Introduction to Quantum Field Theory
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Introduction to Quantum Field Theory V.G. Kiselev University of Freiburg, Germany
Ya.M. Shnir University of Cologne, Germany
A.Ya. Tregubovich Institute of Physics, National Academy of Sciences, Minsk, Belarus
Edited for English by M.J. Lilley and C.J. Houghton
CRC PR E S S Boca Raton London New York Washington, D.C.
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Contents ...
Preface
XII~
I THE PATH INTEGRAL IN QUANTUM MECHANICS
1
1 Action in Classical Mechanics 1.1 The Variational Principle and Equations of Motion . . . 1.2 A Mathematical Note: The Notion of the Functional . . 1.3 The Action as a Function of The Boundary Conditions . 1.4 Symmetries of the Action and Conservation Laws . . . .
3 3 6 9 13
2 The Path Integral in Quantum Mechanics 2.1 The Green Function of the Schrodinger Equation . . . . 2.2 The Path Integral . . . . . . . . . . . . . . . . . . . . . . 2.3 The Path Integral for Free Motion . . . . . . . . . . . . . Free Motion: Straightforward Calculation of the Path Integral . . . . . . . . . . . . . . . . . . . . Free Motion: Path Integral Calculation by the Stationary Phase Method . . . . . . . . . . . . . 2.4 The Path Integral for the Harmonic Oscillator . . . . . . 2.5 Imaginary Time and the Ground State Energy . . . . . .
17 17 21 25
3 The Euclidean Path Integral 3.1 The Symmetric Double Well . . . . . . . . . . . . . Quantum Mechanical Instantons . . . . . . . . . . . The Contribution from the Vicinity of the Instanton Trajectory . . . . . . . . . . . . . . . . . . . Calculation of the Functional Determinant . . . . . 3.2 A Particle in a Periodic Potential. Band Structure . 3.3 A Particle on a Circle . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . .
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26 27 31 33
CONTENTS
vi
I1 INTRODUCTION TO QUANTUM FIELD THEORY 4 Classical and Quantum Fields 4.1 From Large Number of Degrees of Freedom to Particles . 4.2 EnergyMomentum Tensor . . . . . . . . . . . . . . . . . 4.3 Field Quantization . . . . . . . . . . . . . . . . . . . . . Canonical Quantization . . . . . . . . . . . . . . . . . . . Quantization via Path Integrals . . . . . . . . . . . . . . 4.4 The Equivalence of QFT & Statistical Physics . . . . . . 4.5 Free Field Quantization: From Fields to Particles . . . . Momentum Space . . . . . . . . . . . . . . . . . . . . . . Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . ZeroPoint Energy . . . . . . . . . . . . . . . . . . . . . Elementary Excitations of the Field . . . . . . . . . . . .
5 Vacuum Energy in (p4 Theory 5.1 Casimir Effect . . . . . . . . . . . . . . . . . . . . . . . . Simple Calculation of Casimir Energy . . . . . . . . . . . Casimir Energy: Calculation via Path Integral . . . . . . 5.2 Effective Potential of (p4 Theory . . . . . . . . . . . . . . Calculation of U, (p) . . . . . . . . . . . . . . . . . . . The Explicit Form of Ueff. . . . . . . . . . . . . . . . . Renormalization of Mass and Coupling Constant . . . . . Running Coupling Constant, Dimensional Transmutation and Anomalous Dimensions . . . . . . . . . . . . Effective Potential of the Massive Theory . . . . . . . . . 6 The Effective Action in (p4 Theory 6.1 Correlation Functions and the Generating Functional . . 6.2 Z [ J ] W [ J ]and Correlation Functions of the Free Field . The Classical Green Function . . . . . . . . . . . . . . . Correlation Functions . . . . . . . . . . . . . . . . . . . . 6.3 Generating Functionals in Theory . . . . . . . . . . . . . . . . . . . . . ................ (p4 Theory Generating Functionals: Expansion in X . . . . . . . . . Generating Functionals: the Loop Expansion . . . . . . . 6.4 Effective Action . . . . . . . . . . . . . . . . . . . . . . . Expansion of the Functional Determinant . . . . . . . . .
.
CONTENTS
vii
7 Renormalization of the Effective Action 7.1 Momentum Space . . . . . . . . . . . . . . Explicit Form of the Diagrams . . . . . . . 7.2 The Structure of Ultraviolet Divergencies . 7.3 PauliVillars Regularization . . . . . . . . Calculation of Integrals . . . . . . . . . . . About Dimensional Regularization . . . . 7.4 The Regularized Inverse Propagator . . . . Analytic Continuation to Minkowski Space 7.5 Renormalization . . . . . . . . . . . . . . . Renormalization of Mass . . . . . . . . . . Renormalization of the Coupling Constant Renormalization of the Wave Function . . 7.6 Conclusion . . . . . . . . . . . . . . . . . .
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8 Renormalization Group
8.1 Renormalization Group . . . . . . . . . . Renormalization Group Equation . . . . General Solution of RG Equation . . . . Explicit Example . . . . . . . . . . . . . 8.2 Scale Transformations . . . . . . . . . . Scale Transformations at the Tree Level GellMann  Low Equation . . . . . . . . 8.3 Asymptotic Regimes . . . . . . . . . . .
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9 Concluding Remarks 9.1 Correlators in Terms of I'[(p] . . . . . . . . . . . . 9.2 On the Properties of Perturbation Series . . . . . On the Loop Expansion Parameter . . . . . . . . On the Asymptotic Nature of Perturbation Series 9.3 On (p4 Theory with Large Coupling Constant . . The Cases d = 2 and d = 3: SecondOrder Phase transitions . . . . . . . . . . . . . . The Cases d = 4: Possible Triviality of (p4 Theory 9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
I11 MORE COMPLEX FIELDS AND OBJECTS 10 Second Quantisation: From Particles to Fields 10.1 Identical Particles and Symmetry of Wave Functions . . 10.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . OneParticle Hamiltonian . . . . . . . . . . . . . . . . . Creation and Annihilation Operators . . . . . . . . . . . Total Hamiltonian . . . . . . . . . . . . . . . . . . . . . The Field Operator . . . . . . . . . . . . . . . . . . . . . Result: Recipe for Quantisation . . . . . . . . . . . . . . 10.3 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . Oneparticle Hamiltonian . . . . . . . . . . . . . . . . . Creation and Annihilation Operators . . . . . . . . . . . ManyParticle Hamiltonian . . . . . . . . . . . . . . . . . Field Operator . . . . . . . . . . . . . . . . . . . . . . . 11 Path Integral For Fermions 11.1 On the Formal Classical Limit for Fermions . . . . . . . 11.2 Grassmann Algebras: A Short Introduction . . . . . . . . 11.3 Path Integral For NonRelativistic Fermions . . . . . . . Classical Pseudomechanics . . . . . . . . . . . . . . . . . Path Integral Quantisation . . . . . . . . . . . . . . . . . 11.4 Generating Functional For Fermionic Fields . . . . . . . 11.5 Coupling of the Dirac Spinor and the (p4 Scalar Fields . . Loop Expansion and Diagram Techniques . . . . . . . . . Analysis of Divergences . . . . . . . . . . . . . . . . . . . 11.6 Fermion Contribution to the Effective Potential . . . . . 12 Gauge Fields 12.1 Gauge Invariance . . . . . . . . . . . . . . . The Basic Idea . . . . . . . . . . . . . . . . Example of a Globally Invariant Lagrangian Example of a Locally Invariant Lagrangian . Lagrangian of Gauge Fields . . . . . . . . . 12.2 Dynamics of Gauge Invariant Fields . . . . . Equations of Motion . . . . . . . . . . . . . The YangMills Equations . . . . . . . . . . The Total Energy . . . . . . . . . . . . . . . Gauge Freedom and Gauge Conditions . . . 12.3 Spontaneously Broken Symmetry . . . . . .
....... ....... ....... ....... ....... . . . . . . . ....... ....... ....... ....... .......
CONTENTS
12.4
12.5 12.6 12.7
ix
Vacuum and its Structure . . . . . . . . . . . . . . . . Goldstone Modes and Higgs Mechanism . . . . . . . . Elimination of Goldstone Modes. Goldstone Theorem . Examples . . . . . . . . . . . . . . . . . . . . . . . . . Quantization of Systems With Constraints . . . . . . . Primary Constraints . . . . . . . . . . . . . . . . . . . On Constrained Mechanical Systems . . . . . . . . . . Secondary Constraints . . . . . . . . . . . . . . . . . . The Matrix of Poisson Brackets . . . . . . . . . . . . . First and Second Order Constraints . . . . . . . . . . . Quantization . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . Hamiltonian Quantization of YangMills Fields . . . . Quantization of Gauge Fields: FaddeevPopov Method ColemanWeinberg Effect . . . . . . . . . . . . . . . .
. .
. . .
. .
. . . . . . .
.
13 Topological Objects in Field Theory 13.1 Kink in l l Dimensions . . . . . . . . . . . . . . . . . 13.2 A Few Words about Solitons . . . . . . . . . . . . . . . . 13.3 Abrikosov Vortex . . . . . . . . . . . . . . . . . . . . . . GinzburgLandau Model of Superconductivity . . . . . . Nontrivial Solution . . . . . . . . . . . . . . . . . . . . . AharonovBohm Effect . . . . . . . . . . . . . . . . . . . A Few Words about Topology and an Exotic String . . . Vortex Solution in Other Contexts . . . . . . . . . . . . 13.4 The 't HooftPolyakov Monopole . . . . . . . . . . . . . Magnetic Properties of the Solution . . . . . . . . . . . . Lower Boundary on the Monopole Mass . . . . . . . . . Dyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Few Words About the Topology . . . . . . . . . . . . Do Monopoles Exist? . . . . . . . . . . . . . . . . . . . . 13.5 SU(2) Instanton . . . . . . . . . . . . . . . . . . . . . . Nontrivial Solution . . . . . . . . . . . . . . . . . . . . . On the Vacuum Structure of YangMills Theory . . . . . 13.6 Quantum Kink . . . . . . . . . . . . . . . . . . . . . . . Quantum Correction to the Mass of the Kink . . . . . . Physical Contents of Fluctuations around the Kink . . . Elimination of Zero Mode . . . . . . . . . . . . . . . . . Generating Functional . . . . . . . . . . . . . . . . . . .
+
CONTENTS
X
A Some Integrals and Products A.l Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . A.2 Calculation of & (l  &) . . . . . . . . . . . A.3 Calculation of
j$ln(l
. X)
. . . . . . . . . . . .
0
cc
A.4 Calculation of J cc
ln(1
+ z2)
. . . . . . . . . . . . .
A.5 Feynman Parametrization . . . . . . . . . . . . . . . . .
B Splitting of Energy Levels in DoubleWell Potential C Lie C.l C.2 C.3
Algebras Elementary Definitions . . . . . . . . . . . . . . . . . . . Examples of Lie Algebra . . . . . . . . . . . . . . . . . . The Idea of Classification. LeviMaltsev Decomposition . The Adjoint Representation . . . . . . . . . . . . . . . . Solvable and Nilpotent Algebras . . . . . . . . . . . . . . Reductive and Semisimple Algebras . . . . . . . . . . . . 3.4 Classification of Complex Semisimple Lie Algebras . . . . The Cartan Subalgebra. Roots . . . . . . . . . . . . . . Properties of Roots. CartanWeyl Basis . . . . . . . . . Cartan Matrix. Dynkin Schemes . . . . . . . . . . . . . . Compact Algebras . . . . . . . . . . . . . . . . . . . . .
Index
432
To our parents
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Preface W e can understand the effectiveness of mathematics: it is the language of physics and any language is eflective in expressing the ideas of its subject. Field theory, in m y opinion, is also a language that we have invented for describing fundamental systems with many degrees of freedom. R. Jackiwl These days, a student looking for a textbook on Quantum Field Theory (QFT) has to choose from a frighteningly large amount of literature. A new textbook on the subject has to be well motivated. Our motivation in writing this book is to explain Quantum Field Theory by concentrating on the basic physical ideas which are common to its many applications. As far as possible we have tried to be concise, we have tried to avoid the words 'it can be shown' and we have tried to present QFT in a way which is independent of any particular application to statistical or elementary particle physics. We believe this makes QFT easier for the student to understand. QFT is the mathematical tool for many physical disciplines, including elementary particle physics, solid state physics and phase transitions. Typically, a student learns QFT in many different contexts and because of this the student is forced both to struggle with physics and QFT at the same time and to learn essentially the same material in many different ways. Similarly, a researcher who wants to apply fieldtheoretical methods to her or his own work has to go through a book about statistical or 'R. Jackiw, 'The unreasonable effectiveness of quantum field theory', prcprint MITCTP2500 (January 1996), hcpth/9602122.
xiii
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PREFACE
elementary particle physics in order to find useful methods even if their problem has nothing to do with these disciplines. We believe that it is helpful to study field theory in a contextindependent way prior to making use of it in specific physical disciplines. Later on, when studying physics, the student will be better able to see the crossdisciplinary links based on common mathematics and common physical ideas. This approach is already traditional for university courses on the equations of mathematical physics. This book is based on lecture notes from a course designed by V.G. Kiselev in 1989/90 for advanced fourthyear undergraduate students of the Byelorussian University specialising in theoretical physics. All three of us have given the course at different times. The goal of this book is to explain those features of quantum and statistical field systems which result from their fieldtheoretic nature and, therefore, are common to different physical contexts. Among these features are renormalisation, effective interactions, running coupling constants and anomalous scaling dimensions. We try both to supply the reader with practical tools to carry out calculations and to discuss the meaning of the results, highlighting their interdisciplinary nature when appropriate. This book is addressed to advanced graduate students and postdoctoral researchers who specialize in theoretical physics as well as to researchers in other fields who would like to apply fieldtheoretic methods to their work. We also hope the book will be useful to lecturers designing an effective interdependence of theoretical physics course structure or lecturing field theory. We assume that the reader is familiar with some common mathematical analysis, the basic ideas of quantum mechanics and some specific topics in quantum mechanics, i.e. the Schrodinger equation, potential wells, and the harmonic oscillator. A knowledge of statistical and elementary particle physics is not necessary to understand the core of the book. However our discussions of the results obtained will be more interesting for the reader who has some knowledge about these fields. Our compromise between the restriction of finite book volume and the large amount of information on fields was made as follows. First, we use the simplest possible models to illustrate the properties of the field systems, but we try to avoid oversimplifications. This explains the use of different versions of 44 theory in the main part of the book2. Sec2As we deal only with perturbativc analysis, we do not face the problem of the socalled triviality of this model. We comment on this highly nontrivial property in the concluding rcmarks t o part 11.
PREFACE
xv
ond, we try to discuss the features of real systems as soon as we have obtained appropriate results. Third, we try to avoid abstract mathematical constructions. We prefer to start discussions on the grounds of common sense and physical intuition, solve an illustrative problem, and then venture into more general and strict conclusions. The book is structured around carefully selected problems which are solved in detail. Normally we solve one problem of this kind in each chapter. Including these calculations in the book is useful in two ways. Firstly they provide examples of practical calculations and secondly the calculations themselves are the basis for the less strict discussions of related topics which follow. The main object for study and construction in this course is the effective action (its analogue in statistical physics is the free energy). This allows us to discuss different properties of field systems from a single point of view. The basic technical tool applied throughout the book is the path integral calculated via the loop expansion. At the same time, we try to present the reader with an idea of different, equivalent, formalisms used in field theory. The aim is both to make the application of this course to a specific physical purpose easier, and to supply a link to other books where, for example, the operator formalism dominates. Let us mention some topics not included in this book though related to it. The most important among them is the LSZ reduction formula. We do not want to venture into the domain of elementary particle physics, which really begins as soon as this formula is derived. Neither do we consider specific problems in statistical physics. Although they are a hot topic in modern quantum field theory we do not pay much attention to anomalies. We do give an idea of what they are. To study anomalies in more detail is beyond the scope of our book. The same is true of topological objects in field theory. We discuss instantons in quantum mechanics in order to gain some experience in working with path integrals, rather than with the intention of generalizing them later on to instantons in gauge theories. We study topological objects briefly in the last chapter and direct the reader to more specialised literature. We do not go beyond the oneloop level in our calculations. We pay more attention to the general properties of the perturbation series, namely, to its asymptotic nature. In most of the book, we are concerned with the Euclidean formulation of the theory. The analytic continuation to Minkowski space is discussed in a special section. The book is divided into three parts, Part I and I1 are a crashcourse
xvi
PREFACE
in QFT and form the core. In part I we introduce the Feynman path integral and its Euclidean counterpart in quantum mechanics. Part I1 is an introduction to quantum field theory. It begins with the transition to the continuous limit of a microscopic model of a crystal. We then show that the field excitations are particles and obtain for the first time the divergent vacuum energy (chapter 4). We discuss its meaning in statistical physics and quantum field theory, and show that variation of the vacuum energy gives rise to the Casimir effect and to the effective potential (obtained explicitly in the +4 model). Then we perform the first renormalisation. This is done in a rather intuitive way with a serious discussion in order to make the idea as clear as possible (chapter 5). Chapter 6 is devoted to a more rigourous study. We formulate the main problem of field theory, i.e. the calculation of different mean values and correlation functions. We build the effective action as a value which contains the desired information in the most condensed form and is an observable physical quantity. We explore the loop expansion in order to obtain the leading corrections to the bare action (or energy in statistical physics). In chapter 7, we analyse the singularity structure and perform a renormalisation of the theory. The general analysis is illustrated with a detailed calculation of a twopoint correlation function at the oneloop level. In chapter 8, we study the scaling properties of the effective action (renormalisation group). In chapter 9, we summarise the solution to the problem formulated in chapter 6 and make concluding remarks to part 11. Additional chapters 1013, which are collected in part 111, may be independently linked to parts I and 11. In chapter 10, we perform the second quantization of the field starting with Schrijdinger equations for individual particles. This complements chapter 4 where particles were introduced as the elementary excitations of the field. In the chapters 11 and 12, we present fermion and gauge fields respectively. In the chapter 13, we study topologically nontrivial objects of field theory. We are pleased to acknowledge many people who helped and encouraged us in our long work on this book. They were Lev Tomilchick and Evgeniy Tolkachev without whom we would not have started lecturing in 1989; Lev Komarov who helped to form the idea of those lectures, Andrey Listopad who prepared his lecture notes for our further work, former graduate students Igor Boukanov, Dmitry Mogilevtsev, Dmitry Novikov, and Igor Tsvetkov who were our first readers. We are grateful to our colleagues Andriano Di Giakomo, Alexei Kornyshev, Per Osland, Murray Peshkin and Ruedi Seiler for reading preliminary versions of the manuscript and useful comments. We are thankful to Matthew Lilley
PREFACE
xvii
and Conor Houghton who worked on the manuscript as the editors for English. We are deeply indebted to our colleagues from Institute of Physics, National Academy of Sciences, Minsk, Belarus in which this book has been conceived and mainly written. Two of us (V.K. and Ya.S.) would like to thank the Alexander von Humboldt Foundation and Belarussian Foundation for Fundamental Researches for support. During various stages of our work on this project Ya.S. has enjoyed the hospitality of the theoretical physics groups at the ICTP, Trieste, University of Bergen, TU Berlin and DAMTP, University of Cambridge. He would like to thank Nick Manton and the Royal Society for support during 19971998. V.K. thanks Institute of Medicine, Research Center Julich at which a substantial work on the manuscript has been done. Our special thanks go to our students of all years who attended our lectures. Their reaction taught us how to teach QFT. Their questions helped us enormously, some of which gave rise to problems included in this book. We are grateful to our wives for sharing our enthusiasm and for their kind patience even when our work lasted far beyond office hours. The authors. Julich, Cambridge, Minsk. October 1998
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References [l]A textbook in which QFT is present in both t h e contexts of particle and condensed matter physics in equal depth: J. ZinnJustin, Quantum Field Theory and Critical Phenomena, 3rd ed., Oxford, University Press, Oxford 1996. The following books are devoted mainly to QFT in the context of particle physics. [2] M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, AddisonWesley, Reading MA 1995. [3] S. Weinberg, The Quantum Theory of Fields, v.1,2, Cambridge University Press, Cambridge 1996. [4] B. Hatfield, Quantum Field Theory of Point Particles and Strings, AddisonWesley, Redwood CA 1992. [5] M. Kaku, Quantum Field Theory: a Modern Introduction, Oxford University Press, New York 1993. [6] L.S. Brown, Quantum Field Theory, Cambridge University Press, Cambridge 1992. [7] L.H. Ryder Quantum Field Theory, Cambridge University Press, 1986, 2nd ed. 1996. [8] C. Itzykson, J.B. Zuber, Quantum Field Theory, McGrawHill, 1980. [g] J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics and Relativistic Quantum Fields, McGrawHill, 1964 and 1965. [l01 L.D. Faddeev and A.A. Slavnov, Gauge Fields: an Introduction to Quantum Theory, 2nd ed., Addison Wesley, 1991. 
[l11 P. Ramond, Field Theory: A Modern Primer, 2nd ed., Addison Wesley
Loriman, 1988. xix
REFERENCES [l21 K. Huang, Quarks, Leptons and Gauge Fields, Singapore, World Scientific, 1982. [l31 R.J. Rivers, Path Integral Methods in Quantum Field Theory, Cambridge University Press, Cambridge 1987. [l41 F.J. Yndurain, Relativistic Quantum Mechanics and Introduction to Field Theory, SpringerVerlag, New York, 1996. The following books which are devoted to QFT, are mainly oriented to a p plications in statistical physics. [l51 A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover Publications, 1975. [l61 This book which is based on solutions of problems is a good supplement to the above reference [15]: L.S. Leviton and A.V. Shytov, Green Functions: Theory and Practice, published by Moscow Physical Technical Institute, Moscow 1997 (in Russian), to be translated and published in English by Princeton University Press. [l71 V.N. Popov, Functional Integrals and Collective Excitations, Cambridge, University Press, 1987. [l81 D.J. Amit, Field Theory, Renormalization Group, and Critical Phenomena, World Scientific, 1984. [l91 A.Z. Patashinsky and V.L. Pokrovsky, Fluctuation Theory of Phase Bansitions, Oxford, Pergamon Press 1979. [20] A.M. Tsvelik, Quantum Field Theory in Condensed Matter Physics, Cambridge University Press, 1995, 1996. [21] S.K. Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, MA 1976. [22] A.N. Vasiliev, Functional Methods in Quantum Field Theory and Statistical Physics, Gordon & Breach, Amsterdam 1998. The following books are devoted to the method of path integral in various contexts. [23] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGrawHill, 1965. [24] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, 2nd ed. World Scientific, Singapore, 1995.
Part I
THE PATH INTEGRAL IN QUANTUM MECHANICS
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Chapter 1 Action in Classical Mechanics 1.1
The Variational Principle and Equations of Motion
In the middle of 17th Century the French mathematician Fermat introduced the variational principle of geometrical optics. Behind this principle is the beautiful notion that Nature always chooses the simplest and easiest way to act. Three centuries later, following work by such outstanding scientists as Maupertuis, Euler, Lagrange, and Hamilton, this idea has become one of the main tools in the investigation of Nature. These days there is no doubt that the variational principle and the action formalism reflect a deep property of Nature. For a long time, however, in spite of some of the great work that had been done on this concept, it was considered an abstract mathematical construction. The action principle may seem abundant in classical mechanics. Indeed, although the action is a functional which takes a value for any trajectory, only the unique trajectory which is the solution of the Newtonian equations of motion is realized in classical mechanics. The conceptual depth of the formalism became clearer after the work of Noether (1918), which showed the connection between action symmetries and conservation laws, and clearer still after Feynman's functional formulation of quantum mechanics which developed Dirac's ideas. The action has a central role because it determines both the classical and quantum dynamics of physical systems. The universality of this approach is a clue for building future theories of fundamental interactions. According to current ideas, the structure of such a theory is defined by the fundamental action, although it may be a very nontrivial problem to make theoretical predictions from the form of the action. Not only invaluable for a
4
CHAPTER 1. ACTION IN CLASSICAL MECHANICS
'Theory of Everything', the action formalism is useful in other domains of physics. It makes manifest a close relationship between quantum and statistical physics, thus helping to elaborate a common point of view on various physical theories. We try to take this point of view throughout this book. Our first step is to recall the main facts of the classical mechanics of point particles. The simplest case is the motion of a pointlike onedimensional particle under an external force f . In this case the equation of motion is mx = f ( X ) , (1.1) where dots denote derivatives with respect to time. This equation can be rewritten in a more general form which permits generalisation to much more complex systems. For this purpose it is convenient to introduce the socalled Lagrange function C or Lagrangian which depends on system coordinate X and velocity X = dxldt. The Lagrangian is equal to the difference of the kinetic 7 and potential V energies C = I  v . For a correct introduction of the Lagrangian it is necessary that there exists a potential V which generates the force f : f = dV/dx. In our case
Now equation of motion (1.1) can be rewritten as
Problem: Check by direct substitution that Lagrangian (1.3) really gives equation (1.1). The quantity p = d C l d x is called the momentum. In example (1.3) it is, of course, p = mx. The substance of the Lagrange idea is that the equation of motion is the condition for a stationary point of a functional called the action which is constructed from the Lagrangian of the system. Let us discuss this approach in more detail. Let us forget for a while about the equation of motion and consider a generic trajectory x ( t ) beginning at a point xi at time tl and ending at a point xa at time t2. A few such trajectories are presented in fig. 1. The question is which trajectory will be chosen by a real particle? In other words, which one obeys the equation of motion?
1.l. THE VARIATIONAL PRINCIPLE
t1
Figure 1
The answer is as follows. The desired trajectory, denoted :(t) called the classical trajectory, is a stationary point of the action
and
: = To prove this, let us consider a bunch of trajectories close to ~ ( t )x(t) ~ ( t ) Sx(t), where Sx(t) is an arbitrary small deviation which is also called the variation. All possible Sx(t) obey the condition
+
because the initial and the final points of the trajectory are fixed and do not vary. The condition that ~ ( t is) a stationary point of the action means that SS, the variation of the action which is proportional to the first power of 62, vanishes. Thus
up to first order terms in 62 (here we have introduced the standard nbtation S[x] stressing that action depends on the form of trajectory x(t) which is variable in infinitelydimensional functional space). Using
6
CHAPTER 1. ACTION IN CLASSICAL MECHANICS
the definition (IS), we obtain
After integration of the first term by parts, 6 s takes the form
as condition (1.6) holds at the ends of the trajectory. The variation 6 s must vanish for any 6x(t), so the extremal trajectory ~ ( t must ) satisfy equation (1.4). A generalisation of these arguments to the case of a system with s degrees of freedom is straightforward. The Lagrangian in this case is a function of s coordinates xi(t) and s velocities i i ( t ) i = 1 , 2 . . . S, and the variation of the action with respect to all of these leads to the set of s equations
1.2
A Mathematical Note: The Notion of the Functional
In this section we would like to discuss briefly the mathematical background of some notions used in this book. We would like to point out from the very beginning that we shall not provide mathematically rigourous proofs, preferring instead to present the ideas in practical use. As mentioned above, the action defined by expression (1.5) is a number for each specific function x(t). This number is the value of the integral (1.5) for the given trajectory. Such an object is called a functional of x(t). A functional is a generalisation of the function of many variables. It is defined on the infinitedimensional space of all trajectories x(t). Note that a function of another function is not regarded as a functional because it reduces to a composite function in the usual sense. Let us define the functional derivative 6/6x(t), already used in the derivation of the equation of motion (1.4). Let us divide the motion time
7
1.2. THE NOTION O F THE FUNCTIONAL
interval T = tbta into N equal parts, using a discrete set of time points1
ti: where i = 0 , 1 , 2 . . . N , x ( t o ) = X , and x ( t N ) = 26. NOWwe have the discrete set of variables xi = X ( & )for i = 1 , 2 , 3 . . . N  1 which represent the trajectories. The boundary points are fixed. The discretised version of the action (1.5) reads
The stationary point of S N obeys the standard requirement d S ~ / d x = i 0. These equations take the form
The expression in the square brackets tends to X in the continuum limit At + 0. In other words, in this limit we get immediately the equation of motion (1.4) multiplied by At inherited from the integral sum (1.12). Of i At. course, it is convenient to divide all partial derivatives d S ~ / d x by Thus we come to the definition of the variational or functional derivative.
Here it is assumed that the point ti is always chosen to be the nearest to
t. The introduced operation is convenient for the derivation and representation of many useful relations. For example, equation (1.4) can be written in the form 6s
@q=O
and the first variation of the action can be written analogously to the first differential of a function of many variables:
GS[z(t)]= lim dSN = lim ~ t  0 ato
Nl
dSN dxdsi
,_,
6s 6x(t)dt
.l tb

.
(1.16)
t,
'To avoid confusing notation, we use the subscripts a and b for the initial and the end points of the trajectories when a subdivision of the time interval is involved. Otherwise, we use the labels 1 and 2 as in the previous section.
CHAPTER 1. ACTION IN CLASSICAL MECHANICS
8
Note that the definition (1.14) can be rewritten in another form in terms of the &function: S[x(t)
+ a 6(t  t')]  SIX(~)]) .
For the proof, it is sufficient to use the representation of the 6function in terms of the Kronecker symbol Sij: 6(t  t')
=
lim
at0
LAta i j
,
where the points ti and t j are the nearest to t and to t' correspondingly. Expression (1.17) implies that the expansion in powers of the small parameter E is valid although the expressions are singular. This is actually a shortcut of a correct mathematical procedure, which requires us to rewrite (1.17) using a representation of the &function 6(t  t')
= lim f,(t

t')
(1.19)
P0
with a continuous function f,, then consider the limit E = 0 in (1.17) and only after that, set p = 0. Let us give a few examples of functional derivative calculations for various functionals F. 1. F[x] = C we get
=
const. Then, according to definition (1.14) or (1.17),
2. F[x] = Jx(t)dt; In this case
6F[x(t)1 6~(t')
{/
1
= lim E0
6
(x(t)
+ ~ 6 (t t')) dt

/
x(t)dt
}
3. F[x] = J G(t, tl')x(t)dt, where G(t, t") is a function of two variables (t" is a free parameter in this example). Taking the derivative, analogously to the above cases, we get
=
J
lim L{ ~t ~ ( t9'), (x(t)
ato
+ a t a(t  t'))dt

/
~ ( tt"), x(t)dt}
1.3. T H E BOUNDARY CONDITIONS
9
4. FIX]= x ( t ) . Here F is a function rather than a functional. It can however be written in the form of the previous example: F [ x ]= J 6(t1 t)x(t1)dt'.Then = 6(t  t')
Sx (t') The same expression follows directly from definition (1.17) at F = x ( t ) . Its sense is the same as that of the expression dxi/dxj = 6ij in the case of conventional partial derivative calculation. It should be pointed out that until now we have not specified the kind of stationary point which yields the classical trajectory. Is it a minimum or a maximum? We shall return to this question in the following chapters.
Problem: Show that for the harmonic oscillator, % realizes the minimum of the action only at t2  t l < nlw. At larger times % corresponds to a saddle point.
1.3 The Action as a Function of the Boundary Conditions Suppose that we found a classical trajectory ~ ( tobeying ) the boundary conditions x ( t l ) = X I , x(t2)= 5 2 . The value of the action calculated on ~ ( is t )a function of these parameters: S[%(t)] = S ( x 2 ,x1,t2,t l ) . Let us find the action as a function of boundary conditions for two important examples.
1. Free motion:
=I
S[x(t)]
tz
1
pgdt
.
(1.24)
tl
In this case the trajectory is a straight line % = vt. Therefore,
where v should be expressed in terms of the boundary conditions v = ( x 2 x l ) / ( t 2 t l ) . This gives
'To avoid ambiguities one should always distinguish between action considered as a function, and as a functional. As a rule, it is clear from the context.
CHAPTER 1. ACTION IN CLASSICAL MECHANICS
10
2. Harmonic oscillator:
+
Classical trajectories have the form :(t) = xl cos wt (xl/w) sin wt. Here W = (k/m)lI2 and xl is the initial velocity. Expressing x1 in terms of x2 using the condition :(t2) = X Z , we get:

mw 2sinw(t2  tl)
[(X:
+ X:)
cos w(t2  tl)  2x1x2]
.
This function evidently tends to (1.26) for W + 0. It may come as a surprise that this classical formula contains important information about the wave functions and energy levels of the quantum harmonic oscillator. We shall show this in the next chapter. The function S(x2,xl, t2,tl) possesses some useful properties which we would now like to consider. Let us differentiate the free action (1.26) with respect to x2. We get m(x2  xl)/(t2 tl) which is simply the momentum on the classical trajectory. Analogously, the derivative with respect to t2 ]. coincides with the energy of the particle m(x2  ~ ~ ) ~ / [ 2( tt ~2 ) ~Of course, this does not happen by chance. Let us find the change in the action caused by the small variations x2 + x2 Ax2 and XI + XI Axl with the motion time unchanged (see fig. 2 in which Axl = 0). It is clear that the modification of the boundary conditions leads to Ax(t). Here a modification of the classical trajectory :(t) + :(t) Ax(t) is not an arbitrary quantity. It is determined by Axz and Axl. Fortunately, it is not necessary to find Ax(t). It is sufficient to write down the action variation as in the derivation of the equation of motion (1.4) or, equivalently, to repeat formulas (1.8)  (1.9). This gives
+
+
+
It should be noted that in contrast to (1.9), the first term here is no longer zero because we have varied the boundary conditions. However, obeys equation (1.4), the second term in (1.29) vanishes. Thus we as : find t2 dL (1.30) SS = AX~ = p(t2)Ax2 p(tl)Axl .
ax
t1
1.3. THE BOUNDARY CONDITIONS
0
fl
Figure 2
Here p = d L l 8 x is the classical momentum of the system. For the Lagrangian (1.3) it equals p = m x . Let us write down the obtained expressions explicitly:
These relations can be immediately generalised to systems with many degrees of freedom. Problem: Find these generalised expressions.
Let us now find 8S/dt2 and d S / 8 t l . For this purpose we consider the small variations tl + tl Atl and t2+ t2+At2at unchanged X I = x ( t l ) and x2 = x(t2) (see fig.3 where Atl = 0 ). It is convenient to express the action in the form
+
S(xl,x2,tl
+ At,,t2 + A t 2 ) = =
j!
tl+Atl
tz+Atz
/
L(%+ A x , ,
+ Ab)dt
(1.32)
t~+At, tz+Atz
Ldt+
/
t2
Ldt+j?Ldt tl
Here A x ( t ) is the change of the classical trajectory at time t due to the given change in boundary conditions. This quantity can be neglected in
CHAPTER 1. ACTION IN CLASSICAL MECHANICS
Figure 3
the first two terms because of the smallness of the interval of integration. The change in the final integral can be found in the same way as in the case of XI and x2 variation. Finally then, we conclude that the action change up to firstorder terms takes the form
It can be seen in fig.3 that Ax2 = x(t2)At2. Analogously Axl = x(tl)Atl. Taking into account the fact that : obeys the equation of motion, we finally obtain
where is the Hamiltonian of the system. In the particular case of a point particle, the Lagrangian is defined by expression (1.3). Substitution of (1.3) into (1.35) gives H = mx2/2 V = 7 V which is simply the total energy of the mechanical system.
+
+
1.4. SYMMETRIES OF THE ACTION
1.4
13
Symmetries of the Action and Conservation Laws
If all parameters of a system are independent of time, then all points in time are equivalent. Let us consider the relation between this trivial statement and the energy conservation law. Consider a trajectory x(t) beginning at XI at time tl and ending at 2 2 at time t2, and consider a new boundary condition differing from the old one by a small time interval At, so that tl t tl At and tz + t2 At. If the system's parameters do not depend explicitly on time, the new trajectory will simply be x(t At) and the substitution t + t At in the integral (1.5) shows that the action remains unchanged. On the other hand, as follows from (1.34), the action invariance means that H(tl) = H(&). As the considered trajectory is arbitrary (but classical), the Hamiltonian is constant along any classical trajectory. That is, total energy is conserved. The connection between the invariance of the action under spatial translations and conservation of momentum can similarly be established. Let us make a shift of a classical trajectory xl + xl Ax and x2 + x2 Ax. If the Lagrangian does not depend on X, the action along the trajectory will not change and as a result of (1.31) we get p(tl) = p(t2). The fact that the trajectory is arbitrary implies the momentum conservation law. Note that the translational invariance of the Lagrangian allows a dependence only on X. In the simplest example (1.3) this corresponds to free motion. In fact the scope of systems for which the momentum conservation law is valid is much wider because X can be thought of as any generalised coordinate describing the system. In the particular cases considered here we have actually proven the Noether theorem which relates action invariances with conservation laws. Generally speaking, the action functional possesses the following fundamental property: the action invariance under each oneparameter transformation leads to the existence of a certain conserved quantity '.
+
+
+
+ +
+
31t should be pointed out that the substance of this theorem is rcally deeper than this wellknown statement. Thus it is possible that the action invariance with respect t o a continuous transformation does not lead to any conservation law if the corresponding variation of the action is idcntically zero (the socalled Noether identity). This case is the subjcct of the second Noether theorem [l]which is very important in gauge theories. Examplcs of such theories will be considered in chapter 12.
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Bibliography For a detailed review of the connection between symmetries and conservation laws see E. Noether, Nachr. Ges. Wzss., Gijttingen (1918)) 171; E. Hill, Rev. Mod. Phys., v 23 (1950), 253; J. Fletcher, Rev. Mod. Phys., v 32 (1961) 65. There are many good textbooks on classical mechanics to which the readers can refer. For example, H. Goldstein, Classical Mechanics, AddisonWesley, Reading, Mass., 1980. The modern point of view on the Hamiltonian mechanics is presented by V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1980.
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Chapter 2 The Path Integral in Quantum Mechanics 2.1
The Green Function of the Schrodinger Equation
As is well known, the evolution of quantum mechanical systems is described by the Schrodinger equation
with an initial condition for the wave function $(X, tl) at t = tl. For the sake of simplicity we consider first onedimensional systems. The classical motion of such systems is determined by a single space coordinate X. From a mathematical point of view, we are dealing with the Cauchy problem, so the general solution can be constructed by means of the Green function G(x, xl, t , tl) . This function is also called the propagator or resolventa of the Schrodinger equation. It is simply the wave function of a particle which is situated at the point X = X I at the initial time t = tl so that
Here the operator H acts on the variable X. The solution of the Schrodinger equation with an arbitrary initial condition +(X,tl) can be represented
CHAPTER 2. THE PATH INTEGRAL
18
as a superposition of all waves of amplitude initial time t = tl:
$(X,
tl) 'irradiated' at the
Throughout this book only stationary Hamiltonians are considered. The Green function in this case can be represented as an expansion in the normalised eigenfunctions $,(X) of the Hamiltonian H for which the relation H$,(%) = E,$,(x) holds. Thus,
This last relation can be verified by substituting it into (2.2), remembering that each term of the series obeys the Schrodinger equation. The initial condition is fulfilled by virtue of the consistency of the eigenfunction set $,(X): X$n(x)$:(Y)
= a(x

Y)
.
(2.5)
n
It follows from (2.4) that G(x, xl, t, tl) depends on time through the difference t  tl. This is an immediate consequence of the invariance of the system under translations. Let us calculate the Green function of a free quantum particle of mass m as the simplest example. The corresponding Hamiltonian is
Its eigenfunctions are the plane waves $(X) = eRpX, where p is the eigenvalue of the momentum operator. The energy of the particle is E = p2/(2m). The momentum as well as the energy take values in a continuous infinite range (we say that their values fill a continuum). In order to perform the summation over n on the right hand side of (2.4) one should make the spectrum discrete and take the continuum limit in the final expression. For this purpose we close the coordinate line X to form a ring, that is we assume X to vary in a large but finite interval 0 5 X 5 L so that the wave function is periodic in space $(O, t) = $(L, t). Then the normalised eigenfunctions take the form
2.1. T H E GREEN FUNCTION
and the discretised values of the energy and momentum are
where n = 0 , f l , f 2 . . .. Now we can use formula (2.4) to evaluate the Green function:
It follows from (2.8) that the momentum increments by Ap = 2nfilL as n changes by unity. This means that at large L the general term of the sum in (2.9) is a slowly varying function of n. This enables us to regard (2.9) as an integral sum. Multiplying and dividing the right hand side of (2.9) by A p and taking the limit L + CO we get
Calculating this Gaussian integral using the formula +m
J dz exp (az2 + bz) = 00
a exp
() b2 4a
'
(2.11)
(its derivation, as well as some other properties of Gaussian integrals, is given in Appendix A.1), we obtain finally
Note the classical action (1.26) appearing in the exponent. In principle, we could just write this down making sure of its validity by a straightforward substitution into (2.2). Our aim here was to show how the summation over n transforms into integration over p. This procedure will be widely used throughout this book. The obtained expression (2.12) enables us to derive the de Broglie relations. Let us consider a free particle starting from the point xl at time tl. Its wave function is G(x, xl, t, tl) and it looks like a wave whose amplitude and the phase evolve according to (2.12). Suppose that the
CHAPTER 2. THE PATH INTEGRAL
20
particle is detected at the point X at time t. Let the local coordinate and time in the detector's reference frame be Ax and At respectively, and consider the case when they are small: Ax << X and At << t  tl. Let us expand the argument of the exponent in (2.12) in powers of the small ratios Ax/(x  X,) and At/(t  tl). This gives
i m(x 
h 2(t  t,)
 xl) i m(x + tii m(x Ax(t tl) h 2(t  tl) 
l
A t + ...
Keeping only the firstorder terms in Ax and At, we conclude that the wave function of the particle looks locally like a plane wave with wave number k and cyclic frequency W such that k = p/h
and
W
= Elfi,
(2.14)
where p and E are the momentum and energy of a classical particle which starts motion at the point xl at time tl and reaches the point X at time t. Physically, the smallness of the neglected terms requires that the classical momentum and energy change essentially over distances and times much larger than Ax and At respectively. Let us now return to the general properties of the Green function of the Schrodinger equation, and discuss an analogue of the Huygens principle which states that $(X, t) can be regarded as a result of the superposition of all waves irradiated at any intermediate time t2. Indeed, as was noted, formula (2.3) gives such a representation of the wave function for the initial time t = tl. As the Schrodinger equation is firstorder in time, we can take the wave function $(X, t2) at a time t2 as a new initial condition and find $(X, t) for t > t2 by solving this equation. The fact that this should not change the result imposes the following property
for any initial function $(xl,tl). Thus it must be that
The property (2.16) can be directly checked by making use of the representation (2.4). For this purpose it is sufficient to substitute (2.4) into (2.16) and make use of the orthonormality of the eigenfunctions: S dx$k (X) $: (X) = &,.
2.2. THE PATH INTEGRAL
2.2
The Path Integral
Here we derive one more useful representation of the Green function G(x,xl,t, tl). Let us subdivide the interval between tl and t into N (where N is very large) small parts At (see fig. 4) and apply formula (2.16) as many times as there are intermediate time points. The beginning of the ith interval is the point ti, i = 1 , 2 , .. . N. We denote the corresponding coordinate X by xi. Then,
This expression does not yet provide any new information. To gain it, we must find G in the limit At + 0 or N + oo. For infinitesimal At we have for G(x, X I , t At, t)
+
G(x, xl, t
+ At, t) = G(x, xl, t, t )  !HG(X, h
21,t, t
) ~ t.
(2.18)
For a particle in the potential V(s) , the Hamiltonian H is
Such a choice of Hamiltonian enables us to get rid of some additional complications connected with ordering of the coordinate and momentum operators. Using the relation G(x, X I , t, t) = 6(x  xl), and the Fourier representation of the deltafunction
we obtain G(x, xl, t
+ At, t)
M
6(x  21) 
(2.21)
As At + 0, the terms in square brackets can be regarded without reduction in accuracy as an expansion of an exponential function:
[l 
(g+ v(x))]
= exp
[F (g+
~(x))] .
(2.22)
CHAPTER 2. THE PATH INTEGRAL This results in G(x, XI,t
+ At, t)
(2.23)
Substituting this into (2.17), we obtain an expression G(x, X I , t, t ~ )=
lim
N+m
/
~ P N dpi dp2 dx2dz3.. . dxN... 27rK27rK 2nfi
(2.24)
where X N + ~= X. This formula merits some discussion. First note that there are N integrations over momenta, and one fewer over coordinates in (2.24) because the points xl and X N + ~= X are fixed by the boundary conditions. Thus the total dimension of the space over which the integration is taken is 2N  1. The points of this space are the sets 2 2 , . . . XN;PI,.. .PN. In the continuum limit N + CO these sets become x(t) and p(t) respectively. This means that the integration in the continuum limit is taken over the set of all trajectories in the system's phase space. Second, it should be emphasised that X and p appearing in (2.24) are ordinary functions rather than operators. It can also be easily checked that the argument of the exponent in (2.24) reduces to the classical action Slp(t), x(t)] = J dt(px  (p, X)) in the limit N + oo. This rather transparent physical meaning of expression (2.24) enables us to rewrite it in a symbolic form as an integral over the infinitedimensional phase space
Such a form does not contain, however, information concerning the measure of the functional integration, namely, the coefficients (27rfi)l and 'It is not so transparent at the mathematical level because a more rigourous definition of the path integral is conjectural. Nevertheless, all specific physical calculations are carried out at a large but finite value of N. The limit N = co is taken a posteriori.
2.2. THE PATH INTEGRAL
dt Figure 4
the fact that there is one fewer integration over coordinates than over momenta. The representation of the Green function obtained here is called the path integral (PI), or the functional integral, or the Feynman integral, or simply the integral over trajectories. It is the central object in modern quantum field theory (QFT). The use of the PI is often more convenient and clearer than other methods. Note that we did not use any specific form of the Hamiltonian in deriving expressions (2.24), (2.25). Only the fact that the operator H ( @ ,X) has the eigenvalues H(p, X) identified it with the classical Hamiltonian. Now let us make use of the quadratic dependence of H on p, equation (2.19), in order to perform the integration over momenta in (2.24). Evaluating Gaussian integrals
we get G(%,X I , t, tl)
=
l
N+m
d x d . . . dxN
CHAPTER 2. THE PATH INTEGRAL
where the symbolic form of the PI has been used again. This calculation becomes rather complicated if the Hamiltonian depends on momenta nonquadratically. The general recipe is to use (2.24) for construction of a socalled 'effective action' which is the functional appearing in the exponent in an expression of type (2.27) after the integration over momenta and taking the continuum limit. Fortunately the Hamiltonians in quantum theory are, as a rule, quadratic in momenta, so we restrict ourselves to such systems. Let us discuss further expressions (2.25) and (2.27). Formula (2.27) states that the probability of finding the particle located at the point X at time t, having been located at the point xl at time tl, is a sum of the contributions due to all trajectories with starting and finishing points X I , tl and xz,tz respectively. Each trajectory x(t) contributes a weight factor eiSIx(t)ll". One may say that the particle moves simultaneously along all trajectories with given end points, and the superposition of all weighting factors results in the actual wave function. Note the difference between the role played by the action in the classical theory and in the quantum one. In classical theory, the action results in a unique classical trajectory: a single point in the functional space. In quantum theory, we deal with all possible trajectories: that is the whole functional space. Also from (2.25) and (2.27), there follows a transparent way in which classical and quantum mechanics are connected with each other. Let us consider a macroscopic system. This means that the action on its typical classical trajectory is much greater than the fundamental constant 6. In this case the integrands of (2.25) and (2.27) oscillate rapidly so that only a bunch of trajectories providing almost stationary S give a significant contribution to the transition amplitude. The central trajectory of this group satisfies the condition 6s = 0. Thus in the limit S / h + ca the main contribution to the transition amplitude is given by the classical trajectory. Then, qX, xl, t, tl) e i ~ ( x , x ~ , t 3 t d l ~ (2.28) where S is the action on the classical trajectory. This is the key statement because it gives rise to a method of approximate calculation of transition amplitudes which will be used extensively. Below, a more exact formulation of (2.28) will be considered.
2.3. THE PATH INTEGRAL FOR FREE MOTION
25
Note that expression (2.28) leads to the generalised deBroglie relation (2.14). To show this we take small variations of X and t : X + X As and t t t At and expand the action in (2.28) in these variations. Taking into account the definitions of the classical momentum p (1.31) and energy E (1.34), we may write
+
+
Here p is the canonical momentum conjugate to coordinate X . This equation is justified for large classical action S [ x ( t ) ]>> h, which is the case for the macroscopic trajectories, for example, of a particle in a bubble chamber. We would like also to note that the starting point for expressions (2.25) and (2.27) for the Green function was the Schrodinger equation (2.2). We can easily show that this equation follows from from (2.25) and (2.27). It is sufficient to use property (2.16) which is true for functional integrals by construction and to apply the mathematics of (2.21)(2.24) in reverse order. Consequently, representations (2.25) and (2.27) are completely equivalent to Schrodinger equation quantum mechanics. In this book, we prefer the PI formalism because it enables a more straightforward generalisation to quantum field theory. We shall not consider the mathematical matters related to the path integral. The path integral will rather be used as a tool, considered at the physical level of strictness.
2.3
The Path Integral for Free Motion
Here we consider examples of PI calculation for some simple systems. We start with the simplest case  free motion. The action in this case is
0
As all times for the free motion are equivalent, we have chosen t I = 0 and the motion time equal to T. Eventually, the Green function for this problem will be derived three times. First, we need the exact solution of the Schrodinger equation obtained in equation (2.12). Let us rewrite it here in the current notation:
Second, we would like to make sure that this expression really follows from PI (2.27). Thirdly, the stationary phase method will be applied to the calculation in order to gain an insight into this method itself.
CHAPTER 2. THE PATH INTEGRAL
26
Free Motion: Straightforward Calculation of the Path Integral To perform the functional integration and find the Green function
we return to the discrete set of times ti and coordinates xi = x(ti) according to definition (2.27). Then for different Ati we have
This is a chain of Gaussian integrals over dxi for each of which formula (2.11) should be applied. Integration over dx2 gives
1 + exp{~[xr(Atl

t
252
X? +  + Atl +At,
(a":, 2 ) 
l

(2.34)
Substituting into (2.33), we obtain
exp { E
h
[
(53  ~ 1 ) '
2(At,
+ At,)
+
(x4 ~ 2At3
3
+...l) )
~
We see that integration over the intermediate coordinate x2 does not change the form of the PI being reduced to the replacement of Atl and At2 by the time interval Atl At2. That is property (2.16) is valid. It is so because each elementary Green function in the product in (2.33) is exact in the case of free motion. Now it is clear that the subsequent integrations over dx3,dx4,.. . dxN give expression (2.31). The above described method of direct PI calculation is applicable only for free motion because the presence of a potential V(x) prevents evaluation of the infinitely long chain of integrals as in (2.34). However,
+
2.3. T H E PATH INTEGRAL FOR FREE MOTION
27
a useful method of numerical evaluation of the PI is based on a form similar to that defined in (2.24) or (2.27). It is called simulation on the spacetime lattice. The accuracy of this method is restricted by the power of contemporary computers and sometimes it is not large enough to make reliable physical predictions. Analytical methods are traditionally powerful and helpful if one wants to gain insight into the physics of a given system. However, they require us to make some approximations among which the stationay phase approximation plays an important role. Its advantage is the possibility of a straightforward generalisation to QFT.
Free Motion: Path Integral Calculation by the Stationary Phase Method Let us start with an illustrative example of the following ordinary integral: I = dxexp ( f (X)) , (2.36)
I
where f (X) is a smooth function. It is evident that the largest contribution to the integral is given by the region around the point xo where the minimum of f occurs. The point obeys the equation f'(xo) = 0. Expanding f (X) in a series around xo we get
because the integral reduces to the Gaussian one. To establish the applicability condition of such an estimate we note that the main contribution to the Gaussian integral is given by the region for which z  xo ( f " ( ~ ~ ) ) ~ Let ' / ~us . substitute this value into the third and fourthorder terms (that is, the leading neglected terms). The requirement of the smallness of these terms gives
These conditions are valid if the function f and all its derivatives are large. Let us define a new function g so that f(x) = g(x)/h with a
CHAPTER 2. THE PATH INTEGRAL
28
small parameter fi such that g and all its derivatives are of order unity. Conditions (2.38) then take the form << 1 and fi << 1. Accounting for the corrections to (2.37) leads to a contribution which contains the coefficient of the thirdorder term squared. Therefore, fi turns out to be the small expansion parameter. An example of the application of this formula is the asymptotic form of the gammafunction
the socalled Stirling formula . It follows from the condition f l ( t o ) = 1 z / t o = 0 that to = z is the minimum of f and
+
according to (2.37). This method is called the Laplace method or the saddle point approximation. Its analogue for integrals with oscillating exponential function I = d x exp ( i f ( X ) )
/
is called the stationary phase approximation. The specific feature of the latter case is that the smallness of the contributions from the region far away from xo is due to extremely fast oscillations of the integrand rather than the absolute smallness of the exponential function. Therefore all formulae remain valid up to the replacement of the ordinary unity by the imaginary one. Now let us consider the path integral. First we find the trajectory x ( t ) corresponding to the stationary action S S [ x ( t ) ]= 0. This is the classical . we consider the set of trajectories which deviate trajectory ~ ( t )Then from the classical one by a small variation 6 x , i.e. x ( t ) = ~ ( t )6 x . As the boundary conditions are fixed, 6 x ( 0 ) = S x ( T ) = 0. Expansion of the action in a Taylor series around :(t) gives
+
where the shortened expression for the action variations has been used. Before we go on evaluating the PI we clarify this expression and introduce some other widely used abbreviations:
2.3. THE PATH INTEGRAL FOR FREE MOTION
It is often convenient to integrate by parts taking into account the zero boundary conditions for the variation 6x. As a rule, we deal with Lagrangians quadratic in x so that all higher derivatives are equal to zero. Now, let us turn to the expansion (2.42). As this expansion is performed near the classical trajectory, SS = 0. In the considered case of the free motion (2.30) all action variations higher than the second, S'S are zero. For the second variation we have
Here we have integrated by parts and introduced an obvious abbreviation for the second time derivative. Performing the described expansion in PI (2.27), we get the Green function of the free motion
=
exp
' """
x02]
2T
/
V6x(t) exp
[gf
dthx(d:)6~
1
.
Thus we have simplified the problem. The classical action takes care of the boundary conditions while Sx(t) (which is the new variable of functional integration) obeys the zero boundary condition. However, the form of the integrand remains unchanged. As before, the main difficulty is that the variations 6x(t) at neighbouring times are not independent of each other because of the time derivative (compare with (2.27)). To solve this problem let us expand Sx(t) in a series of eigenfunctions $,(t) of the operator 8; (d/dt)2 appearing in the second action variation (2.44). This is possible because of the zero boundary conditions for Sx(t):
Here C, are the coefficients of the expansion. The explicit forms of the eigenfunctions and the corresponding eigenvalues X, are: $. (t) =
rnt sin 7
and
X,
r2n2 n = l , 2 , 3 . .. T2
=
. (2.47)
30
CHAPTER 2. THE PATH INTEGRAL
Substituting into (2.44), remembering the orthonormality of the eigenfunctions $,(t), we get
Thus PI (2.45) can be rewritten in the form
A nontrivial operation here is the replacement of the functional integration variable Sx(t) in (2.27) by the set of coefficients C,. A is the Jacobian of this transformation: it is rather difficult to evaluate. Moreover, it diverges in the continuum limit. On the other hand, the Fourier transformation (2.46) is linear. Therefore the Jacobian is a constant which does not depend on the coefficients C,. This transformation reduces the integral to a product of an infinite number of onedimensional Gaussian integrals of the following type:
Using this formula we get finally G(x,xO,T) = exp
[i F,
 x,)~]
2T
An
/%X
K
The Jacobian A is still undetermined. Its calculation and even the correct definition appears to be a serious mathematical problem. Nevertheless, comparing (2.51) with G(x, xo, T ) in (2.12) we obtain immediately
This last expression enables us to calculate the PI when the straightforward calculation (as used in the free motion case) is not possible. One more remark about terminology. The substitution of the integration variable Sx(t) + C, enables us to diagonalise the second variation of
2.4. THE PATHINTEGRAL FOR THE HARMONIC OSCILLATOR the action, S2S. To make this clearer let us consider a finitedimensional integral (2.53) I = dxldxa.. . dxN exp (ziMijxj) ,
I
where Mij is an arbitrary, in general, nondiagonal symmetric matrix i, j = 1 , 2 . . . N. It is evidently an analogue of the discretised action. The integral is most easily evaluated in the diagonal basis. In other words, new variables xi = Oijxj must be introduced so that the matrix OMOT is diagonal. This is a direct analogy of what we have done in (2.48).
2.4
The Path Integral for the Harmonic Oscillator
The next important example is the Green function of the harmonic oscillator  a particle of mass m moving in the potential V(x) = k2x2/2 (fig. 5)
where W = (k/m)'f2 is the angular frequency. To calculate the PI (2.54), let us use the method of stationary phase. It should be noted that the result must be exact because the action is quadratic in X, i.e. SnS = 0 for all n 3. The recipe for the calculation will be the same as that in the case of free motion:
>
l. Find the classical trajectory ~ ( t and ) the value of its action;
2. Expand the functional of action in the neighbourhood of the classical trajectory;
) zero 3. Perform functional integration over Sx(t) = x(t)  ~ ( t with boundary conditions. This leads to the following expression G(X,xo,T ) = exp [;~[Z(t)l]
JD
e x
t
t
  w2)~x(t)
CHAPTER 2. THE PATH INTEGRAL
Figure 5
where S[z(t)] is defined in (1.28). Now, as was done for the free motion, it is necessary to turn to the Fourier components of Sx(t). To do that it is sufficient to note that the eigenfunctions of the operator 8:  w2 are the same as those of 8: and the corresponding eigenvalues differ from (2.47) by a constant.
Repeating the argument of the previous section we obtain
[i s[z(~)]] A n [g (G )]' CO
G ( r, XO, T) = erp
 w2
(2.57)
n=l
2 ~ can e now easily answer the question of the previous chapter: 'Does the trajectory correspond to the maximum or to the minimum of the harmonic oscillator action?'. As is seen from (2.48), T < r / w corresponds to positive second variation of the action. Hence, the classical trajectory corresponds to the minimum of the action. If r / w < T < 2 ~ / wthen the Fourier coefficient C: has a minus sign and gives a negative contribution to the action. This means that the action is maximal in the direction determined by Cl and minimal in other directions. Further growth of T leads to the appearance of a greater number of negative eigenvalues X ., At T = n n / w the eigenvalue X, goes to zero which means that there are many classical trajectories distinguished by the value of C, (the kinetic focus). This property is exact in the case of the harmonic oscillator.
2.5. IMAGINARY TIME AND THE GROUND STATE ENERGY
=
2iT26 fi d F fi n r
exp [ i ~ ~ z ( tA) ~ ] n=l
(l

n=l
n27r2
The Jacobian A of the transformation x(t) + C, is the same as for the free motion because the eigenfunctions of the operator 8:  w2 are the same functions (2.47). Thus A is given by expression (2.52), and the above expression differs from the free particle Green function (2.51) only by the factor n ( l  $)'l2 which is given by the well known formula m
x2
n(1)=
n=l
sin x x
(see Appendix A.2). Making use of this equality, the Jacobian (2.52), and the explicit form of the classical action S(x, xo, T)
=
mw [(x2 2 sin wT
+ X;)
cos W T 2xxo]
,
(2.59)
from the first chapter (1.28), we obtain the desired final expression for the Green function:
((xi
2rih sin(wT)
Of course, in the limit (2.51).
2.5
W + 0
+ x ( ~ )cos ~W ) T

~ Z ~ X ( T ). ) ]
this gives the free particle Green function
Imaginary Time and the Ground State Energy
To know the Green function is equivalent to knowing the complete solution of the Schrodinger equation. This function contains all the information about the eigenfunctions and eigenvalues. We discuss here how to extract this information. Suppose we know (exactly or approximately) the Green function G. In this book, we shall calculate it via its PI representation (2.27) while the representation (2.4) will be used to extract the information about the energy states and the wave functions. It follows from (2.4) that this can be done by taking the Fourier transformation of G with respect to time (note that the Fourier integral will already be quite nontrivial for the case of harmonic oscillator (2.60)).
CHAPTER 2. THE PATH INTEGRAL
34
Let us consider a simpler problem, namely to find the ground state energy and wave function of the harmonic oscillator. For this purpose let us consider (2.4) and (2.60) at imaginary values of the time variable t2  tl T = ZT and equate them. We get
+
+
$ O ( X ) $ ~ * ( X O ) ~$l(~)$~(~O)eEIT1h ~~~/~ ...
/,
=
27rh sinh(wr) exp
{
m 26 sinh(w7)
(2.61)
+
[(x2 xi) cosh(wr)  2x0x]
We see that as T + oo the leading contribution to the lefthand side is given by the ground state because E. is the minimal energy. On the c o s h w ~G expwr for righthand side, we use the fact that sinh wr WT >> l. This gives
E e x p =
( 7)exp (g (xi + x2))
(z)
'l4 exp
(G)(z) (S) ( 7) mxo
'l4 exp
exp
(g)
Here, & , ( X )= (11~/7rh)l/~ exP and E. = h / 2 are the correct ground state wave function and energy of harmonic oscillator. This is a remarkably simple method of calculating the ground state wave functions and energies: it is sufficient to take imaginary time in the PI transition amplitude and send it to infinity. In expressions of the type A(x, xO)exp(Brlh) the quantity B is to be interpreted as the ground state energy and A(x, xo) as the product of the corresponding wave functions. In quantum field theory, the ground state is called the vacuum. It plays an especially important role in a great number of physical problems because it often defines the system's properties. h many cases the calculation of the PI can be essentially simplified if we turn to imaginary time in equation (2.27). We shall demonstrate this for a system with the action
Now, let us rotate the integration contour in the t' complex plane by 7r/2. Then t' = ir where T is real. The action takes the form
2.5. IMAGINARY TIME AND T H E GROUND STATE ENERGY 35
The quantity SE is called the Euclidean action. It is real, and differs from the initial action in the sign of the potential. Substituting (2.64) into (2.27) we obtain the imaginary time PI which is also called Euclidean:
=
lim
N+m
exp {:AT
dx2dxi . . . dxN
5[7 i=l
J xi)
+
}
If the potential is bounded below, it is natural to expect that this functional integral is convergent. The reader may doubt the legitimacy of the rotation leading to expression (2.65). In general it is not a simple matter, but for our purposes, it is sufficient to regard the Euclidean form of PI (2.65) as a Green function of the imaginary time Schrodinger equation. The latter is simply the heat conduction equation:
To obtain the Euclidean PI, we must introduce the Green function GE of this equation as in (2.2) and repeat the steps leading from equation (2.2) to PI (2.27). Continuing this analogy, we can get the modified version of formula (2.52):
From the beginning of the 1970's, functional integration has been one of the main instruments of vacuumstate exploration, and has made possible a considerable success in this field. The Euclidean path integral plays the central role in calculating socalled nonperturbative effects, a simple example of which will be given in the next chapter.
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Bibliography [l] The first papers on path integral formulation of quantum mechanics
were P.A.M. Dirac, Phys. Zeit. d. Sow. Union, 3 (1933) 64; R.P. Feynman, Rev. Mod. Phys., 20 (1948) 367. [2] The first textbook on PI has been R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGrawHill, 1965. [3] The following books are devoted to the formalism of PI H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physzcs, 2nd ed, World Scientific, Singapore, 1995. [4] A brief introduction and many examples of practical use of the PI can be found in R.P. Feynman, Statistical Mechanics: a Set of Lectures, Reading, Mass., Benjamin, 1972.
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Chapter 3 The Euclidean Path Integral In this chapter, we consider the Euclidean path integral calculation for some model examples: namely the quantummechanical problems of finding the ground state energy and the wave function. The reason for this choice is that the method of solution of these problems can be generalised directly to QFT. The key feature of the approach to be considered is the use of nontrivial classical trajectories called instantons to evaluate the Euclidean PI. We will go into detail in order to explore this calculation method. It will be the final time in this book that we compare these results with those obtained by the standard methods of quantum mechanics.
3.1
The Symmetric Double Well
For the first example, let us consider the motion of a particle described by the onedimensional action:
Here V(x)is the potential with symmetric minima at X = a and X = a, depicted in fig. 6. A potential of this form describes, for example, the motion of a nitrogen atom in an NHS ammonia molecule. The nitrogen atom can be located on either side of the plane formed by the three hydrogen atoms. The states are physically equivalent, but different. This results in an observable peculiarity of the molecule spectrum. For the sake of simplicity let us set the particle mass m and Planck constant fi to unity. This means the following. As it is known, the units of all physical quantities can be expressed as combinations of only three
CHAPTER 3. T H E EUCLIDEAN PATH INTEGRAL
Figure 6
quantities which are assumed to be fundamental. Usually these are mass, length and time. Let us denote as [m],[l],[t],[E],[S]the units of these values, energy, and action respectively. For the latter two quantities, we have [E] = [m][1I2/[tl2and [S]= [m][1I2/[t]. The condition m = F, = 1 means that: (i) [m],[S],[t] are chosen as fundamental quantities, i.e. [l] = ( [ ~ ] [ t ] / [ m ] ) (ii) ~ / ~the ; unit of mass is chosen to be m (rather than, for instance, kilogram), and the unit of action is h. Practically, this means that any physical quantity f , with dimension [m]"" [SIus[tIvt, enters all expressions as f/(mum  F,"S). In order to return to the usual units, we must substitute this ratio for f in all expressions. We shall do this at the end of the chapter. Let us recall some results for the doublewell potential. It is easy to find the particle ground state in the potential V ( x )in the framework of the semiclassical (or WKB) approximation, if the height of the barrier separating the two potential wells is much greater than the ground state energy in each well. In this case, the particle wave function is localised in the vicinity of one of the minima and only weakly penetrates into the subbarrier region. The particle is localised in one of the wells and the influence of the other well is negligible. In this approximation, the ground state of the particle in the left well is described by the ground state wave function of the harmonic oscillator $L E $ r ( x a), localised z$ r ( x  a). near the left minimum. Analogously, for the right well The energy of both states is w / 2 (remember that F, = 1).
+
3.1. THE SYMMETRIC DOUBLE W E L L
41
Nevertheless, it is generally speaking incorrect to neglect the presence of the other well. This is because quantum tunnelling between the wells causes a particle located, say, in the left well to penetrate into the right one, and vice versa. Thus the states localised in a single well are not stationary. The correct wave functions are the even and odd ~uper~ositions:~
where $L, $R > 0. The corresponding states are not degenerate. Their energies are
Thus the previously degenerate energy levels in the left and right well are split. The value of the splitting is exponentially small and is defined by the parameter
where W(E) is the abbreviated action on the classical trajectory:
(these formulae are obtained in Appendix B). Let us consider the evolution of a particle initially (t = 0) located in the left well. Expanding the initial wave function in the Hamiltonian eigenfunctions (3.2), we obtain $(t) =
l
Jz [$o

exp (iEot)
= $L cos(A . t)
+ $R
+
$1
exp (  i E ~ t ) ]
sin(A . t ) .
(3.6)
It follows from this equation that the wave function oscillates between the two wells with frequency A. It is obvious that tunnelling (penetration of the particle through the barrier), can be neglected only if the system is observed over a time interval which is much less than l / A . 'It should be pointed out that this tunnelling restores the initial discrete symmetry of the ground state, which is violated if the particle is localised at a given minimum of the potential.
CHAPTER 3. T H E EUCLIDEAN PATH INTEGRAL
42
Quantum Mechanical Instantons Let us now reproduce the solution of the problem discussed in the previous section by calculation of the Euclidean PI: G(+a, a, T) at T t
CO.
Z* =
I
VX(T)exp {S[X(T)])
(3.7)
Note that the Euclidean Green function and action
are written without the subscript 'E'. Such subscripts as well as the adjective 'Euclidean' of action, trajectories and time will be omitted up to the end of this chapter. It should be pointed out here that we are considering simultaneously two different functional spaces: one for the transitions from X = a to X = a and one for the transitions from X = a to X = a. Let us specify the potential V(x) as the quartic polynomial:
Expanding V(x) as a series in the neighbourhood of
X =
f a , we get:
where w2 = V1'(a) = 8Xa2 is the oscillation frequency. The higher power terms in the expansion (3.10) can be neglected if the quartic term X(x a)4 is small compared with the second order term in the region X l/& where the oscillator wave function differs substantially from zero. This gives a validity criterion for the harmonic oscillator approximation:
Note that this condition also follows from the requirement that V(0) >> w/2. We evaluate the PI (3.7) by finding all the classical trajectories which lead from X = a to X = +a in time T + CO (we recall that we shall calculate two separate values of the Green function which differ from one another in the second argument: 2 2 = a or xz = a). A nonvanishing contribution to the result is clearly made only by those trajectories which
3.1. T H E SYMMETRIC DOUBLE WELL
Figure 7
correspond to a finite action in the limit T + CO.The first such trajectory is apparent, namely x(t) = a. Trivially, the particle remains at the minimum point of V(x). The action on this trajectory is zero. To find the nontrivial trajectories, it is necessary to solve the classical equation of motion
Its solution is especially simple to find when we note that it describes the motion of a point 'particle' with coordinate x in the inverted potential V(x) (see fig. 7). It is clear that in order not to acquire an infinite action as T + CO the particle must spend almost the whole time near one of the peaks of the potential. Thus the particle motion along the trajectory to be found starts at the point X = a at T = 0 with very small (in the limit T + CO, zero) velocity. An instability caused by this initial infinitesimal perturbation grows slowly and results in a fast transition through the region around X = 0, after which the particle stops slowly at the other peak X = a, reaching it at time T = T. The 'energy' of the particle moving in such an inverted potential is conserved:
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
Figure 8
and hence
dx
(3.14)
The solution we are interested in can be found in the limit T = cm, by setting E = 0. Indeed in this case the action on the trajectory leading from X = a to X = a at time T is independent of T itself:
(3.15) Here we used relation (3.13) for E = 0. All other values of E" give trajectories with infinite action in the limit T + cm. Let us find the explicit form of the classical trajectory, denoted by ~ ( .t Integrating ) equation (3.I d ) , we obtain: X
rTO=
dx
1
a
from which it follows that X(T) =a
tanh
w ( r  TO) 2
X
atanh 
,
(3.16)
3.1. THE SYMMETRIC DOUBLE WELL
Figure 9
This trajectory is usually called an instanton (fig. S), and the integration constant TO is its centre the time at which:(rO) = 0. It is important that the action (3.15) does not depend on 70. The characteristic size of the instanton in time is W' << T. The analogous trajectory corresponding to the transition from X = a to X = a is called an antiinstanton. It is clear from symmetry that the classical trajectory leading from X = a to X = a must be an odd function of time. In particular, the instanton should be centred at TO = 0 (and slightly distorted because of the finiteness of T). It is the unique solution of equation (3.13) at finite T. Any other trajectory close to the instanton with TO # 0 is not an exact solution of equation (3.12), but the deviation from an exact solution is small and approaches zero in the limit T + oo. To see this, let us consider the quantity eS on the set of trajectories labelled by TO (see fig. 9). The unique maximum of this function is at the point ro= 0. Other values of 70 give nonzero derivative deCS/d~0# 0 but it is of order eWAT , where AT is the distance between the trajectory centre and the nearest of the boundary points T = 0 or T = T. Thus the To estimate the derivative, it is necessary t o express the initial velocity v0 of the particle in terms of 70. For this purpose we restrict ourselves t o quadratic terms in the expansion of the potential and write X(T) = a (vo/w) sinh(w7). Taking into account only small deviations from the instanton trajcctory we can check that v: (aw)' cxp(WT). This is obviously thc estimate for the energy E corresponding t o the transition for the finite time T, and a t the same time, it is the scale for the first
+

