ (τ ) + iζ x˙ µ (τ ).

(19.377)

If we now form the integral, using the Grassmann formula (7.379), Z

dτ

we find

Z i dζ ˙ dζ h iXµ (τ )DX µ (τ ) = dτ i x(τ ˙ )+iζ θ˙µ (τ ) [iθµ (τ ) + iζ x˙ µ (τ )] , (19.378) 2π 2π Z

dτ −x˙ 2 + iθµ θ˙µ ,

(19.379)

which proportional to the free part of the action (19.349). As a curious property of differentiations in superspace we note that D 2 X µ (τ ) = ix˙ µ (τ ) − ζ θ˙µ (τ ),

D 3 X µ (τ ) = −θ˙µ (τ ) − ζ x¨(τ ),

(19.380)

such that the kinetic term (19.378) can also be written as dζ Xµ (τ )D 3 X µ (τ ). 2π The interaction is found from the integral in superspace −

Z

dτ

(19.381)

dζ µ A (X(τ ))DX(τ ) 2π Z dζ = i dτ [Aµ (x(τ )) + i∂ν Aµ (x(τ ))θν (τ )] [iθµ (τ ) + iζ x˙ µ (τ )] , (19.382) 2π which is equal to i

Z

dτ

i dτ Aµ (τ ) x(τ ˙ ) + Fµν θµ (τ )θν (τ ) , 2 thus reproducing the interaction in (19.349). The action in superspace can therefore be written in the simple form −

A[X] = i

Z

Z

dζ M e dτ − Xµ (τ )D 3 X µ (τ ) + Aµ (X(τ ))DX(τ ) . 2π 2 c

(19.383)

1340

19.7.2


Local Invariance

A larger class of supersymmetry transformations exists for the action without gauge fixing which is the sum of q the free part (19.347) and the interacting part (19.348). Absorbing again the factor h ¯ /2M into the Grassmann variable θµ (τ ), and rescaling in addition h(τ ) by a factor 1/c, the reparametrization-invariant action reads Z

¯ p, θ, h] = A[x,

S 0

(

"

#

h(τ ) e 2 dτ −px˙ + p − A − M 2 c2 2M c M e µ µ ν ˙ + iθµ (τ )θ (τ ) − ih(τ ) Fµν (x(τ ))θ (τ )θ (τ ) . (19.384) 2 c

Let us now compose the action from invariant building blocks. For simplicity, we ignore the electromagnetic interaction. In a first step we also omit the mass term. The extra variable h(τ ) requires an extra Grassmann partner χ(τ ) for symmetry, and we form the action A¯1 [x, p, θ, h, χ] =

Z

0

S

(

)

i h(τ ) 2 M dτ −px˙ + p + iθµ (τ )θ˙µ (τ )+ χ(τ )θµ (τ )pµ (τ ) .(19.385) 2M 2 2

This action possesses a local supersymmetry. If we now perform τ -dependent versions of the supersymmetry transformations (19.374) δxµ = iα(τ )θµ , δh = iα(τ )χ,

δθµ = α(τ )p, δχ = 2α(τ ˙ ).

δp = 0, (19.386)

If we integrate out the momenta in the path integral, the action (19.385) goes over into Z

A¯1 [x, θ, h, χ] =

0

S

x˙ 2 i M dτ − + iθµ (τ )θ˙µ (τ ) + χ(τ )θµ (τ )x˙ µ (τ ) , (19.387) 2h(τ ) 2 2h(τ ) (

)

where a term proportional to χ2 (τ ) has been omitted since it vanishes due to the nilpotency (7.375). This action is locally supersymmetric under the transformations δxµ = iα(τ )θµ ,

δθµ =

δh = iα(τ )χ,

α(τ ) i x˙ − χ θµ , h(τ ) 2

δχ = 2α(τ ˙ ).

(19.388)

We now add the mass term AM

1 =− 2

Z

0

S

dτ h(τ )Mc2 .

