Introduction to Quantum Field Theory

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

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

.

. .

. .

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 Energy-Momentum 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 . . . . . . . . . . . . . . . . . . . . . . . . Zero-Point 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 Pauli-Villars 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 . . . . . . . . . . . . . . . . . .

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

. . . . .

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

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 Gell-Mann - Low Equation . . . . . . . . 8.3 Asymptotic Regimes . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

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: Second-Order Phase transitions . . . . . . . . . . . . . . The Cases d = 4: Possible Triviality of (p4 Theory 9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

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

. . . . . . . .

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

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

...

v111

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Particle Hamiltonian . . . . . . . . . . . . . . . . . Creation and Annihilation Operators . . . . . . . . . . . Total Hamiltonian . . . . . . . . . . . . . . . . . . . . . The Field Operator . . . . . . . . . . . . . . . . . . . . . Result: Recipe for Quantisation . . . . . . . . . . . . . . 10.3 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . Oneparticle Hamiltonian . . . . . . . . . . . . . . . . . Creation and Annihilation Operators . . . . . . . . . . . Many-Particle 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 Non-Relativistic 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 Yang-Mills 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 Yang-Mills Fields . . . . Quantization of Gauge Fields: Faddeev-Popov Method Coleman-Weinberg 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 . . . . . . . . . . . . . . . . . . . . . . Ginzburg-Landau Model of Superconductivity . . . . . . Nontrivial Solution . . . . . . . . . . . . . . . . . . . . . Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . A Few Words about Topology and an Exotic String . . . Vortex Solution in Other Contexts . . . . . . . . . . . . 13.4 The 't Hooft-Polyakov 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 Yang-Mills 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 Double-Well Potential C Lie C.l C.2 C.3

Algebras Elementary Definitions . . . . . . . . . . . . . . . . . . . Examples of Lie Algebra . . . . . . . . . . . . . . . . . . The Idea of Classification. Levi-Maltsev 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. Cartan-Weyl 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 field-theoretical 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 MIT-CTP-2500 (January 1996), hcp-th/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 context-independent 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 cross-disciplinary 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 fourth-year 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 field-theoretic 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 field-theoretic 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 so-called 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 one-loop 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 crash-course

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 two-point correlation function at the one-loop 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 10-13, 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 1997-1998. 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. Zinn-Justin, 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, Addison-Wesley, 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, Addison-Wesley, 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, McGraw-Hill, 1980. [g] J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics and Relativistic Quantum Fields, McGraw-Hill, 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, Springer-Verlag, 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, McGraw-Hill, 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 17-th 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 point-like 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 so-called 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 infinitely-dimensional 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 infinite-dimensional 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 = tb-ta 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 at-o

N-l

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 6-function in terms of the Kronecker symbol Sij: 6(t - t')

=

lim

at-0

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 short-cut 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)

P-0

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 E-0

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)

at-o

+ 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 first-order 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 one-parameter 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 well-known 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 so-called 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, Addison-Wesley, 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 one-dimensional systems. The classical motion of such systems is determined by a single space coordinate X. From a mathematical point of view, we are dealing with the Cauchy problem, so the general solution can be constructed by means of the Green function G(x, xl, t , tl) . This function is also called the propagator or resolventa of the Schrodinger equation. It is simply the wave function of a particle which is situated at the point X = X I at the initial time t = tl so that

Here the operator H acts on the variable X. The solution of the Schrodinger equation with an arbitrary initial condition +(X,tl) can be represented

CHAPTER 2. THE PATH INTEGRAL

18

as a superposition of all waves of amplitude initial time t = tl:

$(X,

tl) 'irradiated' at the

Throughout this book only stationary Hamiltonians are considered. The Green function in this case can be represented as an expansion in the normalised eigenfunctions $,(X) of the Hamiltonian H for which the relation H$,(%) = E,$,(x) holds. Thus,

This last relation can be verified by substituting it into (2.2), remembering that each term of the series obeys the Schrodinger equation. The initial condition is fulfilled by virtue of the consistency of the eigenfunction set $,(X): X$n(x)$:(Y)

= a(x

-

Y)

.

(2.5)

n

It follows from (2.4) that G(x, xl, t, tl) depends on time through the difference t - tl. This is an immediate consequence of the invariance of the system under translations. Let us calculate the Green function of a free quantum particle of mass m as the simplest example. The corresponding Hamiltonian is

Its eigenfunctions are the plane waves $(X) = eRpX, where p is the eigenvalue of the momentum operator. The energy of the particle is E = p2/(2m). The momentum as well as the energy take values in a continuous infinite range (we say that their values fill a continuum). In order to perform the summation over n on the right hand side of (2.4) one should make the spectrum discrete and take the continuum limit in the final expression. For this purpose we close the coordinate line X to form a ring, that is we assume X to vary in a large but finite interval 0 5 X 5 L so that the wave function is periodic in space $(O, t) = $(L, t). Then the normalised eigenfunctions take the form

2.1. T H E GREEN FUNCTION

and the discretised values of the energy and momentum are

where n = 0 , f l , f 2 . . .. Now we can use formula (2.4) to evaluate the Green function:

It follows from (2.8) that the momentum increments by Ap = 2nfilL as n changes by unity. This means that at large L the general term of the sum in (2.9) is a slowly varying function of n. This enables us to regard (2.9) as an integral sum. Multiplying and dividing the right hand side of (2.9) by A p and taking the limit L -+ CO we get

Calculating this Gaussian integral using the formula +m

J dz exp (-az2 + bz) = -00

a exp

(-) b2 4a

'

(2.11)

(its derivation, as well as some other properties of Gaussian integrals, is given in Appendix A.1), we obtain finally

Note the classical action (1.26) appearing in the exponent. In principle, we could just write this down making sure of its validity by a straightforward substitution into (2.2). Our aim here was to show how the summation over n transforms into integration over p. This procedure will be widely used throughout this book. The obtained expression (2.12) enables us to derive the de Broglie relations. Let us consider a free particle starting from the point xl at time tl. Its wave function is G(x, xl, t, tl) and it looks like a wave whose amplitude and the phase evolve according to (2.12). Suppose that the

CHAPTER 2. THE PATH INTEGRAL

20

particle is detected at the point X at time t. Let the local coordinate and time in the detector's reference frame be Ax and At respectively, and consider the case when they are small: Ax << X and At << t - tl. Let us expand the argument of the exponent in (2.12) in powers of the small ratios Ax/(x - X,) and At/(t - tl). This gives

i m(x -

h 2(t - t,)

- xl) i m(x + -tii m(x Ax-(t tl) h 2(t - tl) -

l

A t + ...

Keeping only the first-order terms in Ax and At, we conclude that the wave function of the particle looks locally like a plane wave with wave number k and cyclic frequency W such that k = p/h

and

W

= Elfi,

(2.14)

where p and E are the momentum and energy of a classical particle which starts motion at the point xl at time tl and reaches the point X at time t. Physically, the smallness of the neglected terms requires that the classical momentum and energy change essentially over distances and times much larger than Ax and At respectively. Let us now return to the general properties of the Green function of the Schrodinger equation, and discuss an analogue of the Huygens principle which states that $(X, t) can be regarded as a result of the superposition of all waves irradiated at any intermediate time t2. Indeed, as was noted, formula (2.3) gives such a representation of the wave function for the initial time t = tl. As the Schrodinger equation is first-order in time, we can take the wave function $(X, t2) at a time t2 as a new initial condition and find $(X, t) for t > t2 by solving this equation. The fact that this should not change the result imposes the following property

for any initial function $(xl,tl). Thus it must be that

The property (2.16) can be directly checked by making use of the representation (2.4). For this purpose it is sufficient to substitute (2.4) into (2.16) and make use of the orthonormality of the eigenfunctions: S dx$k (X) $: (X) = &,.

2.2. THE PATH INTEGRAL

2.2

The Path Integral

Here we derive one more useful representation of the Green function G(x,xl,t, tl). Let us subdivide the interval between tl and t into N (where N is very large) small parts At (see fig. 4) and apply formula (2.16) as many times as there are intermediate time points. The beginning of the i-th interval is the point ti, i = 1 , 2 , .. . N. We denote the corresponding coordinate X by xi. Then,

This expression does not yet provide any new information. To gain it, we must find G in the limit At + 0 or N -+ oo. For infinitesimal At we have for G(x, X I , t At, t)

+

G(x, xl, t

+ At, t) = G(x, xl, t, t ) - !HG(X, h

21,t, t

) ~ t.

(2.18)

For a particle in the potential V(s) , the Hamiltonian H is

Such a choice of Hamiltonian enables us to get rid of some additional complications connected with ordering of the coordinate and momentum operators. Using the relation G(x, X I , t, t) = 6(x - xl), and the Fourier representation of the delta-function

we obtain G(x, xl, t

+ At, t)

M

6(x - 21) -

(2.21)

As At -+ 0, the terms in square brackets can be regarded without reduction in accuracy as an expansion of an exponential function:

[l -

(g+ v(x))]

= exp

[-F (g+

~(x))] .

(2.22)

CHAPTER 2. THE PATH INTEGRAL This results in G(x, XI,t

+ At, t)

(2.23)

Substituting this into (2.17), we obtain an expression G(x, X I , t, t ~ )=

lim

N-+m

/

~ P N dpi dp2 dx2dz3.. . dxN-... 27rK27rK 2n-fi

(2.24)

where X N + ~= X. This formula merits some discussion. First note that there are N integrations over momenta, and one fewer over coordinates in (2.24) because the points xl and X N + ~= X are fixed by the boundary conditions. Thus the total dimension of the space over which the integration is taken is 2N - 1. The points of this space are the sets 2 2 , . . . XN;PI,.. .PN. In the continuum limit N -+ CO these sets become x(t) and p(t) respectively. This means that the integration in the continuum limit is taken over the set of all trajectories in the system's phase space. Second, it should be emphasised that X and p appearing in (2.24) are ordinary functions rather than operators. It can also be easily checked that the argument of the exponent in (2.24) reduces to the classical action Slp(t), x(t)] = J dt(px - -(p, X)) in the limit N -+ oo. This rather transparent physical meaning of expression (2.24) enables us to rewrite it in a symbolic form as an integral over the infinite-dimensional phase space

Such a form does not contain, however, information concerning the measure of the functional integration, namely, the coefficients (27rfi)-l and 'It is not so transparent at the mathematical level because a more rigourous definition of the path integral is conjectural. Nevertheless, all specific physical calculations are carried out at a large but finite value of N. The limit N = co is taken a posteriori.

2.2. THE PATH INTEGRAL

dt Figure 4

the fact that there is one fewer integration over coordinates than over momenta. The representation of the Green function obtained here is called the path integral (PI), or the functional integral, or the Feynman integral, or simply the integral over trajectories. It is the central object in modern quantum field theory (QFT). The use of the PI is often more convenient and clearer than other methods. Note that we did not use any specific form of the Hamiltonian in deriving expressions (2.24), (2.25). Only the fact that the operator H ( @ ,X) has the eigenvalues H(p, X) identified it with the classical Hamiltonian. Now let us make use of the quadratic dependence of H on p, equation (2.19), in order to perform the integration over momenta in (2.24). Evaluating Gaussian integrals

we get G(%,X I , t, tl)

=

l

N+m

d x d . . . dxN

CHAPTER 2. THE PATH INTEGRAL

where the symbolic form of the PI has been used again. This calculation becomes rather complicated if the Hamiltonian depends on momenta non-quadratically. The general recipe is to use (2.24) for construction of a so-called 'effective action' which is the functional appearing in the exponent in an expression of type (2.27) after the integration over momenta and taking the continuum limit. Fortunately the Hamiltonians in quantum theory are, as a rule, quadratic in momenta, so we restrict ourselves to such systems. Let us discuss further expressions (2.25) and (2.27). Formula (2.27) states that the probability of finding the particle located at the point X at time t, having been located at the point xl at time tl, is a sum of the contributions due to all trajectories with starting and finishing points X I , tl and xz,tz respectively. Each trajectory x(t) contributes a weight factor eiSIx(t)ll". One may say that the particle moves simultaneously along all trajectories with given end points, and the superposition of all weighting factors results in the actual wave function. Note the difference between the role played by the action in the classical theory and in the quantum one. In classical theory, the action results in a unique classical trajectory: a single point in the functional space. In quantum theory, we deal with all possible trajectories: that is the whole functional space. Also from (2.25) and (2.27), there follows a transparent way in which classical and quantum mechanics are connected with each other. Let us consider a macroscopic system. This means that the action on its typical classical trajectory is much greater than the fundamental constant 6. In this case the integrands of (2.25) and (2.27) oscillate rapidly so that only a bunch of trajectories providing almost stationary S give a significant contribution to the transition amplitude. The central trajectory of this group satisfies the condition 6s = 0. Thus in the limit S / h --+ ca the main contribution to the transition amplitude is given by the classical trajectory. Then, qX, xl, t, tl) e i ~ ( x , x ~ , t 3 t d l ~ (2.28) where S is the action on the classical trajectory. This is the key statement because it gives rise to a method of approximate calculation of transition amplitudes which will be used extensively. Below, a more exact formulation of (2.28) will be considered.

2.3. THE PATH INTEGRAL FOR FREE MOTION

25

Note that expression (2.28) leads to the generalised de-Broglie relation (2.14). To show this we take small variations of X and t : X -+ X As and t -t t At and expand the action in (2.28) in these variations. Taking into account the definitions of the classical momentum p (1.31) and energy E (1.34), we may write

+

+

Here p is the canonical momentum conjugate to coordinate X . This equation is justified for large classical action S [ x ( t ) ]>> h, which is the case for the macroscopic trajectories, for example, of a particle in a bubble chamber. We would like also to note that the starting point for expressions (2.25) and (2.27) for the Green function was the Schrodinger equation (2.2). We can easily show that this equation follows from from (2.25) and (2.27). It is sufficient to use property (2.16) which is true for functional integrals by construction and to apply the mathematics of (2.21)-(2.24) in reverse order. Consequently, representations (2.25) and (2.27) are completely equivalent to Schrodinger equation quantum mechanics. In this book, we prefer the PI formalism because it enables a more straightforward generalisation to quantum field theory. We shall not consider the mathematical matters related to the path integral. The path integral will rather be used as a tool, considered at the physical level of strictness.

2.3

The Path Integral for Free Motion

Here we consider examples of PI calculation for some simple systems. We start with the simplest case - free motion. The action in this case is

0

As all times for the free motion are equivalent, we have chosen t I = 0 and the motion time equal to T. Eventually, the Green function for this problem will be derived three times. First, we need the exact solution of the Schrodinger equation obtained in equation (2.12). Let us re-write it here in the current notation:

Second, we would like to make sure that this expression really follows from PI (2.27). Thirdly, the stationary phase method will be applied to the calculation in order to gain an insight into this method itself.

CHAPTER 2. THE PATH INTEGRAL

26

Free Motion: Straightforward Calculation of the Path Integral To perform the functional integration and find the Green function

we return to the discrete set of times ti and coordinates xi = x(ti) according to definition (2.27). Then for different Ati we have

This is a chain of Gaussian integrals over dxi for each of which formula (2.11) should be applied. Integration over dx2 gives

1 -+ exp{~[xr(Atl

---

t

-252

X? + -- + Atl +At,

(a":, 2 ) --

l

--

(2.34)

Substituting into (2.33), we obtain

exp { E

h

[

(53 - ~ 1 ) '

2(At,

+ At,)

+

(x4 ~ 2At3

3

+...l) )

~

We see that integration over the intermediate coordinate x2 does not change the form of the PI being reduced to the replacement of Atl and At2 by the time interval Atl At2. That is property (2.16) is valid. It is so because each elementary Green function in the product in (2.33) is exact in the case of free motion. Now it is clear that the subsequent integrations over dx3,dx4,.. . dxN give expression (2.31). The above described method of direct PI calculation is applicable only for free motion because the presence of a potential V(x) prevents evaluation of the infinitely long chain of integrals as in (2.34). However,

+

2.3. T H E PATH INTEGRAL FOR FREE MOTION

27

a useful method of numerical evaluation of the PI is based on a form similar to that defined in (2.24) or (2.27). It is called simulation on the space-time lattice. The accuracy of this method is restricted by the power of contemporary computers and sometimes it is not large enough to make reliable physical predictions. Analytical methods are traditionally powerful and helpful if one wants to gain insight into the physics of a given system. However, they require us to make some approximations among which the stationay phase approximation plays an important role. Its advantage is the possibility of a straightforward generalisation to QFT.