46
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
value of the action remains very close to S (3.15) until 70 is separated from the boundary points by a time interval greater than l/w. It is reasonable to approximate the curve in fig9 by a rectangle of height exp(S) and width T. Such an approximation means that we are taking into account any instanton trajectory (3.17) with action (3.15) corresponding to the value of the parameter 70 being in the interval [0,T]. Their contribution to the PI (3.7) is
/ die T
Zl = Const exp(S)
= Const
T exp(S)
.
(3.18)
0
where Const is not yet defined. Let us go on searching for classical trajectories which give finite contribution to 2%(3.7). Strictly speaking, there are no other classical solutions for finite T. Nevertheless there exist a class of trajectories which deviate from the classical one by an exponentially small value when T + cm.These are socalled multiinstanton trajectories. They look like chains of interchanging instantons and antiinstantons with centres at (fig. 10). Thus the amplitude of the transition the points T,('), T ; ~ .). . from X = a to X = a includes a contribution from all multiinstanton trajectories with an odd total number of instantons and antiinstantons. Analogously, the contribution to the amplitude from X = a to itself is given by the trajectories with an even total number of instantons and antiinstantons including the trivial one. If all the (anti)instantons are separated from each another by a time interval much greater than the characteristic instanton size llw, the total action of the ninstanton configuration is merely n S where S is the oneinstanton action. The ninstanton action does not depend on the positions of (anti) instantons in the limit when they are far from each other. It is clear nevertheless that instantons and antiinstantons must alternate with each other along the time axis. In other words
rin)
Integration over all possible positions of (anti)instantons with this restriction gives
variation of the action as well as for other observables vanishing in the limit T ,m.
3.1. T H E SYMMETRIC DOUBLE W E L L
47
To calculate this integral, we extended the intergration range for each 7;) to 0 5 ~,(i)5 T and divided by n!. The latter is the number of domains which are equivalent to (3.19), but differs by the ordering of 7ii). Strictly speaking, the action of n instantons is independent of their positions only if all of them are far enough away from each other. Otherwise the total action must include the terms describing the instanton  antiinstanton interaction. However, the measure of such configurations, (the integration region in the integral over the centre positions) is negligible if there are not too many instantons. For example, the contribution to the integral (3.20) made by trajectories with a tight instanton  antiinstanton pair (formed by any of the instantons) is of order ~ n  l
n Tn

w(n  l)!
wT n!
.
(3.21)
This is much less than (3.20) when n << WT. We shall discuss this condition in more detail below. The ninstanton trajectory contribution to the transition amplitude (3.7) is then
It is convenient to divide and multiply it by the harmonic oscillator transition amplitude for large Euclidean time:
(compare this with (2.60)). Then
The constant K, introduced here has the property Kn = Kn, where K corresponds to the singleinstanton contribution. We postpone the proof of this fact until the next section in order to analyse the resulting expression for G ( f a , a, T ) . Summation of (3.24) over all even n gives the transition amplitude from X = a to itself in time T
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
Figure 10
Similarly, for the transition amplitude from we obtain:
X =
a to
=
ZO
exp
k=O
=
a in time T
~ 2 & 1
03
G(a, a,T)
X =
g e x p
[F]
(2k
+ l)!
sinh [ ~ e x p (  ; ) T ] .
(3.26)
In these formulae, we have substituted S + S/fL and G + ~ ( & / m ) lin/ ~ order to restore the conventional dimensions of physical quantities. It should be noted that eqs. (3.25) and (3.26) can be expressed in the form
where the notation
A has been introduced.
= fL~~'l"
3.1. T H E SYMMETRIC DOUBLE W E L L
Recall the interpretation given to expressions of this type:
Comparing this with (3.27), we deduce that we have found the first two terms of this sum. These two terms represent the contribution of the ground state which is split into two sublevels with energies
As can be seen from (3.27), the wave functions of these sublevels are really the symmetric and antisymmetric combinations of the oscillator ground state wave functions in the left and right wells.
where 40 ( f a ) = (mwlnfi)'l4. We can perform a transition to real time in expressions (3.25) and (3.26) by means of the substitution T = it. The form of the result clearly coincides with (3.6). This (together with the analysis of the eq. (3.27) ) shows that the PI reproduces correctly the structure of the ground state split into two sublevels and gives the correct exponent in the splitting energy (3.4) and (3.5). We can easily check this by calculating the integral in (3.5) with E = 0. To complete the calculation, we have to find the unknown constant K in formulae (3.25)  (3.27). We shall do this in the next section, where we make sure that the PI calculation exactly reproduces formula (3.4).
The Contribution from the Vicinity of the Instanton Trajectory To find the constant K appearing in expressions (3.25), (3.26) and (3.28), it is necessary to integrate over the vicinities of classical trajectories in the PI (3.7). We do this analogously to the previous chapter, for each classical ) the trajectory :(T) we turn to integration over Sx(t) = x(t)  ~ ( t with z S[:] +b2S, action expanded up to the second order terms in &X,S[X(T)] where
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
(for the sake of simplicity we again set m = Ti = 1). From a technical point of view, the calculation of this contribution is the same as in the above considered case of the harmonic oscillator. We find the eigenfunctions x,(T) and eigenvalues X, of the operator H in expression (3.32):
and expand the trajectory variation Sx(r) in x,(r): 6x(r) = Xi Cixi(r). The contribution of the vicinity of the ninstanton trajectory Z to the PI takes the form of a product of an infinite number of independent Gaussian integrals:
This expression should be regarded in a symbolic sense because the product over i diverges. Usually, the infinite number of the multipliers 271. are included in the normalisation constant A, and the integration is written as a functional determinant. The determinant is defined (similarly to that of finite dimensional matrices) as the product of all eigenvalues;
Let us find first the functional determinant for the case of the trivial trajectory n = 0. On this trajectory x(r) = a and, therefore, H = d2/dt2 W'. The eigenfunctions and eigenvalues of this operator are
+
The calculation of the determinant is performed in the same way as in the previous chapter in the case of real time. The result can be obtained by substituting T t iT and x0 = X = 0 into (2.62):
G(a, a, T) =
E
(
 exp 
W:)
3.1. T H E SYMMETRIC DOUBLE WELL
51
Let us start by integrating over the vicinity of the oneinstanton trajectory. This problem cannot be solved in a straightforward way, because one of the eigenvalues X, is equal to zero. Thus the amplitude (3.35) is infinite. The eigenfunction corresponding to the zero eigenvalue is called the zero mode. Let us prove the existence of the zero mode in the spectrum of fluctuations around the instanton trajectory. It follows from equation (3.12) that 5 V'(:) = 0 , (3.38)
+
or, taking the derivative of (3.38) with respect to
T
Comparing this with (3.33) we see that xo = Ax(r) does indeed correspond to the zero eigenvalue of H. To find the constant A, recall that condition (3.13)
is satisfied on the instanton trajectory. This condition gives the following form of the instanton action:
Thus A
=
l/& and the normalised zero mode takes the form xo =
%l&. The physical meaning of the zero eigenvalue in the spectrum of H becomes clear if we note that the addition of the zero mode (multiplied by a small coefficient CO)to :(r) results in a translation of the instanton by dro = co/&along the time axis:
Of course, the action is unchanged by this translation. We might guess the existence of the zero mode simply by looking at fig. 9 where it was clear that the action did not depend on the instanton position. The calculation made above was necessary only in order to find the normalisation constant for xo and to derive using (3.42) the proportionality coefficient between COand the instanton displacement dr0. It is clear now that the integration over the coefficient COin (3.34) is not Gaussian. It corresponds to formula (3.20) with n = 1 and therefore
52
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
the multiplier ( 2 . r r / ~ ~ ) in  ~ (3.34) /~ must be omitted. Taking into account these properties of the zero mode we can express the oneinstanton contribution to the transition amplitude as follows:
Here the notation det' is introduced to remind us that X0 is excluded from the product of H eigenvalues, and the coefficient N is the same as in (3.34). Let us normalise this expression with respect to the trivial trajectory contribution:
+
l
det' [  & / d ~ ~ V"(Z(T))] det [d2/d~2 w2]
+
(3.44) lj2
es
and comparing it with (3.25) and (3.26), we see that the quantity K defining the energy splitting A (3.28) is
S
+
1
w2 det' [d2/,r2 V"(Z(T))] det[d2/i2+w2] L/2
"Iw&[
In this expression, the multiplier w2 was introduced to make the ratio of the two determinants dimensionless. It demonstrates that the quantity K has the correct dimension W (c.f. (3.4)). Before starting to calculate the ratio of the functional determinants, we make the following two observations. First, by turning to formula (3.45) we have reduced the normalisation coefficient N. Let us recall that the coefficient contains the Jacobian A corresponding to change of integration variables X(T) + Ci.It is not obvious that the values of A for the determinants in the numerator and denominator of (3.45) are the same because the eigenfunctions of the corresponding operators are different. A reason for this would be the fact that these two operators differ from each other only in the region whose size is of order the instanton size l/w << T (although the authors do not have a strict proof available). Therefore one of the conclusions to be drawn from the considered PI calculation would be a verification of the N reduction in formula (3.45) by comparison of the result with that obtained from the Schrijdinger equation. Second, we did not consider zero modes for multiinstanton trajectories. Generalisation of the result obtained for n = 1 is simple. There are n zero modes among normal modes, or that is among the fluctuations
3.1. T H E SYMMETRIC DOUBLE W E L L
Figure 11
around an ninstanton trajectory. As in the oneinstanton case, these zero modes correspond to independent translations of the instanton centres. Every such mode contributes a factor ( ~ / 2 7 r ) 'J/ ~d r to the integral. Now the statement about the factorisation of the preexponential term K used in the evaluation of the expressions (3.25) and (3.26) seems to be 'partially proved'. It remains to check that it is also valid for the functional determinant. We do this by explicit calculation of the functional determinant in the next section.
Calculation of the Functional Determinant Let us begin our calculation of the functional determinants (3.43) and (3.45) with an evaluation of the eigenfunctions and eigenvalues of the operator H for the oneinstanton trajectory. It is necessary for this purpose to solve the following equation:
Substituting the explicit form of :(t) (3.17) into V 1 ' ( x )= 4Xa2+ 12Xx2, we get:
54
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
Fortunately, this equation is well known. The potential here is called the modified PeashleTeller potential (see fig. 11) and appears in some nuclear physics problems. We proceed, first finding the bound state wave functions. Consider the asymptotic form of ~ ( r with ) an arbitrary value of the parameter X, = X when T , koo. Here H = $ w2 and we seek a solution of (3.47) in the form x(d = f , (3.48) where n = d a . Substituting this into equation (3.47) we get
+
or, introducing the new variable J = tanh $w(r  T
~ )
This can be recognised as the hypergeometric equation if we note that the highest power of the polynomials multiplying the derivatives of order 2, 1, 0 are 2, 1 and 0 respectively. In order to transform this equation to the standard form, let us make one more change of variable z = (l+J)/2. Equation (3.50) becomes
We shall seek a solution of this equation as a series in z: f = CnBnzn. In terms of the B,, equation (3.51) takes the form
Collecting the coefficient of zn, redenoting summation indices where necessary, we obtain:
This expression results in a set of recurrence relations for the coefficients
3.1. THE SYMMETRIC DOUBLE WELL
Assuming B. = 1, we find:
All higher coefficients are zero. Simple algebraic transformations give the solution of equation (3.47) with arbitrary K :
Consider the asymptotic behaviour of this solution. If then + 1 and P(K, 1) = 1 . Thus X ( ~ M ) e"lTT~l at 1 + 1, and
T  TO
P ( & ,1 ) =
(K (K
+
m. If now
 W ) ( &
+
W)(K

T
T  TO +
TO +
w/2)
+w/2)
m,
(3.57) +m, then
(3.58)
'
The other independent solution of equation (3.47) can be chosen as X (  X ) , except for some special values of X considered below. We have solved equation (3.47), but have not found the eigenvalues yet. They are those values of X for which the solution of equation (3.47) goes to zero at T = 0 and T = T. For the discrete eigenvalues, this can be replaced by the requirement that the eigenfunctions go to zero as T  7 0 + & m . The related error is exponentially small and vanishes in the limit T + m. This condition is already satisfied at T  T O + m. At T  7 0 + +m,.it is obeyed only if the polynomial P ( K ,1 ) tends to zero. This gives the values X = X , which correspond t o the discrete spectrum. There are only two such eigenvalues: X. = 0 for K = W and X I = $w2 for K = w/2. The first eigenvalue corresponds to the already known zero mode. The form of the function xo follows from (3.56)
1 anh W 2
1'
( T  r0)
=
1 2 cosh2 ~
W ( T TO)
, (3.59)
and coincides with the already known zero mode up to the normalisation factor. Thus there is only one discrete level XI = which gives a contribution to the ratio of the determinants (3.45). Let us now consider the contribution of the continuum eigenvalues. The quickest way of finding the corresponding solution of equation (3.47)
iw2
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
56
is to perform the analytic continuation of solutions (3.56) and (3.58) to the value K = i k . Then X takes the form X = w2 k2 > w2 and the solution in the asymptotic region T  TO + fCO reads
+
for
T 70
t
CO
and
for T  TO + CO. Here
Sk
is defined as
+
+
e& = ( k i w )( k i w / 2 ) ( k  iw)(k  iw/2)
This is the phase shift acquired by the wave function of a particle scattered in the potential V"(:). An interesting property of this potential eik(TT~), the is that it is reflectionless. There is no reflected wave potential effect being reduced to the additional phase shift Sk of the transmitted wave relative to the incident wave. Now it is easy to construct the wave functions obeying the zero boundary conditions at r = 0 and T = T. Subtracting from solutions (3.60),(3.61)its own complex conjugate, we get the following solution of equation (3.47) which vanishes at T = 0: X(')
=
{
sin k r sin(kr
+ &)
for 0 5 T << 7 , for << 7 5 T
.
We find the eigenvalues by applying the requirement that this function be zero at T = T.4 In terms of the wave vector k, they are
3Both the reflectionless character of the potential and the exact solvability of equation (3.47) are due t o the existence of an operator which transforms the solutions etkT of the free Scluodinger equation into the solutions of (3.47). This operator contains ~ T ~ ) There. only derivatives with respect to T and terms proportional t o tanh $ w ( fore the action of this operator cannot crcate a reflected wave epZkT'.More information about such transformations (which are based on the supersymmetry property of the given system) can be found in the rcview by L.E. Gendenstein and I.V. Krive [5] 41t should be pointed out that the continuous spectrum does not appear in the case of finite T. Nevertheless the truly discrcte levels are easily distinguishable from those which correspond t o the continuum  the former are separated from each other by thc interval w2, the latter by I/T~.


3.1. THE SYMMETRIC DOUBLE W E L L
57
corresponds to the eigenvalues of the operator in the where k?) = 2 ~ n / T denominator of (3.45). Let us now find the contribution of the continuum levels to the determinant (3.45). We must calculate the quantity
In the limit T + oo, we can neglect the terms of order 1/T2.This gives
to avoid a cumber(where we have used the notation S(k) instead of some s ~ b s c r i p t ) ~The . fact that this expression depends on n via k?) suggests that we should perform the integration over k = k?) instead of the summation over n. To this end, we note that an increase of n by unity corresponds to an increase of k by TIT. Then
This expression takes the following form upon changing the integration variable k to X = k/w and integrating by parts:
Substituting here the expression for the derivative d&/dx
which follows from (3.62). We find the explicit form of the continuum levels contribution to the logarithm of the functional determinant (See 5Thcre is a certain inconsistency in notation here. Thc eigenvalues in (3.64) are labelled by the subscript n running from n = 1. At the same time we have used n = 0 , 1 , 2 . . . in formula (3.34) in order to reserve the common notation X. for the zero mode.
58
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
Appendix A.4 for details of the calculation of the integrals appearing here)
The remaining contribution of the discrete levels to the ratio of functional determinants is:
where the unit factor stays instead of the absent X. . Gathering all the factors appearing here we obtain finally
This leads in turn to the following expressions for the quantity K (3.45) and for the value of the papameter of the energy splitting A (3.28):
and
(recall that the instanton action is S = w3/(12X)). In order to complete the calculation of the energy splitting, we have to show that the ratio of the functional determinants for the multiinstanton trajectories factorises into a product of oneinstanton contributions. This is obvious for the contribution of the discrete level Al. To see it for the continuum level contribution, we recall that the potential V" of each instanton results in an additional phase shift of the eigenfunction x,(r). Thus the total phase shift is the sum of the oneinstanton contributions, and the statement is proven. Our final result for A, equation (3.74), still appears different from expression (3.4). There is a W(w/2) in (3.4) instead of S and the preexponential factors in (3.4) and (3.74) differ from each other. Only the main exponential factors are identical because W(0) = S. Indeed, S(T) = W(E)  ET on the classical trajectory, where E and S(T) are the energy and action as functions of the period of classical motion in the upsidedown potential (fig. 7). It is apparent that E e"' when T t CO (see footnote 2 on page 45 ) and hence S = S(CO) = W(0).

3.1. THE SYMMETRIC DOUBLE WELL
59
We show in Appendix B that the difference between the preexponential factor in (3.74) and that in (3.4) is exactly that required to compensate the deviation of S = W(0) from W(w/2). Thus expressions (3.4) and (3.74) are identical. Note that the calculation of the determinant ratio turned out to be the central problem in the evaluation of the preexponential factor in the energy splitting. It is worth considering one more method of calculation which is also simpler, more elegant, and appears to be more convenient in many specific problems. The essence of this method is as follows. Let l4 and H2 =  d 2 / d ~ 2 V2 with there be two operators H1 = d2/d? the potentials V~(T)and V~(T)having the same asymptotic behaviour as T + m . Consider two corresponding equations of the stationary Schrodinger type:
+
+
where 0 5 7 5 T. Let xl(r), ~ ~ (be7 the ) solutions of these equations r obeying the boundary conditions ~ ~ ( =0 ~) ~ ( =0 0) and d ~ ~ ( 0 ) l d= dx2(0)ldr. Then the following equality is valid:
To prove this statement we note that the functions ~1 (T) and ~2 (T) grow exponentially as T + m unless X takes certain values. These values, at which xl(T) = 0 or x2(T) = 0, are the eigenvalues of the operators HI and H2 respectively. Hence the numerator and denominator of the expressions on the right and left hand sides of (3.76) go to zero at the same values of X. Let us now consider the ratio of the right and left hand sides of (3.76), which we call R(X), as a function of a complex variable X. It follows from the above reasons that this function has neither poles nor zeroes in the complex plane of X. Furthermore, both the right and left hand sides of (3.76) tend to unity as X + m . This is because when X + m we can neglect the potentials both by solving equations (3.75) and by finding eigenvalues of the operators H1 and H2. Indeed by making use of perturbation theory we can make sure that the corresponding first order corrections go to zero if X + cm. Thus we conclude that R(X) has neither poles nor zeroes on the finite part of the complex plane and goes to unity when X = m , and hence R is equal to unity for all X. Let us evaluate the determinant ratio in (3.45) using the above theo= V/'(:), = w2 and the right hand side of (3.76) rem. In this case
v
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
where K = d m (recall that x(T), x0(7) are solutions of the equations (d2/dr2+V"X)x(r) = 0 and (  d 2 / ~ 2 + ~ 2  X ) x 0 ( r=) 0 see (3.56), (3.58)). To find the determinant ratio, it is necessary to set X = 0, but before doing that, we have to extract the zero mode contribution from the left hand side of (3.76). If X # 0 this contribution is equal to X. As the functional determinant is the product of the eigenvalues, it is sufficient to divide (3.76) by X in order to remove the zero mode contribution. Then expanding K = Jw2X FZ W  X/(2w), we get
+
V"(:)] detl[d2/d7' det [d2/d72 w2]
+
=
l ( K  W  2 lim X+O X (K+ w)(n w/2)
+
1 12w2
 .
(3.78)
Thus we have again obtained result (3.72), but via a method where the calculation of integral (3.70) was not necessary. In conclusion, let us formulate a condition under which the obtained results are valid. Our first assumption was that the contribution of the fluctuations around the classical trajectory could be taken to be Gaussian. This is possible if S/fi >> 1. In the system of units where fi = 1, this condition takes the form (3.11) a = X/w3 << 1. One more approximation was used. In deriving formulae (3.25) (3.26) we assumed that the ninstanton action was S, = nS. This is the dilute instanton gas approximation. This assumption is correct if the characteristic instanton size W' is much less than the average distance between neighbouring instantons Tin, i.e. n << W T . Then for large values of n, the instantons overlap and the expression for nth term in formulae (3.25) and (3.26) is not applicable. Nevertheless it would not affect the result if the terms with n W Tdid not contribute to the sums (3.25) and (3.26). In order to find out if this is the case, we must find the terms in (3.25) and (3.26) which give the greatest contribution to the sum
Applying Stirling formula n! (n/e)n to n! (see (2.40)), we get an estimate ( e a . Tin)" for the nth term of (3.79). Taking the derivative with respect to n, we find that the greatest contribution to this expression is made by the terms with n = nm,, = A . T. The trajectories with such
3.2. PERIODIC POTENTIAL
61
number of instantons give the maximal contribution to the transition amis called plitude (3.25), (3.26). For this reason the quantity A = n,,,/T density of instantons. Using its explicit form (3.4), we come to the validity criterion for expressions (3.25), (3.26) in the form exp(w3/12X) << 1. It is obviously fulfilled if a = X/w3 << 1. Finally, we would like to make one more note. The above introduced quantity a is usually called the coupling constant. This quantity is the dimensionless parameter of the perturbation theory. This theory gives corrections to the energy and the wave functions of the particle localised in one of the potential wells in the form of a series in powers of a , in which the zeroth order term corresponds to a pure harmonic oscillator. We have obtained the correction to the ground state energy which turned out to be proportional to exp(Const/a) . A remarkable fact is that this expression is not analytic at a = 0. Indeed all the derivatives dn exp(l/a)/dan vanish at this point. This means that it is impossible to reproduce the above obtained result by calculating the perturbative corrections to the oscillator energy for the wave function localised in one of the wells. In this sense it is reasonable to call the method and the result obtained here nonperturbative. The method used here has a more specific name semiclassical approxzmation which emphasises the use of the nontrivial classical trajectories. Indeed we managed to catch the nonperturbative energy splitting thanks to the fact that we had already taken into account the nontrivial multiinstanton trajectories in the zeroth approximation. At the next order, it was necessary to include the corrections due to the integration over the vicinity of the multiinstanton trajectories. This was possible only when we neglected higher variations of the action, which was correct under the condition a << 1. Thus both the perturbation theory and the semiclassical approximation are valid under the same condition, but they give information about the essentially different contributions to the energy levels in the double well potential. We would like to note that any other method of PI calculation which does not require smallness of the interaction constant is named nonperturbative as well. 

3.2
A Particle in a Periodic Potential. Band Structure
Let us consider one more quantummechanical problem: finding the spectrum of a particle in a periodic potential. This problem is of particular interest because such a potential is a starting approximation in the description of the movement of electrons and holes in the periodic crystal
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
Figure 12
field. Let us recall the known solution of the problem. The periodic potential has the property V(x) = V(x 2 m ) , n = 0, fl , f2 . . . (see fig. 12). Here X is supposed to be dimensionless  this can be done using some parameter a, such that xa has the dimension of length. We can consider V(x) = 1 cos X as an example, but all the results to be obtained below are true for any potential with one deep minimum in the period. As in the case of the double well potential (3.9), we can expand V(X)near one of the minima X = X,, assuming that the ground state energy is less than the barrier height:
+
where w2 = V"(x,)/a2 is the vibration frequency. The tunnelling effects between neighbouring minima are small in the case under consideration, as well as for potential (3.9). Neglecting the tunnelling gives an infinitely degenerate ground state. This reflects the fact that the particle can find itself in any minimum of the potential. A nonzero tunnelling probability removes the degeneracy, and the corresponding wave functions are linear combinations of nonperturbed wave functions xo(x) at every minimum:
3.2. PERIODIC POTENTIAL
63
where the parameter 8 labels different variants of such a superposition. The coefficients Cn(0) in (3.81) should be adjusted so that the states ~0 are eigenfunctions of the translation operator X + X 2 ~ because , this operator commutes with the Hamiltonian. This implies the Bloch theorem:
+
xo(X + 2
~ =) eisXO(x)
,
(3.82)
and hence C,($) = einO.It is easy to check that equation (3.82) gives the eigenfunctions with eigenvalues
Thus the ground state is split into an infinite number of sublevels, known as band. We can find A explicitly for a given V(x) using the standard quantum mechanical methods. Our aim is to reobtain expressions (3.82) and (3.83) for the case of small tunnelling probability, by the instanton method. Let us calculate the probability amplitude of a particle transition from the minimum x(0) = X, to another one x(T) = X, in Euclidean time T:
As above, we find the stationary points of the action which dominate this amplitude. These are, of course, the multiinstanton trajectories. The oneinstanton trajectory in this case corresponds to the transition from the nth minimum to the ( n + 1)th and the antiinstanton trajectory brings the particle back. There is exactly one zero mode in the fluctuation spectrum in the vicinity of each instanton. This follows, as in the previous case, from the action independence from the instanton position. There is only one essential difference between the case under consideration from that of the double well potential. The restriction that the instantons and antiinstantons follow each other in an alternate order is now relaxed. What is important is the difference between the number of instantons k and antiinstantons k', k  k' = m  n z 1. This is defined by the spacing of the points X,, X,. Thus taking into account all trajectories with k instantons and k' antiinstantons, we can write in analogy with equations (3.25), (3.26):
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
64
where Zo is the oscillator trajectory contribution which is the normalisation factor as above, and the quantity K is defined by (3.45):
(in dimensionful units). The Kronecker symbol 6kk',lis included into sum (3.85) in order to restrict the difference between the numbers of instantons and antiinstantons. If we take into account that 2T
d6'
6u,~ = /gexp (i(k  k'
 1)0),
then the summations over k and k' in (3.85) decouple and can be done separately:
2.rr
= 20
J,
e'
[email protected]
A.T exp ( e g X ) enp ( e
A.T
"B_ZT;)
0
where we again used the splitting parameter A (3.28). By substitution of the explicit form of the oscillatory amplitude (3.23), we finally obtain:
(3.89) To find the energies and the wave functions we are interested in, it is necessary to interpret this formula as an expansion of G in the Hamiltonian eigenfunctions (3.29). This yields the energy of the lowest energy band in the correct form
Thus we obtain a continuous energy band of width A, which is parametrised by the angle 0 5 6' 5 27r. It can easily be seen from (3.89) that the corresponding wave functions in fact are the Bloch waves:
3.3. A PARTICLE ON A CIRCLE
3.3
A Particle on a Circle
Let us consider another interesting case of tunnelling which is commonly used t o illustrate instanton calculus in &CD [3, 61. Let the particle considered above move on a circle instead of on the infinite line. This implies periodic boundary condition, that is, an identification of the points X = 0 and X = 27r. In the problems considered above the minima of the potential V(x) has been physically distinguished. Starting with an infinite number of states in the minima of V(x), we have obtained an infinite number of their superpositions and these have formed the energy band. In contrast, there is now a single minimum of V(%).This means we shall not find any energy band. Instead we will find a nonperturbative correction to the ground energy. We can guess the answer if we impose the periodic boundary condition on the wave function (3.91). This implies 9 = 0 in (3.90). Let us see how the instanton technique leads to this result. In the dilute gas approximation the shape of the instanton as well as the functional determinant are not sensitive to the boundary conditions. It is only the summation over the number of instantons (3.85) which needs to be modified. The instanton is now a trajectory which runs round the circle such that the initial and final points are identical. Thus there is no restriction on the number of instantons and antiinstantons in (3.85). We now omit the symbol 6&k',l in (3.85) and easily perform two independent summations over the number of instantons and antiinstantons. This results in formula (3.90) with 9 = 0. With this result in mind, we can speculate that if we impose a quasiperiodic boundary condition (3.91) with a specific value of 9 then we obtain as the result one level (3.90) from the band (3.89) selected by the given value of 9. Interestingly, quasiperiodic boundary conditions on a circle are not nonsense. A system with such boundary conditions does exist. Suppose that the circle is a real circular wire in threedimensional space with a charged particle moving on it and that there is a solenoid which crosses the circle plane somewhere inside the circle shown in fig. 13. If the solenoid is thin enough, then there is no magnetic field on the circle. There is, however, an effect due to the vectorpotential A. As the particle moves round the circle, it acquires a phase factor exp ( i e
f A ~ Z )= exp (iem)
,
where 2 is the length element on the circle, e is the charge of the particle, and @ is the total magnetic flux through the surface spanned on the circle.
CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL
Figure 13
Thus we have 6 = e a . Let us make appropriate changes to the instanton calculations. We start with the classical Hamiltonian H of the system in real time which takes the form H(p,x) = (p  eA)2/2 V(x), where p is the canonical momentum and A is the component of A tangential to the circle. The velocity dxldt is now dxldt = dHldp = p  eA. Performing the Legendre transformation to obtain the Lagrangian L = px  H , we come to dx L =   +eA  V(x) . 2l (dx)2 dt dt
+
After turning to imaginary time, the Euclidean action (denoted now as So)takes the form
The last term here is a total derivative. Let us assume for simplicity that A does not depend on X (otherwise we have to apply a gauge transformation to make A an Xindependent function). This term then takes the form
2 ~ e A= e a , and the value [x(T)  x(0)]/27r is equal to the where 8 number of turns of a given trajectory around the circle. This term is called the 6term. It does not affect the classical equation of motion, but
3.4. CONCLUSIONS
67
it changes the quantum properties of the system which are sensitive to the value of action. Indeed, the action of an instanton SI now takes the form S1 iQ while the action of an antiinstanton is SI  iB. Performing the independent summation over the numbers of instantons and antiinstantons, we obtain the expected form of the groundstate energy:
+
where B is now a fixed parameter. This contrasts with (3.91). The generalisation of this ground state in YangMills theory is called the Bvacuum.
3.4
Conclusions
The examples considered in this chapter have their own value in quantum mechanics and solid state physics. Besides, they may serve as simple models of nontrivial vacuum structure of quantum field theories (the term vacuum is used to denote the ground state of the quantum fields). Though simple, such models possess the key feature of much more complex systems tunnelling between different vacuum states. The most common approximation to describe tunnelling in field systems is to use the instanton formalism. Our analysis may help to understand its basic idea, the main calculation steps, and the restricted applicability of this method. In the next chapter, we start studying the quantum mechanics (and statistical physics) of fields. These are systems with an infinite number of degrees of freedom. We shall use the path integral as a working tool rather than a subject of investigations. The experience gained in this part will be used in quantum field theory calculations. 
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Bibliography The BohrSommerfeld quantization rule has been obtained in the PI formalism by R. Dashen, B. Hausslacher and A. Neveu, Phys. Rev. D10 (1975) 4136. It was V.N. Gribov who realized the connection between the instanton trajectories and the tunnelling transitions in quantum mechanics. In this chapter we drawn on the review by A.I. Vainshtein, V.I. Zakharov, V.A. Novikov and M.A. Shifman, Sov. Phys. Usp. 25 (1982) 195. Other good reviews are the following two papers S. Coleman, The Uses of Instantons, Lectures given at the 15th Course of the International School of Subnuclear Physics, Erice, Italy, 1977; Reprinted also in: S. Coleman, Aspects of Symmetry, Cambridge, University Press, 1985.
J. ZinnJustin, The Principles of Instanton Calculus: a few Applications, Lecture delivered at Les Houches Summer School 1982. Published in Recent Advances in Field Theory and Statistical Mechanics, eds. J.B. Zuber and R. Stora, North Holland, 1984. L.E. Gendenstein and I.V. Krive, Sov. Phys. Usp. 28 (1985) 645. R. Rajaraman, Solitons and Instantons in Quantum Field Theory, NorthHolland Publ. Co., Amsterdam, 1992.
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Part I1
INTRODUCTION TO QUANTUM FIELD THEORY
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Chapter 4 Classical and Quantum Fields The main purpose of this book is to introduce the reader to the characteristic features of quantum systems with very large or, in the limit infinitely large number of degrees of freedom. Such systems are called quantum fields. As it will be shown below, there is a transparent analogy between the Euclidean form of quantum mechanics and statistical physics for reason we discuss statistical properties of the fields as well. In this chapter we consider some simple examples of physical fields and related subjects such as the variational principle and quantization rules. There exist two main recipes of quantization: canonical formalism and PI formalism. We shall discuss both, but use mainly the latter in the following chapters. As a simple example we shall consider small field fluctuations near a minimum of the field energy. In the language of quantum mechanics, it will give excitations above the ground state. As it will be shown below, these excitations look like particles. It should be stressed that neither particle physics, nor condensed matter physics, nor any other branch of physics where the methods of quantum field theory (&FT) are used is the subject of this book. The aim of this book is to study the corresponding formalism as it is, especially those features which are independent of (or only slightly dependent on) the physical context. At the same time, we shall use examples from particle physics and other disciplines for the sake of illustration. 
4.1
From Large Number of Degrees of Freedom to Particles
We start with a toy model of a mechanical system consisting of a large number s of small balls (material points, or 'atoms1) of mass m which can
74
C H A P T E R 4. CLASSICAL AND QUANTUM FIELDS
move along a straight line and are connected to each other by springs (fig. 14). It is clear that the ith ball vibrates near its equilibrium point xi, i = 1,2,. . . S, s + oo. Take the points xi to be situated along the X axis with spacing Ax and the ith ball's deviation from its equilibrium point $i. Each ball is held near its equilibrium point by a corresponding spring with potential energy W1(q$) (shown vertical1 in figure 14, although only the horizontal force component is essential). Another kind of spring mediates an interaction between the neighbors with potential energy W2((4i+l &)/A%). Here A s is introduced to define W2 in a more convenient way1. .. Taking into account these preliminary remarks we can write down the action of the model in the following form: ,
The dynamics of the system are described by a set of s EulerLagrange equations
or, explicitly,
where prime denotes as usual the derivative of a function with respect to its explicitly written argument. We can also use the Hamiltonian formalism introducing s canonical conjugate momenta pi = dL/a& = m&. The corresponding Hamiltonian is:
We shall consider longwave excitations in this system. That is, excitations such that the characteristic length at which $ changes significantly is much greater than the distance between neighboring atoms 'It is worth mentioning that models of this kind possess rather nontrivial properties despite their simplicity. If the potential energy is quadratic with a small cubic term then the model becomes the so called FemiPastaUlam system [l]whose evolution is one of the first nonlinear problems solved on a computer. It was surprisingly observed that there was no equipartition of energy among the modes of this nonlinear system.
4.1. PARTICLES
Figure 14

Ax. We can call these waves 'macroscopic7 while the values in (4.1) (4.4) are 'microscopic'. From the macroscopic point of view, the microscopic values are small, and it is reasonable to take the limit Ax + 0 (the continuum limit) in which the quantities m, Wl and W2tend to zero so that the mass density p = lima,,o m/Ax and the energy density of a simultaneous displacement of all atoms U = lirnaz,O Wl/Ax as well as W2/Axremains finite. Then the action (4.1) takes the form

where
Thus in the continuum limit the set of the discrete canonical variables 4% becomes a field #(X,t), which is a continuous canonical variable defined for any coordinate X . It is also convenient to introduce a Lagrangian density L such that L = JdxL as is done in (4.5). The equations of
76
CHAPTER 4. CLASSICAL AND QUANTUM FIELDS
motion in the continuum limit can be derived from (4.3) by taking the limit Ax L: 0:
It is obvious that as S + m the system of equations (4.3) becomes a partial differential equation (4.7). This equation can of course be derived by variation of the action (4.5). For this purpose let us consider small arbitrary variations 64(x, t) of the field $(X, t ) , for which S$(X,t ) = 0 for t = 0, t = T, and X L: fm , and set the first action variation to zero:
For the sake of compactness of the formulae here the notation d, r a/dx is used. Integrating by parts and taking into account that the field variations are zero on the boundaries of the integration volume we get:
The arbitrariness of the variation 64 leads to the EulerLagrange equations (4.7). Let us make our toy model (4.5) more realistic to describe a crystal. For this purpose 4 should be thought of as the atom displacement in the elementary cell. Hence 4 must be at least a threedimensional vector, or more exactly 3ndimensional vector where n is the number of atoms in the elementary cell. The Lagrangian becomes a function of all the derivatives di4 &b/dxi.As the displacements are measured from the corresponding equilibrium points of atoms, the deformation energy T must have vanishing first derivatives at these points. Furthermore, there are a lot of cases where relative deformations of the elementary cells are small. In such cases we can restrict ourself to the leading term in the expansion of the deformation energy, which is
The next terms can be easily taken into account if necessary. The tensor of second derivatives in (4.10) can be diagonalized. Let us consider for
4.2. ENERGYMOMENTUM TENSOR
77
simplicity only the simple elementary cells for which the eigenvalues of this tensor are the same and equal t o U. Then the action takes the form
where the usual rule of summation over repeated indices is assumed and the subscripts labeling different components of 4 are omitted in formula (4.11) for the sake of simplicity. Equation (4.11) is a starting point in the description of vibrations of a crystal which look like particles called phonons upon quantization. In many cases, especially in particle physics it is convenient to set the coefficients in front of the first and second terms of the integrand of (4.11) to unity. In particle physics, it makes manifest the necessary Lorenz invariance of the theory. Then the action (4.11) can be written in the form (4.12) S = d4x [i(a,cp(X))2  UMX))]
1
which will be widely used below. Note that the subscript p = 0,1,2,3 of the 4vector X, is omitted and summation over p in the first term is assumed. In other according to the formula ( 8 , ~ )=~ (docp)2words we use the metric (1,1,1,1). The simplest case of the onecomponent field cp which is a Lorenz scalar will be further discussed up to chapter 9. By varying the action (4.12) we get the following equation of motion:
Here the d'Alambert operator = d,d, = 8; has been introduced. If U = 0, this equation describes waves with unit velocity. In particle physics this is the speed of light; in solid state physics it may be the sound velocity.
4.2
EnergyMomentum Tensor
The physical processes described by the action (4.12) occur equally at any point of spacetime. If the system is situated in an external coordinate dependent field, this invariance is lost. According to the Noether theorem such invariance leads to the conservation of the field energy and momentum. Indeed, infinitesimal displacements
78
CHAPTER 4. CLASSICAL AND QUANTUM FIELDS
change the Lagrangian density
On the other hand if the Lagrangian does not depend on coordinates then where Sp = p ( x , + E,)  ( ~ ( 5 , = ) E,~"(P(x,). One can combine these equations using the equations of motion (4.13) to obtain
&L
(W v v >(0))
= ~ , d , aL 
By the arbitrariness of c,, it follows that
where
is the conserved energymomentum tensor of the field p. Integrating the conservation law (4.19) over the whole space, we obtain the explicit form for the four conserved quantities
where P, = J dx 0: is the relativistic energymomentum vector. We can make sure that PO really is the field energy by comparing it with the Hamiltonian defined in classical mechanics for the field p ( x ) . Substituting action (4.12) into the definition (4.19), we obtain the explicit form of the energymomentum tensor of the scalar field:
This expression gives for the energy
It is clear that this quantity is positive if the selfinteraction potential U ( p ) is. The field momentum is
4.3. FIELD QUANTIZATION
79
Let us note that the energymomentum tensor (4.21) is symmetric, although the initial formula (4.19) does not guarantee such symmetry generally. Besides this, the tensor is not uniquely defined by (4.18), because any divergenceless quantity can be added to O,,. A detailed discussion of these two types of energymomentum tensors (which are called metrical and canonical respectively) and the transformations mentioned here can be found elsewhere (see e.g. [2]). We skip this subject and turn t o quantization of the system described by the action (4.12).
4.3
Field Quantization
In this section we consider the quantization procedure for the system whose classical action is given by (4.12). As was already mentioned in this chapter there are two recipes for quantization. The canonical formalism takes operators with some fixed commutation relations in correspondence to dynamical variables. The other method uses the path integral. Although the main tool of this book is the PI, we discuss the canonical formalism in this section.
Canonical Quant ization Let us turn again to our toy model (4.1). As it has a finite number of degrees of freedom it can be immediatly quantized by making use of the Schrodinger equation for the wave function \Ir(4l, $2 . . . 4,; t ) depending on S canonical coordinates q5i and time:
Here %! is the Hamiltonian operator derived from the classical Hamiltonian function by substitution of the canonical coordinate and momentum operators therein. The explicit form of the canonical operators depends on which representation is used. In particular, it is convenient to use the coordinate representation which presupposes that the wave function depends on &. In this case the action of the operator & reduces to a multiplication by +4i and the operators pi are defined as follows:
The same quantization recipe can be also formulated as a set of canonical commutation relations which do not depend on the choice of repre
80
CHAPTER 4. CLASSICAL AND QUANTUM FIELDS
sentation. It is obvious that in the coordinate representation
because of (4.25). As the commutation relations are invariant with respect t o unitary transformations this expression is valid in any representation. Similar commutation relations hold for all other pairs of canonically conjugate variables. It should be pointed out that in quantizing system (4.1) in terms of 4i and pi we are not faced with the operator ordering problem mentioned in the second chapter, because Hamiltonian (4.4) does not contain 'dangerous' products of coordinates and momenta. Now let us take the continuum limit Ax + 0. It is convenient to introduce a set of new canonical momenta Q = n / A x , the densities of the old ones. Then the Hamiltonian (4.4) takes the form
The commutation relations (4.26) which in terms of form
[v.3,Pz. l =  '
&j 2AX
and $ take the (4.28)
l
become [v(xl),$(x)]= iS(xX')
(4.29)
.
Here the Sfunction appears as the limit of Sq/Axi for Axi is in agreement with its functional definition
t
0, which
If instead of the Schrodinger picture we use the Heisenberg one, where the time dependence is transfered from the wave functions to the operators, the commutation relations for p = $& and n = v/& are
These equaltime canonical commutation relations are the key idea of quantum field theory. They define, as in quantum mechanics, an algebra of operators acting upon the field wave function 9.As before this wave function can be interpreted as a vector in a linear space describing a state of the given physical system. This quantity Q[p(x,t)] is clearly a functional of p(x). In the coordinate representation the field operators act
4.3. FIELD QUANTIZATION
81
on Q[cp(x,t)] as multiplication. The action of the momentum operators is defined by the relations
in accordance with the commutation relations (4.31) (compare with (4.25)) Thus the quantum field evolution corresponding to action (4.12) is completely defined by the solution of the Schrodinger equation for the wave functional
This equation is essentially more complicated than its quantummechanical analogue. At the same time, the PI in &FT does not look so frightening compared with that of quantum mechanics.
Quantization via Pat h Integrals Analogously to the derivation of the Feynman formula (2.27), let us write down the probability amplitude for the transition between the initial cp(x, 0) and final cp(x, T) field configuration in time T:
In this paragraph, the Planck constant is restored. The quantity G is the Green function (or more strictly functional) of the Schrodinger equation (4.33). It can be expanded as a series of the eigenfunctions a, corresponding to the Hamiltonian of equation (4.33) as is done in quantum mechanics:
Here E, is the field energy in the state labeled by the index a. The derivation of formula (4.34) is competely equivalent to that of expression (2.27). Nevertheless let us point out that the PI over the fields cp(x, t) is thought of as the limit of the product of conventional integrals for each point of a discrete spacetime lattice. Then, after calculation of these integrals the continual limit should be taken. We shall use another definition of the PI below. The system is placed in a large, but
82
CHAPTER 4. CLASSICAL AND QUANTUM FIELDS
finite 4dimensional volume VT. The Fourier components of the field then become discrete, and we integrate over each of them and take the continuum limit thereafter. The necessity of such tricks is due to the rather formal character of the expressions (4.33), (4.34) and (4.35). A mathematically correct definition of these expressions is problematic. Nevertheless, the practical calculations presuppose (as a rule) that the system to be described is the limiting case of a large but finite number of degrees of freedom. Sometimes it is literally the case, e.g. for crystals. This makes possible a direct generalization of all quantum mechanical relations (particulary the formulae (4.33), (4.34), and (4.35)) to &FT. It was shown in the second chapter that the PI approach is completely equivalent to the Schrodinger approach. It enables us to regard equation (4.34) as a postulate which yelds the quantization rule without reference to the Schrodinger equation. Further we assume that all results concerning the PI in quantum mechanics are true in &FT2. If it is necessary to specify the mathematical meaning of the integral (4.34), we shall do it explicitly in the given case. Let us consider the analytic continuation of expression (4.34) in Euclidean space. The effectiveness of this method was shown in chapter 3, at least for the ground state energy calculation. The substitution T = ZTE transforms the Green function of equation (4.33) into
The same substitution in (4.34) gives the Euclidean form of the PI:
where the Euclidean action SE is
Euclidean time here is assumed to be simply an extra spatial coordinate r xq, which varies from 0 to TE. The metric of this space is trivial a;. any $vector squared is (a,)2 = a: a; a:
+ + +
20ne should keep in mind that a direct application of formula (2.27) or its QFT version (4.34) is impossible if there are cyclic coordinates in the theory. Such a situation takes place in gauge theories whose quantization scheme demands modifications [3]. We discuss this problem in the chapter 11.
4.4. THE EQUIVALENCE O F QFT AND STATISTICAL PHYSICS83
4.4
The Equivalence of QFT and Statistical Physics
Note that the action (4.38) is formally similar to the three dimensional field energy (4.22)
where i = 1,2,3. The difference is in the dimension of the integral and the dimension of the gradient appearing in it. This analogy makes it possible to establish a correspondence between &FT and statistical physics for a field system. To do this let us consider the central quantity of statistical physics, the partition function:
Here the summation is taken over the complete set of quantum states, as in expression (4.36), kB is the Boltzman constant, 0 is temperature, and F is called the free energy. It can be seen that if we assume in (4.36) that
then the exponential factors in (4.36) and in (4.40) are the same. The only difference between these expressions is the presence of the wave functionals corresponding to the initial and final field configurations in (4.36), (4.37). To get rid of them let us impose the periodic boundary conditions cp(xo,0) = p(x, TE) in the propagator (4.37) and perform an integration of (4.37) over cp(x,TE) at each point X (that is a functional integration in threedimensional space). After this the wave functionals disappear from (4.37) because of the normalization integral
which generalizes the usual quantummechanical normalization condition. Problem: Write explicitly the quantum mechanical analog of this formula.
84
CHAPTER 4. CLASSICAL A N D QUANTUM FIELDS Thus we obtain
The meaning of the derived expression is simple. For the quantummechanical propagator (4.36) it is necessary to fix the initial and the final field configuraion before integrating over the field values at each intermediate time point. The initial and the final field configurations may differ from each other. As distinct from this case, both configurations in statistical physics are identical. In other words it is the same point in the function space over which the integration should be done. This means that the evaluation of the partition function presupposes all fields to be defined on a 4cylinder R3 X S1where the perimeter of the circle S1 is r = h / k B 8 . Thus, the expression for the partition function takes
where integration is taken over all periodic field configurations on the time interval 0 r < TE = ti/ksO. As a rule, this commonly known fact is not reflected explicitly in the written form of expressions of type (4.44). It should be noted also that when evaluating partition function (4.44) the numbers of integrations over the canonical coordinates and momenta are equal in the Hamiltonian formulation of the PI similar to (2.25). Expression (4.44) makes it possible to trace out the transition from the quantum to the classical partition function as the temperature increases. According to (4.41), I9 + oo if T + 0. If T becomes less than any characteristic oscillation period of the system, the main contribution to the integral is given by static field configurations. More strictly, we must expand the field in a Fourier series in time
<
and substitute it into (4.38). This gives
4.4. THE EQUIVALENCE OF QFT AND STATISTICAL PHYSICS Now it is apparent that (independently of the specific form of U) the action becomes very large in the limit T + 0 for all terms but those corresponding to n = 0, i.e. to the static configurations. Hence the expression takes the form of the partition function for a classical system
In the last expression the functonal integration is taken over all field configurations in threedimensional Xspace and the argument of the exponent is the field energy divided by k ~ 0 . The above reasoning implicitly assures the covergence of all formal series appearing in the theory. However, this is not the case in quantum field theory. We shall study divergences in the following chapters, and now we formulate (without a proof) a theorem which enables us to obtain the correct classical limit. It is the socalled decoupling theorem [4]. First, let us introduce the necessary concepts. As can be seen from (4.46), the contribution to the action due to p, for n # 0 is large, being of the order of SE/T  1/T2. Conversely, the typical energy of the timeindependent mode p" remains finite in the limit T + oo. The modes p, for n # 0 are called the hard sector of the theory, and cpo is called the soft sector. The theorem asserts that functional integration over the hard sector of the theory (with characteristic energy Ehard)leads only to a renormalization of the soft sector (characteristic energy ESoft)with a relative accuracy (Es,ft/Eha,d)2 provided the soft sector is renormalizable by itself. The renormalization means in fact a change of the numerical values of the parameters, say those of U, and the renormalizability means that all divergencies can be collected into the same parameters. We shall consider these properties in much more detail in the following chapters.
Problem: Obtain an expression for the free energy of the onedimensional harmonic oscillator at finite temperature 0 by both (i) a straightforward summation over the oscillator energy levels in definition (4.40) and (ii) by applying the above described procedure to formula (2.61). Explain the result for kBO << fiw. Problem: Consider the same problem for the anharmonic oscillator with an arbitrary potential V ( x )and show that for kBO >> fiw the result
86
CHAPTER 4. CLASSICAL AND QUANTUM FIELDS
coincides, a s could be expected, with the classical expression
where xo is the minimum point of V(x);
Problem: Consider the first quantum correction to the free energy of the harmonic and anharmonic oscillators using the technique developed in the previous chapter. What are the limiting forms of the expression at high and low temperatures?
4.5
Free Field Quantization: from Fields to Particles
Let us consider small oscillations of the field (4.12) near the equilibrium point of U(cp). We may expand U(cp) in powers of the deviation from this point and neglect all powers higher than the second. Without loss of generality, we may assume the minimum of U(p) to be located at the point cp = 0. Then the action (4.12) takes the following form
The field described by this action is called the free field. As will be seen below, this model is as important in quantum field theory as the harmonic oscillator is in quantum mechanics.
Momentum Space The action (4.49) can easily be diagonalized. To do this, we must represent the field as Fourier integral over the set of spatial plain waves. For correct calculations the system must be placed in a box of a large but finite size, imposing suitable boudary conditions upon the field and sending the size of the box to infinity in the final expressions. The natural boundary conditions are those which require the vanishing of the field on the boundaries. Periodic boundary conditions are also convenient. They correspond to the field being defined on a threedimensional torus. As will be shown below, the results of all calculations do not depend on the specific form of the boundary conditions in the limit of infinite volume. Let the coordinates xl, za, X Q vary between zero and L1, La, L3 respectively, and let us choose periodic boundary conditions. The functions
4.5. FREE FIELD QUANTIZATION obeying these conditions are
1
f
n
2nn1x1 2rn2x2 2rn3x3 { +ip + i L3 } Jv exp z L1 L2 n
, (4.50)
where ni = 0, +l,l f 2 . . .. They form an orthonormal set. It is apparent that (4.50) corresponds to the usual expression
where all possible values of the wave vector p' components are numerated by indices nl, n2, ns which we shall denote the sake of brevity by n. In this way, the space of p' vectors becomes discrete. The volume of an / Vis .possible to formulate a rule elementary cell in this space is ( ~ T ) ~ It for replacing the summation over n1, 722,723 with integration over F i n the limit of infinite volume:
It is important that the integrand F($ really depends on 6its change should be small for any change by unity of nl,n 2 or n3. Let us examine now the independence of the integral in (4.52) of the form of boundary conditions. If zero boundary conditions are chosen, the corresponding Fourier harmonics take the form
Now, the labels ni take only integral values ni = 1 , 2 , . . .. If these functions are used rather than (4.50) then the rule (4.52) takes the form
where the integration is taken over the part of the whole space where all components of the vector $are positive. As a rule, the integrand depends only on even powers of p', so the integration in (4.54) can be extended to the whole space by including a factor 112 for each dimension. This brings us back to formula (4.52). As for functions (4.50) and (4.53),we assume them to be orthonormal:
J d 3 x f c i ,a)w,ail = c,,
88
CHAPTER 4. CLASSICAL AND QUANTUM FIELDS
This is convenient in the case of the discrete sum. In the continuum limit when the relevant quantity is an integral of type (4.52), it is more worthwile to normalize the functions by &functions so that
For this change in the normalization of the functions (4.50), (4.53), it is sufficient t o omit the factor in (4.50), (4.53). Then, in the right hand side of (4.55) we get VS,,,,, which in the continuum limit gives the right hand side of (4.56). In conclusion we would like to note that we shall call @space momentum space because the vector @in (4.50), (4.53) becomes the momentum of particles upon quantization as will be shown below.
Normal Modes Let us diagonalize the action (4.49). To do so, we use the Fourier representation3 in spatial plane waves (4.51)with the normalization (4.56)
It follows from reality of the field that
cp(@,t)=F*($,t)
.
(4.58)
Subtituting expansion (4.57) into action (4.49) we get
Using the orthogonality property (4.56) of the plane waves, we integrate over one of the momenta:
3We hope that notation of the field itself and its Fourier transformation by the same symbol cp will not mislead the reader, as long as the argument X or p is written explicitly.
4.5. FREE FIELD QUANTIZATION
89
For the sake of strictness we turned here again to the discrete set of momenta by placing the system into a box of finite volume V (cf. the rule (4.52)). It can be seen that action (4.49) takes the form of a sum of an infinite number of independent harmonic oscillators each having mass 1/V and frequency W ( $ . They are called the normal modes of the field. Only the pairs p($, t) and p($, t) = p*(p', t) are not independent in the expression for the integrand of (4.60). They correspond to real coordinates which represent sine and cosine Fourier transformations of the field: p, (g, t) =
1
d3xp(i,t)
[email protected])
and
p c ( a t) =
d3xp(i,t) COS($)
.
(4.61) It is apparent that
Note that each pair of p, and p, appears in the sum twice: a t the momenta and &. Thus a naive count of the number of oscillators gives one degree of freedom per &. It is clear that further quantization of the free field (4.63) can be done in any of the many known ways for the harmonic oscillator. For example, we can use the creation and annihilation operators for each mode or construct the coherent states that are, in some sense, the closest to the classical motion of the system. Here, we would just like to note that the field wave function can be written down as a product of oneparticle oscillator wave functions, because of the independence of the normal modes, and proceed to a classification of the energy levels of the quantizeed free field.
ZeroPoint Energy Let us consider the ground state of the free field which is also called the vacuum and is usually denoted by 10). In this state, all the normal modes are not excited, i.e. all oscilators are in the ground state. Nevertheless,
90
CHAPTER 4. CLASSICAL AND QUANTUM E'IELDS
the vacuum energy is not automatically equal to zero because each oscillator in the ground state possess an energy w / 2 and hence, the total field energy is
This energy is called the zeropoint energy. It is quite clear that integral (4.64) diverges and we get on the face of it, an absurd result: the vacuum has an infinite energy! However, we can get rid of this infinity by recalling that energy is . ~ means that only deviations of defined up to an additive ~ o n s t a n t This energy from its vacuum value, rather than its absolute value, are physically significant. Thus we shall subtract the infinite constant (4.64) from all observable energies. It should be pointed out that this is the first example of divergences appearing in quantum field theory. The recipe for the subsequent elimination of such divergences (called renormalization) will be considered in the following chapters. The idea of subtracting an infinite quantity from all key expressions of the theory should cause difficulties, at least because all expressions for observable values will be obtained in the form of a difference of two or more divergent quantities of type (4.64). Such an operation is, of course, mathematically incorrect. To understand it better, let us discuss in more detail the origin of the infinity of the zeropoint energy. Integral (4.64) obviously diverges at large momenta. If the field system describes a crystal, then there are no momenta larger than some maximal value. Indeed, atom oscillations with period less than the elementary cell size = .rr/a are absent in are meaninless. Thus components with Ipl > p,, the sum (4.64). Then the the energy E. (4.64) becomes large, but finite. We can attempt to measure this energy by heating the crystal from a temperature close to absolute zero until evaporation. The result is equal to the difference between the gas energy in the final state and the total energy expended. Unfortunately, an elementary length such as the lattice constant is not known in particle physics. However we can hope that at very large W 41t is true as long as the gravitatinal interaction is not taken into account. Thus we have to hope that the as yct incomplete theory of quantum gravity, the ultimate insite into the structure of matter, will yield a convergent vacuum energy. 5At least it has not yet been discovered. Howevcr, there is a spatial scale called the Planck scale a t which any theory which does not take into account quantum gravity is incorrect. This can easily be seen if we recall that the increase in the particle mass leads t o a corresponding decrease in its Compton radius RC = hlmc
4.5. FREE FIELD QUANTIZATION
91
there are modes which do not appear in (4.64), cancelling the divergence in the expression for Eo. Theoretical constructions with a fundamental length attract a lot of attention at present but all of them seem to be far from completion. Despite this, modern field theories enable us to make precise predictions for a wide class of phenomena. This is a consequence of the renormalizability of the corresponding theories, in other words, the independence of the physics at low energies on the arrangement of the theory at high energies. It is sufficient in a theory possessing this property to bound the p interval in (4.64) and related expressions as if we can dealing with a crystal rather than with continuous spacetime. After this, we make all the necessary calcultions and subtractions of the + co in the final results. The quantities like (4.64), taking the limit p,, renormalizability of the theory ensures the independence of the result on the specific method of imposing the condition p < p,,. In the subsequent chapters we consider this procedure in detail.
Elementary Excitations of the Field Now let us consider elementary excitations of the free field, that is the excitations of a single oscillator in the system (4.60). Let this correspond to a cell in momentum space. It follows from (4.60) that we have to take into account the excitation in the cell 6 as well. Both oscillators have the same frequency w(k) = (z2 and the energy of the first excited level is 3w(k)/2. The energy of the system, therefore differs from the vacuum energy by
z
+
This relation coincides with that for the energy of a relativistic particle and not, of course, by chance. The field of mass m and momentum + excitations of the Fourier component 6 do carry momentum k and energy w ( z ) . This enables us to identify them with particles, usually called field quanta. To verify this statement, let us make sure that the momentum of the excited field really is equal to z. First, note that this state is degenerate because excitations corret ) and t) have the same energy. To build these sponding to states we use the creation and annihilation operators in the form follow
z,
and incrcase in its Schwarzschield radius Rs = 2Gm/c2. At the value m (FLc/G)~/~ 2.2  10W5g = 1 . 2 . ~ o ~ ' G c v / c ~ called , the Planck mass, RC RP = (FLG/C~)'/~= 1.6 . loW"cm. This is the Planck length scale.
 mp Rs
=
92
CHAPTER 4. CLASSICAL AND QUANTUM FIELDS
ing from (4.63):
a! (g)
1
=
(q
*J
(Z)
+ iw (Z)pf (Z))
.
is the operator of the Here the index f takes values s and c, and canonical momentum conjugate to the variable cpf(Z). At the classical level this quantity takes the form
as follows from (4.63). In the quantum theory, the operator qf obeys the standard commutation relation
This gives the (also standard) commutation relations for af and a): [as(Z), a! (Z)] = 1 and
[a,(c), a:(;)]
=
l
.
(4.69)
All other commutators are zero. The same is true of the operators af and a!, corresponding to different cells of the momentum space. Note that for taking the continuum limit it is often convenient to make a minor change of the momentum operator normalization by introduction of n = qf ( Z ) V / ( ~ X )Then, ~. the commutator (4.68) reads
In the continuum limit, the right hand side of this relation becomes iSffS(Za  ,&). It is natural also to include the normalization factor ~  ~ / ~ / ( 2inn the ) ~ definition /~ of the annihilation and creation operators (4.66). This turns the right hand side of (4.70) into a representation of the Sfunction. This normalization is employed in many QFT books, but it is preferable for our purposes to use definitions (4.66) and (4.67). Let us build the excited states by acting with the creation operators on the vacuum state. For this purpose we assume for a while that the states are labeled in such a way that we can distinguish sine and cosine
4.5. FREE FIELD QUANTIZATION
93
oscillators excited states of the momentum L: In,(k), n,(k)). It can easily be checked that these states are eigenstates of the energy operator
whose eigenvalues are
(the energy of the first excitation is just w(L)). In the above notation the vacuum state can be written [OS,0,) = to), and for the lowlying excited states we obtain
l1,,Oc) =a:(L) 10) and
I0,,1,) =a:($) 10) .
(4.73)
Any linear combinations of these states obviously has the same energy W
(L).
We find now the field momentum in these states. To do this it is useful to represent the spatial components of the momentum operator (4.23) in terms of the Fourier transformation of field:
In the last line we have added to the integral the same expession with @' repleced by F, and divided the sum by 2. This makes it obvious that 9 is real. Turning to the discrete momentum space, we express p($, t ) as the real part of p($, t ) = v,($,t ) icp,(g t ) and the canonically conjugate momentum
+
Substituting for rlf and cpf the operators in the Schrodinger representation (4.68) we can easily convert this classical expression to its quantum analog. Note that the ordering problem does not appear in this way because (4.75) consists only of products of commuting operators. Thus, we
94
CHAPTER 4. CLASSICAL AND QUANTUM FIELDS
can treat (4.75) as an operator in the Schrodinger representation (where qf and cpf become timeindependent). Expressing the above quantities in terms of creation and annihilation operators (4.66), we obtain finally the field momentum operator in the form
Let us recall some important properties of the operators a and at for each given value of p', (we omit for brevity the argument pa):
It is clear that In,, n,) are not the eigenstates of the operator alas  aia, (whose average value and hence the mean field momentum  in these states is zero). The eigenstate of aLa,  a:a, is a superposition 
These states are created by the action of the following creation operator on the vacuum: 1 I + k ) =  (af r i a : ) 10) =at(&) 10) .

Jz
The corresponding annihilation operator (hermitian conjugate to at) takes the form 1 (4.80) a ( f k) =  (a, f ia,) .