(19.389)

This needs a supersymmetric partner to compensate the variation of Am under (19.388). A5 =

i 2

Z

0

S

dτ θ5 (τ )θ˙5 (τ ) + Mcχ(τ )θ5 (τ ) . h

i


1341


Indeed, add to (19.388) the transformation δθ5 = Mc α(τ ),

(19.391)

we see that the sum AM + A5 is invariant. Adding this to (19.385), we obtain the locally invariant canonical action ¯ p, θ, θ5 , h, χ] = A[x,

Z

0

S

(

dτ −px˙ +

i M h h(τ ) 2 h(τ ) p − Mc + i θµ (τ )θ˙µ (τ ) + θ5 (τ )θ˙5 (τ ) 2M 2 2

i + χ(τ ) [θµ (τ )pµ (τ ) + Mcθ5 (τ )] . 2

(19.392)

Notes and References Relativistic quantum mechanics is described in detail in J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964, relativistic quantum field theory in S.S. Schweber, Introduction to Relativistic Quantum Field Theory, Harper and Row, New York, 1962; J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 1965; C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York, 1985. The individual citations refer to [1] For the development and many applications see the textbook H. Kleinert, Gauge Fields in Condensed Matter , World Scientific, Singapore, 1989; Vol. I, Superflow and Vortex Lines (Disorder Fields, Phase Transitions); Vol. II, Stresses and Defects (Differential Geometry, Crystal Melting) (wwwK/b1, where wwwK is short for (http://www.physik.fu-berlin.de/~kleinert). [2] See Vol. I of the textbook [1] and the original paper H. Kleinert, Lett. Nuovo Cimento 35, 405 (1982) (ibid.http/97). The theoretical prediction of this paper was confirmed only 20 years later in S. Mo, J. Hove, A. Sudbø, Phys. Rev. B 65, 104501 (2002) (cond-mat/0109260); Phys. Rev. B 66, 064524 (2002) (cond-mat/0202215). [3] R.P. Feynman, Phys. Rev. 80, 440 (1950). [4] There are basically two types of approach towards a worldline formulation of spinning particles: one employs auxiliary Bose variables: R.P. Feynman, Phys. Rev. 84, 108 (1989); A.O. Barut and I.H. Duru, Phys. Rep. 172, 1 (1989); the other anticommuting Grassmann variables: E.S. Fradkin, Nucl. Phys. 76, 588 (1966); R. Casalbuoni, Nuov. Cim. A 33, 389 (1976); Phys. Lett. B 62, 49 (1976); F.A. Berezin and M.S. Marinov, Ann. Phys. 104, 336 (1977); L. Brink, S. Deser, B. Zumino, P. DiVecchia, and P.S. Howe, Phys. Lett. B 64, 435 (1976); L. Brink, P. DiVecchia, and P.S. Howe, Nucl. Phys. B 118, 76 (1976). These worldline formulations were used to recalculate processes of electromagnetic and strong interactions by M.B. Halpern, A. Jevicki, and P. Senjanovic, Phys. Rev. D 16, 2476 (1977); M.B. Halpern and W. Siegel, Phys. Rev. D 16, 2486 (1977); Z. Bern and D.A. Kosower, Nucl. Phys. B 362, 389 (1991); 379, 451 (1992);

1342

19 Relativistic Particle Orbits M. Strassler, Nucl. Phys. B 385, 145 (1992); M.G. Schmidt and C. Schubert, Phys. Lett. B 331, 69 (1994); Nucl. Phys. Proc. Suppl. B,C 39, 306 (1995); Phys. Rev. D 53, 2150 (1996) (hep-th/9410100). For many more references see the review article in Ref. [5].