Free Motion: Path Integral Calculation by the Stationary Phase Method Let us start with an illustrative example of the following ordinary integral: I = dxexp (- f (X)) , (2.36)

I

where f (X) is a smooth function. It is evident that the largest contribution to the integral is given by the region around the point xo where the minimum of f occurs. The point obeys the equation f'(xo) = 0. Expanding f (X) in a series around xo we get

because the integral reduces to the Gaussian one. To establish the applicability condition of such an estimate we note that the main contribution to the Gaussian integral is given by the region for which z - xo ( f " ( ~ ~ ) ) ~ Let ' / ~us . substitute this value into the third- and fourth-order terms (that is, the leading neglected terms). The requirement of the smallness of these terms gives

These conditions are valid if the function f and all its derivatives are large. Let us define a new function g so that f(x) = g(x)/h with a

CHAPTER 2. THE PATH INTEGRAL

28

small parameter fi such that g and all its derivatives are of order unity. Conditions (2.38) then take the form << 1 and fi << 1. Accounting for the corrections to (2.37) leads to a contribution which contains the coefficient of the third-order term squared. Therefore, fi turns out to be the small expansion parameter. An example of the application of this formula is the asymptotic form of the gamma-function

the so-called Stirling formula . It follows from the condition f l ( t o ) = -1 z / t o = 0 that to = z is the minimum of f and

+

according to (2.37). This method is called the Laplace method or the saddle point approximation. Its analogue for integrals with oscillating exponential function I = d x exp ( i f ( X ) )

/

is called the stationary phase approximation. The specific feature of the latter case is that the smallness of the contributions from the region far away from xo is due to extremely fast oscillations of the integrand rather than the absolute smallness of the exponential function. Therefore all formulae remain valid up to the replacement of the ordinary unity by the imaginary one. Now let us consider the path integral. First we find the trajectory x ( t ) corresponding to the stationary action S S [ x ( t ) ]= 0. This is the classical . we consider the set of trajectories which deviate trajectory ~ ( t )Then from the classical one by a small variation 6 x , i.e. x ( t ) = ~ ( t )6 x . As the boundary conditions are fixed, 6 x ( 0 ) = S x ( T ) = 0. Expansion of the action in a Taylor series around :(t) gives

+

where the shortened expression for the action variations has been used. Before we go on evaluating the PI we clarify this expression and introduce some other widely used abbreviations:

2.3. THE PATH INTEGRAL FOR FREE MOTION

It is often convenient to integrate by parts taking into account the zero boundary conditions for the variation 6x. As a rule, we deal with Lagrangians quadratic in x so that all higher derivatives are equal to zero. Now, let us turn to the expansion (2.42). As this expansion is performed near the classical trajectory, SS = 0. In the considered case of the free motion (2.30) all action variations higher than the second, S'S are zero. For the second variation we have

Here we have integrated by parts and introduced an obvious abbreviation for the second time derivative. Performing the described expansion in PI (2.27), we get the Green function of the free motion

=

exp

' """

-x02]

2T

/

V6x(t) exp

[gf

dthx(-d:)6~

1

.

Thus we have simplified the problem. The classical action takes care of the boundary conditions while Sx(t) (which is the new variable of functional integration) obeys the zero boundary condition. However, the form of the integrand remains unchanged. As before, the main difficulty is that the variations 6x(t) at neighbouring times are not independent of each other because of the time derivative (compare with (2.27)). To solve this problem let us expand Sx(t) in a series of eigenfunctions $,(t) of the operator -8; (d/dt)2 appearing in the second action variation (2.44). This is possible because of the zero boundary conditions for Sx(t):

Here C, are the coefficients of the expansion. The explicit forms of the eigenfunctions and the corresponding eigenvalues X, are: $. (t) =

rnt sin 7

and

X,

r2n2 n = l , 2 , 3 . .. T2

=-

. (2.47)

30

CHAPTER 2. THE PATH INTEGRAL

Substituting into (2.44), remembering the orthonormality of the eigenfunctions $,(t), we get

Thus PI (2.45) can be rewritten in the form

A nontrivial operation here is the replacement of the functional integration variable Sx(t) in (2.27) by the set of coefficients C,. A is the Jacobian of this transformation: it is rather difficult to evaluate. Moreover, it diverges in the continuum limit. On the other hand, the Fourier transformation (2.46) is linear. Therefore the Jacobian is a constant which does not depend on the coefficients C,. This transformation reduces the integral to a product of an infinite number of one-dimensional Gaussian integrals of the following type:

Using this formula we get finally G(x,xO,T) = exp

[i F,

- x,)~]

2T

An

/%X

K

The Jacobian A is still undetermined. Its calculation and even the correct definition appears to be a serious mathematical problem. Nevertheless, comparing (2.51) with G(x, xo, T ) in (2.12) we obtain immediately

This last expression enables us to calculate the PI when the straightforward calculation (as used in the free motion case) is not possible. One more remark about terminology. The substitution of the integration variable Sx(t) + C, enables us to diagonalise the second variation of

2.4. THE PATHINTEGRAL FOR THE HARMONIC OSCILLATOR the action, S2S. To make this clearer let us consider a finite-dimensional integral (2.53) I = dxldxa.. . dxN exp (ziMijxj) ,

I

where Mij is an arbitrary, in general, non-diagonal symmetric matrix i, j = 1 , 2 . . . N. It is evidently an analogue of the discretised action. The integral is most easily evaluated in the diagonal basis. In other words, new variables xi = Oijxj must be introduced so that the matrix OMOT is diagonal. This is a direct analogy of what we have done in (2.48).

2.4

The Path Integral for the Harmonic Oscillator

The next important example is the Green function of the harmonic oscillator - a particle of mass m moving in the potential V(x) = k2x2/2 (fig. 5)

where W = (k/m)'f2 is the angular frequency. To calculate the PI (2.54), let us use the method of stationary phase. It should be noted that the result must be exact because the action is quadratic in X, i.e. SnS = 0 for all n 3. The recipe for the calculation will be the same as that in the case of free motion:

>

l. Find the classical trajectory ~ ( t and ) the value of its action;

2. Expand the functional of action in the neighbourhood of the classical trajectory;

) zero 3. Perform functional integration over Sx(t) = x(t) - ~ ( t with boundary conditions. This leads to the following expression G(X,xo,T ) = exp [;~[Z(t)l]

JD

e x

t

t

- - w2)~x(t)

CHAPTER 2. THE PATH INTEGRAL

Figure 5

where S[z(t)] is defined in (1.28). Now, as was done for the free motion, it is necessary to turn to the Fourier components of Sx(t). To do that it is sufficient to note that the eigenfunctions of the operator -8: - w2 are the same as those of -8: and the corresponding eigenvalues differ from (2.47) by a constant.

Repeating the argument of the previous section we obtain

[i s[z(~)]] A n [g (G )]' CO

G ( r, XO, T) = erp

- w2

(2.57)

n=l

2 ~ can e now easily answer the question of the previous chapter: 'Does the trajectory correspond to the maximum or to the minimum of the harmonic oscillator action?'. As is seen from (2.48), T < r / w corresponds to positive second variation of the action. Hence, the classical trajectory corresponds to the minimum of the action. If r / w < T < 2 ~ / wthen the Fourier coefficient C: has a minus sign and gives a negative contribution to the action. This means that the action is maximal in the direction determined by Cl and minimal in other directions. Further growth of T leads to the appearance of a greater number of negative eigenvalues X ., At T = n n / w the eigenvalue X, goes to zero which means that there are many classical trajectories distinguished by the value of C, (the kinetic focus). This property is exact in the case of the harmonic oscillator.

2.5. IMAGINARY TIME AND THE GROUND STATE ENERGY

=

2iT26 fi d F fi n r

exp [ i ~ ~ z ( tA) ~ ] n=l

(l

-

n=l

n27r2

The Jacobian A of the transformation x(t) + C, is the same as for the free motion because the eigenfunctions of the operator -8: - w2 are the same functions (2.47). Thus A is given by expression (2.52), and the above expression differs from the free particle Green function (2.51) only by the factor n ( l - $)-'l2 which is given by the well known formula m

x2

n(1--)=-

n=l

sin x x

(see Appendix A.2). Making use of this equality, the Jacobian (2.52), and the explicit form of the classical action S(x, xo, T)

=

mw [(x2 2 sin wT

+ X;)

cos W T- 2xxo]

,

(2.59)

from the first chapter (1.28), we obtain the desired final expression for the Green function:

((xi

2rih sin(wT)

Of course, in the limit (2.51).

2.5

W -+ 0

+ x ( ~ )cos ~W ) T

-

~ Z ~ X ( T ). ) ]

this gives the free particle Green function

Imaginary Time and the Ground State Energy

To know the Green function is equivalent to knowing the complete solution of the Schrodinger equation. This function contains all the information about the eigenfunctions and eigenvalues. We discuss here how to extract this information. Suppose we know (exactly or approximately) the Green function G. In this book, we shall calculate it via its PI representation (2.27) while the representation (2.4) will be used to extract the information about the energy states and the wave functions. It follows from (2.4) that this can be done by taking the Fourier transformation of G with respect to time (note that the Fourier integral will already be quite nontrivial for the case of harmonic oscillator (2.60)).

CHAPTER 2. THE PATH INTEGRAL

34

Let us consider a simpler problem, namely to find the ground state energy and wave function of the harmonic oscillator. For this purpose let us consider (2.4) and (2.60) at imaginary values of the time variable t2 - tl T = -ZT and equate them. We get

+

+

$ O ( X ) $ ~ * ( X O ) ~$l(~)$~(~O)e-EIT1h -~~~/~ ...

-/,

=

27rh sinh(wr) exp

{-

m 26 sinh(w7)

(2.61)

+

[(x2 xi) cosh(wr) - 2x0x]

We see that as T --+ oo the leading contribution to the left-hand side is given by the ground state because E. is the minimal energy. On the c o s h w ~G expwr for right-hand side, we use the fact that sinh wr WT >> l. This gives

E e x p =

(- 7)exp (-g (xi + x2))

(z)

'l4 exp

(-G)(z) (-S) (- 7) mxo

'l4 exp

exp

(-g)

Here, & , ( X )= (11~/7rh)l/~ exP and E. = h / 2 are the correct ground state wave function and energy of harmonic oscillator. This is a remarkably simple method of calculating the ground state wave functions and energies: it is sufficient to take imaginary time in the PI transition amplitude and send it to infinity. In expressions of the type A(x, xO)exp(-B-rlh) the quantity B is to be interpreted as the ground state energy and A(x, xo) as the product of the corresponding wave functions. In quantum field theory, the ground state is called the vacuum. It plays an especially important role in a great number of physical problems because it often defines the system's properties. h many cases the calculation of the PI can be essentially simplified if we turn to imaginary time in equation (2.27). We shall demonstrate this for a system with the action

Now, let us rotate the integration contour in the t' complex plane by -7r/2. Then t' = -ir where T is real. The action takes the form

2.5. IMAGINARY TIME AND T H E GROUND STATE ENERGY 35

The quantity SE is called the Euclidean action. It is real, and differs from the initial action in the sign of the potential. Substituting (2.64) into (2.27) we obtain the imaginary time PI which is also called Euclidean:

=

lim

N-+m

exp {:AT

dx2dxi . . . dxN

5[7 i=l

J xi)

+

}

If the potential is bounded below, it is natural to expect that this functional integral is convergent. The reader may doubt the legitimacy of the rotation leading to expression (2.65). In general it is not a simple matter, but for our purposes, it is sufficient to regard the Euclidean form of PI (2.65) as a Green function of the imaginary time Schrodinger equation. The latter is simply the heat conduction equation:

To obtain the Euclidean PI, we must introduce the Green function GE of this equation as in (2.2) and repeat the steps leading from equation (2.2) to PI (2.27). Continuing this analogy, we can get the modified version of formula (2.52):

From the beginning of the 1970's, functional integration has been one of the main instruments of vacuum-state exploration, and has made possible a considerable success in this field. The Euclidean path integral plays the central role in calculating so-called nonperturbative effects, a simple example of which will be given in the next chapter.

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Bibliography [l] The first papers on path integral formulation of quantum mechanics

were P.A.M. Dirac, Phys. Zeit. d. Sow. Union, 3 (1933) 64; R.P. Feynman, Rev. Mod. Phys., 20 (1948) 367. [2] The first textbook on PI has been R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965. [3] The following books are devoted to the formalism of PI H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physzcs, 2nd ed, World Scientific, Singapore, 1995. [4] A brief introduction and many examples of practical use of the PI can be found in R.P. Feynman, Statistical Mechanics: a Set of Lectures, Reading, Mass., Benjamin, 1972.

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Chapter 3 The Euclidean Path Integral In this chapter, we consider the Euclidean path integral calculation for some model examples: namely the quantum-mechanical problems of finding the ground state energy and the wave function. The reason for this choice is that the method of solution of these problems can be generalised directly to QFT. The key feature of the approach to be considered is the use of nontrivial classical trajectories called instantons to evaluate the Euclidean PI. We will go into detail in order to explore this calculation method. It will be the final time in this book that we compare these results with those obtained by the standard methods of quantum mechanics.

3.1

The Symmetric Double Well

For the first example, let us consider the motion of a particle described by the one-dimensional action:

Here V(x)is the potential with symmetric minima at X = a and X = -a, depicted in fig. 6. A potential of this form describes, for example, the motion of a nitrogen atom in an NHS ammonia molecule. The nitrogen atom can be located on either side of the plane formed by the three hydrogen atoms. The states are physically equivalent, but different. This results in an observable peculiarity of the molecule spectrum. For the sake of simplicity let us set the particle mass m and Planck constant fi to unity. This means the following. As it is known, the units of all physical quantities can be expressed as combinations of only three

CHAPTER 3. T H E EUCLIDEAN PATH INTEGRAL

Figure 6

quantities which are assumed to be fundamental. Usually these are mass, length and time. Let us denote as [m],[l],[t],[E],[S]the units of these values, energy, and action respectively. For the latter two quantities, we have [E] = [m][1I2/[tl2and [S]= [m][1I2/[t]. The condition m = F, = 1 means that: (i) [m],[S],[t] are chosen as fundamental quantities, i.e. [l] = ( [ ~ ] [ t ] / [ m ] ) (ii) ~ / ~the ; unit of mass is chosen to be m (rather than, for instance, kilogram), and the unit of action is h. Practically, this means that any physical quantity f , with dimension [m]"" [SIus[tIvt, enters all expressions as f/(mum - F,"S). In order to return to the usual units, we must substitute this ratio for f in all expressions. We shall do this at the end of the chapter. Let us recall some results for the double-well potential. It is easy to find the particle ground state in the potential V ( x )in the framework of the semiclassical (or WKB) approximation, if the height of the barrier separating the two potential wells is much greater than the ground state energy in each well. In this case, the particle wave function is localised in the vicinity of one of the minima and only weakly penetrates into the sub-barrier region. The particle is localised in one of the wells and the influence of the other well is negligible. In this approximation, the ground state of the particle in the left well is described by the ground state wave function of the harmonic oscillator $L E $ r ( x a), localised z$ r ( x - a). near the left minimum. Analogously, for the right well The energy of both states is w / 2 (remember that F, = 1).

+

3.1. THE SYMMETRIC DOUBLE W E L L

41

Nevertheless, it is generally speaking incorrect to neglect the presence of the other well. This is because quantum tunnelling between the wells causes a particle located, say, in the left well to penetrate into the right one, and vice versa. Thus the states localised in a single well are not stationary. The correct wave functions are the even and odd ~uper~ositions:~

where $L, $R > 0. The corresponding states are not degenerate. Their energies are

Thus the previously degenerate energy levels in the left and right well are split. The value of the splitting is exponentially small and is defined by the parameter

where W(E) is the abbreviated action on the classical trajectory:

(these formulae are obtained in Appendix B). Let us consider the evolution of a particle initially (t = 0) located in the left well. Expanding the initial wave function in the Hamiltonian eigenfunctions (3.2), we obtain $(t) =

l

Jz [$o

-

exp (-iEot)

= $L cos(A . t)

+ $R

+

$1

exp ( - i E ~ t ) ]

sin(A . t ) .

(3.6)

It follows from this equation that the wave function oscillates between the two wells with frequency A. It is obvious that tunnelling (penetration of the particle through the barrier), can be neglected only if the system is observed over a time interval which is much less than l / A . 'It should be pointed out that this tunnelling restores the initial discrete symmetry of the ground state, which is violated if the particle is localised at a given minimum of the potential.

CHAPTER 3. T H E EUCLIDEAN PATH INTEGRAL

42

Quantum Mechanical Instantons Let us now reproduce the solution of the problem discussed in the previous section by calculation of the Euclidean PI: G(+a, -a, T) at T --t

CO.