Jz
A straightforward calculation shows that a ( f i)and at(&) form the standard algebra
while all other commutators are zero. Thus, we have obtained the field excitations I f to a definite value of the momentum
@ / & ) = f iG)~ f
+
z) which correspond (4.82)
and energy u ( k ) = (i2+2)1/2 independently for each component of the Fourier decomposition (4.57). The values of the energy and momentum suggest that we should identify the normal mode excitations with particles.
4.5. FREE FIELD QUANTIZATION There are higher excited states which correspond to a large number of particles in the system. There are two cases to be distinguished: the excitation of a large number of oscillators in different cells of the momentum space, and a strong excitation of one mode. The first case corresponds to a large number of particles bearing different momenta. The second one may be identified with the quasiclassical limit with a large number of particles in a single mode. For example, the coherent states of one of the normal modes describe a quasiclassical field with minimal uncerSuch states arise in laser radiation where there are tainity +(i)~~(i). a large number of photons, the quanta of the electromagnetic field, in a few normal modes while the total energy is macroscopically large [5]. To sum up, we have constructed particles as vacuum excitations in a quantized field system. The energy of the vacuum state is infinite but the particle energies measured above that of the vacuum are finite. It may appear that the vacuum energy is fictitious and physically senseless. Nevertheless, this is not the case because the presence of external fields, or a change in the boundary conditions leads to a deviation of the vacuum energy from its initial value, which can be observed experimentally. The Casimir effect considered in the next chapter is a phenomenon of this kind.
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Bibliography [l] E. Fermi, J. Pasta and S.M. Ulam, Studies in nonlinear problems, Tech. Rep., LA1940, Los Alamos Sci. Lab. (Also in Collected Papers of Enrico Fermi, vol.11, 1965, p.978, Chicago University Press).
[2] L.D. Landau and E.M. Lifshits, Course of Theoretical Physics. v.2: The Classical Theory of Fields, 4th ed., London, Pergamon, 1975.
[3] The ideas of constrained systems quantization discussed here were developed by Dirac in P.A.M. Dirac, Canad.J.Phys. 2 (1950) 129; 3 (1951) 1. For a general review see, for example, N.Kh. Ibragimov, Transformation Groups applied to Mathematical Physics, Dordrecht, Reidel, 1985. [4] T . Appelquist and J. Carazzone, Phys. Rev., D11, (1975) 2856. [5] A.M. Perelomov, Generalized Coherent States and their Applications, Berlin, SpringerVerlag, 1986.
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Chapter 5 Vacuum energy in p4 theory Let us return to a result derived in the previous chapter: the vacuum state energy density of the free field is expressed by a divergent integral
where W($
=
Jm.
As was discussed in the previous chapter in section 4.5, the vacuum energy E. is not an observable quantity. We can detect only deviations of the field energy from Eo. Usually, such deviations are due to particles (the field excitations in the background of the vacuum) which carry a finite energy. Another way of observing variations of E. is to change slightly the function W ( $ (5.2). Such a variation can result from, for example, changed boundary conditions or switching on an external field. This does cause finite observable energy variations. Famous examples of such phenomena are the Casimir efSect and the Lamb shift l . In this chapter, we shall consider the Casimir effect in a simplified model. In the second part of this chapter, we shall calculate the energy of the spatially homogenous field configuration, the socalled efSective potential of the cp4 model. These are the first examples of the calculation program formulated in section 4.5. We shall restrict the integration (summation) in momentum space in expression (5.1) and then remove this restriction 'The Lamb shift of the levels of a hydrogenlike atom 2Sflz and 2PllT: can be understood as a consequence of the fact that the radial probability distributions corresponding to different Dirac wave functions have noncoinsiding maxima. The electrons occupying 2Sl/z level are 'closer' to the nucleus than 2Pl/z electrons, and hence their interaction with thc background Coulomb field is stronger than that of 2Pl12 . However in the exact calculation some other factors should be taken into account.
100
CHAPTER 5. VACUUM ENERGY IN
ip4 THEORY
when physically meaningful quantities are obtained. As we shall show below the resultant values remain finite. It should be emphasized that in this introductory chapter we try to simplify calculation techniques as much as possible, sometimes applying common sense instead of rigorous procedures. We shall give a more solid justification for the method used here in the next chapters.
5.1
Casimir Effect
The vacuum energy manifests itself in a fact which seems at first sight to be paradoxical. It has been experimentally verified that two neutral conducting plates attract each other. This effect was theoretically predicted by H.Casimir in 1948 [l],and can be explained only by considering the zero point energy fluctuations, whose spectrum is changed if the vacuum is located between the plates rather that in infinite space. Let us consider the Casimir effect for the case of a free scalar field in (1+1) dimensions. The action of the model is
Simple Calculation of Casimir Energy For the sake of simplicity let us consider at first the case m = 0 and quantize the field in the interval between the plates (the plates reduce to two boundary points in our onedimensional space). The 'plates' impose zero boundary conditions for p:
The eigenfunctions which diagonalize the action are 
where k,, = m / L , n = 1 , 2 . . . because of the boundary conditions (compare this quantity with that of chapter 4). Substituting (5.5) into (5.3) we find the frequency of the kth mode:
It is worth comparing this with the analogous result for the field defined on a circle, where the points X and X L are identified. For this
+
5.1. CASIMIR EFFECT case, the eigenfunctions (5.5) take the form
It is possible, of course, to use the real sin and cosine components. The periodicity condition cp(x, t ) = p(x nL, t ) for this case requires that k n = 2 7 r n / L , n = 0 , f l , f 2 .... Let us return to the field defined on the interval. The vacuum energy takes the form
+
As we might expect, the last sum diverges badly. So we need a trick to extract from this divergent energy that part of it which corresponds to the interaction of the plates. The trick is to supress the sum at large n, i.e. at large momenta. It is called regularization. Consider the following convergent sum instead of (5.8)
which becomes (5.8) in the limit a a7r/L, we write
+
0. Introducing the notation
t =
Summing the geometric progression we obtain
This expression diverges at t + 0 as expected. Its expansion in a Laurent series in the vicinity of the point E = 0 is
In terms of the initial regularization parameter a , this vacuum energy takes the form L 7r E % m 2 24L+O(;) Now we must interpret this expression  it is reasonable to consider its limiting form at L + m. It is obvious that in this limit the plates do
102
CHAPTER 5. VACUUM ENERGY IN q4 THEORY
not interact. Therefore everything that survives at L + CO is the zero point energy, whose density should be subtracted from E,, regardless of the distance between the plates. This is exactly the first term on the right hand side of (5.13). It is proportional to the onedimensional volume L and this gives the energy density 1/(27ra2). As expected, it diverges at a + 0. So the observable interaction energy of the plates A E is defined by (5.13) with the first term on the right hand side omitted. It is
We can now set a = 0. This is trivial because A E does not depend on a. Thus this expression is our final result. It should be emphasized that the obtained result depends sensitively on the form of the boundary conditions [2]. The Casimir energy of the same field with periodic boundary conditions gives A E = x/(6L). This is four times as large as the previous result, because the change of boundary conditions leads to a doubling of the number of modes (k, takes both positive and negative values), and the frequency of each mode. For the more realistic case of the electromagnetic field, the Casimir energy is A E = 7r2S/(720L3) where S is the area of the conducting plates on which boundary conditions of zero tangential field are satisfied. It should also be pointed out that a change of boundary conditions may lead to a change in sign of the Casimir energy. For example, for a scalar field defined on a sphere of radius R, A E = 0.09235/(2R) corresponding to a repulsion (a tendency for the sphere to inflate) rather than an attraction. This reflects a general tendency of the Casimir energy to be positive for more compact manifolds, though there is no strict relation to their topology. It is worth mentioning that the Casimir effect must be accounted for in the theory of modern nanosize physical devices, such as the atomic force microscope used to investigate solid surfaces. The core of this device is a microscopic needle with an atomicsize point. It scans the surface at separation of a few Angsroem, driven by piezocrystals. The force of its interaction with the surface is measured. The Casimir effect contributes to this force because of the small system size. Let us return now to the scalar field defined on the interval. Note that we were lucky to have used the cutoff parameter a. Indeed we have could choosen another one, say E, which was used temporarily up to equation (5.12). This expression for the Casimir energy would then be required instead of (5.13). This is not so straitghtforward, because all the terms in (5.12) are proportional to 1/L, which makes it impossible to recognize
5.1. CASIMIR EFFECT
103
the constant vacuum energy density therein. Moreover, the second term in (5.12) changes its form depending on the choice of cutoff parameter. For example, a replacement c2 + c2/(1 E') leads to an additional Eindependent term lr/(2L) in (5.12). We can believe that regularization by the multiplier exp(ak,) in expression (5.13) is advantageous because it is done in terms of the momentum k, rather than the number of an oscillator mode n. This may be preferable because k,, is an experimentally measurable quantity. However this method is hardly selfconsistent. So let us find the Casimir energy once again in a more rigorous way, making use of the PI.
+
Casimir Energy: Calculation via Path Integral Let us now consider a massive scalar field with action (5.3) and, using the recipe deduced in chapter 3 and generalized for the case of QFT in chapter 4, find the ground state energy. For this purpose we take a P I of type (4.37) for the model (5.3):
where
.
,
In the limit T 4 oo,the main contribution to Z is given by the state with minimum energy, i.e. by the vacuum state. Equation (5.15) then takes the form , (5.17) Z=Nexp{EOT) where the preexponential factor does not depend on T. Let us fix the boundary conditions at r = 0 and T = T. We presume Z to be the Euclidean amplitude of a transition from the configuration p(x) = 0 into the same one, i.e. p(x, 0) = p(x, T) = 0. The final result does not depend on the boundary conditions in the limit T + oo. From now on, the calculations have much in common with those performed in chapter 3. To evaluate integral (5.15), let us diagonalize the action SE (5.16). An integration by parts gives T
St.=
2
L
J d r / d ~ p ( z , L p(x, r ) , T)
0
0
(g)+ 2
where L

rn2 .
104
CHAPTER 5. VACUUM ENERGY IN
q4 THEORY
Now we expand the field in a series in the eigenfunctions of the operator L such that L&,~(X, r) = Xn,l$n,l (X, 7):
Here $,,J(x, r) are the familiar functions (4.53) which in this case can be written in the following form:
The corresponding eigenvalues are
It can easily be seen that substitution of (5.19) in (5.18) makes the action diagonal:
The functional integral (5.15) then becomes a product of onedimensional integrals (see (2.57)):
(5.23) In the last equality the product of all the eigenvalues of the operator L is called the functional determinant of L, analogously to the case of finitedimensional matrices and to the model considered in chapter 3. Substituting the explicit form of the eigenvalues into the last expression, we obtain r212 r2n2 (5.24) ++m2 L2 lI2 T2
l
We have to choose now the order of calculation of the products over n and over 1. Let us consider both possibilities, starting with
where the notation
W:
+ m2 has been introduced.
= x2n2/L2
5.1. CASIMIR EFFECT
105
The expression in the square brackets is similar to the result of the PI calculation for the harmonic oscillator (2.55), (2.57). Therefore
in the limit T
+

cm.Thus the PI (5.23) takes the form exP {
 ~ w ~ T=}exp {T
n
Consequently the vacuum energy is
This is simply expression (4.64) for the vacuum energy. In the massless case it leads to result (5.13) for the Casimir energy. To check this result, let us change the order of the calculation of the product in (5.23). Introducing the notation v: = 7r212/T2 m2 we can rewrite (5.23) in the form
+
Note that the Euclidean form of the PI and the boundary conditions are symmetric in the coordinates X and T . Hence transposing the products must lead only to a transposition of W , and vz:
This case differs from equation (5.27) by the finiteness of L: sinh(vlL) cannot be replaced by exp{vlL)/2. At the same time, the condition T + CO makes the summation over 1 equivalent to the integration over k 17r/T:
Z

03
1 e x p {  T 27r . Jdkin 27r sinh(v(k)L) v(k) 0
}
where v(k)
=
d
w
(5.31) According to interpretation (5.17), which should be given to expressions of this type, the ground state energy is Eo(L) =
J"dk27r[ ~ + l n a + l n 0
(1  e2"L
(5.32)
106
CHAPTER 5. VACUUM ENERGY IN
(p4
THEORY
The Casimir energy can be obtained from this equation in two steps. First, note that the term l n x contributes to the normalisation factor N in (5.17). From a physical point of view, it is a constant in the energy (though divergent) which does not depend on L. Thus it does not affect the force between the plates which is the derivative dEoldL. Second, the vacuum energy density found at infinite separation of the plates, must be subtracted. To this end, we must find the value Eo(L)/L in the limit L t m, multiply it by L, and subtract the result from (5.32). This gives
In the limit of small L (compared with l l m ) expression (5.33) takes the form (a detailed calculation of the integral over t is presented in Appendix A.3)
Thus we have reproduced the previously obtained result (5.14). At large L (mL >> l ) , integral (5.33) can be estimated by the saddle point method (see e.g. [3] and our example of estimating the integral (2.36))2. We expand the exponent in (5.33) near k = 0:
Retaining only the first two terms of this expansion, we can see that the . such values resulting integral in (5.33) converges at k  ( m / ~ ) ' / ~At of k the last term in (5.35) is of order O(mL)l, which is negligible. Expanding the logarithm in (5.33), we reduce the integral to Gaussian form:
'Strictly speaking the method of integral estimation by using the maximum of the integrand's exponent is called the Laplace method. Sometimes the contour of integration has to be shifted into the complex plane in order to find such a maximum. In this case the method is usually called the saddle point approximation. An analogous method for oscillating exponential functions is called the stationary phase approximation.
5.2. EFFECTIVE POTENTIAL OF cp4 THEORY
107
Thus the interaction vanishes exponentially at large distances. It should be noted that this is a general property of all massive theories. Let us finally note that we subtracted one divergent quantity from another one when we derived formula (5.33) from (5.32). Such an operation is mathematically ambiguous. More strictly, we must make the integrals convergent (regularize them), for example, by replacing the u p per infinite integration limit with a large, but finite quantity A called the cutoff parameter. This is known as cutoff regularization. The vacuum energy denoted now as E,$(L) acquires a dependence on A. For any fixed A, there is no problem subtracting the vacuum energy from E,$(L). The Casimir energy obtained in this way depends on A only marginally via a term exp{AL). Thus the final result obtained in the limit A + oo coincides with formula (5.33).
5.2
Effective Potential of
p4 Theory
For further study of the vacuum energy we need a scalar field defined by the action (4.12) in 3+1 dimensions:
The higher powers of cp make the potential U(cp) anharmonic (the theory is no longer free). In the simplest case
(see fig.15). This model is called (p4 theory. It is widely used in statistical physics as well as (in a more complicated form) particle physics. Let us now pose the following question: how can we (at least in a thought experiment) measure the selfinteraction potential U(cp)? It is clear that if cp = 0 is a minimum of the potential U(cp) then the potential is the energy density of a field deviating from cp = 0 homogeneously over the whole space. In order to be able to monitor such deviations, let us introduce by hand an additional auxiliary field interacting with cp, so that the action becomes
The physical meaning of the quantity J depends on the nature of the field cp. In the simplest mechanical model illustrated in fig. 14 on page 75
CHAPTER 5. VACUUM ENERGY IN q4 THEORY
Figure 15
J is thought of as an external force. If cp were a vector and described the magnetization of a ferromagnet, then J would have the sense of an external magnetic field. In quantum electrodynamics, the interaction between the electromagnetic field and an external source is described by a term of the same form: jf'A,, where A, is the Cpotential (the field) and j, is the $current. It was J. Schwinger who proposed the introduction of such auxillary fields in order to investigate the energy spectrum of a given model [4]. The quantity J is called the source or current in analogy with quantum electrodynamics. Applying a spatially homogenous J , we can obtain a homogeneous field cpo # 0 as the solution of the equation of motion which takes the simplest form U'(cp0) = J . (5.40) The energy density of the whole system is at first sight simply w(J) = U(cp0)  J p o . U(cp0) can be extracted by measuring w(J) and subtracting from it the value of Jcpowhere J is known (being applied by the experimentalist) and cpo is measured. This procedure is true only in the case of the free field, or when quantum corrections are negligible. Indeed, for any value of J, we generate a new vacuum state at cp = c p o Its energy has a contribution from all fluctiations (5.1). The form of the eigenvalues W ( $ in this formula is slightly different for different po. This results in additional variations of the vacuum energy as a function of cpo. In any experiment the total
5.2. EFFECTIVE POTENTIAL OF
(p4
THEORY
109
energy is measured, which is the sum of the classical part U(po) and the quantum contribution. This sum is called the eflective potential Ueff(p) which differs from U(p) by quantum corrections. We shall calculate of this quantity in the next section.
Calculation of U, (9) To calculate Ueff(p),it is useful to apply the above proposed method of evaluation of the ground state energy. Analogously to equations (5.15), (5.17), let us introduce imaginary time and calculate the amplitude Z for a transition from the vacuum field configuration, p = p0 defined by the value of J, to itself over the infinitely large Euclidean time T :
Here p = 1,2,3,4, integration in the Euclidean action is performed over a large volume V and time interval [0,T]. We assume that J be constant everywhere within the integration $volume. We must also define a quantity (p), the average field corresponding to the given value of J . In general, it does not coincide with po. The quantity (p) as a function of J is
It can be written in a more compact form as follows:
Unfortunately, only the Gaussian integrals can be calculated exactly. The surprising fact is that such integrals, together with the expansion in powers of small parameters, make it possible to solve a large number of problems. In particular, we are going to evaluate (approximately) the integral in the right hand side of (5.41). For this purpose we use the functional generalization of Laplace's method, i.e. we find the maximum of the exponent in (5.41) and (5.42), expand it near the maximum, and perform a Gaussian integration over the vicinity of the maximum.
CHAPTER 5. VACUUM ENERGY IN p4 THEORY
110
The maximum of the exponent in (5.41) is reached for the constant field po (5.40). Let us use a new variable ~ ( x=) p ( x )  p. of functional integration in integral (5.41). The field ~ ( x obeys ) the zero boundary conditions. Expanding the action in the exponent of (5.41) in powers of x ( 4 we get
/ o x ( x ) exp { [6sE

1 1 d l x ~ 2!d~ 2 s ~ d3sE 3!
+
+
+ 4!
The expansion terminates at the fourth term because the action does not contain powers of the field greater than (p4. The sum of the first variation of the action and the term J X in expansion (5.44) is zero, because the field po is chosen to obey the classical equation of motion 6 s = J which takes the form (5.40) in our case. The second variation of the action reads
This expression is the Euclidean form of the action of the free field (4.49) up to the factor 112 and the substitution of U1'(po)for m2. To calculate integral (5.41) approximately, we neglect the higherorder terms S 3 S and ~ S4SE in (5.44). This is the socalled semiclassical approximation. It is valid when the main contribution to the functional integral is given by a set of trajectories close to the classical one. The effective smallness of X enables us to neglect higher powers of X . We shall find below a strict criterion for the applicability of this approximation. The problem now is to calculate the integral
subject to the zero boundary conditions imposed on X(.) We do this in the same way that led from (5.15) to (5.23) and (5.24): we diagonalize d2sEperforming the integration over the Fourier components of the field. This results in 22
= N [det
( (a,)2 + ~ " ( ( p ~ ) ) ]  l ' ~ .
The explicit form of the functional determinant is entirely analogous to (5.24):
5.2. EFFECTIVE POTENTIAL OF
(p4
THEORY
l11
where L1, L2, L3 are the dimensions of the box in which the system is placed  L1L2L3 = V . Before we calculate quantity (5.48), it is worth expressing Ueff in terms of it. Substituting (5.47) in (5.44) and in (5.41) we get
where the current J = U' according to equation (5.40). This expression reduces to the potential U(po) if we neglect the quantum corrections, which are the difference between p0 and (p), and the determinant term in (5.49). Let us now account for these corrections in the first nonvanishing approximation. Their smallness makes it possible to combine the first and second terms on the right hand side of (5.49) in U((p)) and to neglect the difference between p0 and (p) in the functional determinant. Thus we obtain finally
It should be noted that it is the quantities (p) and Ueff((p)), rather than p0 and U(po), which are experimentally observable, and so these quantities which have physical sense. We hope that all divergences, such as those in the functional determinant, cancel. To make the expressions more compact, we omit in what follows the angular brackets in the variable (p),denoting the quantum average of the scalar field in the presence of an external current by p.
The Explicit Form of Ueff To calculate the determinant in (5.50) we begin with expression (5.48). Taking the logarithm of both sides of (5.48), and replacing the summations by integrations according to the repeatedly used rule (4.52):
 this is an exact transformation in the limit of infinite V and T. The problem reduces to the calculation of the following integral:
CHAPTER 5. VACUUM ENERGY IN (p4 THEORY
112
It is convenient to perform the integration in 4D spherical coordinates. To do so, we need the formula for the total solid angle in 4dimensional space. For a derivation, let us consider the following auxiliary Gaussian integral in ndimensional space:
(5.53) The same integral can be calculated in another way:
according to the definition of the !?function. Thus the desired solid angle
This gives R4 = 27r2 for n = 4. Expression (5.52) takes the following form in spherical coordinates:
Introducing a new integration variable X to the form
= k2/U1' transforms
the integral
Integral (5.57) obviously diverges a t the upper limit and we come again to the divergence of the quantum corrections to classical quantities. Let us apply the simplest method of divergence elimination, namely cut08regularization. We restrict the upper integration limit with a very large but finite L,, = A >> 1. This enables us t o calculate the value of AV(cp), after which we return to the discussion of the physical sense of this procedure (see also the discussion in section 4.5). Taking the regularized integral (5.57), we obtain AU(p)
=
AZ/U"
m[ / (Ut1)
0
dxxln(1
A2/U"
+ X) +
/ 0
dzx ln U"]
(5.58)
5.2. EFFECTIVE POTENTIAL OF p4 THEORY
113
An expansion of the logarithm
results in
where all the terms which vanish in the limit A
+ cm
are omitted.
Renormalization of Mass and Coupling Constant For the sake of simplicity we consider the massless version of theory (5.38): 1 ~ 4!
m = 0 then U ( p ) = 
1 pand~ U1'(p)= Xp2 2
,
(5.61)
although the main conclusions of this section also hold in the more general case m # 0. Let us try to understand the meaning of expression (5.60) some terms of which diverge. First of all, it is necessary to note that the first two terms do not depend on the field p, or on any parameters of the theory. So these terms can be interpreted as the vacuum energy in the absence of the field. They should be discarded as in the calculation of the Casimir energy. However, even after this there are still two terms which diverge quadratically and logarithmically. They vanish only for the case of free field. We should be puzzled about the last result because a quantum correction must be small by the calculation procedure, yet it is infinite! This problem, which was realized in the 1930s at the very beginning of QFT, indicates a very peculiar feature of the theory. In order to see it better, let us analyze once again our calculation procedure. The starting
114
CHAPTER 5. VACUUM ENERGY IN p4 THEORY
point was the classical expression for the field energy U(p) = Xp4/4!. Then we evaluated the quantum corrections AU(p) (5.60) due to the selfinteraction of the field. The crucial point is that the parameters m and X appearing in the Lagrangian (5.38) were understood as the mass and coupling constant of the field without quantum corrections. But at the quantum level, this is physically meaningless because it is impossible to measure the classical and quantum parts of the energy separately. Only their sum is experimentally observable and hence a physical sense can be attached only to this. In particular, we might suggest that it was not such a good idea to drop the mass term imZy2in the classical action (compare (5.38) and (5.61)), because one more term quadratic in p appears from the quantum correction anyway:
This resembles a phenomenon which occurs in solids: the interaction of an electron with a crystal lattice leads to a change in its effective mass m (e.g. a large mass m means that it is difficult to accelerate the electron because it is strongly coupled to the deformation of the crystal lattice). There are two essential differences from the present case: (i) in solids there always exists a natural maximal momentum because of the finite size of the elementary cell. Then all corrections of the type (5.60) are finite including the effective mass; (ii) the electron can be taken away from the crystal. Then it is possible to measure its mass m. in the vacuum and see the difference from m. Let us call m. the bare electron mass. It is worth while to make the same kind of distinction between the parameters of p4 theory. Let us call the those quantities which enter the classical action in the exponent in the path integral (5.41) the bare quantities. We shall label them by a zero subscript, m0 and Xo. Thus the expression for the potential (5.38) takes the form U = m;p2/2+ X;p4/4!. A particle of the 9field cannot be extracted from the field. The bare mass mo, therefore, is not observable. The observable mass m is its sum with = m: 6m2. all corrections. In our case, this is m2 = m; XoA2/(26~2) The quantum corrections to the bare mass are divergent because there is no minimal length in the theory, like the elementary cell size for the case of solids. A possible solution of this problem is to include the divergences into m ~which , may diverge because it is not observable. The divergence of m. should be adjusted in order to cancel the divergence in the quantum corrections and thus make finite the observable mass m. Then we may hope that the divergences of the effective potential (and
+
+
5.2. EFFECTIVE POTENTIAL OF (p4 THEORY
115
other quantities) are only caused by the use of m" (and X") in the expressions. If this is the case, then the effective potential is free of divergences. To make this clear, we must express it in terms of the observables m and X. Such a procedure is called renormalzzation. It should be noted that such uncerimonious manipulations of infinities became commonly used first in the 1950s. Before this, they shocked most physicists as well as mathematicians. P.A.M. Dirac, who first calculated the radiative corrections to the electron energy and discovered them to be infinite, said that in mathematics one could neglect a quantity only if it was infinitely small but by no means if it was infinitely large and one wanted merely to get rid of it [5]! Nevertheless, the renormalization of &FT has became a standard tool, successfully compared experimentally with the 8digit precise predictions of quantum electrodynamics. There is one more possibility of infinity cancellation which has attracted a great deal of attention since seventies. In some cases the quantum corrections due to Fermi and Bose fields cancel each other. For such a cancellation to occur, there should be special relations between the number and charges of the Bose and Fermi field. These relations are provided by a special symmetry called supersymmetry. Even in supersymmetric models, the problem of getting rid of all divergences in QFT remains open though significant progress has been achieved. We shall not consider this research field in the present book. Let us renormalize the effective potential. It is possible to define the physical mass as in analogy with
m;
=
Urr(p)/ 'p=o
.
(5.63)
If we still wanted to deal with the massless theory, then we would require m rather than m" to be zero. Thus the most troublesome quadratic infinity (i.e.  A2) is hidden due to renormalization of the physical mass, for which a zero value is assigned. However, there is one more, namely logarithmic infinity in the expression of the effective potential
so an additional renormalization is necessary. To perform it, let us first note that the effective potential (5.64) is the sum of the classical term of order X" and the correction of order X;. The smallness of the correction
116
CHAPTER 5. VACUUM ENERGY IN
(p4
THEORY
implies the smallness of X. (we forget for a while about the divergence of X. e.g. because of the regularization). We are interested in the first nonvanishing quantum corrections  it is beyond the accuracy of our approximation to account for the quantum effects in the correction term in (5.64). The difference between X. and X is such an effect. Thus we can replace X. with X in this term, which we do in all expressions below. A more rigorous procedure taking more care over the divergence of the neglected terms would require us to perform the renormalization at each order of the expansion in X. We do not discuss this here. Note that the divergent part of the quantum correction to U e f f is a polynomial in cp of the same order as the potential in the classical action. To make this clear, we rewrite (5.64) in the form
Here, we have introduced an arbitrary parameter with the dimensions of mass in order to present ln(Xcp2/A2)as a sum of the constant ln(XM2/A2), and the finite pdependent term ln(Xv2/M2). We may attempt to declare the expression in the square brackets in (5.65) to be an observable finite quantity X. It is necessary, however, to deal with a more strict definition of the coupling constant. Indeed, any change in the divergent term (e.g. a variation of M , such a s the replacement M + 2M) affects the finite terms. The definition can be made by a generalization of the corresponding expression for the bare coupling constant in (5.38) and (5.61):
However in doing this we cannot take the limit cp = 0 because of the logarithmic divergence of the derivative of (p4 In cp, rather we should take a nonzero value cp = M. This is usually called the renormalization point. The coupling constant becomes in this way a function of M. Thus
Now, let us rewrite the effective potential (5.64) in terms of this physical coupling constant X(M) (5.67). It gives finally
5.2. EFFECTIVE POTENTIAL OF cp4 THEORY
This is the desired expression for the effective potential of the massless p4 theory. The expansion parameter is X and the omitted terms are of order X3. Thus all divergence in Uef disappeared after the renormalization of the mass and the coupling constant. Now it is possible to take the limit A + oo or, in other words, to lij? the regularization. This justifies the neglection of the negative powers of A in expression (5.60). After this, the effective potential (5.68) is indeed Aindependent. In other words, the properties described by potential (5.68) do not depend on the structure A. Such a property could be, for of the theory at large momenta p example, the fundamental minimal length of order l/A in a crystal or an unknown theory of all fundamental interactions. Even if this occured, it would not change the form of the effective potential. It should be emphasized that by no means every theory possesses such a property. Only for a few theories is it possible to eliminate all divergences by means of renormalization of a finite number of parameters. Those theories for which it is possible are called renormalixable. It is an interesting fact that the renormalization procedure which seemed to be an artificial trick at the early stage of the evolution of QFT has become a routine technique now. Furthermore, it became generally accepted from the beginning of the 1970s that only renormalizable theories can pretend to the role of consistent theories of elementary particles.
Running Coupling Constant, Dimensional Transmutation and Anomalous Dimensions Let us discuss further the effective potential (5.68) derived above. It is a nontrivial fact that however we started with the massless field, a constant M possessing the dimensions of mass appears in the final formula. In the classical action, the only dimensionful value was the field, which is a variable rather than a parameter. This resulted in a scaling law. Indeed, let cp(2,t ) obey the classical equation of motion 6S[cp] = 0. It can easily be checked (at least for the case of action (4.12) with potential (5.61)) that the function Slcp(s2, st), where S is an arbitrary constant, is also a solution of this equation. We say that there is no mass (or, equivalently, length) parameter in the theory, therefore any solution can be scaled with an arbitrary S. The presence of the dimensionful parameter M in the theory means that there is a special scale of mass, and hence of length. The generation of the mass parameter M by the quantum corrections in
CHAPTER 5. VACUUM ENERGY IN
v4
THEORY
Figure 16
QFT is called dimensional transmutation. As we shall show, the scaling law still holds in this case, but it takes a rather nontrivial form. There is another peculiarity of the effective potential under consideration: it has a nonzero minimum at cp = cpmin obeying
X In M2
327r2 11X +and, therefore, 3 3
  
(see fig. 16). The minimum of the effective potential corresponds to the field which can exist in the absence of the external current, because in this case X J e f f / d c p= J = 0. If SO, the mean field in the vacuum is cpmin or +,in. Let us note in advance that this phenomenon, called dynamical symmetry breaking, should be distinguished from spontaneous symmetry breaking which occurs at the classical level by the special choice of the potential e.g., U(cp) = X((p2  cp&i,)2. We shall discuss spontaneous symmetry breaking later. Here, we would like just to note that both cases give the same result: cpmin # 0 becomes the true vacuum and all excitations above it acquire a mass proportional to Icpol. There are two possible realisations of the system (with cp = fcpmin) since its action is an even function of cp. However it would be premature to conclude that dynamical symmetry breaking does take place in the considered case. The minimum in formula (5.68) is beyond the scope of its applicability. Indeed it ap
5.2. EFFECTIVE POTENTIAL O F (p4 THEORY
119
pears from the balance of two terms: the classic one proportional to X and the quantum one proportional to X2. But expression (5.68) is derived under the assumption of smallness of the coupling constant i.e., X2 << X << 1. The second term in (5.68) remains a small correction only if I X ln(p/M) I<< 1. Hence expression (5.68) is valid only in the vicinity of the renormalization point p = M . The minimum p = p,i, is out of this range. As we shall see below, this minimum disappears in a more accurate approach3. Let us analyse further the dimensional transmutation. From a technical point of view, its origin is clear when evaluating the effective potential, we regularized the divergent integral (5.57) by assigning an upper integration limit km,, = A >> (U")'/'. It was this momentum cutoff which introduced a large massscale into the theory. After the renormalization, it was eventually replaced by the finite parameter M , the renormalization point. This poses the question: is this mass generation inherent to the theory or may it be an artifact of the calculation procedure? In attempting to answer this question, we note that the physically consistent result should not depend on the choice of the renormalization point. Let this take another such point M' # M. The effective potential should be the same but the value of coupling constant (5.67) changes. (p;X, M ) and hence, Let this be X' = X(M1). Then Uef(p;X', M') = Ueff 
as follows from equation (5.68). Here, as was done above, we do not distinguish X and X' in the correction term, which is of the order of X'. We can rewrite (5.69) in the form
which leads to the following condition:
31t will be shown in the chapter 12 that dynamical symmetry breaking does take place in thc model where thc field (o interacts with a vector (gauge) field. This is thc socallcd Coleman Weinberg effect [ 6 ] .
120
CHAPTER 5. VACUUM ENERGY IN cp4 THEORY
where a new constant A with the dimension of mass has been introduced4. It is very large at X << 1 because from (5.71)
It is important to note that the value of this constant does not depend on the choice of the renormalization point and thus represents the fundamental mass scale of the theory. Let us make one more remark on the dimensional transmutation. The fact that taking quantum corrections into account introduces a mass scale reflects the sequence of operations we have made in calculating the effective potential. We started with the classical action with m = 0 and then we found that the quantum correction depended on the mass parameter M. Such a sequence is in some sense just the opposite to that which takes place in nature. Physically observable quantities include the sum of all quantum corrections. A system is considered to be classical if all such corrections are negligible. From this point of view, the specific mass scale, A, disappears in the classical limit. The task of &FT is to restore the observables to the full scale range, using the classical limit as the zero order approximation. Recall that formulae (5.69), (5.68) are valid only if the second term on the right hand side of (5.69) is much smaller than the first one, so that only a finite interval around the renormalization point can be considered. Nevertheless, the interval of applicability of (5.69) can be expanded for the following reason. If the value of the field cp = M' does not satisfy the condition X(M) ln(cp/M) << 1, then we can choose another renormalization point, say M", which does. The value of the effective potential is in both cases the same. We are assuming here that we can choose the classical part of the potential in different ways by choosing different values of X. That is, we can assert as 'classical' some proportions of the total value of X depending on the renormalization point M . In order to do this, we shall consider infinitesimal variations of the renormalization point in (5.69) M' = M+dM. Introducing a new variable t = 1n(M1/M), we obtain 3h2 (t) dt X(t dt) = X(t) 1 16r2
+
dX dt
 = p(X),
+
where
3X2 P(X) = . 167r2
4There is no relation between this constant and the cutoff parameter.
5.2. EFFECTIVE POTENTIAL OF (p4 THEORY
Figure 17
This is the simplest example of the socalled renormalization group equation (it can easily be checked that the transitions from one renormalization point to another possess the group property). Integration of this equation gives the coupling constant behavior for finite changes of M :
Returning to the initial notation, we obtain the expression for X(M) in the form
The function X(Mf) in (5.76) is called the running coupling constant. It is the value of X which is observable as the coupling constant when cp = M'. If it is chosen, the quantum correction to the effective potential is zero at cp = M' and small in the vicinity of this point. It is still necessary for the applicability of the calculation method that X << 1. Formula (5.69) represents the first two terms of the expansion of (5.76) in powers of X(M) ln(Mf/M) << 1. Formula (5.76) makes it possible to reach the point M' = 0. It follows from (5.76) that X(0) = X(M)/(l oo) = 0 or, in other words, the interaction vanishes at M = cp, = 0 and the calculation method is valid
+
122
CHAPTER 5. VACUUM ENERGY IN q4 THEORY
up to cp = 0. To obtain the effective potential it is sufficient to recall that X is its fourth derivative. Integrating (5.76) four times with the initial conditions U;jf(0) = U;ff(0) = Ukff(0) = 0 we can obtain the improved expression for Ueff(p). It has a minimum at cp = 0, because X(p) is everywhere positive. This procedure does not help us to extend the result to large values of X because expression (5.76) has a pole at M' = A. Near this pole, the value of X is large and our calculation method fails. This pole is called the Landau pole. It indicates the region in which the theory becomes strongly coupled. The classical form of the effective potential is no longer a reasonable approximation there. Let us discuss another interesting consequence of formula (5.76)  a nontrivial form of the scaling law. The classical action S[cp(~, t ) ]=
d 4 ~ ( k ( a u ~+) $291)
is invariant with respect to the scale transformations:
for an arbitrary value of the dimensionless parameter S . This scaling law is trivial in the sense that it follows from the metric dimension of the variables. In particular, the coupling constant does not vary under such transformations because it is dimensionless. But at the quantum level, the metric and scaling dimensions of X become different. As follows from (5.76), the transformation cp + p' = scp results in the following change in the coupling constant:
Thus the coupling constant acquires a scaling dimension y = 3X/(16n2), known as anomalous dimension. We have encountered an interesting phenomenon: some symmetries of the classical action (in this specific case it is the scale invariance of the coupling constant) are destroyed by the quantum corrections to this action, an anomaly in the quantum field theory. It should be emphasized that all these peculiarities of the cp4 model are generic to more realistic theories. Thus the analogous solution of
5.2. EFFECTIVE POTENTIAL OF
(p4
THEORY
Figure 18
the renormalization group equation in quantum electrodynamics (QED) gives the following expression for the running coupling constant e2/(47r):
The intensity of the electromagnetic interaction increases with M', as in (p4 theory, and experiments at the modern colliders confirm this conclusion. Thus for the energy of colliding electrons and positrons of order Mz = 90 MeV the coupling constant becomes e2(MZ)/47r = 11128, rather than e2/47r = 11137 as it is for energies smaller than the electron mass (formula (5.80) is valid only for higher energies, fig.17). It is apparent that there is also a Landau pole, i.e. the mass scale in QED at which the coupling constant becomes very large. It is
which is incredibly far away. It is much greater than the Planck scale FZ 1O1%eV, where any known QFT breaks down due to the uncertainties in the structure of space and time. Note that QED becomes part of the unified theory of electroweak interactions at the much lower energy scale of order Mz.
124
CHAPTER 5. VACUUM ENERGY IN
THEORY
Nevertheless, Nature does give us the possibility of seeing what happens as we approach the Landau pole. Quantum chromodynamics, which is widely accepted to be the correct theory of strong interactions at high energies, is an example of such a theory. The behavior of the running coupling constant is just the opposite of that in QED and (p4: it decreases as M' grows (see fig.18)
The Landau pole of the theory is in the low energy region AQcD = 180 MeV. The classical Lagrangian of QCD describes free quarks and gluons bearing the color charge. The intensity of the strong interaction decreases at high energies, so that quarks barely interact at short distances, a property called asymptotic freedom. On the other hand, the theory is not able to describe from first principles strong interaction physics at energies of order AQcDand below. It is well known experimentally, however, that the excitation spectrum changes drastically in this region. There exist only hadrons, which are the colorless (white) bound states of two or three quarks. The coupling constant of the hadron interactions is not small. The problem of quark confinement has persisted already for a few decades. It is beyond the scope of this book to discuss it in more detail, though recent progress connected with the paper by N. Seiberg and E. Witten [7] has revived hope for its solution.
Effective Potential of the Massive Theory Let us now consider a more general case of nonzero mass. In this case
and we take m2 > 0. The calculation of Ueff for this case does not differ in practice from the massless case. Furthermore, the singularity at p + 0 disappears because (5.83) does not vanish anywhere, so the renormalization conditions can be chosen simply at p = 0. For example, we can take U:ff(0) = m2 and U,";f(0) = X . The other case m2 < 0 leads to an interesting problem. The reason is that U1'(p) < 0 for p2 < 21m21/X. This results in the divergence of the Gaussian functional integral (5.46). We may try to obtain Ueff(p) as the analytic continuation from the domain where Url(p) > 0. But in doing this, the effective potential acquires an imaginary part for those cp for which U" < 0. So let us discuss this problem in more detail.
5.2. EFFECTIVE POTENTIAL O F (p4 THEORY
125
At first, we should note that the above discussed method of measuring U e f f in the presence of the external current J (5.44) fails for U(cp)" < 0 , because the expression U Jcp has no minima in this region. The interpretation of the imaginary part of the effective potential was given by Erick J. Weinberg and Aiqun Wu [8].Here we give only the result of this work without the proof. It was shown that the effective potential should be defined as an analytic continuation from the region where U(cp)I1 > 0. The real part of the potential is, as before, the energy density of the spatially homogeneous field configuration with the average value cp. Such a state is unstable. For suppose it is created as an initial configuration. Then the evolution of the field results in growing field components with nonzero wavelengths. The field becomes more and more inhomogeneous. It was shown that the imaginary part of U e f f is equal to the time increment of this instability. The reason for the instability is clear. Let the field cp, such that U(cp)" < 0 , solve equation (5.40). Such a value of cp is a maximum of U(cp), e.g. cp = 0 in fig. 15 for m2 < 0. This point corresponds to the equilibrium which is unstable with respect to small perturbations to the right or left of the maximum. According to the result of paper [8], the effective potential contains information about the initial stage of the instability development in the region U(cp)I1< 0. Let us find the final state of the system initially released in the unstable region U(cp)" < 0. It depends on what physical meaning is attached to cp(x). It is worth while to consider first the case of the constant field average 1 (5.84) $90 = cp(x)hx
+
vl
in the process of the instability development. This condition is fulfilled, for example, if cp(x) is the concentration of a solution. In this case the integral in (5.84) is defined by the total amount of the dissolved substance. We shall now show that the equilibrium state here is a stratified mixture consisting of a number of spatial regions with different values of cp. In a ferromagnet such regions are called domains. In the considered case the mixture consists normally of salt crystals in a moderately concentrated solution. Let us prove as a lemma that all states with constant cpo are unstable if Utf ( P O ) < 0. An example is given in fig.19. By contrast, the homogeneous states with U z f f( p o ) > 0 are stable with respect to the stratification, at least when the corresponding fluctuations are small. Suppose that a trial stratified state consist of a mixture of regions with different cp values: the value cpl occupying the volume fraction and the value cpz
CHAPTER 5. VACUUM ENERGY IN cp4 THEORY
B
P1 PO P2
P
Figure 19
occupying the volume fraction 1  c. To preserve the constant average value of cp it is necessary to require @pl (1  <)vz = cpo Assume that it is possible to neglect the contribution of the boundary regions, where the field changes continuously between cpl and cpz. This is so if the domains' characteristic size is large compared with the boundary thickness, and their form is simple (not fractal). Then the average energy density takes the form
+

Ueff = CUeff(~1)+ (1  <)Ueff(~z) . (5.85) Geometrically, (see fig.19) this is the ordinate of the intersection point of the secant AB and the vertical line cp = cpo. If the curve Ueff(cp) is < Ueff( p o )The further apart cpl and cpz are, the lower convex then the energy. What is the minimal energy of the mixture at fixed cpo? It is reached when the secant AB becomes the tangent (see fig.20). Thus it is defined as the solution of the following system of equations
ueff
for unknown cpl, cpz and J. Thus the states with the average field in the
5.2. EFFECTIVE POTENTIAL OF (p4 THEORY
Figure 20
interval (p1 < cpo < (p2 (fig.20) correspond to the mixture of two phases: one with the average value (pl and the other one with (p2. Outside this interval the homogeneous states are possible. To find the meaning of the quantities (p1 and 9 2 , let us note that, as follows from (5.86), their values does not change if we add a term + Ueff j(p. We can choose linear in cp to the effective potential: Ueff j such that the minima of Uef in fig.20 are degenerate (this corresponds to J = 0 in (5.86)). Then (p1 and (p2 are the field values at these minima. As can be seen in fig.20 there always exist intervals (p1 < (p < (p3 and (p4 < (p < (p2 where Uzf > 0. Hence the homogeneous states corresponding to these intervals are stable with respect to small fluctuations. At the same time, larger fluctuations make the states unstable, as is seen in fig.20. Such states are called metastable. They generally have an exponentially large but finite lifetime (for more details on this subject see the comprehensive papers by J.S. Langer [g], [10]). The homogeneous states are absolutely unstable in the interval (p3 < cp < 9 4 . The transition of such a state to the stable one, which is a mixture of two different phases, is called spinodial decomposition. Now let us consider the system for which the field average is not fixed. It could be a sample of ferromagnet in an external magnetic field J, or a selfinteracting scalar field in the early Universe. For such systems, an equilibrium domain vacuum structure is possible only if the effective potential minima at cpl and (p, are degenerate. Otherwise, the domains
+
128
CHAPTER 5. VACUUM ENERGY IN
THEORY
with the lower value of Ueff would expand to fill the whole space. This is equal to can be easily understood if we note that the quantity Ueff the pressure on the boundary between the two phases.
Problem: Find the effective potential for the scalar field (5.37), (5.38) for m2 < 0 at finite temperature. The problem can be simplified if we consider only the high temperature limit. Find the first nontrivial term in the expansion in powers of 1/kB8. Find also the critical temperature. Is this calculation reliable?
Bibliography [l] H.B.G. Casimir, Kon. Ned. Akad. Wetensch. Proc. 51 (1948) 793.
[2] V. Mostepanenko and N.N. Trunov, The Casimir Eflect and its Applications, Oxford, Clarendon Press, 1997. [3] J. Mathews and R.L. Walker, Mathematical Methods of Physics, 2nd ed., Benjamin/Cummings Pub. Co., 1970. [4] J. Schwinger, Particles, Sources, and Fields, Reading, MA., Addison Wesley, 197073. [5] P.A.M. Dirac, Directions in Physics, New York, Wiley, 1978.
[6] S. Coleman and E.Weinberg, Phys. Rev. D7 (1973) 1888. [7] N. Seiberg and E. Witten, Nucl. Phys., B426 (1994) 19. [8] E.J. Weinberg and Aiqun Wu, Phys. Rev., D36 (1987) 2474. [g] J.S.Langer, Annals of Phys., 41 (1967) 108. [l01 J.S.Langer, Annals of Phys., 54 (1969) 258.
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Chapter 6 The Effective Action in Theory
(p4
The scalar field quantization scheme developed in the previous chapter yielded reasonable results, although it was not mathematically consistent. In this chapter we attempt to construct a more rigorous calculation method. This method is approximate, valid at small coupling constant. Let us begin by asking a question: what physical quantities is it meaningful to calculate in QFT? The answer to this question is rather simple: only average quantities are observable in both QFT and statistical physics, although the nature of field fluctuations is different. In QFT, the average of a nonstationary quantity A[cp(x,t,)] at time t, takes the form
This implies that the initial and final field configurations must be fixed and, if necessary, subsequent integration over these configurations with the corresponding wave functional may be performed. As an example, let us check that expression (6.1) indeed gives the quantum mechanical average, provided an integration over the field with the vacuum wave functional at the initial and the final times is implied. We split the 'It should be pointed out that vacuum averages are the fundamental quantities in particle physics. The correspondcncc between them and the transition amplitudes is cstablished by the LehmanSymanzikZimmermann formula ( L S Z reduction formula) which is beyond the scope of this book (scc e.g. the books [l,21).
132
CHAPTER 6. THE EFFECTIVE ACTION IN
(p4
THEORY
integral into two parts: one for the interval 0 < t < t, and one for t, < t < T. Each part can be represented as an expansion in the energy operator eigenfunctions (4.35). Then we integrate over the initial and final field configurations with the vacuum wave f ~ n c t i o n a l . ~The normalization integral Z comprises all the above quantities excluding A itself. Then according to (6.1)
This is the standard quantummechanical average. The statistical analogue of (6.1)takes the form
(6.3)
6.1
Correlation Functions erat ing Functional
and the Gen
Which average values are of interest? Most obvious is the average value of the field (cp(x))itself. The meaning of this value may be different as dictated by the context of the problem. It could be the magnetization of a sample or ion density in a plasma. Important physical information is also contained in the product of two field functions at different points ('p(x)cp(y)).The temporal components of X and y may be different, equal, or be absent altogether as they typically are for statistical physics. Such an average is called the correlation function, or correlator. Sometimes, for reasons explained below, it is called the Green function3. 2The analogue of such an integration in onedimensional quantum mechanics is j'dxy!~(x). . .. 3Such a large number of terms: correlator, propagator, Green function, corresponding to the same theoretical object (cp(x)cp(y)), is not a result of terminological
6.1. CORRELATION FUNCTIONS
133
Let us consider some examples. Let cp have two components p+ and cp, which represent the density of positive and negative ions in a plasma. The correlation functions W++ = (p+(x)cp+(y)) and W + = (p+(X)+ (y)) are proportional to the probabilities of finding two positive ions and two ions of different signs respectively at the points X and y. It is clear that the Coulomb interaction decreases this probability for ions of the same sign, and increases it for ones with opposite charges. Manypoint or manyparticle correlators (e.g., (cp(x)cp(y)cp(z))etc.) as well as twopoint ones are often the subject of interest. It is an important fact that the knowledge of all possible correlators enables the reconstruction of the partition function, and hence, complete information about the system. Let us construct a consistent method to calculate the one and manypoint correlation functions. From now on, we restrict ourselves to the Euclidean variant of field theory. This makes it possible to obtain information about the ground state of the system and its statistical properties. In order to get information about particles (excited states of the field in the real time), we must analytically continue the results from Euclidean to Minkowskian space (this continuation will be discussed). Below we omit the index 'E' for all quantities in Euclidean space. In principle, we can calculate the correlation functions using the definition (6.1) or (6.3) every time we need them. It is more convenient to perform the functional integration only once, in order to obtain a quantity which contains complete information about the correlators. This is called the generating functional. To construct the simplest one, let us consider the PI in the presence of a source J ( x , t ) :
Here we have assumed h = 1 (or k B T = 1) again. The meaning of this amplitude is the same as for the effective potential: the system is probed by its interaction with an arbitrary external current (or source) J(x). The only restriction implied on the function J ( x ) is that it must vanish at lxol + *m. An obvious advantage of expression (6.4) is that we can extract from it any correlator by taking the functional derivative with respect to J(x). Indeed, as follows from (6.4),
confusion taking place in this book. It merely reflects the situation in the scientific litcraturc. Thc twofield corrclation function appears in a largc number of contexts, in each having a specific name.
134
CHAPTER 6. THE EFFECTIVE ACTION IN
(p4 THEORY
This expression can be simplified by introducing the quantity W [ J ]a s follows: Z[J]= eUWLJ] . (6.8) It is not difficult to identify W as the field theoretical analogue of the free energy F in statistical physics. We recall that the latter is defined by the relation
By making use of W [ J ]we can obtain all the correlators in the same way as (6.5)  (6.7). In particular,
Another quantity which is often used is
This is called the exact propagator or the correlation function, denoted also as (cp(x)cp(y)),. It follows from the definition of G that G(x,y) =
G(Y, 4. Substituting (6.8) into (6.6) we obtain the relation between G(x, y) and (cp(x)cp(y)):
Thus the quantity
describes the mutual influence of the fields at the points X and y, which goes to zero for statistically independent (cp(x)) and (cp(y)). This is clear also from definition (6.11).
6.2. CORRELATORS OF THE FREE FIELD
135
Mathematically, (6.13)means that g does not contain the so called disconnected part, or terms which can be factorized as a product of multipliers depending only on X and on y. This property of including only the connected part is true4 also for higher functional derivatives of W [ J ] . For this reason W [ J ] is called the generating functional of connected Green functions while Z [ J ]is called simply the generating functional.
6.2
Z[J]W , [ J ]and Correlation Functions of the Free Field
Let us find the explicit form of the generating functional of the scalar field defined by the Euclidean action
Generating functional (6.4)reads
normalized so that Z[O]= 1. It is convenient to shift the functional variable p(x) = po(x) +X(%) so that the new one gives the maximum of the integrand at ~ ( x=) 0. The function po(x) must obey the Euclidean analog of the EulerLagrange equations:
(6.16) This nullifies the firstorder terms in X and the integral (6.15)takes the form
where we have integrated by parts (applying the zero boundary conditions). 4We do not prove this statement here (see, e.g. [l,61). Thc reader can check it for low ordcr corrclators. Problem: Do this for up to the fourth derivative of W.
136
CHAPTER 6. THE EFFECTIVE ACTION IN
(p4
THEORY
The rest of the integral is Gaussian and does not depend on the external current:
This can be included in the normalization constant as it does not affect averages (6.3) or (6.5)  (6.7). So Z[J] = ~ [ d e t (  a :
+ m')]'"
exp
{i1
d4~J(x)(po(~)}
The analogy between the present calculations and our consideration of the harmonic oscillator in quantum mechanics is evident.
The Classical Green Function It is convenient to express the solution of equation (6.16) in terms of a Green function which obeys the KleinGordon equation
G(x  X') is the scalar field Green function, which depends only on the difference X  X' because of the translation invariance of the theory. To find its explicit form we must take the Fourier transform
which gives after the Fourier transformation of (6.20)
and finally
where p is the Euclidean momentum. Note that the analytic continuation of this expression into Minkowski space (which we do not need yet) can be performed by the rotation of the integration contour about p4 by an angle  ~ / 2 so that p4 = ipo
6.2. CORRELATORS OF THE FREE FIELD
137
(this leads to the Feynman propagator, which is the most often used in particle physics.) It is an important fact that equation (6.20) has a unique solution. We recall that there are several solutions of this equation in Minkowski space, depending on the choice of the corresponding integration contour over p. which enclosed the poles of l/(pi   m'). The solution of equation (6.16) can be written in the form
where the compact notation '*' for the convolution operation has been introduced. The expressions for generating functionals (6.19) and (6.8) take the form Z[J]
=
N' exp
{lJ
d4xd4r'J(x')G(x  I) J(x)}
Here we have omitted the normalization constant N' in Z and the corresponding term  In N' in W. Now we would like to point out some useful properties of the Green function G(x  y). First, equation (6.20) means that
Here G'(X, y) is the inverse propagator which acts on any function in the functional space as follows
and particularly
Besides that, we can easily see that the classical part of action (6.14) can be represented in the form
138
CHAPTER 6. THE EFFECTIVE ACTION IN
THEORY
Figure 21
Correlation Functions Having found generating functional (6.25) or (6.26), we can find any vacuum average of the theory with the action (6.14) by functional differentiation. For example,
6.2. CORRELATORS OF THE FREE FIELD
Figure 22
It follows from the second relation that for the free theory
Although these expressions appear cumbersome, their structure is in fact transparent, enabling us to use a simple graphic representation. The Feynman diagram technique associates a graphic element with the basic objects in the integrals (6.31)  (6.33), the current and propagator (fig. 21). The propagator G(x  y) is depicted as an interval with end points at X and y. The source J(x) is denoted as a cross. The junction of two graphic elements means the convolution of the corresponding functions. The dummy variables over which the convolution integration is performed are not labeled, while the explicit arguments of the whole convolution are. These rules make it possible to manipulate expressions of type (6.31)(6.33) by means of the graphic objects. The integral corresponding to any given diagram can be reconstructed unambiguously. For example, the diagrams for equations (6.26) and for (6.31)  (6.33) are depicted in fig. 21 and fig. 22 respectively. 5Diagrams in Minkowski space are sometimes regarded for the sake of clearness as a representation of the temporal evolution. This reflccts the fact that the propagator is simply the amplitudc for the propagation of a particle from one point to another. From this point of view, the second diagram in fig. 21 describes particle creation by the external source at the point X', propagates to the point X. The third diagram in
CHAPTER 6. THE EFFECTIVE ACTION IN
140
(p4
THEORY
We note a characteristic feature of all the diagrams considered, which is seen e.g. in fig. 22 : correlations of more than two fields are absent (only pairs of points are connected by a line). Thus the correlator ((p4) can be expressed in terms of (pp). This may be regarded as a criterion of the free theory. All its correlators are expressed in terms of G and G * J . If the external current is switched off (J(x) = 0), then all correlators are zero except those which are constructed from the twopoint Green = G(x  y) only. This means that all the manyfunction (cp(x)cp(y))I particle correlators are disconnected (this is apparent if we take a look at fig. 22).
J=,
It is convenient to store the information about correlators in a more compact form, namely to deal only with the generating functional for connected diagrams and to reconstruct the rest if necessary. A suitable quantity for this purpose is W[J]. It has the form (6.26) represented in fig. 21. We see that W[J] is constructed only from the one graph which generates all the diagrams in fig. 22. It should be noted that the generating functional Z [ J ] (6.25) can be represented graphically too. Indeed, as follows from (6.31)  (6.33), the correlators are simply the coefficients of the expansion of Z[J] in a functional Taylor series:
or, graphically as shown in fig. 23.
Z[J]
=
I
+ XX
+ 1/2!
+1/3! >tJ<
W
+...
W Figure 23
fig. 21 depicts creation a t X', propagation t o X where it is absorbed by J(x). However, the consistent use of this graphic language is impossible without introducing somchow the unusual notion of travelling backward in time.
6.3. GENERATING FUNCTIONALS IN
6.3 (p4
(p4
THEORY
Generating Functionals in
ip4
141
Theory
Theory
The free field considered in the previous subsection is often not applicable to real physical situations. Usually it plays the role of the initial approximation, analogous to that of the harmonic oscillator in quantum mechanics. The simplest theory including interaction terms is (p4 theory (5.37)(5.38) in which the selfinteraction term Uint = XP4/4! is added to the Lagrangian (6.14). The Euclidean action of the model is
where
1
X
l
2
2
~ ( c p= ) m2cp2+cp4= m cp +Ui,t(cp) . (6.37) 2 4! 2 This system was considered in the previous chapter for the case m = 0. The cases: m2 > 0 and m2 < 0 are also of interest. In the first case, m2 > 0, the potential has a minimum at cp = 0 (see fig. 15). This is the average value of the field in the vacuum state. For m2 < 0 the potential U(cp) has two symmetric minima at cp = f ( 6 ~ m ' ) ~=I f~ a (fig. 15). Hence two vacuum states with the average field value cp = f a are possible. The actual choice between the two values f a may be random, or follow from the history of the system. The choice when made violates the cp tt  p symmetry of the action (6.36), (6.37). This phenomenon is called spontaneous symmetry breaking. The potential of model (6.37) coincides with that of the phenomenological LandauGinzburg theory of phase transitions if m2 is regarded as a function of temperature (m2(T)changes sign at some critical value T = T,). This is why models of type (6.36)(6.37) are widely used in physical applications.
Generating F'unctionals: Expansion in X Let us consider model (6.36)  (6.37) in the case m2 > 0  the case m2 < 0 will be examined later. We need to find all correlators of the theory with action (6.36). It is sufficient for this purpose to evaluate the generating functional
142
CHAPTER 6. THE EFFECTIVE ACTION IN ip4 THEORY
As mentioned before, there is a lack of tools for dealing with such integrals. Actually, the only thing we can do is to evaluate the Gaussian integrals considered in the previous section. We may hope that for small X all observables differ only slightly from their values in the free theory. If this is really the case then we can find them in the form of a power series in X. Let us proceed with this idea and develop a perturbation theory to evaluate Z ( J ) (6.38) for X << 1. First, we expand the exponent of Xv4/4! in (6.38):
Every term of this sum results in a PI of type (6.15) with a prefactor (xn):
exp
1 { /d4x[$(&99)2 + rn2ip2 2
 ~(x)p(x)])
As discussed above, any power of cp(x) can be obtained by differentiation of the exponent in (6.39) with respect to J(x):
Substituting this into (6.39) and factoring out the variational derivative from the integral, we obtain the following compact expression:

he accuracy of this statement is far from evident. There are theories for which it is impossible to define the coupling constant, such that it would be small for all field values under consideration. Quantum chromodynamics, thc main candidate for the description of strong interactions, is an important cxample.
6.3. GENERATING FUNCTIONALS IN
v4
THEORY
143
Figure 24
using relation (6.25). The function of the operator 6 / 6 J ( x ) is understood in the sense of the corresponding Taylor expansion. To first order in X, this formula results in
Calculating subsequently four variational derivatives with respect to J ( x ) we get
As for the free case, this expression can be represented in graphic form. To do this, we note that the Green function G ( 0 ) = G ( x  X) should be represented as a closed loop, shown in fig. 24. Expression (6.43) is depicted graphically in fig. 25. Now let us write out the expression for the generating functional to first order in X:
144
CHAPTER 6. THE EFFECTIVE ACTION IN p4 THEORY
Figure 25
The corresponding expression for W [ J ]is
It is clear that only the first term in (6.44) and fig. 25 does not vanish at J = 0. Its value is proportional to the volume S d4x r R. Along with the higher order terms, which do not depend on the current J , this is called the vacuum fluctuation contribution to the total field energy. Often only the part of the total energy which exceeds that of the vacuum is interesting. Then the vacuum part is subtracted, as was done in calculating the Casimir energy and effective potential.7 The dropping of the vacuum diagrams corresponds to the normalization Z [ J ]+ Z [ J ] / Z [ O ] . Exactly four lines are always brought to the point which corresponds to the multiplier X (such points are called vertices) as seen in fig. 25. This is a simple consequence of the fact that Uint Xp4 and enables us to omit the coefficient X/4! in our calculation. The coefficients of the terms in (6.43) and in fig. 25 are called the symmetry factors or symmetry coeficients, as they can be calculated combinatorically. To show this, we split for a while the vertex point X into four different points, X I , . . . x4, and consider the differentiation of generating functional (6.25) once more. Taking the first derivative with respect to J ( x l ) , the second with respect to J ( x z ) etc., the differentiation of exp J * G * J ) with respect to J ( x i ) gives the factor J G ( x i  yi) J(yi)dyi at each step. Besides this, the differentiation of the preexponential factor gives a term proportional to G ( x i  x j ) for each xj with j < i. It follows from this procedure that every term of the

{l
7Note that the vacuum diagrams cannot be omitted so easily in the presence of gravitation.
6.3. GENERATING FUNGTIONALS IN (p4 THEORY
145
Figure 26
expression has a unitary coefficient. The terms differ from each other by the number of propagators G, which varies from zero to its maximal value for each type of diagram. As for the terms with the same number of G's, they can be distinguished by the manner in which the propagators connect the points xi. Such terms are physically identical. Every connection variant appears only once. Now we can identify the points zi corresponding to the same vertex again. After this some diagrams become indistinguishable. The number of those is equal to the number of these identical terms, that is to the number of possible connections between the given points. To illustrate this, let us reproduce the coefficients in (6.43) and in fig. 25. The diagram without any propagators appears with unit weight. The coefficient 6 = 4.312 of the second term is equal to the number of ways of selecting two indistinguishable points from four (the two indistinguishable points are the arguments of one of the propagators). The indistinguishability of a pair of points means that each pair is taken into account only once (rather than twice). The same logic requires the coefficient 112 when selecting two pairs of points out of four points, as two loops (see fig. 25) are identical: 3 = (4.3/2)/2. The remainder of the calculation by functional differentiation of Z [ J ] is straightforward. For example, the twopoint Green function is
It is evident that the first term in (6.44) gives the free propagator G(x y) after the differentiation. Then dropping the terms which vanish at J = 0, we obtain
depicted graphically in fig. 26.
146
CHAPTER 6. THE EFFECTIVE ACTION IN cp4 THEORY
Note that we can get the same result by making use of the connected Green function generating functional W [J](6.45), rather than Z[J].
Generating Functionals: the Loop Expansion Now we consider model (6.36)  (6.37) for m2 < 0. In this case it is convenient to rewrite the potential (6.37) in the form
so that the minimum of U(p) is zero. Any attempt to apply the above method will fail because of a pole in expression (6.23) for m2 < 0. It is possible to understand the reason for this, and some other difficulties, without going into the details of the theory. Indeed the free theory (6.14) with potential U = m2cp2/2 is a very bad initial approximation because this potential looks quite different from the double well one (6.48). It would not be reasonable to expect any correct results for model (6.36), (6.48) taking so unsuitable first approximation. To improve the initial approximation it is worth while to account for more information about the structure of the theory. For this purpose we find the field configuration cpo(x) at which the integrand e  S + J * ~in (6.4) reaches its maximum for arbitrary J(x), expand the action near this trajectory and integrate over the deviation cp  cpo using perturbation theory if necessary. This technique is a generalization of the Laplace or saddle point method for the estimation of ordinary integrals (see e.q. [3]). Let us use it for calculation of Z in (6.4). As the method is applicable for the general form of the potential, we use the generic form of the action (6.36), taking (6.48) for the sake of illustration. We start with the definition of 90. In order to minimize S  J * cp, it must obey
which for the action (6.36) takes the form
where U' dU/dcp. Supposing that we know the solution cpo[J] as a functional of J, we introduce a new variable of functional integration ~ ( x =) ~ ( x ) cpo(x)
6.3. GENERATING FUNCTIONALS IN
(p4
THEORY
147
and expand the action in X : 1
1
S[cp]J*cp=S[yo]J*cpo+6SJ*X+S2~+63~+... 2! 3! . (6.51) The terms of the expansion for action (6.36) are:
and similarly for the higher order terms. For the case of (p4 theory, all higher variations of order greater than four are zero. Now we note that cpo has the property SS  J * X = 0 (see (6.49)). Therefore the first order term vanishes and integral (6.4) takes the form
Here a different representation is used to represent the action variation, in a form intermediate between (6.51) and (6.53)  (6.54). The quantity S2S/SY2is equal to S(xX') multiplied by the operator in square brackets in (6.53), and 63S/6cp3 is simply U1"(cpo). Let us consider the exponential function in the classical action in (6.55) before proceeding with the calculation of the PI. If this PI is neglected, then the expression for Z represents the generating functional in the classical approximation. Taking the first variation with respect to the current J we obtain
This is not an unexpected result. It does not however take into account fluctuations around cpo. To find the contribution of fluctuations, we must evaluate integral (6.55). It cannot be calculated exactly. Let us consider only the case when the main contribution to the integral is given by a small vicinity
148
CHAPTER 6. THE EFFECTIVE ACTION IN
(p4
THEORY
of the classical trajectory cpo(x). This is the semiclassical approximation. The conditions for the validity of this assumption will be considered later. The smallness of the considered region means that X is small compared with cpo in the region giving the main contribution to the integral. Thus we neglect all powers of X greater than two, hoping that the Gaussian integral which is left in the expression
provides a good approximation for Z . This Gaussian integral was calculated above (6.18). Thus the expression for Z takes the form
The functional determinant in this expression cannot be included into the normalization constant as was done in (6.19) because it depends explicitly on the field, and hence on the current via U1'(cpo).This determinant cannot be found for arbitrary fields. We shall consider its calculation in this and in the next chapter for fields close to the vacuum. Now we would like to emphasize some disadvantages of the proposed method for the calculation of correlators, and to improve it by the introduction of one more generating functional.
6.4
Effective Action
Consider expression (6.58). It could hardly be of practical use because it is generally impossible to express p o ( x ) in terms of J ( x ) or, in other words, to solve the nonlinear equation (6.16). Nevertheless, J can easily be found as a function of p,. It would be convenient if the field was the independent variable rather than the current. Let us attempt to engineer this. It could be two different values of this functional variable: po or (cp) determined by (6.10). The latter is preferable because it is an observable quantity, in contrast to cpo. The fact that ( p ) can be expressed in terms of the functional derivative of W with respect to the current, makes it possible to apply the standard method of Legendre transformations to construct the quantity depending on the field. The term 'the quantity depending on the field' should probably be explained in more detail. The field and the current are dependent on
6.4. EFFECTIVE ACTION
149
each other by virtue of equation (6.16). An arbitrary functional F can be formally represented to depend either on the field, or on the current, or even on a mixture of both. In the latter case its differential is
where the field is assumed to be an independent variable. We say that F depends on the field if the right hand side of expression (6.59) does not contain S J / S p or, in other words, its value does not change as the functional dependence J [ p ]changes8 Now let us introduce a new functional J? by means of the functional Legendre transformation:
q v ) ]= W [ J ]+ J d 4 x ~ ( x ) ( p ( x )=) W [ J ]+ J ( x ) * ( p )
.
(6.60)
This is an immediate generalization of the ordinary Legendre transformation used in classical mechanics. The functional l? is constructed so that its derivative SI'ISJ (6.60) be zero. It follows from this that F depends on the field ( p ) and, therefore, it is reasonable to express the functional in terms of ( p ( x ) ) .It follows from (6.60) also that
One more variation with respect to the field gives
according to (6.29). Indeed,
Now let us find I'[(p)]in the saddle point approximation. We start with expression (6.58) rewritten in terms of W [ J ]as
'The geometrical meaning of the Legendre transformation is explained in the book [41.
CHAPTER 6. THE EFFECTIVE ACTION IN
150
(p4 THEORY
Here the Planck constant h is r e ~ t o r e d As . ~ is intuitively evident (it will be proven in the next chapter) this approximation is valid if the dimensionless parameter = (characteristic value of action)/h >> 1. The corrections to the classical part (which is the first two terms on the r.h.s. of (6.64)) are called quantum corrections, as suggested by the appearance of the Planck constant. According to (6.60) we obtain from (6.64):
The second term in this expression is of the same order in h as the last one, because (p) differs from po only if quantum corrections are taken into account. Thus m the relation
is valid to order 0(h2), we get finally
This is the desired expression for the effective action in the oneloop approximation. Let us sum up the properties of r[(p)] which we have found so far:
r[(p)] is a functional of the average field ( ~ ( x )which ) is an observable quantity; a
a
r[(cp)]determines the average field evolution in space and time (see eq.(6.61)) in the same way as the classical action does for the classical field evolution. To realize this it is sufficient to compare (6.61) and (6.49). For J = 0 we have the evolution of the unperturbed selfinteracting field; r[(cp)] coincides with the classical action S[p]in the limit h (see eqs. (6.56) and (6.67)).
+
0
'All the h's in the integral over X are absorbed by the normalization factor, as in
(6.18).
6.4. EFFECTIVE ACTION
151
These properties explain the name of the functional r[(p)]  the efective action. The term 'effective' implies that r[(cp)] contains all quantum corrections and 'action' emphasizes its role in the dynamics. It follows from the relation between l? and W that the effective action gives complete information about all correlators, that is it provides the full description of a quantum system. It is clear that such a fundamental quantity cannot be calculated exactly for a general case1'. Below we consider a method of calculation of I'[(q5)]. To simplify the notations we shall omit the angular brackets writing simply l?[$].
Expansion of the Functional Determinant Even the first quantum corrections to the effective action are difficult to find, because the functional determinant in (6.67) cannot be calculated for a generic field. However we can easily evaluate it for a spatially constant field cp = q5 as was done in the previous chapter. Let us now find r[cp]for an arbitrary field in the small vicinity of q5. We shall represent the , q5 is a constant field (not necessary field in the form cp = q5 ~ ( x )where small) and ~ ( xis) a small coordinatedependent quantity. The expansion U1'(cp) in (6.67) in powers of X reads
+
u ~ P ( x ) )= U"(+)
+ u(~~)x(x)+ 51~ Z v ( q 5 ) ~=( ~on($) ) 2 + u.,(x(x))
.
(6.68) This is an exact relation for the potential cp4 (6.48) for which
In other theories the expansion may need to be continued. It is useful to rewrite lndet in (6.67) in the form lndet A = TrlnA, the trace of the logarithm of A. This relation is apparent for finitedimensional matrices which can be diagonalized. The determinant of such a matrix is the product of the corresponding eigenvalues and Tr In A is the sum of their logarithms. Actually, it is not necessary to diagonalize the matrix A because both the trace and the determinant are invariant 1°A specific cxprcssion for I' may be extremely complicated. Even the argument (9)may turn out to be inadequate for a real system. For example, the simplest macroscopic description of a superconductor is realized in terms of Cooper pairs of electrons. At the same timc, at short distances individual electrons must be observable. A similar, but more complicated situation arises in quantum chromodynamics where baryons arc observcd at large distances and the quarks they consist of at short distances.
152
CHAPTER 6. THE EFFECTIVE ACTION IN (p4 THEORY
quantities of unitary transformations. Of course, the generalization of these quantities to infinitedimensional matrices (i.e. for the linear o p erators in (6.67)) turns out to be nontrivial. It is assumed in this book that there exists a regularization making the matrix finitedimensional. All calculations are carried out with the regularized matrix, after which the regularization is removed. We consider this procedure in detail in the next chapter. Here we merely suggest that the formal expressions like det A are welldefined. The quantum corrections to the effective action (6.67) can be rewritten in the following form
The form and the properties of the Green function introduced here coincide with those of (6.20)  (6.36) up to the substitution m2 + U1'(q5), as $ is constant. It may appear attractive to apply the formula ln(AB) = 1nA In B to the last expression in (6.70). However, it is incorrect for noncommuting operators A and B. The situation is saved by taking the trace: Tr ln(AB) = Tr ln A Tr In B. This can be understood if we notice that for noncommuting A and B ln(AB) differs from In A In B by an infinite series consisting of the commutators of A and B of increwing complexity. The trace of any commutator is zero because trace is invariant under cyclic permutations. Thus
+
+
Now, using the expansion ln(1
+
+ X) = X  ;x2 + 5x3 + . . ., we obtain
The convolutions appearing here are more detailed representations of the nth power (G * Uint)" of the operator G * Uint for n = 1,2,3. . .. We
6.4. EFFECTIVE ACTION
Figure 27
can easily write down the explicit form of these expressions in the coordinate representation, where each operator is determined by its kernel, depending on two variables G(x  y) and Uint(x(x))G(x y). The kernel of a product of operators is a convolution of the kernels of each factor, and taking the trace closes the chain of operators to form a ring. Thus the explicit form for, say, the third term in (6.72) is
To deal with such integrals it is convenient to use the language of Feynman diagrams. The graphic representation of expression (6.72) is depicted in fig.27 where the bold points denote Uint(x). Thus the problem of finding the effective action is solved, in the sense that it is represented as a sum
where the classical action depends on the constant field component, and the correction rt,is a functional of the small variable component ~ ( x ) . U"($)] as for the parameters entering depends on $ via Tr ln[d2 the expression for the propagator. The classical action reduces to S[$]= W ( $ ) because $ is constant (recall that R = J d4x). Together with U"($)] contained in !?E, it gives the effective the term (1/2)Tr ln[d2 potential Ueff(4) found in the previous chapter. The rest of rh is represented in the form of the expansion in powers of Uint(x(x)) given in (6.72) and fig. 27. It can easily be rewritten as a series in X because Uint(x(x)) contains only X and x2. Such a series is
+
+
154
CHAPTER 6. THE EFFECTIVE ACTION IN
(p4
THEORY
Figure 28
the standard form for the effective action. In the case of constant (and not small) 4 we have
Here l?(n),which is called the vertex function or just the vertex, depends on n spatial points and the nth term of the expansion is
In order to find the explicit form of the first terms of this expansion, we must substitute the expression for Uint (6.69) into (6.72). Then the nth term of the series is a sum of n + l terms with the power of X ranging from n to 2n. To depict this graphically, we take in correspondence to each term in Uint its graphic symbol as shown in fig. 28. Then the final expansion for the effective action takes the form graphically shown in fig. 29. Let us discuss the structure of the first terms in r[v]in view of expansion (6.75). r(l)is
Here the first term contains the external current J(+), which determines the field $ according to (6.61). The second term corrects it to J(p). The to (6.75) vanishes if we define the decomposition of contribution of )'(?l the total field cp into the constant and variable components 4 and ~ ( x ) , so that R4 = J d4xcp.Then J d4xX(x)= 0. In other words, X does not contain the Fourier component with zero wave number. Note that in this case, one of the infinite number of degrees of freedom of the field cp(x) is ) particular, this assigned to 4 and all others are accumulated in ~ ( x In
6.4. EFFECTIVE ACTION
r[@]= S[@]+
45 Trln (I + G *UinJ= S[@]
Figure 29
+
decomposition implies that variation with respect to cp = @ X means differentiation with respect to @ and variation with respect to ~ ( x ) . The second term of expansion (6.75) contains the inverse propagator of the field X , as follows from (6.62). It takes the form
The first term here is the classical one. Similar terms containing U"'(@) and U" appear also in the expressions for 1'(3) and respectively. The ~ ( ' 4 ) 'with s n 2 5 are entirely due to quantum corrections. Note that the correction (6.76) in (6.75) belongs to the the nth order of perturbation theory in X. However the graphs for all r(")'s look like closed loops. We shall consider the relation between perturbation theory and loop expansion, as well as that of I'[cp] and W [ J ]in , the following chapters where the explicit form of the oneloop diagrams will be obtained. A common feature of the diagrams in fig 29 is that they are oneparticle irreducible, that is none of them can be broken into two disconnected parts by cutting one line. This property holds in all orders of the loop expansion. We shall discuss it in the chapter 9 considering the exact Green functions. Thus F is the generating functional of oneparticle irreducible diagrams. The method of computing the quantum corrections considered here is applicable in principle to expansions around any given field configuration
156
CHAPTER 6. THE EFFECTIVE ACTION IN
(p4
THEORY
4(x) which may be spacedependent. Typically, such a configuration is called the background field. The difference from the considered case is that flueffin (6.75) should be replaced by S[+],which has in general no relation to the effective potential. G(x  y) in (6.78) should be replaced by G(x, y) calculated for the given background field. The fundamental difficulty of this method is that the form of the propagator in the case of the variable background field is difficult to find. Only a few known background field configurations are able to be simply calculated. Let us mention in conclusion that there is one more way to expand the effective action for the case of fields (not necessary small) slowly varying in space:
Here Z contains the term S,,S(x  y) which is the contribution of the classical action. This expansion, which is called the gradient expansion, can be rewritten in a simpler way in momentum space, where it is an expansion in powers of the wave number p. Convergence of the gradient expansion should be controlled in each specific case.
Bibliography [I] The LSZ reduction formula is derived in many textbooks, see, for example, C. Itzykson, J.B. Zuber, Quantum Field Theory, McGrawHill, 1980. [2] In the framework of the P I formalism, the LSZ formula is derived in L.H. Ryder, Quantum Field Theory, 2nd ed., Cambridge University Press, Cambridge 1996. [3] J. Mathews and R.L. Walker, Mathematical Methods of Physics, 2nd ed., Benjamin/Cummings Pub. Co., 1970. [4] V.I. Arnold, Mathematical Methods of Springer, 1980.
Classical Mechanics,
[5] E.J. Weinberg and Aiqun Wu, Phys. Rev., D36 (1987) 2474. [6] J. ZinnJustin, Quantum Field theory and Critical Phenomena, 3rd ed., Oxford University Press, Oxford 1996.
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Chapter 7 Renormalizat ion of the Effective Action In the previous chapter, we derived the effective action of (p4theory (6.36) in the rather symbolic form as an infinite sum of diagrams (equations (6.75)  (6.78) with the graphical representation given in fig. 29). In this chapter, we shall learn how to calculate the contribution of the diagrams. We shall analyze the explicit form of the corresponding integrals, some of which will appear to diverge, and find out the source and the structure of the divergences. After the renormalization, we shall get a finite result for the oneloop propagator and discuss the structure of the next terms of the effective action which apper in fig. 29.
7.1
Momentum Space
It is convenient to use the momentum representation for explicit calculation of the Feynman diagrams.' To this end, let us introduce the fourdimensional Fourier representation of the field ~ ( x ) :
'The quantum mechanical term 'momentum space' for the Fourier representation is due t o the fact that X is the wave function of the field quanta with definite momentum (see, chapter 4 or [5]).
160
CHAPTER 7. RENORMALlZATION
Since ~ ( xis) real, X(k) = x*(k). Sometimes, ~ ( xis) represented in the explicitly real form
An important property of Fourier harmonics is their orthogonality:
Let us substitute (7.2) into the classical action expanded as a power series , to (6.51): S[4 ~ ( x )= ] S[4]+ S S S2S S3S S4S. in ~ ( x )similar ) at k = 0. We set this The first term here is proportional to ~ ( k taken component to zero, as explaned after equation (6.77). Using the notation U''($) = m2, we find the following expression for the second order terms (6.53):
+
+
+
+
Performing the integration over k', we obtain
The kernel in (7.5) is simply the Fourier transform of the inverse propagator Gl (X, X') (6.27):
+
Gl (k, k') = ( 2 ~ ) ~ 6 ( kk')(k2
+ m2)
.
(7.7)
The measure of the integration over k contains the factor l / ( 2 ~ i )in~ troduced in eq.(7.2). Thus the symbolic equality G * G' = 1 reads explicitly d4k' G(k, kl)G'(k', k") = S(k  k") . (7.8) Here, the propagator G(k, k') takes the following form, in accordance with (6.23): 6(k k') G(k, k') = k2+m2 .
+
7.1. MOMENTUM SPACE
161
As the quantities (7.7) and (7.9) actually depend on just one argument, it is more convenient to use the notations
We shall call them the momentumspace propagator and inverse propagator respectively. Now let us express the third and fourth variations of the action in terms of the Fourier transform ~ ( k ) :
The regularity of the structure of expressions (7.5),(7.11) and (7.12) can easily be understood. It should be noted that the presence of a 6function of the sum of all momenta follows from the translational invariance of the theory, and that of the background field 4. Indeed it is the fact that U"($) does not depend on coordinates that leads to the diagonalization of the second variation of the action which, in turn, results in the obtained simple expressions for 6% and b4S. Let us show that it follows from the translational invariance of the theory that not only the action but also an arbitrary term of the Taylor expansion of the effective action (6.75) has the form
analogously to (7.5), (7.11) and (7.12). For this purpose we displace the ) X ( X + a ) everywhere. system by the vector a, that is we replace ~ ( xwith
162
CHAPTER 7. RENORMALIZATION
+.
The integrand in (7.13) acquires a phase factor exp{i(kl . .+k,)a). The translational invariance means that the effective action does not change. As X is arbitrary, this is possible only if kl+. . .+k, = 0 everywhere within the integration area, giving 6(kl . . . k,) in (7.13). All specific features of a given field system, besides translational invariance are contained in I'(,)(kl, k2,.. . k,; 4). Therefore these functions are the subject of an interest. They are usually called the vertex functions or vertices.
+ +
Explicit Form of the Diagrams Let us turn to the sequence of oneloop diagrams making up the first quantum correction to the effective action (see fig. 29 and expressions (6.77) and (6.78)). Taking the Fourier representation of all quantities in the analytic expression of the diagrams, we can easily obtain the contribution to the effective action in the form (7.13). We first consider the simplest graph (6.77) in fig. 29, which is often called the 'tadpole ':
where
An analogous procedure applied to the first term in (6.78) (this is the third diagram in fig. 29 usually called the 'Jish') results in:
7.1. MOMENTUM SPACE
Figure 30
where
It is clear that integration over the spactial variables X,y, . . . , corresponding to each vertex of an arbitrary diagram gives a Sfunction of the sum of all momenta as in previously considered cases. To find the sign of a momentum appearing in the argument of the 6function, we use the general form of the Fourier transform of the propagator as a function of two arguments:
Then all momenta in the exponent here and in the eq.(7.2) have the coefficient f i . These are incoming momenta. Analogously, momenta having the coefficient i are outgoing (they are absent in the considered diagram). As G(pl, p,) 6(pl p,) (see (7.9)), the integration over pz leads to the substitution p2 + pl at one of the vertices connected by the propagator. Thus the same momentum p1 corresponding to the propagator is incoming at one vertex and outgoing at another one. This is illustrated by arrows coming to and going from a vertex. The diagrams obeying (7.15) and (7.17) together with this additional rule are depicted in fig. 30. Note also that k2 = kl because of the Sfunction in (7.16). In the following calculations, we shall concentrate on the integrals which are the convolutions of the Green functions without any further
+
C H A P T E R 7. RENORMALIZATION
Figure 31
coefficient. Let us denote them as A("),where n is the number of vertices. Each diagram is a product of one of the A(") with the vertex factors (which are X X 2 or XaX) and the common coefficient. For example, equations (7.15) and (7.17) can be written as C(') = XA(l)/2 and C(') =  ~ ' q $ ~ A ( ' ) / 2respectively. The explicit form of A(") can easily be sketched:
1 [(P +
+ m2]. . . [ ( p+ kl + k2 + . . . kn_')2 +
m 2 ~'
A(") is a symmetric function of its arguments, subject to the condition that the deltafunction S(kl + kz + . . . + k,) is included in the integration. A(") actually depends only on n  1 arguments, but we shall keep the notation (7.19) which better reflects the graphic structure of A("). The only exception is A('),which is a constant with no argument. , expansion displayed in fig. 29 contains two more Besides ~ ( ' ) ( k )the diagrams of the same type, which differ by the number of external legs. They can be expressed in terms of A(2)( k l ,k2 k3) and A(2)( k ~ k~, k3 k4). In concluding this section, let us write out the first terms of expansion (6.75):
+
+
+
The term 0(fi2) in the last expression is rather symbolic because we can suppose f i = 1. It implies the smallness of the second order in
7.2. THE STRUCTURE OF ULTRAVIOLET DIVERGENCIES
165
the quantity which plays the role of the loop expansion parameter when h = 1. As will be shown below, this is X. The sum of all corrections in (7.20) 0(h2) (7.21) C C(1)
+
+
is called the selfenergy part (of the propagator). The threepoint vertex function has the form
The fourpoint vertex function is
and similar for higherorder terms. The obtained expressions are still only symbolic because of the divergence of A(') and A ( ~ at ) p + oo. This fact is by no means surprising for us, as we have already encountered such divergences in calculating the effective potential in chapter 5. It was shown that the effective potential becomes finite after renormalization of the mass and coupling constant. In the next sections we shall study the structure of the divergences, regularize the divergent integrals, and renormalize the parameters in the classical action.
7.2
The Structure of Ultraviolet Divergences
Let us consider the divergent terms in the effective action shown in fig. 29 and expressed by formulae (7.20), (7.22) and (7.23). Only the contributions of A(') and appear to be divergent. All other oneloop diagrams containing three or more propagators converge. An important fact is that A(l) and the divergent part of A(2),denoted as Ag), do not depend on external momenta. This is obvious for ~ ( l ) The kindependence of )A : can be shown by an expansion of the integrand in (7.19) for n = 2 at p + oo. The leading term of this expansion has the form l/p4 and does not depend on k. However, we must be careful when operating with infinite expressions, so we take the derivative of A(2)with respect to k for a more rigorous proof:
d
p+k r ) 4 [p2 m2][(p k)2
+
+ + m212
.
CHAPTER 7. RENORMALIZATION
166
The convergence of this integral proves the statement. Analogously, we ): does not depend on the value of 4 entering the mass can show that A definition m2 = U"(4). Substituting the divergent constants A('), Ag) into (7.20), (7.22) and (7.23), we obtain
+ I [h
+
?X2A(2)])x(x)% 3! 2 finite terms.
(7.25)
This shows that 0
0
Divergences appear only in the coefficients of x2,x3 and x4. Strictly also diverges, but the chosen definition speaking the coefficient of of X provides J d4x X = 0; Expression (7.25) contains only two types of divergences because the structure of the divergent coefficients of x3 and x4 is the same.
Problem: Show that if Jd4x ~ ( x # ) 0, then the structures of the divergences in the coefficients of X and x2 are identical too. Recall that it is the effective action, rather than the classical one, that describes physical processes. It is clear as well that there is no experiment which could measure separately different terms belonging to the same coefficient. Therefore the quantities m;,,,,
XphYs
=
m2
=
X
1 1 + AA(')  (X+)~A:) + finite terms, 2 2 J X2AE) + finite terms, 2