[5] C. Schubert, Phys. Rep. 355, 73 (2001); G.V. Dunne, Phys. Rep. 355, 73 (2002); [6] As a curiosity of history, Schrödinger invented first the relativistic Klein-Gordon equation and extracted the Schrödinger equation from this by taking its nonrelativistic limit similar to Eqs. (19.44) and (19.45). [7] In particle physics, the Chern-Simons theory of ribbons explained in Section 16.7 was used to construct path integrals over fluctuating fermion orbits: A.M. Polyakov, Mod. Phys. Lett. A 3, 325 (1988). For more details see C.H. Tze, Int. J. Mod. Phys. A 3, 1959 (1988). [8] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). English translation available at wwwK/files/heisenberg-euler.pdf. J. Schwinger, Phys. Rev. 84, 664 (1936); 93, 615; 94, 1362 (1954). [9] A. Vilenkin, Phys. Rev. D 27, 2848 (1983). [10] See the internet page http://super.colorado.edu/~michaele/Lambda/links.html. [11] The path integral of the relativistic Coulomb system was solved by H. Kleinert, Phys. Lett. A 212, 15 (1996) (hep-th/9504024). The solution method possesses an inherent supersymmetry as shown by K. Fujikawa, Nucl. Phys. B 468, 355 (1996). [12] E.A. Uehling, Phys. Rev. 49, 55 (1935). [13] T. Kinoshita (ed.), Quantum Electrodynamics, World Scientific, Singapore, 1990. [14] For a first attempt see H. Kleinert, Lettere Nuovo Cimento 4, 285 (1970) (wwwK/24). New developments can be traced back from the recent papers I.I. Kogan and D. Polyakov, Int. J. Mod. Phys. A 18, 1827 (2003) (hep-th/0208036); D. Juriev, Alg. Groups Geom. 11, 145 (1994); R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde, Comm. Math. Phys. 185, 197 (1997).



Quandoquidem inter nos sanctissima divitiarum maiestas Since the majesty of wealth is most sacred with us Juvenal, Sat. 1, 113

20 Path Integrals and Financial Markets An important field of applications for path integrals are financial markets. The prices of assets fluctuate as a function of time and, if the number of participants in the market is large, the fluctuations are pretty much random. Then the time dependence of prices can be modeled by fluctuating paths.

20.1

Fluctuation Properties of Financial Assets

Let S(t) denote the price of a stock or another financial asset. Over long time spans, i.e., if the data are recorded at low frequency, the average over many stock prices has a time behavior that can be approximated by pieces of exponentials. This is why they are usually plotted on a logarithmic scale. This is best illustrated by a plot of the Dow-Jones industrial index over 60 years in Fig. 20.1. The fluctuations of the index have a certain average width called the volatility of the market. Over

Figure 20.1 Periods of exponential growth of price index averaged over major industrial stocks in the United States over 60 years [?].

1343

1344

20 Path Integrals and Financial Markets

a 500

S&P 500

100

b Volatility σ × 103

5.0

1.0 1984

1986

1988

1990

1992

1994

1996

Figure 20.2 (a) Index S&P 500 for 13-year period Jan. 1, 1984 — Dec. 14, 1996, recorded every minute, and (b) volatility in time intervals 30 min (from Ref. [?]).

long times, the volatility is not constant but changes stochastically, as illustrated by the data of the S&P 500 index over the years 1984-1997, as shown in Fig. 20.2 [?]. In particular, there are strong increases shortly before a market crash. The theory to be developed will at first ignore these fluctuations and assume a constant volatility. Attempts to include them have been made in the literature [?]–[?] and a promising version will be described in Section ??. The distribution of the logarithms of the volatilities is approximately normal as illustrated in Fig. 20.3.

1.0 0.8

300-min data log-normal fit Gaussian fit

0.6

Probability ×10−3 0.4 0.2 0.0 0.000

0.001

0.002

0.003

0.004

Volatility σ

Figure 20.3 Comparison of best log-normal and Gaussian fits to volatilities over 300 min (from Ref. [?]). H. Kleinert, PATH INTEGRALS

20.1 Fluctuation Properties of Financial Assets

1345

An individual stock will in general have larger volatility than an averge market index, especially when the associated company is small and only few shares are traded per day.