Z* =

I

VX(T)exp {-S[X(T)])

(3.7)

Note that the Euclidean Green function and action

are written without the subscript 'E'. Such subscripts as well as the adjective 'Euclidean' of action, trajectories and time will be omitted up to the end of this chapter. It should be pointed out here that we are considering simultaneously two different functional spaces: one for the transitions from X = -a to X = a and one for the transitions from X = -a to X = -a. Let us specify the potential V(x) as the quartic polynomial:

Expanding V(x) as a series in the neighbourhood of

X =

f a , we get:

where w2 = V1'(a) = 8Xa2 is the oscillation frequency. The higher power terms in the expansion (3.10) can be neglected if the quartic term X(x a)4 is small compared with the second order term in the region X l/& where the oscillator wave function differs substantially from zero. This gives a validity criterion for the harmonic oscillator approximation:

Note that this condition also follows from the requirement that V(0) >> w/2. We evaluate the PI (3.7) by finding all the classical trajectories which lead from X = -a to X = +a in time T -+ CO (we recall that we shall calculate two separate values of the Green function which differ from one another in the second argument: 2 2 = a or xz = -a). A non-vanishing contribution to the result is clearly made only by those trajectories which

3.1. T H E SYMMETRIC DOUBLE WELL

Figure 7

correspond to a finite action in the limit T -+ CO.The first such trajectory is apparent, namely x(t) = -a. Trivially, the particle remains at the minimum point of V(x). The action on this trajectory is zero. To find the nontrivial trajectories, it is necessary to solve the classical equation of motion

Its solution is especially simple to find when we note that it describes the motion of a point 'particle' with coordinate x in the inverted potential -V(x) (see fig. 7). It is clear that in order not to acquire an infinite action as T + CO the particle must spend almost the whole time near one of the peaks of the potential. Thus the particle motion along the trajectory to be found starts at the point X = -a at T = 0 with very small (in the limit T + CO, zero) velocity. An instability caused by this initial infinitesimal perturbation grows slowly and results in a fast transition through the region around X = 0, after which the particle stops slowly at the other peak X = a, reaching it at time T = T. The 'energy' of the particle moving in such an inverted potential is conserved:

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

Figure 8

and hence

dx

(3.14)

The solution we are interested in can be found in the limit T = cm, by setting E = 0. Indeed in this case the action on the trajectory leading from X = -a to X = a at time T is independent of T itself:

(3.15) Here we used relation (3.13) for E = 0. All other values of E" give trajectories with infinite action in the limit T + cm. Let us find the explicit form of the classical trajectory, denoted by ~ ( .t Integrating ) equation (3.I d ) , we obtain: X

r-TO=

dx

1

-a

from which it follows that X(T) =a

tanh

w ( r - TO) 2

X

atanh -

,

(3.16)

3.1. THE SYMMETRIC DOUBLE WELL

Figure 9

This trajectory is usually called an instanton (fig. S), and the integration constant TO is its centre- the time at which:(rO) = 0. It is important that the action (3.15) does not depend on 70. The characteristic size of the instanton in time is W-' << T. The analogous trajectory corresponding to the transition from X = a to X = -a is called an anti-instanton. It is clear from symmetry that the classical trajectory leading from X = -a to X = a must be an odd function of time. In particular, the instanton should be centred at TO = 0 (and slightly distorted because of the finiteness of T). It is the unique solution of equation (3.13) at finite T. Any other trajectory close to the instanton with TO # 0 is not an exact solution of equation (3.12), but the deviation from an exact solution is small and approaches zero in the limit T + oo. To see this, let us consider the quantity e-S on the set of trajectories labelled by TO (see fig. 9). The unique maximum of this function is at the point ro= 0. Other values of 70 give nonzero derivative deCS/d~0# 0 but it is of order e-WAT , where AT is the distance between the trajectory centre and the nearest of the boundary points T = 0 or T = T. Thus the To estimate the derivative, it is necessary t o express the initial velocity v0 of the particle in terms of 70. For this purpose we restrict ourselves t o quadratic terms in the expansion of the potential and write X(T) = -a (vo/w) sinh(w7). Taking into account only small deviations from the instanton trajcctory we can check that v: (aw)' cxp(-WT). This is obviously thc estimate for the energy E corresponding t o the transition for the finite time T, and a t the same time, it is the scale for the first

+

-

46

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

value of the action remains very close to S (3.15) until 70 is separated from the boundary points by a time interval greater than l/w. It is reasonable to approximate the curve in fig9 by a rectangle of height exp(-S) and width T. Such an approximation means that we are taking into account any instanton trajectory (3.17) with action (3.15) corresponding to the value of the parameter 70 being in the interval [0,T]. Their contribution to the PI (3.7) is

/ die T

Zl = Const exp(-S)

= Const

T exp(-S)

.

(3.18)

0

where Const is not yet defined. Let us go on searching for classical trajectories which give finite contribution to 2%(3.7). Strictly speaking, there are no other classical solutions for finite T. Nevertheless there exist a class of trajectories which deviate from the classical one by an exponentially small value when T + cm.These are so-called multi-instanton trajectories. They look like chains of interchanging instantons and anti-instantons with centres at (fig. 10). Thus the amplitude of the transition the points T,('), T ; ~ .). . from X = -a to X = a includes a contribution from all multi-instanton trajectories with an odd total number of instantons and anti-instantons. Analogously, the contribution to the amplitude from X = -a to itself is given by the trajectories with an even total number of instantons and anti-instantons including the trivial one. If all the (anti-)instantons are separated from each another by a time interval much greater than the characteristic instanton size llw, the total action of the n-instanton configuration is merely n S where S is the one-instanton action. The n-instanton action does not depend on the positions of (anti-) instantons in the limit when they are far from each other. It is clear nevertheless that instantons and anti-instantons must alternate with each other along the time axis. In other words

rin)

Integration over all possible positions of (anti-)instantons with this restriction gives

variation of the action as well as for other observables vanishing in the limit T -,m.

3.1. T H E SYMMETRIC DOUBLE W E L L

47

To calculate this integral, we extended the intergration range for each 7;) to 0 5 ~,(i)5 T and divided by n!. The latter is the number of domains which are equivalent to (3.19), but differs by the ordering of 7ii). Strictly speaking, the action of n instantons is independent of their positions only if all of them are far enough away from each other. Otherwise the total action must include the terms describing the instanton - anti-instanton interaction. However, the measure of such configurations, (the integration region in the integral over the centre positions) is negligible if there are not too many instantons. For example, the contribution to the integral (3.20) made by trajectories with a tight instanton - anti-instanton pair (formed by any of the instantons) is of order ~ n - l

n Tn

--

w(n - l)!

wT n!

.

(3.21)

This is much less than (3.20) when n << WT. We shall discuss this condition in more detail below. The n-instanton trajectory contribution to the transition amplitude (3.7) is then

It is convenient to divide and multiply it by the harmonic oscillator transition amplitude for large Euclidean time:

(compare this with (2.60)). Then

The constant K, introduced here has the property Kn = Kn, where K corresponds to the single-instanton contribution. We postpone the proof of this fact until the next section in order to analyse the resulting expression for G ( f a , -a, T ) . Summation of (3.24) over all even n gives the transition amplitude from X = -a to itself in time T

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

Figure 10

Similarly, for the transition amplitude from we obtain:

X =

a to

=

ZO

exp

k=O

=

-a in time T

~ 2 & 1

03

G(a, -a,T)

X =

g e x p

[-F]

(2k

+ l)!

sinh [ ~ e x p ( - ; ) T ] .

(3.26)

In these formulae, we have substituted S + S/fL and G -+ ~ ( & / m ) lin/ ~ order to restore the conventional dimensions of physical quantities. It should be noted that eqs. (3.25) and (3.26) can be expressed in the form

where the notation

A has been introduced.

= fL~~-'l"

3.1. T H E SYMMETRIC DOUBLE W E L L

Recall the interpretation given to expressions of this type:

Comparing this with (3.27), we deduce that we have found the first two terms of this sum. These two terms represent the contribution of the ground state which is split into two sublevels with energies

As can be seen from (3.27), the wave functions of these sublevels are really the symmetric and antisymmetric combinations of the oscillator ground state wave functions in the left and right wells.

where 40 ( f a ) = (mwlnfi)'l4. We can perform a transition to real time in expressions (3.25) and (3.26) by means of the substitution T = it. The form of the result clearly coincides with (3.6). This (together with the analysis of the eq. (3.27) ) shows that the PI reproduces correctly the structure of the ground state split into two sublevels and gives the correct exponent in the splitting energy (3.4) and (3.5). We can easily check this by calculating the integral in (3.5) with E = 0. To complete the calculation, we have to find the unknown constant K in formulae (3.25) - (3.27). We shall do this in the next section, where we make sure that the PI calculation exactly reproduces formula (3.4).

The Contribution from the Vicinity of the Instanton Trajectory To find the constant K appearing in expressions (3.25), (3.26) and (3.28), it is necessary to integrate over the vicinities of classical trajectories in the PI (3.7). We do this analogously to the previous chapter, for each classical ) the trajectory :(T) we turn to integration over Sx(t) = x(t) - ~ ( t with z S[:] +b2S, action expanded up to the second order terms in &X,S[X(T)] where

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

(for the sake of simplicity we again set m = Ti = 1). From a technical point of view, the calculation of this contribution is the same as in the above considered case of the harmonic oscillator. We find the eigenfunctions x,(T) and eigenvalues X, of the operator H in expression (3.32):

and expand the trajectory variation Sx(r) in x,(r): 6x(r) = Xi Cixi(r). The contribution of the vicinity of the n-instanton trajectory Z to the PI takes the form of a product of an infinite number of independent Gaussian integrals:

This expression should be regarded in a symbolic sense because the product over i diverges. Usually, the infinite number of the multipliers 271. are included in the normalisation constant A, and the integration is written as a functional determinant. The determinant is defined (similarly to that of finite dimensional matrices) as the product of all eigenvalues;

Let us find first the functional determinant for the case of the trivial trajectory n = 0. On this trajectory x(r) = -a and, therefore, H = -d2/dt2 W'. The eigenfunctions and eigenvalues of this operator are

+

The calculation of the determinant is performed in the same way as in the previous chapter in the case of real time. The result can be obtained by substituting T --t -iT and x0 = X = 0 into (2.62):

G(-a, -a, T) =

E

(

- exp --

W:)

3.1. T H E SYMMETRIC DOUBLE WELL

51

Let us start by integrating over the vicinity of the one-instanton trajectory. This problem cannot be solved in a straightforward way, because one of the eigenvalues X, is equal to zero. Thus the amplitude (3.35) is infinite. The eigenfunction corresponding to the zero eigenvalue is called the zero mode. Let us prove the existence of the zero mode in the spectrum of fluctuations around the instanton trajectory. It follows from equation (3.12) that -5 V'(:) = 0 , (3.38)

+

or, taking the derivative of (3.38) with respect to

T

Comparing this with (3.33) we see that xo = Ax(r) does indeed correspond to the zero eigenvalue of H. To find the constant A, recall that condition (3.13)

is satisfied on the instanton trajectory. This condition gives the following form of the instanton action:

Thus A

=

l/& and the normalised zero mode takes the form xo =

%l&. The physical meaning of the zero eigenvalue in the spectrum of H becomes clear if we note that the addition of the zero mode (multiplied by a small coefficient CO)to :(r) results in a translation of the instanton by dro = co/&along the time axis:

Of course, the action is unchanged by this translation. We might guess the existence of the zero mode simply by looking at fig. 9 where it was clear that the action did not depend on the instanton position. The calculation made above was necessary only in order to find the normalisation constant for xo and to derive using (3.42) the proportionality coefficient between COand the instanton displacement dr0. It is clear now that the integration over the coefficient COin (3.34) is not Gaussian. It corresponds to formula (3.20) with n = 1 and therefore

52

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

the multiplier ( 2 . r r / ~ ~ ) in - ~ (3.34) /~ must be omitted. Taking into account these properties of the zero mode we can express the one-instanton contribution to the transition amplitude as follows:

Here the notation det' is introduced to remind us that X0 is excluded from the product of H eigenvalues, and the coefficient N is the same as in (3.34). Let us normalise this expression with respect to the trivial trajectory contribution:

+

l

det' [ - & / d ~ ~ V"(Z(T))] det [-d2/d~2 w2]

+

(3.44) -lj2

e-s

and comparing it with (3.25) and (3.26), we see that the quantity K defining the energy splitting A (3.28) is

S

+

1

w2 det' [-d2/,r2 V"(Z(T))] det[-d2/i2+w2] -L/2

"Iw&[

In this expression, the multiplier w2 was introduced to make the ratio of the two determinants dimensionless. It demonstrates that the quantity K has the correct dimension W (c.f. (3.4)). Before starting to calculate the ratio of the functional determinants, we make the following two observations. First, by turning to formula (3.45) we have reduced the normalisation coefficient N. Let us recall that the coefficient contains the Jacobian A corresponding to change of integration variables X(T) -+ Ci.It is not obvious that the values of A for the determinants in the numerator and denominator of (3.45) are the same because the eigenfunctions of the corresponding operators are different. A reason for this would be the fact that these two operators differ from each other only in the region whose size is of order the instanton size l/w << T (although the authors do not have a strict proof available). Therefore one of the conclusions to be drawn from the considered PI calculation would be a verification of the N reduction in formula (3.45) by comparison of the result with that obtained from the Schrijdinger equation. Second, we did not consider zero modes for multi-instanton trajectories. Generalisation of the result obtained for n = 1 is simple. There are n zero modes among normal modes, or that is among the fluctuations

3.1. T H E SYMMETRIC DOUBLE W E L L

Figure 11

around an n-instanton trajectory. As in the one-instanton case, these zero modes correspond to independent translations of the instanton centres. Every such mode contributes a factor ( ~ / 2 7 r ) 'J/ ~d r to the integral. Now the statement about the factorisation of the pre-exponential term K used in the evaluation of the expressions (3.25) and (3.26) seems to be 'partially proved'. It remains to check that it is also valid for the functional determinant. We do this by explicit calculation of the functional determinant in the next section.

Calculation of the Functional Determinant Let us begin our calculation of the functional determinants (3.43) and (3.45) with an evaluation of the eigenfunctions and eigenvalues of the operator H for the one-instanton trajectory. It is necessary for this purpose to solve the following equation:

Substituting the explicit form of :(t) (3.17) into V 1 ' ( x )= -4Xa2+ 12Xx2, we get:

54

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

Fortunately, this equation is well known. The potential here is called the modified Peashle-Teller potential (see fig. 11) and appears in some nuclear physics problems. We proceed, first finding the bound state wave functions. Consider the asymptotic form of ~ ( r with ) an arbitrary value of the parameter X, = X when T ---, koo. Here H = -$ w2 and we seek a solution of (3.47) in the form x(d = f , (3.48) where n = d a . Substituting this into equation (3.47) we get

+

or, introducing the new variable J = tanh $w(r - T

~ )

This can be recognised as the hypergeometric equation if we note that the highest power of the polynomials multiplying the derivatives of order 2, 1, 0 are 2, 1 and 0 respectively. In order to transform this equation to the standard form, let us make one more change of variable z = (l+J)/2. Equation (3.50) becomes

We shall seek a solution of this equation as a series in z: f = CnBnzn. In terms of the B,, equation (3.51) takes the form

Collecting the coefficient of zn, re-denoting summation indices where necessary, we obtain:

This expression results in a set of recurrence relations for the coefficients

3.1. THE SYMMETRIC DOUBLE WELL

Assuming B. = 1, we find:

All higher coefficients are zero. Simple algebraic transformations give the solution of equation (3.47) with arbitrary K :

Consider the asymptotic behaviour of this solution. If then + -1 and P(K, -1) = 1 . Thus X ( ~ M ) e-"lT-T~l at 1 + 1, and

T - TO

P ( & ,1 ) =

(K (K

+

-m. If now

- W ) ( &-

+

W)(K

-

T

T - TO +

TO +

w/2)

+w/2)

-m,

(3.57) +m, then

(3.58)

'

The other independent solution of equation (3.47) can be chosen as X ( - X ) , except for some special values of X considered below. We have solved equation (3.47), but have not found the eigenvalues yet. They are those values of X for which the solution of equation (3.47) goes to zero at T = 0 and T = T. For the discrete eigenvalues, this can be replaced by the requirement that the eigenfunctions go to zero as T - 7 0 + & m . The related error is exponentially small and vanishes in the limit T + m. This condition is already satisfied at T - T O + -m. At T - 7 0 + +m,.it is obeyed only if the polynomial P ( K ,1 ) tends to zero. This gives the values X = X , which correspond t o the discrete spectrum. There are only two such eigenvalues: X. = 0 for K = W and X I = $w2 for K = w/2. The first eigenvalue corresponds to the already known zero mode. The form of the function xo follows from (3.56)

1 anh -W 2

1'

( T - r0)

=

1 2 cosh2 ~

W ( T TO)

, (3.59)

and coincides with the already known zero mode up to the normalisation factor. Thus there is only one discrete level XI = which gives a contribution to the ratio of the determinants (3.45). Let us now consider the contribution of the continuum eigenvalues. The quickest way of finding the corresponding solution of equation (3.47)

iw2

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

56

is to perform the analytic continuation of solutions (3.56) and (3.58) to the value K = i k . Then X takes the form X = w2 k2 > w2 and the solution in the asymptotic region T - TO -+ fCO reads

+

for

T -70

--t

-CO

and

for T - TO + CO. Here

Sk

is defined as

+

+

e& = ( k i w )( k i w / 2 ) ( k - iw)(k - iw/2)

This is the phase shift acquired by the wave function of a particle scattered in the potential V"(:). An interesting property of this potential e-ik(T-T~), the is that it is reflectionless. There is no reflected wave potential effect being reduced to the additional phase shift Sk of the transmitted wave relative to the incident wave. Now it is easy to construct the wave functions obeying the zero boundary conditions at r = 0 and T = T. Subtracting from solutions (3.60),(3.61)its own complex conjugate, we get the following solution of equation (3.47) which vanishes at T = 0: X(')

=

{

sin k r sin(kr

+ &)

for 0 5 T << 7 , for << 7 5 T

.

We find the eigenvalues by applying the requirement that this function be zero at T = T.4 In terms of the wave vector k, they are

3Both the reflectionless character of the potential and the exact solvability of equation (3.47) are due t o the existence of an operator which transforms the solutions etkT of the free Scluodinger equation into the solutions of (3.47). This operator contains ~ T ~ ) There. only derivatives with respect to T and terms proportional t o tanh $ w ( fore the action of this operator cannot crcate a reflected wave epZkT'.More information about such transformations (which are based on the supersymmetry property of the given system) can be found in the rcview by L.E. Gendenstein and I.V. Krive [5] 41t should be pointed out that the continuous spectrum does not appear in the case of finite T. Nevertheless the truly discrcte levels are easily distinguishable from those which correspond t o the continuum - the former are separated from each other by thc interval w2, the latter by I/T~.