(7.26)
7.2. THE STRUCTURE OF ULTRAVIOLET DIVERGENCIES
167
rather than m and X are really measurable. Like all observables, these must be finite quantities. Hence the parameters m2 and X entering the initial functional integral (6.38) must diverge, in order to cancel the infinities in the effective action (7.25). Such quantities are called bare quantities. Below we assign them the subscript '0' in order to distinguish them from the corresponding physical quantities having no subscript. It is convenient also to introduce the differences 4 X0 4 X 1 2 X 2 = m2 1  1 2 2 5 m X and X = X (7.27) Sm 2 2 4! 4! 4! which are called counterterms and should be determined to the same accuracy as all other quantities. Thus they are equal to zero if quantum corrections are neglected. In the oneloop approximation, one should know the counterterms to the first order in h. The fact that the effective action (7.25) is a generating functional guarantees the absence of divergences in the expression for any correlator, provided the result is expressed in terms of m2 and X. The procedure for the determination of the counterterms and reexpressing all results in terms of the physical parameters is called renormalization. It should be noted that in order to compute the finite terms in (7.25) we must define the difference of infinite quantities in a mathematically rigorous manner. This can be done by a choice of the regularization of the theory, one of many variants of which will be considered in the next section. Thus we conclude that it is sufficient to use two counterterms for the renormalization of cp4 theory at the oneloop level. It is known that the number of counterterms necessary at any order of the loop expansion is not larger than three [6]. Thus the (p4 model belongs to a class of theories which require a finite number of counterterms. Such theories are called renormalizable. Let us show that theories with the selfinteraction term cpn do not possess this property for n > 4 in fourdimensional space. As in the cp4 model, in such theories there are only two divergent diagrams ) logawith one loop: A(') (diverging quadraticaly) and A ( ~(diverging rithmically). However the maximal number of legs attached to diagram A(2)is now 2(n  2). To cancel the corresponding divergence, we have to introduce counterterms up to (p2(n2). The field exponent in this term coincides with n only if n = 4. When n > 4 then 2(n  2) > n. For example, in (p5 theory, a divergence cp6 appears, requiring a cp6 counterterm. Taking this into account in the higher terms of the loop expansion leads in turn to divergences corresponding to greater n. The set of all counterterms appears to be infinite, although it is finite at each order of the loop expansion (or perturbation theory). a X 4
168
CHAPTER 7. RENORMALIZATION
Problem: Prove (to one loop) that in 1+1 dimensions a theory with an arbitrary selfinteraction potential U(cp) is renormalizable. When analyzing the divergence structure in (7.25), we did not pay attention to the effective potential, which should be renormalized too (see chapter 5). Let us fix an arbitrary field value 4o = const and expand the effective potential in a power series in 4  40. We can easily check by comparison of expressions (5.64) and (7.25) that the divergence structure of Ueff(4) coincides with that found in this chapter. This follows from the fact that the divergent part of the effective action does not depend on momenta. The only difference between these two cases is that Ueff (4) depends on the field component corresponding to zero momentum, which is absent in ~ ( x )Therefore there is also a term ~ " ' ( 4 ~ ) A ( ~)40)/2. (4 One more specific term in the expansion of the effective potential is Ueff(4o). This is simply the vacuum energy density of the background field 40. It should be regularized and subtracted as was done in chapter 5. The fact that in order to cancel four divergences in the coefficients of X , . . . x4 it is sufficient to renormalize only two constants, is by no means accidental. It can be made obvious by expanding the effective action at 4o = 0. In this case only the even powers of the field appear and the total number of divergences is three: the zero point energy which is constant, and the coefficients of 42 and 44. The divergence structure obtained by the expansion at an arbitrary point 4o is a result of the rearrangement of expansion terms after the substitution 4 + 4 X. It should be pointed out that the expansion at +o = 0 is not applicable for practical purposes because all the diagrams starting at A(2) diverge at p = 0. These are called infrared divergences. As was shown in chapter 5, this results in a logarithmic singularity of the type 44In+ in the effective potential. In particular, this makes it impossible to define the selfinteraction constant, as X = U" (0).
+
7.3
PauliVillars Regularization
The first step in the systematic elimination of the infinities is to make a regularization of the theory. We dealt with divergent integrals in chapter 5, where the simplest regularization scheme was used the upper limit of the integration over momentum was replaced by a large but finite value km,, = A. This was the most straightforward and natural way but it was not particularly convenient to carry out. Here we consider a 
7.3. PA ULIVILLARS REG ULARIZATION
Figure 32
more analitic method of PauliVillars regularization. Let us add a term describing another scalar field @ to initial action (6.36). Let the field @ has a very large mass M :
If we try to compute the corresponding generating functional we must also perform the integration over the field @. The second variation of the action near the configuration cp = cp(x), @ = 0 is
The result of the Gaussian functional integration over the field cp is the functional determinant (6.58) to the power 112. Let us assume that the field cP obeys hypothetical integration rules which lead to a functional determinant of the type (6.58), but to the power +1/2. Then the regularized effective action (6.67) reads
1 r[p] = S[cp] S R [ ~51n det
+

+ [g+ U"(cp)] 1 In det [a: + UN(p)+ M'] + 0(h2). 2

(7.30)
Now each term in the expansion of the first determinant corresponds to an analogous term in the expansion of the second one (with M 2 added in the propagator) taken with the opposite sign. Thus instead of the 'It should be emphasized that this method is by no means unique. There are a number of alternative regularization schemes. The most popular are dimensional regularization [l]and regularization on a lattice [2].
CHAPTER 7. RENORMALIZATIOfl
170
tadpole diagram in fig. 30 we get the diagram difference shown in fig. 32, where the dashed line corresponds to the PauliVillars field. Explicitly, the regularazed diagram takes the form
The contribution of the PauliVillars field cP softens the asymptotic behavior of the divergent diagrams at large momenta. The quadratic divergence cancels in the difference of two tadpoles (fig. 32), while the logarithmic divergence persists. The fish diagram in fig. 30 becomes finite. All finite diagrams do not change their values in the limit M t cm because M only appears in the denominators of the corresponding integrands. Taking the limit M + cm is referred to as taking 08the regularization. The PauliVilars fields are called sometimes the regulators. The regulators' contribution to the divergent diagrams is negligible for any finite integration range, but for p  M this contribution regularizes the divergence. In order to make all divergent diagrams finite, it is necessary to add a few more regulator fields cPi with masses M,:
A coefficient Ci is assigned to each @i. C, determines the power of the determinant produced by the integration over cPi. Then the sum of all contributions to the tadpole leads to the integral
Expanding the integrand for Iarge p we get
To make the integral convergent, we require ==l i
and
C C ~ M :. = ~ ~(7.35) 2
7.3. PAULIVILLARS REGULARIZATION
171
Under these conditions, the integral (7.33) behaves as J d l ~ I I I p 1at~ p + oo and is obviously convergent. Thus we conclude that the contribution of the regulator compensates for all divergences. Note that although the regulator integration rules look somewhat artificial, we can regard them as a rough illustration of one of the most fruitful ideas of contemporary high energy physics. This idea is based on the fact that fermion fields really give the minus sign at Trln Gl (this will be shown in chapter 11) although the explicit form of the propagator is different. One of the important research areas in particle physics today is the attempt to construction a theory free of all ultraviolet divergences, as the result of cancellation of bosonic and fermionic contributions. For this to occur, there must be a special relationship between the numbers of bosons and fermions (as well as between their charges and other quantum numbers). These are present in theories which are invariant under a special transformations which transforms bosons into fermions and vice versa. This property is called supersymmetry. The modern approach to supersymmetry is described in many textbooks, for example [3].
Calculation of Integrals For the subsequent calculation of diagrams, it is useful to find the regularized form of the integral
For z = 1, it corresponds to the tadpole diagram and for z = 2, to the fish diagram with zero external momenta. For z > 2 it converges and does not require any regularization. Using the PauliVillars regularization, we must compute differences of type (7.31), but it is more convenient to operate with simpler expressions like (7.36). We could find the regularized expression for (7.36) by making use of an intermediate regularization, say the cut off regularization, and then taking off the regularization in the difference Iz(m2) C,Iz(m2 M:). Here, though, we choose a more elegant and instructive way. Let us calculate the integral Iz(m2)for z > 2, where it converges, and then continue it analytically to the whole complex plane of the variable z. The analytic continuation of a difference is unique and coincides with the difference of the analytic continuations of each term. Therefore the CiIz(m2 difference of the analytical continuations of Iz(m2) and M?) coincides with the regularized integral (7.36), because the latter is convergent in the points of interest z = 1 and z = 2.
xi
+
xi
+
CHAPTER 7. RENORMALIZATION
172
To compute integral (7.36) for z > 2 it is convenient to choose spherical coordinates in Euclidean fourdimensional space, so we need an expression for the total solid angle 0 4 . It was calculated in chapter 5 (see (5.55) in a space of arbitrary dimension n:
and in particular, Cl4 = 2n2. Now we write integral (7.36) for z > 2 as
2n2
03
J (W4,
I, (m2)=  p3dp where t
= p2/m2.
1
dtt
(m2)2z
v J m 0
,
(7.38)
Using the definition and properties of the Bfunction
we obtain I z ( m 2 )in the form
the analytic continuation of I, (m2). Let us now evaluate the integral at z = 1
+ E where E ,0.
This diverges as E + 0. The regularized expression reads
7.3. PAULIVILLARS REGULARIZATION
173
Condition (7.35) imposed upon the coefficients Cicancels all the singular terms and we get
Expanding the logarithm for M: >> m2 using condition (7.35) and denoting C CiIn M: 5 lnA2, we finally obtain the expression which depends 2
explicitly on the regulator masses:
Now we find Iz(m2)for z = 2

+ E. In this case
1 (l~lnm~~)+O(~) &16.rr2
.
The regularized integral is
where condition (7.35) and the definition of the parameter A have been used.
About Dimensional Regularization The idea of the substitution of the divergent integrals for their analytic continuations with respect to a certain parameter is actually an independent method of regularization. It is widely applicable for the evaluation of differences of divergent integrals. Such differences appear only in the
CHAPTER 7. RENORMALIZATION
174
final result e.g., 6I +Sm2, ,(4)++SX (compare with (7.25) and (7.26)). It should be pointed out that analytic continuation leads to a finite result in any case, even if the theory is nonrenormalizable, but it has no physical sense then. Usually a method which is close to one expressed by (7.36) is used in the diagram technique. It is called dimensional regularization. All divergent integrals are evaluated in a space of noninteger dimension 4E. This means that we must take the spherical coordinate system in kspace with volume element R4,L3'dk. The diagrams are expressed in terms of B or l?functions (for a detailed analysis of dimensional regularization see, for example, [l]and references therein).
7.4
The Regularized Inverse Propagator
Let us now find the explicit form of the regularized inverse propagator (7.20). There are two integrals that contribute to the propagator: the ( k , which can be tadpole diagram (fig. 30) given by (7.36), and ~ ( ~ ) k), reduced to the form (7.36) by Feynman parametrization based on the identity (See Appendix A.5)
+
1
1 S(x1 . . .X,  1) = (n  l)! dxldx2.. . dx, a l a 2 . .. a n n (a1x1+ ~ 2 x 2 . . . anxn),
J
+
'
(7.47)
This gives for n = 2 1
=
a142
1
Jdx
1
[alx
+ ~2.1 x)I2
(7.48)
(This relation can easily be checked by immediate integration). Using (7.48), we can write I(k) in the form: 1
k)
=
1 J dx J ( 2 7 ~ )[p2 ~ + m2 + ((k + p)2 + m2  (p2 + m2))xI2 n "
'
(7.49) It is straightforward to transform this integral to the form
where p' = p 12(k2x(l X)
+ +X. Thus the inner integral in A(2)(k,k) is reduced to + m2) which is defined by (7.36). The regularized form
7.4. THE REGULARIZED INVERSE PROPAGATOR
175
of this integral is given in equation (7.46). This results in the following k): form of A(~)(Ic,
The integral over
X
can easily be calculated:
= 2+ln(a21)+a1n
a+l al'
+
where t = 22  1 and the parameter a = 1 4m2/k2 > 0 have been introduced. Thus taking into account (7.51)' (7.52) and restoring the interaction constant Xa at the vertices, we finally obtain the fish diagram contribution (7.17):
Collecting the loop corrections (7.53) and = X11(m)/2 given by eq. (7.15), we obtain the following expression for the regularized inverse propagator (7.20) as a function of k2:
+
Recall that m; = m2 6m2 is the bare mass squared. The difference between m and m0 is of order O(ti), so it can be neglected in the
CHAPTER 7. RENORMALIZATION
176
quantum correction (it should, though, be taken into account at the next order of the loop expansion). Therefore only m appears in the expression for C('). A proper renormalization of mo in the leading term in (7.54) requires an exact definition of mass. In order to give one, we shall go back to Minkowski space where the real particle exists.
Analytic Continuation to Minkowski Space We return now to Minkowski space. In order to do this, we must analitically continue the obtained expression (7.54) to the imaginary axis of x4: x4 = it. This implies a rotation of x4 in its complex plane: x4 + 1x41eia where a is the angle of the rotation3. The values a = 0 and a = 7r/2 correspond to Euclidean and Minkowski spaces respectively. In order to keep all Fourier integrals convergent, we have to rotate simultaneously the integration contours in the complex plane of the variable k4. This is done by a rotation through an angle a: k4 = lk41eia. After this a (i = 1,2,3) is replaced vector k squared in Euclidean space k2 = ki by ki k: k&, which is minus this vector k squared in Minkowski space.
+
+ +:
The Classical Propagator Let us consider the analytic continuation of the propagator (6.23). According to the proposed recipe, we obtain
Traditionally the exponential function in this expression is written in the form ecibX"where kpxp = koxO kixi, and the values xO= XO, xi = xi, kO = ko, and ki = ki, correspond to the time, length, energy, and momentum respectively (this correspondence is shown below). In order to write equation (7.55) in this notation, it is sufficient to change the sign of the integration variable ko. Then we obtain
During the analytic continuation, the contour of integration may not cross the singular points of the integrand. Thus this continuation followed 3Note that the correct sign in front of it is Fixed by the sign of the wave function evolution in quantum mechanics: $ cc ePiEt. This must correspond to the Euclidean ~ ~ ~ . form e
7.4. THE REGULARIZED INVERSE PROPAGATOR
177
Figure 33
by the substitution ko + ko defines uniquely the integration contour in as depicted in fig. 33 left. the vicinity of the poles at ko = f The same integral is obtained if we make an infinitesimal displacement of the poles up or down from the real axis and integrate entirely along the axis:
JG
where E = +O (fig. 33 right). In order to attach a physical meaning to the poles of the propagator, ko = f let us perform the integration over ko in (7.57). For xO= t > 0, we close the integration contour in the lower halfplane, and
JG,
the integral is determined by the residue at
h,= Jfm2:
JG.
This is simply the propagator of a scalar field where w = quantum (oneparticle excitation, chapter 4). In a box of side L, this function takes form:
where ~ ( xt ), = G e i ( "  w t ) is the normalized solution of the KleinGordon equation. This solution describes the field oscillation mode which
CHAPTER 7. RENORMALIZATION
178
corresponds to one particle. The particle wave function depends on time through the factor ei"t. Thus W is the particle energy, and we conclude that the position of the propagator pole at k2 = m2 determines the mass of the particle. This conjecture has been derived independently of the explicit form of the propagator in momentum space. Only the structure of its poles was important. Therefore the conjecture is also valid for the exact propagator
G. The Oneloop Propagator Let us find now the explicit form of the propagator in Minkowski space, substituting k2 = Ik21ei?r into (7.54). Then,
where
a = arctan
4

1
using the fact that the phase of logarithm argument in the numerator is a and that of the denominator is n  a. Hence the regularized propagator (7.54) takes the form
This expression is real at l k21 < 4m2. If the external momentum lkI2 > 4m2, then C(')( Ik21) here acquires an imaginary part. The quantity k = 2m is called the twoparticle threshold (of the reaction). Let us clarify this point, at last on a naive level. Suppose that there is another scalar field B interacting with cp, and having the mass M.
7.5. RENORMALIZATION Problem: Write the action of such a theory. The quantity C(')(k2)is a particular contribution to the proper energy part of the B particle propagator. At M < 2 m the propagator has an imaginary part, shifting the pole away from the real axis. This means that the amplitude of propagating B quanta, which is proportional to Im c ( ~ ) (  M z ) describing ~} attenuation acquires the factor exp of the oneparticle amplitude. The reason for this attenuation is that the particle can decay into two pfield quanta if M > 2m. The same conclusion could be reached in a more rigorous way, based on the relationship between Im and the amplitude for the creation of two p quanta. Such relations, called dispersion relations, can be derived from the analytic properties of the propagator and other amplitudes. To do so, however, is beyond the scope of this book (see [7]).
{
7.5
Renormalization
Renormalization of Mass Let us return to the physical mass m in expressions (7.54) and (7.20). According to relation (7.26), m cancels some divergent terms in the effective action, together with some of the finite terms. Other finite terms depend on the exact definition of m. This definition follows from the physical context of the problem, two main possibilities for which are discussed below. In particle physics it is most natural to define m via the relation w2 = i2 +m2.This can be regarded as a constraint imposed on the fourdimensional vector k . The resultant threedimensional surface is called the mass shell. As was shown above, the energy factor ePiwt follows from the pole of the propagator. Thus it is the position of the propagator pole in the complex plane of k2 which is the particle mass squared. This is the usual definition of mass in particle physics. It is applicable to the exact propagator, beyond the loop expansion. Let us apply this definition to the oneloop propagator (7.62). This results in the condition
m2
+ m; +
+ ~ ( ~ ) ( =m0 ~. )
(7.63)
Here m is to be measured experimentally,and m0 found from this equation. Substituting m: from (7.62) into (7.54) and (7.62), we find the inverse propagator in the form
6l ( k 2 )= k2 + m2+
( k 2 ) ~ ( ~ ) (  m . ~ ) (7.64)
CHAPTER 7. RENORMALIZATION
180
C(') is absent here because of its independence of k (for this reason we could neglect it from the very beginning). The contribution of C(2)(k2) ~ ) , does has taken the form of the difference ~ ( ~ ) (k~~( ~) ) (  m which not contain any divergences. The reason for this is the independence of the divergent part of A(2)(k) on k (see equation (7.24)). We can make this relation manifest by expanding C(k2) in powers of the external momentum near the point k2 = m2:
The only divergent term here is C(m2), which cancels on the righthand side of (7.64). Let us write down the renormalized inverse propagator, taking into account the explicit form of c(~)(Ic'):
2
JsJs5) arctg

for k2
<0
It is clear here that the correction term has relative order A2q52 Am2 if q5 a. The selfcoupling X is, therefore, the dimensionless parameter determining the smallness of the corrections. Let us consider the calculation of the propagator (7.66) in another physical context. Suppose that we study the statistical properties of the field cp. Then r(cp) is the free energy in the presence of an external current J. The physical meaning of r(cp)depends on that of cp. If, for example, cp is the local magnetization, then J is the magnetic field. In statistical physics the system 'sits' in threedimensional space so it is not meaningful to consider it in full Minkowski space. In contrast, the typical experimental problem is to measure the system's response to a homogeneous external field. In the above example, it is the magnetic susceptibility of the homogenous magnetic field which is proportional to
7.5. RENORMALIZATION
181
U:; ((P). Thus the appropriate definition of m is m2 = U:; ((P). The value of U:; ((P) is m: + C(0). Problem: Show this. Therefore the propagator, expressed in terms of the new m, does not coincide with the derived above expression (7.64) taking now the form
In the explicit expression (7.66) for the correlator, the third term is now 2 instead of .rr/fi Problem: Show this.
Renormalizat ion of the Coupling Constant We would like to be able to extract finite terms from the expression for the coupling constant X (7.26). The divergent terms are produced by the above considered diagram A(2)but with four tails. The main idea is the same that we have used for the renormalization of the mass: it is necessary to point out the method of measuring the coupling constant. This case differs from the previous one by the absence of an obvious definition like 'the pole of the propagator'. Therefore let us choose a specific set of four momenta kl = ky, . . . k4 = kf and assume by definition that X = rc4)(k;. . . I$') . (7.68) It is apparent that only three momenta are independent, because of the deltafunction in (7.13). The convenient choice of the momentum configuration depends, of course, on the context of the problem. In particle physics coupling constants are measured in scattering experiments, so all momenta lie on the mass shell k: = . . . k: = m2. This decreases the number of independent parameters to 12. Lorenzinvariance eliminates 6 more parameters, and 4 parameters are excluded by energymomentum conservation. Thus we are left with only 2 independent parameters of F ( ~ ) .Let two particles bearing momenta kl and k2 acquire momenta ky and h after a collision. The independent variables are commonly written in the form of 4vectors squared s = (kl k2)2, the invariant mass, which is equal to the total collision energy squared in the center of mass reference frame, the transfered momentum squared. and t = (k3 
+
182
CHAPTER 7. RENORMALIZATION
In the context of statistical physics it is meaningful to define X at zero external momenta. This is the definition used in chapter 5. Thus we fix a suitable configuration of momenta and define X according to (7.68). Then the expression for the vertex function takes the form
where all the oneloop correction terms are denoted r f ) , taking into account definition (7.68), namely
The expression in square brackets in (7.69) does not contain any divergences. This follows from the independence of A:) on external momenta. As follows from the explicit form of the corresponding diagrams, the value of these terms is of order X2.
Problem: Check this for the diagrams contributing to fig. 29.
rf)shown in
In other words, the relative value of the correction is of order X.
Problem: Check that this is true for the corrections to l?(3) shown in fig. 29 as well. Finally we would like to emphasize once again that any result for quantum corrections containing the parameters X and m2 is not physically meaningful until the precise definition of these parameters is given.
Renormalization of the Wave Function In the previous subsections, we derived the expression for the effective action r(p)in terms of the physical mass and the coupling constant. All divergences were canceled by the bare values of these parameters. It was necessary only to give accurate definitions of the physical parameters to fix the finite part of the effective action. Such definitions are formulated in terms of observables and therefore are related to the effective action as a whole. This, when applied to the loop expansion, means that it is necessary to adjust the values of the bare constants at each order, so
7.5. RENORMALIZATION
183
that the physical parameters remain both unchanged and equal to their values at the classical level (where they coincide with the experimental data). Let us discuss this problem in more detail in the context of particle physics. The observable objects of this theory are particles freely propagating far from each other4 and representing the asymptotic states of the field. A particle of mass m and momentum has energy w = (i2+m2)l/'. The normalized wave function has the form
and obeys the KleinGordon equation (6.20). The propagator corresponding to this equation is given by expression (6.23). It is intuitively clear that a single particle must propagate as if it were free. Its propagation amplitude is the exact propagator evaluated at the momentum lying on the mass shell. This function must coincide with the free propagator (6.23). Note that we have already used such arguments when considering mass renormalization. It should be pointed out that the relationship between the asymptotic and exact states of the field is very subtle. The reader can find a detailed analysis of this in the book [4]. The main result of such an analysis is the LSZ reduction formula, which makes it possible to substantiate the above statement in a rigorous way. Let us now attempt to justify the coincidence of the exact propagator given by (7.64) with the free one given by (6.23). Returning to the positionspace representation, we obtain
d" E(t,f,O) =  i J w m
dk,,
Here we have changed the sign of the integration variable ko in Minkowski space, in order to write the expression in a standard form. Calculating the integral we have
4Sec, however, the discussion in chapter 5, page 124 and footnote on page 142.
184
CHAPTER 7. RENORMALIZATION
For the free propagator the residue is equal to 1/(2w), which is exactly the normalization factor of the wave function (7.71). Nevertheless this is not true for the oneloop propagator (7.66):
where the residue in the complex plane of the variable k2 appears in the second term, and the constant
has been introduced. For the oneloop propagator (7.66)
Thus quantummechanical corrections change the wave function normalization (this is a typical result for any perturbative calculations). The wave function of the oneparticle excitations is not the field X, but
xphys)has the correct value of the residue at k2 = whose propagator (xphys m2. Now we can replace the field X by xphYs in all expressions, including the expansion of the effective action (6.75). This is called renormalization of the wave function. Then the nth term of the expansion takes the form 2factors should be taken into account, for example, in the definition of X. Consistent consideration of 2factors can be found in most of quantum field theory textbooks. It should be noted that the field renormalization is equivalent to the introduction of a counterterm (2;l  l)(dp)'/2 into the action. In (p4 theory, this term is finite at one loop. Thus wave function renormalization is not connected with ultraviolet divergences of the theory. The context of the problem under consideration fix if we must to make this renormalization. For example, in quantum electrodynamics the electromagnetic field interacts with the fermion current proportional to 89.
7.6. CONCL USION
Figure 34
Therefore the contribution of the 'fish' diagram to the divergent part of the effective action is proportional (at large momenta) to k2. In this case, the wave function renormalization is absolutely necessary, because the result diverges without it. In pure $4 theory, there is a divergence proportional to lc2, as in QED. It appears at the second order of the loop expansion from the 'sunset' diagram (fig. 34). We refer the reader to the book [l]for detailed calculations.
7.6
Conclusion
In the present and previous chapters we established the structure of the oneloop effective action I'[cp] in terms of a Taylor series (6.75) close to a ] in principle, been determined for small constant field $. F[$ ~ ( x )has, ) p(x)  4 by expression (6.75), along with figure 29. deviations ~ ( x = A calculation of the terms quadratic in ~ ( x was ) carried out completely, with the result (7.66). The structure of the Xcorrections to cubic and ) quartic terms was analyzed. Higher terms of the expansion in ~ ( x can be found by making use of the Feynman substitution (7.48) and formula (7.40). These terms are finite at large momenta and are purely quantum: there are no such terms in the classical action. In the performed analysis, we did not assume the smallnes of the background field $. This value parametrizes the line in the whole functional space of $ where the oneloop effective action is known. On this line, the value of I' is the effective potential times the fourdimensional volume. In the vicinity of this line, I' can be found approximately as described above. A physically preferable point is $ = f a , which corresponds to the ground state in the absence of an external current.
+
186
CHAPTER 7, RENORMALIZATION
The next chapter is devoted to a discussion of the obtained results and of the methods of calculation.
Bibliography [l]P. Ramond, Field Theory: A Modern Primer, 2nd ed., Addison Wesley Loriman, 1988.
[2] K.G. Wilson and J. Kogut, Phys. Rept. 12C (1974) 75. [3] J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, 1983. [4] C. Itzykson, J.B. Zuber, Quantum Field Theory, McGrawHill, 1980. [5] L.H. Ryder, Quantum Field Theory, 2nd ed., Cambridge University Press, Cambridge, 1996. [6] J. ZinnJustin, Quantum Field theory and Critical Phenomena, 3rd ed., Oxford University Press, Oxford 1996. [7] H.M. Nussenzveig, Causality and Dispersion Relations, Academic Press, London 1972.
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Chapter 8 Renormalization Group 8.1
Renormalization Group
Renormalization Group Equation Because the effective action is observable its value must not depend on the way it is calculated. In this chapter, we shall study an important consequence of this obvious statement. The effective action is invariant with respect to the renormalization point, that is, the value of external momenta at which the theory parameters are defined. In the previous chapter, the value of such points were k2 = m2 in (7.64), k2 = 0 in (7.67) and kl = k r , . . . k4 = k? in the definition (7.68). In chapter 5, we used a definite value of the field cp rather than momentum when renormalising the effective potential. These were particular cases of the general situation: the effective action depends on dimensional quantity (or quantities), that is on a renormalization point (or points). Let us denote the set of such points by p. While l? has an explicit dependence on p, there should not be an overall dependence on it. The explicit dependence must be compensated by an implicit dependence on p through the theory parameters: the mass, the coupling constant and the field renormalization constant. Examples of such dependences are given by (5.67), (7.67), (7.69). The fact that l? has no overall dependence on p can be written as a differential equation:
where, by convention, the differentiation is with respect to ln p rather than p because, as we have already seen, this is a typical l? dependence on the renormalization point.
190
CHAPTER 8. RENORMALIZATION GROUP
To obtain a more explicit form of equation (8.1) we assume p to be the unique point at which all of parameters are defined: the mass squared m2 according to (7.64), the wave function renormalization constant Z, according to (7.75) and the coupling constant X according to (7.68) and (7.77). The renormalization procedure makes it necessary to clarify our notation. The bare field cpo does not depend on p and the renormalised field cpphys = (PO does (here, we extend definition (7.77) to the total field including the background as well as fluctuations). In this chapter cp,h, is called the field and the quantities are defined as the coefficients of the functional expansion in this field (7.78). Below we shall omit the subscript phys for the sake of compactness. Equation (8.1) takes the form
.
.
where l n p is considered as the variable. The dependence of theory parameters on the renormalization point is most transparently manifested in the case of the coupling constant X. Here, we use the calculation results and the definition of X (7.68) from the previous chapter. First, let us make the definition (7.68) more specific. For this purpose let us fix the renormalization point of I'(4)(k1. . . k4) by choosing a set of momenta kl = kp, . . . k4 = kp (see expression (7.68) and the discussion thereafter). We shall call the change of the renormalization point the change of the overall scale of momenta. That is, the renormalization point at an arbitrary p is defined as kl = . . k4 = p i r , where @" = kY/Ikr l . . . k r = kf/IkfI are unit vectors. Because of (7.68) and (7.78) the renormalised expression for the vertex function is
&F.
We emphasise that the function f introduced here depends only on the ratio of the momenta to the mass. The dependence of the mass and the constant Z, on p is not as straightforward. It depends on the context of the problem. If the subject of interest is the propagation and the interaction of quanta of the cp field, then the most reasonable definition of m2 is as the pole of propagator (7.64) and Z, is defined as the constant normalising the residue at this
8.1. RENORMALIZATION GROUP
191
pole to unity (7.75). In this case the renormalization point is fixed: k2 = m2. Let us give an example in which the definition of m2 and 2, is not so obvious. Let the field cp describes the local magnetisation or another order parameter of a sample near a secondorder phase transition. The effective action in this case is the free energy of the system. Strictly speaking, the system is threedimensional and one has to reproduce the treatment of the previous chapter for this case. One should have four dimensions only in the case of very low temperature when quantum effects are essential but we can hardly imagine the system which exhibits a secondorder phase transition at such a temperature. Thus let us return to equation (8.2) and discuss the meaning of m2 and Z, depending on p despite the incorrect dimension. For this case, the experimentally observable quantity is the correlator of the order parameter fluctuations (cp(x)cp(y)) in kspace (this can be measured, for example, by neutron scattering). At the tree level it is nothing but k2 m2, where m' is the correlation length. A more exact expression taking account of fluctuations is G(k2) (7.64) or G(k2)/Z, if the field normalisation is controlled. It describes the observable shape of G(k2) better if m2 and 2, are properly chosen. They can be specified, for example, as follows. The propagator is required to take the classical form k2 m2 near a point p:
+
+
G'(p2) = p2
+ m2,
and
d dk2 ~'(k~)l
k2=$
=
1 .
(8.4)
One can regard these conditions as a set of equations with respect to m2 and 2 . The explicit form of the propagator (7.64) with the calculated value of C(k2)as well as the experimentally measured values of G(k2) and dG(k2)/k2should be substituted therein. Thus the parameter definition does not differ from that of X (7.68). It should be noted here that the same conditions can be also applied in particle physics in the region far from the propagator pole. Such an application of equation (8.2) will be discussed below. Let us consider equation (8.2) for the vertex functions l?(") appearing in the expansion of I' (see (6.75)) in powers of the field xphys (7.77). This expansion substituted into (8.2) results in
This is the renormalixation group (RG) equation. It is a cp4analogue of the CallanSymanzilc equation in QED (see, e.g. [l]).
192
CHAPTER 8. RENORMALIZATION GROUP
Let us comment on the origin of this name. It is not difficult to understand that the set of the parameters transformations corresponding to all possible changes of p makes up a group. The transformation from p1 to p2 and then from p2 to p3 leads to the same value of parameters as the direct transformation from p1 to p3, otherwise, the observables depend on the method of calculation. The unit element is the trivial transformation leaving p unchanged.
General Solution of RG Equation An effective action must satisfy the RG equation (8.2) to be renormalization invariant, that is to be independent of variations of p. The RG equation also makes it possible to extend the region where perturbative calculations are possible. An example of this use of the RG equation was given in chapter 5 (eq. (5.74)). We recall that the effective potential was obtained and renormalised near a point p = M. We could not apply the result far away from this point because the oneloop correction, proportional to ln(4/M), became large. Applying the RG equation enabled us to improve the result as 4 t 0 and to show that at large 4 the difficulty was not removable. Let us now consider that procedure in a more general form. For this purpose let us fix an arbitrary renormalization point p0 and change variable from p to' t = ln(p/po). We shall write l?(") in the form r(")(kl, . . . k,; X, m, p") where the last argument of l?(") indicates the renormalisation point. As a rule, l? has the form of the classical action plus a correction depending, like (8.3), on the dimensionless ratios ki/pO and ki/m. Considerable deviations of k from p0 may make the correction terms in large and thus violate the applicability conditions of the method. For example, this is the case for (8.3) as (kl k2)2+ 00. The situation can be improved if one includes the large correction terms in the classical action.
+
Problem: Show that it can always be done in a renormalisable theory. Technically this means that one should recalculate the corrections for new m, 2, and X taking a new value of p for the renormalization point. The only requirement imposed upon this new renormalization point is that the corresponding ki have to be close to the values of interest. This extends the scope of the approximate calculations, but there arises one more problem to solve: how can one express the theory parameters at one renormalization point, say p, in terms of their values at another ('old')
8.1. RENORMALIZATION GROUP
193
point, say p o This problem is tackled by solving eq. (8.5). Let us consider the general solution of (8.5) in more detail. First we introduce the following notation for the coefficients appearing in (8.5):
1dln 2, 7(X) = Z a t l , = , The function (8.7) is called the 0function, and 7 (8.8) is called the anomalous dimension of the field. According to (8.6)  (8.8) all these functions only depend on p0 via m and X. It is clear that one can vary the value of p, in (8.6) (8.8) without changing the functional forms of these relations. Now, let us consider the solution of (8.2). It can be found by the method of characteristics. The value of F(") is known in the hyperplane t = 0 in the 4dimensional space (t, m, Z,,X). We need F(") all over the space. Let us first consider the case n = 0. From (8.5) with n = 0 we get 
dt

at
ar(O) + am2adt + 0dt a~ dJy)

=0
We see that the differential dt and the differentials dm2 = adt
and
dX = 0dt
,
(8.10)
which depend on it determine a vector field in the subspace (t, m2,X) along which F(') does not change. Integrating the equations (8.10), the are found: lines of constant value of "'(?l m"
z(t)
and
X
= X(t)
(8.11)
(an example is shown in fig. 35). These functions are called the running mass and the running coupling constant respectively. Any function which is constant along these lines is a solution of equation (8.9). The unique solution we need is fixed by the boundary condition in the plane t = 0. To find F(O)(t,X) at an arbitrary point (t, m2,X) it is necessary to draw a line (8.11) (i.e. the characteristics of eq. (8.9)) up to an intersection with the plane t = 0
CHAPTER 8. RENORMALIZATION GROUP I
Figure 35
(fig. 35). The value of l?(') at this point is the desired solution of (8.9). Let us write it down in the form
(
r(0) m2
,X
7
)
t = r(0)( s ( 0 ) m 0)
7
(8.12)
where s ( 0 ) , X(0) are determined by the conditions s ( t ) = m2 and X(t) = X. To avoid misleading notation, let us recall that X and m2 denote the theory parameters whose numerical values are to be chosen to describe a specific physical system. X(p) and m2(p) denote the same values, but with explicit reference to the point p at which the parameters should be measured. At the same time, m2 and X are the characteristics of equation (8.5) which cross the point m2(p), X(p). The solution of the RG equation, taken in the form (8.12), makes apparent the renorminvariance of The effective action depends on its arguments m2,X, t via the combinations s ( 0 ) and X(0) and these are renorminvariant by construction. Any change of t results in a change of m2(t) and X(t) which still belongs to the same characteristic line. Now let us solve eq. (8.5) at n f 0. It is straightforward to check that

Thus the problem is solved in general. Let us proceed further with an explicit example using the results of the previous chapter.
8.1. RENORMALIZATION GROUP
Explicit Example Let us apply the method that has been developed to the vertex function theory (6.36), (6.37). For this purpose we need (8.3) in the expressions (7.64), (7.66) and (8.3). We start rewriting the formula (7.64) in the form
+ +
where Z, is expanded in powers of X : Z, = l 6 . . . and the higher terms in X are neglected. We have also omitted the label ' ( 2 ) ' on C. An application of conditions (8.4) gives
where the prime on C means differentiation with respect to k 2 . The substitution of m2 expressed in terms of m 2 ( p ) according to (8.16) leads to an inverse propagator in the form
By differentiation of expression (8.15), we find the anomalous dimension of the field: y=2 ~ 2 ~ " ( p 2 ) (8.18) up to firstorder terms in X. A simple way to obtain expressions for the aand pfunctions is to differentiate these equations (7.64) and (8.3) with respect to p taking account of the fact that the left hand sides of them do not depend on p :
The next step should be to solve the set of equations (8.10). However, the complicated form of C (7.66) and of the unknown function f means there is no real hope of analytic solution. Let us simplify the problem by considering only large momenta k2 >> m 2 , the so called deeply Euclidean region. At such large k2 the expression (7.66) for C reduces to
196
CHAPTER 8. RENORMALIZATION GROUP
That leads to and in the limit k 2 +m. Of course, such a simple result is not obtained by chance. Let us show that as k 2 + CO the functions a , p and y are determined by the contributions of the divergent diagrams only. To this end, we examine fig. 29 and expression (7.19). All the external momenta appear in the denominator of the corresponding matrix elements. Therefore, all the convergent diagrams (7.19) tends to zero as k 2 t cm.The only contribution to the propagator and the vertex functions turns out to come from the divergent 'fish' diagram A:) (7.25). Let us consider the propagator as a function of the external momentum k as k 2 + CO. In this limit we can neglect the mass and write
It is clear that the integrand reaches its asymptotic value llp4only for p2 >> k 2 while the presence of k in the denominator eliminates the infrared divergence of the approximating integral S d4plp4. Thus the divergent part of A(2)takes the form
where A is the cutoff parameter which is replaced by p after renormalization. This expression determines completely the effective action for k 2 >> m 2 . In particular, it results in the second term in the right hand side of (8.3) leading to p= 3X2 16n2 for k 2 >> m'. Thus the functions a, 3 !, and y at large momenta are determined by the structure of divergences and do not depend on the details of the renormalization procedure. It explains, for example, the vanishing of y: the corresponding diagrams converge. It should also be noted that expression (8.25) coincides with formula (5.74) derived for the massless theory ( p 4 . To find the characteristics of the RG equations we have to find s ( t ) and X(t) as solutions of eqs. (8.10) with the boundary conditions s ( 0 ) =
8.2. SCALE TRANSFORMATIONS
197
m2 and X(0) = X. From the second equation in (8.10) we obtain an implicit expression for X(t):
It gives, by virtue of the boundary conditions,
which is the same as (5.76) except p replaces M. Similarly we can obtain an expression for the running mass:
For k; >> m', this running mass does not have an essential effect on the effective action for the following reasons. It gives a subdominant contribution to the inverse propagator (8.17) as compared with the leading term k 2 . There in no dependence on m2 in 1'(4), and all F(") with n > 4 are negligible. Therefore, the most relevant quantity in the RG equation in the limit being considered are the running coupling constant and the corresponding pfunction.
8.2
Scale Transformations
The renormalization point p is a dimensionful quantity, that is why it enters all dimensionless expressions through the ratios k i / p and m / p If it is possible to neglect the mass then k / p is the only argument of any dimensionless function depending on the momenta. In this case a simultaneous change of all momenta, by say, a factor s is equivalent to a change of p by a factor S'. The renorminvariance condition (8.2) is then the condition requiring invariance with respect to such transformations, which are called scale transformations. They are discussed here, starting with those in the classical theory.
Scale Transformations at the Tree Level Let us consider the Euclidean action
(p4
theory in a ddimensional space specified by the
198
CHAPTER 8. RENORMALIZATION GROUP
Let us take an arbitrary field configuration cp(x) and consider scaling its spacetime dimensions by S', that is, let us consider a configuration cp(sx). A similarity law occurs in the system (8.29) if one can keep the action unchanged by rescaling the field and the theory parameters: cp t zPp, m + zmm and X + zxX. For example, if cp(x) was a solution of the equations of motion with parameters m and X then p(s'X) would also be a solution with parameters zmm and zxX. In order to derive the similarity law, let us substitute the transformed field and parameters into the action (8.29) and replace the integration variable by y = sx. This leads to
At the classical level only the equations of motion are relevant. They are insensitive to a constant factor in S. Therefore, the invariance of the action requires that the relation between the coefficients in (8.30) is unchanged. This gives z, = s and z,zx = s2. In quantum theory, the value of S matters. The smaller S is, the more important fluctuation corrections are. Therefore, the theory invariance with respect to scale transformations requires the coefficients of all terms of (8.30) to be equal to unity. This gives
Thus it is seen that all quantities here are transformed according to their metric dimensions (the dimension of the field is mdl2' and that of the coupling constant is m47). It does not seem to be surprising because the action as well as any other physical quantities must contain all its terms with the right dimension. That is why the change in the quantities according to their dimensions results in a common constant factor which must be unity for the action. Relation (8.31) is just a trivial implication of this fact. A nontrivial problem appears if one tries to establish the similarity law for different solutions of the same theory. The restriction 'the same theory' means that the parameters m and X remain invariant under scale transformations. It follows from (8.31) that m invariance is impossible in a space of arbitrary dimension. One can only consider the massless 'Thc similarity law make it possible t o simulatc thc behaviour of large objects (e.g. ships or planes) by carrying out experiments with rcduccd copies of them.
8.2. SCALE TRANSFORMATIONS
199
theory or the situation when the mass is negligible. The last of equalities (8.31) imposes an additional restriction: the scale invariance of the action is possible only in the case d = 4 if zx = 1. Thus let us turn to the case d = 4 and consider scale invariance in the quantum theory.
GellMann  Low Equation Let us derive the equation describing the vertex function change under scale transformations at large momenta, that is if the terms of order m/ki give a negligibly small contribution to the effective action. We express the equivalence of the momentum scale change ki + slc, to the renormalization point change p + s p l p in the form of the following equation for the vertex functions: ( ~ k. .~. ~, k , ;
(
p) = s4nr(n). . . , X
,
)
.
(8.32)
S
Here, the factor s4ntakes account of the metric dimension of l?(") (as an illustration, it can be applied to (8.3) for ki >> m). The nontriviality of the quantum field theory comes partly from the fact that the effective action inevitably contains an additional mass parameter, namely the renormalization point p. Therefore, the equality (8.32) is not yet the solution of the problem. It is necessary to express X(p) and &(p) in terms of p l s . To do so, let us consider an infinitesimal scale transformation with the parameter s = l +ds. We differentiate (8.32) with respect to s and exclude dr'ldp using (8.5). This yields the equation
which determines the behaviour of vertex functions under the momenta scale transformation. It is called the GellMann  Low equation [2]. The solution to this equation, obtained in the same way as (8.13), can be written in the form r'CrL) (Skl, . . . Sk,,,; X (p) , p)
(8.34)
200
CHAPTER 8. RENORMALIZATION GROUP
Thus we came to a rather nontrivial scale transformation law. The coupling constant and the field X do not change according their metric dimensions under infinitesimal scale transformations, but rather as sflIXX and SYX respectively. In addition to their natural dimensions 4  n the vertex functions acquire the anomalous dimension y for each external line (i.e. l?(") K snY). The effective action itself remains invariant under such transformations. The rule for parameter transformations under scale change is called the similarity law. A system that does not change its behaviour under scale transformations of its parameters is said to have scale invariance or scaling. Let us recall that the result obtained, eq. (8.34), is true only at large momenta. In the region k,2 m2 one cannot neglect the mass dependence, and the vertex functions dependence on external momenta becomes much more complicated (see, for example expression (7.66)). In the next section we shall obtain an explicit form of (8.34) and briefly discuss different possibilities for the coupling constant behaviour at large momenta.
8.3
Asymptotic Regimes
The renormalization group equation (8.5) as we11 as the GellMann  Low equation (8.33) obtained in the limit kg/m2 + oo are exact. Nevertheless, the key quantities P and yfunctions cannot, as a rule, be found beyond a version of the perturbation theory. The so called exactly solvable models are exceptions. The search for, and the investigation of, such models is a subject beyond of the scope of this book (see, e.g. [3]). Let us now use the results of the oneloop calculations: the vanishing of y and the expression (8.25) for the Pfunction at large momenta. Expression (8.27) takes the following form under scale transformations:
where the corresponding momentum scale k is taken as the renormalization point. Substituting this into (8.3) and (8.34) at n = 4, we obtain
A . , ,
(8.36) This equality is trivial if s is close to unity. In this case the expansion of the first term in A ln s leads to the replacement of Ic, by ski in the
8.3. ASYMPTOTIC REGIMES
201
second term. For finite Xlns, the result (8.36) is nontrivial. One can see that in the (p4 theory the coupling constant grows as energy increases (correspondingly, as the space scale of the field configuration decreases). Let us have a look at the previous examples from a mathematical point of view. The order of the first quantum correction is not exactly X, but X Ins as can be seen, for example from (8.3). At large S, this makes the convergence of the resulting series worse. Actually, the results of our calculations are terms of the expansion in powers of two parameters: X and In S. The renormalization group equation (or the related GellMann  Low equation) enables one to perform a summation in In S. However, the coefficients in (8.36) remain firstorder in X. It follows from expressions (8.35) and (8.36) that X becomes small for fields slowly varying in space. Formally X + 0 at k + 0. Such behaviour is true until k >> m. On the other hand, for quickly varying fields the coupling constant becomes large. Expression (8.35) diverges at
This singularity is called the Landau pole. The value of A is very large if X(ko) is small. Evidently, expression (8.35) is not applicable near the Landau pole because the loop expansion (or any other variant of the perturbation theory in X) fails because of the large value of the coupling constant. This is often called the nonperturbative region. Unfortunately, the pfunction in this region is not known. Let us just discuss possible scenarios for the pfunction behaviour starting from the point p = 0 at X = 0 (that is, in the absence of the interaction). For instance, it can happen that: 1.
p(X) remains positive at any values of X.
Due to (8.35) this means that the coupling constant grows monotonously, getting very large near the Landau pole. This is the simplest scenario. Such behaviour characterises the QED coupling constant and both of the electroweak interactions. It should be noted that the Landau pole goes to infinity if p(X) increases as X or slower as X + oo;
2. The pfunction is positive only at small values of the coupling constant, turns to zero at X = XI, and remains negative after that (fig. 36). The point X = X1 is called the fixed point because if the coupling constant coincides with XI at some k, it remains equal to it at all energy scales. Let us expand the pfunction near this
CHAPTER 8. RENORMALIZATION GROUP
Figure 37
If the value of the derivative dP/dX at the fixed point is negative as is shown in fig. 36, then an increase in energy for the region X < X1 leads to an increase in X and so it approaches Al. If X > X1 then the Pfunction is negative and the coupling constant decreases as energy increases, again approaching to Al. Such a fixed point is called as ultraviolet stable point, because X(k) tends asymptotically to X1 as k + oo. Theories with such a Pfunction behaviour seem to be very interesting because the perturbation theory never fails if XI is small;
3. The Pfunction is negative at any X, having the form P X2 for X ,0 (fig. 37). Although this scenario has no relation to (p4, we consider it for the sake of completeness. According to the above arguments, the fixed point is X = 0. The theory with such Pfunction is called asymptotically free. An example of such a theory is QCD, the theory describing strong interactions. The Landau pole in this case lies about 180 MeV, that is the meson mass scale. At larger energies, that is, at smaller distances, the observable states are quarks and gluons, which are the analogues of the electrons and photons of QED. At lower energies, the observable states are mesons and nucleons whose QCDcharge is zero ('white' states). Quarks and gluons, their constituent parts, cannot be separated to distances greater than the characteristic size of mesons. This
8.3. ASYMPTOTIC REGIMES
Figure 38
phenomenon is called confinement. Even now there is no complete theory of this phenomenon. This example shows how nontrivial a theory with a large coupling constant (and a complicated group of internal symmetry) can be. 4. The pfunction is negative at small values of the coupling constant, turns to zero at a point X = X1 and increases monotonously after that, remaining positive (fig. 38). As one can see in fig. 38, the derivative dp/dX is positive at the fixed point Al. It means that X decreases as energy grows, moving away from the fixed point. Such a fixed point is called an infrared stable point, because X(k) tends to X1 asymptotically as k t 0.
In principle, one can imagine a theory with a ,Bfunction having a couple of fixed points (see fig. 39). In this case if X < X I , then dp/dX < 0, and the first fixed point of the theory is ultraviolet stable. If X > Xz, then dp/dX > 0 and the second fixed point is infrared stable. The value of X will monotonously increase with energy (or tend to the next ultraviolet stable point if there is one). It is worth while here to make one further note concerning the scaling properties of the theory near a fixed point. It is clear that in this region @(X) M X  X1 and the anomalous dimension is y M $X1). Due to this
204
CHAPTER 8. RENORMALIZATION GROUP
fact, one can calculate the exponential factor in (8.34):
On the other hand, it follows from (8.10) that dX/(X1  X) z Ins and instead of (8.34) we get . . . k,; XI, p). r(")(skl,. . . sk,; X, p) = S4n(1+~(Xl))r(n)(k1,
(8.40)
This means that near a fixed point XI # 0 the simple scale invariance of the theory is restored, although with an anomalous dimension of the field.
Bibliography [l]C.G. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Comm. Math. Phys. 18 (1970) 227.
[2] M. GellMann and F.E. Low, Phys. Rev. 95 (1954) 1300. [3] Z. Horvath and L. Palla, Conformal Field Theories and Integrable Models, Springer, BerlinNew York, 1997. [4] the modern formulation of the renormalization group approach is due to K.G. Wilson, Phys. Rev. B4 (1971) 3174. The renormalization group is presented in many textbooks. See for example J. ZinnJustin, Quantum Field theory and Critical Phenomena, 3rd ed., Oxford University Press, Oxford 1996. C. Itzykson, J.B. Zuber, Quantum Field Theory, McGrawHill, 1980. [5] Books devoted to this subject are J.C. Collins, Renormalization: an Introduction to Renormalization, the Renormalization Group and the Operator Product Expansion, Cambridge University Press, Cambridge, 1984. 0.1. Zavialov, Renormalized Quantum Field Theory, Kluwer Academic Publishers, 1990. For the renormalization group in the theory of critical point see K.G. Wilson and J. Kogut, Phys. Rept. 12C (1974) 75.
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Chapter 9 Concluding Remarks In chapter 6 we set the problem of calculating the correlation functions in quantum field theory. The subsequent chapters were devoted to the solution of this problem in the framework of the loop expansion. What is the parameter of this expansion? Why is it so called? How restricted is its domain of applicability? Is the cp4model really the simplest field theoretic one? In this chapter we try to answer these questions, but first we would like to calculate the correlators of the type (6.5)  (6.7). We shall express them through the effective action (introduced in chapter 6) which contains complete information about all the Green's functions of the theory in the most compact form.
9.1
Correlators in Terms of r[p]
We know the effective action r[cp]. We would like to express the correlation functions given by (6.5) (6.7) in terms of I'[cp]. We are only interested in the connected parts of the correlators which constitute the generating functional W (6.8) because one can easily reconstruct disconnected parts knowing the connected ones and, more importantly, because the connected parts describe mutual effects of the field at different spacetime points (see, expression (6.13)). These mutual effects are the aspect of greatest interest. Therefore we set the problem of expressing the connected correlators through the oneparticle irreducible ones. Let us consider 
CHAPTER 9. CONCLUDING REMARKS
etc. The first of the these averages is the average field. This is the functional argument of I'[p]. In what follows, we shall omit the angular brackets when writing this quantity. The second line determines the dressed field propagator (6.11). According to (6.62) this function is inverse to in the sense of (6.29). Let us write down the corresponding relations once again:
where d y ~ ( X' , y ) G ( y , z )
= 6(x z)
.
(9.6)
All other connected correlators beginning from (9.3) can be expressed These are the functional through G ( x ,y ) and the vertex functions I'(")[cp]. derivatives of I'[cp]with respect to the field, they are defined by (6.75), (6.76). To show that, one should simply go from (9.2) to (9.3) and then to (9.4) by repeated differentiation. Let us find the 3point function (9.3). The substitution of the second variation from (9.2) leads immediately t o
In order to express the right hand side in terms of 1'(2)= Gl let us vary the identity GP' * G = 1 with respect to the current and convolve it with G , multiplying from the right. This gives
where X' and y' run over all space. Using the definition of the exact propagator (6.11), we represent the variation with respect to the current in the form
Figure 40
Thus we finally obtain
appears as a result of the variation where the 3point vertex function of the expression Gl = 1'(2)with respect to the field. This expression has a clear graphical representation given in fig. 40 where the figure on the left hand side symbolizes (cp(z)p(y)cp(x)),, the circle with three points illustrates and the thickened lines depict the exact propagators. The integration variable is denoted by the junction points of different graphic elements. Now, the calculation of the 4point correlator (9.4) is not a problem. There appear four terms after the differentiation of expression (9.10) with respect to the current. In the first term, the vertex function differentiated according to the rule (9.9) gives l?(*) and one more external leg. In each of the three other terms one of propagators in (9.10) is differentiated. When doing this, the second part of equality (9.10) is used as a differentiation rule. The result is exhibited in fig. 41 where the integration points X', y' etc. are not shown. One can easily note that the last three terms differ from each other only by permutations in the joint scheme of the external points to the internal ones. The contributions of these terms to W become equal upon convolutions with four fields 4 x 1 . . . P(.).
CHAPTER 9. CONCLUDING REMARKS
Figure 41
Problem: Write the corresponding integrals. The calculation of higherorder correlators can be carried out in a similar way. We emphasize that all the relations considered here are exact. Note that fig. 40 and fig 41 give a hint how to prove that l? generates only oneparticle irreducible diagrams  l? can be obtained from W by cutting all oneparticle propagators in the corresponding Green functions.
9.2
On the Properties of Perturbation Series
In this section we answer the question about the expansion parameter of the perturbation series and of the loop expansion. We shall also discuss the difference between these two expansions and the convergence of the corresponding series.
On the Loop Expansion Parameter Let us write down our main quantity, namely the path integral for the theory p4 in the most general form:
{
Z [ J ]= J D [ ~ ( exp ~)I
1 2 2 rl ~ c ~ x ( i ( t ~ p )p ~+ p3 3!
+
+ 4!")} p (9.11)
9.2. ON THE PROPERTIES OF PERTURBATION SERIES
211
The current term Jcp (6.4) is absent here. Let us assume that in this expression cp denotes the deviation from the solution GS/Scp = J . The current appears in (9.11) implicitly through m2 and 7. The most straightforward method of calculation of (9.11) is to expand the exponent containing anharmonic terms in a series and then calculate the resulting Gaussian path integrals with polynomial preexponential factors. The calculation of integral (6.38) in chapter 6 is an example. Let us find the order of smallness of an arbitrary term of this series, i.e. of a diagram containing v3 vertices with three lines coming out and v4 vertices with four lines coming out. We denote this quantity &(v3,Q). However, this estimate is not suitable It is clear that &(v3,v4) Xw47w3. because the constant 7 has the dimension of mass. For a more accurate estimate we make the action in (9.11) d'imensionless by means of the substitution X = z,t, cp = X,$. The constants z, and z, should be adjusted so that the coefficients at 4 in all terms of the action become identical to each other. This imposes three conditions on two parameters, thus the problem in general has no solution. Let us relax the restrictions, requiring only three coefficients to be equal: the 4' and of 44.This gives coefficients of N
1
x =  Em
and
m
cp=4
v5
,
and the action in (9.11) takes the form
where
C denotes the following parameter combination
Problem: Make the action dimensionless in a space of arbitrary dimension d. Make sure that X in the common coefficient in front of the integral is then replaced by the dimensionless combination X/m4d. Now we turn to the functional integration over the field 4, using the action (9.13). All perturbation theory integrals give only numerical factors. To estimate the common factor multiplying them, it is worth recalling that the quadratic terms in the action are proportional to 1 / X , that corresponds to the inverse propagator. Therefore, each line of the diagram contributes a factor X. The fourline vertex gives 1 / X and the
CHAPTER 9. CONCLUDING REMARKS
212
three lines gives
where l is the number of lines. It can be expressed in terms of the number of vertices if one counts all lines at all vertices and divides this by two to account only one time for each line. This gives l = (3v3 4v4)/2. This number is integer since v3 is even. Finally we get
+
Problem: Perform a similar analysis of the definite integral
<
Let us consider different possible scales. We take as the first example the theory (9.11) with 7 = 0 and m2 > 0. Switching on a spatially homogeneous current leads to a shift of the potential minimum to a point cpo(J). A typical current value (without any special smallness or greateness) is that which causes cp m. In this case, 7 = Uttt(cp) Am, 6 and &(v3,v4) = Xv4+"3,that is, each vertex adds one power of the coupling constant. One faces a more complicated situation at large values of the background field cp,. One can reach such values either by increasing J or even, in the case of spontaneous symmetry breaking by keeping it zero. Let us consider the latter example. Let the potential be p2Q,2/2 XQ4/4! with p2 > 0. Expanding the field Q> = Q,o cp near the minimum 4 = p m one comes to an action of type (9.11) with the following W
<
N
+
+
values of the parameters: n2= 2p2, 7 = m m . We see that large values of the background field Q,  m / f i lead to the large parameter 7 that gives  1 in (9.16). The consequence is that the diagrams of the same order in the coupling constant really have different orders of magnitude. For example, a few diagrams of the order X and X2 shown in fig. 42 belong to different orders of the perturbation theory (from the first to the fourth order). Let us show that the order of any diagram (including those with external lines) is determined by the number, L, of independent closed loops of the graph: E(%, v4) = XL'. It is obviously so for the diagrams in fig. 42. In the general case, note that L is equal to the number of
c
9.2. ON THE PROPERTIES OF PERTURBATION SERIES
213
Figure 42
independent integrations over momenta. To count these, it is sufficient to recollect that each propagator (the total number of which is I) gives just one integration (see (6.21), (6.23)). Each vertex (the total number is v = v3 v4) contributes a Sfunction of all momenta entering it (a detailed analysis of this fact was done for a particular case in section 7.1). All the Sfunctions result in one common &function expressing the energymomentum conservation for the external lines (7.13). The remaining v  1 &functions decrease the number of integrations to
+
which is the number of diagram loops. Comparing this expression with (9.15), we obtain the desired relation in the form
In theories with spontaneous symmetry breaking C 1. Otherwise, < 1 at not very large J, that causes further dependence of the diagram order on the type of vertices it contains. To find the formal loop expansion parameter which does not depend on we turn to dimensional units, i.e. we introduce llfi into (9.11). Then, as can easily be seen from (9.13), one should replace X by fiX while remains unchanged. That leads to
Thus the Planck constant is the loop expansion parameter. We already used this fact, which was checked to the first order in fi, when the effective action was introduced in chapter 6 (see expression (6.65)). We saw also
CHAPTER 9. CONCLUDING REMARKS
214
that the oneloop correction to the effective action (6.72) contained all orders of the coupling constant. It should be stressed that both the expansion in the coupling constant and the loop expansion are variants of the same perturbation theory approach. They differ from each other only by the way diagrams are arranged according to their order of smallness. The origin of perturbation theory is the expansion of the exponential function containing the third and the fourth powers of the field in the Taylor series near a properly chosen point. The point can be taken in different ways. In the next section we shall discuss some principle restrictions on the accuracy of the perturbation theory itself.
On the Asymptotic Nature of Perturbation Series In this section we shall not obtain any exact statements about the convergence of the perturbative expansion. Instead we model it with the definite integral
and present some evidence that the petrurbation series of this integral is asymptotic.
Let us expand the expression ecXX4in (9.20) in a series and integrate each term individually. This gives
The series obtained evidently diverges because of the r(2n),which grows faster than n!. At the same time, the first term of this series, I ( X ) z 1  X/2 3A/8 . . ., improves considerably the accuracy of the zeroth 1for X << 1. That can be verified by numerorder approximation I ( X ) ical calculation. Thus it would not be wise to disregard the possibility of making use of diverging series such as this one. In fact, all the impressively precise calculations in QED are done using this method. Let us look closely at the structure of the series (9.21). The reason for its divergence is clear. The exponential factor ePXx4in (9.20) is very small as X + CO. However, this smallness is a result of the cancellation of very large terms in the corresponding Taylor expansion.
+
+
9.2. ON THE PROPERTIES OF PERTURBATION SERIES
215
We destroyed this subtle compensation by expanding the exponential function in (9.20) and taking the limit X + CO as is required by the integral. As a result, the nth term of the series in the second part of (9.21) has a maximum at r = @ which is absent in the original function. The value of the maximum becomes of the order unity at n % 4 e ~ ' / ~ / XThus . one should expect that the series does not approximate the initial integral beyond n 1/X. Let us discuss this problem in more detail. Let us find the common term C,(X) of the series as n >> 1. To estimate it, we use the Stirling formula for the Ffunction and factorial