20.1.1

Harmonic Approximation to Fluctuations

To lowest approximation, the stock price S(t) satisfies a stochastic differential equation for exponential growth ˙ S(t) = rS + η(t), S(t)

(20.1)

where rS is the growth rate, and η(t) is a white noise defined by the correlation functions hη(t)i = 0,

hη(t)η(t′ )i = σ 2 δ(t − t′ ).

(20.2)

The standard deviation σ is a precise measure for the volatility of the stock price. The squared volatility v ≡ σ 2 is called the variance. The quantity dS(t)/S(t) is called the return of the asset. From financial data, the return is usually extracted for finite time intervals ∆t rather than the infinitesimal dt since prices S(t) are listed for certain discrete times tn = t0 + n∆t. There are, for instance, abundant tables of daily closing prices of the market S(tn ), from which one obtains the daily returns ∆S(tn )/S(tn ) = [S(tn+1 ) − S(tn )]/S(tn ). The set of available S(tn ) is called the time series of prices. For a suitable choice of the time scales to be studied, the assumption of a white noise is fulfilled quite well by actual fluctuations of asset prices, as illustrated in Fig. 20.4.

S(ω)

ω [sec−1 ]

Figure 20.4 Fluctuation spectrum of exchange rate DM/US$ as function of frequency in units 1/sec, showing that the noise driving the stochastic differential equation (20.1) is approximately white (from [?]).

1346


For the logarithm of the stock or asset price1 x(t) ≡ log S(t)

(20.3)

this implies a stochastic differential equation for linear growth [?, ?, ?, ?] x(t) ˙ =

S˙ 1 − σ 2 = rx + η(t), S 2

(20.4)

1 rx ≡ rS − σ 2 2

(20.5)

where

is the drift of the process [compare (18.389)]. A typical set of solutions of (20.4) is shown in Fig. 20.5. 6 5 4 3 2 1

xS (t) ≡ log S(t)

2

4

6

8

10

t

-1

Figure 20.5 Behavior of logarithm of stock price following the stochastic differential equation (20.3).

The finite differences ∆x(tn ) = x(tn+1 )−x(tn ) and the corresponding differentials dx are called log-returns. The extra term σ 2 /2 in (20.5) is due to Itô’s Lemma (18.395) for functions of a stochastic variable x(t). Recall that the formal expansion in powers of dt: dx 1 d2 x 2 dS (t) + . . . dS(t) + dS 2 dS 2 " #2 ˙ ˙ S(t) 1 S(t) = dt − dt2 + . . . S(t) 2 S(t)

dx(t) =

(20.6)

may be treated in the same way as the expansion (18.406) using the mnemonic rule (18.409), according to which we may substitute x˙ 2 dt → hx˙ 2 idt = σ 2 , and thus "

˙ S(t) S(t)

#2

dt → x˙ 2 (t)dt = σ 2 .

(20.7)

The higher powers in dt do not contribute for Gaussian fluctuations since they carry ˙ higher powers of dt. For the same reason the constant rates rS and rx in S(t)/S(t) 2 ˙ and x(t) ˙ do not show up in [S(t)/S(t)] dt= x˙ 2 (t)dt. 1

To form the logarithm, the stock or asset price S(t) is assumed to be dimensionless, i.e. the numeric value of the price in the relevant currency. H. Kleinert, PATH INTEGRALS

1347


In charts of stock prices, relation (20.5) implies that if we fit a straight line through a plot of the logarithms of the prices with slope rx , the stock price itself grows on the average like hS(t)i = S(0) erS t = S(0)herx t+

Rt 0

dt′ η(t′ )

i = S(0) e(rx +σ

2 /2)t

.