-

-

3.1. THE SYMMETRIC DOUBLE W E L L

57

corresponds to the eigenvalues of the operator in the where k?) = 2 ~ n / T denominator of (3.45). Let us now find the contribution of the continuum levels to the determinant (3.45). We must calculate the quantity

In the limit T + oo, we can neglect the terms of order 1/T2.This gives

to avoid a cumber(where we have used the notation S(k) instead of some s ~ b s c r i p t ) ~The . fact that this expression depends on n via k?) suggests that we should perform the integration over k = k?) instead of the summation over n. To this end, we note that an increase of n by unity corresponds to an increase of k by TIT. Then

This expression takes the following form upon changing the integration variable k to X = k/w and integrating by parts:

Substituting here the expression for the derivative d&/dx

which follows from (3.62). We find the explicit form of the continuum levels contribution to the logarithm of the functional determinant (See 5Thcre is a certain inconsistency in notation here. Thc eigenvalues in (3.64) are labelled by the subscript n running from n = 1. At the same time we have used n = 0 , 1 , 2 . . . in formula (3.34) in order to reserve the common notation X. for the zero mode.

58

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

Appendix A.4 for details of the calculation of the integrals appearing here)

The remaining contribution of the discrete levels to the ratio of functional determinants is:

where the unit factor stays instead of the absent X. . Gathering all the factors appearing here we obtain finally

This leads in turn to the following expressions for the quantity K (3.45) and for the value of the papameter of the energy splitting A (3.28):

and

(recall that the instanton action is S = w3/(12X)). In order to complete the calculation of the energy splitting, we have to show that the ratio of the functional determinants for the multi-instanton trajectories factorises into a product of one-instanton contributions. This is obvious for the contribution of the discrete level Al. To see it for the continuum level contribution, we recall that the potential V" of each instanton results in an additional phase shift of the eigenfunction x,(r). Thus the total phase shift is the sum of the one-instanton contributions, and the statement is proven. Our final result for A, equation (3.74), still appears different from expression (3.4). There is a W(w/2) in (3.4) instead of S and the preexponential factors in (3.4) and (3.74) differ from each other. Only the main exponential factors are identical because W(0) = S. Indeed, S(T) = W(E) - ET on the classical trajectory, where E and S(T) are the energy and action as functions of the period of classical motion in the upside-down potential (fig. 7). It is apparent that E e-"' when T t CO (see footnote 2 on page 45 ) and hence S = S(CO) = W(0).

-

3.1. THE SYMMETRIC DOUBLE WELL

59

We show in Appendix B that the difference between the pre-exponential factor in (3.74) and that in (3.4) is exactly that required to compensate the deviation of S = W(0) from W(w/2). Thus expressions (3.4) and (3.74) are identical. Note that the calculation of the determinant ratio turned out to be the central problem in the evaluation of the pre-exponential factor in the energy splitting. It is worth considering one more method of calculation which is also simpler, more elegant, and appears to be more convenient in many specific problems. The essence of this method is as follows. Let l4 and H2 = - d 2 / d ~ 2 V2 with there be two operators H1 = -d2/d? the potentials V~(T)and V~(T)having the same asymptotic behaviour as T + m . Consider two corresponding equations of the stationary Schrodinger type:

+

+

where 0 5 7- 5 T. Let xl(r), ~ ~ (be7 the ) solutions of these equations -r obeying the boundary conditions ~ ~ ( =0 ~) ~ ( =0 0) and d ~ ~ ( 0 ) l d= dx2(0)ld-r. Then the following equality is valid:

To prove this statement we note that the functions ~1 (T) and ~2 (T) grow exponentially as T -+ m unless X takes certain values. These values, at which xl(T) = 0 or x2(T) = 0, are the eigenvalues of the operators HI and H2 respectively. Hence the numerator and denominator of the expressions on the right and left hand sides of (3.76) go to zero at the same values of X. Let us now consider the ratio of the right and left hand sides of (3.76), which we call R(X), as a function of a complex variable X. It follows from the above reasons that this function has neither poles nor zeroes in the complex plane of X. Furthermore, both the right and left hand sides of (3.76) tend to unity as X + m . This is because when X + m we can neglect the potentials both by solving equations (3.75) and by finding eigenvalues of the operators H1 and H2. Indeed by making use of perturbation theory we can make sure that the corresponding first order corrections go to zero if X -+ cm. Thus we conclude that R(X) has neither poles nor zeroes on the finite part of the complex plane and goes to unity when X = m , and hence R is equal to unity for all X. Let us evaluate the determinant ratio in (3.45) using the above theo= V/'(:), = w2 and the right hand side of (3.76) rem. In this case

v

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

where K = d m (recall that x(T), x0(7) are solutions of the equations (-d2/dr2+V"-X)x(r) = 0 and ( - d 2 / ~ 2 + ~ 2 - X ) x 0 ( r=) 0 -see (3.56), (3.58)). To find the determinant ratio, it is necessary to set X = 0, but before doing that, we have to extract the zero mode contribution from the left hand side of (3.76). If X # 0 this contribution is equal to -X. As the functional determinant is the product of the eigenvalues, it is sufficient to divide (3.76) by -X in order to remove the zero mode contribution. Then expanding K = Jw2-X FZ W - X/(2w), we get

+

V"(:)] detl[-d2/d7' det [-d2/d72 w2]

+

=

l ( K - W - 2 lim X+O -X (K+ w)(n w/2)

+

1 12w2

- .

(3.78)

Thus we have again obtained result (3.72), but via a method where the calculation of integral (3.70) was not necessary. In conclusion, let us formulate a condition under which the obtained results are valid. Our first assumption was that the contribution of the fluctuations around the classical trajectory could be taken to be Gaussian. This is possible if S/fi >> 1. In the system of units where fi = 1, this condition takes the form (3.11) a = X/w3 << 1. One more approximation was used. In deriving formulae (3.25) (3.26) we assumed that the n-instanton action was S, = nS. This is the dilute instanton gas approximation. This assumption is correct if the characteristic instanton size W-' is much less than the average distance between neighbouring instantons Tin, i.e. n << W T . Then for large values of n, the instantons overlap and the expression for n-th term in formulae (3.25) and (3.26) is not applicable. Nevertheless it would not affect the result if the terms with n W Tdid not contribute to the sums (3.25) and (3.26). In order to find out if this is the case, we must find the terms in (3.25) and (3.26) which give the greatest contribution to the sum

Applying Stirling formula n! (n/e)n to n! (see (2.40)), we get an estimate ( e a . Tin)" for the n-th term of (3.79). Taking the derivative with respect to n, we find that the greatest contribution to this expression is made by the terms with n = nm,, = A . T. The trajectories with such

3.2. PERIODIC POTENTIAL

61

number of instantons give the maximal contribution to the transition amis called plitude (3.25), (3.26). For this reason the quantity A = n,,,/T density of instantons. Using its explicit form (3.4), we come to the validity criterion for expressions (3.25), (3.26) in the form exp(-w3/12X) << 1. It is obviously fulfilled if a = X/w3 << 1. Finally, we would like to make one more note. The above introduced quantity a is usually called the coupling constant. This quantity is the dimensionless parameter of the perturbation theory. This theory gives corrections to the energy and the wave functions of the particle localised in one of the potential wells in the form of a series in powers of a , in which the zeroth order term corresponds to a pure harmonic oscillator. We have obtained the correction to the ground state energy which turned out to be proportional to exp(-Const/a) . A remarkable fact is that this expression is not analytic at a = 0. Indeed all the derivatives dn exp(l/a)/dan vanish at this point. This means that it is impossible to reproduce the above obtained result by calculating the perturbative corrections to the oscillator energy for the wave function localised in one of the wells. In this sense it is reasonable to call the method and the result obtained here nonperturbative. The method used here has a more specific name semiclassical approxzmation which emphasises the use of the nontrivial classical trajectories. Indeed we managed to catch the nonperturbative energy splitting thanks to the fact that we had already taken into account the nontrivial multi-instanton trajectories in the zeroth approximation. At the next order, it was necessary to include the corrections due to the integration over the vicinity of the multi-instanton trajectories. This was possible only when we neglected higher variations of the action, which was correct under the condition a << 1. Thus both the perturbation theory and the semiclassical approximation are valid under the same condition, but they give information about the essentially different contributions to the energy levels in the double well potential. We would like to note that any other method of PI calculation which does not require smallness of the interaction constant is named nonperturbative as well. -

-

3.2

A Particle in a Periodic Potential. Band Structure

Let us consider one more quantum-mechanical problem: finding the spectrum of a particle in a periodic potential. This problem is of particular interest because such a potential is a starting approximation in the description of the movement of electrons and holes in the periodic crystal

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

Figure 12

field. Let us recall the known solution of the problem. The periodic potential has the property V(x) = V(x 2 m ) , n = 0, fl , f2 . . . (see fig. 12). Here X is supposed to be dimensionless - this can be done using some parameter a, such that xa has the dimension of length. We can consider V(x) = 1- cos X as an example, but all the results to be obtained below are true for any potential with one deep minimum in the period. As in the case of the double well potential (3.9), we can expand V(X)near one of the minima X = X,, assuming that the ground state energy is less than the barrier height:

+

where w2 = V"(x,)/a2 is the vibration frequency. The tunnelling effects between neighbouring minima are small in the case under consideration, as well as for potential (3.9). Neglecting the tunnelling gives an infinitely degenerate ground state. This reflects the fact that the particle can find itself in any minimum of the potential. A nonzero tunnelling probability removes the degeneracy, and the corresponding wave functions are linear combinations of nonperturbed wave functions xo(x) at every minimum:

3.2. PERIODIC POTENTIAL

63

where the parameter 8 labels different variants of such a superposition. The coefficients Cn(0) in (3.81) should be adjusted so that the states ~0 are eigenfunctions of the translation operator X + X 2 ~ because , this operator commutes with the Hamiltonian. This implies the Bloch theorem:

+

xo(X + 2

~ =) eisXO(x)

,

(3.82)

and hence C,($) = einO.It is easy to check that equation (3.82) gives the eigenfunctions with eigenvalues

Thus the ground state is split into an infinite number of sublevels, known as band. We can find A explicitly for a given V(x) using the standard quantum mechanical methods. Our aim is to re-obtain expressions (3.82) and (3.83) for the case of small tunnelling probability, by the instanton method. Let us calculate the probability amplitude of a particle transition from the minimum x(0) = X, to another one x(T) = X, in Euclidean time T:

As above, we find the stationary points of the action which dominate this amplitude. These are, of course, the multi-instanton trajectories. The one-instanton trajectory in this case corresponds to the transition from the nth minimum to the ( n + 1)th and the anti-instanton trajectory brings the particle back. There is exactly one zero mode in the fluctuation spectrum in the vicinity of each instanton. This follows, as in the previous case, from the action independence from the instanton position. There is only one essential difference between the case under consideration from that of the double well potential. The restriction that the instantons and anti-instantons follow each other in an alternate order is now relaxed. What is important is the difference between the number of instantons k and anti-instantons k', k - k' = m - n z 1. This is defined by the spacing of the points X,, X,. Thus taking into account all trajectories with k instantons and k' anti-instantons, we can write in analogy with equations (3.25), (3.26):

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

64

where Zo is the oscillator trajectory contribution which is the normalisation factor as above, and the quantity K is defined by (3.45):

(in dimensionful units). The Kronecker symbol 6k-k',lis included into sum (3.85) in order to restrict the difference between the numbers of instantons and anti-instantons. If we take into account that 2T

d6'

6-u,~ = /gexp (i(k - k'

- 1)0),

then the summations over k and k' in (3.85) decouple and can be done separately:

2.rr

= 20

J,

e-'l@

A.T exp ( e g X ) enp ( e

A.T

-"B_ZT;)

0

where we again used the splitting parameter A (3.28). By substitution of the explicit form of the oscillatory amplitude (3.23), we finally obtain:

(3.89) To find the energies and the wave functions we are interested in, it is necessary to interpret this formula as an expansion of G in the Hamiltonian eigenfunctions (3.29). This yields the energy of the lowest energy band in the correct form

Thus we obtain a continuous energy band of width A, which is parametrised by the angle 0 5 6' 5 27r. It can easily be seen from (3.89) that the corresponding wave functions in fact are the Bloch waves:

3.3. A PARTICLE ON A CIRCLE

3.3

A Particle on a Circle

Let us consider another interesting case of tunnelling which is commonly used t o illustrate instanton calculus in &CD [3, 61. Let the particle considered above move on a circle instead of on the infinite line. This implies periodic boundary condition, that is, an identification of the points X = 0 and X = 27r. In the problems considered above the minima of the potential V(x) has been physically distinguished. Starting with an infinite number of states in the minima of V(x), we have obtained an infinite number of their superpositions and these have formed the energy band. In contrast, there is now a single minimum of V(%).This means we shall not find any energy band. Instead we will find a non-perturbative correction to the ground energy. We can guess the answer if we impose the periodic boundary condition on the wave function (3.91). This implies 9 = 0 in (3.90). Let us see how the instanton technique leads to this result. In the dilute gas approximation the shape of the instanton as well as the functional determinant are not sensitive to the boundary conditions. It is only the summation over the number of instantons (3.85) which needs to be modified. The instanton is now a trajectory which runs round the circle such that the initial and final points are identical. Thus there is no restriction on the number of instantons and anti-instantons in (3.85). We now omit the symbol 6&k',l in (3.85) and easily perform two independent summations over the number of instantons and anti-instantons. This results in formula (3.90) with 9 = 0. With this result in mind, we can speculate that if we impose a quasiperiodic boundary condition (3.91) with a specific value of 9 then we obtain as the result one level (3.90) from the band (3.89) selected by the given value of 9. Interestingly, quasiperiodic boundary conditions on a circle are not nonsense. A system with such boundary conditions does exist. Suppose that the circle is a real circular wire in three-dimensional space with a charged particle moving on it and that there is a solenoid which crosses the circle plane somewhere inside the circle shown in fig. 13. If the solenoid is thin enough, then there is no magnetic field on the circle. There is, however, an effect due to the vector-potential A. As the particle moves round the circle, it acquires a phase factor exp ( i e

f A ~ Z )= exp (iem)

,

where 2 is the length element on the circle, e is the charge of the particle, and @ is the total magnetic flux through the surface spanned on the circle.

CHAPTER 3. THE EUCLIDEAN PATH INTEGRAL

Figure 13

Thus we have 6 = e a . Let us make appropriate changes to the instanton calculations. We start with the classical Hamiltonian H of the system in real time which takes the form H(p,x) = (p - eA)2/2 V(x), where p is the canonical momentum and A is the component of A tangential to the circle. The velocity dxldt is now dxldt = dHldp = p - eA. Performing the Legendre transformation to obtain the Lagrangian L = px - H , we come to dx L = - - +eA- - V(x) . 2l (dx)2 dt dt

+

After turning to imaginary time, the Euclidean action (denoted now as So)takes the form

The last term here is a total derivative. Let us assume for simplicity that A does not depend on X (otherwise we have to apply a gauge transformation to make A an X-independent function). This term then takes the form

2 ~ e A= e a , and the value [x(T) - x(0)]/27r is equal to the where 8 number of turns of a given trajectory around the circle. This term is called the 6-term. It does not affect the classical equation of motion, but

3.4. CONCLUSIONS

67

it changes the quantum properties of the system which are sensitive to the value of action. Indeed, the action of an instanton SI now takes the form S1 iQ while the action of an anti-instanton is SI - iB. Performing the independent summation over the numbers of instantons and antiinstantons, we obtain the expected form of the ground-state energy:

+

where B is now a fixed parameter. This contrasts with (3.91). The generalisation of this ground state in Yang-Mills theory is called the B-vacuum.

3.4

Conclusions

The examples considered in this chapter have their own value in quantum mechanics and solid state physics. Besides, they may serve as simple models of nontrivial vacuum structure of quantum field theories (the term vacuum is used to denote the ground state of the quantum fields). Though simple, such models possess the key feature of much more complex systems tunnelling between different vacuum states. The most common approximation to describe tunnelling in field systems is to use the instanton formalism. Our analysis may help to understand its basic idea, the main calculation steps, and the restricted applicability of this method. In the next chapter, we start studying the quantum mechanics (and statistical physics) of fields. These are systems with an infinite number of degrees of freedom. We shall use the path integral as a working tool rather than a subject of investigations. The experience gained in this part will be used in quantum field theory calculations. -

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Bibliography The Bohr-Sommerfeld quantization rule has been obtained in the PI formalism by R. Dashen, B. Hausslacher and A. Neveu, Phys. Rev. D10 (1975) 4136. It was V.N. Gribov who realized the connection between the instanton trajectories and the tunnelling transitions in quantum mechanics. In this chapter we drawn on the review by A.I. Vainshtein, V.I. Zakharov, V.A. Novikov and M.A. Shifman, Sov. Phys. Usp. 25 (1982) 195. Other good reviews are the following two papers S. Coleman, The Uses of Instantons, Lectures given at the 15th Course of the International School of Subnuclear Physics, Erice, Italy, 1977; Reprinted also in: S. Coleman, Aspects of Symmetry, Cambridge, University Press, 1985.

