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 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 > 1. In the system of units where fi = 1, this condition takes the form (3.11) a = X/w3 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 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  > 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
> 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 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
1). To calculate specific values of the 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 > 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 > 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 > 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.
+
+
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
+
+
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