(20.8)

This result is, of course, a direct consequence of Eq. (18.405). The description of the logarithms of the stock prices by Gaussian fluctuations around a linear trend is only a rough approximation to the real stock prices. The volatilities depend on time. If observed at small time intervals, for instance every minute or hour, they have distributions in which frequent events have an exponential distribution [see Subsection ??], whereas rare events have a much higher probability than in Gaussian distributions. They have heavy tails of in comparison with the extremely light tails of Gaussian distributions. This was first observed by Pareto in the 19th century [?], reemphasized by Mandelbrot in the 1960s [?], and investigated recently by several authors [?, ?]. The theory needs therefore considerable refinement. For this purpose we shall introduce beside the heavy power-like tails also the so-called semi-heavy tails, which drop off faster than any power, such as a e−x xb with arbitrarily small a > 0 and large b. We shall see later in Section ??, that semi-heavy tails of financial distribution may be derived as a consequence of volatility fluctuations. Before we come to such more refined models we shall fit the data phenomenologically with various non-Gaussian distributions and explore the consequences.

20.1.2

L´ evy Distributions

Following Pareto and Mandelbrot we may attempt to fit the distributions of the price changes ∆Sn = S(tn+1 ) − S(tn ), the returns ∆Sn /S(tn ), and the log-returns ∆xn = x(tn+1 ) − x(tn ) for a certain time difference ∆t = tn+1 − tn approximately with the help of Lévy distributions [?, ?, ?, ?]. For brevity we shall, from now on, use the generic variable z to denote any of the above differences. The Lévy distributions are defined by the Fourier transform ˜ λ2 (z) ≡ L σ

Z

∞

−∞

dp ipz λ e Lσ2 (p), 2π

(20.9)

with h

i

Lλσ2 (p) ≡ exp −(σ 2 p2 )λ/2 /2 .

(20.10)

˜ For an arbitrary distribution D(z), we shall write the Fourier decomposition as ˜ D(z) =

Z

dp ipz e D(p) 2π

(20.11)

and the Fourier components D(p) as an exponential D(p) ≡ e−H(p) ,

(20.12)

1348


where H(p) plays a similar role as the Hamiltonian in quantum statistical path ˜ integrals. By analogy with this we shall also define H(z) so that ˜ ˜ D(z) = e−H(z) .

(20.13)

An equivalent definition of the the Hamiltonian is

P (z) P (z)

1 + λ ≈ 2.7

1+λ≈4 1/x1+λ

z/σ

z/σ

Figure 20.6 Left: Lévy tails of the S&P 500 index (1 minute log-returns) plotted against z/δ. Right: Double-logarithmic plot exhibiting power-like tail regions of the S&P 500 index (1 minute log-returns) (after Ref. [?])

e−H(p) ≡ he−ipz i.

(20.14)

For the Lévy distributions (20.9), the Hamiltonian is 1 H(p) = (σ 2 p2 )λ/2 . 2

(20.15)

The Gaussian distribution is recovered in the limit λ → 2 where the Hamiltonian becomes simply σ 2 p2 /2. For large z, the Lévy distribution (20.9) falls off with the characteristic power-law ˜ λ2 (z) → Aλ2 L σ σ

λ . |z|1+λ

(20.16)

This power falloff is the above discussed heavy tail of the distribution (also called power tail , Paretian tail , or Lévy tail ). The size of the tails is found by approximating the integral (20.9) for large z, where only small momenta contribute, as follows: ˜ λ2 (z) ≈ L σ

Z

∞

−∞

dp ipz 1 e 1 − (σ 2 p2 )λ/2 2π 2

λ , |z|1+λ

(20.17)

dp′ ′λ σλ p cos p′ = sin(πλ/2) Γ(1 + λ). π 2πλ

(20.18)

→ Aλσ2

z→∞

with Aλσ2 = −

σλ 2λ

Z

0

∞


1349


The stock market data are fitted best with λ between 1.2 and 1.5 [?], and we shall use λ = 3/2 most of the time, for simplicity, where one has 3/2

Aσ2 =

1 σ 3/2 √ . 4 2π

(20.19)

The full Taylor expansion of the Fourier transform (20.10) yields the asymptotic series ˜ λσ2 (z) = L

∞ X sin πλ (−1)n Z ∞ dp σ λn pλn (−1)n+1 σ λn 2 cos pz = . (20.20) Γ(1 + nλ) 1+λ n nπ n! π 2 n! 2 |z| 0 n=0 n=0 ∞ X

This series is not useful for practical calculations since it diverges. In particular, it is unable to reproduce the pure Gaussian distribution in the limit λ → 2.