J. Zinn-Justin, The Principles of Instanton Calculus: a few Applications, Lecture delivered at Les Houches Summer School 1982. Published in Recent Advances in Field Theory and Statistical Mechanics, eds. J.-B. Zuber and R. Stora, North- Holland, 1984. L.E. Gendenstein and I.V. Krive, Sov. Phys. Usp. 28 (1985) 645. R. Rajaraman, Solitons and Instantons in Quantum Field Theory, North-Holland Publ. Co., Amsterdam, 1992.

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Part I1

INTRODUCTION TO QUANTUM FIELD THEORY

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Chapter 4 Classical and Quantum Fields The main purpose of this book is to introduce the reader to the characteristic features of quantum systems with very large or, in the limit infinitely large number of degrees of freedom. Such systems are called quantum fields. As it will be shown below, there is a transparent analogy between the Euclidean form of quantum mechanics and statistical physics for reason we discuss statistical properties of the fields as well. In this chapter we consider some simple examples of physical fields and related subjects such as the variational principle and quantization rules. There exist two main recipes of quantization: canonical formalism and PI formalism. We shall discuss both, but use mainly the latter in the following chapters. As a simple example we shall consider small field fluctuations near a minimum of the field energy. In the language of quantum mechanics, it will give excitations above the ground state. As it will be shown below, these excitations look like particles. It should be stressed that neither particle physics, nor condensed matter physics, nor any other branch of physics where the methods of quantum field theory (&FT) are used is the subject of this book. The aim of this book is to study the corresponding formalism as it is, especially those features which are independent of (or only slightly dependent on) the physical context. At the same time, we shall use examples from particle physics and other disciplines for the sake of illustration. -

4.1

From Large Number of Degrees of Freedom to Particles

We start with a toy model of a mechanical system consisting of a large number s of small balls (material points, or 'atoms1) of mass m which can

74

C H A P T E R 4. CLASSICAL AND QUANTUM FIELDS

move along a straight line and are connected to each other by springs (fig. 14). It is clear that the i-th ball vibrates near its equilibrium point xi, i = 1,2,. . . S, s + oo. Take the points xi to be situated along the X axis with spacing Ax and the i-th ball's deviation from its equilibrium point $i. Each ball is held near its equilibrium point by a corresponding spring with potential energy W1(q$) (shown vertical1 in figure 14, although only the horizontal force component is essential). Another kind of spring mediates an interaction between the neighbors with potential energy W2((4i+l &)/A%). Here A s is introduced to define W2 in a more convenient way1. .. Taking into account these preliminary remarks we can write down the action of the model in the following form: ,

The dynamics of the system are described by a set of s Euler-Lagrange equations

or, explicitly,

where prime denotes as usual the derivative of a function with respect to its explicitly written argument. We can also use the Hamiltonian formalism introducing s canonical conjugate momenta pi = dL/a& = m&. The corresponding Hamiltonian is:

We shall consider long-wave excitations in this system. That is, excitations such that the characteristic length at which $ changes significantly is much greater than the distance between neighboring atoms 'It is worth mentioning that models of this kind possess rather nontrivial properties despite their simplicity. If the potential energy is quadratic with a small cubic term then the model becomes the so called Femi-Pasta-Ulam system [l]whose evolution is one of the first nonlinear problems solved on a computer. It was surprisingly observed that there was no equipartition of energy among the modes of this nonlinear system.

4.1. PARTICLES

Figure 14

-

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

-

where

Thus in the continuum limit the set of the discrete canonical variables 4% becomes a field #(X,t), which is a continuous canonical variable defined for any coordinate X . It is also convenient to introduce a Lagrangian density L such that L = JdxL as is done in (4.5). The equations of

76

CHAPTER 4. CLASSICAL AND QUANTUM FIELDS

motion in the continuum limit can be derived from (4.3) by taking the limit Ax L: 0:

It is obvious that as S + m the system of equations (4.3) becomes a partial differential equation (4.7). This equation can of course be derived by variation of the action (4.5). For this purpose let us consider small arbitrary variations 64(x, t) of the field $(X, t ) , for which S$(X,t ) = 0 for t = 0, t = T, and X L: fm , and set the first action variation to zero:

For the sake of compactness of the formulae here the notation d, r a/dx is used. Integrating by parts and taking into account that the field variations are zero on the boundaries of the integration volume we get:

The arbitrariness of the variation 64 leads to the Euler-Lagrange equations (4.7). Let us make our toy model (4.5) more realistic to describe a crystal. For this purpose 4 should be thought of as the atom displacement in the elementary cell. Hence 4 must be at least a three-dimensional vector, or more exactly 3n-dimensional vector where n is the number of atoms in the elementary cell. The Lagrangian becomes a function of all the derivatives di4 &b/dxi.As the displacements are measured from the corresponding equilibrium points of atoms, the deformation energy T must have vanishing first derivatives at these points. Furthermore, there are a lot of cases where relative deformations of the elementary cells are small. In such cases we can restrict ourself to the leading term in the expansion of the deformation energy, which is

The next terms can be easily taken into account if necessary. The tensor of second derivatives in (4.10) can be diagonalized. Let us consider for

4.2. ENERGY-MOMENTUM TENSOR

77

simplicity only the simple elementary cells for which the eigenvalues of this tensor are the same and equal t o U. Then the action takes the form

where the usual rule of summation over repeated indices is assumed and the subscripts labeling different components of 4 are omitted in formula (4.11) for the sake of simplicity. Equation (4.11) is a starting point in the description of vibrations of a crystal which look like particles called phonons upon quantization. In many cases, especially in particle physics it is convenient to set the coefficients in front of the first and second terms of the integrand of (4.11) to unity. In particle physics, it makes manifest the necessary Lorenz invariance of the theory. Then the action (4.11) can be written in the form (4.12) S = d4x [i(a,cp(X))2 - UMX))]

1

which will be widely used below. Note that the subscript p = 0,1,2,3 of the 4-vector X, is omitted and summation over p in the first term is assumed. In other according to the formula ( 8 , ~ )=~ (docp)2words we use the metric (1,-1,-1,-1). The simplest case of the onecomponent field cp which is a Lorenz scalar will be further discussed up to chapter 9. By varying the action (4.12) we get the following equation of motion:

Here the d'Alambert operator = d,d, = 8; has been introduced. If U = 0, this equation describes waves with unit velocity. In particle physics this is the speed of light; in solid state physics it may be the sound velocity.

4.2

Energy-Momentum Tensor

The physical processes described by the action (4.12) occur equally at any point of space-time. If the system is situated in an external coordinate dependent field, this invariance is lost. According to the Noether theorem such invariance leads to the conservation of the field energy and momentum. Indeed, infinitesimal displacements

78

CHAPTER 4. CLASSICAL AND QUANTUM FIELDS

change the Lagrangian density

On the other hand if the Lagrangian does not depend on coordinates then where Sp = p ( x , + E,) - ( ~ ( 5 , = ) E,~"(P(x,). One can combine these equations using the equations of motion (4.13) to obtain

&L

(W v v >(0))

= ~ , d , -aL --

By the arbitrariness of c,, it follows that

where

is the conserved energy-momentum tensor of the field p. Integrating the conservation law (4.19) over the whole space, we obtain the explicit form for the four conserved quantities

where P, = J dx 0: is the relativistic energy-momentum vector. We can make sure that PO really is the field energy by comparing it with the Hamiltonian defined in classical mechanics for the field p ( x ) . Substituting action (4.12) into the definition (4.19), we obtain the explicit form of the energy-momentum tensor of the scalar field:

This expression gives for the energy

It is clear that this quantity is positive if the self-interaction potential U ( p ) is. The field momentum is

4.3. FIELD QUANTIZATION

79

Let us note that the energy-momentum tensor (4.21) is symmetric, although the initial formula (4.19) does not guarantee such symmetry generally. Besides this, the tensor is not uniquely defined by (4.18), because any divergenceless quantity can be added to O,,. A detailed discussion of these two types of energy-momentum tensors (which are called metrical and canonical respectively) and the transformations mentioned here can be found elsewhere (see e.g. [2]). We skip this subject and turn t o quantization of the system described by the action (4.12).

4.3

Field Quantization

In this section we consider the quantization procedure for the system whose classical action is given by (4.12). As was already mentioned in this chapter there are two recipes for quantization. The canonical formalism takes operators with some fixed commutation relations in correspondence to dynamical variables. The other method uses the path integral. Although the main tool of this book is the PI, we discuss the canonical formalism in this section.

Canonical Quant ization Let us turn again to our toy model (4.1). As it has a finite number of degrees of freedom it can be immediatly quantized by making use of the Schrodinger equation for the wave function \Ir(4l, $2 . . . 4,; t ) depending on S canonical coordinates q5i and time:

Here %! is the Hamiltonian operator derived from the classical Hamiltonian function by substitution of the canonical coordinate and momentum operators therein. The explicit form of the canonical operators depends on which representation is used. In particular, it is convenient to use the coordinate representation which presupposes that the wave function depends on &. In this case the action of the operator & reduces to a multiplication by +4i and the operators pi are defined as follows:

The same quantization recipe can be also formulated as a set of canonical commutation relations which do not depend on the choice of repre-

80

CHAPTER 4. CLASSICAL AND QUANTUM FIELDS

sentation. It is obvious that in the coordinate representation

because of (4.25). As the commutation relations are invariant with respect t o unitary transformations this expression is valid in any representation. Similar commutation relations hold for all other pairs of canonically conjugate variables. It should be pointed out that in quantizing system (4.1) in terms of 4i and pi we are not faced with the operator ordering problem mentioned in the second chapter, because Hamiltonian (4.4) does not contain 'dangerous' products of coordinates and momenta. Now let us take the continuum limit Ax -+ 0. It is convenient to introduce a set of new canonical momenta Q = n / A x , the densities of the old ones. Then the Hamiltonian (4.4) takes the form

The commutation relations (4.26) which in terms of form

[v.3,Pz. l = - '

&j 2-AX

and $ take the (4.28)

l

become [v(xl),$(x)]= -iS(x-X')

(4.29)

.

Here the S-function appears as the limit of Sq/Axi for Axi is in agreement with its functional definition

-t

0, which

If instead of the Schrodinger picture we use the Heisenberg one, where the time dependence is transfered from the wave functions to the operators, the commutation relations for p = $& and n = v/& are

These equal-time canonical commutation relations are the key idea of quantum field theory. They define, as in quantum mechanics, an algebra of operators acting upon the field wave function 9.As before this wave function can be interpreted as a vector in a linear space describing a state of the given physical system. This quantity Q[p(x,t)] is clearly a functional of p(x). In the coordinate representation the field operators act

4.3. FIELD QUANTIZATION

81

on Q[cp(x,t)] as multiplication. The action of the momentum operators is defined by the relations

in accordance with the commutation relations (4.31) (compare with (4.25)) Thus the quantum field evolution corresponding to action (4.12) is completely defined by the solution of the Schrodinger equation for the wave functional

This equation is essentially more complicated than its quantum-mechanical analogue. At the same time, the PI in &FT does not look so frightening compared with that of quantum mechanics.

Quantization via Pat h Integrals Analogously to the derivation of the Feynman formula (2.27), let us write down the probability amplitude for the transition between the initial cp(x, 0) and final cp(x, T) field configuration in time T:

In this paragraph, the Planck constant is restored. The quantity G is the Green function (or more strictly functional) of the Schrodinger equation (4.33). It can be expanded as a series of the eigenfunctions a, corresponding to the Hamiltonian of equation (4.33) as is done in quantum mechanics:

Here E, is the field energy in the state labeled by the index a. The derivation of formula (4.34) is competely equivalent to that of expression (2.27). Nevertheless let us point out that the PI over the fields cp(x, t) is thought of as the limit of the product of conventional integrals for each point of a discrete space-time lattice. Then, after calculation of these integrals the continual limit should be taken. We shall use another definition of the PI below. The system is placed in a large, but

82

CHAPTER 4. CLASSICAL AND QUANTUM FIELDS

finite 4dimensional volume VT. The Fourier components of the field then become discrete, and we integrate over each of them and take the continuum limit thereafter. The necessity of such tricks is due to the rather formal character of the expressions (4.33), (4.34) and (4.35). A mathematically correct definition of these expressions is problematic. Nevertheless, the practical calculations presuppose (as a rule) that the system to be described is the limiting case of a large but finite number of degrees of freedom. Sometimes it is literally the case, e.g. for crystals. This makes possible a direct generalization of all quantum mechanical relations (particulary the formulae (4.33), (4.34), and (4.35)) to &FT. It was shown in the second chapter that the PI approach is completely equivalent to the Schrodinger approach. It enables us to regard equation (4.34) as a postulate which yelds the quantization rule without reference to the Schrodinger equation. Further we assume that all results concerning the PI in quantum mechanics are true in &FT2. If it is necessary to specify the mathematical meaning of the integral (4.34), we shall do it explicitly in the given case. Let us consider the analytic continuation of expression (4.34) in Euclidean space. The effectiveness of this method was shown in chapter 3, at least for the ground state energy calculation. The substitution T = -ZTE transforms the Green function of equation (4.33) into

The same substitution in (4.34) gives the Euclidean form of the PI:

where the Euclidean action SE is

Euclidean time here is assumed to be simply an extra spatial coordinate r xq, which varies from 0 to TE. The metric of this space is trivial a;. any $-vector squared is (a,)2 = a: a; a:

+ + +

20ne should keep in mind that a direct application of formula (2.27) or its QFT version (4.34) is impossible if there are cyclic coordinates in the theory. Such a situation takes place in gauge theories whose quantization scheme demands modifications [3]. We discuss this problem in the chapter 11.

4.4. THE EQUIVALENCE O F QFT AND STATISTICAL PHYSICS83

4.4

The Equivalence of QFT and Statistical Physics

Note that the action (4.38) is formally similar to the three dimensional field energy (4.22)

where i = 1,2,3. The difference is in the dimension of the integral and the dimension of the gradient appearing in it. This analogy makes it possible to establish a correspondence between &FT and statistical physics for a field system. To do this let us consider the central quantity of statistical physics, the partition function:

Here the summation is taken over the complete set of quantum states, as in expression (4.36), kB is the Boltzman constant, 0 is temperature, and F is called the free energy. It can be seen that if we assume in (4.36) that

then the exponential factors in (4.36) and in (4.40) are the same. The only difference between these expressions is the presence of the wave functionals corresponding to the initial and final field configurations in (4.36), (4.37). To get rid of them let us impose the periodic boundary conditions cp(xo,0) = p(x, TE) in the propagator (4.37) and perform an integration of (4.37) over cp(x,TE) at each point X (that is a functional integration in three-dimensional space). After this the wave functionals disappear from (4.37) because of the normalization integral

which generalizes the usual quantum-mechanical normalization condition. Problem: Write explicitly the quantum mechanical analog of this formula.

84

CHAPTER 4. CLASSICAL A N D QUANTUM FIELDS Thus we obtain

The meaning of the derived expression is simple. For the quantummechanical propagator (4.36) it is necessary to fix the initial and the final field configuraion before integrating over the field values at each intermediate time point. The initial and the final field configurations may differ from each other. As distinct from this case, both configurations in statistical physics are identical. In other words it is the same point in the function space over which the integration should be done. This means that the evaluation of the partition function presupposes all fields to be defined on a 4-cylinder R3 X S1where the perimeter of the circle S1 is r = h / k B 8 . Thus, the expression for the partition function takes

where integration is taken over all periodic field configurations on the time interval 0 r < TE = ti/ksO. As a rule, this commonly known fact is not reflected explicitly in the written form of expressions of type (4.44). It should be noted also that when evaluating partition function (4.44) the numbers of integrations over the canonical coordinates and momenta are equal in the Hamiltonian formulation of the PI similar to (2.25). Expression (4.44) makes it possible to trace out the transition from the quantum to the classical partition function as the temperature increases. According to (4.41), I9 + oo if T -+ 0. If T becomes less than any characteristic oscillation period of the system, the main contribution to the integral is given by static field configurations. More strictly, we must expand the field in a Fourier series in time

<

and substitute it into (4.38). This gives

4.4. THE EQUIVALENCE OF QFT AND STATISTICAL PHYSICS Now it is apparent that (independently of the specific form of U) the action becomes very large in the limit T + 0 for all terms but those corresponding to n = 0, i.e. to the static configurations. Hence the expression takes the form of the partition function for a classical system

In the last expression the functonal integration is taken over all field configurations in three-dimensional X-space and the argument of the exponent is the field energy divided by k ~ 0 . The above reasoning implicitly assures the covergence of all formal series appearing in the theory. However, this is not the case in quantum field theory. We shall study divergences in the following chapters, and now we formulate (without a proof) a theorem which enables us to obtain the correct classical limit. It is the so-called decoupling theorem [4]. First, let us introduce the necessary concepts. As can be seen from (4.46), the contribution to the action due to p, for n # 0 is large, being of the order of SE/T -- 1/T2. Conversely, the typical energy of the time-independent mode p" remains finite in the limit T -+ oo. The modes p, for n # 0 are called the hard sector of the theory, and cpo is called the soft sector. The theorem asserts that functional integration over the hard sector of the theory (with characteristic energy Ehard)leads only to a renormalization of the soft sector (characteristic energy ESoft)with a relative accuracy (Es,ft/Eha,d)2 provided the soft sector is renormalizable by itself. The renormalization means in fact a change of the numerical values of the parameters, say those of U, and the renormalizability means that all divergencies can be collected into the same parameters. We shall consider these properties in much more detail in the following chapters.