We see that for X << 1 the terms of the series at first decrease as n grows, but then, beginning from some n = no, they increase again. Let us find no under the assumtion that it is large. For this purpose we differentiate the second factor in (9.23) with respect to n:
Equating that to zero, we find no = 1/(4X). Substituting n = no into (9.23), we get an estimate for the smallest term of the series (9.21):
Thus the smaller X is, the later the divergence of the series manifests itself and the smaller the minimal term is. It is natural to expect that the quantity (9.25) is an estimate of the maximal accuracy which can be achieved when summing the series (9.21). Let us state here without a proof that this is precisely the case [14]. The best approximation for the exact expression can be obtained by summing series (9.21) up to the terms with numbers of order no. Besides that, the exact expression contains the term of type econst/x which, of course, cannot be expanded in Taylor series near X = 0 (all derivatives of this function vanish at X = 0 ) . In quantum field theory such contributions are called nonperturbative as opposed to the perturbative ones which are terms of the perturbation series.
CHAPTER 9. CONCLUDING REMARKS
Figure 43
Figure 44
Let us show that nonperturbative terms do indeed exist. This will, at the same time, be a proof of the statement that the radius of convergence of series (9.21) is zero. It follows from the fact that I ( X ) (9.20) has a cut along the negative part of the real axis on the complex plane of the variable X. This cut begins at the point X = 0. I ( X ) has an imaginary part on the edges of the cut, which in the case X t 0, takes the form
on the upper and the lower edge respectively. To show this, let us consider the analytic continuation of I ( X ) , (9.20), on the X complex plane to values X < 0. We define the change to negative X as a rotation X = IXleia, with the phase a varying from 0 to T . It is evident that integral (9.20) diverges for A < 0 but this divergence no longer has anything to do with the function I(X). The correct integral representation of the analytic continuation I(X) can be derived from the integral (9.20) by rotation of the integration contour in the X complex plane so that the integral remains convergent for all values of the phase cu including a = T . This can easily be shown. Let us consider the integral (9.20) in the complex plane of X. By convention, we will refer to the sets of directions in which the integrand vanishes (goes to infinity) for 1x1 + oo on sectors of convergence (divergence). The divergence sectors of the integral are depicted in fig. 43 by the bold solid lines, and the integration
9.2. ON THE PROPERTIES OF PERTURBATION SERIES
217
contour coincides with the real axis. Formula (9.20) defines the analytical function I(X) in the region 7r/2 < argX < 7r/2. Let us rotate the contour by the angle P so that it occupies the position pictured in fig. 43 by the slanted dashed line. The value of the integral will not change after that. However, it now defines the function I(X) in the region 7~12 p < arg X < 7r/2 ,B. Thus two analytical functions defined by the integrals over both the contours in fig. 43 coincide with each other ,f3 < argX < n/2. Hence, by in the intersection of the regions 7r/2 virtue of the theorem about the uniqueness of the analytic continuation, two integrals define the same analytic function I(X). The statement for an arbitrary phase a can be proved in an analogous way by repeating the above arguments several times.
+
+
+
After the continuation of X to negative values of X = IXlem",he integration contour in (9.20) takes the position shown in fig. 44 by the dashed line. The points z = 0 and z = fl/mare saddle points of the integrand. They are marked off by circles in the figure. The integrand equals to unity at z = 0, falls monotonously as it approaches these points along the real axis and then grows to infinity. Its value a t z = f1 1 0 is exp {1/(4X)) and determines the imaginary part of I(IXlei"). To extract the expression ImI(lXlei") from the total integral, we take the integration contour as shown in fig. 44 by the solid line. Then, the integrand is real in the interval from z = l/& to z = l/&. The vicinity of the peak at X = 0 gives the main contribution to ReI(lXlei") = 1. At the same time, the imaginary part of I(IX(ei") is due to the parts of the contour lying beyond the real axis. The main contribution to ImI(lXlei*) is given because the contour comes out of by the regions close to r = f these points in the direction of the steepest descent which is parallel to the imaginary axis. Taking account for this contribution in the Gaussian approximation we come to formula (9.26).
l/m
Now we consider the change of sign of X when going to the lower In this case the rotation of the integration halfplane: X = contour in (9.20) takes place in the anticlockwise direction opposite to that shown in fig. 44. Therefore, the sign of the imaginary part of (9.26) is reversed. Thus we have shown that X = 0 is the branching point of I(X). Consequently, the series in powers of X cannot converge near this point. It should also be noted that (9.26) contains the same exponential factor as the accuracy estimate (9.25). The example considered here serves as a good model for the situation in QFT. To demonstrate the divergence of the perturbation series and the existence of the nonperturbative part In Z let us make analytically
CHAPTER 9. CONCLUDING REMARKS
Figure 45
continue (9.11) to X < 0 keeping m2 > 0. After that, the potential in (9.11) takes the form shown in fig. 45. The alternation of the sign of X does not affect the properties of the perturbation series near the point cp = 0. At X sufficiently small one can use the vacuum state cp = 0 and study crosssections of particles scattering on this background. However, the vacuum state itself is metastable. That is, it has a finite lifetime. Sooner or later, either quantum fluctuations or thermal ones lead to the appearance of an embryo of a new phase, i.e. to the nucleation of a bubble such that the energy density inside it is less than U(0). The size of the bubble will grow quickly. A firstorder phase transition occurs. A similar process was already discussed in chapter 5, where some references were also given. According to papers [l],[2] the free energy of a metastable state has an imaginary part proportional to ieconst~x.This fact can easily be understood if we recall that the metastable states in quantum mechanics are characterized by an imaginary part in their energy. Technically, these arguments are similar to the above example but with X corresponding to a field coordinate describing the growth of the new phase bubble. Finally, we can conclude that the convergence radius of the perturbative series is zero. The nonperturbative terms become essential to all orders when the expansion parameter of the perturbation series is not small. This can result in very nontrivial properties of the theory such as those discussed in the next sections.
9.3. ON
9.3
(p4
THEORY WITH LARGE COUPLING CONSTANT
219
On (04 Theory with Large Coupling Constant
Here we only say a few words about the cp4 model with large coupling constant and address the reader to the literature. The system behaviour depends drastically on the number of spatial dimensions d (in which time may be also included). Let us discuss separately the two most interesting cases.
The Cases d = 2 and d = 3: SecondOrder Phase Transitions The case d = 3 corresponds to numerous systems which occur in statistical physics. The case d = 2 describes interfaces (for example, a crystal surface). In both numbers of dimensions we find secondorder phase transitions. Although an impressive progress has been made in the last decades these field of research remain a challenge. The literature is huge. We can refer, for example, to [3, 4, 5 , 6, 71 and references therein. Let us have a brief look at the phase transition from the point of view of the fieldtheoretical models discussed in chapters 69. Consider the free energy F[cp]. The possible relation of the field cp to realistic physical values was discussed in chapter 6. Practically, there is a much larger variety of systems which can be described in terms of a field theory. In the theory of phase transitions the field is commonly called the order parameter. For small variation of cp and small gradients dcp, one can expand the free energy in a Taylor series whose first relevant terms are
where d = 2,3. This obviously coincides with the (p4 model. The parameters m2 and X are functions of temperature 8. A value 8, for which m2(8) = 0 is called the critical point at which a secondorder phase transition occurs. Note that the dimensionless coupling constant is X/m2 and X/m for d = 2 and d = 3 respectively. It goes to infinity as 8 + 8,. Also, for any fixed momentum k, the dimensionless ratio k/m becomes infinite, thus recalling the largeenergy asymptotic regime discussed in chapter 8. Experiment shows that very different systems exhibit universal scaling behaviours in their critical points with the anomalous dimensions si of the correlation functions. The values of si depend on d, on the number of the field components and on their symmetry. The same is true
220
CHAPTER 9. CONCLUDING REMARKS
of the scaling of all thermodynamic quantities which are proportional to (Q 0,)"~with specific values of fi as Q + Q,. In the theory of the critical point the vi as well as the si are called critical exponents. Problem: Expand m2 near the critical point:
m2 = a(0  Q,),
(9.28)
a > 0 and find the critical exponent of the order parameter below the critical point where m2 < 0. The answer is 112 while the experimental value is 0.33  0.34 in d = 3 (see [6] and references therein) and 118 in d = 2'. The reason for such a discrepancy is that only the bare value of m2 may be regular at 8 = Q,. Correspondingly, (9.27) with (9.28) is the form of the bare (microscopic) energy E rather than of F. The difference between E and F is due to the fluctuations. These cannot be neglected at large coupling constant. The strong fluctuations govern the system properties near the critical point, as confirmed by the experiment. The simplest form of the free energy (9.27) with (9.28) is the basis of phenomenological theory of the critical point developed by Landau (see, e.g. [g]) which does not account for fluctuations. This theory explained the origin of the nonanalytical behaviour of thermodynamic variables near the critical point (for example, the order parameter is identically zero above Q,, but nonzero below it). Although it captures qualitative features of phase transitions, Landau theory is quantitatively incorrect near the critical point where the coupling constant becomes large and one cannot neglect fluctuations. It is beyond the scope of this book to discuss the beautiful and powerful methods developed for realistic calculation of the critical exponents.
The Cases d = 4: Possible Triviality of
(p4
Theory
Let us consider the case d = 4 which corresponds to elementary particle physics. For d = 4 X is dimensionless. Thus the mass is not relevant for the coupling constant, which is just X. As we have seen in chapters 5, 7, and 8, the (P4 theory has a Landau pole at large momenta. Surprisingly, it may follow from this fact that the (p4 theory is trivial (see [l01 and reference therein). This property implies that (i) the correlation functions of the field for odd numbers of points are zero and (ii) the multipoint correlation functions of even numbers of points are simply products of the twopoint correlators G ( x  y), as in the case of free l ~ h i is s the value for the exactly solvable Ising model. See, for example, [g] and references thercin
9.4. CONCLUSION
221
theory (equation(6.33) and fig. 22), except G need not coincide with the treelevel twopoint correlator. This is suggested by numerical experiments on supercomputers (simulations on the lattice as were mentioned in chapter 6) and by analytical approaches. The problem is still investigated and the final answer is not yet available. The triviality of (p4 theory was proven in d 5. It is known not to be trivial in d 3. The intermediate case of d = 4 turned out to be the most difficult one. This case is of practical interest for elementary particle physics because a (p4 field is an important ingredient of the unified theory of weak and electromagnetic interactions often called the Standard Model. The triviality of this sector would spoil the composition of the model. It is suggested that a theory which includes a p4 sector may be nontrivial at small X and trivial for large values of X. Although the Standard Model is very successful, the scalar particle corresponding to the pfield (Higgs particle) has not been discovered yet and so must have a mass value r n ~ 89.3 GeV at 95% confidence level [ll]. A requirement for nontriviality of the theory imposes an upper bound (about r n ~ 900 GeV [12]) on the mass of this particle.2 Let us note that the possible triviality of the (p4theorydoes not affect the content of this book. Our aim was to present the method of loop expansion and to discuss the basic properties of the field systems. We used the (p4 model as a simple example. If the theory is trivial, this only shows how nontrivial the nonperturbative properties of field systems can be.
>
>
9.4
<
<
Conclusion
This chapter completes the part of the book in which we have attempted to study the general features inherent in any quantum field theory using the example of the (p4 model. In the next chapters we shall consider in more details a quantization procedure, which leads logically to fermionic fields. Both these and gauge fields will be quantized. In addition, we shall discuss interesting classical objects which appear in nonlinear field theory.
2Note that the experiments sensitive to the presence of the Higgs particle suggest that its mass might be not much higher that 80 GeV [13].
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Bibliography [l]J.S. Langer, Ann. of Phys. 41 (1967) 108.
[2] S. Coleman, Phys. Rev. D15 (1977) 2929; Erratum: ibid D16 (1977) 1248. [3] K.G. Wilson and J . Kogut, Phys. Rep. 12C (1974) 75. [4] S.K. Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, MA 1976. [5] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover Publications, 1975. [6] A.Z. Patashinskij and V.L. Pokrovskij, Fluctuation Theory of Phase Transitions, Oxford, Pergamon Press 1979. [7] J. ZinnJustin, Quantum Field Theory and Critical Phenomena, 3rd ed. Oxford, University Press, 1996. [8] F.R.S. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982. [g] L.D. Landau, E.M. Lifshits and L.P. Pitaevskii, Course of Theoretical Physics. v.5: Statistical Physics. Pt.1, 3rd ed., London, Pergamon, 1980. [l01 D.J.E. Callaway, Phys. Rep. 167C (1988) 241. [l11 S. de Jong, Higgs Search at LEP, talk at the 23rd Rencontres de Moriond (Electroweak Interactions and Unified Theories), Les Arcs, France, March 1998; C. Caso et al, The European Physical Journal C3 (1998) 1.
[l21 R. Dashen and H. Neuberger, Phys. Rev. Lett. 50 (1983) 1897.
224
BIBLIOGRAPHY
[l31 D. Treille, Searching for New Particles at Existing Accelerators, talk a t the ICHEP'98, Vancouver, B.C., Canada, July 2329 1998, Canada (http://ichep98.triumf.ca). [l41 A. Erdelyi, Asymptotic Expansions, Dover, 1956.
Part I11
MORE COMPLEX FIELDS AND OBJECTS
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Chapter 10 Second Quantization: from Particles to Fields It was shown in the chapter 4 that fluctuations of quantum fields are particles. The phonons in the solid state physics are a classical example: they are quanta of the crystal lattice vibrations. However, electrons or other elementary particles seem to be logically more primary than any field. Formally, one may introduce a field that upon quantization gives rise to the particles which really exist in our World. The field that possess the desired properties can be used as a theoretical object justified by the comparison of its predictions with experiment. This is possible in principal, but it is not strightforward. In this chapter, we would like to describe another method of second quantisation in which the elementary particles are considered as primary objects and the quantum field as a result. This approach allows us not only to complete logically the second quantisation presented in chapter 6, but also to introduce fermionic fields in a very natural way.
10.1
Identical Particles and Symmetry of Wave Functions
Elementary particles of the same type are identical. This is a standard principle one can find in any textbook on quantum mechanics. Actually, its meaning is related to the definition of the notion elementary particle. In order to distinguish different interacting particles one has to label them. This is only possible if these particles have an internal structure and if this is the case they are not elementary. Let us stop the discussion of this fundamental property here and consider its main consequences for
228
CHAPTER 10. SECOND QUANTIZATION
the quantum mechanics of manyparticle systems. Let us formulate the property of indistinguishability as a condition on the twoparticle wave function. The interchange of two particles does not change any observables. Thus, as a result of the interchange, the wave function of a twoparticle system can only get a phase factor em". Two consecutive interchanges is the same as none because two interchanges bring the particles back to their initial states. Thus eZi"" = 1, i.e. ei"" = 1 or ei"" = 1. The first case corresponds to bosons and the second to fermions. Let us denote the twoparticle wave function as 9 ( x l , x2) where the coordinate xl corresponds to the first particle and xa to the second one. Then 9(x2,xl) = *(g1, x2) for the boson system and 9(x2,xl) = Q(xl, x2) in the case of two fermions. Consider N identical nonrelativistic particles. The Hamiltonian is
where
and so on. Here all the summation indices take the values 1. . . N , and the terms with two or more coinciding indices in multiple sums are excluded. For example, the term v(~)(x,,,sal, xaz) that could appear in we ~ , , xal) in Formally we can include in V(') and the term v ( ~ ) ( x xal, require, as a condition, that the potential is equal to zero if any two of its arguments coincide:
Further, the potentials v('")(x,, . . . , X,,) tations of their arguments.
are symmetric under all permu
IHowever, in two space dimensions new possibilities arise. This is a special case because any rotation of two particles around each other (which is the way to interchange them) is characterized by only one angle. This allows arbitrary values of or, for example, for particles moving in a plane orthogonal to an external magnetic field. The objects with noninteger or are called anyons [l,21.
10.1. IDENTICAL PARTICLES
229
The operator <(l) is the socalled oneparticle Hamiltonian. It is given by the sum of the conventional SchrGdinger operators H:') of noninteracting particles of mass M in an external potential v(')(x). As one example one can consider the motion of electrons in metal. In this case v ( ~ ) ( x is ) a potential of the ionic crystal lattice. The next term <(2) describes the twoparticle interaction with the potential energy V('). For example, it can be the Coulomb repulsion of two electrons. The combinatorial factor 1/2! appears to avoid the double counting of each couple of identical interacting particles. This operator is the socalled twoparticle Hamiltonian. defined by (10.4) is the threeparticle HamiltoThe next term nian. The potential V(" describes the modification of the twoparticle interaction energy due to the presence of the third particle. This situation is a bit more complicated. As a relevant example, one can consider the correction to the interaction of two atoms due to the presence of the third one. As the third atom approaches, the electron shells of the initial two atoms get perturbed and this changes the interaction between them. Of course, the separation of these three atom into a twobody system and an external perturbing atom is fairly symbolic when all three atoms are identical. The factor 1/3!, corresponding to the number of permutations of three elements, is introduced in (10.4) to count the contribution from each three particle configurations only once. Let us recall the example of electrons in metal. The presense of an electron affects the lattice polarization. This effect is, in general, nonlinear. Thus, in this simple case, an electron changes the dielectic permittivity of the matter between two other interacting electrons. As a result the energy of their interaction changes and the corresponding correction is V("). The meaning of the other terms in (10.1) could be considered in the same manner. There are many physical situations where particles are less coupled to each other than to a background potential and the character of interaction between them is mainly onetoone. Then the expansion (10.1) is fast converging and we can restrict ourself to consideration of only the first a few terms. In this chapter we consider a very convenient formalism which takes into account the symmetry of the wave function under interchange of particles. Let Q(xl, x2 . . . ,xN) be an Nparticle wave function and let the system be described by the Hamiltonian (10.1). One can introduce the two particle interchange operator by defining
This operator commutes with
6 because its action is just
a renaming of
CHAPTER 10. SECOND QUANTIZATION
230
the summation indices in expressions (10.3)  (10.4). Thus 'Paband the Hamiltonian should have a common set of eigenfunctions. Let us find them. To do this, we choose a complete set of orthogonal normalizable functions &(X), i = 0,1,2 . . .. A common choice is to consider the planewave basis qbi(z)= eikz where the momenta take all possible values allowed by the boundary conditions in, say, a potential box. We shall use the expansion
&
Each of the multiparticle basis functions &l (xl)giz(x2). . . qiN(xN) describes a state where the first particle is in the state il, the second one is in the state i2 and so on. It is obvious these functions are not symmetric under particle interchange. More precisely, the interchange of two particles, indicated by the numbers a and b in the functions (10.7), means that the arguments interchange as X, + xb, while the state labels i, and ibremain unaffected. Of course this operation is equivalent to the interchange of indices ia + ib by the unchanged arguments. Thus we have to construct basis functions which are symmetric with respect to such operations. We do this first for bosonic systems.
10.2
Bosons
Let the N particles under consideration be bosons, i.e.
for any a and b. Our goal is to symmetrize the set of the basis functions (10.7) and calculate the matrix elements of the Hamiltonian in the new basis. To symmetrize the functions &l (xl)qbi2(x2). . . +iN (XN) we have to perform all permutations of indices, sum up the results and introduce a normalization factor. The correct expression for the wave function takes the form
Here Zn1..,ni...n3is a normalization factor. Since in the sum over permutations only interchanges of different states are included, all terms in
10.2. BOSONS
231
(10.9) are orthogonal to each other. Thus Z is just the number of configurations in the sum. Let us find it. One can introduce the socalled ith state occupation number ni;this is the number of times the term Gi appears in the product +i,( x ~ ) (+ x 2~). .~. +iN ( x N ) (Each time, of course, with a new argument X,). This number is the number of particles with a given state number i. Its possible values are ni = 0 , 1 , . . . N . It is apparent that 00
xni=N . (10.10) i=O Let us expresss 2,,,.,, i,.,n,,., in terms of the occupation numbers. This is the number of all particle permutations N ! divided by the number of permutations of the particles in identical states:
Formula (10.9) is rather cumbersome. It would be more convenient not to write all the arguments X , because only the occupation numbers are really meaningful. Let us use representation independent notation
for the state described by the wave function (10.9). This notation contains only information about the occupation numbers of each states. Knowing these numbers allows the whole wave function (10.9) to be restored. The states (10.12) are orthogonal if the occupation numbers are different for at least one singleparticle state. We shall use, for a while, the representation (10.9) in order to derive the rules of operation on the manyparticle states. Our goal is to undestand how the Hamiltonian (10.1) acts on the functions (10.9). Let us start with the oneparticle Hamiltonian (10.2).
OneParticle Hamiltonian Let us denote by H$) the matrix elements of the oneparticle Hamiltonian (10.2):
The operator H(') acts on the basis function
232
CHAPTER 10. SECOND QUANTIZATION
Consider the action of the oneparticle Hamiltonian (10.2) on the function (10.9). This operator transforms the first term in (10.9) to
and so on. Obviously the number of terms in the wave function increases, but the number of factors gi in each term is still the same. The only changes to the occupation numbers come from the change of one lltj to qhi and this change gives rise to a factor H,. Thus
Here the Bii are as yet unknown coefficients. This form should be supplemented with a convention that any term with at least one negative occupation number is zero. Indeed, if in the initial state ni = 0 then the coefficient Hij does not appear by the action of the operator f i ( l ) . To calculate the coefficients Bij we track down the contributions to a matrix element Hi,. Let us consider an arbitrary term in (10.9). It contains nj factors $ j which differ by their argument. Only these functions contribute to Hi,under the action of the '$l). Each term gives rise to nj new terms proportional to Hij in which $ j are replaced by $i with unchanged argument. Thus the total number of terms proportional to the H,. in the whole of the wave function is Znj where Z is defined by (10.11). As before these terms differ by all possible permutations of states, but a part of them is now superfluous if ni > 0. Indeed, (xl, 2 2 . . . xN) inherit all the terms proportional to H,(:) in ";I(')@ the interchanges between the initial state i in (10.9) and identical states generated by the Hamiltonian action. Such interchanges are forbidden because we permutate only different states. Let us find the number of these obsolete transmutations. The total number of terms differing by permutation of the states i and j in the initial function is ,,,,,,,,,,i,,.
As we already noted, this number is multiply by nj by the action of the Hamiltonian. At the same time the correct number of permutations
10.2. BOSONS
233
between ni + 1 particles in the state i and n j  1 particles in the state j (generated as a result of the action of E(') in (10.16)) is (ni
(ni + n j ) ! + l)!(nj l ) ! 
Multiplying (10.17) by nj and then dividing by (10.18) we find the number of the obsolete transmutations to be ni+l. This is the additional in front of the matrix element coefficient to the factor l/,/=
H::'. Let us write this coefficient in the form
where Z,l...712+1...7L3~... is the correct normalization factor for the wave function after the action of 'Id1). Thus we have found the coefficients Bi, and (10.16) now takes the form
Although this notation is already compact, it is only the first step towards a very convenient formalism specially created to describe manyparticle wave functions.
Creation and Annihilation Operators To develop this formalism, let us introduce operators ai and a! according to the definitions
(note the square root is taken of the bigger of the new and the old state i occupation numbers). Traditionally the operators ai and a! are written without a hat. Operator ai decreases the state i occupation number by one. The occupation numbers of all other states do not change. The action of ai gives zero if ni = 0. This operator is the socalled annihilation operator of a particle in the state i. Operator a! increases the state i occupation number by one. This operator is called the creation operator.
234
CHAPTER 10. SECOND QUANTIZATION
It is easy to see that the operators introduced here are Hermitian conjugates (this has already been reflected in the notation a f ) :
In the last term here the operator af acts to the left on the state i with occupation number mi and 6,, is the Kronecker symbol. It follows from the definitions (10.21), (10.22) and the orthogonality of the states (10.12) that [ai,a;] = . (10.24) Let us introduce one more definition. The product fii alai is called the particle number operator or occupation number operator of the state i because (10.25) a f a i ( .. . n i . . .) = nil.. . ni . . .) .
>
In accordance with its meaning, ni 0. Formally, this property is secured by the zero coefficient when ai acts on the state ni = 0. In other words, it is impossible to construct a state with a negative occupation number by any consecutive action of the annihilation operator. Using expression (10.20) let us write the oneparticle Hamiltonian 'H(') in terms of the annihilation and creation operators. This yields
This formula turns to be especially simple if the wavefunctions & ( X ) , n = 0 , 1 , 2 . . . are the eigenfunctions of the oneparticle Schrodinger o p erator H('). In this case H(;) = Eibij and (10.26) takes the form
The meaning of this expression is quite obvious. The Hamiltonian of a system of noninteracting particles is equal to the sum of the energies of the oneparticle states multiplied by the corresponding occupation numbers. Note that the latter are operators and the numbers of particles in any state are not fixed. Let us investigate the question of the completeness of the set of all states with all possible occupation numbers (and variable total number
10.2. BOSONS
235
of particles N). These states form the socalled Fock space. The annihilation and creation operators act on this space. Obviously N is increased by one under the action of the creation operator (10.22) and there is no upper bound on its value. As was mentioned above, there is a restriction on the lowest possible value of N because all occupation numbers ni 2 0. Thus by consecutive action of the annihilation operators, one can obtain a single state without any particles. Note that the Fock space without this state is incomplete with respect to the action of the annihilation operators. Let us denote this state as
This is the vacuum state or vacuum. F'uther action of the annihilation operator on this state gives zero while one can construct all states by acting with appropriate creation operators on the vacuum. It would be impossible to describe a system with variable numbers of particles without including the vacuum state in the complete set. However, it is impossible to write a wave function of the vacuum state in analogy with (10.9).
Total Hamiltonian Let us consider the action of the twoparticles Hamiltonian (10.3) on the wave functions (10.9), (10.12). In principle, we have to repeat the combinatorial considerations of section 10.2. However, such an approach is getting rather cumbersome and awkward. Let us apply a more formal approach using the already derived result (10.26). Let us expand the twoparticle interaction potential in (10.3) on a complete set of functions wi(x) (this set has nothing to do with functions (10.7)):
Let us consider the functions wi as the oneparticles Hamiltonian, then, applying the relation (10.26), we obtain
) ~ (wj),, ~ are the matrix elements of wi(x) and wj(x) sandHere ( w ~ and wiched between the oneparticle basis functions. In order to rewrite this
CHAPTER 10. SECOND QUANTIZATION
236
expression in its final form, we have to express it in terms of the matrix of the operator V ( ~ ) ( X x2) ~ , which are elements
vL~~,
A substitution of this expression into (10.30) results in
It should not be forgotten to check whether this expression is unique. One could change the order of aial and aka, in (10.30). This would give rise to two additional sums in (10.32) which included the commutators [al,ak] and [a,, ail. According to (10.24) these commutators are proportional to Ssymbols. The additional terms in (10.32) would, therefore, include matrix elements of the twoparticle potential with coinciding indexes from the first and the second pairs. One of two such terms would
Note it has the form of a correction to the oneparticle Hamiltonian. As follows from the middle expression in (10.31) and because of the completeness of the eigenfunction set $i, the summation over l results in this additional term being proportional to v ( ~ ) ( x , x ) .This is zero according to (10.5). A generalization of the procedure considered above for the interaction potentials of higher number of particles is obvious:
The meaning of these relations is rather clear. An nparticle Hamiltonian annihilates n particles in all ossible states and create n particles in other states with an amplitude v ; ~ ~ ~a.,.kn., , ~ ~ ~ ~ ~
P
The Field Operator Let us introduce an operator
10.2. BOSONS
237
This operator, when it acts on the vacuum, creates a particle at a point X. Let us check it:
In the coordinate representation the oneparticle state wave function which appears in (10.36) is Gi(y). Exploiting the completeness of the set we see that the righthand side of the expression (10.36) is just S(Y 4 . An operator 03
+(~)=X$i(x)ai
,
(10.37)
i=o
conjugate to +?(X), is called a field operator. Its action is to annihilate a particle at a point X:
Here we have used the action of +(X) on oneparticle states:
It follows immediately from the definitions (10.35), (10.37) and the commutation relations (10.24) that
We are going to reexpress the Hamiltonians (10.26), (10.32) and, generally, (10.34) in terms of the field operator. To this end we invert the expressions (10.35) and (10.37) as
and substitute these relations into (lO.26), (10.32) and (10.34). This results in the following form for the oneparticle Hamiltonian
CHAPTER 10. SECOND QUANTIZATION
238
It follows from the definition of the matrix element (10.13) that the expression in squared brackets is S(x1  x ~ ) H ( ~ ) ( xThen ~ ) . we obtain, finally, A(')= d x ~ t ( x ) ~ ( 1 ) ( x ) 8 (.x ) (10.44)
/
In the same way we arrive at the following expression for the twoparticle Hamiltonian
The generalization of these formulae for manyparticle Hamiltonians is obvious. Let us now write the equation of motion for the timedependent field operator in the Hamiltonian representation. We start from the basic equation d (10.46)
[email protected](X, t ) = ["ri,$ ( X , t ) ] ,

at
substitute in it the expansion (10.1)  (10.4) for f i in the nparticle Hamiltonians and use the results (10.44) (10.45) along with the commutation relations (10.40). The result is the operator equation 
d ih$(X, t ) = H(')@(x,t )
at
+ J d y ~ t ( yt ,) ~ ( ~ ) (y )x8 ( y ,t ) 8 ( x ,t ) + . . .
+A / dy, . . .dyn,8t(yl, t ). . . 8t(yn, ,t ) (n l ) !
(10.47)

v ( ~ ). .(. yn)8(y1, Y~ t ) . . . 8(ynYn,, t ) 8 ( Z lt ) + . . . where the operator the compact form
a
fi(l) acts on the variable X .
at
S
A
[email protected](X, t ) =
S$+
,
This can be written in
f i [ 8 ( X 1t ) ,@ ( X , t)]
(X,t)
The meaning of this equation is discussed in the next section.
Result: Recipe for Quantization One can view the results of the previous section as giving a formal prescription for manyparticle system quantization. Since it follows from expressions (10.35) and (10.37), one has to expand the oneparticle wave function in a basis set and to declare the expansion coefficients to be the annihilation and creation operators with the commutation relations
10.2. BOSONS
239
(10.24). As a result, the oneparticle wave function is promoted to an operator. In general, a transition from cnumbers to operators (or qnumbers) is called quantization. This is why the procedure described here is called the second quantization (the primary quantization has been done when we wrote the Schrijdinger oneparticle equation). To complete this construction in a logical way let us formulate the theory of a classic field such that upon quantization it gives the manyparticle states described above. Let us denote such a field by @(X, t). The corresponding Hamiltonian can be written as an expansion in powers of @ in analogy with (10.1), (10.44)  (10.45). The difference now is that the Hermitian conjugation is replaced with complex conjugation and both the Hamiltonian and the field are cnumbers written without the hat symbols. Note that this Hamiltonian does not depend on the time derivative of the field dQ/dt. However, it depends on both @ and Q+. We can hope that the momentum canonically conjugate to Q is @t (the canonical momentum is necessary in Hamiltonian dynamics). This is indeed the case if one starts with the action
[email protected](x t) a[@(x t), , @*(X, t)] . ih d ~ d t Q * (,)st , at Problem: Check that this is correct and this action gives rise to the Hamiltonian 7l applying the standard procedure (1.35).
J
The dynamical equation of this system takes the form
This coincides with (10.48) up to the replacement of the quantum field with the classical one. This equation is first order in time. As well as the case considered, such equations are also used to describe relaxation processes in a manyparticles system. For example P! could be a diffusing particle density (if, of course, we drop the factor i on the lefthand side of the equation (10.50)). If the field 9 is real such an equation cannot be derived from any variational principle. Let us avoid going into a more detailed discussion and just note that in relativistic quantum mechanics the Schrodinger equation is second order in time as dictated by the Lorenz invariance of the theory (see, e.g. [3]). In this case the action (10.49) and the equations of motion (10.50) as well as (10.48) take the familiar form with second order time derivatives. In this sense the expression (10.1) can be consider as an analogue of the expansion of the effective action (6.75) in powers of the field.
CHAPTER 10. SECOND QUANTIZATION
240
To sum up, by second quantization the linear part of the classical field equations corresponds to the Schrodinger equation describing the oneparticle states. The nonlinear terms describe interaction between the particles of a manyparticle system.
Problem: Compare this to the results of section 4.5.
10.3
Fermions
Suppose the particles under consideration are fermions. Then the expressions (10.1) (10.4) for the Hamiltonians are valid as is the wave function expansion (10.7). The real difference is that the multiparticle wave function has t o change its sign under the interchange of any two iN in equation (10.7). Thus the particles. Therefore, AiIi,,,.iN = &ilia... wave function describing the N particles in the states il,i 2 .. . iN can be represented as a determinant 
This formula is the fermionic counterpart of (10.9). It is obvious that two particles cannot occupy identical states because the corresponding wave function then equals zero. This is socalled Pauli exclusion principle. As before, we use the short notation for manyparticle states (10.12) taking into account that the fermionic occupation number can take only two values: ni = 0 , l . Note that we need an additional rule about the ordering of the the states i. For instance, they can appear in the determinant (10.51) in order of increasing number. The specific forms of the ordering rule does not matter, but there must be one.
OneParticle Hamiltonian Let us define the action of the oneparticle Hamiltonian (10.2) on the wavefunction (10.51). Expression (10.16) is valid for both the bosonic and the fermionic cases and all we need now is to recalculate the coefficients B,,. First, note that Bij is zero if ni = 1 or nj = 0. If this is not the case, then the action of the Hamiltonian transforms the state number i in the determinant (10.51) into the state number j . The created state j should be moved to its correct place in the determinant. As a result the
10.3. FERMIONS
241
where KiJ wave function gets an additional transposition factor is number of the occupated states between the initial ( i ) and the final j1
( j ) states: KG
C nl. Thus
=
k i t 1
m
=
C ~ i : ) (  l ) " ~ " n ~ , n. .ln, .i S l , . . . n,

l . . .)
. (10.52)
i,j=O
Creation and Annihilation Operators Let us introduce creation ( c ) and annihilation ( e t ) operators to simplify the formula (10.52). Let us define
Here Koi is the number of occupied states between the ground state i = 0 and the state number i . It is easy to see that these operators are Hermitian conjugate. Let us find the commutation relations of ci and c!. It is obvious that c: = (C:)' = 0. The operators ci and cj for i # j do not commute because the result of their action depends on the number of previously created particles. According to the definitions (10.53) and (lO.54), we obtain cicj = cjci. All these properties can be written down in the form of the following anticommutation relations
and
{&4} =0 In the same way we find cicj = cici if i # j . To find the commutation relation between operators ci and c! let us write down all the nonzero results of the action of cicf and c!ci operators:
Thus we find {ci, c l } = 1. Finally, collecting all this information into a single relation we come to
We see that the properties of the fermionic creation and annihilation operators result in their commutation relations being formulated in terms of anticommutators. This contrasts with the bosonic case.
242
CHAPTER 10. SECOND QUANTIZATION
ManyPart icle Hamiltonian We can rewrite the expression (10.52) via the operators ci and c; as
This appears the same as the bosonic manyparticle Hamiltonian (10.26) up to the replacement of a with c. The reason for this is that this formula does not rely on the commutation relations. For the same reason all nparticles fermionic Hamiltonians have forms analogous to (10.26), (10.32), and (10.34). It also remains true that the order of creation and annihilation operators in the nparticle Hamiltonian is unimportant provided condition (10.5) is valid.
Field Operator Finally let us introduce the field operator
and it conjugate
cc
In analogy with the bosonic system these operators create or annihilate a oneparticle state at a point X. It follows from (10.55) that these operators anticommute
If we try further to exploit the analogy with the bosonic system, our next step should be to find a classical field such that, upon second quantization, it gives rise to the operators considered above. This is important for the introduction of the fermionic path integral. However there is no analogy with the boson case at this point. The reason is that the bosonic fields commute with any noncoincident values of the arguments (see (10.40)). In the transition to the classical quantities one could neglect the noncommutativity at one point. In contrast, the fermionic operators do not commute at any point. Their classical limit would contain rather exotic objects, this is to say anticommuting cnumbers. Sometimes this problem, that is the absence of such a numbers in classical analysis (as
10.3. FERMIONS
243
well as on experimental device scales) is treated as an absence of a classical limit of the fermionic fields. In the next chapter we discuss this problem in more detail and present a consistent construction of the path integral over fermionic field.
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Bibliography [l]F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144.
[2] F. Wilczek (ed.), Fractional Statistics and Anyon Superconductivity, World Scientific, Singapore, 1990.
[3] J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics, McGrawHill, 1964.
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Chapter 11 Path Integral For Fermions In this chapter we include fermions in the path integral quantisation scheme. It cannot be done in as straightforward a way as in the bosonic case because the Pauli principle has no classical analogue. Below, we construct the path integral for fermions, starting with the anticommutativity of the fermions creation and annihilation operators deduced in the previous chapter.
11.1
On the Formal Classical Limit for Fermions
The PI quantisation of bosons relies on the notion of classical trajectories. To build the PI for fermions, we must first define the notion of the classical motion of fermions. The way to tackle this problem is to compare the standard quantisation schemes for bosons and fermions, to try to understand the differences between them and to try to realize those differences in the framework of the PI formalism. Canonical quantisation operates with the full set of canonically conjugate variables qj, pk; j , k = 1,. . . N. These obey the classical conditions
(here {F, G) is the Poisson bracket of F and G, [F, G] will denote the commutator and [F, G]+ the anticommutator of operators F, G). The quantisation reduces formally to replacing the Poisson brackets with commutators:
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
248
A function F(q,p) of two canonical variables becomes on quantisation the operator F(&$)which is the same function of canonical observables j , k = 1,. . . N . This correspondence means that the Poisson q p bracket of two functions F and G is replaced by a commutator
{F,G)

Xi [F, G
] .
In the bosonic case we can construct annihilation and creation operators for any pair of canonically conjugate observables &, pk
and
[&,&
=
tjk .
(11.5)
Taking the classical limit h + 0 in (11.2) gives the obvious result that classical dynamical variables commute with each other and, thus, can be described by conventional cnumbers. Let us now attempt to follow the same logic in reverse order for the fermionic case. As we saw in chapter 10, the Pauli principle leads to the requirement that the annihilation and creation operators and t in this case must obey the anticommutation relation
In analogy with (11.4), we can construct fermionic coordinates and moand %p such that menta
i,
[ia,iP~+=[.ira,.irpl+=o;
1 i ~ , i i p l + = i h 6 , ~.
(11.8)
c,
If we assume that these 'classical' analogues and TO exist then the limit fL + 0 leads to an unexpected conclusion: E, and ~p must anticommute
=O [ m p ] += [ ~ a , ~=p [~L +K ~ I +
.
(11.9)
'This procedure is ambiguous because the order of the noncommuting & and ?ji may be different. One can present several operators F corresponding t o the same classical function. Thus there may be several quantisation schemes. This matter is, however, out of the scope of the present book, we address the reader t o [l]for detailed investigation of this problem. 'In quantum field theory this requirement follows from positiveness of the field energy. This statement is known as the spinstatistics theorem (see e.g. [ 6 ] ).
11.2. GRASSMANN ALGEBRAS: A SHORT INTRODUCTION
249
Thus E, and TO cannot be conventionalnumber valued. In order to proceed in constructing the PI, we have to accept anticommuting numbers, these are called Grassmannian or Grassmann numbers and furnish the so called Grassmann algebra. We first consider their mathematical properties and then the corresponding classical mechanics.
11.2
Grassmann Algebras: A Short Introduction
In this section we describe the most important properties of Grassmann algebras without rigourous mathematical proofs, these can be found in, e.g., PI. Let us consider a set of generating elements {J,?), j = 1, , n such that
t j b + G 5 j =a7,
(11.10)
.
A set consisting of elements f where
(11.11)
and fiI,.,i, are complex (or real) numbers is called a Grassmann algebra G(n, C ) (or G(n, R)) with the basis of generating elements (Q . . . &) over complex (or real) numbers. We shall also use the notation G,, or simply G, if it is clear from the context which Grassmann algebra is being discussed. The decomposition (11.11) cannot be infinite because 1: = 0. The maximal m for which fi,,.,,,# 0 for some il . . . i, is called the degree of the element f : m = deg( f ) . One can prove [2] that this definition does not depend on the choice of the basis (G . . . &). For any elements f and g of 4 the following property is apparent: deg(f g) = deg(f ) deg(g). Elements whose decomposition (11.11) contains only terms with even k are called even elements. Analogously, the ones containing only terms with odd k are called odd elements. It can easily be established that G is split into the direct sum of the linear subspace G+ C G consisting of all even elements and 4 C G consisting of all odd ones. Thus it is
+
250
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
straightforward to check the following:
3) if f , g ~ G  , then f g =  g f 4) if f
;
€G+
E G + , ~ E G  , then f g = g f E G 
,
for a fixed basis. Mathematicians would say that these properties mean G is a Z2graded associative algebra. One can introduce the Grassmannian parity a(f) for any even or odd element f as follows: 0, f even 1, f o d d
.
Thus for two elements with a definite parity we have
In the case of a complex Grassmann algebra with an even number of generating elements one can introduce a complex basis ql, v;, . . . , %, instead of the real basis . . . , Jz, This is constructed as follows:
cl,
where cl, c2 are conventional complex numbers. Let us briefly discuss the generalisation of Lie algebras (see Appendix C) to the fermionic case. To do so, one has to replace the associative product in the definition of the Grassmann algebra by the Liean product. This gives the definition of a Lie superalgebra, that is, given a Z2graded linear space L = L. @ L1; a(L0) = 0, a(L1) = 1, we define the Lie product [X, y] of elements X and y with a definite parity so that
(
[X,
[y, %]l
+ (l)"(~)"(") [y, [X, X]] + ( l)
a ( z ) a ( y ) [%, [X,
Then, if the linear space L obeys properties (11.12), that is
y]] = 0 .
11.2. GRASSMANN ALGEBRAS: A SHORT INTRODUCTION 251 for any even elements Ei and odd elements Oj, it is called a Lie superalgebra. A useful construction aiming to describe both bosonic and fermionic field is the so called Berezin algebra. This is an algebra with elements (11.11) where the coefficients fil,,,ik are some functions of, say, m real or complex variables X I , . . . , X,. Any function of z l , . . . ,X,; J1,. . . ,Jn, considered as a Taylor series, is an element of the corresponding Berezin algebra:
+ C,iz filiz(x1,. . . ,X,) Eillik+ . . .
.
(11.17) Let us briefly discuss analysis on Berezin algebra. It is quite natural to define left and right derivatives with respect to the anticommuting variables as follows:
It follows from (11.18) that
and, in the general case,
&
where denotes that the corresponding variable, &, is absent. As any function of (1, . . . , Jn is polylinear in Jil one can see that repeated differentiation with respect to an anticommuting variable always gives zero:
and so the operator d/d& is nilpotent:
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
252
Moreover, one can easily verify that in the general case
It is reasonable to introduce, formally, differentials dti as elements anticommuting with & and with each other:
One can use (11.19), (11.20) to check that for any two functions f and g with a definite parity the following relations hold:
a
 (fg)
ati
=
(& f )
g
+ (1)""'
f
(&

g)
7
(11.23)
Furthermore, for functions of arbitrary parity the following relations are valid:
where summation over repeated indices is assumed. As follows from (11.23), if f is odd then the left and right derivatives coincide with each other:
As the derivative operator under consideration is nilpotent, there is no inverse operation. Hence, indefinite integral over anticommuting variables cannot be defined in the usual way. Furthermore, there is no notion of distance between two elements of the Grassmann algebra, or, in other words, we cannot define an appropriate topology and so it is not
11.2. GRASSMANN ALGEBRAS: A SHORT INTRODUCTION 253
reasonable to speak about definite integrals because it is impossible to construct the corresponding integral sum. Nevertheless, one can introduce integration axiomatically as an algebraic operation. Let us discuss what properties we would like such an operation to possess. We proceed from the requirement that the Grassmannian integration must be close to the usual one in some sense. For instance, since we usually deal in quantum theory with integrals with infinite limits, it is natural to require that the integral we are constructing is invariant with respect to shifts of the integration variable. In the case of one anticommuting variable, J, this requirement reads
+
Since a function of one variable is linear, f (J) = a bJ, we immediately obtain a formal integration rules for Grassmannian variables
These rules look somewhat artificial. Let us consider a physical example so that we can make sure that they work and help in obtaining physically meaningful results. Namely, let us calculate the partition function for Fermi particles using integration over Grassmannian variables. In the simplest case of a twolevel system, there are two states to), 11) and a pair of anticommuting annihilation and creation operators 2 , i.t obeying
The operators and the state vectors can be realized as 2 X 2 matrices and twodimensional column vectors as follows:
and
Let us recall first the calculation for the case of the bosonic oscillator (discussed in chapter 4). We need the partition function written in terms
CHAPTER 11. PATH INTEGRAL FOR FERMZONS
254
of both a discrete and a continuous complete system in the space of states, denoted by In) and lz) respectively. These states obey the conditions
where dp(z) is a properly chosen integration measure. The continuously labelled states can be, for example, the so called coherent states; these are defined as follows:
where z is a complex number. For this case dp(z) = dzdz*/n d(Re z) d(Im z)/n. The partition function can be written in terms of both types of states:
where p is the inverse temperature and E, is the nth energy level. Using the Harniltonian H = ~ ( 6 t h 112) and neglecting the 'vacuum' part of the energy, w/2, we obtain the wellknown formula
+
C exp(P
nw)
=
Sp (exp(pwhth))
n
Let us calculate, by analogy with the bosonic case, the partition function for the Fermi oscillator one in the following form
3Such states minimise the uncertainty relation for coordinate and momentum o p erators constructed from G and G+, i.e.
(
J2
(A?), (Ap), = h/2 where ( AA
), = ( Z ~ A ~~ (Z( z) ~ A ~ z ) )These ~ . states are widely used in quantum optics [3] and for
obtaining of exact solutions of Schrodinger equation in the case of quantum systems with symmetries [4].
11.2. GRASSMANN ALGEBRAS: A SHORT INTRODUCTION 255
where, as in the bosonic case, the states
I()
are the eigenstates of c:
Let us find the eigenstates. One can easily see that E must be a Grassmannian variable because of the nilpotence of c. Thus the states to be found are linear combinations of 10) and I l) with complex Grassmannian coefficients < , c . It is straightforward to check that the desired states have the form
and that they obey the completeness relation
This gives
Here we have used the relation
which follows from the idempotence of eft:
The result (11.33) is correct since it follows from a direct calculation of the statistical sum (11.28) which includes only two terms. The unusual properties of Grassmannian integration manifest themselves in the unusual transformation laws for the integrand under the substitution of integration variables, namely,
256
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
as opposed to the conventional formula
Formula (11.34) can be proved easily. For this purpose we note that the most general transformation of Grassmannian variables is linear
where summation over j is assumed. If the algebra has N generating elements then, following (11.11),(11.25), only terms proportional to &,,. . . ,tinsurvive in the integral over d& . . . d&, where ik runs from l to n rather than to N. Thus the integral takes the form
where &il...i, is the LeviCivita symbol (i.e. the antisymmetric tensor) and account has been taken of the antisymmetry of fil...i, with respect to transpositions of any two indices. We omit the argument X for the sake of brevity. The integral over dql . . . dqrLreads
Comparison of (11.36) and (11.37) proves (11.34). There are more general transformations of integration variables for integration over a mixture of the Grassmannian variables and the usual variables. In this case
where sdet(J) is called the superJacobian or superdeterminant of the integration variables transformation: sdet ( J ) = det
1$
(
= det (A  BD'C) det (D')
11.2. GRASSMANN ALGEBRAS: A SHORT INTRODUCTION 257
with
We address the reader to the book 121 for details. The final subject of this section is the calculation of Gaussian integrals over Grassmannian variables and their Fourier transformation. Let us consider the integral
I=
J
dc1 . . .den exp
( 1(A<)
,
(11.39)
where ( = (&,. . . , I n ) and, without lose of generality, A can be thought of as an antisymmetric nondegenerate matrix. Of course, the nondegeneracy implies A has even order n = 2m. To find the integral it is sufficient to note that the antisymmetric matrix A can be brought by an orthogonal transformation to the blockdiagonal form
A=diag
[(
OX1
0' l ) . . .
(
OAn
',)l
0
,
where all X's are real. Because of the orthogonality of the transformation the corresponding Jacobian is unity and we have
where does not depend on <. In the case of integration over complex Grassmannian variables, A is an antiHermitian matrix and can be represented as A = i H , where H is Hermitian.. The matrix H can be diagonalised by a unitary transformation. Thus we get
+ v*<+ v<*)= det A exp (v*A'v)
dFld<; . . . d<,,d
,
(11.42) which differs from its commutative analogue exp (.*A. 7'r
+ a'z + az*) = (det A)'
exp ( a * ~  ' a )
258
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
Note that the result of integration in (11.42) can be obtained by substituting the solution of the equations
into the exponent. One can interpret this as an analogue of saddlepoint method for anticommuting variables. The Fourier transformation of Grassmannian variables is completely analogous to the usual case:
Here the order of d& and f (C) is important because of the noncommutativity The components of f are
where 6,
=i12 =
{
1, l even i, lodd
It can be verified that the inverse transformation is
Therefore, the function
plays the role of the deltafunction in the Grassmannian analysis.
11.3
Pat h Integral For NonRelat ivist ic Fermions
Now we have all the necessary mathematical tools for constructing the quantummechanical amplitude in the form of a path integral. As in the case of a conventional bosonic particle we need the kernel of the evolution operator of our quantummechanical system. When evaluating
11.3. PATH INTEGRAL FOR FERMIONS
259
this quantity we would like to obtain classical dynamical variables rather then operators representing the kernel in the form of an infinite integral product (cf. the derivation of the path integral in chapter 2). In the bosonic case this is quite natural because these dynamical variables are the usual canonical coordinates and momenta, but in the fermionic case these quantities are shown to be anticommuting. Thus it is natural first to consider the anticommuting analogue of classical mechanics which we shall call pseudomechanics.
Classical Pseudomechanics Let us consider a Lagrangian function which depends both on even dynamical variables qi, ii and on odd ones i,:
c,,
In any meaningful theory the Lagrangian L must be an even element so that
An arbitrary variation of the Lagrangian has the form
Introducing even and odd momenta
one can rewrite it as follows:
Variation of (11.49) in view of (11.51) gives equations of motion
As in the standard classical mechanics, one can obtain the relation between the action and the momenta where the former is regarded as a function of the final time tb and of the coordinates qi (tb),[a (tb):
260
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
Here and below the index b is omitted. Introducing the symbol H for the derivative of S with respect to the final time
we get the generalised Legendre transformation
The minimal action principle gives the equations of motion in terms of the analogue of the Hamiltonian function H:
These equations of motion look very similar to the usual ones, however a more attentive analysis reveals an essential difference. Because the Grassmannian coordinates are nilpotent, they can only appear in the Lagrangian in bilinear form. This appears to be analogous to classical mechanics where higher powers of Qi are absent. As L(q, Q, J, is linear in each of and &, in the case when the kinetic part of the Lagrangian is diagonal, that is when it has the form ti, the matrix of second derivatives ~ld~/di,&$ll is identically zero. Hence, one cannot exclude ( a from the expression for H . Thus even in the simplest case of the onedimensional Grassmannian dynamical system, hamiltonization of the system is impossible. Such systems are called singular. Using (11.56), (11.57) one can conclude that for any function of dynamical variables F(qi,pi;t,,n,,t) the time derivative can be written
i)
4,
where the following generalisation of the Poisson brackets has been introduced: {G, F ) p B =
d F dG d F C (dG 8%api api aqi

+
Defined in this manner the Poisson brackets algebra turns out, after quantisation, to be a Lie superalgebra (11.15)) (11.16).
11.3. PATH INTEGRAL FOR FERMIONS
261
Let us now consider a few simple examples of dynamical systems and their quantisations. The Lagrangian of a onedimensional system, which must be even and real, is (11.60)
In this case the equation of motion for the anticommuting variable is trivial:
<=o
.
In the 2dimensional case the most general Lagrangian is (11.61) It is more convenient to rewrite it in terms of the complex variables
i
7rn
=
I
'I*,
i 2Q
(ll .62)
TV*= 
l
whereupon it takes the form
The equation of motion for
For V2= W
= const
gives
we have the periodic solution ~ ( t= ) eiUt~ ( 0 )
with period T
= 2.rrlw.
Hence, the 'energy' is
m 2
E =  V ~ + K + W ~ * ~
.
It follows from (11.64) that the Grassmannian part
is itself a constant. An attempt to use the usual semiclassical quantisation rule J = dq7rrI dq*7rn*= E n T = nS (11.66)
+
262
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
leads to
E, = n h . For fermions, the only possible values of n are 0 and 1. On the other hand, canonical quantisation gives the fermionic part of the energy operator in the form 1 HF = ~ ~ w (c ) ~c (11.68) 2 ' where c, ct are creation and annihilation operators (11.7) which obey (11.6). Taking into account (11.62), their explicit forms are
Since we cannot take large n as for the quasiclassical limit for bosons, one cannot neglect the vacuum term h / 2 and so (11.67) is wrong. This is the reason why the classical limit of Fermi systems is only a formal one: one cannot take the limit of large n together with h , 0. In the case of a 3dimensional Grassmann algebra the most general Lagrangian is
where V,. = l.$, i, j = 1,2,3. If we take rotation group then l& must be of the form
to be a vector under the
which gives
where S is a vector product
It can easily be checked that
so S becomes the spin operator after quantisation. If we assume that the last term in (11.70) is the spinorbital term i.e., V = pq A v , where p is the gyromagnetic ratio, it leads to the following equations of motion:
11.3. PATH INTEGRAL FOR FERMIONS
263
Introducing the mechanical and orbital angular momenta M = mq A v, L = q A p we get
Thus the total angular momentum is conserved, as it must be. Nevertheless, there are some strange features of such pseudomechanics. First, one can see from (11.72) that in the presence of the spinorbital interaction, q k , pi are not just real numbers as we would expect. Second, since anticommuting numbers are not observable, there must be a procedure which forms a correspondence between any even Grassmannian dynamical variable and a real number. The only possibility is to take the average: (f) =
/
P(<)f (C)
,
(11.73)
where p(<) is a distribution obeying the following natural requirements: 1. The average of any even real Grassmannian element is a real number; 2. (l)= l;
3. For any even element f , (f f *)
>0 .
The first requirement gives for p
where a, ci are real numbers. The second requirement leads to a = 1. Substitution of p(<) into (11.3) gives (S) = c, however we can see that the third requirement demands c = 0. This is one more reason why Grassmannian mechanics is not applied to the real word. Although it acquires a physical meaning only after the quantisation, it can be successfully used for constructing path integrals for fermions.
Path Integral Quantisation To construct the path integral amplitude one should know two things: the classical Hamiltonian of the system under consideration and the evolution equation in the Schrodinger representation. The first of these has just been cleared up but the second is rather unclear: we do not know what corresponds to the Schrodinger equation if we replace all spatial coordinates with their anticommuting analogues. To tackle this problem one should generalise the naive concept of quantisation used in chapter 2.
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
264
Let us return to bosons and consider again the method for constructing their path integrals. The amplitude (X,,, tblx,, t,) represented in chapter 2 as a path integral is nothing but the kernel of the evolution operator
where ~ ( t bta) , denotes the evolution operator: ~ ( t bta) , / X , t,) = / X , tb). It is important to emphasise that the basis in which the kernel is calculated is not significant. We can use, e.g., the basis of coherent states (11.27). In this basis, the analogue of the formula (2.17) for the kernel takes the form
where N
+m,
At = (tb  t,)/N, and
+
(11.76)
~ ( tAt,t) = exp
for an infinitesimal At, where H is the Hamiltonian of the system under consideration. It is convenient to represent H in the normal form H(&+,6) in which the creation operator is situated to the right of the annihilation one. Then, the matrix element of H between two coherent states can easily be calculated:
The function H(z*,z) is called the normal synbol of the operator H. Taking account of the form of the scalar product
following from (11.27), using (11.75), and taking the limit N obtain an expression for the kernel in the form
t
m , we
11.3. PATH INTEGRAL FOR FERMIONS
265
where zk is assumed to be a function of t so that xk = x(tk) and At (tb  ta)/N. This limit is by definition the path integral
=
where the dot denotes the time derivative. Because x*(t) and z(t) are regarded as independent variables, the boundary conditions should be prescribed for x* at tb and for z at t,. The coherent state representation is useful for constructing the path integral of fermions. As mentioned above, the Lagrangians (11.60), (11.61) and (11.69) which we would like to quantise are singular: they are linear in velocities and, therefore, coordinates and momenta cannot be regarded as independent dynamical variables. This makes it impossible to integrate out the momenta in the path integral as was done in the bosonic case in chapter 2. Thus the path integral for fermions can only be defined in the whole phase space of the system. To do this we use fermionic coherent states of the type (11.31). For our purposes, it is convenient to normalise them
where Q is an element of twodimensional Grassmann algebra, with the scalar product
We define the path integral as the kernel
of the evolution operator in the coherent states basis corresponding to the Hamiltonian H(??, E) written down in the normal form. Here Et, ? are the fermionic creation and annihilation operators. Using the completeness relation (11.32) one can easily show that G possesses a property analogous to (11.75):
=/(g
)
do,*doJ
G'(&, tb; QN,to
+ NAt) . . . G(Ql, ta + At; O,
t,).
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
266
It should be noted that in order to apply this formula one has to extend the Grassmann algebra to 2N 2 dimensions. Making use of expressions (11.81) and (11.83) leads to an expression which is completely analogous to (11.78) up to a substitution z + B and this in turn leads to the desired expression for the fermionic path integral:
+
Here the integral is thought of as an integral over an infinitedimensional Grassmann algebra. Assuming 0 and O* to be
one can easily recognise in the integrand the following Lagrangian:
L=
& (t)e (t)  e*(t)e (t) 22
 H(O*(t), 8(t)) = it(  H ( t )
.
Thus the path integral takes the form
Note that any discrete finitedifference approximation to the path integral has to include an even number of integrations, as it should for a phase space functional integral. If the Lagrangian also depends on conventional coordinates and velocities, as is the case for the Lagrangians (11.60), (11.61), (11.69), they should be included in the quantisation scheme. It is clear that the path integral over bosonic degrees of freedom can be constructed independently, because the result of integration over the fermionic variables is a functional of the bosonic trajectory x(t). Thus the mixed amplitude
can be represented as follows:
11.4. GENERATING FUNCTlONAL
267
Thus the quantisation of fermions leads to a functional integral over an infinitedimensional Grassmann algebra. The techniques of integration over anticommuting dynamical variables makes it possible to consider interacting fields of different statistics. We consider an example below.
Generating Functional For Fermionic Fields
11.4
The generalisation of expressions (11.84), (11.85) to the case of a system with an infinite number of degrees of freedom is rather straightforward. The Lagrangian of a free Dirac field can be written in terms of Grassmannian variables in Minkowski space as follows:
where and MOis the bare mass. This form of the Lagrangian meets the requirement of Lorentz covariance. The details can be found in any textbook on quantum field theory, e.g., [5, 61. The function $(X) is a column of four anticommuting components $ a ( ~ )so that at each point of Minkowski a,p = 1,2,3,4 is space the Grassmann algebra with basis ($, G;), defined. The bispinors $ furnishes the (1/2,1/2) Lorentz group repreyiyj, sentation space with the standard form of the generators basis "/~yk, where the y, matrixes may be chosen, for example, in the following form:
Here ok are Pauli matrices and I is the 2 X 2 unit matrix. In Euclidean space the Dirac conjugation should be replaced by the Hermitian one that corresponds to the replacement of the Lorentz group by SO(4). The Lagrangian takes the form
where yo = To, yk = 2Yk and summation over p forms the Euclidean scalar product. The generalised minimal action principle gives free Dirac equations in the Euclidean space: (
7
+ M OX
)
= 0,
?Wt(ypdP

MO)= 0 .
(11.91)
268
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
Let us recall that, as follows from (11.90), (11.91), the action vanishes on the classical solution. If $ vanishes at spacial infinity, one can give the action functional a more convenient form by integrating by parts:
where $t and $ should be considered as independent. A straightforward generalisation of (11.M), (11.85) gives rise to the following generating functional:
zht,d= J ~ $ ~ ( x ) W ( ~x )X {sE[l/it,d] P + rlt * d + dt *I)} , (11.93) where 17t,7 are external spinor sources. It should be noted that the path integral over fermions is essentially Gaussian because of the nilpotence of anticommuting fields $. At the same time, there are theories which include such fermion selfinteractions4 as ($t$)2. To include such terms in the functional approach we introduce integration over an auxiliary scalar field which has no kinetic term. Using the Grassmannian integration rules we obtain z[rlt,rll = N exp(rlt * S o * 17) where
,
(11.94)
is an (infinite) normalisation constant and the linear operator5
is determined by its kernel
with the Fourier transform
[email protected]) = (ip
ifi + MO + MO)' = p2 + M,"
where i i a,y,. Note that in the Euclidean theory the integral in (11.95) is not ambiguous because the integrand has no poles on the real axis. 4Although such term makes the theory nonrenormalisable in four dimensions, such models can be of interest in, say, two dimensions or can be considered as effective theories. 5We denote this operator, which is the fermionic propagator, with the same root letter S as is used for the action because this is a traditional notation. To avoid confusion, we retain the subscript 'E' on the Euclidean action.
11.4. GENERATING FUNCTIONAL
269
As in the scalar case, we can calculate average values of any function of dynamical variables according to the rule
where
Here the left arrow denotes the left variational derivative. We must avoid a common confusion with the form of expression (11.97). ~2 is acting to the right like any other operator, but while calculating the result of its action, one has to differentiate Z from left to right. We see that, as in the scalar case, all correlators of the free theory can be expressed in terms of the twopoint function
Nontrivial correlators appear after an interaction is introduced. The simplest type of interaction is the Yukawa coupling with a free scalar field. The total Lagrangian in this case is
Generalising formula (11.93) in view of (11.87) we can write the generating functional in the form
where the action SE corresponds to the Lagrangian (11.99). A further analogy with the scalar case is helpful. For instance, it is easy to derive the generalised formula for the perturbation expansion in the case of a theory with the interaction Lagrangian Lint((plGt1g):
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
270
where Zo is the corresponding free generating functional and account has been taken of (11.97), (11.98). By virtue of (11.94) one can immediately write down the expression for Zo:
where Go and Soare the free scalar and spinor propagators respectively. The generalisation of the formula for correlators is also rather straightforward:
where AI, A~ are defined by (11.98) and
For the theory with Lagrangian (11.99) formula (11.101) gives
Let us calculate the first perturbative correction for the bosonic propagator G(xl  22).
Expanding Z = Zo
+ 21 + . . . in g and writing
we obtain to the first order in g
z 1 = ( 1  9 ~ ) Z 0 = ( 1  9 ~ 1 [ ~ , 1 ) ~ , 720 1) , where
11.4. GENERATING FUNCTIONAL

*      Y
X
So(xvl
Figure 46
and
( G O *J ) , =
d 4 ~ ~ o ( ~  z ) J ,( t )
Applying (11.105) t o (11.106) we see that the propagator and so we have to calculate
21
(11.108)
gives no corrections to
The last term in (11.111) modifies (11.106) as follows: 2 2 =
1 ( 1  ss1 [J, v', v1 + , s 2 ~ 2 ) S
,
(11.112)
where
B 2 [ J vt1 , VI =
J d4x J d4y P [ J ;
X,
~)c(v+~ v;Y )
Y I [ C ( v; V~~
+Tr SO(.  y)So(y  X ) ] (11.113) with
P [ J ; x , y ]= G o ( x  Y ) + ( G o * J),(Go* J ) , . Substituting (11.112) into (11.105), taking account of (11.107) and (11.113) and keeping second order terms yields the following form of the propagator:
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
Figure 47
Figure 48
It is much more convenient to represent such expressions by diagrams. As depicted in figure 46, we represent the fermionic propagator with a dashed line. The coupling constant is represented by a vertex with two fermionic outgoing lines and one bosonic one. Other rules for drawing diagrams are the same as in the purely bosonic case. For instance, each vertex with an internal line corresponds to one integration. For example, the expression for Zo is represented graphically in fig. 47. The propagator, which is the connected part of (11.114), is depicted in fig. 48. Of course, some of the diagrams diverge. The next step should be regularisation of these diagrams followed by their renormalization. Unfortunately, for reasons given below, the theory appears to be nonrenormalisable. The simplest theory of interacting Dirac spinors and scalars looks somewhat more complicated.
11.5
Coupling of the Dirac Spinor and the io4 Scalar Field
In the previous section, we have considered an example of perturbative calculation in the theory with Yukawa coupling of scalars and Dirac
11.5. COUPLED FIELDS
273
spinors. Here we develop the loop expansion for a system of interacting fermions and bosons. It is reasonable to take a renormalisable model. One such model is a system of selfinteracting scalars coupled to Dirac spinors. The Lagrangian is
L
1
X
=5(%~(~))2+z(~2(~)a2)2+$t(~)(~r~~+~o)~(~)+
(11.115) Sometimes this simple renormalisable model is called the SLAC bag model [716. It can also be regarded as a particular case of the so called amode?.
Loop Expansion and Diagram Techniques Let us consider the generating functional (11.100) with this Lagrangian (11.115). The maximum of the exponent in the integrand in (11.100) is provided by a field configuration which obeys the following conditions:
Expanding all fields near this classical solution and keeping terms up to secondorder in deviations, we obtain the generating functional Z[J, rlt, v] = exp( W [J,rlt, 71) in the form
where u(x) is the column
K: is the 5 X 5 operator matrix with kernel
'In the 70s there were attempts to study qualitative aspects of confincment using this model. In this context scalar and spinor fields correspond to mesons and quarks respectively. One has to distinguish this model from a morc phenomenological and more succcssful approach called the MIT bag model [8] 7amodel may include some other terms, e.g. pseudosclar interaction g L p ( x ) $ I t ( x ) ~ 5 $ I or ( x )selfinteracting triplct of bosons.
274
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
where the subscript ' X ' indicates that the corresponding operator acts upon functions of this variable, and the blockdiagonal elements of this matrix are
E?'
=
EX =
1 2
(a; + ~ ~ ~ c p ~ ( X ;) ) ) yPap + MO+ 990 ( X ) . 
(11.122) (11.123)
The first multiplier in (11.119) is determined by the classical contribution
One cannot calculate the Gaussian integral in (11.119) because of cross terms due to the offdiagonal elements in the block matrix (11.121). To tackle this problem it is necessary to make the matrix K: blockdiagonal. This is done by introducing a new variable
As in (11.108)  (11.110) the star product in the last term of (11.124) denotes a convolution. Since the shift term in (11.124) is a constant in the function space, the Jacobian of the transformation is unity and (11.119) takes the form
Here the operator
is determined by its kernel
where S ( x  y) is the kernel of the operator E'. Now the integration in (11.125) can be easily performed. The result for W is
+
1 In det K^(cpo, 2

+A, Go)  In det
As in the scalar case, we conclude that it is more convenient to deal with the effective action l? than with W because I' depends on average fields (cp(x)) = 4 ( x ) , ( + ( X ) ) = < ( X ) , and ( @ ( X ) ) = < + ( X ) which may
11.5. COUPLED FIELDS
Figure 49
enter experimentally measurable quantities. In the present case defined as follows:
r
is
+
Recollecting thats 4  cpo ti and recognising in the sum 'Wo current terms', the first two terms of the Taylor expansion of r' near the point (vo, $0) (cf. chapter 6), we obtain the first quantum correction in the form:
$A,
r'[q5,C',
,
C] = S[$,lt,(l + 1 lndet g(q5,C',
()

lndet E($) . (11.130)
$Ll
We shall quantise near the classical solutions cpd (X), ( X ) , $Jcl (X) obeying the equations of motion (11.116)  (11.118) with zero right hand sides. This means that the average field should be represented as a sum of the classical solution and a small deviation:
From our experience of instanton calculus in quantum mechanics (see chapter 3) we expect that the calculation of determinants in the case of a nontrivial classical background may be rather complicated. For this reason we take the trivial vacuum background pcl(x) = a,
(d (X) = v =
const
,
where a is the minimum of the selfinteraction classical potential. Furthermore, we put the vacuum expectation value of the fermionic field to 8 0 f course, the same cstirnate is true for the differences C 
and ~t