20.1.3

Truncated L´ evy Distributions

Mathematically, an undesirable property of the Lévy distributions is that their fluctuation width diverges for λ < 2, since the second moment 2 ˜ λ2 (z) = − d Lλ2 (p) σ 2 = hz 2 i ≡ dz z 2 L σ dp2 σ −∞ p=0

∞

Z

(20.21)

is infinite. If one wants to describe data which show heavy tails for large log-returns but have finite widths one must make them fall off at least with semi-heavy tails at very large returns. Examples are the the so-called truncated Lévy distributions [?]. They are defined by ˜ (λ,α) L σ2 (z) ≡

Z

dp ipz (λ,α) e Lσ2 (p) = 2π

∞

−∞

Z

∞

−∞

dp ipz−H(p) e , 2π

(20.22)

with a Hamiltonian which generalizes the Lévy Hamiltonian (20.15) to i σ 2 α2−λ h (α + ip)λ + (α − ip)λ − 2αλ 2 λ(1 − λ) (α2 + p2 )λ/2 cos[λ arctan(p/α)] − αλ = σ2 . αλ−2 λ(1 − λ)

H(p) ≡

(20.23)

The asymptotic behavior of the truncated Lévy distributions differs from the power behavior of the Lévy distribution in Eq. (20.17) by an exponential factor e−αz which guarantees the finiteness of the width σ and of all higher moments. A rough estimate of the leading term is again obtained from the Fourier transform of the lowest expansion term of the exponential function e−H(p) : ˜ (λ,α) L σ2 (z)

≈ →

z→∞

h io dp ipz n 1 − s (α + ip)λ + (α − ip)λ e −∞ 2π sin(πλ) e−α|z| λ e2sα Γ(1 + λ) s 1+λ , π |z| λ

e2sα

Z

∞

(20.24)

1350


where σ 2 α2−λ . s≡ 2 λ(1 − λ)

(20.25)

The integral follows directly from the formulas [?] Z

dp ipz Θ(z) e−αz e (α + ip)λ = , 2π Γ(−λ) z 1+λ

∞

−∞

Z

∞

−∞

dp ipz Θ(−z) e−α|z| e (α − ip)λ = ,(20.26) 2π Γ(−λ) |z|1+λ

and the identity for Gamma functions2 1 = −Γ(1 + z) sin(πz)/π. Γ(−z)

(20.27)

The full expansion is integrated with the help of the formula [?] Z

∞

−∞

dp ipz e (α + ip)λ (α − ip)ν 2π

= (2α)λ/2+ν/2

1 |z|1+λ/2+ν/2

1 W(ν−λ)/2,(1+λ+ν)/2 (2αz) z > 0, Γ(−λ) for  1   z < 0,  W(λ−ν)/2,(1+λ+ν)/2 (2αz) Γ(−ν)     

(20.28)

where the Whittaker functions W(ν−λ)/2,(1+λ+ν)/2 (2αz) can be expressed in terms of Kummer’s confluent hypergeometric function 1 F1 (a; b; x) of Eq. (9.45) as Γ(−2κ) xκ+1/2 e−x/2 1 F1 (1/2 + κ − δ; 2κ + 1; x) Γ(1/2 − κ − δ) Γ(2κ) + x−κ+1/2 e−x/2 1 F1 (1/2 − κ − δ; −2κ + 1; x), Γ(1/2 + κ − δ)

Wδ,κ (x) =

(20.29)

as can be seen from (9.39), (9.46) and Ref. [?]. For ν = 0, only z > 0 gives a nonzero integral (20.28), which reduces, with W−λ/2,1/2+λ/2 (z) = z −λ/2 e−z/2 , to the left equation in (20.26). Setting λ = ν we find ∞