Problem: Obtain an expression for the free energy of the one-dimensional harmonic oscillator at finite temperature 0 by both (i) a straightforward summation over the oscillator energy levels in definition (4.40) and (ii) by applying the above described procedure to formula (2.61). Explain the result for kBO << fiw. Problem: Consider the same problem for the anharmonic oscillator with an arbitrary potential V ( x )and show that for kBO >> fiw the result

86

CHAPTER 4. CLASSICAL AND QUANTUM FIELDS

coincides, a s could be expected, with the classical expression

where xo is the minimum point of V(x);

Problem: Consider the first quantum correction to the free energy of the harmonic and anharmonic oscillators using the technique developed in the previous chapter. What are the limiting forms of the expression at high and low temperatures?

4.5

Free Field Quantization: from Fields to Particles

Let us consider small oscillations of the field (4.12) near the equilibrium point of U(cp). We may expand U(cp) in powers of the deviation from this point and neglect all powers higher than the second. Without loss of generality, we may assume the minimum of U(p) to be located at the point cp = 0. Then the action (4.12) takes the following form

The field described by this action is called the free field. As will be seen below, this model is as important in quantum field theory as the harmonic oscillator is in quantum mechanics.

Momentum Space The action (4.49) can easily be diagonalized. To do this, we must represent the field as Fourier integral over the set of spatial plain waves. For correct calculations the system must be placed in a box of a large but finite size, imposing suitable boudary conditions upon the field and sending the size of the box to infinity in the final expressions. The natural boundary conditions are those which require the vanishing of the field on the boundaries. Periodic boundary conditions are also convenient. They correspond to the field being defined on a three-dimensional 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 6-its 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) sin@)

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.

Zero-Point 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 zero-point 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 zero-point 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 space-time. 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 S-function. 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 low-lying 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 time-independent). 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 non-linear problems, Tech. Rep., LA-1940, 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, Springer-Verlag, 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 so-called 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 hydrogen-like 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 one-dimensional 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 k-th 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 one-dimensional 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 atomic-size 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 cut-off 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 cut-off 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 self-consistent. 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 one-dimensional 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 finite-dimensional 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"dk-27r[ ~ + l n a + l n 0

(1 - e-2"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 cut-off parameter. This is known as cut-off 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 self-interaction 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 so-called 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 4-dimensional space. For a derivation, let us consider the following auxiliary Gaussian integral in n-dimensional 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 self-interaction 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 9-field 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 8-digit 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 p-dependent 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 A-independent. 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 S-lcp(s2, st), where S is an arbitrary constant, is also a solution of this equation. We say that there is no mass (or, equivalently, length) parameter in the theory, therefore any solution can be scaled with an arbitrary S. The presence of the dimensionful parameter M in the theory means that there is a special scale of mass, and hence of length. The generation of the mass parameter M by the quantum corrections in

CHAPTER 5. VACUUM ENERGY IN

v4

THEORY

Figure 16

QFT is called dimensional transmutation. As we shall show, the scaling law still holds in this case, but it takes a rather nontrivial form. There is another peculiarity of the effective potential under consideration: it has a nonzero minimum at cp = cpmin obeying

X In ---M2

327r2 11X +and, therefore, 3 3

- - ---

(see fig. 16). The minimum of the effective potential corresponds to the field which can exist in the absence of the external current, because in this case X J e f f / d c p= J = 0. If SO, the mean field in the vacuum is cpmin or +,in. Let us note in advance that this phenomenon, called dynamical symmetry breaking, should be distinguished from spontaneous symmetry breaking which occurs at the classical level by the special choice of the potential e.g., U(cp) = X((p2 - cp&i,)2. We shall discuss spontaneous symmetry breaking later. Here, we would like just to note that both cases give the same result: cpmin # 0 becomes the true vacuum and all excitations above it acquire a mass proportional to Icpol. There are two possible realisations of the system (with cp = fcpmin) since its action is an even function of cp. However it would be premature to conclude that dynamical symmetry breaking does take place in the considered case. The minimum in formula (5.68) is beyond the scope of its applicability. Indeed it ap-

5.2. EFFECTIVE POTENTIAL O F (p4 THEORY

119

pears from the balance of two terms: the classic one proportional to X and the quantum one proportional to X2. But expression (5.68) is derived under the assumption of smallness of the coupling constant i.e., X2 << X << 1. The second term in (5.68) remains a small correction only if I X ln(p/M) I<< 1. Hence expression (5.68) is valid only in the vicinity of the renormalization point p = M . The minimum p = p,i, is out of this range. As we shall see below, this minimum disappears in a more accurate approach3. Let us analyse further the dimensional transmutation. From a technical point of view, its origin is clear when evaluating the effective potential, we regularized the divergent integral (5.57) by assigning an upper integration limit km,, = A >> (U")'/'. It was this momentum cut-off which introduced a large mass-scale into the theory. After the renormalization, it was eventually replaced by the finite parameter M , the renormalization point. This poses the question: is this mass generation inherent to the theory or may it be an artifact of the calculation procedure? In attempting to answer this question, we note that the physically consistent result should not depend on the choice of the renormalization point. Let this take another such point M' # M. The effective potential should be the same but the value of coupling constant (5.67) changes. (p;X, M ) and hence, Let this be X' = X(M1). Then Uef(p;X', M') = Ueff -

as follows from equation (5.68). Here, as was done above, we do not distinguish X and X' in the correction term, which is of the order of X'. We can rewrite (5.69) in the form

which leads to the following condition:

31t will be shown in the chapter 12 that dynamical symmetry breaking does take place in thc model where thc field (o interacts with a vector (gauge) field. This is thc so-callcd Coleman- Weinberg effect [ 6 ] .

120

CHAPTER 5. VACUUM ENERGY IN cp4 THEORY

where a new constant A with the dimension of mass has been introduced4. It is very large at X << 1 because from (5.71)

It is important to note that the value of this constant does not depend on the choice of the renormalization point and thus represents the fundamental mass scale of the theory. Let us make one more remark on the dimensional transmutation. The fact that taking quantum corrections into account introduces a mass scale reflects the sequence of operations we have made in calculating the effective potential. We started with the classical action with m = 0 and then we found that the quantum correction depended on the mass parameter M. Such a sequence is in some sense just the opposite to that which takes place in nature. Physically observable quantities include the sum of all quantum corrections. A system is considered to be classical if all such corrections are negligible. From this point of view, the specific mass scale, A, disappears in the classical limit. The task of &FT is to restore the observables to the full scale range, using the classical limit as the zero order approximation. Recall that formulae (5.69), (5.68) are valid only if the second term on the right hand side of (5.69) is much smaller than the first one, so that only a finite interval around the renormalization point can be considered. Nevertheless, the interval of applicability of (5.69) can be expanded for the following reason. If the value of the field cp = M' does not satisfy the condition X(M) ln(cp/M) << 1, then we can choose another renormalization point, say M", which does. The value of the effective potential is in both cases the same. We are assuming here that we can choose the classical part of the potential in different ways by choosing different values of X. That is, we can assert as 'classical' some proportions of the total value of X depending on the renormalization point M . In order to do this, we shall consider infinitesimal variations of the renormalization point in (5.69) M' = M+dM. Introducing a new variable t = 1n(M1/M), we obtain 3h2 (t) dt X(t dt) = X(t) -1 16r2

+

dX dt

- = p(X),

+

where

3X2 P(X) = ---. 167r2

4There is no relation between this constant and the cut-off parameter.

5.2. EFFECTIVE POTENTIAL OF (p4 THEORY

Figure 17

This is the simplest example of the so-called renormalization group equation (it can easily be checked that the transitions from one renormalization point to another possess the group property). Integration of this equation gives the coupling constant behavior for finite changes of M :

Returning to the initial notation, we obtain the expression for X(M) in the form

The function X(Mf) in (5.76) is called the running coupling constant. It is the value of X which is observable as the coupling constant when cp = M'. If it is chosen, the quantum correction to the effective potential is zero at cp = M' and small in the vicinity of this point. It is still necessary for the applicability of the calculation method that X << 1. Formula (5.69) represents the first two terms of the expansion of (5.76) in powers of X(M) ln(Mf/M) << 1. Formula (5.76) makes it possible to reach the point M' = 0. It follows from (5.76) that X(0) = X(M)/(l oo) = 0 or, in other words, the interaction vanishes at M = cp, = 0 and the calculation method is valid

+

122

CHAPTER 5. VACUUM ENERGY IN q4 THEORY

up to cp = 0. To obtain the effective potential it is sufficient to recall that X is its fourth derivative. Integrating (5.76) four times with the initial conditions U;jf(0) = U;ff(0) = Ukff(0) = 0 we can obtain the improved expression for Ueff(p). It has a minimum at cp = 0, because X(p) is everywhere positive. This procedure does not help us to extend the result to large values of X because expression (5.76) has a pole at M' = A. Near this pole, the value of X is large and our calculation method fails. This pole is called the Landau pole. It indicates the region in which the theory becomes strongly coupled. The classical form of the effective potential is no longer a reasonable approximation there. Let us discuss another interesting consequence of formula (5.76) - a nontrivial form of the scaling law. The classical action S[cp(~, t ) ]=

d 4 ~ ( k ( a u ~+) $291)

is invariant with respect to the scale transformations:

for an arbitrary value of the dimensionless parameter S . This scaling law is trivial in the sense that it follows from the metric dimension of the variables. In particular, the coupling constant does not vary under such transformations because it is dimensionless. But at the quantum level, the metric and scaling dimensions of X become different. As follows from (5.76), the transformation cp -+ p' = scp results in the following change in the coupling constant:

Thus the coupling constant acquires a scaling dimension y = 3X/(16n2), known as anomalous dimension. We have encountered an interesting phenomenon: some symmetries of the classical action (in this specific case it is the scale invariance of the coupling constant) are destroyed by the quantum corrections to this action, an anomaly in the quantum field theory. It should be emphasized that all these peculiarities of the cp4 model are generic to more realistic theories. Thus the analogous solution of

5.2. EFFECTIVE POTENTIAL OF

(p4

THEORY

Figure 18

the renormalization group equation in quantum electrodynamics (QED) gives the following expression for the running coupling constant e2/(47r):

The intensity of the electromagnetic interaction increases with M', as in (p4 theory, and experiments at the modern colliders confirm this conclusion. Thus for the energy of colliding electrons and positrons of order Mz = 90 MeV the coupling constant becomes e2(MZ)/47r = 11128, rather than e2/47r = 11137 as it is for energies smaller than the electron mass (formula (5.80) is valid only for higher energies, fig.17). It is apparent that there is also a Landau pole, i.e. the mass scale in QED at which the coupling constant becomes very large. It is

which is incredibly far away. It is much greater than the Planck scale FZ 1O1%eV, where any known QFT breaks down due to the uncertainties in the structure of space and time. Note that QED becomes part of the unified theory of electroweak interactions at the much lower energy scale of order Mz.

124

CHAPTER 5. VACUUM ENERGY IN

THEORY

Nevertheless, Nature does give us the possibility of seeing what happens as we approach the Landau pole. Quantum chromodynamics, which is widely accepted to be the correct theory of strong interactions at high energies, is an example of such a theory. The behavior of the running coupling constant is just the opposite of that in QED and (p4: it decreases as M' grows (see fig.18)

The Landau pole of the theory is in the low energy region AQcD = 180 MeV. The classical Lagrangian of QCD describes free quarks and gluons bearing the color charge. The intensity of the strong interaction decreases at high energies, so that quarks barely interact at short distances, a property called asymptotic freedom. On the other hand, the theory is not able to describe from first principles strong interaction physics at energies of order AQcDand below. It is well known experimentally, however, that the excitation spectrum changes drastically in this region. There exist only hadrons, which are the colorless (white) bound states of two or three quarks. The coupling constant of the hadron interactions is not small. The problem of quark confinement has persisted already for a few decades. It is beyond the scope of this book to discuss it in more detail, though recent progress connected with the paper by N. Seiberg and E. Witten [7] has revived hope for its solution.

Effective Potential of the Massive Theory Let us now consider a more general case of nonzero mass. In this case

and we take m2 > 0. The calculation of Ueff for this case does not differ in practice from the massless case. Furthermore, the singularity at p + 0 disappears because (5.83) does not vanish anywhere, so the renormalization conditions can be chosen simply at p = 0. For example, we can take U:ff(0) = m2 and U,";f(0) = X . The other case m2 < 0 leads to an interesting problem. The reason is that U1'(p) < 0 for p2 < 21m21/X. This results in the divergence of the Gaussian functional integral (5.46). We may try to obtain Ueff(p) as the analytic continuation from the domain where Url(p) > 0. But in doing this, the effective potential acquires an imaginary part for those cp for which U" < 0. So let us discuss this problem in more detail.

5.2. EFFECTIVE POTENTIAL O F (p4 THEORY

125

At first, we should note that the above discussed method of measuring U e f f in the presence of the external current J (5.44) fails for U(cp)" < 0 , because the expression U Jcp has no minima in this region. The interpretation of the imaginary part of the effective potential was given by Erick J. Weinberg and Ai-qun Wu [8].Here we give only the result of this work without the proof. It was shown that the effective potential should be defined as an analytic continuation from the region where U(cp)I1 > 0. The real part of the potential is, as before, the energy density of the spatially homogeneous field configuration with the average value cp. Such a state is unstable. For suppose it is created as an initial configuration. Then the evolution of the field results in growing field components with nonzero wavelengths. The field becomes more and more inhomogeneous. It was shown that the imaginary part of U e f f is equal to the time increment of this instability. The reason for the instability is clear. Let the field cp, such that U(cp)" < 0 , solve equation (5.40). Such a value of cp is a maximum of U(cp), e.g. cp = 0 in fig. 15 for m2 < 0. This point corresponds to the equilibrium which is unstable with respect to small perturbations to the right or left of the maximum. According to the result of paper [8], the effective potential contains information about the initial stage of the instability development in the region U(cp)I1< 0. Let us find the final state of the system initially released in the unstable region U(cp)" < 0. It depends on what physical meaning is attached to cp(x). It is worth while to consider first the case of the constant field average 1 (5.84) $90 = cp(x)hx

+

vl

in the process of the instability development. This condition is fulfilled, for example, if cp(x) is the concentration of a solution. In this case the integral in (5.84) is defined by the total amount of the dissolved substance. We shall now show that the equilibrium state here is a stratified mixture consisting of a number of spatial regions with different values of cp. In a ferromagnet such regions are called domains. In the considered case the mixture consists normally of salt crystals in a moderately concentrated solution. Let us prove as a lemma that all states with constant cpo are unstable if Utf ( P O ) < 0. An example is given in fig.19. By contrast, the homogeneous states with U z f f( p o ) > 0 are stable with respect to the stratification, at least when the corresponding fluctuations are small. Suppose that a trial stratified state consist of a mixture of regions with different cp values: the value cpl occupying the volume fraction and the value cpz

CHAPTER 5. VACUUM ENERGY IN cp4 THEORY

B

P1 PO P2

P

Figure 19

occupying the volume fraction 1 - c. To preserve the constant average value of cp it is necessary to require @pl (1 - <)vz = cpo Assume that it is possible to neglect the contribution of the boundary regions, where the field changes continuously between cpl and cpz. This is so if the domains' characteristic size is large compared with the boundary thickness, and their form is simple (not fractal). Then the average energy density takes the form

+

-

Ueff = CUeff(~1)+ (1 - <)Ueff(~z) . (5.85) Geometrically, (see fig.19) this is the ordinate of the intersection point of the secant AB and the vertical line cp = cpo. If the curve Ueff(cp) is < Ueff( p o )The further apart cpl and cpz are, the lower convex then the energy. What is the minimal energy of the mixture at fixed cpo? It is reached when the secant AB becomes the tangent (see fig.20). Thus it is defined as the solution of the following system of equations

ueff

for unknown cpl, cpz and J. Thus the states with the average field in the

5.2. EFFECTIVE POTENTIAL OF (p4 THEORY

Figure 20

interval (p1 < cpo < (p2 (fig.20) correspond to the mixture of two phases: one with the average value (pl and the other one with (p2. Outside this interval the homogeneous states are possible. To find the meaning of the quantities (p1 and 9 2 , let us note that, as follows from (5.86), their values does not change if we add a term -+ Ueff j(p. We can choose linear in cp to the effective potential: Ueff j such that the minima of Uef in fig.20 are degenerate (this corresponds to J = 0 in (5.86)). Then (p1 and (p2 are the field values at these minima. As can be seen in fig.20 there always exist intervals (p1 < (p < (p3 and (p4 < (p < (p2 where Uzf > 0. Hence the homogeneous states corresponding to these intervals are stable with respect to small fluctuations. At the same time, larger fluctuations make the states unstable, as is seen in fig.20. Such states are called metastable. They generally have an exponentially large but finite lifetime (for more details on this subject see the comprehensive papers by J.S. Langer [g], [10]). The homogeneous states are absolutely unstable in the interval (p3 < cp < 9 4 . The transition of such a state to the stable one, which is a mixture of two different phases, is called spinodial decomposition. Now let us consider the system for which the field average is not fixed. It could be a sample of ferromagnet in an external magnetic field J, or a self-interacting scalar field in the early Universe. For such systems, an equilibrium domain vacuum structure is possible only if the effective potential minima at cpl and (p, are degenerate. Otherwise, the domains

+

128

CHAPTER 5. VACUUM ENERGY IN

THEORY

with the lower value of Ueff would expand to fill the whole space. This is equal to can be easily understood if we note that the quantity -Ueff the pressure on the boundary between the two phases.