&.
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
Figure 50
zero: v = 0. Then, the kernels of  Y)
K(x, Y) =
K^ and D^ take the form
(a: + m; + ~ ( x ) ) 2g2 t t ( x ) s ( x  Y)C(Y),
+
where Ml = MO ga. Let us start with the simpler calculation of the oneloop corrections which is given by lndet D^. It is possible to apply the same techniques as in the case of the pure (P4 model (see chapter 6):
l
Here S1 is the kernel of the operator D1 and 1 denotes the unit operator. The first term does not contain ~ ( x and ) contributes to the effective potential. Expanding the logarithm, we get for the second term
If we represent S1with an internal dashed line, we obtain the graphic representation (11.136), given in figure 49. The calculation technique for In det g is basically the same:
where
11.5. COUPLED FIELDS
277
Figure 51
Along with (11.135) the first term here contributes to the effective potential. By analogy with (11.136) we get for the second term in ( 11.137)
Substituting expression (11.138) for R, we obtain the second part of the oneloop corrections to the effective action. As the result is very long, we prefer to present it in diagrammatic form. For this purpose we use and an internal dashed line a directed external dashed lines for and with a cross for the S which is contained in R. Expressing the Green function S in terms of Sl one can express the line with a cross in terms of crossless and external bosonic lines:
The symbolic form of this relation is depicted in figure 50. Accounting for (11.139), (11.140) and (11.138) leads to the series of diagrams shown in figure 51 in which all diagrams up to the third order are included. Substituting the expression for R (11.138) into (11.139) and making Fourier transformation of all multipliers in the integrand of (11.139), we come to the following Feynman rules.
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
278
1. External lines:
(a) ~ ( k corresponds ) to a solid line; (b) Jt(q) and J(q) correspond to directed dashed lines with opposite directions. 2. Internal lines: (a) &(g) corresponds to an internal dashed line; (b) Go(k) corresponds to a internal solid line.
3. Vertices: (a) A vertex with three solid lines gives the multiplier Xa; (b) A vertex with four solid lines gives the multiplier X/2; (c) A vertex with one solid line and two dashed lines gives the multiplier fig. 4. Integration over all internal momenta is assumed, as is summation over all spinor indices.
5. Each vertex gives the multiplier ( 2 ~times ) ~ the Sfunction of all incoming and outgoing momenta expressing the energy conservation law.
6. Each fermion loop appears with the additional multiplier 1.
Analysis of Divergences Let us analyse the divergent diagrams additional to those of the pure (p4 theory considered in chapters 5 and 7. To do this in a general way, let us consider an arbitrary theory of interacting scalars and fermions in d dimensions. Let there be s types of vertices such that the kth vertex has Bkbosonic and Fk fermionic lines. We consider an arbitrary diagram containing L loops, Ib internal bosonic lines, If internal fermionic lines, Vk vertices of the kth type, E6 external bosonic lines and Ef external fermionic lines. The integral representing the diagram contains L ddimensional integrations over momentum p of Ib scalar propagators each of which gives the contribution p2 on the asymptotic and If fermionic ones giving p1. Thus the integral behaves at large momenta as pD where
11.5. COUPLED FIELDS
279
and is called the superficial degree of divergence (SDD). If D = 0 the diagram diverges logarithmically, if D > 0, it diverges as Let us now express the SDD in terms of the number and type of the external lines and vertices. For this purpose we make use of the Euler relation for a planar graph which relates the total number of vertices to the number of loops and internal lines:
(cf. chapter 9). On the other hand, if one counts the sum over all k of VkBkit is equal to the total number of external lines plus twice the number of the internal ones, this is because each internal line joins two vertices and thus is counted twice. Therefore
Analogously,
C VkFk = E f +21f
.
(11.144)
k
Eliminating L, Ib and I f from (11.141)  (11.144) we get
D
=
d
dl d2 Eb  Ef 2 2
where the relation
is used. It should be pointed out that negative D does not prove convergence of the diagram. The Weinberg theorem states that any diagram of an arbitrary theory converges if and only if D is negative both for the whole diagram and for any its subdiagrams. A subdiagram is a diagram derived from the initial one by cutting some internal lines. Of course, we can more easily establish whether a given diagram diverges or not using (11.141) rather than (11.145). Nevertheless, the last formula is useful for analysing the number and types of divergencies in the whole theory. One can easily see that if the kth term in the sum in (11.145) is positive then the SDD increases with the number of vertices of this type and that means that the theory is nonrenormalisable. Let us consider a few examples. For the purely scalar theory considered in chapter 7 , E f = Fk = 0 and the problem reduces to deciding
280
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
the sign of the expression B (d/2  1)  d. In this way we conclude that the only divergent diagrams for d = 2 are tadpolesg and so in two dimensions any scalar theory is renormalisable. In three dimensions the maximal power n of cp is 6, for d = 4 it is 4 and there are no renormalisable scalar theories for n > 6. In theories of selfinteracting scalars and Dirac spinors, there are three types of vertices B = 3, F = 0; B = 4, F = 0 and B = l, F = 2. As the first two cases have been already considered in the previous chapters, we have only to analyse the latter. For this vertex
This formula does not contain V which gives hope that the theory is renormalisable. The formula enables one to find all new divergent oneloop diagrams. Here they are: 1. The tadpole Ef = 0, Eb= 1,D = 3 (fig.52a). It vanishes since it is ) is assumed to be zero. proportional to the integral of ~ ( xwhich
2. The fermionic 'fish' Ef = 0, Eb= 2, D = 2 (fig.52b).
5. The fermionic proper energy diagram for Ef = 2, Eb = 0, D (fig. 52e).
=
1
6. The vertex diagram Ef = 2, El, = 1,D = 0 (fig.52f). There is one more important conclusion following from (11.145). It is not difficult to check that the coefficients of Vj in the third term of (11.145) are nothing but the dimensions of the corresponding coupling constants in units of length, or, in other words, in energy units with the opposite sign. Thus we can reformulate the main conclusion of this section in a more convenient form: the theory is nonrenormalizable if at least one of its coupling constants has a negative dimension in units of energy. It should be emphasised that the opposite statement is not true. For instance, the simplest theory with Yukawa coupling described by Lagrangian (11.99) has dimensionless coupling constant g. Nevertheless, there is a logarithmically divergent diagram with four bosonic legs (the same as that in fig. 52d) that requires a counterterm proportional to 'Such divergences can be removed by means of normal ordering of the Lagrangian. They are called divergences of normal ordering.
11.6. EFFECTIVE POTENTIAL
Figure 52
which is absent in the bare Lagrangian. It is clear that in the case of (p4 coupled with Dirac fermions all the divergent diagrams require counterterms of types which are already present in the bare Lagrangian (11.115). Hence, this theory is renormalisable. Thus we come to the conclusion that the dimensional analysis of the coupling constants of the theory is a preliminary 'negative' test for the theory. A constant with negative dimension in units of energy is a guarantee of nonrenormalisability of the theory. Conclusion of renormalisability can be made only after structure analysis of the counterterm Lagrangian.
11.6
Fermion Contribution to the Effective Potential
Here we would like to analyse the first term in (11.135) and in (11.137). These two terms determine the oneloop effective potential of the theory. Let us start with the more general situation of nonzero (but constant) = v. In this case the vacuum expectation value of the fermion field oneloop corrections to the effective potential can be written down as follows: 1 = lndet(8; m:  2y2vt,!?1u) lndet El , (11.147) 2 The symbol 'det' in the last term of (11.147) presupposes taking the determinant in both the functional space and the fourdimensional linear 
+
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
282
+
space. If the M1 included in D1 is constant (since Ml = MO gcpd, it is constant if cp,~ = const), these two operations are independent and one can take the fourdimensional determinant first and the functional afterwards. Doing that, we obtain ln det
6 = In det(%d, + Ml) = 2
where the formula det(a,y,
+ M;)
~ln(8; r
+ b) = (b2

,
(11.148)
(11.149)
a')'
has been used. Here a,, b are arbitrary complex numbers. Representing 91 as I h
& = E;1 = D ~ ( D ~ D ~, )  ~ where
A
D1 = y,d,
+M1
,
we get for AUef 1 AUef =  ln det[ebcf 2
+ 2g2(a,d,  P)]  25 ln det ef 
,
(11.150)
where
Note that a firstorder differential operator appears in the first term of (11.150). This leads to an imaginary term in the logarithm when it is calculated in the momentum representation (see chapter 5). This in turn means that the effective potential has an imaginary part or, in other words, indicates an instability of the bosonic vacuum. Thus we come to the conclusion that the nonzero vacuum expectation value of the fermion field destroys the scalar vacuum [17]1°. We do not consider this phenomenon and restrict ourselves to the case v = 0. That calculation reduces to the case investigated in chapter 5. Problem: Obtain the regularised expression for Ueff using the results of chapter 5. 1°So called chiral fermions, for which vty,v = 0, are an exception to this.
11.6. EFFECTIVE POTENTIAL
283
In the interest of presenting various techniques let us use another method to calculate the functional determinant (see e.g. [H]). Consider an arbitrary operator A^ with discrete spectrum
For the determinant of
A^, we obtain the expression
where the function
m
C&)
Anz
=
(11.155)
n= l
is called the <function of the operator A^. The determinant calculation is based on a representation of CA in terms of a Green function related to the operator A^. Consider the fundamental solution of the so called heat equation for operator A^:
The solution G, called the heat kernel, takes the form
where f,, are the eigenfunctions of
A^ such that
(d is the space dimension). It is straightforward to check using (11.158)
(11.159)
Let
If G is known, one can obtain < A and hence the desired determinant. A^ be A^=~;+C , (11.160)
where C is a function of the constant average field. For this case the heat equation (11.156) can easily be solved in the same way as for the
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
284
solution of the Schrodinger equation (2.12). (Problem: do it.) The result
Substituting this form into (11.159) with d = 4 results in
As we could expect (see chapter 5), this expression diverges like the 4volume. Thus we obtain a regularised expression for the density of In det A^: 1 In det (8; VT

C2 3 + C) = + In C) 32r2 ( 2 
.
This equation looks awkward because of the logarithm of the dimensionful quantity C (whose dirnensionality is l/length2 = mass2). The appearance of In C means that the answer depends on an arbitrary constant with the dimension of mass, the unit of mass in which C is expressed. The origin of this is the logarithm of dimensionful X, which appears in (11.154). The mathematical reflection of this fact is the %dependent dimensionality of CA(%)in (11.155). The arbitrary constant in the result (11.163) is not harmful because only the differences of functional determinants enter the final answers. In such differences, the ambiguity disappears. However, it is reasonable (and common) to write the arbitrary constant explicitly rather than to refer to the unit of mass. Let us denote such a constant by A. It can be introduced in (11.155) in order to make that expression dimensionless. To do so, we multiply both sides of (11.155) by A2" and differentiate A2"CA(z)instead of CA(%)in (11.154). This modifies the results (11.162) and (11.163) to
and 1 
lndet(8:
VT correspondingly.
C2 + C) = 32n2
3
+ 1"
)A2C
Problem: How to use (5.60) to check this result?
(11.165)
11.6. EFFECTIVE POTENTIAL
285
Now we have to substitute into (11.165) first Cband then Cfto derive the desired effective potential. Here it is worth recalling that the parameters m0 and Ml appearing in the expressions for Cb and Cfdepend on the average field 4 as follows:
For the sake of simplicity we take the value zero for MO and for the parameter a in U(4). After that we come to the following form of the effective potential
( l l.166) Defining the physical parameters as
we get m2 = 0 and
Then, the effective potential expressed in terms of physical constants reads [l91
Here the coupling constant g remains unrenormalised. Note that the contribution due to the interaction with fermions has a negative sign and decreases the scalar vacuum energy as 4 grows [17]. It follows from (11.167) that change in the renormalization point M leads to the following change in X:
Thus we come to the following ,&function:
286
CHAPTER 11. PATH INTEGRAL FOR FERMIONS
which, according to equations (8.10) and (8.11), determines the running coupling constant. We see that the sign of the fermionic contribution to the effective potential and to the pfunction is opposite to that of scalar bosons. The one loop corrections in (11.168) and the righthand side of (11.170) turn to zero at X = 4g2. This does not imply supersymmetry yet (cf. the discussion in chapter 5 and 7). Beyond a specific relation between the coupling constants, the supersymmetry requires tightly correlated forms of the bosonic and fermionic sectors [20]. It is remarkable that because of the supersymmetry of the model the higherorder corrections do not contribute to the pfunction. Thus the oneloop result remains exact in all orders of the loop expansion. This was shown by using instanton methods [21, 241 and by referring to the perturbation theory [22] and by using symmetry arguments [23]. Moreover, the supersymmetric oneloop pfunction in general does not vanish. It turns out to be zero only if there is an additional (conformal) symmetry of the model1' (see for example 1241). In conclusion, let us note that the <function regularisation, although effective and elegant, can be applied only after the analysis of divergences is already done. The reason for this is the same as for the dimensional regularisation discussed in section 7.3. We can also get nice finite results for a nonrenormalizable theory because the structure of divergences is hidden by the method of regularisation. Besides that, the problem of obtaining exact solutions of the heat equation (11.156) is not always as easy as in the case considered. In this connection we would like to note that the general expression for the heat kernel (11.156) is nothing but the quantummechanical propagator of a fictitious particle moving in the potential Uf'(x). To solve the heat or Schrodinger equation with this potential is unavoidable in any method for calculating the functional determinant.
''A famous example of such a model in four dimensions is the N=4 SUSY YangMills model [25] which is a supersymmetric analogue of &CD.
Bibliography [l]F.A.Berezin, The Method of Second Quantisation, New York, N Y ,
Academic Press, 1966. [2] F.A.Berezin, Introduction to Superanalysis, Dordrecht, Reidel, 1987. [3] A.M. Perelomov, Generalized Coherent States and their Applications, Berlin, SpringerVerlag, 1986. [4] M.A. Markov (ed.), Squeezed and Correlated States of Quantum Systems , Commack, NY, Nova Science, 1993. [5] J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics and Relativistic Quantum Fields, McGrawHill, 1964 and 1965. [6] C. Itzykson, J.B. Zuber, Quantum Field Theory, McGrawHill, 1980. [7] W. Bardeen et. al., Phys. Rev. D11 (1975) 1094. [8] A. Chodes et. al., Phys. Rev. D9 (1974) 3471. [g] The first systematic consideration of the PI quantisation of fermions by means of Grassmannian variables and of the consequent formal classical limit is in the papers by R. Casalbuoni, Nuovo Cimento 33 A (1976) 115; 389. [l01 Grassmannian variables have been introduced in quantum mechanics by A.F. Berezin and M.S. Marinov, Ann. Phys. (NY) 104 (1977) 336. [l11 Grassmannian variables have been introduced in field theory by L.Brink, P. Di Vecchia and P.Howe, Nucl. Phys. B136 (1977) 76.
[l21 Some earlier Grassmannian variables were introduced in quantum physics in connection with the idea of supersymmetry: D.V. Volkov and V.P. Akulov, Phys. Lett. B46 (1973) 109.
288
BIBLIOGRAPHY
[l31 The very idea of an extention of the Poincari: group by adding fermionic generators was put forward by T.A. Gol'fand and E.P. Likhtman, Sou. Phys. JETP Lett. 13 (1971) 323. The modern formulation of SUSY was given by G.Wess and B.Zumino, Nucl. Phys. B70 (1974) 39. [l41 The structure of the effective action via loop expansion was investigated by T.D. Lee and M.Margoulis, Phys. Rev. D11 (1975) 1591. [l51 M.Margoulis, Phys. Rev. D13 (1976) 1621. [l61 It would be instructive to see a practical calculation of the Smatrix elements using the generating functional. An example with an application to pionnucleon scattering in a model with Yukawa coupling of Dirac spinors and scalars without selfinteraction in given in a book by L.H. Ryder , Quantum Field Theory, 2nd ed., Cambridge University Press, Cambridge, 1996. [l71 R. MacKenzie, F. Wilczek and A. Zee, Phys. Rev. Lett. 53 (1984) 2203. [l81 It was J. Schwinger who constructed the effective action for QED and introduced this method of calculating the effective potential: J. Schwinger, Phys. Rev. 82 (1951) 664. For a general review of the zetafunction approach see: E. Elizalde et. a1 Zeta Regularzsatzon Technzques wzth Applzcatzons, Singapore TVolltl Sc ~ r n~tf i c 1994 [l91 ShauJin Chang and TungMow Yan, Phys. Rev. D12 (1975) 3225 [20] J.Wess and J.Bagger, Supersymmetry and Supergravity, Princeton University Press, 1983. [21] V. Novikov, M. Shifman A. Vainstein and V. Zacharov, Nucl. Phys. B229 (1983) 381. [22] M. Grisaru, W.Siege1 and M. Rocek, Nucl. Phys. B159 (1979) 429. [23] N. Seiberg, Phys. Lett. B318 (1993) 469. [24] M. Shifman, Prog. Part. Nucl. Phys. 39 (1997) 1. [25] K. Intriligator and N. Seiberg, Nucl. Phys. Proc. Suppl. 45BC (1996) 1.
Chapter 12 Gauge Fields In this chapter we consider gauge fields. The simplest example of a gauge field is the electromagnetic field. A straightforward generalization of the boson quantization considered above is not possible for these fields, because, like fermion systems, gauge theories are singular. A theory is called singular if there are velocities which cannot be expressed in terms of coordinates and momenta and so the standard Hamiltonian representation is not plausible. The quantization of such fields require a more serious modification than was necessary for the case of fermions. Let us start by recalling some facts about classical gauge theory.
12.1
Gauge Invariance
The Basic Idea The observables in quantum mechanics do not depend on the phase of the wave function, but do depend on the phase difference between two subsystems. From the point of view of locality it is not elegant for an effect to result from a phase difference between two particles on the opposite sides of the Earth. It would be more consistent with our experience with special relativity if the wave function phase could be varied independently in different spatial locations. That implies the possibility of multiplying the wave functions by a factor with a coordinate dependent function a ( x ) ,without changing any observables. Such a symmetry is called local in contrast with a global one which only allows the same phase transformations at all space points. There are both kinds of symmetry in Nature. A famous example is the local gradient invariance of electrodynamics. In electrodynamics, all observables are invariant under so called gauge transformations which are the simultaneous multiplication of the wave
290
CHAPTER 12. GAUGE FIELDS
functions of all particles by a factor exp[iea(x)] where the integer e characterizes the particle species and is called the electric charge (measured in units of the elementary charge). Attempts to explain the great variety of elementary particles have lead to a generalization of the notion of charge1. It was observed that there are groups of hadrons whose members behave identically under the strong interaction and can only be distinguished by other interactions, e.g., the electromagnetic one. Such groups are called multiplets. The simplest example is a couple consisting of a proton and a neutron. The wave function of this pair can be written as a twocomponent column:
where $p and 1CI, are Dirac spinors describing proton and neutron respectively. The Lagrangian for the strong interaction has to be invariant under all possible rotations mixing $p and $,: f + U f where U is any element of the corresponding group; SU(2) in the case being considered. The space spanned by $p and $, is called the isotopic space and the symmetry with respect to transformations in it is isotopic symmetry. Generally, the symmetry group is a Lie group. Those are groups in which any element U can be represented in the form U = exp(iakAk), where k = 1. . . N, a,+are some numbers, and Ak are operators called generators. The generators must form a closed algebra
called a Lie algebra where fkl, are structure constants. The eigenvalues of Ak are the generalized charges of a given group. The well established gauge symmetries are those of electrodynamics (with the group U(l)), a part of weak interactions (with the group2 SU(2)), and the strong interactions (with the group SU(3)).
Example of a Globally Invariant Lagrangian As an example, let us consider a theory which is invariant with respect to a gauge group G. Let the model include a multiplet of Nf Dirac lPhysics of elementary particles is beyond the scope of this book. In this chapter, we consider only the quantixation procedure which is necessary for applications. The books in elementary particle physics are numerous, see, e.g., 13, 4, 51. 21n fact, the weak and electromagnetic interactions are mixed in a combined group SU(2) X U(1). We do not discuss this further.
12.1. GAUGE INVARIANCE
291
spinors interacting with a multiplet of Nb charged scalars by the Yukawa coupling. The Lagrangian of such a system can be written in the form (we start here in Minkowski space)
where 9 and
are columns
y, are the direct products of the conventional ymatrices by the unit matrix of the dimension Nf and summation over n is assumed. M is a mass matrix. We assume for simplicity it is proportional to the unit matrices acting on the Dirac spinors and isotopic indices 3. V(@) is a selfinteraction potential and A, ,n = l , . . . N6 are Nf x Nf matrices providing the coupling of bosons and fermions, &, = yoALyo. The requirement of renormalizability restricts the potential V(@)in four dimensions to polynomials of fourth order. A simple form is
which is the natural generalization of the doublewell potential for one scalar field. Each multiplet transforms under a unitary irreducible finitedimensional representation of G. Isotopic symmetry of the Lagrangian means that it is invariant under the transformations
where U and S are matrices of the corresponding representations:
and g is the coupling constant. The Nf X Nf Hermitean matrices T, and the N6 X Nb Hermitean matrices L, are the generators of the Lie algebra4 g in the Nf and Nbdimensional representations of G respectively:
31sotopic symmetry presupposes that, in the absence of other interactions, the masses of particles with the same isospin are equal t o each othcr. 4The necessary information about Lie algebra is summurized in appendix C.
CHAPTER 12. GAUGE FlELDS
292
W,, a, are real parameters and N is the dimension of g. Such an invariance of the Lagrangian (12.1) is called a global gauge invariance. To provide this invariance the matrices A, must obey the condition
Example of a Locally Invariant Lagrangian Let us make the Lagrangian (12.1) locally invariant5 under transformations ( l 2 . 3 ) , (12.4) If we substitute a point dependent transformation into (12.1),we obtain
The term B, violates the gauge invariance of the Lagrangian. To save the gauge invariance it is necessary to introduce a new field which compensates the invariancedestroying term B,. It is obvious that this compensating field must be gvalued like B,. Let us introduce such a field as follows: a,, + D, = a,  igAIL(x) . (12.7)
A,, the element of G is determined by a set of functions AE(x) so that A,(x) = A: (x)T, if D, is acting on Q and A,(x) = AE(x)L, in D,@. The functions AE(x) are called the gauge fields. The constant g is introduced by analogy with electrodynamics where the minimal coupling is 8, + 8,  ieA,(x). One can easily check that the Lagrangian
where
is gauge invariant if the transformation law of the nonAbelian gauge potential is 1 A; = U A , U ~ ,(a,u)ut . (12.12) 29
+
5 ~ h i symmetry s do not yield a new conservation laws rather it results in modifications of the system dynamics as described below in this chapter. Such situations are the subject of the second Noether theorem (see also the footnote on page 13).
12.1. GAUGE INVARIANCE
Lagrangian of Gauge Fields The price paid for making the Lagrangian (12.1) gauge invariant is the introduction of a new gauge fields. These fields remain auxillary or unphysical as long as there are no corresponding kinetic terms. Let us construct such a term. Besides elegance, the main reason for such an extension of the system is the experimental success of the resulting theories. The kinetic term must fulfil the following conditions:
1. It must have the same dimension as all other terms. The simplest choice is to take it quadratic in d,Az(x); 2. It must be both a Lorentz and a gauge invariant quantity;
3. It must coincide with the Lagrangian of the free electromagnetic field if the gauge group is chosen to be U(1). These requirements lead to the following Lagrangian for the gauge fields:
where G,, is a nonAbelian analogue of the field tensor; the specific normalization of the generators:
is assumed. We represent G,, in the form
G,,
= F,,
+ KPV ,
(12.14)
where
F;,
= &A;

d,AI",
and K,, is an additional term which we shall find below. This term must be zero if G is an Abelian group. It follows from (12.13) and the gauge invariance that G,, is transformed according to the adjoint representation of the group G
This restricts essentially the class of gauge groups to be used. Indeed, the adjoint representation of an arbitrary Lie group, U, is not unitary in
CHAPTER 12. GAUGE FIELDS
294
general (see appendix C). However the unitarity condition is one of the fundamental restrictions on physically selfconsistent theories. Thus we must restrict ourselves to semisimple groups (besides the trivial case of the group U(1)). For semisimple algebras, the generators of the adjoint representation are Hermitean because of the complete antisymmetry of the structure constants, this provides unitarity. Any semisimple Lie algebra is a direct sum of some number of simple ones. Thus, the corresponding generators can be represented in the blockdiagonal form and the symbol g must be, in general, a diagonal matrix constant on each component of the direct sum. This follows from the fact that each orthogonal term of the direct sum can be independently multiplied by a scale factor. Therefore, it is reasonable to introduce this factor explicitly. In other words, in the case of semisimple gauge group there may be just as many charges g as there are terms in the decomposition of the group in the direct sum of simple terms. In what follows we shall suppose the gauge group to be simple and compact6. Let us derive G,, using (12.14), (12.16). A; in (12.12) reads A:, = A,,+B,
,
(12.17)
where A, = UA,U~ and B,, is defined by (12.6). In this notation
F;, = UF,,,ut
+ [B,, A,]  [B,, A,,] + (llig) [B,,, B,]
.
(12.18)
On the other hand,
Combining (12.18) and (12.19) we find the combination which has the proper transformation properties for the adjoint representation:
Thus K,, = ig [A,, A,]. Finally the field strength tensor is
or in component notation
6Noncompact algebras results in unphysical states such as those with negative norm. We do not consider such algebras here.
12.1. GAUGE INVARIANCE
Expression (12.21) can be also represented in the form
It is easy to check that this strength tensor is zero if A, is a pure gauge, i.e. if A, = (d,U)Ut. Let us discuss some properties of the strength tensor. Geometrical Meaning of t h e Field Strength Tensor Gauge transformations result in infinitely many representations of a given field configuration. Thus, we must pay special attention to the quantities which are invariant under gauge transformation because only those can be observable. There is a known analogy with differential geometry where parallel transport of vectors is a basic concept. Let us define the parallel transport of q so that Q is the subject of parallel transport if it is a covariant constant:
For finite parallel transport along a smooth curve x(t) we get Q(x) = T exp
dt
where xo = x(to), X = x(tl) and T denotes the time ordering. We can regard this as a definition of the path (P)ordering appearing here because of the obvious fact that A, does not commute with itself at different points X. Then the tensor G,,, being the commutator [D,,, D,], can be interpreted as the curvature. Indeed, for an infinitesimal closed contour defined by the vectors Sa,, 6,B, we get
where do," is the infinitesimal element of a surface C. It is just the geometrical definition of the curvature 2form. Thus, the A1;(x) play the role of the connection coefficients. For a detailed review of the geometric interpretation of the gauge fields see [ll].Formula (12.25) to be used to express the Pexponent for a finite closed curve C in (12.24) in terms
CHAPTER 12. GAUGE FIELDS
296
of G,,. Triangulating the surface C bounded by C by means of small surfaces with area elements do," and applying (12.25) one obtains a nonAbelian analogue of Stokes theorem
Dual Strength Tensor and Topological Current From expression (12.23) follows the 'kinematic' equation of motion for G,, which in Abelian case corresponds to the second pair of the Maxwell equations. Indeed, if the potential A, is differentiable everywhere then the Jacobi identity for covariant derivatives is E,,,
[D", [DP,D"]
=0
.
This gives, together with (l2.23), the Bianchi identity for the field strength tensor (12.27) DpG,v+DpGvp+DvGw=O , or just
D~G,, = o , where G,, denotes the tensor dual to G,,
The term T~(G,,G,,) could also be included in the Lagrangian, but we shall show that it is a 4divergence. Indeed, 1 T~(G,,G,,) 2
=
4
F,,F,,

2igF,,[A,, A,]
(12.30)
where F,, is defined by (12.15). Evaluating each term inside the brackets
12.1. GAUGE INVARIANCE
and we obtain finally
using the cycle permutation symmetry in the second formula and Jacobi identity in the third one. In this expression, the vector k p is called the topological current
or, in components,
This terms is usually omitted in the Lagrangian as it does not affect the equations of motion. However, there are cases when this term is physically relevant, for example, if we consider gauge theory in a nonsimply connected area7. Besides that, as we shall see in the next chapter, the integral of over 3dimensional space gives a conserved quantity called the topological charge. It provides a topological classification of classical solutions of pure gauge theories.
The Total Lagrangian Thus, the correct gauge invariant Lagrangian of the system under consideration reads L = Lf Lb+ Lbf Lg , (12.34)
+
+
where Lf , Lb, Lb are determined by (12.9) (12.9) and L, is given by (12.13). Finally, let us summarize the transformation laws for the field variables appearing in (12.34) under an infinitesimal gauge transformation parametrised by wa(x). Following from (12.3), (12.12) they are: 
6Gr(x)
=
gfabcwb(x)G?(x)
,
(12.36)
71n this case one cannot turn to a 3sphere of infinite radius when applying Stokes theorem because the boundary of a volume appearing in this theorem consists of a t least two parts and one of them is not this sphere. Hence, one cannot drop the current k , by requiring its 'ingredients' t o fall off sufficiently quickly a t infinity. The requirement of vanishing k,, on all parts of the boundary can as a rule, only be fulfilled trivially (k,, = 0).
CHAPTER 12. GAUGE FlELDS
12.2
Dynamics of Gauge Invariant Fields
Equations of Motion Let us find which equations of motion correspond to the Lagrangian (12.34). Taking respectively variations of the G, 9, @ t , @ and A, components, we obtain
Here j, is the current of matter fields, consisting of the fermionic and bosonic parts
where the bold letters stand for isotopic vectors. Note that, in contrast to the Maxwell theory, the gauge covariant current j i is not conserved. We can also construct the conserved current
12.2. DYNAMICS OF GAUGE INVARIANT FIELDS
299
but its gauge transformation law is not covariant. On the other hand, the nonAbelian current is covariantly conserved, because according to (12.43)
The YangMills Equations Equations (12.39)  (12.43) together with (12.28) give a complete description of the system with Lagrangian (12.34). The set of equations encoded in (12.43) are called the YangMills equations. Let us consider these equations in detail. For this purpose we represent equations (12.28), (12.43) in 3dimensional form. To do that we need the following convention. We introduce two kinds of bold letters with greek and latin indices. The first ones (for example, A,) denote a Lie algebra valued component of a Lorentz vector, the second ones (e.g. Ab) correspond to a threedimensional spatial vector Ab with an additional isotopic index b. By analogy with Maxwell electrodynamics we introduce colored electric and magnetic fields
or, in the form with explicit group indicies:
In this 3dimensional notation we can see a onetoone analogy with the expressions for the Lagrangian and the topological term in Maxwell electrodynamics. Indeed,
It is easy to see that the dual tensor G," can be obtained from GP" by the discrete dual transformation
as was the case in the Abelian theory.
CHAPTER 12. GAUGE FIELDS
300
Using (12.50), (12.51) we can get from (12.43)
where the charge density is p,(x) = j:(x). An analogous transformation of (12.28) leads to the second pair of equations
VB, rot E,
+
aBa

at
+ gfabcAbBc =
+ g f,b,(A;~, + AbA E,)
=
0 ,
(12.55)
0 .
(12.56)
Although these equations resemble the usual Maxwell equations of electrodynamics, there are two essential differences. First, they include the potential along with the electric and magnetic fields. Second, the divergence of B , is not zero as was in the Abelian theory:
As the right hand side term is generally speaking not zero there are solutions with nonzero magnetic charge in nonAbelian gauge theories. We consider such a solution in the next chapter.
The Total Energy Let us now obtain the expression for the energy of the system. To do that we must apply the canonical formula for the Hamiltonian density
where U denotes the set of all dynamical variables, A is a collective index labeling Lorentz and isotopic degrees of freedom as well as the species of a field, and T A = d L / 8 u A is the canonical momentum. First we should calculate the momenta corresponding to the entire set of the dynamical variables of the theory: 1. for T : n = ( i / 2 ) y o q ; 2. for Q : n = (i/2)GYo; 3. for @t : n = [email protected]; 4. for cP : n = ([email protected])?; 5. for A,: n, =E,, 7Fjl = 0 ,
12.2. DYNAMICS OF GAUGE INVARIANT FIELDS
where the Dirac matricies result we get
74
and
yk
301
are defined in (11.89). Using this
where j," is defined by (12.44)(12.46) and
where a k = y 4 y k and P = 7 4 . Expressing dA,/dt from (12.50) we have for the last three terms:
where
'FI,
=
1 (E:
+ BE)
(12.62)
.
The last term vanishes because of the first Yang Mills equation (12.53). Thus, neglecting the divergence which vanishes after integration over the volume, we get finally 
where the terms of the sum are determined by (12.58) (12.62).

(12.60) and
Gauge Freedom and Gauge Conditions Thus we have derived gauge covariant equations of motion (12.39)(12.43), (12.28) for the system being considered. These equations obey the fundamental physical requirement that there is no prefered basis in the isotopic space of the matter fields. However these equations have infinitely many solutions labelled by a functions of spacetime points. Indeed, any solution can be gauge transformed without changing the physical state of the described system. It is useful to restrict the class of possible solutions in order to make them unique (or at least countable) for a given state of the system. In other words, only one representative should be taken
CHAPTER 12. GAUGE FIELDS
302
from each class of gauge equivalent field configurations. To this purpose, we impose an additional condition on the gauge and matter fields:
+
where A and denote the gauge and the matter fields. (12.64) is called the gauge condition. As the gauge group contains N parameters, it is clear that the number of gauge fixing conditions for each fixed X must be N as well, i.e. Y must be gvalued. In order to fix all N parameters w,(x) of the gauge transformations, Y(A, 4, X) should be a nondegenerate function8: det
1
lpO
S Y , ( A ~ ( ~+U(x), ) , X) Sub(Y
This condition means that the operator
has no zero modes in its spectrum. In component notation the expression (12.66) has the following form:
To find variational derivatives of the field variables in (12.67) we must use expressions (12.35)(12.35) for the infinitesimal transformations of Az(x) and &(X):
Thus x b takes the form
Let us present the most widely used gauges in electrodynamics and pure YangMills theory: his fixes small gauge transformations. However, there may exist many solutions t o equations (12.64) with significantly different fields. This does not affect the usual perturbative calculations in quantum field theory.
12.2. DYNAMICS OF GAUGE INVARIANT FIELDS
1. d,Ag
= 0,
the Lorentx gauge9;
2. VA,
= 0,
the Coulomb gauge;
3. n,Az
= 0,
nf'n,
303
= 1, the axial gauge;
4. At = 0, the temporal (or Hamiltonian) gauge;
5. D,AE = 0 , where D, = 8,  igB, with is a fixed external field B,(x). This gauge is called background gauge; 6. D,Ag
+ Eq5,
= 0,
where ( is a parameter, the 't Hooft gauge.
The gauge operator Fahfor the Coulomb gauge does have zero modes [12]. This follows simply from the fact that the equation TA, = 0 has many solutions for a given G,,. Fortunately, the Coulomb gauge is unambiguous for perturbative calculations, i.e. for small Ag(x). Let us As follows from (12.70), it reads write the explicit form of
xb.
where A denotes the Laplacian with derivatives with respect to y. An attempt to generalize the PI quantization procedure to gauge fields by direct analogy with scalars encounters a serious difficulty. For example the 'naive' expression for the Euclidean generating functional in electrodynamics
diverges because the operator of the second variation of action has zero modes. This follows from the obvious expression