Z

−∞

dp ipz 2 1 1 e (α + p2 )ν = (2α)ν/2 1+ν W0,1/2+ν (2α|z|). 2π |z| Γ(−ν)

(20.30)

Inserting W0,1/2+ν (x) =

s

2z K1/2+ν (x/2), π

(20.31)

we may write Z

∞

−∞

2

dp ipz 2 e (α + p2 )ν = 2π

2α |z|

!1/2+ν

√

1 K1/2+ν (α|z|). πΓ(−ν)

(20.32)

M. Abramowitz and I. Stegun, op. cit., Formula 6.1.17. H. Kleinert, PATH INTEGRALS

1351


For ν = −1 where K−1/2 (x) = K1/2 (x) = Z

∞

−∞

q

π/2xe−x , this reduces to

dp ipz 1 1 −α|z| = . e e 2 2 2π α +p 2α

(20.33)

Summing up all terms in the expansion of the exponential function e−H(p) : ˜ (λ,α) L σ2 (z)

2sαλ

≈ e

Z

∞

−∞

∞ in X dp (−s)n h 1+ (α + ip)λ + (α − ip)λ eipz (20.34) 2π n! n=1

(

)

yields the true asymptotic behavior ˜ (λ,α) L σ2 (z)

→

z→∞

λ )sαλ

e(2−2

Γ(1 + λ)

sin(πλ) e−α|z| s 1+λ , π |z|

(20.35)

which differs from the estimate (20.24) by a constant factor (see ?? for details) [?]. Hence the tails are semi-heavy. In contrast to Gaussian distributions which are characterized completely by their width σ, the truncated Lévy distributions contain three parameters σ, λ, and α. Best fits to two sets of fluctuating market prices are shown in Fig. 20.7. For the S&P 500 index we plot the cumulative distributions P< (z) =

Z

z

−∞

′ ˜ (λ,α) dz L σ2 (z ), ′

P> (z) =

Z

z

∞

′ ˜ (λ,α) dz ′ L σ2 (z ) = 1 − P< (z), (20.36)

for the price differences z = ∆S over ∆t = 15 minutes. For the ratios of the changes of the currency rates DM/$ we plot the returns z = ∆S/S with the same ∆t. The plot shows the negative and positive branches P< (−z), and P> (z) both plotted on the positive z axis. By definition: P< (−∞) = 0, P< (0) = 1/2, P< (∞) = 1, P> (−∞) = 1, P> (0) = 1/2, P> (∞) = 0.

(20.37)

The fits are also compared with those by other distributions explained in the figure captions. A fit to most data sequences is possible with a rather universal parameter λ close to λ = 3/2. The remaining two parameters fix all expansion coefficients of Hamiltonian (20.23): 1 1 1 1 H(p) = c2 p2 − c4 p4 + c6 p6 − c8 p8 + . . . . 2 4! 6! 8!

(20.38)

The numbers c2n = −(−1)n H (2n) (0) are the cumulants of the truncated Lévy distribution [compare (3.584)], also denoted by hz n ic . Here they are equal to hz 2 ic = c2 = σ 2 , hz 4 ic = c4 = σ 2 (2 − λ)(3 − λ)α−2 , hz 6 ic = c6 = σ 2 (2 − λ)(3 − λ)(4 − λ)(5 − λ)α−4, .. . Γ(2n − λ) 2−2n hz 2n ic = c2n = σ 2 α . Γ(2 − λ)

(20.39)

1352


P > (±z)

P > (±z)

Path Integrals in Quantum Mechanics, Statistics and Polymer Physics

Quantum Mechanics and Path Integrals

Quantum Mechanics and Path Integrals

Path integrals in quantum mechanics, statistics, polymer physics, and financial markets

Path integrals in physics. Stochastic processes and quantum mechanics

Path integrals in physics. Stochastic processes and quantum mechanics

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