Problem: Find the effective potential for the scalar field (5.37), (5.38) for m2 < 0 at finite temperature. The problem can be simplified if we consider only the high temperature limit. Find the first nontrivial term in the expansion in powers of 1/kB8. Find also the critical temperature. Is this calculation reliable?

Bibliography [l] H.B.G. Casimir, Kon. Ned. Akad. Wetensch. Proc. 51 (1948) 793.

[2] V. Mostepanenko and N.N. Trunov, The Casimir Eflect and its Applications, Oxford, Clarendon Press, 1997. [3] J. Mathews and R.L. Walker, Mathematical Methods of Physics, 2nd ed., Benjamin/Cummings Pub. Co., 1970. [4] J. Schwinger, Particles, Sources, and Fields, Reading, MA., Addison Wesley, 1970-73. [5] P.A.M. Dirac, Directions in Physics, New York, Wiley, 1978.

[6] S. Coleman and E.Weinberg, Phys. Rev. D7 (1973) 1888. [7] N. Seiberg and E. Witten, Nucl. Phys., B426 (1994) 19. [8] E.J. Weinberg and Aiqun Wu, Phys. Rev., D36 (1987) 2474. [g] J.S.Langer, Annals of Phys., 41 (1967) 108. [l01 J.S.Langer, Annals of Phys., 54 (1969) 258.

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Chapter 6 The Effective Action in Theory

(p4

The scalar field quantization scheme developed in the previous chapter yielded reasonable results, although it was not mathematically consistent. In this chapter we attempt to construct a more rigorous calculation method. This method is approximate, valid at small coupling constant. Let us begin by asking a question: what physical quantities is it meaningful to calculate in QFT? The answer to this question is rather simple: only average quantities are observable in both QFT and statistical physics, although the nature of field fluctuations is different. In QFT, the average of a nonstationary quantity A[cp(x,t,)] at time t, takes the form

This implies that the initial and final field configurations must be fixed and, if necessary, subsequent integration over these configurations with the corresponding wave functional may be performed. As an example, let us check that expression (6.1) indeed gives the quantum mechanical average, provided an integration over the field with the vacuum wave functional at the initial and the final times is implied. We split the 'It should be pointed out that vacuum averages are the fundamental quantities in particle physics. The correspondcncc between them and the transition amplitudes is cstablished by the Lehman-Symanzik-Zimmermann formula ( L S Z reduction formula) which is beyond the scope of this book (scc e.g. the books [l,21).

132

CHAPTER 6. THE EFFECTIVE ACTION IN

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THEORY

integral into two parts: one for the interval 0 < t < t, and one for t, < t < T. Each part can be represented as an expansion in the energy operator eigenfunctions (4.35). Then we integrate over the initial and final field configurations with the vacuum wave f ~ n c t i o n a l . ~The normalization integral Z comprises all the above quantities excluding A itself. Then according to (6.1)

This is the standard quantum-mechanical average. The statistical analogue of (6.1)takes the form

(6.3)

6.1

Correlation Functions erat ing Functional

and the Gen-

Which average values are of interest? Most obvious is the average value of the field (cp(x))itself. The meaning of this value may be different as dictated by the context of the problem. It could be the magnetization of a sample or ion density in a plasma. Important physical information is also contained in the product of two field functions at different points ('p(x)cp(y)).The temporal components of X and y may be different, equal, or be absent altogether as they typically are for statistical physics. Such an average is called the correlation function, or correlator. Sometimes, for reasons explained below, it is called the Green function3. 2The analogue of such an integration in one-dimensional quantum mechanics is j'dxy!~(x). . .. 3Such a large number of terms: correlator, propagator, Green function, corresponding to the same theoretical object (cp(x)cp(y)), is not a result of terminological

6.1. CORRELATION FUNCTIONS

133

Let us consider some examples. Let cp have two components p+ and cp-, which represent the density of positive and negative ions in a plasma. The correlation functions W++ = (p+(x)cp+(y)) and W + = (p+(X)+ (y)) are proportional to the probabilities of finding two positive ions and two ions of different signs respectively at the points X and y. It is clear that the Coulomb interaction decreases this probability for ions of the same sign, and increases it for ones with opposite charges. Many-point or many-particle correlators (e.g., (cp(x)cp(y)cp(z))etc.) as well as two-point ones are often the subject of interest. It is an important fact that the knowledge of all possible correlators enables the re-construction of the partition function, and hence, complete information about the system. Let us construct a consistent method to calculate the one and manypoint correlation functions. From now on, we restrict ourselves to the Euclidean variant of field theory. This makes it possible to obtain information about the ground state of the system and its statistical properties. In order to get information about particles (excited states of the field in the real time), we must analytically continue the results from Euclidean to Minkowskian space (this continuation will be discussed). Below we omit the index 'E' for all quantities in Euclidean space. In principle, we can calculate the correlation functions using the definition (6.1) or (6.3) every time we need them. It is more convenient to perform the functional integration only once, in order to obtain a quantity which contains complete information about the correlators. This is called the generating functional. To construct the simplest one, let us consider the PI in the presence of a source J ( x , t ) :

Here we have assumed h = 1 (or k B T = 1) again. The meaning of this amplitude is the same as for the effective potential: the system is probed by its interaction with an arbitrary external current (or source) J(x). The only restriction implied on the function J ( x ) is that it must vanish at lxol + *m. An obvious advantage of expression (6.4) is that we can extract from it any correlator by taking the functional derivative with respect to J(x). Indeed, as follows from (6.4),

confusion taking place in this book. It merely reflects the situation in the scientific litcraturc. Thc two-field corrclation function appears in a largc number of contexts, in each having a specific name.

134

CHAPTER 6. THE EFFECTIVE ACTION IN

(p4 THEORY

This expression can be simplified by introducing the quantity W [ J ]a s follows: Z[J]= eUWLJ] . (6.8) It is not difficult to identify W as the field theoretical analogue of the free energy F in statistical physics. We recall that the latter is defined by the relation

By making use of W [ J ]we can obtain all the correlators in the same way as (6.5) - (6.7). In particular,

Another quantity which is often used is

This is called the exact propagator or the correlation function, denoted also as (cp(x)cp(y)),. It follows from the definition of G that G(x,y) =

G(Y, 4. Substituting (6.8) into (6.6) we obtain the relation between G(x, y) and (cp(x)cp(y)):

Thus the quantity

describes the mutual influence of the fields at the points X and y, which goes to zero for statistically independent (cp(x)) and (cp(y)). This is clear also from definition (6.11).

6.2. CORRELATORS OF THE FREE FIELD

135

Mathematically, (6.13)means that g does not contain the so called disconnected part, or terms which can be factorized as a product of multipliers depending only on X and on y. This property of including only the connected part is true4 also for higher functional derivatives of W [ J ] . For this reason W [ J ] is called the generating functional of connected Green functions while Z [ J ]is called simply the generating functional.

6.2

Z[J]W , [ J ]and Correlation Functions of the Free Field

Let us find the explicit form of the generating functional of the scalar field defined by the Euclidean action

Generating functional (6.4)reads

normalized so that Z[O]= 1. It is convenient to shift the functional variable p(x) = po(x) +X(%) so that the new one gives the maximum of the integrand at ~ ( x=) 0. The function po(x) must obey the Euclidean analog of the Euler-Lagrange equations:

(6.16) This nullifies the first-order terms in X and the integral (6.15)takes the form

where we have integrated by parts (applying the zero boundary conditions). 4We do not prove this statement here (see, e.g. [l,61). Thc reader can check it for low ordcr corrclators. Problem: Do this for up to the fourth derivative of W.

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CHAPTER 6. THE EFFECTIVE ACTION IN

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The rest of the integral is Gaussian and does not depend on the external current:

This can be included in the normalization constant as it does not affect averages (6.3) or (6.5) - (6.7). So Z[J] = ~ [ d e t ( - a :

+ m')]'"

exp

{i1

d4~J(x)(po(~)}

The analogy between the present calculations and our consideration of the harmonic oscillator in quantum mechanics is evident.

The Classical Green Function It is convenient to express the solution of equation (6.16) in terms of a Green function which obeys the Klein-Gordon equation

G(x - X') is the scalar field Green function, which depends only on the difference X - X' because of the translation invariance of the theory. To find its explicit form we must take the Fourier transform

which gives after the Fourier transformation of (6.20)

and finally

where p is the Euclidean momentum. Note that the analytic continuation of this expression into Minkowski space (which we do not need yet) can be performed by the rotation of the integration contour about p4 by an angle - ~ / 2 so that p4 = -ipo

6.2. CORRELATORS OF THE FREE FIELD

137

(this leads to the Feynman propagator, which is the most often used in particle physics.) It is an important fact that equation (6.20) has a unique solution. We recall that there are several solutions of this equation in Minkowski space, depending on the choice of the corresponding integration contour over p. which enclosed the poles of l/(pi - - m'). The solution of equation (6.16) can be written in the form

where the compact notation '*' for the convolution operation has been introduced. The expressions for generating functionals (6.19) and (6.8) take the form Z[J]

=

N' exp

{lJ

d4xd4r'J(x')G(x - I) J(x)}

Here we have omitted the normalization constant N' in Z and the corresponding term - In N' in W. Now we would like to point out some useful properties of the Green function G(x - y). First, equation (6.20) means that

Here G-'(X, y) is the inverse propagator which acts on any function in the functional space as follows

and particularly

Besides that, we can easily see that the classical part of action (6.14) can be represented in the form

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CHAPTER 6. THE EFFECTIVE ACTION IN

THEORY

Figure 21

Correlation Functions Having found generating functional (6.25) or (6.26), we can find any vacuum average of the theory with the action (6.14) by functional differentiation. For example,

6.2. CORRELATORS OF THE FREE FIELD

Figure 22

It follows from the second relation that for the free theory

Although these expressions appear cumbersome, their structure is in fact transparent, enabling us to use a simple graphic representation. The Feynman diagram technique associates a graphic element with the basic objects in the integrals (6.31) - (6.33), the current and propagator (fig. 21). The propagator G(x - y) is depicted as an interval with end points at X and y. The source J(x) is denoted as a cross. The junction of two graphic elements means the convolution of the corresponding functions. The dummy variables over which the convolution integration is performed are not labeled, while the explicit arguments of the whole convolution are. These rules make it possible to manipulate expressions of type (6.31)(6.33) by means of the graphic objects. The integral corresponding to any given diagram can be reconstructed unambiguously. For example, the diagrams for equations (6.26) and for (6.31) - (6.33) are depicted in fig. 21 and fig. 22 respectively. 5Diagrams in Minkowski space are sometimes regarded for the sake of clearness as a representation of the temporal evolution. This reflccts the fact that the propagator is simply the amplitudc for the propagation of a particle from one point to another. From this point of view, the second diagram in fig. 21 describes particle creation by the external source at the point X', propagates to the point X. The third diagram in

CHAPTER 6. THE EFFECTIVE ACTION IN

140

(p4

THEORY

We note a characteristic feature of all the diagrams considered, which is seen e.g. in fig. 22 : correlations of more than two fields are absent (only pairs of points are connected by a line). Thus the correlator ((p4) can be expressed in terms of (pp). This may be regarded as a criterion of the free theory. All its correlators are expressed in terms of G and G * J . If the external current is switched off (J(x) = 0), then all correlators are zero except those which are constructed from the two-point Green = G(x - y) only. This means that all the manyfunction (cp(x)cp(y))I particle correlators are disconnected (this is apparent if we take a look at fig. 22).

J=,

It is convenient to store the information about correlators in a more compact form, namely to deal only with the generating functional for connected diagrams and to re-construct the rest if necessary. A suitable quantity for this purpose is W[J]. It has the form (6.26) represented in fig. 21. We see that W[J] is constructed only from the one graph which generates all the diagrams in fig. 22. It should be noted that the generating functional Z [ J ] (6.25) can be represented graphically too. Indeed, as follows from (6.31) - (6.33), the correlators are simply the coefficients of the expansion of Z[J] in a functional Taylor series:

or, graphically as shown in fig. 23.

Z[J]

=

I

+ XX-

+ 1/2!

+1/3! >t--J<

W

+...

W Figure 23

fig. 21 depicts creation a t X', propagation t o X where it is absorbed by J(x). However, the consistent use of this graphic language is impossible without introducing somchow the unusual notion of travelling backward in time.

6.3. GENERATING FUNCTIONALS IN

6.3 (p4

(p4

THEORY

Generating Functionals in

ip4

141

Theory

Theory

The free field considered in the previous subsection is often not applicable to real physical situations. Usually it plays the role of the initial approximation, analogous to that of the harmonic oscillator in quantum mechanics. The simplest theory including interaction terms is (p4 theory (5.37)-(5.38) in which the self-interaction term Uint = XP4/4! is added to the Lagrangian (6.14). The Euclidean action of the model is

where

1

X

l

2

2

~ ( c p= ) -m2cp2+-cp4= -m cp +Ui,t(cp) . (6.37) 2 4! 2 This system was considered in the previous chapter for the case m = 0. The cases: m2 > 0 and m2 < 0 are also of interest. In the first case, m2 > 0, the potential has a minimum at cp = 0 (see fig. 15). This is the average value of the field in the vacuum state. For m2 < 0 the potential U(cp) has two symmetric minima at cp = f ( 6 ~ m ' ) ~=I f~ a (fig. 15). Hence two vacuum states with the average field value cp = f a are possible. The actual choice between the two values f a may be random, or follow from the history of the system. The choice when made violates the cp tt - p symmetry of the action (6.36), (6.37). This phenomenon is called spontaneous symmetry breaking. The potential of model (6.37) coincides with that of the phenomenological Landau-Ginzburg theory of phase transitions if m2 is regarded as a function of temperature (m2(T)changes sign at some critical value T = T,). This is why models of type (6.36)-(6.37) are widely used in physical applications.

Generating F'unctionals: Expansion in X Let us consider model (6.36) - (6.37) in the case m2 > 0 - the case m2 < 0 will be examined later. We need to find all correlators of the theory with action (6.36). It is sufficient for this purpose to evaluate the generating functional

142

CHAPTER 6. THE EFFECTIVE ACTION IN ip4 THEORY

As mentioned before, there is a lack of tools for dealing with such integrals. Actually, the only thing we can do is to evaluate the Gaussian integrals considered in the previous section. We may hope that for small X all observables differ only slightly from their values in the free theory. If this is really the case then we can find them in the form of a power series in X. Let us proceed with this idea and develop a perturbation theory to evaluate Z ( J ) (6.38) for X << 1. First, we expand the exponent of Xv4/4! in (6.38):

Every term of this sum results in a PI of type (6.15) with a prefactor (xn):

exp

1 {- /d4x[$(&99)2 + -rn2ip2 2

- ~(x)p(x)])

As discussed above, any power of cp(x) can be obtained by differentiation of the exponent in (6.39) with respect to J(x):

Substituting this into (6.39) and factoring out the variational derivative from the integral, we obtain the following compact expression:

-

he accuracy of this statement is far from evident. There are theories for which it is impossible to define the coupling constant, such that it would be small for all field values under consideration. Quantum chromodynamics, thc main candidate for the description of strong interactions, is an important cxample.

6.3. GENERATING FUNCTIONALS IN

v4

THEORY

143

Figure 24

using relation (6.25). The function of the operator 6 / 6 J ( x ) is understood in the sense of the corresponding Taylor expansion. To first order in X, this formula results in

Calculating subsequently four variational derivatives with respect to J ( x ) we get

As for the free case, this expression can be represented in graphic form. To do this, we note that the Green function G ( 0 ) = G ( x - X) should be represented as a closed loop, shown in fig. 24. Expression (6.43) is depicted graphically in fig. 25. Now let us write out the expression for the generating functional to first order in X:

144

CHAPTER 6. THE EFFECTIVE ACTION IN p4 THEORY

Figure 25

The corresponding expression for W [ J ]is

It is clear that only the first term in (6.44) and fig. 25 does not vanish at J = 0. Its value is proportional to the volume S d4x r R. Along with the higher order terms, which do not depend on the current J , this is called the vacuum fluctuation contribution to the total field energy. Often only the part of the total energy which exceeds that of the vacuum is interesting. Then the vacuum part is subtracted, as was done in calculating the Casimir energy and effective potential.7 The dropping of the vacuum diagrams corresponds to the normalization Z [ J ]-+ Z [ J ] / Z [ O ] . Exactly four lines are always brought to the point which corresponds to the multiplier X (such points are called vertices) as seen in fig. 25. This is a simple consequence of the fact that Uint Xp4 and enables us to omit the coefficient X/4! in our calculation. The coefficients of the terms in (6.43) and in fig. 25 are called the symmetry factors or symmetry coeficients, as they can be calculated combinatorically. To show this, we split for a while the vertex point X into four different points, X I , . . . x4, and consider the differentiation of generating functional (6.25) once more. Taking the first derivative with respect to J ( x l ) , the second with respect to J ( x z ) etc., the differentiation of exp J * G * J ) with respect to J ( x i ) gives the factor J G ( x i - yi) J(yi)dyi at each step. Besides this, the differentiation of the pre-exponential factor gives a term proportional to G ( x i - x j ) for each xj with j < i. It follows from this procedure that every term of the

-

{l

7Note that the vacuum diagrams cannot be omitted so easily in the presence of gravitation.