1 1  4/ ~ t , ( x ) d ~ x =   A2 , * K , , r A , ,
The operator
A
where
K,,,=6,,8:+Q3,. (12.73)
g,, vanishes on all 4gradients:
where f is an arbitrary smooth function. This difficulty makes it necessary to return to the more fundamental definition of PI in the Hamiltonian form. But in order to use the Hamiltonian formalism we must gAs we shall see, this condition could be used as a constraint. However, the separation of constraints and gauge conditions is conventional because both groups enter the complete set of the secondorder constraints equally  see section 12.4.
CHAPTER 12. GAUGE FIELDS
304
express the velocities A, in terms of the canonical momenta xP. As mentioned above, in this case is identically zero (see page 300). Because it is impossible to define unambiguously four independent variables related by only three equations, the 'hamiltonization' of the system is impossible. To tackle this problem we must consider the quantization of systems with singular Lagrangians. As will be shown in the next sections, this results in the quantization of systems with constraints. Before venturing into this procedure (section 12.4) it is necessary to better understand the classical background of quantization that is the structure of vacuum.
12.3
Spontaneously Broken Symmetry
Vacuum and its Structure In this section we consider the properties of classical vacuum in more detail. A general statement, which is applicable to vacuum itself, is that the gauge theories have unphysical degrees of freedom. We shall count here the physical degrees of freedom and find the correspondent mass terms. Upon the quantization, the former gives rise to observable particle species with the masses given by the latter. Remaind that vacuum is a static solution of equations of motion with the minimal possible energy. As it follows from expression (12.63), the vacuum configuration is = 0, A", 0, Q, = cPo1 where Q. is the minimum of V(@). For the sake of simplicity we do not take into account the fermion field and consider only the gauge fields interacting with a neutral multiplet of scalars. As the potential V(@) is assumed to be gauge invariant, for a nontrivial gauge group the vacuum is necessarily degenerate, because any ~ the minimum of gauge transformed function @;(X) = U ( X ) @remains the potential and thus it also is a vacuum. As simple example, we can consider a gauge group which rotates cPo in isotopic space. These rotations do not change the potential V(@) which depends on the length of the vector a o . Thus for a nontrivial gauge group we deal with a continuous set M of degenerate vacua. This set is called the vacuum manzfold. Let us study the structure of M. In many cases the geometry of M is obvious. For example, for a rotated isotopic vector and V(@) depending on its length, the vacuum manifold is a sphere of the correspondent dimension. In this case the rotations of the vacuum isovector on this sphere are equivalent to gauge transformations. Let us analyze this correspondence in detail. If for any element U of the group G representation @h = [email protected] # @o, M is topologically equivalent to G. This means that there is onetoone cor
12.3. SPONTANEOUSLY BROKEN S Y M M E T R Y
305
respondence between all elements in G and the points of M. In a more general case there can be group elements h leaving @o unchanged.
Problem: Prove that the elements h form a subgroup in G. We shall denote the subgroup containing all such elements by H. In mathematics this subgroup is called the invariant subgroup and in physics H is sometimes called also the little group. As all points of M are equivalent, the little groups of any point on M is isomorphic to H. Thus M should be equivalent to all group elements which do not leave a given vacuum unchanged. Mathematically, this is formulated as a statement that each point of M corresponds to an equivalence class with respect to H and therefore M is topologically equivalent to G I H .
Goldstone Modes and Higgs Mechanism Remind that the mass of quanta of a field fixed by the potential term in the Lagrangian, such as m2a2(cf. the dicussion in chapter 4 for the case of onedimensional scalar field). As the variations of the multicomponent scalar field on M do not change the field energy, the corresponding excitations are massless upon quantization. They are called the Goldstone modes. In gauge theories changes of an isovector in direction defined by these modes is equivalent to infinitesimal gauge transformations. As the latter are unphysical, we can expect that the Goldstone modes cannot be observable. We show below that this is really the case and the gauge field components, related to these modes, become massive. Let us denote the generators of H by re, a = 1 , . . . , No and the generators of G / H by T,, i = 1 . . . , K z N  No, where N and No are dimensions of G and H respectively. The condition of invariance of Q. under the transformation from H gives the following property of the group generators: T,@~
= 0,
a
=
1,.. . , N o ,
T,Qo # 0,
J' =
This leads to an important property of V(@)at @ an infinitesimal transformation
l,...K = ao.
.
(12.74)
Let us perform
where 6wa(x) are infinitesimal parameters. Using the gauge invariance of V and expanding it around (Do, we get
CHAPTER 12. GAUGE FIELDS
where all T, have disappeared because of (12.74). As 6wj(x) are arbitrary, we conclude that the matrix
has K eigenvectors &j) = TjGo with zero eigenvalues. These vectors are linearly independent and they are the Goldstone modes we are looking for. This is obvious that the number of these modes is exactly the dimension of G/ H. Let us analyze the mass spectrum of the theory. We decompose a generic variation of the scalar field as
where 4 is a superposition of all Goldstone modes and cp is an isovector which is orthogonal to 4. Let the dimension of isotopic space be Nb. Then q5 and cp consisted of K and Nb  K components of the fields q5i and cp, respectively. In these notations the matrix p takes the form
where the matrix block made of elements p,, has only positive eigenvalues. Let us find the masses of other fields. In terms of the small variation A:(%), &(X),and cp,(x) the linearized equations (12.41), (12.43) take the form
where Mjk = 2g2(T'@O)n([email protected])n does not have negative eigenvalues10 and the Lorentz gauge dpA;(x) = 0 has been assumed. We see that there are K massless Goldstone modes while the corresponding gauge fields, which transform according to GIH, acquire masses. Other components 1°1f X is an eigenvalue such that (Tiao,Tjao)uj= Xui then (uiTi Q0)' therefore X > 0.
= Xu2 and
12.3. SPONTANEOUSLY BROKEN S Y M M E T R Y
307
of the scalar field are in general massive. The generators on the invariant subgroup H remains massless. Before we show that the Goldstone modes are unphysical, let us pay attention to the terminology. While the theory we consider is gauge invariant, the vacuum state is not invariant under transformations from GIH. Neither are equations (12.78) and (12.79), which depend from the explicit form of the vacuum. This effect of nontrivial minima of V(@) is called the spontaneous symmetry brealcing. We shall say that the symmetry is broken from G to H by the vacuum expectation value of a. Note that the mass matrix Mjk of the gauge fields is written in terms of the vacuum expectation value of the scalar field. This value is transformed under the gauge transformation in contrast to a constant mass matrix which we could write in the Lagrangian by hands in order to give masses to the gauge fields. This difference is crucial from point of view of the renormalizability of the theory which we do not consider here (see, e.g., [3, 41). The mechanism of mass generation of the gauge fields considered here (and elimination of the Goldstone modes) is called the Hzggs mechanism.
Elimination of Goldstone Modes. Goldstone Theorem As we have mentioned, the Goldstone modes are very related to the gauge transformations. This relationship is known as Goldstone theorem. It asserts that for any isotopic vector @(X)there exists a unitary gauge transformation Uo(x), such that
To prove this statement we consider a real function on the group G, which is defined as f (U) = (Qo,U. (x)@(x)). Here the external brackets stand for the scalar product. This function f (U) is, of course, continuous on G. Besides that, we consider compact Lie algebras of gauge symmetry, that provides G to be a compact manifold. Then the set of f values is also compact on the real axis and f must have at least one extremum. Let us denote it by Uo(x). Thus first variation of f vanishes in some vicinity of U,: 6wU(x)(@o,TaUo(x)@(x))= 0 , that leads, because of hermicity of T, and arbitrariness of SW", to
CHAPTER 12. GAUGE FIELDS
308
This gives the statement of the theorem. The gauge (12.80) in which the Goldstone modes are explicitly eliminated, is called the u n i t a r y gauge. Note the concervation of total number of degrees of freedom. Each massive vector field have one (longitudinal) component more that the massless ones. Totally there are K 'new' components after the spontaneous symmetry breaking, which compensate K eliminated Goldstone modes.
Examples Let us consider two examples of spontaneous symmetry breaking in theories, describing respectively (i) a charged scalar field interacting with electromagnetic field, and (ii) a neutral triplet of scalar fields, interacting with nonAbelian SO(3) gauge field. The Lagrangian of the first model is
In this case G = U(1), H = 1, the isotopic space is twodimensional and K = 1, No = 0. Thus there is one Higgs field and one Goldstone mode. Then after spontaneous symmetry breaking the photon must acquire a mass. Let us show this explicitly. For this purpose we represent p in polar coordinates cp(x) = p(x) exp[ia(x)]. Problem: Show that the field a(x) is the Goldstone mode. Unphysical massless field a(x) can be eliminated by the gauge transformation
The gauge symmetry breaks spontaneously by nonzero vacuum expectation value po = a. Thus we obtain for the small excitations A,(x) and [(X) = p(x)  a the equations
where, as usual, the Lorentz condition has been used. Now we deal with one Higgs field with the mass m~ = 2 a a and one massive vector field with the mass mv = $&a. The Goldstone mode disappears having been transformed into the longitudinal component of the gauge field.
12.3. SPONTANEOUSLY BROKEN SYMMETRY
309
This model is a good phenomenological theory of superconductivity. In this model cp(x) is a collective macroscopic wave function describing an electron pair correlated in the momentum space l l . The parameter a in the theory is a function of temperature: a = a(T). Near some critical point T, a good approximation is a2(T) = k(T,  T), where k is some coefficient. If T > T,, a2 < 0 and there is a unique minimum of the potential. If T < T,, a2 > 0 and the vacuum of the model becomes degenerated. Using (12.82) we obtain in the case of static p(x) the following equation of the magnetic field in a semiconducting media
This means that the magnetic field falls off exponentially within the media. Thus an external field does not penetrate into a superconducting sample beyond a thin boundary layer whose thickness is about mvl. This is socalled Meissner eflect. It the context of field theory, we can say that the broken gauge symmetry of the vacuum state in a superconducting media leads to massive photons. The second example is the isotopic triplet of scalar fields interacting with SO(3) gauge fields. The Lagrangian of the system is
where a2 > 0, @' = (41,q52, 43) and the corresponding generators are i = 1,2,3. The potential V($) is invariant under the (Ti),b = gauge rotations from the group G = SO(3). Any isotopic vector 4 is invariant under rotations around 4 axis. Therefore, H = S 0 ( 2 ) , which is isomorphic to U(1), the isotopic space is threedimensional and K = 2, No = 1. Hence there are one Higgs field, two Goldstone modes and two of three gauge fields acquire mass as a result of the spontaneous symmetry breaking. We can chose the unitary gauge such that 4 is always parallel to the third axis. Then taking q5T = (0,O, a+cp(x)), we can express D,& and D,,q!12 in terms of A:L, that gives for (12.84) the expression
llOf course, correctness of the introduction of such a function must be confirmed by the microscopic theory of superconductivity. Such theory was constructed by J. Bardeen, L.N. Cooper, J.R. Schrieffer and N.N. Bogolubov [14], [15].
CHAPTER 12. GAUGE FIELDS
310
Thus in this case the Higgs particle has the mass m& = 8Xa2 and two vector fields acquires the masses m$ = aZg2because of the Higgs mechanism.
12.4
Quantization of Systems With Constraints
In this section, we shall mostly study the classical dynamics of finitedimensional constrained systems. Having gained the necessary experience, we shall proceed with the quantization of the gauge field1'. The reader who does not want to follow a tedious constraint classification may proceed directly to section 12.6 where a shorter although more formal method of quantization is presented.
Primary Constraints Let us consider a classical system with N degrees of freedom described by a Lagrangian L(q, Q, t ) . Denoting q, = v,, a = 1. . . N , we get equations z L(q, v, t ) in the form of motion for the Lagrangian
L ( ~ will ) be called the extended Lagrangian in what follows. Let the system be singular, i.e., det
I ::yb1
 = 0
.
Let the rank of this determinant (called the Hessian) be n and let us arrange the minor of the maximal rank in the upper left corner of the Hessian for all v, and q,. Thus, the dynamical variables are split into two sets
and
''Note that we are following here the approach developed by P. Dirac [17]. An alternative and more simple description, based on Darboux theorem, was presented by L. Faddeev and R. Jackiw [23].
12.4. Q UANTIZATION OF SYSTEMS WITH CONSTRAINTS
where det
~~~
I 1 
3 11
o
We shall assume the rank of (12.87) to be constant within a time interval including the initial moment. Without loss of generality we can assume the last m1 lines of (12.87) to consist of zeroes, this can always be achieved by canonical transformation1? As the matrix is symmetric, the last m1 columns vanish as well. Hence, the extended Lagrangian can be assumed to be a linear function in all X, with coefficients independent of v:
where L. does not depend on X. The last formula enable us to express we get vi in terms of q and p: vi = fi(q, p). Substituting this into for the Hamiltonian the following expression
where
Thus, the equations of motion take the form
where {., .) stand for usual Poisson brackets. An arbitrary function of dynamical variables g(q,p, t ) obeys the evolution equation
The last group of equations in (12.95) can be interpreted as a set of ml constraints imposed upon the system. Therefore, they are called the primary constraints. Like the velocities X,, they play the role of Lagrangian multipliers. This terminology is explaned in the next section. 130f course, this transformation depends on time, but in the case of constant rank this does not matter.
312
CHAPTER 12. GAUGE FIELDS
On Constrained Mechanical Systems Let us consider as an example a roller coaster. As a rigid body, the car has six degrees of freedom q,, a = 1. . . 6 (three positions and three angles), but its motion is actually constrained to the prescribed trajectory by five constraints. We denote them as f,(q,) = 0, a = 1.. .5. These functions are responsible for the complexity of motion even though the Lagrangian Lo(&, q,, ) is a very simple one describing the motion of a rigid body in a gravitation potential. The equations of motion take the form
There are two ways to treat such a system. First, we can solve the constrains to express all coordinates in terms of one (say the length along the rails) and then substitute the resulting expressions into the first equation. This is logically straightforward and seems suitable for the rollercoaster, but may be difficult in other cases. The other way to solve the system (12.97) is to benefit from simplicity of L. by finding the solution to the equations of motion and imposing the constraints on these solutions. This is a selfconsistent procedure because an extended Lagrangian L ( ~ = ) L X, f, results in equations (12.97) upon the requirement that it is stationary with respect to all X, (cf. Lagrangian (12.91)):
+
In this way we must solve the equation of motion for unconstrained La1 , the resulted trajectory into this Lagrangian, grangian ~ ( ~ substitute then solve equations (12.98) to obtain the explicit constraints specific to the trajectory we have found.
Problem: Find in such a way the motion of a point sliding down on a slope with zero initial velocity. Use vertical and horizontal coordinates in which the initial position is zero. Note that, in general, the constraints may depend on both the coordinates and the velocities.
Secondary Constraints It is clear that the constraints must be conserved quantities:
12.4. QUANTEATION OF SYSTEMS W I T H CONSTRAINTS
313
If the matrix of the Poisson brackets of the constraints is nonsingular on the surface of constraints, i.e. if det Cla8 # 0, we can express all X, using (12.99) and thus reduce the Hamiltonian H to a function of q, p. There is no reason to expect such a happy ending and so we must assume that the matrix Clap is singular. Let its rank be pb We assume it to be constant in some vicinity of the initial conditions on the constraint surface. Then, the set of linear equations ClaauP = 0 , Substitution of u("(q, p) has v = ml pl independent solutions d k ) ( q p). into (12.99) gives (
p
)=u k ( q , )
p ,H} = 0 ,
kl
=
l , . . . , V1
.
(12.100)
Some of the conditions (12.100) may be fulfilled identically, but not all of them. Let us choose those which are functionally independent of the primary constraints and of each other. These conditions can be regarded as new constraints. They are called secondary constraints. Let the number of the new constraints be m2. They must also be conserved in time, so we can carry out the same procedure once again. As the system is described by a finite number of dynamical variables and the equations of motion are supposed to be noncontradictory, there must be a finite number of steps after which we obtain m = ml +m2 . . m, independent constraints such that ml are the primary constraints and the others are the secondary ones. Note that the secondary constraints are determined up to a linear combination of the primary ones. In general, two system of constraints result in the same dynamics if they can be obtained from each other by a nonsingular linear transformation. We shall call such constraint systems equivalent. Thus we have found the complete set of constraints. We would like to express all X, in terms of the physical coordinates and momenta using this set. To this end, we must extend the set of X, to Xk and solve the correspondingly extended set of equations (12.99). Let us analyse this procedure in detail.
+.
The Matrix of Poisson Brackets There is a cross classification of constraints according to weither they are primary or secondary and weither they are first or second order. Let us explain the latter. Consider the matrix of Poisson brackets wkl
= { Q k , (D1)
,
k, l = l , . . . , m .
(12.101)
CHAPTER 12. GAUGE FIELDS
314
Let rank W),_, = S . As for Rap, we assume that s = const on the constraint surface in a finite vicinity of the initial conditions. The same argument as that applied when considering the Hessian (12.87) leads to the conclusion that without loss of generality we can assume the matrix to have the maximal rank minor placed in the upper right corner with the last m  s lines zero. Because of the antisymmetry of w k l the elements of the last m  s columns are zeroes as well. The structure of w k ~on the constraints surface leads to the following general structure:
where at, 1 = l, . . . , S, forming the maximal rank minor have been denoted by v,, a = l , .. . , S , {a) means terms proportional to the constraints and the subscript of the Poisson brackets denotes the dimension of the corresponding block. We do not need to consider higher powers of a. Indeed, the Poisson bracket of any such quantity is proportional t o cP, thus it vanishes on the surface of constraints. Thus the expression {a)can be thought of as a linear combination of the constraints. Taking account of this one can rewrite (12.102) in the form (12.103) where the notation been introduced.
xi i = 1 , . . . , m  s for @l,
1 = s + 1,
m  s has
First and Second Order Constraints Due to the arguments above, the

{xi,xj>
{G>
xi obey the relations {Xi,
v,)
 {W
(12.104)
The Poisson bracket of any ~ i ( q , p with ) any constraint results in a linear combination of constraints. Such functions of dynamical variables are called first order quantities. For instance, ~ ~ ( 9are , ~called ) first order constraints. The constraints cp,(q,p) are called the second order constraints. Thus all constraints are split into two groups:
12.4. QUANTIZATION OF SYSTEMS W I T H CONSTRAINTS
315
Let us first consider the case when only second order constraints are present, i.e. s = m (m even). This means
In this case we can express all X , from the conservation conditions analogous to (12.99). Substituting these expressions into (12.92) we get
It should be noted here that {Q1,H I ) must vanish on the constraints surface because of the energy conservation requirement and (12.96):
and therefore the second term in (12.105) is quadratic in constraints and thus can be neglected. It also follows from the structure of the secondary constraints that they can be added to the Hamiltonian and so it can be written as (12.106) H = HI XkQk
+
Thus, m dynamical variables can be expressed from the set of equations @l (q, p) = 0 so we arrive at the system with N  m/2 degrees of freedom. These ideas can be rigorously formulated in the two following statements. Their proofs are straighforward, but rather tedious and we omit them. The first statment reads: (I) Given a maximal set of m second order constraints there exists a canonical transformation leading to the set of pairs of canonically conjugate variables (v, m), = (X,p), 77" = ( K , T ) , dim m = m, such that the constraints take the form 170 = 0. The phase space of two systems with equivalent constraints 770 and ijo are reduced to canonical sets with the same 77. Thus the dimension of the phase space of the system is reduced to 2N  m where N is the number of degrees of freedom. The procedure of phase space reduction comprises of two steps: first a maximal set of constraints is sought and secondexpression for some dynamical variables is found using these constraints. The resulting Hamiltonian describes completely the dynamics of the system. Let us now return to the general case when det W vanishes on the surface of constraints. It is impossible to express all X, in the way it was done in the previous case. m  s of them remain indefinite. This obvious
CHAPTER 12. GAUGE FIELDS
316
fact can be expressed in the second statement: (11) Given a set of m constraints, s of which are second order ones, there exists a canonical transformation leading to the following set of canonically conjugate variables: ( r ] , [,v1)where r] = (X, p), r]' = ( K ' , m),110 = ( K , T ) such that the pairs r], r]o and (E, d)(dim r]' = m, dim T' = dim [ = m  S) consist of canonically conjugate variables and the constraints are equivalent to the set r]' = 0, where qo are second order constraints and K' are the first order ones. Thus in this case m  s coordinates are neither dynamical variables nor included in the constraints. They are absolutely arbitrary. Physical quantities must not depend on any arbitrary functions. Therefore, to make the Hamiltonian description of the system dynamics sensible one has to introduce additional 'artificial' constraints. Let them be
where gi(x,p) are functionally independent. Hence, the extended matrix of Poisson brackets (;I = I/{&, 6)II where 6 = (a,F) takes the following form on the constraint surface (compare it with (12.103)) :
It is easy to see that if all F, are independent, (;I is nonsingular. Indeed, turning to the canonical variables according to statement (11) we get
Therefore, det({Fi, xj)l det
# 0 and det (;I reduces to
(;I =
det
I{cp, v)l det2({Fi,Xj)(
.
(12.109)
As cp are second order constraints, both multipliers in the right hand side of (12.109) do not vanish, nor does the total determinant. Thus, complementing m constraints with ms additional ones Fi (g,p), which we call gauge conditions, we come to the dynamical system with second order constraints. To apply the twostep procedure described
12.4. QUANTIZATION OF SYSTEMS W I T H CONSTRAINTS
317
above to this system, we must make sure that the choice of gauge conditions does not affect the physics described by the corresponding physical Hamiltonian. Indeed, let us take another gauge conditions determined by the set of independent functions lj(x,p) analogously to (12.108). Then, it is possible to express 2iin terms of Fj because, by the above assumption, the corresponding Jacobian does not vanish in some vicinity of the initial conditions. Thus, two sets of constraints v' and ij' (in the notations of the statement (11)) are equivalent and the two corresponding Hamiltonians have the same physical sector according to the statement (I). In other words, two sets of canonical variables distinguished by gauge conditions are connected by a canonical transformation and this guarantees that the same physics is being described.
Quantization The above information about constrained classical dynamics enables us to quantize systems with singular Lagrangians. Let us first consider systems with second order constraints. It is now clear that to construct a quantum transition amplitude in the form of a path integral we must integrate only over trajectories lying in the physical sector of the phase space. This can be done by introducting into the integrand a delta function excluding the unphysical coordinates of the phase space. Let us use the canonical variables of statement (I). Thus, we get
where all indices labeling the physical coordinates and momenta xi(t), p j ( t ) , and labeling the constraints14 have been omitted. The symbol S(%) is thought of as a product of Sfunctions on each constraint. This reduces the functional integration to the physical sector of the phase space. As this physical sector may be a rather complicated surface in functional space, it is reasonable not to resolve the constraints before the functional integration and thus preserve the possibility of choosing the most appropriate canonical variables. Thus, we have to write the PI (12.110) in terms of arbitrary canonical variables in the whole phase space (Q, P) (which includes both physical and unphysical variables, denoted now by '"et ( 4 2
us recall that q0 consists of the pairs of canonically conjugated variables
( t ) ,.ir,( t ) ) .
CHAPTER 12. GAUGE FIELDS
318
Q and P) and the corresponding second order constraints 9., The reexpression of the Sfunction in (12.110) in terms of new arguments results in the Jacobian15 det J = det I)dp/dm)l.Namely,
where 'T' stands for transposition. Using the antisymmetry of the determinant under column transposition and multiplying the matrices in (12.111) we reduce 6(%) to the expression
Thus, the final expression for PI takes the form
If there are m  s first order constraints xi, the path integral (12.113) should be modified in view of statement (11) and (12.109). It gives, together with (12.I l l ) , the following formula:
Formulae (12.113), (12.114) which solve the problem of the quantization of constrained systems can be directly generalized to the field theory. For this purpose the usual derivatives in Poisson brackets should be replaced by variational ones. Namely, for a set of field variables ua(x) and canonically conjugate momenta pa(x) and two functionals F[u] and G[u] the Poisson brackets are the continual limit of those for the discrete case:
{F,G)=/
dx
SF
6G
6ua (X) 6Pa (X)
6G
Spa (X) 6ua (X)
)
.
(12115)
Unlike the bracket in the discrete case, this Poisson bracket can be an operator. The expression for the PI in the field theory can be built in a way analogous to (12.113), (12.114) up to the difference expressed by (12.115). We shall write it explicitly for the examples considered below. 15A new variable in the &function means geometrically a new parametrization of the constraints surface.
12.4. QUANTIZATION OF S Y S T E M S WITH C O N S T R A I N T S
319
Examples The Proca Lagrangian is an example of a second order constrained system: l m2 L = F p   A AP . (12.116) 4 p" 2 P There is a unique primary constraint Q1 = po. The canonically conjugate variable is X = &Ao. Other canonically conjugate variables are doAi,pi = FOi = Ei. { ~ o A P ( x ) , P " ( Y )=>g,J(x  Y ) . Taking account of the expression for the canonical momentum the Hamiltonian density takes the following form:
The conservation condition for
is
According to the general recipe, the first secondary constraint is
We can check that there are no other constraints. The matrix of Poisson brackets (12.101) takes the form
Thus, we can see that there is a second order set of constraints. The conservation condition for Q2 enables us to determine X:
Now it is possible to determine physical and unphysical canonical pairs of variables 77 and 770 according to the statement (I). To do so let us note that {Q1,a 2 ) = m2. The quantities which appeared in statement (I) take the following explicit forms:
CHAPTER 12. GAUGE FIELDS
320
Finally we get the following expression for the physical Hamiltonian:
The second example is the free electromagnetic field. In this case the Lagrangian is given by
By analogy with the previous example we obtain the set of constraints
For the Hamiltonian density we have
It is clear that the matrix W in this case is singular (m2 = 0 in equation (12.118)). Therefore free electrodynamics without a source is a theory with two first order constraints. We denote first order constraints as + X . Let us find a variable canonically conjugate to Q2. This quantity must obey the equation
This is equivalent to
which gives f (X)
=
AldiAi(x)
,
where A is the Laplacian. In order to obey any selfconsistent boundary conditions, this operator must be invertible, that provides existence of the solution f (X). Thus, we have to find two gauge conditions for two first order constraints. Let the first one be
We can easily check that it is nondegenerate (see the end of the previous section). The second condition follows from the conservation of Fl:
12.4. QUANTIZATZON OF SYSTEMS W I T H CONSTRAINTS
321
and it is convenient to take the second gauge condition in the form
Thus, the complete set of constraints and gauge conditions is
The extended matrix of Poisson brackets is
As LZt is evidently nonsingular with detk
= det A2
,
(12.129)
the set of constraints 6 is of the second order. Now we can construct canonical sets of variables following the statement (11):
where A; and denote two independent 3dimensional transverse components of A and p:
Thus expressing old variables in (12.124) in terms of those that have been found, we obtain for the physical Hamiltonian
where
F; =&A;  8 , ~ ; . Taking account of the definition electric and magnetic field we get for the Hamiltonian 1 (12.135) ~ p h y s = (E;+H:) . 2 Thus, we conclude that the 'hamiltonization' of the phase space of electrodynamics, as a theory with constraints, leads, just as in the case of the Lagrangian formalism, to the first pair of Maxwell equations. The second equation from this pair (the Gauss law) appears in this framework as a secondary constraint V p r VE = 0.
322
CHAPTER 12. GAUGE FIELDS
12.5
Hamiltonian Quantization of Yang Mills Fields

In this section we apply the general foramlism described above to the quantization of gauge fields. Let us start with Abelian electrodynamics. As mentioned in the previous section, electrodynamics is a first order constrained theory. Therefore, to construct the generating functional Z [ J ]we must use the formula (12.114), where the complete system of constraints is determined by (12.123), (12.125), (l2.126), and (12.127). This gives in Euclidean space (t + it, v0 + iv4 for any 4vector v,):
where p = 1,2,3,4. Let us perform integration over four momenta p,L. It is convenient to split the momentum pk into longitudinal (pllk)and transverse (plk) components correspondingly. Taking into account the Sfunctions of momenta, we have the following form of Z[J]
where we exploit the possibility granted by S(&&) of adding the longuitudinal component of Ak. Performing Gaussian integration over the transverse momenta, we obtain the final expression for the generating functional:
z[J] = J DA,(x)S(&Ai)S(A4) det(A) exp (SE + JPt A,)
, (12.138)
where
1 SE= 
/
d4x~,,(x)F,,(z) . 4 det(A) in (12.138) does not depend on A, so it gives a normalization constant (generally speaking, infinite) and can be omitted. Although the variable A4 is not a dynamical one, the term J4A4is retained in order to provide covariance. Of course it does not affect any Green functions. The expression obtained for Z[J] is gauge invariant, i.e. it does not depend
12.5. HAMILTONIAN QUANTIZATION
323
on the choice of gauge condition. We shall see this when we quantize the nonAbelian gauge fields. For instance, we can choose the Lorentz gauge d,A, = 0 instead of the Coulomb gauge. Note that now the second variation operator of the action has become invertible because of the gauge conditions. Thus, in the Lorentz gauge, it is
which leads to the well known expression for the free Euclidean propagator A
Dpv(k)= K,;; (k) = 
k2
Let us now consider YangMills theory with a nonAbelian gauge group. It is more convenient to carry out the Hamiltonization procedure in the Minkowski space and then to perform a Wick rotation when constructing the PI. Therefore, we shall use expression (12.13) along with the list of canonical momenta in Minkowski space given on page 300. Thus we get the set of primary constraints
In analogy with electrodynamics we have for the Hamiltonian density
The requirement that the primary constraints are conserved gives the secondary constraints: = {p:,
H)
=
(Dipi)a G X; = 0
.
We can easily check that 1;1; = 0 and {xl, ~ 2 =) 0 and so we are dealing with first order constraints. Therefore, we have t o supply these constraints using two gauge conditions. The first is
The second gauge condition follows from the conservation of the first one:
where the operator the form
X^"b is determined by (12.71) and can be rewritten in
xah= A
.
CHAPTER 12. GAUGE FIELDS Taking into account the relation
we can obtain the following pairs of canonically conjugate variables, the first and the second of which are unphysical:
E,"
= A," , b
G =  ( X  ) A, , physical variables: A y A , ^ l a b
nl
7 ~ 2=
p",,
= p;
,
(Dipi)" X = 1,2
,
,
(12.144) (12.145) (12.146)
where the index X labels independent components of transverse vectors. As in the case of electrodynamics one can check that the set of constraints
is second order. Indeed, the extended matrix of Poisson brackets is equal to
(12.148) It is a 4N X 4N matrix ( N is the dimension of the gauge group) written in the form of a 4 X 4 block matrix. It is evidently triangular, so we can easily calculate its determinant:
In the framework of perturbation theory, at least the operator X"" is invertible and the kernel of its inverse Mub(x,y) can be represented in the form of a formal series in g as the solution of the integral equation
Nevertheless, as it was mentioned on page 303, X^ can possess zero modes for large fields At (X) and so the operator M(%,Y) does not exist. This problem is out of the scope of the present book and we shall suppose that the expression for the PI to be constructed is correct in this gauge only in the framework of perturbation theory.
12.5. HAMILTONIAN Q UA NTIZATION
325
To quantize YangMills theory let us use formula (12.114). In analogy with electrodynamics, after the Wick rotation (see formula (12.136)), we get the following expression in Euclidean space:
x
detli2{@,@}exp
[S
1 d4x(ip~~ pp2 2~
The integration over momenta is entirely analogous to the integration over momenta in case of the electrod~namics. The last &function in (12.151) gives a contribution of detI Xab. Taking this into account we get
where
S,
SE
=
S
d4xGE, (x)G;, (X)
An essential difference between (12.152) and (12.138) is that d e t ( F b ) cannot be factored out of the integral in (12.152) and omitted. This is on A:. It is worthwhile to note (cf. (12.71)) that because F"epends
where AU is the result of the infinitesimal gauge transformation with parameters a. Although it was shown in the previous section that the PI does not depend on the choice of gauge conditions, let us check this explicitly. Let Fa(A) be an arbitrary nondegenerate set of gauge conditions. We would like to prove that the PI can be written as
Z [ J ]=
DAE(x) det
6(F,) exp(S,
+ J * A)
.
(12.154)
Because if (12.153) this is true in the particular case of the Coulomb gauge and Z [ J ]has the same form if any other nondegenerate gauge conditions F" are chosen. Introducing the notation
326
CHAPTER 12. GAUGE FIELDS
and evaluating the Sfunction in the integrand of (12.154) as follows
we conclude that expression (12.154) is gauge invariant. With this form of the generating functional, we can forget about calculating the Poisson brackets every time a change of the gauge is needed. The expression (12.154) could also be written with an additional term in the Lagrangian quadratic in gauge conditions. Such terms can be added to simplify calculations because they do not affect any results. For the sake of the development of the perturbative or loop expansion it would be convenient to represent the determinant in the preexponential factor in (12.154) in the form of a path integral. As the determinant appears in it with the power +l,it must be an integral over Grassmannian variables. Namely,
where the Grassmannian fields %(X),$(X) are the so called ghost fields which were first introduced by Faddeev and Popov. It should be noted that ghosts are auxiliary fields and do not describe any physically real thing. Therefore, they can be assumed to possess some of the contradictory properties necessary for the calculation procedure. For instance, it follows from (12.155) that ghosts are scalars although they are represented by Grassmannian fields. As a rule, eab(x,y) is diagonal in the variables X ,y so that one integration in (12.155) is cancelled by Sfunction. After that (12.154) becomes
where S is the sum of the gauge field action S, and the ghost action S g h corresponding to the Lagrangian
Note that the action in (12.156) is not gaugeinvariant. Indeed, it was our aim to remove the puregauge degrees of freedom from the functional integration. However, the action has a hidden symmetry which becomes manifest if we agree to transform the ghost fields in the same way as the
327
12.5. HAMILTONIAN QUANTIZATION
fermionic fields during gauge transformations. This symmetry is called BRSTsymmetry. It is helpful, for example, for the derivation of the generalized Ward identities [5]. Now we can establish the Feynman rules for pure YangMills theory. As the ghost Lagrangian depends on the gauge, we must first choose the gauge fixing conditions. Let us choose the Lorentz gauge d,AE = 0. In this gauge the ghost Lagrangian is
We also need the propagator in this gauge. For this purpose it is convenient to introduce the term quadratic in d,AE into the Lagrangian. The total Lagrangian then takes the form
The Fourier transform of the gauge field propagator is
The case P + 1 corresponds to the Feynman propagator, when P + 0, it is the Landau propagator. Thus, the Feynman rules in the Lorentz gauge read: 1. External lines.
Figure 53
Figure 54
(a) Fourier transforms of ghost fields q*(k) and ~ ( k correspond ) to oppositely directed dotted lines as shown in fig. 53
(b) AE(k) corresponds to A; as shown in fig. 54 2. Internal lines. There are two propagators in the theory: (a) the ghost propagator d i 2 + 6ab/k2 shown in fig. 55 left; (b) the gauge field propagator D$(k) is determined by (12.160) and is graphically represented in fig. 55 right.
CHAPTER 12. GAUGE FIELDS
328
Figure 55
Figure 56
Figure 57
3. Vertices. There are three kinds of vertices, describing gauge fields coupling with ghosts as well as the cubic and quartic selfinteraction of the gauge fields: (a) the term g f"dhrl*ad,(A$rlb)gives k, components of 4momentum in the Fourier representation so that we have for the contribution of a ghostgauge vertex g fabCk,= fig. 56; (b) the cubic selfinteraction term
can be symmetrized because of its cyclic symmetry in a, b, c and antisymmetry in p, v so that the contribution of the vertex equals
and is graphically represented by fig. 57. (c) The contribution of the quartic selfinteraction term
12.5. HAMILTONIAN QUANTIZATION
Figure 58
takes the form
where, as in the previous case, the cyclic symmetry in b, c, d and U,p.c~ has been used. This expression corresponds to fig. 58 Let us also say a word about the axial gauge. The general expression for it is F"[A] = n,AE = 0 , (12.161) where n, is a fixed unit vector. We have for
6
Therefore the ghost Lagrangian is especially simple
Since it does not contain the coupling with gauge fields, the integral over ghosts in the axial gauge can be included in the normalization constant as in the case of electrodynamics. However, this advantage is balanced by a rather complicated structure for the gauge field propagator. In analogy with (12.159) we have
L
= L,
1 + L,,, + (n,A;)2 2P
,
(12.163)
CHAPTER 12. GAUGE FIELDS
330
and the Fourier transform of the propagator is
6p,k2k,,k,+npn,,
P
12.6
Quantizat ion of Gauge Fields: FaddeevPopov Met hod
Thus we have quantized gauge fields using the canonical procedure of extracting the physical sector from the total phase space and constructing the PI over the physical variables and subsequently integrating over momenta. There is another method of constructing the PI which does not requires integration in the phase space and turns out to be more elegant although more formal. Let us consider the 'naive' expression for the Euclidean PI: (12.165) Z = DAE (X)eS.
J
The integration here is performed over all possible field configurations. They fall into two classes. First, there are configurations which are distinguishing by different values of physical observables such as electric and magnetic fields. For each such configuration, there is a class of fields which can be obtained by general gauge transformations. It is called the group orbit of a given field. As the action on a given orbit is a constant, the PI (12.165) diverges because of the extra integration over the gauge group at each space point. To make the PI meaningful, we must restrict the integration to physically distinguished field configurations. To do this in a gauge invariant way, let us consider a quantity
where F is a set of gauge conditions and the measure D U ( x ) is the product of the integration measure on a gauge group orbit for each space point: n
where d p ( U ) is the invariant measure on the group. At each point x j it
12.6. FADDEEVPOPOV METHOD is given by the expression
where a is the set of group parameters, NG is the dimension of the group and MG is a specific function determining the measure for the given Lie group. A[A] is gauge invariant. Indeed,
=
D ~ ( u ( x ) (4))6F[AUu1]) u~ = A'[A]
,
where the main property of the invariant measure has been used. Inserting the identity as l = A'A into the integrand of (12.165), we get
Now one can use the gauge invariance property of A[A]. Making the gauge transformation U'(X), we come to the expression
The last factor here is the volume of the gauge group multiplied by itself as many times as there are X'S, i.e. 'continuum' times. It diverges badly and causes the divergence of the PI. Since Z have been factorized and the divergent factor does not depend on A,",(x)the divergence can be omitted in all expressions by including it in the normalization. To obtain the final expression for Z we have to find A. As we see from (12.167), the actual integration variables in A' are a,(x) while the &function sets the gauge conditions F[AU] to zero. Let us make a functional transformation of the integration variable in (12.166) from aa(x) to F,(x). This is possible if the mapping {W,} + {F,} is singlevalued. In other words, it is possible if the gauge conditions extract exactly one representative in each class of gauge equivalent fields A,",16. This results in
A' [A] =
l
1 I 1 IF=,
DF,(X)S(F[A~]) det S F (4 6ab(y)
= det
SFa (X) 6ab(y)

160f course, for this t o be the case we are assuming the F, are singlevalued functions of At.
CHAPTER 12. GAUGE FIELDS
332
Substitution of this expression into (12.168) and omission of the infinite normalization factor transforms (12.168) into our final expression (12.154) with J = 0. Although this method, proposed by Faddeev and Popov, is rather formal, it enables us to visualize the geometrical meaning of the gauge field quantization and the origin of the divergence in the PI. We see that the removal of divergence in the PI is equivalent to turning from integration over all fields including gauge equivalent ones to integration over only gauge nonequivalent ones. Each gauge class is represented in the integral exactly once. Speaking more formally, we must integrate over all orbits of the gauge group l7 rather than over gauge fields themselves. Let us illustrate this statement with a finitedimensional example. Let f (X) be a function such that
Let us consider a 3dimensional integral
It can be easily seen that it diverges. Now let us introduce the hyperbolic coordinates
After that +m
I=
d s s 2 ~ ( x ) Jdo 0
,
C
where F(x) f (s2), the E denotes that the integration is over the surface of the hyperboloid x2  Y2  z2 = s2 and do = sinhv dvdcp is the surface element on the unit hyperboloid. The divergence originates in the integration over the hyperboloid surface since this has infinite area. To make I convergent one has to drop the integral over do. What does it mean geometrically? As the function f (s2) is constant on a hyperboloid, it is invariant under hyperbolic rotation. This forms the group G = SO(2, 1) and is an analogue of the Lorentz group in 2 1 dimensions. It is determined by three parameters, two of them correspond to
+
17Formally defined, the orbit of a group G acting on a topological space X is the set { g . X ; any g E G) for any fixed X from X.
12.7. COLEMANWEINBERG EFFECT
333
hyperbolic rotations and one corresponds to the usual Euclidean rotation. As cp and v parametrize the usual rotations in the (y, z ) plane and hyperbolic ones in (2,X) plane, they can be regarded as two parameters of SO(2,l). The third parameter, which we denote by U, corresponds to hyperbolic rotations in (X,y) plane. If it is fixed, we obtain the subspace of the group topologically equivalent to the unit hyperboloid. Thus, I can be rewritten in the form
where dp(g) denotes the invariant measure on SO(2,l) and g is an arbitrary element of it acting on the point X. As F(x) is invariant we can factor it out of J dp(g). Now, it is clear that to make I finite means to integrate only over the orbits1' of the group SO(2,l). Of course, this example is not an exact analogue of the FaddeevPopov procedure because in this case we get infinity because of noncompactness of the invariance group. In the case of gauge fields the invariance group is compact but the divergence appears because the product of its finite volume with the volume of 4dimensional space is infinitelg .
12.7
ColemanWeinberg Effect
Including interaction with gauge fields modifies the theory of matter fields in an essential way. In this section we consider the simple example of the electrodynamics of selfinteracting scalars with spontaneously broken symmetry. It would be interesting to find out what the difference is between the effective potential of this theory and of pure cp4 theory. The Euclidean Lagrangian of the theory is
where the complex scalar field cp(x) describes charged scalars, A, is the usual electromagnetic potential and D, = 8,  ieA,(x). We assume the gauge condition 8,A, = 0 and add the last term in (12.170) for more convenient functional integration in what follows. The potential V(cp) has the socalled 'mexican hat' form:
18Not over all orbits but rather over special ones obeying the condition U = 0. ''An adequate mathematical language for the correct description of this situation would be the theory of fiber bundles.
334
CHAPTER 12. GAUGE FIELDS
which leads the spontaneous symmetry breaking. Before we start evaluating the effective potential, let us put the Lagrangian (12.170) in a more convenient form. For this purpose we represent cp as p(x) = . Without the gauge field the field a(x) would describe a massless scalar exitation (the Goldstone mode or Goldstone particle). In the presence of the gauge field, a(x) can be removed from the scalar field by a gauge transformation
Thus the Lagrangian takes the form
where a scaling transformation p + p/& has been made. The last term shows that the gauge field has acquired a mass proportional to e(p). To calculate the effective potential, we have to consider the vacuumtovacuum transition amplitude
We are looking for the vacuum energy in the state
As in the case of the effective potential for interacting bosons and fermions considered in chapter 11, this means that there is only one coupling constant X to be renormalized. The counter terms to e do not show up. In one loop, the integral (12.172) is Gaussian:
Here the fields are combined in one column +JI! = ( p , A)T for breavity is a blockdiagonal matrix consisting of a onedimensional and a and fourdimensional block:
6
12.7. COLEMANWEINBERG EFFECT where 2 2 2 2 X0 mb = (34' 6 a2), mg=e$ , and the electromagnetic propagator is chosen in the Feynman form (P = 1). Integration over one component of the electromagnetic field disappears due to the &function in the integrand and we obtain
We are already familiar with such expressions (see chapters 5 and 11) and can immediately write down the expression regularized by, e.g., Cfunction. Let us consider the simplest case a = O in what follows. Then
V,,,
(4)=
Xo
4
X%i54
3e444
e2$'
+ 256712(In12?:)+64111(InF:) 
'
(12.175) The coupling constants X. and e are independent. Let us consider e4 where the most interesting effects take place. In this the range X range we can neglect the term proportional to X:. Defining the coupling constant as (12.176) h(M) = we net
The renormalized expression for the effective potential takes the form
This effective potential differs essentially from that found in chapter 5 for pure (p4 theory because the second term need not be smaller than the first one. Thus the minimum of Vefffound at
lies within the validity range of the loop expansion (remember the expressions considered are valid when X2 << e4 X). The symmetry of the classical Lagrangian with a = 0 is broken and the gauge field acquires a mass e&,. One calls this the ColemanWeinberg eflect. Also used are the terms dynamic symmetry breaking and dynamic mass generation to
336
CHAPTER 12. GAUGE FIELDS
distinguish them from their spontaneous counterparts caused by a # 0. As was shown in the original paper [25], the effects found at one loop remains valid upon a renormalization group improvement of the effective potential. Using (12.179) one can express the effective potential in terms
In this form, the couple of dependent parameters M and X(M) is replaced by the vacuum expectation value of the scalar field. The mass of scalars in this vacuum is
The mass of the vector field is m: = e2+Li,. As in the case of spontaneous symmetry breaking both the scalar and the vector fields acquire a mass. However, dynamic mass generation, being a oneloop effect, results in a small mass of the scalar field:
If the tree level scalar mass is not zero (a2 > 0) then the corrections to Vef from the gauge field are small compared with the leading term. Dynamical mass generation (more complicated than that considered here) is believed to occur in QCD where the bare masses of light quarks U and d are assumed to be zero while the physical ones are small, but finite. The ColemanWeinberg effect makes it possible to estimate a lower bound for the mass of Higgs particle in the Standard Model of electroweak interactions. There are three gauge vector bosons in this theory: the oppositely charged W' with mass m w and the neutral Z0 with mass m z . Other features of the interaction between the gauge bosons and the Higgs field look similar to those considered above. The vacuum expectation value of the Higgs field is fixed because it gives the experimentally known masses of the vector bosons: mw = ewq50 and m z = ezq50 where ew and ez are the corresponding coupling constants (we shall refer to them as e, where v = W+,W, 2).At tree level, the mass of the Higgs particle m~ is arbitrary. In particular, it can go to zero in proportion to the scalar coupling constant X. However, this is not the case when the oneloop corrections are taken into account because the vector fields then generate a mass of the order of e:+o. Let us consider this in more details.
12.7. COLEMANWEINBERG EFFECT
337
To this puprose, we calculate the effective potential of the scalar field as described above. We again neglect the oneloop correction from the scalar loop because we expect to balance the classical terms in the scalar potential against the oneloop correction from the vector field. The effect of fermions is not accounted for because apart from tquark, they are all lighter than the gauge bosons. In order to discuss the basic idea of limiting the Higgs particle mass, we consider only the gauge boson contribution. The regularized expression for Kff(4)takes a form similar to (12.180):
where the term 312 appearing in (12.175) is now included in A and subscript 0 labels the bare quantities. The renormalized m: is the position of the pole of the scalar propagator (chapters 7,8) (k2 = m&). In general, it differs from However, the difference is negligible in the case considered because the oneloop correction in the inverse propagator depends on the ratios k2/mw and k2/mz which are small. We thus take the renormalization condition KTf(+') = m.: The other condition for X can be the same as above, but it is convenient to express the effective potential in terms of 4' which is experimentally known. This is in fact (4') = 0. The equivalent to the second renormalization condition result is
where C)
It may now appear that we may give any value to m f ~ However, . there is the additional obvious requirement that the vacuum q5 = 4' is stable, that is the point 4 = 4' provides the global minimum of the effective potential2'. The value of the effective potential at the minimum 4 = q50
201f it is not so, the vacuum state decays via nucleation of bubbles inside which the energy density is lower [26]. This is nothing but a conventional firstorder phase transition.
338
CHAPTER 12. GAUGE FIELDS
The lower bound on m~ results from the requirement Kf (40) < Kf (0) = 0. The restriction takes the form
Taking into account that
where cr. = 11137 and Ow is a parameter of the theory, we get finally
Independent estimates give $0 e 248 GeV and sin2Ow e 0.22. Hence 6.8 GeV. the desired quantity is r n H m i n
Problem: Consider the effect of the heavy topquark with mass mt 175 GeV on the bound (12.184) using the results of chapter 7. Remember that, as we have already mentioned at the end of chapter 9, the current ~ 89.3 GeV at experimental bound on the Higgs particle mass is r n 2 95% confidence level [28].
Bibliography [l]The term 'gauge field' was firstly introduced by H. Weyl, Ann. d.Phys 59 B (1919) 101.
[2] NonAbelian gauge fields were introduced by C.N. Yang and R. Mills, Phys. Rev. 96 (1954) 191. [3] L.B. Okun', Leptons and Quarks, NorthHolland, Amsterdam, 1982. [4] C. Itzykson and J.B. Zuber, Quantum Field Theory, McGrawHill, 1980. [5] B. Hatfield, Quantum Field Theory of Point Particles and Strings, Redwood City, AddisonWesley, 1992. [6] Historically the first book on the unitary symmetry of elementary particles was probably M. GellMann and Y. Ne'eman, The Eightfold Way, New York, W.A. Benjamin, 1964. [7] A good short introduction to the SU(3)classification of elementary particles is given in K. Huang, Quarks, Leptons and Gauge Fields, World Scientific, Singapore, 1982. [8] For quark models of hadrons and SU(3) symmetry see also O.W. Greenberg and C.A. Nelson, Phys. Rep. 32 C (1977) 71. [g] The theory of SU(3) representations can be found in W. Miller, Symmetry Groups and Their Applications, New York, Academic Press, 1972. [l01 A good introduction into Lie algebras is R. Gilmorel, Lie Groups, Lie Algebras and Some of Their Applications, New York, WileyInterscience, 1974.
340
BIBLIOGRAPHY
[l11 G.L. Naber, Topology, Geometry, and Gauge Fields: Foundations, New York, SpringerVerlag, 1997.
[l21 Gribov ambiguity was discovered by V.N. Gribov, Nucl. Phys. B139 (1978) 1. [l31 The connection between the Gribov ambiguities and topological charge was established by R. Jackiw, I. Muzinich and C. Rebbi, Phys. Rev. D17 (1978) 1576. [l41 J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [l51 N.N. Bogolubov, Sou. Phys. J E T P 11 (1947) 23. [l61 For a review of the theory of superconductivity see e.g. J.R. Schrieffer, Theory of Superconductivity, Benjamin, New York, 1964. [l71 The theory of constrained system quantization discussed here was developed by Dirac in P.A.M. Dirac, Canad. J. Phys. 2 (1950) 129; ibid 3 (1951) 1. [l81 The method of quantization of constrained systems was applied to the quantization of nonAbelian gauge theories by N. Christ, A. Guth and E. Weinberg, Nucl. Phys. B114 (1976) 61. [l91 The FaddeevPopov procedure and ghosts were introduced in L.D. Faddeev and V.N. Popov, Phys. Lett. B25 (1967) 29. [20] A completely gauge covariant quantization scheme was constructed in the following papers: E.S. Fradkin and G.A. Vilkoviski, Phys. Lett. B55 (1975) 244; I.A. Batalin and G.A. Vilkoviski, Phys. Lett. B69 (1977) 309; ibid B102 (1981) 27; I.A. Batalin and E.S. F'radkin, Phys. Lett. B122 (1983) 157; ibid B128 (1983) 308. [21] For the loop expansion of gauge fields see C.G. Callan, R.F. Dashen and D.J. Gross, Phys. Rev. D17 (1978) [22] A good textbook on this subject is A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems, Rome, Academia Nazionale dei Lincei, 1976.
BIBLIOGRAPHY
341
[23] L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60 (1988) 1692; R. Jackiw, (Constrained) Quantization Without Tears, in the collection Diverse Topics in Theoretical and Mathematical Physics, Singapore, World Scientific, 1995, p. 367. [24] B. DeWit and J. Smith, Field Theory in Particle Physics, Amsterdam, NorthHolland, 1992. [25] The ColemanWeinberg effect was discovered in S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [26] The mechanism of vacuum decay in quantum field theory was understood in the following two papers: M.B. Voloshin, I.Yu. Kobzarev and L.B. Okun, Sou. J. Nucl. Phys. 20 (1975) 644 and S. Coleman, Phys. Rev. D15 (1977) 2929. Independently, the same approach had been already developed in the context of quantum physics of condensed matter by I.M. Lifshitz and Yu. Kagan, Sou. Phys. JETP, 32 (1972) 206. The theory of nucleation in the firstorder phase transitions in classical systems was created by J.S. Langer, Ann. Phys. 41 (1967) 108; ibid 54 (1969) 258. [27] The estimate of the lower bound for the mass of the Higgs particle was independently obtained by S. Weinberg, Phys. Rev. Lett. 36 (1976) 294 and by A.D. Linde, Sou. Phys. JETP Lett. 23 (1976) 64. [28] There are many other ways to bound m ~They . can be found in the Reviews of Particle Physics which are regulary published by the Particle Data Group (http://pdg.lbl.gov). When this book was written, the most recent reference was C. Caso et al, The European Physical Journal C3 (1998) 1.
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Chapter 13 Topological Objects in Field Theory
In the previous chapters, we studied the quantization of small excitations above the vacuum state. The latter, a spacetimeindependent field configuration, was the simplest solution of the classical equations of motion. In this chapter, we shall call such solutions trivial. The nontrivial solutions are other field configurations which obey the equation of motion. Depending on the context, they can describe either objects with finite energy or processes (such as tunneling). The stability of such objects is guaranteed by conservation laws, which in many cases have a topological origin. This explains the title of this chapter. We consider in this book only theories with a weak coupling constant. Let us denote the coupling by a and let the mass of the field quanta be m. Then the typical mass of a classical object is as large as m/a >> m and the typical action of a processes is l / a >> 1. Some of the topological objects have been observed in real systems (kinks and vortices), other are presently purely theoretical constructions (magnetic monopoles). There are reasons to believe that QCD has a nontrivial vacuum structure due to fluctuations which are dominated to some accuracy by topological objects. In this chapter we consider a few topological objects in successively increasing number of space dimensions and make some remarks about their classification and quantization.
344
13.1
CHAPTER 13. TOPOLOGICAL OBJECTS
Kink in 1 + 1 Dimensions
Let us first consider (p4 theory in one spatial dimension. The Lagrangian in Minkowski space reads
where ~ V((p) = ~ ( ( p a2)2 The equation of motion takes the form
We are interested in finiteenergy solutions of this equation. The field energy is
At first we shall search for static solutions, so that
Obviously, the solution to this equation minimizes E. There are two trivial solutions of this equation (the classical vaccua): (p = f a which are the minima of the potential energy V((p). The third solution (p = 0 is the local maximum of V((p) and thus it results in a finite energy density, causing the total energy of this configuration to be infinite. This is of no interest for physical applications. The solution we are looking for interpolates between the vacuum (p = a at X = 00 and (p = a at X = +m. In the context of a onedimensional system it is called a kznk. The other solution, with boundary ) f a , is called an antikink, and can always be obconditions ( p ( ~ o o= tained from the kink by a spatial reflection. The solution we need was already found in chapter 3 when we considered tunneling in quantum mechanics (cf p. 45). Recall that we used a mechanical analogy which helped us to find a specific conservation law
from which the solution was obtained by a simple integration. It had the form 1 (13.7) cp(x) = *a tanh  m ( x  xo) , 2 where m2 = 8Xa2 and xo is the kink position. The signs f correspond to kink and antikink respectively. Let us present here another method of solution which can be generalized for multidimensional objects. We rewrite the field energy in the form
The second term in the integrand can be integrated as a full derivative:
where we have used the explicit form of V(cp).Thus the energy takes the form
We see that this energy is bounded below by the first term, because the integral cannot be negative. The minimum of the energy is now provided by a function for which the integrand vanishes for all X. Thus we come again to the condition (13.6) from which the solution (13.7) follows. Its energy is just M =m3 = E (13.11) 12X 12a ' where the dimensionless coupling constant is a r X/m2. This expression has the same form as that obtained for the instanton action in chapter 3 , up to a replacement of W by m. This analysis gives the simplest example of socalled topologically nontrivial boundary conditions. Let us explain this term. The requirement of finite energy can be satisfied if the field at spatial infinity coincides with a classical vacuum. In other words, we must assign a different vacuum state to each infinite spatial point. In our onedimensional example, spatial infinity consists of two points x = fCO and there are two possible vacuum states cp = f a . This mapping is trivial if both infinities are mapped into the same vacuum state and nontrivial if the field is mapped
CHAPTER 13. TOPOLOGICAL OBJECTS
Figure 59
to different vacua in opposite spatial directions. In the latter case the field must cross zero at some point X = xo, indicating a region with positive energy density. In order to descibe this accurately, let us introduce a quantity Q, called topological charge, such that
where k,(x) is the topological current:
The possible values of the topological charge are Q = 0, f l for vacuua, kink, and antikink respectively. The current Ic,(x) is conserved identically (a property which distinguishes topological currents from Noether ones) : d,kP=0 . This description is in fact too complex for such a simple case as the kink in one dimension. However, the logic of this analysis is valid in multidimensional spaces, where it becomes genuinely helpful. The kink can move as a whole like a classical particle. This suggests that we shall think of the kink as a particlelike object. Indeed a Lorentz boosted configuration m (X cp(x,t) = fa tanh 2

vt  X")
,/m
is also a solution of equation (13.3). The energy (13.4) takes the form
13.2. SOLITONS
347
as expected from the Lorentz invariance of the theory. Let us mention in brief some physical systems for which the kink solution is relevant. Let us consider a polyathetilene chain (CH),, (figure 59a) where each line represents an electron and the CH groups are not shown. There are two ground states which are obviously equivalent (figures 59a). The kink and antikink solutions can qualitatively describe the defects shown in figure 59b. These defects carry electric charge and can move along the chain. Interestingly, the electric charge of the kink is 112 of the elementary charge. This is apparent if we count the number of electrons necessary to create two kinks. The answer is one (see figure 59b). The same result is valid in a rigorous analysis of kink solutions in cp" theory coupled with fermions [2] which we do not attempt here. Another example where the cp4 model is quantitatively valid is the LandauGinzburg theory of phase transitions [3]. Let us consider now two or threedimensional systems. Below the transition point, there are two possible phases cp = f a (which we have called vacua above). The field (in this context known as the order parameter) may take the values cp = *a in different spatial regions called domains. The kink then describes the boundaries between these regions. The coordinate X denotes the direction orthogonal to the boundary. The field does not depend on the coordinates y and z in the tangential directions. Such a configuration is called domain wall, and is common in statistical physics. In the context of particle physics this solution was discussed in paper [4] concerning a possible phase transition in the early Universe. It is interesting to note that the domain wall is invariant under Lorentz transformations for motion in the y and z directions: the solution looks exactly the same in all reference systems moving parallel to the wall.
13.2
A Few Words about Solitons
Let us consider one more model which leads to particlelike solutions. The Lagrangian of this theory takes the form (13.1) with the potential
This potential has minima at cp defined as m2 = V U ( 2 m / a ) .
=
2 ~ n / a n,
=
0, fl , f 2 . . . and m2 is
CHAPTER 13. TOPOLOGICAL OBJECTS
348
Problem: Show by expanding V near the minimum that the coupling constant1 of the quantized theory is a2. The corresponding equation of motion
is called the sineGordon equation. The solution to this equation can be found analogously to the (p4 theory considered above.
Problem: Find the static solution. The solution called the soliton for the reasons discussed below takes the form 4 (p(x) = f  a r c t a n e x p m ( x  s o ) , a
which after boosting becomes ~ ( x=)
+a4 arctan exp
X

vt  $0
The plus sign corresponds to a soliton and the solution with the minus sign is called antisolzton. It is clear that (p(x) 2nnla for any integer n is also a solution. Unlike the above considered (p4 model, the sineGordon model has infinitely many degenerate minima. The topological charge is defined as for the (p4 model up to a different normalization chosen in such a way that the soliton and the antisoliton have charges Q = +l respectively. The explicit expression is
+
where kp(x) takes the form
There is a deeper difference between the (p4 and sineGordon models. As distinct from (l3.3), equation (13.16) belongs to the class of completely lThere is a harmless inconsistency in these notations: the coupling constant of the cp4theorywas denoted as a rather then a2.
13.2. SOLITONS
349
integrable equations. This property makes it possible to find all its timedependent solutions. This, however, requires sophisticated mathematical techniques, for example the method of inverse scattering problem, and others which were developed over the last thirty years. This very interesting subject is not considered in this book. We refer the reader to [5], [6], [7]. Here, we only comment on the remarkable properties of these solutions. Let us contrast the solitons, which are called also solitary waves, with ordinary waves. The latter are solutions of linearized equations. For linear equations the superposition principle is valid so that any linear combination of solutions is a solution as well. If we apply a Fourier transformation (t, X) + (W, k) to an equation containing derivatives with respect to time and spatial coordinates, we obtain the dispersion law w(k) and any wave packet with this dispersion law will be a solution of the equation. Therefore each component of an initial wave packet propagates with its own velocity and the packet inevitably looses its initial shape. Solitons, on the other hand, are solutions of nonlinear equations2. Typically, solitons look like a solitary wave which propagates keeping a fixed shape. The energy density of such a solution takes the form €(X vt) where v is a constant velocity vector (this can be considered as the definition of solitary waves although this is only one possible choice). The term 'soliton' is applied to solitary waves which possess an even more remarkable property. They survive collisions with each other, the only effect of collision being a displacement with respect to the initial trajectory. This can be formulated accurately as the following statement for the solution which may consist of many solitons. If the energy density is €(X, t ) , the asymptotic form for such a solution u(x, t) at t + m is
where to(x  vt) is the energy density corresponding to another solution uo(x,t) (typically, it is a single soliton), then its asymptotic form at
where the v,, q,are constant vectors and the 6, are constant scalars. In other words, for the solution u(x, t) the superposition principle is valid in the weak form. If uo(x,t) is interpreted as a pseudoparticle, then u(x, t) can be interpreted as describing the scattering of N such particles. 'In general, a nontrivial topology is not necessary for their stability.
CHAPTER 13. TOPOLOGICAL OBJECTS
350
Let us give a few examples of solutions which describe solitonsoliton scattering. It is sraightforward to check that the following functions are solutions of equation (13.16): p1,1(X,
t) =  arctan (Y
Jm)
v sinh(mx/ cosh(mvt/J~)
sinh(mvt/ J) p l ,  l ( ~t), =  arctan a vc o s h ( m x / J m )
)
(13.20)
.
Let us find the asymptotic form, for example, of the second solution at t + m. It reads (2,t)
tioo
+
4 arctanexp {my[x a

+ v(t + 6/2)])
4 arctanexp (my[x  v(t + 6/2)]) a

,
where y = ( l  v2)1/2 and 6 = (lnv)/(yv). We see that in the infinite past the solution represents a soliton and an antisoliton moving towards each other with velocity v. The asymptotic form in the infinite future takes the form (X,
t)

t'+~
4

arctan exp (my[x
ff
+ v(t

6/2)])
Thus the two asymptotic forms differ from each other only by a phase shift equal to S, and cpl,l(x, t) describes elastic scattering of the soliton and antisoliton. It can be checked analogously that pl,l (X, t) corresponds to the scattering of a soliton on another soliton. There exist also nsolitonic solutions describing the scattering of n solitons and antisolitons, solutions with Q = 0 which can be interpreted as solitonantisoliton bound states. Generally, a given initial configuration evolves as a set of solitons and their bound states, plus some amount of conventional plane waves.
13.3
Abrikosov Vortex
In this section, we consider a topological object, known as the Abrikosov vortex, which is experimentally observed in superconductors. Its counterpat in particle physics is called the NielsenOlesen string [8]. Respecting
13.3. ABRIKOSOV VORTEX
351
hard experimental evidence, we start with a model specification in the context of syperconductivity, find the solution, and then discuss generalizations to more complex (and more hypothetical) systems.
GinzburgLandau Model of Superconductivity In the superconducting state (see, for example, [g]), the electrons in a metal are coupled in pairs, called Cooper pairs, which obey Bose statistics. At low temperatire, they form a condensate p(%) which can be thought of as the wave function of the state occupied by a macroscopic number of pairs (cf. our discussion of quantization of bosons in chapter 10). Then the conventional quantummechanical current gives the macroscopic electric current. The effect of temperature is a reduction in the magnutude of the condensate as the temperature increases from zero to a critical temperature Tc: where the condensate disappears. This is the point of the secondorder phase transition. This description can be summarized in the following phenomenological form of the condensate energy (strictly speaking it is the free energy) [3].
Here cp is a complex scalar field, D, = 8,  ieA,,(x) where Ap(x) is the electromagnetic field, E and B are the corresponding electric and magnetic fields, e is the Cooper pair charge equal to two elementary charges, The parameter a2 depends on temperature and V(cp) = (X/4)(cp*cp~~)~. T such that a = 0 at some T = T,. T, is called the critical temperature. Near this temperature a2 = k(T  T,), where k is a constant, positive coefficient. At temperatures larger than T,, a2 is negative and the potential has the unique minimum cp = 0, for which the superconductivity is absent. At T = T,, the potential V acquires infinitely degenerate minima at lcpl = a (the phase of cp is arbitrary). Below T, the metal is superconducting. Refering to our experience with gauge fields gained in chapter 12, we conclude immediately that in the superconducting state the electromagnetic field acquires a 'mass' m~ = 2ea, while the 'mass' of the cp field is m, = d a . Problem: Check this. Practically this means that small deviations of A, from zero and of Icp( from a decay exponentially as we go inside the bulk superconductor.
352
CHAPTER 13. TOPOLOGICAL OBJECTS
Such deviations may be caused by conditions applied at the boundary of a superconducting sample. It is no surprise that the electric field does not penetrate into any of the conductor. Specifically for a superconductor, the magnetic field does not penetrate into it. The physical reason for this is the current in the surface layer of order l / m A which screens the magnetic field. This screening is perfect because of superconductivity. This is known as Meissner eflect.
Nontrivial Solution We need to find solutions which minimize the energy (13.21) for T < T,. For the energy to be finite, it is necessary that the field cp minimizes V(cp) at infinity. This means that cp(lx1 = m) may take values on the circle Icpl = a. In more mathematical language, we have to map the spatial infinity of 3D space, represented by the sphere S2,into the circle S1. The only trivial solution is possible which is a mapping of the whole S" onto a single point of S1 (for example cp = a). All other trivial mappings can be obtained from this one by gauge transformations. It is clear that the mapping can be nontrivial in twodimensional space, where infinity is also the circle S'. As the spatial infinity point moves around this circle, the value of cp can wind around the circle IcpI = a an integral number of times Q = 0, *l, f2.... Any mapping with Q = 0 is trivial. It can be reduced to, say, cp = a by a gauge transformation. The gauge transformations do not change Q because a contour winding around a circle cannot be continuously shrunk to a point. We shall return to this description below, but now let us consider the solution with Q = 1. In threedimentional space it will describe a configuration which looks like a string because it is independent of one spatial coordinate (say 2 ) . The following boundary condition is obviously nontrivial, with a winding number Q = 1: (13.22) p(r t CO) = aeia , where r and a are the polar radius and the angle in cylindrical coordinates in the twodimensional Xplane . Thus we relate the isotopic orientation of cp(x) with spatial coordinates. The boundary condition (13.22) sets the potential term in the energy (13.21) to zero, but not yet the gradient term. Indeed the field (13.22) has an angular gradient component proportional to llr which results in a logarithmic divergence of the integral (13.21). Thus no nontrivial solution is possible in the pure 2D cpmodel without the gauge field. The situation is cured by the presence of the gauge field Aj ( j = 1,2 and we set Ag = A. = 0). We can find the boundary condition for A,
13.3. ABRIKOSOV VORTEX from the requirement Djp = 0 at r
+
cm.This gives
where e,(x) is the unitary vector tangential to the circle of radius r at the point X. A generalization of (13.22), (13.23) for arbitrary winding number is obvious:
where Q is an integer. Any deviation from an integer value results in a discontinuity of p which leads to an infinite energy. The winding is simply the topological charge. The boundary conditions (13.22), (13.23) result in remarkable electromagnetic properties of the solution. Let us calculate the flux m of the magnetic field in the zdirection. It reads
where B3 is the zcomponent of the magnetic field and S" is the interior of circle of infinite radius, C,. This is the famous condition for flux quantization . It states that the nontrivial solution (which we have not yet found) describes the magnetic field which penetrates the superconductor along a line, the socalled flux tube. In order to describe this object better, let us find the localization of the magnetic field. It cannot be distributed over the whole area S" in the zy plane because the potential (13.23) does not produce any magnetic field for X # 0. Thus the magnetic field should be present near the core of the solution where the fields deviates from the boundary conditions (13.22), (13.23). At some point of this region the field p must be zero. To prove this, we have to shrink the contour C, to a very small one, far from the core. The winding number is Q = 1 for the former and Q = 0 for the latter. As the winding mumber is an integer, the abrupt change of Q can occur only at the moment when the contour crosses the point where p = 0, at which the phase of p is undefined. The nulling of p means that the superconducting state is destroyed. In physical terms, we say that the magnetic field penetrates into the superconductor along the flux tube where the superconductivity is destroyed. The tube 31n dimensional units the flux quantum is
= hcle.
354
CHAPTER 13. TOPOLOGICAL OBJECTS
thickness is a result of the balance between the surface tension of the interface between the normal and the superconducting phases (which tries to compress the tube) and the pressure of fields compressed in the tube. The flux tubes resolved an experimental puzzle  the observation of magnetic field penetration through some superconductors [10],which had been called type I1 superconductors. The flux tubes were theoretically predicted in 1957 by Abrikosov and are called Abrikosov vortices. Let us reproduce without proof the main results of paper [l11 concerning the stability of vortices with arbitrary winding number Q. In analogy with expression (13.10), the linear energy density of the vortex (in particle physics this value is called the string tension) can be expressed as a surface term equal to m i l ( 4 a ) plus an integral term where a = e2/(47r) is the coupling constant. For IQ1 2 this still allows solutions in the form of one vortex with winding number Q or many vortices with unitary topological charge. Which possibility is realised depends on the ratio m , / m ~ . In type I superconductors for which m, < m ~a ,vortex with IQ1 2 2 has lower energy than Q vortices with unitary winding number. The vortices tend to merge together. The situation is reversed . vortices with large in type II superconductors for which m, > m ~ The Q decay into many with Q = 1. As was discovered by Abrikosov [10],the vortices in this case form a regular lattice in their crosssectional plane. In experiments with macroscopic samples the individual vortices might be not resolved. In this case we can measure the average macroscopic field through the superconductor which would be impossible if all the fields were homogemeous. Let us now consider a single vortex with Q = n. Its profile is the solution of the following equations
>
with the asymptotic boundary conditions (13.24). Taking the gauge A. = 0, VA = 0 and the following ansatz in polar coordinates,
we get the equations
13.3. ABRIKOSOV VORTEX where prime denotes the derivative with respect to r . The boundary conditions for p and F follow from (13.24):
Expressing the magnetic field in terms of E ( r )
we obtain from the flux quantization condition the limit of F at r = 0:
Substituting this into (13.29) and taking the limit r + 0 we get the behaviour of p(r) at r + 0: p  rn. Exact solutions of (13.29), (13.30) are not known except for the special case m, = mA [12]. The vortex profile can be found in the limiting case of strongly type I1 superconductors: m, >> m A . In this case p goes to a much more quickly than F falls off at infinity and we can replace p with its asymptotic form in the equation for F . Substituting in (13.30) a instead of p, introducing a new variable y = d a e r and a new unknown function f ( y ) = F ( y ) / y , we get the following equation for f
The solutions are the firstorder Bessel functions with imaginary arguments. The boundary condition selects the MacDonald function Kl ( y ) and we finally get the asymptotic solution first obtained by Nielsen and Olesen [8]:
Now we should check that the difference p(r)  a does tend to zero much more quickly than F ( r ) . Introducing ~ ( r=) p(r)  a and neglecting in (13.29) the terms x ' l r , F"r2 and all terms nonlinear in X , we get for X an estimate X ( r ) eTr'9T) that makes our solution selfconsistent. Using (13.32) we can check that the magnetic field vanishes exponentially a t infinity. We see that the condensate (i.e. the pfield) differs from its bulk value a in a tube of thickness of the order of m;', while the magnetic field is concentrated in a characteristic size m i l .
356
CHAPTER 13. TOPOLOGICAL OBJECTS
AharonovBohm Effect It should be noted that the flux quantization condition requiring the field to be singlevalued is true only if the space is simply connected. This means that any two points of the space can be connected by a continuous curve and any closed curve can be shrinked continuously to a point *. If a function f is singlevalued in some area of a simply connected space, it can be continued to the whole space along paths connecting the points of the area with all other points. The condition for f to be singlevalued requires that for any two points xo and X, continuations of f (xo) along each path connecting these two points must give the same result, f (X). In particular, the continuation along any closed curve going through xo must lead to the initial value f (xo). This is automatically true if any two paths connecting xo and X can be deformed into each other, i.e. all paths are topologically equivalent, or homotopic. For closed loops this means that any such loop can be shrunk to a point. If this is not so, there could be cases when the condition of unambiguously cannot be held. Let us consider the example in which only an external magnetic field is present. Let the magnetic field be nonzero within a tube with the disk Do as the crosssection, and vanish outside. Suppose that the tube is not transparent to the charged field cp, so cp vanishes on its boundary. The magnetic field can be regarded as produced by a long solenoid such that we can neglect the boundary effects as well as the magnetic field outside. Thus it sufficient to consider the field cp only in the crosssectional plane from which the disk Do is cut. This is a nonsimply connected area because all curves are split into different classes: topologically trivial those which can be deformed to a point, and topologically nontrivial those which cannot. Every contour must traverse Do and is characterized by the number of circumventions. In contrast to the Abrikosov vortex, the magnetic field is not created by circular currents, rather by spiral ones (which are also external for the cpfield). Thus there is no quantization condition for these currents and, therefore, there is none for the total magnetic flux. The magnetic field is zero outside the solenoid, whereas the electromagnetic potential is not. The field cp acquires, under a chain of translations along a closed curve C, the phase factor (see also (12.24)) 
41f only the first requirement is fulfilled, the space is called linearly connected. The circle S1is an example of a linearly, but not simply connected space.
13.3. ABRIKOSOV VORTEX
357
If cp is interpreted as the wave function of a particle, then this particle acquires a nonunit phase factor as it goes round the solenoid. In experiments, one measures the relative phase shift of two particles, emitted at a point, coming to a given point from either side of a solenoid. The phase is equal to the magnetic field flux in the solenoid and is experimentally measured. This means that if the particles are detected on a screen behind the solenoid, the phase shift results in an interference pattern which can be registered (say, whith a scincillator screen for the case of electrons). This effect, which was measured many times in different systems over the last thirty years [13],is called AharonovBohm eflect.
A Few Words about Topology and an Exotic String We shall now untroduce some mathematical terminology in order to facilitate the finding of topological objects in higher dimensional spaces. At the end of this section, we shall present one more string solution with a rather exotic property of its topological charge Q. As was stated at the beginning of section 13.3, the finiteness of energy requires us to map continuously the spatial infinity in the given number of dimensions into the vacuum manzfold of the given theory. Spatial infinity in ddimensional space is the sphere Sdl (for the simplest case d = 1 it is reduced to two points). In the theories with a scalar field, the vacuum manifold is defined by the minima of V(cp). A condition for nontrivial solutions to exist is a nontrivial minimum of V(p) at Ipl = a # 0. To keep the energy of such a solution finite, we must use a sensibly chosen gauge field. Otherwise there are dangerous divergences in the gradient term of the energy. If the potential energy is gauge invariant the minimum at Icpl = a is necessarily degenerate, due to the action of the group. Without going into detail, let us restrict our consideration to those vacuum manifolds in which each pair of elements can be transformed into each other by a group transformation" For the case of vortices, the group is U(1), which consists of all rotations on the circle S'. This case is so simple that we have been able to find all mappings without involving topology. In this case all mappings can be labelled by an integer n. More surprising may be the fact that all mappings form a group equivalent to Z, the group of additions on the integers. "n
obvious counterexample is a potential V(cp) which has degenerate minima a t # az.
= a1 and Icpl = a2 with a1
Problem: Show that such a thcory possesses domain walls.
CHAPTER 13. TOPOLOGICAL OBJECTS
Figure 60
Before going further, let us explain the physical content of this statement, namely the rule which gives the topological charge Q12 of a combination of two vortices with charges Q1 and Q2. We might expect that Qlz = Q1 Q2, which is indeed the case, but it is instructive to be proven. Figure 60 gives such a proof. In this figure the cp field of the vortex shown on the right is multiplied by 1, which does not change its properties. It is obvious that the winding numbers of the two vortices are added as we go round the contour shown with a dashed line. Having calculated the total winding number, we can take the contour to infinity, shown schematically with a dashed line. During this distortion, the winding number is conserved as we do not cross the vortex centres. Any two mappings with the same values of Q can be continuously transformed into each other. This is rather obvious for the considered case of the group U(1). In mathematics this becomes a definition: two mappings of a contour into a manifold M are called homotopic if they can be continuously transformed into one another. For mappings a1(6) and ~ ~ ( 6taking ') values in M , and I9 running over the mapped contour (0 I9 2n), this means that there exists a continuous function a(I9,t ) , 0 It 1 such that a(8,O) = a1(19) and a ( 0 , l ) = a2(0). As we are considering the continuous mappings of a circle, we require that all functions take the same values for 8 = 0 and I9 = 2n, that is we consider only closed contours in M. It is important that we are able to define the addition operation on such mappings. Let a1(19) and az(I9) be two closed contours in M . The sum al a 2 is defined as
+
< < <
+
It can be easily checked that such a sum obeys all group axioms. The mapping into a single point plays the role of the unit element and al(8)
13.3. ABRIKOSOV VORTEX
359
is the inverse of a1( 6 ) . An important characteristic of any manifold M is first homotopy group (also called the fundamental group) of M . It characterizes the connectivity of the manifold and is constructed as follows. Let us select a point in M and consider all closed contours which start and end at this point. Then we classify all contours as belonging to classes, such that any two contours in one class are homotopic and those from different classes are not. There always exists a trivial class, equivalent to the contour which never leaves the initial point. The addition operation between the classes can be defined based on the addition operation for contours (13.35). Problem: Prove this. Then all classes form a group which is the first homotopy or fundamental group and which is usually denoted r l ( M ) . The choice of the initial point does not matter in connected manifolds. Problem: Prove this. Otherwise the fundamental group should be constructed for each connected submanifold separately. Problem: Show that nl group of integer numbers).
=
Z
X
Z if M is a torus (Z is the additive
The first homotopic group is called nontrivial if it consists of more than one element. In this case there are contours which cannot be continuously shrunk to a point. Thus a nonsimply connected space is precisely the space with nontrivial first homotopic group. Returning to vortices, we can say that nl(U(1)) = nl(S1) = Z. Here U(1) is the gauge group of the model and the circle S1 is its parameter space, which coincides in this case with the vacuum manifold6. Let us check two more complex groups, namely SU(2) and S 0 ( 3 ) , for the existence of vortices. First we shall show that the manifold of all group transformations of SU(2) has the topology of the sphere S3. TO do this, we use an explicit parametrisation. A conventional sphere S2is described by two angles: GThegauge group can be larger than the vacuum manifold because there are group clemcnts which do not change the elements of M. For example, a rotation around a vector in 3D does not change the vector.
CHAPTER 13. TOPOLOGICAL OBJECTS
Figure 61
<
the azimuthal angle 0 a 5 27r, such that a = 27r is equivalent to a = 0, and the polar angle 0 5 0 X , such that the values 0 = 0 and 0 = 7r correspond to two different points, each of which is the same for all values of a. This is illustrated in figure 61 where (from left to right) are shown: the starting manifold of a and 0 before any convention about the endpoints is made; the result of the identification of a = 2n and cr = 0 which gives the cylinder; and the result of the identification of all points with B = 0 and all points with 0 = 7r which gives the sphere. The sphere SQas one more polar angle 0 5 q 5 7r. This can be visualized if one imagines subsequent sections of S" (with unitary radius) as threedimensional hyperplanes. The sections are, of course, spheres S2 with the radius sinq. We must now show that the group manifold of SU(2) is S3. The elementary representations of this group are spinors, which are tranformed by 2 X 2 unitary matrices. There is a onetoone correspondence between these matrices and rotations in our 3D space, established by the following form of the group element
<
where a' is the vector of Pauli matrices and G is a vector which is the rotation axis in 3D space. The length of this vector W is equal to the rotation angle. The evaluation of the exponential function is simple: +
W
exp ( $ i ~ = cos )2
+ 2. wW + sin W2 U
.
Problem: Prove this equality using the Taylor expansion of the expo= W'. nential function and the fact that
13.3. ABRIKOSOV VORTEX
Figure 62
We see that U(;) = 1 only for W = 47r (a wellknown property of spinors). In order to parametrize all rotations, it is sufficient to use only the interval 0 W 27r. This is because a rotation by an angle 27r U, with v > 0, is equivalent to a rotation by angle U around the vector G, as follows from (13.37).
< <
+
Problem: Show that the correspondence (13.36) really is onetoone using (13.37).
Thus we parametrise this sphere S2by the two angles a and 9, which are defining the direction of vector w', while W , the length of this vector, can be identified with the second polar angle of S3. Indeed W varies from 0 to 27r with an additional condition that each of the end values defines only one rotation, regardless of the values of a and 9. These points are the zero rotation (the 'noth pole' of S3)and the 27r rotation (the 'south pole') respectively. Let us consider now the group SO(3) whose elementary representation is the vectors of fixed radius in 3D. The above parametrization of rotations remains valid, up to one change. For the 3D vectors, a rotation defined by vector w' is equivalent to a rotation by the angle 27r  W around the unit vector w'/w. This is directly related to the fact that unlike the spins a 27r rotation of 3D vectors is identical to the null rotation. Indeed, the north an the south poles of S" are then equivalent, but all pairs of opposite points are equivalent on the sphere. Thus we come to the conclusion that the manifold of SO(3) elements is equivalent to the sphere S3 with opposite points identified. Thus, indeed, is the difference between SO(3) and SU(2), which are locally equivalent (one says 'isomorphic'). The global difference between them is indicated by different fundamental groups. While nl (SU(2)) is trivial (any closed loop on S3can be shrunk to a point, figure 62, left), the group 7r1(S0(3)) is nontrivial. There exist
CHAPTER 13. TOPOLOGICAL OBJECTS
contours which start at point and end at the opposite point (figure 62, right). Two such coutours, traversed one after another, are equivalent to the zero contour because they describe rotation by an angle 27r. This means that nl(SO(3)) = Z2, where Z2 is the group of addition modulo 2. That is Z2 consists of only two elements, 0 and 1, and the group operations are 0 1 = 1 0 = 1 and 1 1 = 0. Thus the element 1 is its own inverse. This can be visualized grafically, as shown in figure 63 where S" is represented by its S2section. The contour shown in figure 63a can be continuously rotated in its own plane by an angle T, as shown in figure 63a to the position shown in figure 63b, and then wrapped around the axis defined by the identified point (figure 63b) up to the position shown in figure 63c). While the position of the contour in figure 63c coincides with the initial one, the direction of the circumvention of the contour is inverted. This proves the statement. We should now investigate whether or not vortices exist in SU(2) and SO(3) gauge theories. Let us consider for simplicity only the pure gage theories without the scalar field. This is possible for the nonAbelian gauge groups, because in this case the gauge field is complex enough to have the necessary property wich was attributed to the scalar field in the above example of Abrikosov vortex. Namely, the vacuum manifold consists of all group elements U which are covariantly constant, that is DU = 0. For such fields, the energy density is zero, as it should be for the vacuum. It follows from the nontriviality of r1(S0(3)) that there is a vortex solution. Its topological charge has two properties which follow from the fact that ~ l ( s O ( 3 )= ) Z2: (i) it only takes the values Q = 1 for a vortex, and Q = 0 for the vacuum; (ii) the vortex is the antivortex of itself, that is two vortices annihilate upon contact because their summed topological charge is zero.
+
+
+
13.3. ABRIKOSOV VORTEX
Figure 64
Vortex Solution in Other Contexts Recall that the kink or domain wall solution in pure cp4 theory described in this chapter has the same profile as the quantummechanical instanton for the quartic doublewell potential. The same is true of the vortex solution. Being essentially the solution of the same equations, it has a different physical meaning depending on the context. It is clear that in a (2+l)dimensional theory it describes a particlelike solution. The same solution embedded in 3 or (3+1)dimensional space describes a static or moving vortex respectively. Let us show that the vortex solution in (l+l)dimensional space is, in fact, the instanton. The Euclidean Lagrangian of the model takes the form
which coincides with (13.21) if we set E = B, = B, = 0. We shall denote as X and xa as T . Here the field tensor F,, is the remainder of its full 4dimensional predecessor. F,, depends now on the twocomponent gauge field A, which produces the onecomponent 'magnetic' field B = XI
tp,qJL. Since cp is complex it is twocomponent and so there are many vacuua in this model. We illustrate this in figures 64a, in which the vacuum cp field is represented by arrows, which indicate its phase. The vacua are the configurations cp = a exp(ia(x)) with arbitrary Xdependent phase a(x). If we consider the system on an infinitely large circle (which implies periodic boundary conditions on spatial infinities), then a ( x ) must obey the restriction a(+oo)  a(oo) = 27rn, for an arbitrary integer n, which classifies all possible nonequivalent vacuum states. We refer to n as the winding number of vacuum.
CHAPTER 13. TOPOLOGICAL OBJECTS
364
As the vortex is essentially twodimensional, it is clear that it must correspond to a process in Euclidean space. This process is a trasition from a vacuum with winding number n to a vacuum with winding number n Q, where Q is the topological charge of the vortex. The proof is actually presented in figure 64, drawn for the case n = 0, Q = 1. Figure 64a shows the two vacua which are the initial and the final states. Figure 64b shows the vortex as defined by the boundary conditions (13.22), (13.23). We are free to make any phase rotations (gauge transformations) which are continuous in the whole space. Using this freedom, we pass from figure 64b to figure 64c in which all the winding number of the vortex is collected in that part of the contour which corresponds to the final state. This proves the statement for the simplest case, and its generalization is obvious. Thus we have the oneinstanton solution. The general framework of the instanton calculus is similar to that discussed in chapter 3, particulary section 3.3.
+
Problem: Perform the summation over the number of instantons in the dilute gas approximation. When is this approximation valid? We shall consider the famous SU(2) instanton in four dimensions in section 13.5.
13.4
The 't HooftPolyakov Monopole
Let us construct a genuinely threedimensional object which is a generalization of the Abrikosov vortex. The space infinity in 3 dimensions is the sphere S2. It is thus easy to construct a nontrivial mapping if the vacuum manifold of the theory is S2 as well. This is so in models with an isotopic triplet of scalar fields:
and a scalar potential in the form
As suggested by the experience gained in the previous section, we need a gauge field to make the energy of the nontrivial solution finite. As the gauge group transformations of a given isotopic vector should cover the whole vacuum manifold S2,the gauge group should not be smaller than
13.4. THE 'T HOOFTPOLYAKOV MONOPOLE
365
SO(3). Thus we use the model described by the following (Minkowski) Lagrangian:
Here AE(x), a reads
=
1 , 2 , 3 is the gauge field, and the covariant derivative
+
D P q= aPP g&,bcA;@
,
are the explicit form of the SO(3) structure constants. The where fundamental and the adjoint representations of SO(3) coincide with each other. Thus the gauge field tensor takes the form
We are looking for static nontrivial solutions of the equations of motion which minimize the energy
The explicit form of the equations is
In order to find the simplest possible solution, we first consider the case A8 = 0 and assume the solution is spherically symmetric. The following boundary condition for the Higgs field is obviously nontrivial:
where r is the radius in the spherical coordinate system. This asymptotic form mixes spatial and isotopic indices. One calls it the hedgehog configuration because each point of the spatialinfinity sphere corresponds to a 3dimensions isovector pointed outwards. Let us bear in mind, however, that all isotopic vectors can be simultaneously rotated without any actual change of the ansatz. We can now find the gauge field necessary to compensation for at infinity. It must obey the condition
CHAPTER 13. TOPOLOGICAL OBJECTS
366
After simple transformations
we get finally the following asymptotic boundary condition (13.44) The solution to equations (13.41), (13.42) with the boundary conditions (13.43)) (13.44) does exist, although it is not known analytically in the general case. Below we shall discuss this issue in brief, but now let us consider the electromagnetic properties of the solution which follow from (13.43), (13.44).
Magnetic Properties of the Solution Let us show that the solution we are constructing is a magnetic monopole. The asymptotic form of Aj gives the following behaviour of the colored magnetic fields
The electric fields are zero. Two combinations of these fields are fixed by the requirement that the angular derivatives are zero on the infinite sphere. There is the third residual field. As suggested by the power behaviour (13.45) of all magnetic fields, the residual one will be proportional to l/r2. This is typical of point electric charges, however, we are now dealing with the magnetic field. This enables us to suggest that the field configuration defined by (l3.43), (13.44) has a magnetic charge. To make this statement more exact, let us first find out what the residual field is. We shall call this field 'electromagnetic' as we would if the considered model described the electroweak interactions. It does not and our treatment is actually applicable to more complex models in which SO(3) or the locally isomorphic7 SU(2) are subgroups of a larger group. We briefly discuss this issue at the end of this section. Having made these excuses, we shall call the residual field 'electromagnetic'. There is no problem in selecting the elecromagnetic field from three gauge fields of SO(3) at each space point where the Higgs field lies in a vacuum state (see also 12.3). Let us visualize & as a 3dimensional vector in isotopic 7The difference in spheres.
./rl
between these two groups is not relevant for mapping of S2
13.4. THE 'T HOOFTPOLYAKOV MONOPOLE
367
space. The group generators are the matrix which perform rotations there is a around three ortogonal axes in this space. For any given generator set such that the third generator performs rotations around $h. If we consider the generators as a matrixvalued vector T~ then this third generator is just
The action of this generator on $h gives zero. Two other generators move on the sphere of vacuum manifold, that is they 'eat' the Goldstone modes of the $ field. These generators correspond to the massive fields (as we discussed in chapter 12, section 12.3) while the electromagnetic one is massless. This is in agreement with another way of selecting Q. We start with an obvious statement that the electric charge of the vacuum is zero. The charge of a specific field is the eigenvalue of an appropriate group generator. Therefore Q is selected by the requirement that its action on gives zero. Let us now apply these ideas to analyse the boundary conditions (13.43), (13.44). The definition (13.46) is valid locally at each point. The key issue is that the fiels inevitably depends on the space point because it is impossible to make a continuous transformation to a constant $h on the whole infinite sphere. We now substitute into G,, the gauge fields decomposed on the sum of Q and two other generators and separate the part depending on Q. The result is the electromagnetic tensor which has the form [14]:
The last term here is due to variations of isotopic space, then
6 in space.
If
6 is constant in
This coincides with the naive expectations. There is no magnetic charges in electrodynamics because the magnetic field B"" is the rotor of A and thus VBem= 0 identically. This is not the case for the electromagnetic tensor (13.47). The magnetic field selected from it obeys the property
368
CHAPTER 13. TOPOLOGICAL OBJECTS
for the boundary conditions (13.43), (13.44). Here X is the radius vector. Thus the solution is indeed a magnetic monopole with charge l l g . It is called the 't HooftPolyakov monopole. More generally, F,, obeys the property that the divergence of its dual tensor does not vanish:
where k p is the topological current introduced in chapter 12 and considered in more details below. For static configurations and At = 0, we get 47r VBem=ko . (13.48) 9
Lower Boundary on the Monopole Mass Here we specify functions which define the monopole shape and obtain an estimate for its mass. According to the asymptotic forms (13.43), (13.44)) we search for spherically symmetric static purely magnetic solutions in the form of an ansatz (called hedgehog ansatz) proposed by t'Hooft and Polyakov [14, 151:
where the requirement of the finiteness of the energy gives for the functions W and F the following boundary conditions:
Substituting (13.49), (13.50) into (13.41), (13.42) and introducing new unknown functions K(r) = 1  F(r) and H(r) = graW(r) we get the equations
13.4. THE 'T HOOFTPOLYAKOV MONOPOLE
Figure 65
which can be solved numerically with the result depicted in figure 65 (where the variable = gar used). The exact solution is known in the case X = 0 (this means that X + 0 at fixed a). This is
<
K ( r ) =  ' H(r) = gar 1 sinh gar tanh gar This solution was independently obtained by Bogomolny and by Prasad and Sommerfield [ll,181 and is called the BPS monopole. We shall now see that there is a reason deeper than pure luck for this solution to be known analytically. Let us obtain a lower bound on the monopole mass by making the following rearrangement of the expression for the field energy: ,
The second term in this expression can be rewritten in the form ~i~kG:~Dk$a=&(&kijG:j$a)
.
Then, after integration of the second term by parts in view of (13.48) we come to the expression
370
CHAPTER 13. TOPOLOGICAL OBJECTS
This is called Bogomolny boundary. Here Q is the value of the topological charge ko. Its value is Q = 1 for the monopole solution discussed above. The lower boundary can be reached if (i) V ( $ )= 0 with fixed a and (ii) the first term in the integrand is zero. The latter condition results in firstorder equations which are much easier to solve. The solution is exactly the (13.53). Thus, for X = 0 (the BPSlimit), the monopole energy is 47ra r n ~ Esps =  = (13.56) (U 9
~ the mass of the gauge fields and a = g2/47r is the electrowhere r n is magnetic coupling constant. Problem: Is it possible to reach the BPS limit in quantized theory (13.39)? Hint: follow the discussion in section 12.7. [35]
Dyons The term dyon denotes an object which has both magnetic and electric charges. To find such a solution, we relax the restriction A; = 0 and search for a solution in the form proposed by Julia and Zee [l91
where J(0) = 0. The equations of motion take the form
This solution carries the electric charge
in addition to the same magnetic charge as that of the monopole.
13.4. THE 'T HOOFTPOLYAKOV MONOPOLE
371
In the Bogomolny limit, the solution to these equations which is called the BPS dyon takes the form:
agr K(r) = sinh gar '
H(r) =
(
tanh agr

1) sinh C
(13.61)
where C is an arbitrary constant. The electric charge of this solution is (4.rrlg)sinh C.
A Few Words About the Topology Let us discuss the solution obtained mathematically. The main idea is to derive the properties of the vacuum manifold from the properties of the gauge group. We have started our search for the finiteenergy nontrivial solution with the statement that the Higgs field asymptotically goes to its vacuum manifold:
where Q, = {plc), A p = { A : ) and a, k are the isotopic indices of the scalar and the gauge field. If the potential energy is gauge invariant, its minima are necessarily degenerate and form the manifold which we have already refered to as the vacuum manzfold. As A, and Q, are supposed to be transformed under an irreducible representation of the gauge group, the vacuum manifold M is determined by the structure of the group manifold. As we have seen above there are transformations in the group which leave a fixed point of M unchanged. Such transformations form a subgroup H in G and M is topologically equivalents to a factor group GIH. Thus in the case of the double well potential and the unique real field p, M = {a, a). For the same potential and complex cp the gauge group is U(1) and M is equivalent to a circle, M 2 S'. If is an isotopic doublet, we have SU(2), H trivial and M 2 S3. In the case of the neutral isotopic triplet G = S0(3), H = U(1) and M = S0(3)/U(1) 2 S2. 'This is valid, of course, if the dimension of the representation considered is greater than the dimension of the generators basis corresponding to the space G I H .
372
CHAPTER 13. TOPOLOGICAL OBJECTS
As the fields at infinity must be in their vacuum state, the boundary conditions imposed on spatial infinity imply a mapping of points of S2 onto the vacuum manifold which is equivalent to the factor group S0(3)/U(l). Such mappings can be classified in the same way as the mappings of S1 considered in the previous section. Using this analogy one can show that the mappings S2+ S2are split into nonintersecting equivalency classes which are characterized by an integer n called the winding number. n is the number of times the mapping covers its image while its arguments cover the first sphere exactly ones. In general, classes the group of equivalent mappings of the sphere S" to an arbitrary topological space X is called the kth hornotopy group and is denoted by 7rk(X). The term 'group' is reasonable beause one can define a product of such mappings in the same way as for paths, but this definition is tedious and will not be used in what follows so we do not quote it. Thus we conclude that the second homotopic group of S2is isomorphic to the additive group of integers 7r2(S2)= Z. However S2is topologically equivalent to SO(3)/U(l), hence 7r2(SO(3)/u(l)) = Z. In general if the symmetry described by the gauge group G is spontaneously broken to its subgroup H , we can find out whether there topologically nontrivial solutions with finite energy in 3 spatial dimensions by calculating 7rz(G/H). This can be proven by the methods of algebraic topology that if G is a connected simply connected Lie group and H is a connected subgroup, then 7r2(G/H) = 7rl(H). This theorem covers a lot of physically relevant cases so that absence or presence of topological solutions in a theory with degenerate vacuum is determined by the first homotopic group of the unbroken symmetry group For instance, if the case of the double well potential for an isotopic doublet is considered, the unbroken symmetry group is trivial and there are no topological solutions. We have used the term 'topological current' above. Let us define it now. As in the Zdimensional case it is a conserved local quantity giving the topological charge after integration over the spatial volume. It is
"
where 8, = $,/)$1, vation condition d,k'
/$l2 =
= @a$a. One can make sure that the conser0 is identically fulfilled. Let us show that the
the case of SO(3) this theorem is not applicable because .rrl(SO(3)) = Z2 = (0,l). This is an essential difference between SO(3) and SU(2), because nl(SU(2)) = .rrl(S" = 0. Geometrically SO(3) differs from SU(2), it is a 3sphere in which all antipodal points are equivalent. Thus, there are two types of nonequivalent contours: diameters and all others.
13.4. THE ' T HOOFTPOLYAKOV MONOPOLE
373
integral of ko over the space volume really gives the topological charge. We have
where d2ai is the area element on the sphere S; which the infinite boundary of space:
(here ul, u2 are internal coordinates on the sphere),
and integration by parts has been taken. Substituting the sphere parametrization xi = xi(ul,u2) we get after some algebra
This integral is just the winding number of the mapping of S; onto the vacuum manifold S2. The analysis of the magnetic properties of the monopole can be generalized for arbitrary topological charge. Both the integral which gives the magnetic charge and the one which gives the energy of BPS monopole are proportional to the topological charge Q. Thus the monopole charge is Q/g and Esps = QmV/a. Note that there are multimonopole solutions ~ D equation ~ q (13.55)). h ~ The of the Bogomolny equation G$  ~ ~ ~ (see energy Qmv/a of these solutions does not depend on the actual configuration been defined by the boundary conditions. This means that the BPS monopoles do not interact with each other. Problem: Try to explain why two BPS monopoles do not interact. Hint: there are more than one longranged interactions, see [20].
Do Monopoles Exist? We have mentioned already that monopoles are absent in the electroweak theory in which the Higgs field is an isodoublet (isospinor) rather than an isovector. The vacuum manifold is then S3 and .rrz(S" = 0. For
374
CHAPTER 13. TOPOLOGICAL OBJECTS
many years the hope of discovering monopoles has been directed towards those created in the hypothetical phase transition in the early Universe at the very high energy scale of unification of electroweak and strong interactions (see, e.g., [21]). During this phase transition, a largemass Higgs field acquires a vacuum expectation value which breaks the initial high symmetry down to the direct product of the electroweak and the color symmetry groups. The group of unified interactions may be large and, in many cases, it has a subgroup necessary for the existence of m o n ~ ~ o l eHowever s ~ ~ . the cosmic monopole has not been discovered even in very sensitive experiments. The present direct experimental limit on the flux of cosmic monopole is at the level of 10l5 c m  2 S' srl [22]. Extensive work has been done to find out the mechanisms of monopole annihilation and their trapping in galactic magnetic fields [23]. The strong current interest in monopoles is due to the idea of explaining the OCD confinement as a dual Meissner eflect [24]. In superconductors the magnetic field does not penetrate inside a sample because the latter is absolutely screened by currents of Cooper pairs. Similarly, it is hypothesized that our vacuum is filled with a condensate of monopoles which are thought of as special fluctuations of the QCD fields. Then the chromoelectric field is screened by monopole currents. Particles with color charge (quarks and gluons) destroy the condensate in analogy with the condensate destruction in the magnetic flux tube in superconductors. In the white state (with zero net color charge) this results in the creation of an effective 'bag' which confines the quarks. If a quark or a gluon is removed from the bag (having been hit in a highenergy collision) then there is a flux of chromoelectric field between the ejected particle and those which remain in the bag. This flux is compressed in a string of color field, as for the magnetic flux in superconductors. Such strings have been observed in highenergy collisions and used for many years to describe light quarkantiquark systems with high angular momenta. The only drawback of this plausible model is the fact that the existence of monopole condensate has not been established in QCD, although remarkable progress has been recently achieved in sypersymmetric QCD 1251.
loSimplyspeaking, the vacuum expectation value in remote spatial regions is formed independently. Thus topological defects at some points are possible. The creation of such defects during phase transitions is observed in some types of liquid crystals [27].
13.5. SU(2) INSTANTON
13.5
SU(2) Instanton
The Higgs field is a necessary component of the monopole solution found in the previous section. Here we construct a nontrivial solution in fourdimensional Euclidean space in pure Yang  Mills theory. This solution is called a BPST or SU(2) instanton [26]. It describes a tunnelling process between two vacua with different topological numbers. The simple example described in section 13.3 is a good model to understand the idea of the BPST instanton. In the previous sections, we discussed the topological aspects of the solution which was obtained solely on the grounds of common physical sense. In contrast, we use here the experience gained above to derive the solution and then discuss the process it describes.
Nontrivial Solution Let us consider the pure Euclidean Yang Euclidean action

Mills field described by the
We shall search for nontrivial solutions with finite action localized in 4dimensional space. The finiteness of the action requires the following asymptotic form of the gauge fields
+ +
where U(x) is an SU(2) local gauge transformation and r2 = X: X; + X;. This boundary condition defines a mapping of the sphere S: of infinite radius in Euclidean space onto the group SU(2). As SU(2) is topologically equivalent to S3, such mappings are classified in the same way as was done for lowerdimensional cases in the two previous sections. All possible mappings fall into classes of equivalence, which form the group 7r3(SU(2))= 7r3(S3).Using the known result of algebraic topology 7rn(Sn)= Z (or just common sense), we conclude that each class is characterized by a winding number n, which is an integer. In other words, each solution of the YangMills equations in 4dimensional Euclidean space is characterized by a topological charge. As in the previous cases let us construct the topological charge density. The corresponding topological current must be rotationally and gauge invariant and be represented in the form of a 4divergence. The X:
CHAPTER 13. TOPOLOGICAL OBJECTS
376
quantity kp determined by (12.32) could be a candidate for it. Let us prove that it is indeed the case. We have
where da, = (1/3!)~,,,dz,dx~dx,. As SU(2) is topologically equivalent to S3,there must be a set of parameters El, J2, which simultaneously are coordinates on the sphere. Parametrizing S& by means of [i we obtain 1 dx, dxp dx, da, = ErVPuEijk &ld<2d<3 . (13.69) 3! ad& ad& ad[,
c3
Substituting this into the integrand and taking into account (l3.68), we get after some algebra
where n appears as the winding number of the mapping S&+ S3,after the replacement of the spatial sphere S& with the group space of SU(2), which is S3.we know from the theory of Lie groups that the expression
represents the invariant measure on SU(2). Its integral over all possible values of the parameters gives the volume of the group space, i.e. the volume of S y i t s 3dimensional 'surface area'):
<,
J d3
~ i=)2a2
~ 2 ,
S3
We finally obtain
Thus the quantity
is indeed the topological charge of the solutions of the YangMills equations.
13.5. SU(2) INSTANTON
377
Let us find an exact finiteaction localized solution of the sourceless YangMills equations D,G,,=O . This is a set of second order equations cubic in the unknown functions. The order of these equations can be reduced by means of the following trick [26],which is analogous to the Bogomolny estimate discussed above. Let us consider the evident inequality
Taking into account (13.67), this results in the following estimate
where SEis the Euclidean action. Therefore the fields obeying the equation G,,, = &G,. (13.74) automatically minimize the action (therefore they also satisfy the full YangMills equations). Such fields are called selfdual . Similarly to the energy of the BPS monopole, the Euclidean action is proportional to the topological charge. It follows from (13.72) that
Let us solve the selfduality equation (13.74). For this purpose we choose the gauge function U(X) in (13.68) in the following form:
where the matrices q,l = (iok, 1) obeying
have been introduced. It is clear that U(x) defines the mapping with winding number n = 1,so that the corresponding solution has unit topological charge. Using this form of U, we obtain
CHAPTER 13. TOPOLOGICAL OBJECTS
378
where qfiu is the selfdual tensor
v,,
= ~(v,vL 4 l u )
(13.80) The obtained asymptotic form suggests the following ansatz for the solution: 1 X, A, =  v , u F f (4 7 (13.81) 9 where the unknown function f ( r ) must obey the boundary conditions f ( 0 ) = 0 in order that the solution be regular at r = 0 and f ( r ) + 0 at infinity because of the energy finiteness requirement. f can be found from the selfduality equations (13.74). For this purpose we calculate the field tensor G,, and its dual G,, using (13.81):
where
X,%pXp . Comparing (13.82) and (l3.83), we conclude that the selfduality equation is fulfilled if 1 f l   f(1 f ) = 0 . (13.84) r2 The solution of (13.84) obeying the correct boundary conditions is I,
= x,v,pxp

where a is an arbitrary constant called the instanton size. It is not surprising that the solution of the classical equations has this ambiguity. It cannot be fixed just because there are no dimensionful parameters in the model. Substituting f from (13.85) into (13.81), we obtain finally the instanton solution. For the field tensor we have
13.5. SU(2) INSTANTON
379
By construction this tensor is selfdual. It can also be formally checked using the properties of qpv Modifying q as follows
we obtain antiselfdual solutions, called antiinstanton.
On the Vacuum Structure of YangMills Theory Let us say a few words about the physical meaning of the instanton. We have already shown (see chapter 3 and section (13.3)) that fourdimensional nontrivial solutions in Euclidean space correspond to tunneling between different classical vacua. We would now like to know what the different vacua in the YangMills theory are. In order to look at the instanton as a process occurring in Euclidean time, it is convenient to segregate time from the spatial coordinates (in some sense this is analogous to the transformation made in figures 64b,c). For this purpose, it is convenient to make a gauge transformation such that the transformed instanton fields obey the condition Aj",= 0. This representation is also useful to apply the analysis of chapter 12, performed under the same gauge condition. For this purpose, we note that by virtue of (13.81), (13.85) the instanton solution can be represented in the form
where the matrix U is determined by (13.76). It is more convenient to represent U in exponential form:
where We are looking for the gauge transformation S(x) after which A. vanishes. This condition can be written as
Searching for the solution of this equation in the form
380
CHAPTER 13. TOPOLOGICAL OBJECTS
we get IT
a(x) =   arctan 2 ((Ix12 ~ ~ 2 ) 1 / 2 )' The asymptopic forms of a are
which leads to the following asymptotic form of S:
where
Thus the spatial components of the transformed instanton field take the form l A ~ ( x ) ( ~ "= " ~S(X)A~ ) (x)(~""~)s'(X) + F(diS)Spl . 29
This gives the following limits at x4
+ foo:
Thus the instanton solution Ai(xq,x)(~""~) interpolates between two pure gauges. These two gauges are not topologically equivalent. It follows from the fact that v(x) belongs t o the gauge class with winding number n = 1 because there is a onetoone correspondence between the parameters of v(x) and coordinates on the sphere S3. This means that the vacuum state of the YangMills fields can be represented in even more forms than stated above. As we know, gauge fields in the vacuum state 10) can be described equivalently in any gauge. Now we are forced to enlarge the class of allowed gauge transformations. Indeed the transformation v(x) applied to topologically trivial vacuum fields transforms them into a configuration with winding number equal to unity. Repeating this many times, that is applying the transformation v,(x) = ( ~ ( x ) ) ~ , we obtain the configurations with winding number n. It is clear from the above considered topological reasons that transformations belonging to different gauge classes cannot be connected by a continuous gauge transformation (v(x) cannot be obtained from the identical transformation by any continuous distortion). This is reflected by the homotopic
13.5. SU(2) INSTANTON classification of gauge transformations. The gauge transformations with nonzero winding number are called large gauge transformations, distinguished from the small ones whose winding number is zero. Thus we reach an important conclusion: the vacuum of pure YangMills theory is infinitely degenerate, consisting of a set of topologically nonequivalent states labeled by an integer number n = 0, &l,k2, . . .l1. Let us examine this statement more closely. We have met a situation like this in the two quantum mechanical problems considered in chapter 3: (i) a particle in a periodic potential and (ii) a particle on a circle. The difference between these situations is clear: in the former there is an observable value (the coordinate X) which is different in different vacua. This makes the vacua physically different. As a result, the ground states in all potential wells are combined in infinitely many superpositions, labelled by an angle Q, forming an energy band. For case (ii), there is only one physical ground state although there are nontrivial paths (instantons) leading from this state around the circle to the same state. As a result, there is still only one physical state which can be thought of as being selected from the energy band of case (i) by a specific value of 8. Which case is an adequate model of the Yang  Mills vacuum? The answer is, in fact, the case (ii). Indeed neither the small nor the large gauge transformations change any physically observable value. Thus the topologically nontrivial vacuum should be understood as physically the same vacuum state as in the case of a particle on a circle. With this idea in mind, we can perform the summation over arbitrary numbers of instantons in the Euclidean amplitude for vacuumtovacuum transitions in order to estimate the nonperturbative instanton effect in the vacuum energy (compare this with section 3.3). Let us consider the vacuumtovacuum amplitude
for a transition from AI to A2, which are just the asymptotic forms of A ~ ( x ) ( ~ " "at~ )x4 t cm and 5 4 t cm respectively. We do not need to specify the gauge condition f (A;). Applying the instanton calculus described in chapter 3, we obtain (see section 3.3)
c c m
z=zon=m m=oo
(TVZl)n+m n!m!
=
2, exp (2TVZl)
,
(13.91)
where Zo is the unimportant contribution of the trivial Euclidean trajectory, Z1 is the instanton contribution, n is the number of instantons, m ''The same is valid for any excited state.
382
CHAPTER 13. TOPOLOGICAL OBJECTS
is the number of antiinstantons, and T and V are the Euclidean time and the spatial volume respectively. In contract to the problem about the periodic potential, there is no restriction on the number of instantons and antiinstantons. The instanton contribution Zl takes the following form in the first two orders of the loop expansion (cf. chapters 3, 6, 12):
Z1=
7
daJ
0
det' (S2Sinst) det (S2S0)
where S2Sin,t and S2S0 are the second variations of the action on the instanton and the trivial background respectively, a prime denotes omission of zero modes, and J symbolizes integration over the zero modes corresponding to the isotopic orientations of the instanton also including the Jacobian of integration over the instanton position and its size a). At the tree level, z1 e  S ~= e8r2/g2 (13.93) The functional determinant was calculated by t'Hooft [36] with the main result that the coupling constant g2 in (13.92) is replaced by its running value at the scale a (see chapter 8). As the YangMills theory is asymptotically free (cf. the discussion in chapter 5 , 8), this means that instantons of small size can be accurately described, but their effect is exponentially suppressed because g is small at short distances. A stronger effect is due to instantons of large size, because g is larger for them (there is a Landau pole at some value of a), but large g destroys the applicability of the loop expansion. Let us stop the discussion at this point and direct the reader instead to the literature [33, 31, 32, 341. Let us instead revise our logic above. We have stated that the vacuum remains the same physical state upon the action of large gauge transformations and actually concluded from this that the physical vacuum 10) is transformed to 10). This is seen in formula (13.91) where the oneinstanton contribution (13.92) is determined only by its action and the subsequent terms of the loop expansion. This requirement is too restrictive: a phaserotated state eielO) describes the same physical state as well. Thus there is nothing to forbid the large gauge transformations resulting in an overall phase factor. We can easily describe such a factor if we add one more term to the action: S + S Q, where Q is the functional of topological charge given in equation (13.72). Such a modification does not affect the classical equations of motion because Q is the integral of a total derivative. Neither it does affect any result obtained within one selected topological class. This term, called the 0term, contribute a factor in Zl in (13.92). The result (13.91) takes the
+
13.6. QUANTUM KINK
383
form
Z = Zo exp (2TVZ1cos 8)
.
(13.94)
Problem: deduce this formula. This formula gives the energy of a vacuum state constructed by superposition of vacua with all winding numbers n taken with weights eine (cf. chapter 3). This state is called the 8 vacuum. These instanton effects are important in QCD. In the presence of fermions, Zl is zero, but the remaining effect is the existance of CPviolating processes [36]. The question remains, what is the value of 8 in our vacuum? The answer can be given experimentally because the crosssection of reactions with PC nonconservation are different, depending on the value of 0 [30, 371. Current experiments imply that (81 < 1OP9 [38]. The question why 8 is so small is known as the strong CP problem. There is one more specific field configuration which can easily be understand by analogy with the particle on a circle. The circle can be thought of as standing vertically in a gravitational field which attracts the particle to the lowest point on the circle. Then a particle positioned at on the top point of the circle obeys the equation of motion, although its equilibrium is unstable: any small deviation from the top point increases and the particle falls down (in real time). In field theory, the stationary field configuration corresponding to the top of the barrier between the vacua is called a sphaleron [39]. It has topological charge 112, and there is a negative mode corresponding to the direction in the functional space in which the energy decreases. This configuration is relevant when the transition between different topological sectors of a theory occurs via thermal activation (over the top of the potential barrier), rather than tunneling. Such processes may have taken place in the early Universe. This intriguing subject is outside the scope of the present book. A comprehensive review of this subject can be found in [40].
13.6
Quantum Kink
The effective action is in principle defined for all field configurations. Practically, we can calculate it only for small vicinity of field configurations whose properties are known at the classial level. We have treated this problem in chapters 4  7 for fluctuations around the constant vacuum configuration cp = a. In this section we consider quantization of small fluctuations around the kink solution in two spacetime dimensions. The set of quantum states of small fluctiations around the kink
CHAPTER 13. TOPOLOGICAL OBJECTS
384
together with the parameters wich describe the kink (e.g. its position) is called the kink sector of the theory. Analogously, there is a vacuum sector. Although kink is the simplest example of nontrivial background for the fluctuations, this physical system gives rise to rather difficult problems mentioned in subsection 13.6. The Lagrangian of the model under consideration is (13.1), (13.2) continued to Euclidean space. The Euclidean action reads
where T +oo and L 4 m are the extends of Euclidean time and space respectively, V(p) takes the form
and a = X/m2 << 1. Consider a stationary kink (13.7) positioned at x~ = 0 (for definiteness, we chose the sign in (13.7)). The value which we shall calculate in this section is the amplitude Z of transitions from the kink at r = 0 to other field configurations:
+
Z
=
S
Vp(t , X) eS[~I.
(13.97)
As the final state, we consider only configurations which are close to the initial kink which we denote now as +(X). Let us first chose the final configuration to be also +(X). Then Z m ecET, where E is the physical energy of this configuration. Below we shall use this formula when computing corrections to the mass of the kink. Let us try to apply the technique of calculation of Z developed in chapters 4  7. We expand the field around the kink: p(x, r) = +(X) ~ ( xr), and keep in the action only the terms up to the second order. This gives 1 S = M T +  JdxdrXZx , (13.98) 2 where the firstorder terms are absent becase obeys the equation of motion and
+
+
We have to find the eigenvalues and eigenfunctions of operator expand ~ ( xr), in the latter:
and
13.6. QUANTUM KINK
385
This brings the action (13.98) in the form of a sum of actions of independent oscillators (cf. chapter 4):
where M = m3/(12X) is the kink mass (13.11). The functional integration should be independently performed over each a,(r). Let us remind the result of chapter 3 where we have found that the spectrum of operator H consists of one zero eigenvalue W: = 0, one discrete level with W; = 3m2/4, and the continuum. The eigenfuctions in the continuum do not have any component in the form of a wave reflected from the potential well of H^. The effect of this potential well is the additional phase factor (3.62) in the outgoing wave. The eigenfunctions in continuum have the asymptotic form defined in formula (3.63) in which T should be now replaced with X . In calulating the path intefgral (13.97), we face a problem resulted from the fact that wo = 0 (the zero mode). The difficulty of this problem increases as we put more specific questions which we subsequently consider in this section. The presence of zero mode is almost irrelevant when we answer our first question about the mass of the kink in the oneloop level. A
Quantum Correction to the Mass of the Kink The energy of the ground state of the system described by the action (13.100) takes the form
where we have introduced the bare kink mass MOwhich is expressed in terms of the bare parameters12 a. or mo. For example, MO= m,3/(12X). The mass of the kink is the difference between this energy and the energy of vacuum (4.64) E. = ~w::)/2, where wiO)are the eigenvalues of the operator
The eigenvalues W, and W?) have been found in chapter 3. Let us reproduce the result here. There is a zero mode WO = 0 and a discrete level 12We do not renormalize the coupling constant X because it is finite in two dimensions.
CHAPTER 13. TOPOLOGICAL OBJECTS
386 W:
= 3m2/4. The continuum consists of levels with
W:
with n 5 2:
In contrast to chapter 3, we now start counting the levels w i O ) at n = 0. In this notation, the number of nodes of the wave function within the interval L12 < X < L / 2 is exactly n for both the fluctuations around the kink and around the vacuum. The phase shift S, takes the form
S,
= 2 arctan
P m

P + 2 arctan 2m
as it follows from (3.62). Thus the correction to the mass of the kink takes the form
where the first term is the contribution of the discrete level n = 1, the second term is the contribution of W?) and W?) in the limit of large L. These two levels are the counterparts of the discrete levels wl and w a . They are not accounted in the sum over levels of continuum because this sum consists of differences W, and wiO)for equal values of n. We calculate the sum in (13.104) in the way similar to the calculations performed in chapter 3 (equations (3.63) (3.72)). We substitute (13.102) into (13.104), replace the sum over n with an integral over p = n7r/L, and make an expansion in S,/L. This results in the following expression: 
Integration by parts with the use of expression (3.69) gives
where we have imposed a cutoff regularization with a large parameter A. As it follows from (l3.103), S, = 27r for p = 0. This value consists of contribution of 7r from each of two terms on the righthand side of
13.6. QUANTUM KINK
387
(13.103). Note that there is a onetoone correspondence between these terms and the discrete levels of the operator H^ because the zero boundary condition at X + cm is fulfilled at ei6k = 0. The value of 6, for p = 0 is in general equal to n multiplied by the number of discrete levels13. Thus this contribution always cancels against the second term on the righthand side of (13.106). This allows us to forget about the contribution of the levels of continuum which do not enter the sum in (13.104) and simultaneously to put So = 0. Substituting S, = 3m/p for p + cm, we obtain the following expression for AM:
(13.107) Let us discuss the renormalization of this result prior to calculating of the integral. We chose the mass of the field quantum m as the second derivative of the effective potential in the vacum p = a:
(see the dicsussion of the renormalization of mass in section 7.5). For this purpose, we do not need to calculate K f f It is sufficient to find its first and second derivatives. V e f fhas been obtained in equation (5.50):
1
K ( p ) = V ( p )+  In det (a2 2LT where d2
+ ~ " ( p ),)
d 2 / 3 r 2+ d 2 / d x 2 . We write V ( p )in the form
where a is the physical vacuum expectation value of the p field, and Sa2 is a divergent counterterm. We calculate the derivatives of the functional determinat in (13.108) using the formula
a
In det
dv
(a2 + ~ " ( p )=) V1"(p)Tr 1 a2 + V1/(cp)
1 3 ~ h i is s easy to inderstand if we note that the lowest level in the continuum in the considered case is already the third level of the system. Consequently, its wave function (3.63) has two nodes which means that the phase shift in (3.63) is 2n when k =0
388
C H A P T E R 13. TOPOLOGICAL OBJECTS
which is easy to prove if we consider the determinant as the product of all eigenvalues of the differential operator. Using the explicit form of the trace in the momentum space (chapter 7), performing one more differentiation over p, and using the explicit form of Veff(5.68), we find that
cf( a ) ( a
+3 ~ 1 ) l 8Xa6 + 4X (6a2 + 3 ~ 1 ) ( 2
= 4Xa (6a2
) =
 
(13.109) 2 4 ~ a. )(13.110) ~ ~ ~
Here AI and A2 stand for the tadpole and the fish diagram respectively. We write both of them without the vertex factors (cf. section 6.4):
We see that one counterterm Sa2 removes the divergency in both (13.109) and (13.110). This feature has been discussed in section 7.2. We use now the first condition (13.108) as the renormalization condition. This gives Sa2 = 3A1. The second condition (13.108) thus defines the physical mass m as
(13.112) The bare mass or the kink should be written in terms of a2 because we have chosen the counterterm to be Sa2:
Expressing a2 in the leading classical term here in terms of physical mass m as defined in (13.112), we get
Substituting MO into (13.101) and using the approximate relation m2 = 8Xa2 in the correction term, we obtain the following expression for M :
where A M is defined in (13.106). The divergence in A M cancels against the divergency in 3mA1. For the correct calculation, we have to find A I which is regularized exactly
13.6. QUANTUM KINK
389
in the same way as A M . That is the interval of integration over the time component p, of Euclidean momentum p is not bounded while the integral over the spacial component p, is taken from A to A. After the integration over p,, AI takes the form
It is now obvious that M is free of divergencies. Performing the residual integrations in 3 m A 1+ A M and using the value of A2 (13.111),we obtain finallv
This result differs from that obtained in [41]and [5]by the contribu. reason is that the theory in tion of the diagramA2 which is 9 m / ( 8 ~ )The [41]and [5]was regularized by the normal ordering wich implies the subtraction of the tadpole contribution from all divergent expressions. As it is seen from (13.112), this means that the parameter m in [41]and [5]is defined as m2 = 8Xa2 where a is the vacuum expectation value. This differs from (13.112) which follows from our definition of m as m2 = V,'(ff(a). Problem: Find the modification of (13.114) for the case when m is defined on the mass shell (section 7.5).
Physical Contents of Fluctuations around the Kink Let us analyse the physical content of some of small fluctuations around the kink. In twodimensional space each eigenvalue W: of operator H^ gives rise to a family of vibrational modes which are the eigenfunctions of K^ with the eigenvalues
where Ic = 1 , 2 . . . (we use now the zero boundary conditions for the fluctuations at r = 0 and r = T). As we knows from chapter 3 , excitations of zero mode such that cp = 4 ( x ) Co$Io(x)describe small displacements of kink: $ ( X ) Co$Io(x)= + ( X X " ) , where xo = c o l a .Correspondingly, the modes (13.115) for n = 0 describe all possible Euclidean trajectories C o ( t )of the kink. The eigenfunction $Il(x) is an antisymmetric function of X . Thus it describes small contractions or stretching of the kink. The
+
+
CHAPTER 13. TOPOLOGICAL OBJECTS
390
modes with n 2 2 describes the plane waves distorted by the potential V"(4)of the kink. We have seen in the previous section (equation 13.104) that the zero mode do not contribute to the correction to the mass of the kink. Let us find what is its contribution to the transition amplitude (13.97). This transition amplitude with the account for all fluctuations in the Gaussian approximation takes the form
where the prefactor is the functional determinant (chapter 6) and v g 2 are the eigenvalues of the fluctuations around the vacuum (they have the same form as (13.115) but wn replaced with wiO)). Performing the product over k results in equation (13.101) as it has been shown in chapter 5, equations (5.24)  (5.28)). The only exception is the zero mode. Its contribution to the functional determinant reads
The exponential factor has already been accounted for in (13.104) while the term 2mT results in the timedependent preexponential factor in 2:
where M is the physical mass of the kink. Thus the zero mode gives the same dependence on time in the prefactor of the transition amplitude as that in the Green function of free moving particle (chapters 2 and 3). Let us analyse this analogy in more detail. For this purpose we consider this transition amplitude from the kink positioned at X = 0 to the kink positioned at X = xo << T. We are interested in terms which depend on T. The classical action is now defined by the length of the Euclidean trajectory which is obviously a straight line: M(T2 M T MxiI(2T). The boundary effect at T = 0 and r = T does not contribute a Tdependent term. By the rotational symmetry of Euclidean space, we can consider the inclined trajectory of the kink as directed along new Euclidean time. Thus the contribution of fluctuations corrects MOto M up to a boundary effect. With the prefactor, the transition amplitude takes the form
+
+
13.6. QUANTUM KINK
391
This corresponds to the propagator of a free particle of mass M which is the kink. This is a particular case of a more general statement: For any trajectory of the kink such that the radius of curvature R, at each point is much longer than l l m , the contribution of fluctuations other than the zero mode results in replacement of MO with M in the action which is M1 where 1 is the total length of the trajectory. The zero mode then describes a particle with mass M and gives the appropriate contribution to the transition amplitude. This property has been formulated and used in an elegant way in [50]. The proof is based on the decoupling theorem [49] formulated on page 85. The hard sector of the theory includes all modes with the eigenvalues of the order of m'. Those are all modes except for the family of zero mode. The zero mode gives rise to the modes of the soft sector with the characteristic scale of eigenvalues 1/Rz. We can expect this having the experience of calculating the Casimir energy in chapter 5. The renormalization of the soft sector is the correction of MOto M. The soft sector is trivially renormalizable because it describes quantum mechanics of a particle which is a finite theory.
Elimination of Zero Mode In the above considered problems we always separated the kink as the background and small fluctuations around it. Much more difficult is to answer questions about interaction between the background and the fluctuations. For example: What is the bremsstrahlung of kink? What is the crosssection of a reaction when the kink absorbs a particle? The technique described below might be useful to answer such questions although we do not give an example. However, it seems to be useless to answer even more difficult questions about processes which involve drastical change in the background field. For example: What is the probability to create a kinkantikink pair in a collision of two particles at very high energy? Although some progress has been achieved, this question remains unanswered in our opinion. We do not consider this question here. Now we proceed with excluding of the zero mode from the path integral. In other words, we should restrict the integration area in the functional space by the subspace orthogonal to xo(x). The situation looks like that in the case of gauge fields quantization, where infinity appears from the integration over gauge equivalent configurations. Therefore, taking account of this analogue one can say that to exclude the zero mode from the path integral it is necessary to extract the integration over the translational shift explicitly and then discard it imposing the orthogonality
CHAPTER 13. TOPOLOGICAL OBJECTS
392 condition
/
(13.116)
dx V(X)XO(X) = 0 .
To do that we represent 'p(x) in view of (13.116) in the form
where q(t) is a collective coordinate describing translations and
Thus we have to extract the integration over the coordinate q(t) defined implicitly. It is a difficult problem because the corresponding Jacobian is unknown. Nevertheless, this difficulty can be overcome if we quantize in the phase space. In this case the Hamiltonian should be expressed in terms of q(t), its canonically conjugated momentum p(t) and the residual phase space variables. It is not difficult to understand the physical sense of p: it corresponds to the motion of the kink center of mass. Thus, the Hamiltonian of the system is
where r is the momentum of the system and
In view of (13.117) it gives
where MO coincides with the classical action on the kink MO = 2Ri/X, 0; = Xa2/3 and
It is obvious that 3t is translational invariant, so it does not depend on q and the momentum p is conserved. Thus, we would like to rewrite Z1 as follows:
2'=
1
Vp(t, x)Vr(t,X) exp [ir * 3 
/
d t ~ ( t ) ].
(13.121)
13.6. QUANTUM KINK
393
The imaginary unity appears in the first exponential term because t is the Euclidean time. It is necessary to rewrite (13.121) in terms of q ( t ) , p(t). For this purpose we decompose the momentum as follows
Thereafter we apply the trick analogous to that of FaddeevPopov procedure (see chapter 12). Let us rewrite (13.121) inserting into the integrand the identity
where Fl , F2 express orthogonality conditions ( l 3 . l l 6 ) , (13.122):
After that, like in FaddeevPopov procedure, we discard the part
giving infinity in (13.121) and get the expression
where the exponential term in the integrand depends implicitly on p ( t ) , q ( t ) but not on q ( t ) because of the translational invariance. To find it one should express no. The momentum .rro can be found by imposing of the condition [email protected]=pq+ d m d. x (13.124)
J
Taking account of the evident relation
J
at
394
CHAPTER 13. TOPOLOGICAL OBJECTS
we express a0 in terms of p:
where (f,g) denotes the integral J dx f (x)g(x). Then, the Hamiltonian in (13.123) takes the form
where
Now, making the variables change X  q(t) + X, nl + n and sending the term zpq to the normalization factor we get for Z
where
i
A^ is an integral operator with the kernel and T + W. Let us analyse the expression obtained. It is seen from (13.130) that elimination of the zero mode results in the effective Hamiltonian depending parametrically on the center of mass momentum of the kink p. Although the Hamiltonian is still quadratic in a, it is nonlocal and nonpolynomial in X' that gives rise to the infinite number of vertices in the theory. Nevertheless, the presence of the small parameter l / M o can simplify further calculation^^^. Performing integration over a 14We recall that the semiclassical approximation is applicable only if the classical action MO is large comparing with other terms.
13.6. QUANTUM KINK
395
we would get the effective Lagrangian and then, making use of the stationary phase approximation, we could arrive at the oneloop effective action. It should be also pointed out that as p appears as a parameter in the expression for 2, translationally shifted configurations are treated as equivalent ones. If one would like to distinguish the shifted configurations, the integration in (13.130) should be supplied with the integration over q(t) and p(t). As an example let us consider the amplitude of the transition vacuum + shifted vacuum i.e., cpo(x) + po(x xo) in the time T. In the lowest approximation one can neglect the terms proportional to (po,X') and ( ~ 1X') , in the first term of (13.128) and we get
that gives
where x;[x, nl =
J dx [;a: + 1 + ~ ( x ) ] 2 ~ t 2
with V(x) determined by (13.120). The boundary conditions are q(0) = X , q(T) = X  so. Integration over q(t), p(t) can be performed exactly and gives the ordinary quantum mechanical propagator of free particle: =
M04 ( 2nT 3) exp ' (FZ),
containing only the second where Z is determined by (13.123) with term. Thus, as one could expect, the integration over the collective coordinate q(t) corresponding to the zero mode gives the account of the steady kink movement.
Generating Functional To obtain the generating functional and the effective action one should integrate over T. It can in principle be done because one can easily get expressions for A' (X, y) and In det A^ in (13.130). However, the structure of the effective Lagrangian we would deal with is rather complicated and
396
CHAPTER 13. TOPOLOGICAL OBJECTS
it is difficult to build the diagram techniques. The complete analysis of the oneloop action in this case is to our mind out of the scope of a standard text book on quantum field theory. We would rather consider some elements of the diagram techniques. It is reasonable to redefine slightly the definition of the generating functional [44]. Let us consider
where J1, J2 are external currents. To calculate Z [ J 1 ,Jz] one can start with the quantity
and then making perturbative expansion by means of
where 'Ho, 'H1 are determined by (13.127), (13.128). Evaluation of Zo has some specific features. After elementary integration over 7r we get
where the notation 'I' implies that the zero mode is excluded from the expansion of the corresponding quantity in eigenfunctions of K determined by (13.99). Now it is useful to represent the term linear in using the identity J ~ * x =J ~ * x , A
A
where
T2 is the integral operator with the kernel
are defined Besides that one has to take into account that g', (?')l on the functional subspace not containing the zero mode. But Sfunction does contain it:
13.6. QUANTUM KINK Therefore, the star product
cannot be equal to S(x  X') but rather to cc
A(x,xl) = S(t  t')
C xn(x)x;(xl).
After that the integration over X can be easily taken. We have
where Gll(x,xl)
=
K^'l(x,xl)
Thus, elimination of the zero mode leads to three types of internal lines in diagrams corresponding to the Green functions (13.140)  (13.141). The infinite number of vertices in the theory can be divided in the following four groups. The first one consists of the usual two meson vertices corresponding to the terms proportional to x3 and x4. Three other groups appear from the expansion of the expression
(1 + (vb, ~ ' ) l M o ) =~ (1+ ( 4 , x ) / ~ o )  ~ in (p;, x1)/M0 powers 'Qy multiplying this expansion by one of three terms coming from the term Ip (T,X')]2. Adding two types of external lines corresponding to X and .rr we get all basic elements for the diagram techniques in this approach and can make perturbative calculations. It should be noted that although the effective Hamiltonian (13.126) is not manifestly Lorentzcovariant, the theory is originally so. Therefore, Lorentzcovariance must be restored at each order of p/Mo. If we restrict ourselves only with the lowest order quantum correction to the kink action in the rest frame, we shall get (13.106) again.
+
I5It is possible because this term is proportional to the small parameter A l l 2 .
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Bibliography [l]Topological objects are considered in many textbooks on field theory. See, e.g., M. Nakahara, Geometry, Topology and Physics, IOP Publishing, Bristol, 1990. K. Huang, Quarks, Leptons and Gauge Fields, World Scientific, Singapore, 1982. L.H. Rider, Quantum Field Theory, 2nd ed., Cambridge University Press, Cambridge, 1996.
[2] R. Jackiw, NonPerturbative and Topological Methods in Quantum Field Theory, in the collection Diverse topics in Theoretical and Mathematical Physics, Singapore, World Scientific, 1995, p.449. See also R. Jackiw and J.R. Schrieffer, Nucl. Phys. B190 (1981) 253, and P.W. Su, J.R. Schriffer, and A.J. Heegeer, Phys. Rev. B22 (1980) 2099. [3] V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fix. 20 (1950) 1064 (in Russian). Reprinted in English in Collected Papers of L.D. Landau, D. Ter Haar (ed.), Pergamon Press, Oxford, 1965. [4] Ya.B. Zeldovich, I.Yu. Kobzarev and L.B. Okun, Sou. Phys. JETP 40 (1974) 1.
[5] R. Rajaraman, Solitons and Instantons, North Holand, 1982. [6] A.C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985. [7] M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. [8] H.B. Nielsen and P. Olesen, Nucl. Phys. B61 (1973) 45. [g] R.P. Feynman, Statistical Mechanics: a Set of Lectures, Reading, Mass., Benjamin, 1972.
400
BIBLIOGRAPHY
[l01 A.A. Abrikosov, Sov. Phys. JETP 5 (1957) 1174. [l11 E.B. Bogomolny, Sou. J. Nucl. Phys. 24 (1976) 449;
[l21 H. de Vega, F.Shaposhnik, Phys. Rev. D14 (1976) 1100. [l31 Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [l41 G. 't Hooft, Nucl. Phys. B79 (1974) 276. [l51 A.M. Polyakov, Sov. Phys. JETP Lett. 20 (1974) 194. [l61 For topological properties of the monopole solution see e.g. J. Arafune, P.G.O. Freund and C.J. Goeble, J. Math. Phys. 16 (1975) 433. [l71 Very good reviews on monopoles are by P. Goddard and D.I. Olive, Rep. Progr. Phys. 41 (1978) 1357; S. Coleman, in The Unity of Fundamental Interactions ed. A. Zichichi, Plenum Press, 1973; The Magnetic Monopoles Fifty Years Later, lectures given at the 19th Course of the International School of Subnuclear Physics, Erice, 1981. [l81 M.K. Prasad and C.H. Sommerfield, Phys. Rev. Lett. 35 (1975) 760. [l91 B. Julia and A. Zee, Phys. Rev. D11 (1975) 2227. [20] N. Manton, Nucl. Phys. B126 (1977) 525. [21] Ya.B. Zeldovich, and M.Yu. Khlopov, Phys. Lett. B79 (1978) 239; A.H. Guth and S.H.H. Tye , Phys. Rev. Lett. 44 (1980) 631. [22] G. Giacomelli and V. Popa, Nucl. Phys., Proc. Suppl. B65 (1998) 275; by MACRO Collaboration, Phys. Lett. B406 (1997) 249. [23] E.N. Parker, Astrophys. J. 160 (1970) 383. [24] S. Mandelstam, Phys. Rept. 23 (1976) 245; G. 't Hooft, Nucl. Phys. B190 (1981) 455. [25] N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19. [26] A.A. Belavin, A.M. Ployakov, A S . Schwarz and Yu.S. Tyupkin, Phys. Lett. B59 (1975) 85.
BIBLIOGRAPHY
401
[27] For a good review of the topological theory of defects in ordered media, see N.D. Mermin, Rev. Mod. Phys. 51 (1979) 591. For a review of the role of defects both a cosmological and condensed matter context see Formation and Interaction of Topological Defects, eds. A.C. Davis and R. Bradenburger, NATO AS1 Series B, Physics, Vol 349, Plenum, New York, 1995. [28] The preexponential factor for the instanton was found by G. tlHooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D14 (1976) 3432. [29] Tunneling between vaccua of YangMills fields was first considered by R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 72. [30] Possible values of 6' were estimated from the experimental constraints on the T noninvariance by F. Wilczek, Phys. Rev. Lett. 44 (1978) 279. 1311 S. Coleman, The Uses of Instantons, Lectures given a t the 15th Course of the International School of Subnuclear Physics, Erice, Italy, 1977; Reprinted also in: S. Coleman, Aspects of Symmetry, Cambridge, University Press, 1985. 1321 J . ZinnJustin, The Principles of Instanton Calculus: a few Applications, Lecture delivered at Les Houches Summer School 1982. Published in Recent Advances in Field Theory and Statistical Mechanics, eds. J.B. Zuber and R. Stora, NorthHolland, 1984. [33] A.I. Vainshtein, V.I. Zakharov, V.A. Novikov and M.A. Shifman, Sou. Phys. Usp. 24 (1982) 195. 1341 T . Schafer and E.V. Shuryak, Rev. Mod. Phys. 70 (1998) 323. About the instanton effects in QCD see D. Diakonov, Chiral QuarkSoliton Model, Lectures given at Advanced Summer School on Nonperturbative Quantum Field Physics, Peniscola, Spain, 26 Jun 1997; hepph/9802298; D. Diakonov, Chiral Symmetry Breaking by Instantons, Talk given at International School of Physics, 'Enrico Fermi', Course 80, Varenna, Italy, 27 Jun  7 Jul 1995; Published in Selected Topics in Nonperturbative QCD, eds. D.Diakonov and A. Di Giacomo, IOS Press, 1996, p 397; hepph/9602375.
402
BIBLIOGRAPHY
[35] V.G. Kiselev and K.G. Selivanov, Phys. Lett. B213 (1988) 165; V.G. Kiselev, Phys. Lett. B249 (1990) 269. [36] G. 't Hooft, Phys. Rept. 142 (1986) 357; Reprinted also in: G. 't Hooft (ed), Under the Spell of the Gauge Principle, Singapore, World Scientific, 1994. [37] Particle Data Group 1998 (http://pdg.lbl.gov). 1381 V. Baluni, Phys. Rev. D19 (1979) 2227: R.J. Crewther, P. Di Vecchia, G. Veneziano and E. Witten, Phys. Lett. B88 (1979) 123. [39] Sphaleron configuration was found by R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D10 (1974) 4138 and rediscovered in the context of electroweak theory by N. Manton, Phys. Rev. D28 (1983) 2019; [40] V.A. Rubakov and M.E. Shaposhnikov, Sou. Phys. Usp. 39 (1996) 461. [41] R. Dashen, B. Hausslacher and A. Neveu, Phys. Rev. D10 (1975) 4136. [42] The sectors of the quantized (p4 theory in the kink background were firstly discussed by J. Goldstone and R. Jackiw, Phys. Rev. D11 (1975) 1486. [43] The collective coordinates method for zero modes elimination was proposed by N.H. Christ and T.D. Lee, Phys. Rev. D12 (1975) 1606, and independently by E. Tomboulis, Phys. Rev. D12 (1975) 1678. 1441 For the path integral approach the method of collective coordinates was adapted by G.L. Gervais and B. Sakita, Phys. Rev. D11 (1975) 2943; G.L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038. [45] For further development of these ideas see G.L. Gervais and A. Jevicki, Nucl. Phys. B110 (1976) 93; ibid B110 (1976) 113; A. Jevicki, Nucl. Phys. B117 (1976) 365.
BIBLIOGRAPHY
403
[46] The considered quantization procedure can be carried out also in other scalar theories. For quantization of sineGordon model see L.D. Faddeev and V.E. Korepin, Phys. Rep. C 42 (1978) 1. [47] The SLAC bag model (see chapter 11) has also an exact static solution in 1+ l dimensions and it can be also quantized. It was done by D.K. Campbell and Y.T. Liao, Phys. Rev. D 1 3 (1976) 2093. [48] On quantization of more complicated objects like BPS monopole see ref. 35 and K. Zarembo, Nucl. Phys. B464 (1996) 73. [49] T. Appelquist and J. Carazzone, Phys. Rev. D11, (1975) 2856. [50] M.B. Voloshin, Sou. J. Nucl. Phys. 42 (1985) 644.
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Appendix A Some Integrals and Products A. 1 Gaussian integrals Gaussian integrals are very important in performing the integration over momenta in eqs. (2.23), (2.24), as well as in many other situations. The simplest such integral is
It is very easy to evaluate, taking into account that
where polar coordinates on the X,y plane have been introduced. Thus
This result also allows us to calculate the integral of the next form:
Let us generalise these relations for the case of integration over an ndimensional volume:
406
APPENDIX A. SOME INTEGRALS AND PRODUCTS
where A;, is a real symmetric matrix of dimension n X n , which can be brought in the diagonal form A' by means of a transformation
where OT is the transposed matrix and OiOT = O;Oi = 1. For the sake of simplicity let us denote the set of coordinates xi via the vector X = ( x l ,x2.. . X,). Then integral (A.5) takes the form
The Jacobian of the transformation from the set of variables X to the new coordinates Y = OX = y,, y2 . . . y, is equal to one because 0'0 = 1. This means that the integral (A.5), in terms of variables Y , reduces to the product of n Gaussian integrals
where ala2 . . . a, are the diagonal elements of matrix A'. Thus
In some applications the Gaussian integral over the complex plane z is defined as follows: IT Tdxdyea(x2+Y2i = . a
J dzdz*eazz*
(A.lO)
00
The generalisation of this integral for n complex variables reads
I

J d ~ d ~ * e  rn~ (* ~ ~A det
=
)l,
(A.ll)
where A is a Hermitian n X n matrix with positive eigenvalues. These relations allow us to define the Gaussian integral in the continuum limit n + m.
A.2
Calculation of
nn (1  A)
Let us prove the formula
x2 n
sin X X
A.2. CALCULATION OF PRODUCTS
407
which have been applied in chapter 2 to calculate the path integral for the case of the harmonic oscillator. We start from the Fourier expansion of the function cos(ax) on an interval [T; T]:
L (a)cos(nx) ,
cos(ax) =
(A.13)
n
where the expansion coefficients are
T
n ) ~ ] + sin[(a an
dx cos(ax) cos(nx) = T
0
(A. 14) In the particular case X = T we may reexpress the function cos(a.rr) (A.13) as an expansion in powers of a :
Evaluating the integral over a from eq. (A.16) from 0 to for the lefthand side
X,
we obtain
and, in the same manner, for the nth term of the righthand side we have
Thus In
(F) = .ln
(1 
g)
This formula coincides with (A.12) upon the substitution
(A.19) TX + X.
408
A.3
APPENDIX A. SOME INTEGRALS AND PRODUCTS 1
Calculation of i $ln(l

X)
0
An integral of the form
appeared in chapter 5 when we considered the Casimir effect (see eq. (5.33)). In order to evaluate it, note that on the interval where the integration variable X is defined, the integrand may be expanded as a
Evaluating this series termbyterm gives
where
is the socalled <function. This function is a series which converges for any real s > l (if s is a complex number the region of convergence is Re s > 1). To calculate specific values of the <function, including <(2), we reexpress the series (A.22) in terms of Bernoulli numbers B,,, which are the power expansion coefficients of some of the elementary functions. For example we can define these numbers as
Expansion of the lefthand side of this expression yields
A.4. CALCULATION OF SOME INTEGRALS
409
Equating powers of X we obtain a chain of equations to define the Bernoulli numbers:
The first few of these numbers are
By the substitution X + 2ix in eq. (A.24) and taking the real part we obtain the following power expansion of X cot X (0 < 1x1 < K): m
xcotx
=
B2n (22)2n y(qn
(an) !
71=0
On the other hand, as it follows from (A.16) 1
a n cot (an) = 1 + 2a2
1
1
+ )
. (A.30)
The r.h.s. of this expression can be rewritten as a power expansion in a which yields
By comparison of this formula with the power series (A.29) we obtain
Thus
C(2)
=n
2 ~= 2
7r2
(A.33)
This is the value of the integral (A.20).
A.4
CO Calculation of , S X dx ln(1 + x2) +a
Let us evaluate the integral
(A.34)
APPENDIX A. SOME INTEGRALS AND PRODUCTS
Figure 66
which has been necessary for calculation of the functional determinant for the quantummechanical instanton (chapter 3). For this calculation, we extend X to a complex variable z such that X = Re x and perform integration in (A.34) along the contour C shown in figure 66. We would like to evaluate first the integral I(a) for a < 1. Note that the integrand is a multivalued function. Thus we cut the xplane as shown in fig. 66. Obviously the integral along the closed contour C consists of four pieces: (i) the integral I(a) along the real axis; (ii) and (iii) the integrals Iz,I3along the different sides of the cut; (iv) the integral over the infinitely large part of the contour. This integral is zero. The function ln(1 z2) on the left side of the cut differs by 27ri from its value on the right side of the cut. Thus
+
where y = Imz. Evaluation of this integral yields
The contour C encloses one simple pole at z = ia. Thus the residue
A. 5. FEYNMAN PARAMETRIZATION theorem gives:
and finally we have 2 I(a) = Ic  Iz  I3=  ln(1  a )
= 21n(l+a)
l a + In l+a
(A.38)
.
This result is valid also for a 2 1. Indeed, it is obvious from the original form (A.34) that I(a) is an analytical function of a in the region Re a2 > 0.
A.5
Feynman Parametrization
Here we prove formula (7.47)
(A.39) Let us perform the integration over X,. Note that by a fixed value of X, none of other variables xk, k = 1 , 2 . . . n  1 can be larger than 1  X because of the Sfunction. This allows us to reduce the ranges of integration over these variables to 0 < xk < 1  X,. Changing the variables t o tk = x k / ( l X,), we transform the righthand side of (A.39) to the following form
Now we take the inner integral over x n / ( l  X,)). This gives:
X,
Thus, the r.h.s. of (A.39) takes the form
(the suitable variable is z =
412
APPENDIX A. SOME INTEGRALS AND PRODUCTS
Thus we have extracted the factor l l a , reducing the number n by one. Repeating this procedure we come to the lowdimensional integral (7.48) which is easy to calculate. This proves the statement.
Appendix B Splitting of Energy Levels in DoubleWell Potential In this appendix we find the value of splitting of energy levels (3.3) in the doublewell potential considered in chapter 3. For the ground state, we obtain expression (3.4). In conclusion, we show that (3.4) coincides with the result of calculations based on instantons (3.74) as it is stated in chapter 3, page 58. Consider the Schrodinger equation for the potential V(x) sketched in fig. 6: 1
As V(x) is an even function of X, the eigenfunctions $, should be even for n = 0 , 2 . . . and odd for n = 1 , 3 . . .. For example, the wave function of the ground state qo in (3.2) is a symmetric nodeless function: $;(0) = 0. The function &(X) is antisymmetric: $1(0) = 0. The functions qL(x) and qn(x) in (3.2) are constructed as follows. Consider the solution of equation (B.l) at continuously increasing parameter E such that $(cc) = 0. When E becomes larger than the energy of the ground state Eo,a node of $(X) moves from X = +cc to finite values. This node reaches its position at X = 0 when E = El. Let us chose a value E = EL such that the node is close to the position of the minimum X = a of V(x) in the right potential well. The obtained function has a maximum in the left potential well and is exponentially supressed in the right well. This function diverges at X + +cc as does any solution of equation (B.l) with 'incorrect' energy E # E,. The function $L(%)is this solution continued to zero in the region X > a. For example, we can put $L(x) = 0 for all X larger than the node in the right well. We normalise this funcx 1. By the symmery, we can define $R(x) tion on unity: J $ L ( x ) ~ ~=
APPENDIX B. SPLITTING OF LEVELS
414
as $R(x) =: $ L (  X ) . Analogous functions can be constructed for each couple of split energy levels. In order to find the difference E L  Eo, we calculate the following : integral [l]
0 =
1
Wo(H  EL)$L  $ L ( H  Eo)$o] d x
cc
1
=  [ E L E0  $ J L ( ~ ) l $ m l I
Jz
,
where we have integrated by parts, used relations (3.2), and the normalisation of $L. Analogously, we find El  E L with the result
where A is the value introduced in (3.3). In order to find $L, we have to use the approximation of harmonic oscillator near the minimuma of V ( x ) and the semiclassical approximation far from these minima. Consider the minumum X = a. Near this minimum 1 1 V ( X= ) w2t2 2 Vu'(a)t3 3! + ... , (B.3)
+
+
where t = X a is the deviation from the minimum and w2 = 8Xa2. We can neglect the third and the higher order terms for J << w2/V"'(a) a. At the same time, the semiclassical approximation is applicable for v ' / v ~<> l / f i . Thus there is a region l / & << J << a where we can match the wave function of the harmonic oscillator to the semiclassical solution of the Schrodinger equation. This should be done for the values of J beyond the turning point which is F. = for the nth energy level. The matching of wave functions is possible only for n2 << w3/X. We consider here only such potentials for which the splitting of the levels is much smaller than the distance between levels in a single well. This is the case only if w3/X >> 1. Note that this is also the condition of validity of the instanton method as it is discussed in chapter 3. We begin calculations with the following form of the normalized wave function of the harmonic oscillator (see, for example, [ l ] ) :
Jm
where H, is the Hermite polinomials:
dn H, ( z ) = ez2e" dzn
.
APPENDIX B. SPLITTING OF LEVELS
415
For z >> 1, H,(z) z (22)". Using this asymptotic form, we obtaing the wave function in the region where it falls off exponentially:
The semiclassical solution $ J W K B of the Schrodinger equation which falls off exponentially from the region of the left potential well X z a to X = 0 takes the following form [l]
Jw),+
where p(x) = and X* = a 6 is the turning point. Using the leading term of expansion (B.3), we calculate the integral and expand the result in the small parameter E/(w[)~. This leads to
(F) E/2w
$JWKB = ~
onst
(wt2)
E/2w114 exp (
1  Z ~ ~ 2 )
.
(B.6)
Comparing (B.6) and (B.4), we find the constant in (B.5) for E = E, = (n 1 1 2 ) ~ :
+
Finally we use formula (B.2) for the parameter of splitting A. The wave function is given by (B.6) with expression (B.7) for Const. This results in the following form of A for the nth couple of split levels:
where we have used the symmetry of V(x). Let us remind that this result is valid for n << w3/X. In our calculations we have not used the explicit form of the potential V(x) which is thus generic. The quantity w3/X should be understood as a rough estimate of the value of the exponent in (B.8). For n >> 1 we apply the Stirling formula to n! in (B.8). This gives the well known result obtained, for example, in [l]:
416
APPENDIX B. SPLITTING OF LEVELS
For the ground state (B.8) gives expression (3.4). Let us prove now that (3.4) coincides with the result (3.74) of chapter 3. As it was stated in chapter 3, the distinction of the preexponential factor in (3.74) from that of (3.4) is exactly the one which is required to compensate the deviation of S = W ( 0 ) from W ( w / 2 ) . This can be shown for the generic form of V ( x ) but such a proof requires to establish a general relation between the functional determinant and the classical action which is beyond the scope of this appendix. Let us just check the statement for the quartic potential (3.9). For this potential the abbreviated action takes the form of an elliptic integral. Instead of citing the theory of these functions, let us find the relation between W (0) and W ( E )for E << V ( 0 ) .We present the integral for W ( E )in the following form:
where D is a point selected in the region where the potential V ( x ) still has the quadratic form, but it is already V ( x ) >> E . We calculate the first integral with the quadratic form of V ( x ) :
In the second integral we make an expansion in E / V ( z ) keeping the terms up to the order of unity. This gives
Substitution of these expressions in (B.9), results in the following formula: W
This formula is a bridge between (3.4) and (3.74). Substitution of W ( E ) for E = w/2 in (3.4) results in (3.74) and vice versa.
Appendix C Lie Algebras In this appendix we briefly consider some of the basic properties of Lie algebras. There is much more information here than we may need, so we can regard this Appendix as a 'crash course' on Lie algebras. We consider the main steps in the classification of Lie algebras. Some of the information is contained in the simple problems appearing in the text. The language used in this appendix is more mathematical than that in other parts of this book.
C.1
Elementary Definitions
A Lie algebra is a linear vector space g (its elements we denote as X , y, . . .) over the real or complex numbers, supplied with a bilinear operation [X, y] E g. This operation is called the Lie bracket or commutator and has the following properties:
2. Jacobi identity:
y] = 0 for any X, y, g is called an Abelian Lie algebra We shall consider only finite algebras, i.e. g is finitedimensional as a vector space. This means that there is a finite basis in the algebra whose elements e,, a = 1,.. . ,N are called the generators of g. They obey the commutation relation [e,, Q] = e, . If
[X,
APPENDIX C. LIE ALGEBRAS
418
The numbers fibare called the structure constants of the algebra. They are antisymmetric in the lower indices. Because of the Jacobi identity the fibmust satisfy
A subspace of g closed with respect to the commutator operation is called a subalgebra of g. For two subalgebras gl and g:! we shall denote by [g1,g:!] the set of all possible commutators [xl,X:!] with XI E gl, X:! E g:!. A subalgebra n of g for which
is called an ideal in g. The set of all elements commuting with any element of g is an example of an ideal. It is called the centre of g. Problem: Prove that if n l , n2 are ideals, then [nl, nz] is an ideal as well. Problem: Prove that if n is a subalgebra in g, then g/n is also a subalgebra if and only if n is an ideal in g. This subalgebra is called a factoralgebra. The existence of subalgebras or ideals in g imposes certain restrictions on its structure constants, for instance, if (el, . . . ,e,) is the basis of a subalgebra, f:b=O a,b<s,c>s . and if it is the basis of an ideal
ei
The direct sum of algebras g = gi is the direct sum of the corresponding linear spaces provided that [gi, gj] = 0. Each term gi is obviously an ideal in g. If g is over the real numbers, we can elementary extend it to an algebra over the complex numbers assuming the coefficients at the basis vectors to be complex. The algebra obtained on such a way is called the complexijicationof g. It is denoted by g" and can be considered , . . . ,ieN). Analogously, as a real algebra with generators (el, . . . , e ~iel, for any complex algebra g, the algebra g' is called the real form of g if (gT>"= g. Of course, it would be preferable to study realization of Lie algebras in the form of a set of linear operators acting on a vector space. A
C.2. EXAMPLES OF LIE ALGEBRAS
419
linear mapping t of Lie algebra into the set of all linear operators on a vector space is called a representation of g if this mapping preserves the commutator structure i.e., t ( [ x ,y ] ) = [ t ( x ) t, ( y ) ] t ( x ) t ( y ) t ( y ) t ( x )for any X, y from g. If this mapping is an isomorphism, the representation is called exact l .
C.2
~ x a m ~ l of e sLie Algebras
Here we consider some examples of the Lie algebras which most often appear in physics. The first example is the Poincare algebra. Its generators are rotations M,, and translations P, in Minkowski space. The commutation relations are
where g,, is Minkowskian metric tensor. It is obvious that {P,) is an ideal in the algebra. Other examples are the four sequences of algebras called the classical complex algebras: 1. The algebra of complex n X n traceless matrices. It is denoted by sl(n, C ) . An alternative notation for this algebra is Anl;
2. The algebra of complex (2n+ 1) X (2n+ 1) antisymmetric matrices. It is denoted so(2n + 1,C ) or B,; 3. The symplectic algebra s p ( n , C ) or C,. It consists of complex n X n matrices X , antisymmetric with respect to the matrix J: xTJ = J x , where
Here 1, denotes the n
X
n unit matrix;
4. The algebra of even dimensional antisymmetric matrices so(2n, C ) or D,. lActually, the study of an arbitrary finitc Lie algebra is reduced t o that of a matrix algebra due t o the Ado theorem which states that any finite Lie algebra has a finite exact representation.
APPENDIX C. LIE ALGEBRAS
420
As an example of a subalgebra of so(p+q, C) one can take the algebra of real (p q) X (p q) matrices z acting in C p + 4 and antisymmetric with respect the matrix
+
+
IxT = XI.This is the pseudoorthogonal algebra of (p, q) type.
C.3 The Idea of Classification. LeviMaltsev Decomposition ,
The Adjoint Representation Of course, the division of Lie algebras into Abelian and nonAbelian algebras is too rough. It would be good to develop a more detailed classification. For this purpose it is useful to find an universal matrix representation of any Lie algebra. Fortunately, such representation exists naturally. The representation space is the algebra itself regarded as a vector space. The matrices of the representation are defined in the following way. For each X E g we take the linear mapping t(x) : g + g such that t(x)y = [X, y]. This representation is called the adjoint representation and is denoted by ad g , t(x) ad X. The matrices ad e, are determined by the structure constant of the algebra: (ad e,): = i fib. Problem: Prove that the set of matrices ad X is indeed a representation of g i.e., [ads, ad y] = ad [X,y]. Of course, the adjoint representation is not exact because the image of the centre of g is zero in adg. Nevertheless, the study of it gives an effective way to classify Lie algebras, at least partially. Let us consider the action of ad g in g. If ad g is irreducible i.e., g has no invariant subspaces with respect to ad g, then the algebra g is called simple. Problem: Prove that the simple algebra has no ideals besides the trivial ones (i.e., besides g itself and (0)). If g is not simple, it has invariant spaces. Let us denote them nk, where dimnk = nk. This means that all matrices of a d g can be simul
C.3. LEVIMALTSEV DECOMPOSITION taneously represented in the quasitriangular form
Here Dk are square nk X nk matrices and K'') are matrices with ni rows and ni columns. This form is universal, but is still too complicated. Therefore, we consider two important particular cases.
Solvable and Nilpotent Algebras The first is the case where all the quasidiagonal blocks are onedimensional (i.e., ni = 1). In other words, the adjoint representation in this case consists of upper triangular matrices. Such algebras are called solvable. Problem: Prove that the oscillator algebra n = {a+a, a', a, l), where at and a are the conventional bosonic creation and annihilation operators, is solvable. Solvable algebras possess some nice properties which are formulated in the following problem. Problem: Using properties of the commutation of triangular matrices, prove the following statements. Let g be a solvable Lie algebra and
g ( i ) is called the commutant of g(''). Then,
1.
C: g(i) is an ideal in g(')
2. The factoralgebra g ( i ) / g ( i  l )is Abelian.
3. g(k)= O for some k . It should be pointed out that the last property in this problem can be assumed as the alternative definition of a solvable Lie algebra i.e., the corresponding two statements are equivalent. We shall not prove this equivalence.
APPENDIX C. LIE ALGEBRAS
422
Problem: Prove using the results of the last problem that if n is a solvable ideal in g and g/n is solvable then, g is solvable. An important class of solvable Lie algebras are those whose adjoint representation consists of the special triangular matrices which have zero diagonal elements. Such algebras are called nilpotent.
Problem: Prove that the HeisenbergWeyl algebra nl = {a+,a, l), where a+ and a are creation and annihilation operators, is nilpotent. Problem: Using properties of the commutation of special triangular matrices prove the following statements. Let g be a nilpotent Lie algebra and (C.4) g(0) = g , g(i) = [g,q i  l ) ] . Then, 1. For any element of g (adx)"
=0
for some n;
3. g(k) = 0 for some Ic. The last property is equivalent to the definition of nilpotent algebra, but we shall not prove it.
Reductive and Semisimple Algebras The second particular case of adjoint representation (C.2) is when all rectangular matrices q") are zeroes. Then the adjoint representation takes the quasidiagonal form with irreducible blocks when adg = diag(D1,. . . , D,).
(c.5)
Of course, each block depends on the concrete element of g, but the form (C.5) is universal. Such algebras are called reductive. In other words, the algebra is called reductive if its adjoint representation is completely reducible 2 .
Problem: Let ad gi = diag(0,. . .O, Di, 0 . . . ,0). Prove that gi is an ideal in g. Particularly, if Di = 0, gi is a subset of the centre of g. 2A set of matrices acting in a vector space is called completely reducible if there is a basis in which they take blockdiagonal form.
C.3. LEVIMALTSEV DECOMPOSITION
423
Using the result of the last problem we can represent a reductive algebra as a direct sum of its centre and a reductive algebra with nonzero Di.Such an algebra is called semisimple.
Problem: Prove that if g is reductive and S is its Abelian ideal, then is a subset of g centre. It follows from the result of the last problem that any semisimple algebra has no Abelian ideals.
S
Problem: Prove the inverse conjecture: if the algebra has no Abelian ideals, it is semisimple. As the blocks Diare irreducible, we may conclude that any semisimple algebra can be represented as a direct sum of simple algebras. Thus, the nontrivial structure to be studied is contained in the simple complex algebras. Of course, the two cases considered above, do not exhaust all possible realisations of Lie algebras. For example, if an algebra g whose adjoint representation is not triangular has a nontrivial solvable ideal which does not coincide with the centre of the algebra, then this algebra is neither reductive nor solvable. Let n be the maximal solvable ideal of g. It is called the radical. Nevertheless, the following statement gives a remarkable route to the classification of all Lie algebras: let g be a Lie algebra and n be its radical. Then, the factoralgebra g/n is semisimple. To prove this we consider the homomorphism p : g + g/n. Let there be a solvable ideal cp(m) in g/n. Then, m contains n, because p(n) = 0. Thus, m/n and n are solvable. Therefore, so is m. But this contradicts the assumption that n is the maximal solvable ideal. In other words, any Lie algebra consists of two subalgebras n and S such that
n is solvable, S is semisimple. Such decomposition of g is called the LeviMaltsev decomposition. Thus, the problem of classification reduces to the following three steps: 1. Classification of semisimple algebras;
2. Classification of solvable algebras; 3. Classification of all mappings of solvable algebras determined by ad S, S semisimple.
APPENDIX C. LIE ALGEBRAS
424
The last two problems have not yet been solved completely. The first one is reduced to that of classifying the simple Lie algebras. The solution of this problem is one of the most beautiful results in the theory of Lie algebras and will be discussed in the next section.
C.4
Classification of Complex Semisimple Lie Algebras
Here we build a zoo of simple (and hence, for semisimple) Lie algebras. First we study some general properties of semisimple algebras which make it possible to reduce the problem of classification to linear algebra of some special sets of vectors.
The Cartan Subalgebra. Roots The main idea is to extract all commutative subalgebras in the adjoint representation. In these subalgebras we can find an element with the minimal possible multiplicity of the eigenvalue zero. V u c h an element is called regular. Then let us take the maximal commutative subalgebra containing a regular element. This subalgebra is called the Cartan subalgebra. Its dimension is called the rank of the algebra. The possibility of the complete description of semisimple Lie algebras is provided by the following result whose proof is beyond the scope of this book. If g is a semisimple Lie algebra and h is its Cartan subalgebra, then all matrices ad h , h E h can be simultaneously diagonalised. This means that all nonzero eigenvalues of all ad h are nondegenerate and each zero eigenvalue has multiplicity r = rankg. Then, for each h from h and any X the relation [h,X] = a(h)x hold. The function a ( h ) is called the root and X is called the root vector. It is clear that for X @ h ~ ( h#) 0 and thus it is labelled by the nonzero eigenvalues of ad h encoded in a ( h ) , so we can denote a given root vector by X,.
Problem: Using the Jacobi identity show that
3 ~ h i eigenvalue s is always present because any operator commutes at least with itself.
C.4. CLASSIFICATION O F ALGEBRAS
425
where a + P # 0 and Nap is a complex number which equals zero if a is not a root.
+P
Problem: Using (C.8) prove that there are no multiple roots p a ,
I P I > 1. It follows from (C.7), (C.8) that all diagonal elements of the matrix ad X, ad xp equal to zero and therefore
In particular, Tr(ad2X,) = 0. In other words, the vectors X,, xg are orthogonal with respect to the bilinear form g(%,y) = (X,y) Tr(ad X ad y), or gab = f&f&. This form is called the KillingCartan form. It is very important for the purpose of studying Lie algebra because of the fundamental Cartan theorem: given Lie algebra is semisimple if and only if its KillingCartan form is nondegenerate. Using this theorem we can see that the root system of any semisimple algebra is symmetric under the discrete transformation a a. Indeed, if it were not so for some a, then that a would be orthogonal to the whole g and thus the KillingCartan form would be degenerate.


Problem: Show that KillingCartan form possesses the property (X,ad zy) = (ad
zx,y)
(C.10)
Problem: Using the result of the previous problem show that if (e,, e,) = 1 then a ( h ) = (h,, h) . (C.ll) In particular this means that (h,, h,) # 0, otherwise, a(h) = 0 which is impossible. Therefore, the KillingCartan form is also nondegenerate on h. Below we shall write (a,P) instead of (h,, hp).
Properties of Roots. CartanWeyl Basis The roots possess other remarkable properties. To find them, we introduce the subspace mp, = {eg Ice,), where q 5 k p and q,p 2 0 are maximal and minimal integers for which k a is a root. Below we use the results of the following two simple problems.
+
+
<
Problem: Prove that mp, is invariant under the action of eh,.
APPENDIX C. LIE ALGEBRAS
426
Problem: Prove that if V is a linear space and V. is its subspace invariant with respect of two matrices AI, A2 then the restriction of the commutator [Al,A2] on V. has zero trace. Because of the last two properties the matrix h, = [e,, e,] is traceless on mp, and therefore
and thus
is an integer. This remarkable property will be used below. Now let us find the product (a,a) using (C.12). We have
where
1 P. = q
C (PP,
lo)'
P
is a positive rational number. Thus the scalar product
is a positive rational number. This fact enables us to calculate all Nap. First, let us turn to some properties of these constants. The first of them is obvious: . (C.14) Nap = Np, The second one follows from the Jacobi identity for e,, ep, e, for a +P
+
y = 0. Indeed, the Jacobi identity gives for this particular case
Combining the last two conditions and the fact that all three roots are not zero, we get
We can see that the elements h,, e+, form a subalgebra in g isomorphic to SU(2) if (a,a) > 0 that is really the case due to (C.13). Irreducible
C.4. CLASSIFICATION OF ALGEBRAS
427
representations of su(2) are well known from the theory of angular momentum. It is clear that e* = f i e + , / Jar a ) corresponds to the ladder operator J+ of this theory. Therefore, ad h,, ade* is a representation of su(2) acting in g. As the weights of su(2) representations are halfinteger going without gaps from j to j where j is the maximal weight (in our case it is from c  q to c p) we conclude that
+
Nap # O , a + p i s aroot.
(C.16)
Now we see that Nap is nothing but the matrix element of e, in the irreducible representation of su(2) under consideration. Using the same techniques as in the theory of angular momentum we have
N$
= P
2
(a, a) .
Thus we see that semisimple Lie algebra is completely defined by its root system. Therefore we can chose the commutation relations in g in the form
where the constants Nap fulfil conditions (C.14)  (C.17). The basis of generators in (C.18) is called the CartanWeyl basis.
Cartan Matrix. Dynkin Schemes Thus to classify semisimple Lie algebras we shall study all possible systems of roots (we shall denote a system of roots by A). As there are r linearly independent h,, there must be r linearly independent roots. The problem is to choose a suitable basis in the root system. For this purpose we separate all a ' s and a's. Let us introduce a real basis {hl, . . . , hz,) so that any element of h is a linear combination of hi with real coefficients. We shall call the root positive in this basis if the first nonzero coordinate is positive. Analogously, a > ,G' if a  P > 0. It follows from the inversion symmetry of the system that exactly one half of all nonzero roots is positive. A positive root which can not be represented as a sum of two other positive roots is called simple. We can see that the set of all simple roots II is a basis in A. Indeed, let us consider the system of all positive roots A+. First we notice that the difference of two simple roots a  p can be neither a positive nor a negative root, because it would contradict their simplicity. Then if a root a from A+ is not simple, a = P y, P, y > 0. Applying this procedure to each term of
+
APPENDIX C. LIE ALGEBRAS
428
this sum we come after finite number of steps to a linear combination of simple roots with positive coefficients. For the set of negative roots Awe have linear combinations of simple roots with integer negative coefficients. Thus the commutative relations of the algebra are determined by its system of simple roots II = {wl, . . . ,W,) or by the Cartan matrix cij
which has integer elements because of (C.12). This statement can be formulated as a theorem we shall not prove here: two semisimple algebras having the same Cartan matrix are isomorphic. We would also like to point out some properties of Cartan matrix. First, denoting the angle between wi and wj by Q, we notice that
Therefore there are four possibilities for wi # *wj: cijcji = 0,1,2,3 that corresponds to Oij = 90°, 120°, 135", 150". System of simple roots can be depicted graphically in the form of a flat graph. For each wi we draw a point joined with other wj by cijcji lines. Such graph is called a Dynkin scheme. Problem: Prove that if wl, w2 are simple roots then, there is a sequence of roots {wl kw2, 0 k  (wl, w2). Problem: Prove that (wi,wj) 5 0. (Use the fact that the combination wi kwj can not be a root for any k < 0.) Problem: Using the properties of the structure constants Nap (C.14)  (C.17) prove that if II = 111 U 112, where Itl, 112are mutually orthogonal systems of simple roots then, g = g, @g2 ([gl,g2] = 0) and lIl,2 are the systems of simple roots of g,,,. If the system of simple roots II can be split into two mutually orthogonal subsystems, we shall call it splitable. We can see that if the algebra g is a direct sum g = gl @gz, its system of simple roots is splitable (and this corresponds to a nonconnected Dynkin scheme). Taking into account this result altogether with the the statement of the last problem, we come to the conclusion that the problem of the classification of complex simple (and hence, semisimple) algebras is reduced to the problem of the classification of all nonsplitable linearly independent rdimensional systems of vectors with nonpositive integer c,., that is, the classification of all possible connected Dynkin schemes. The solution of
+
+
<
C.4. CLASSIFICATION OF A L G E B R A S
c,,
t.
Dn
*    G 4
e6
e7
M_g_o
T
Figure 67
the last problem can be obtained by elementary methods and we shall not quote it here. The result is that the complete set of complex simple Lie algebras consists of four infinite sets of the classical complex Lie algebra~A,(sl(n+ l ,C ) ) ,&([email protected]+ l ,C ) ) ,C,(sp(n, C ) ) , Dn(so(2n,C ) ) , ( n > 2) whose Dynkin schemes are depicted in fig. 67a and five exceptional Lie algebras, gz, fq, e ~e7,, e8 with the Dynkin schemes presented i n fig. 67b. In these pictures the smaller root is black if the line joins two roots of nonequal lengths. The dimensions of the algebras are respectively n ( n + 2 ) , n ( 2 n + l ) , n ( 2 n + l ) , n ( 2 n  l ) , 14, 52,78, 133 and 248.
Compact Algebras Complex simple algebras contain a rich structure of real subalgebras which should be extracted from the basis ( e k , i e k ) where ek are generators of the complex algebra. Thus, the sets e,, e,, h , and ie,, ie,, h, are real subalgebras of g isomorphic to 4 2 ) and s u ( 1 , l ) respectively. We shall not consider the classification of real algebras here (see e.g., [2], [3])but rather say some words on compact algebras. A real Lie algebra g is called compact if its KillingCartan form does not change its sign. One can check that s u ( 2 ) is compact and s u ( 1 , l ) is not. Finally we prove that all structure constants of a compact semisimple Lie algebra can be chosen to be completely antisymmetric. Indeed, let us assume the KillingCartan tensor gab to be positively defined. It follows from the theory of quadratic forms that in this case it can be reduced to the
APPENDIX C. LIE ALGEBRAS
430
unit tensor by an appropriate choice of the basis: constants fabc
gab
=
aa6. Then, the
= g c p f:6
equal to the structure constants f;b. We prove that fabe = fab. Using Jacobi identity we get
Thus, the structure constants are antisymmetric in all indices.
Bibliography L.D. Landau and E.M. Lifshits, Course of Theoretical Physics. 71.3: Quantum Mechanics, 3rd ed., Oxford, Pergamon, 1977. A.O. Barut and R. Raczka, Theory of Group Representations and Applications, World Scientific, Singapore, 1986. M. Gourdin, Basic of Lie Groups, Lie GifsurYvette, Ed. F'rontieres, 1982. D.P. Zelobenko, Compact Lie Groups and their Representations, Providence, American Mathematical Society (AMS), 1973.
Index Abrikosov vortices 350, 354 Action 4 Euclidean 35 Ado theorem 419 AharonovBohm effect 357 Annihilation operator bosonic 94, 234 fermionic 241, 248 Anomalous dimension 122, 193, 200 Anomaly 122 Antiinstanton 379 Antikink 344 Antisoliton 348 Anyons 228 Asymptotic freedom 124 Axial gauge 303
Coherent states 95, 253 fermionic 265 ColemanWenberg effect 119, 336 Continuum 18 Cooper pairs 351 Correlation function, npoint bosonic 133, 207 fermionic 268 Correlator 132 exact 207 Coulomb gauge 303 Coupling constant 60 Counterterms 167 Creation operator bosonic 94, 234 fermionic 241, 248 Critical exponents 219 Critical temperature 351
Background field 156 Background gauge 303 Bands structure 63 Berezin algebra 251 Bianchi identity 296 Bogomolny boundary 370 Bosons 228 BPS limit 370 BPST instanton 375 deBroglie relation 19 BRSTsymmetry 327
Decoupling theorem 85, 391 Degree of divergence 278 Diagonalization 3 1 Dimensional transmutation 118 Domain wall 347 Domains 347 Dynamic mass generation 336 Dynkin scheme 428 Dyon 370 BPS dyon 371
Cartan theorem 425 CartanWeyl basis 427 Casimir effect 100
Effective action 150 Effective potential 99, 109, 281, 335
INDEX Energymomentum tensor 78 Equaltime commutation relations 80 Euclidean time 35 Factoralgebra 418 FermiPastaUlam system 74 Fermions 228, 240 Feynman path integral 23 Feynman diagrams 139 Field 75 elementary excitations 91 operator 237, 242 First homotopy group 359 First order constraints 314 Flux quantization 353 Fock space 235 Functional 6 Functional derivative 7 Functional determinant 50, 104 Functional differentiation 6 Functional integral 23 Fundamental group 359 Gauge field 292 Gauge condition 302 Gauge transformation 289 large 381 small 381 Gaussian integral 19 GellMann Low equation 199 Generating functional 133 Ghosts 326 Global symmetry 289 Goldstone particle 334 Gradient expansion 156 Grassmann algebra 249 Grassman numbers 248 Green function 17, 132 classical 136 
connected 135 exact see Correlator of free field 139 oneparticle irreducible 155, 210 Group orbit 330 Hamiltonian 12 Hamiltonian gauge 303 Heat kernel 283 Hedgehog ansatz 368 Hessian 310 Higgs mechanism 307 Homotopic mappings 358 Homotopy group 372 Huygens principle 20 Instanton 44, 375 density 60 dilute gas approximation 60 multiinstanton trajectories 46 Isotopic space 290 Isotopic symmetry 290 Jacobi identity 296 KillingCartan form 425 Kink 344 Kink sector of theory 384 Lagrangian 4 Lamb shift 99 Landau pole 122, 201 Legendre transformations 148 Lie algebra 290, 417 Abelian 417 adjoint representation 420 structure constants 418 subalgebra 418 Lie group 290
INDEX generator 290, 417 Lie superalgebra 250 Local symmetry 289 Long wave approximation 75 Lorentz gauge 303 Magnetic monopole 366 Mass shell 180 Meissner effect 309, 352 dual 374 Metastable states 127 Momentum 4, 11 Multiplet 290 Nilpotent algebras 422 Noether theorem 13, 78 Nonperturbative methods 61 Occupation number 231 Order parameter 347 (P4 theory
107 Partition function 83 Path integral 23 PeashleTeller potential 53 Pauli exclusion principle 240 Phase transitions firstorder 218 secondorder 219 Phonon 77 Planck scale 90 Primary constraint 311 Proca Lagrangian 319 Propagator 17 Pseudomechanics 258 Regularization 90, 168 cutoff 101, 112 dimensional 173 PauliVillars 168
Renormalization 90, 115, 167 of coupling constant 116, 181 of mass 115, 179 of wave function 182 Renormalization group equation 121, 189 Renormalization point 1l 6 Renorminvariance 194 Running coupling constant 121, 193
a model 273 SU(2) instanton 375 Saddle point approximation 106, 146 Scalar field 77 Scale transformations 198 Scaling 200 Second order constraints 314 Second quantization 239 Secondary constraints 313 Selfdual field 377 Semiclassical approximation 61, 110, 148 Similarity law 200 Singular systems 260 SLAC bag model 273 Solitary waves 349 Soliton 349 Source 133 Sources method 108 Sphaleron 383 Spinodial decomposition 127 Stable point infrared 203 ultraviolet 202 Stationary phase approximation 27 Stirling formula 28 String tension 354 SuperJacobian 256
INDEX Superconductivity 308, 353 Superconductors of types I and I1 354 Superdeterminant 256 Symbol of a normal operator 264 Symmetry breaking dynamical 118, 336 spontaneous 141, 304 Symmetry coefficients 144 0vacuum 57, 383 Temporal gauge 303 Topological boundary conditions 345 Topological charge 297, 346, 348 Topological current 297, 346, 348 Triviality 220 Tunnelling 40 Twoparticle threshold 179
Ultraviolet divergences 165 Unitary gauge 308 Vacuum 34, 89, 235, 304 fluctuations 144 metastable 218 Vacuum manifold 304, 357 Variational principle 5 Vertex 154, 162 Weinberg theorem 279 Winding number 352 YangMills equations 299 (function 283 Zero mode 50, 52 Zeropoint energy 90