6.3. GENERATING FUNGTIONALS IN (p4 THEORY

145

Figure 26

expression has a unitary coefficient. The terms differ from each other by the number of propagators G, which varies from zero to its maximal value for each type of diagram. As for the terms with the same number of G's, they can be distinguished by the manner in which the propagators connect the points xi. Such terms are physically identical. Every connection variant appears only once. Now we can identify the points zi corresponding to the same vertex again. After this some diagrams become indistinguishable. The number of those is equal to the number of these identical terms, that is to the number of possible connections between the given points. To illustrate this, let us reproduce the coefficients in (6.43) and in fig. 25. The diagram without any propagators appears with unit weight. The coefficient 6 = 4.312 of the second term is equal to the number of ways of selecting two indistinguishable points from four (the two indistinguishable points are the arguments of one of the propagators). The indistinguishability of a pair of points means that each pair is taken into account only once (rather than twice). The same logic requires the coefficient 112 when selecting two pairs of points out of four points, as two loops (see fig. 25) are identical: 3 = (4.3/2)/2. The remainder of the calculation by functional differentiation of Z [ J ] is straightforward. For example, the two-point Green function is

It is evident that the first term in (6.44) gives the free propagator G(x- y) after the differentiation. Then dropping the terms which vanish at J = 0, we obtain

depicted graphically in fig. 26.

146

CHAPTER 6. THE EFFECTIVE ACTION IN cp4 THEORY

Note that we can get the same result by making use of the connected Green function generating functional W [J](6.45), rather than Z[J].

Generating Functionals: the Loop Expansion Now we consider model (6.36) - (6.37) for m2 < 0. In this case it is convenient to rewrite the potential (6.37) in the form

so that the minimum of U(p) is zero. Any attempt to apply the above method will fail because of a pole in expression (6.23) for m2 < 0. It is possible to understand the reason for this, and some other difficulties, without going into the details of the theory. Indeed the free theory (6.14) with potential U = m2cp2/2 is a very bad initial approximation because this potential looks quite different from the double well one (6.48). It would not be reasonable to expect any correct results for model (6.36), (6.48) taking so unsuitable first approximation. To improve the initial approximation it is worth while to account for more information about the structure of the theory. For this purpose we find the field configuration cpo(x) at which the integrand e - S + J * ~in (6.4) reaches its maximum for arbitrary J(x), expand the action near this trajectory and integrate over the deviation cp - cpo using perturbation theory if necessary. This technique is a generalization of the Laplace or saddle point method for the estimation of ordinary integrals (see e.q. [3]). Let us use it for calculation of Z in (6.4). As the method is applicable for the general form of the potential, we use the generic form of the action (6.36), taking (6.48) for the sake of illustration. We start with the definition of 90. In order to minimize S - J * cp, it must obey

which for the action (6.36) takes the form

where U' dU/dcp. Supposing that we know the solution cpo[J] as a functional of J, we introduce a new variable of functional integration ~ ( x =) ~ ( x -) cpo(x)

6.3. GENERATING FUNCTIONALS IN

(p4

THEORY

147

and expand the action in X : 1

1

S[cp]-J*cp=S[yo]-J*cpo+6S-J*X+-S2~+-63~+... 2! 3! . (6.51) The terms of the expansion for action (6.36) are:

and similarly for the higher order terms. For the case of (p4 theory, all higher variations of order greater than four are zero. Now we note that cpo has the property SS - J * X = 0 (see (6.49)). Therefore the first order term vanishes and integral (6.4) takes the form

Here a different representation is used to represent the action variation, in a form intermediate between (6.51) and (6.53) - (6.54). The quantity S2S/SY2is equal to S(x-X') multiplied by the operator in square brackets in (6.53), and 63S/6cp3 is simply U1"(cpo). Let us consider the exponential function in the classical action in (6.55) before proceeding with the calculation of the PI. If this PI is neglected, then the expression for Z represents the generating functional in the classical approximation. Taking the first variation with respect to the current J we obtain

This is not an unexpected result. It does not however take into account fluctuations around cpo. To find the contribution of fluctuations, we must evaluate integral (6.55). It cannot be calculated exactly. Let us consider only the case when the main contribution to the integral is given by a small vicinity

148

CHAPTER 6. THE EFFECTIVE ACTION IN

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of the classical trajectory cpo(x). This is the semiclassical approximation. The conditions for the validity of this assumption will be considered later. The smallness of the considered region means that X is small compared with cpo in the region giving the main contribution to the integral. Thus we neglect all powers of X greater than two, hoping that the Gaussian integral which is left in the expression

provides a good approximation for Z . This Gaussian integral was calculated above (6.18). Thus the expression for Z takes the form

The functional determinant in this expression cannot be included into the normalization constant as was done in (6.19) because it depends explicitly on the field, and hence on the current via U1'(cpo).This determinant cannot be found for arbitrary fields. We shall consider its calculation in this and in the next chapter for fields close to the vacuum. Now we would like to emphasize some disadvantages of the proposed method for the calculation of correlators, and to improve it by the introduction of one more generating functional.

6.4

Effective Action

Consider expression (6.58). It could hardly be of practical use because it is generally impossible to express p o ( x ) in terms of J ( x ) or, in other words, to solve the nonlinear equation (6.16). Nevertheless, J can easily be found as a function of p,. It would be convenient if the field was the independent variable rather than the current. Let us attempt to engineer this. It could be two different values of this functional variable: po or (cp) determined by (6.10). The latter is preferable because it is an observable quantity, in contrast to cpo. The fact that ( p ) can be expressed in terms of the functional derivative of W with respect to the current, makes it possible to apply the standard method of Legendre transformations to construct the quantity depending on the field. The term 'the quantity depending on the field' should probably be explained in more detail. The field and the current are dependent on

6.4. EFFECTIVE ACTION

149

each other by virtue of equation (6.16). An arbitrary functional F can be formally represented to depend either on the field, or on the current, or even on a mixture of both. In the latter case its differential is

where the field is assumed to be an independent variable. We say that F depends on the field if the right hand side of expression (6.59) does not contain S J / S p or, in other words, its value does not change as the functional dependence J [ p ]changes8 Now let us introduce a new functional J? by means of the functional Legendre transformation:

q v ) ]= W [ J ]+ J d 4 x ~ ( x ) ( p ( x )=) W [ J ]+ J ( x ) * ( p )

.

(6.60)

This is an immediate generalization of the ordinary Legendre transformation used in classical mechanics. The functional l? is constructed so that its derivative SI'ISJ (6.60) be zero. It follows from this that F depends on the field ( p ) and, therefore, it is reasonable to express the functional in terms of ( p ( x ) ) .It follows from (6.60) also that

One more variation with respect to the field gives

according to (6.29). Indeed,

Now let us find I'[(p)]in the saddle point approximation. We start with expression (6.58) rewritten in terms of W [ J ]as

'The geometrical meaning of the Legendre transformation is explained in the book [41.

CHAPTER 6. THE EFFECTIVE ACTION IN

150

(p4 THEORY

Here the Planck constant h is r e ~ t o r e d As . ~ is intuitively evident (it will be proven in the next chapter) this approximation is valid if the dimensionless parameter = (characteristic value of action)/h >> 1. The corrections to the classical part (which is the first two terms on the r.h.s. of (6.64)) are called quantum corrections, as suggested by the appearance of the Planck constant. According to (6.60) we obtain from (6.64):

The second term in this expression is of the same order in h as the last one, because (p) differs from po only if quantum corrections are taken into account. Thus m the relation

is valid to order 0(h2), we get finally

This is the desired expression for the effective action in the one-loop 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 self-interacting 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 re-write 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 infinite-dimensional 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 finite-dimensional. 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 well-defined. 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 non-commuting 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 n-th 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 re-written 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 n-th 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 n-th 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 n-th 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 one-loop 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 one-particle 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 space-dependent. 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, McGraw-Hill, 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. Zinn-Justin, 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 one-loop 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 four-dimensional 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 G-l (X, X') (6.27):

+

G-l (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 momentum-space 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 6-function 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 one-loop 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 S-function of the sum of all momenta as in previously considered cases. To find the sign of a momentum appearing in the argument of the 6-function, 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 S-function 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 delta-function 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 self-energy part (of the propagator). The three-point vertex function has the form

The four-point vertex function is

and similar for higher-order 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 one-loop 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 k-independence 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 one-loop 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 re-expressing 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 one-loop 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 self-interaction term cpn do not possess this property for n > 4 in four-dimensional 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(n-2). 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 self-interaction 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 self-interaction constant, as X = U" (0).

+

7.3

Pauli-Villars 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 ULI-VILLARS REG ULARIZATION

Figure 32

more analitic method of Pauli-Villars 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 Pauli-Villars field. Explicitly, the regularazed diagram takes the form

The contribution of the Pauli-Villars 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 Pauli-Vilars 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. PAULI-VILLARS 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 G-l (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 Pauli-Villars 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 four-dimensional 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)2-z

v J m 0

,

(7.38)

Using the definition and properties of the B-function

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. PAULI-VILLARS 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., 6-I +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 non-integer dimension 4-E. This means that we must take the spherical coordinate system in k-space 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(a2-1)+a1n-

a+l a-l'

+

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 = lk41e-ia. 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 half-plane, 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 (one-particle 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 e-i"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 One-loop Propagator Let us find now the explicit form of the propagator in Minkowski space, substituting k2 = Ik21e-i?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 two-particle 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 one-particle amplitude. The reason for this attenuation is that the particle can decay into two p-field 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 three-dimensional 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 one-loop 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

6-l ( 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 right-hand side of (7.64). Let us write down the renormalized inverse propagator, taking into account the explicit form of c(~)(Ic'):

2

JsJs5) arctg

-

for k2

<0

It is clear here that the correction term has relative order A2q52 Am2 if q5 a. The self-coupling X is, therefore, the dimensionless parameter determining the smallness of the corrections. Let us consider the calculation of the propagator (7.66) in another physical context. Suppose that we study the statistical properties of the field cp. Then r(cp) is the free energy in the presence of an external current J. The physical meaning of r(cp)depends on that of cp. If, for example, cp is the local magnetization, then J is the magnetic field. In statistical physics the system 'sits' in three-dimensional space so it is not meaningful to consider it in full Minkowski space. In contrast, the typical experimental problem is to measure the system's response to a homogeneous external field. In the above example, it is the magnetic susceptibility of the homogenous magnetic field which is proportional to

7.5. RENORMALIZATION

181

U:; ((P). Thus the appropriate definition of m is m2 = U:; ((P). The value of U:; ((P) is m: + C(0). Problem: Show this. Therefore the propagator, expressed in terms of the new m, does not coincide with the derived above expression (7.64) taking now the form

In the explicit expression (7.66) for the correlator, the third term is now -2 instead of -.rr/fi Problem: Show this.

Renormalizat ion of the Coupling Constant We would like to be able to extract finite terms from the expression for the coupling constant X (7.26). The divergent terms are produced by the above considered diagram A(2)but with four tails. The main idea is the same that we have used for the renormalization of the mass: it is necessary to point out the method of measuring the coupling constant. This case differs from the previous one by the absence of an obvious definition like 'the pole of the propagator'. Therefore let us choose a specific set of four momenta kl = ky, . . . k4 = kf and assume by definition that X = rc4)(k;. . . I$') . (7.68) It is apparent that only three momenta are independent, because of the delta-function in (7.13). The convenient choice of the momentum configuration depends, of course, on the context of the problem. In particle physics coupling constants are measured in scattering experiments, so all momenta lie on the mass shell k: = . . . k: = -m2. This decreases the number of independent parameters to 12. Lorenz-invariance eliminates 6 more parameters, and 4 parameters are excluded by energy-momentum conservation. Thus we are left with only 2 independent parameters of F ( ~ ) .Let two particles bearing momenta kl and k2 acquire momenta ky and h after a collision. The independent variables are commonly written in the form of 4-vectors squared s = (kl k2)2, the invariant mass, which is equal to the total collision energy squared in the center of mass reference frame, the transfered momentum squared. and t = (k3 -

+

182

CHAPTER 7. RENORMALIZATION

In the context of statistical physics it is meaningful to define X at zero external momenta. This is the definition used in chapter 5. Thus we fix a suitable configuration of momenta and define X according to (7.68). Then the expression for the vertex function takes the form

where all the one-loop correction terms are denoted r f ) , taking into account definition (7.68), namely

The expression in square brackets in (7.69) does not contain any divergences. This follows from the independence of A:) on external momenta. As follows from the explicit form of the corresponding diagrams, the value of these terms is of order X2.

Problem: Check this for the diagrams contributing to fig. 29.

rf)shown in

In other words, the relative value of the correction is of order X.

Problem: Check that this is true for the corrections to l?(3) shown in fig. 29 as well. Finally we would like to emphasize once again that any result for quantum corrections containing the parameters X and m2 is not physically meaningful until the precise definition of these parameters is given.

Renormalization of the Wave Function In the previous subsections, we derived the expression for the effective action r(p)in terms of the physical mass and the coupling constant. All divergences were canceled by the bare values of these parameters. It was necessary only to give accurate definitions of the physical parameters to fix the finite part of the effective action. Such definitions are formulated in terms of observables and therefore are related to the effective action as a whole. This, when applied to the loop expansion, means that it is necessary to adjust the values of the bare constants at each order, so

7.5. RENORMALIZATION

183

that the physical parameters remain both unchanged and equal to their values at the classical level (where they coincide with the experimental data). Let us discuss this problem in more detail in the context of particle physics. The observable objects of this theory are particles freely propagating far from each other4 and representing the asymptotic states of the field. A particle of mass m and momentum has energy w = (i2+m2)l/'. The normalized wave function has the form

and obeys the Klein-Gordon equation (6.20). The propagator corresponding to this equation is given by expression (6.23). It is intuitively clear that a single particle must propagate as if it were free. Its propagation amplitude is the exact propagator evaluated at the momentum lying on the mass shell. This function must coincide with the free propagator (6.23). Note that we have already used such arguments when considering mass renormalization. It should be pointed out that the relationship between the asymptotic and exact states of the field is very subtle. The reader can find a detailed analysis of this in the book [4]. The main result of such an analysis is the LSZ reduction formula, which makes it possible to substantiate the above statement in a rigorous way. Let us now attempt to justify the coincidence of the exact propagator given by (7.64) with the free one given by (6.23). Returning to the position-space representation, we obtain

d" E(t,f,O) = - i J w m

dk,,

Here we have changed the sign of the integration variable ko in Minkowski space, in order to write the expression in a standard form. Calculating the integral we have

4Sec, however, the discussion in chapter 5, page 124 and footnote on page 142.

184

CHAPTER 7. RENORMALIZATION

For the free propagator the residue is equal to 1/(2w), which is exactly the normalization factor of the wave function (7.71). Nevertheless this is not true for the one-loop propagator (7.66):

where the residue in the complex plane of the variable k2 appears in the second term, and the constant

has been introduced. For the one-loop propagator (7.66)

Thus quantum-mechanical corrections change the wave function normalization (this is a typical result for any perturbative calculations). The wave function of the one-particle excitations is not the field X, but

xphys)has the correct value of the residue at k2 = whose propagator (xphys -m2. Now we can replace the field X by xphYs in all expressions, including the expansion of the effective action (6.75). This is called renormalization of the wave function. Then the n-th term of the expansion takes the form 2-factors should be taken into account, for example, in the definition of X. Consistent consideration of 2-factors can be found in most of quantum field theory textbooks. It should be noted that the field renormalization is equivalent to the introduction of a counterterm (2;l - l)(dp)'/2 into the action. In (p4 theory, this term is finite at one loop. Thus wave function renormalization is not connected with ultraviolet divergences of the theory. The context of the problem under consideration fix if we must to make this renormalization. For example, in quantum electrodynamics the electromagnetic field interacts with the fermion current proportional to 89.

7.6. CONCL USION

Figure 34

Therefore the contribution of the 'fish' diagram to the divergent part of the effective action is proportional (at large momenta) to k2. In this case, the wave function renormalization is absolutely necessary, because the result diverges without it. In pure $4 theory, there is a divergence proportional to lc2, as in QED. It appears at the second order of the loop expansion from the 'sunset' diagram (fig. 34). We refer the reader to the book [l]for detailed calculations.

7.6

Conclusion

In the present and previous chapters we established the structure of the one-loop effective action I'[cp] in terms of a Taylor series (6.75) close to a ] in principle, been determined for small constant field $. F[$ ~ ( x )has, ) p(x) - 4 by expression (6.75), along with figure 29. deviations ~ ( x = A calculation of the terms quadratic in ~ ( x was ) carried out completely, with the result (7.66). The structure of the X-corrections to cubic and ) quartic terms was analyzed. Higher terms of the expansion in ~ ( x can be found by making use of the Feynman substitution (7.48) and formula (7.40). These terms are finite at large momenta and are purely quantum: there are no such terms in the classical action. In the performed analysis, we did not assume the smallnes of the background field $. This value parametrizes the line in the whole functional space of $ where the one-loop effective action is known. On this line, the value of I' is the effective potential times the four-dimensional volume. In the vicinity of this line, I' can be found approximately as described above. A physically preferable point is $ = f a , which corresponds to the ground state in the absence of an external current.

+

186

CHAPTER 7, RENORMALIZATION

The next chapter is devoted to a discussion of the obtained results and of the methods of calculation.

Bibliography [l]P. Ramond, Field Theory: A Modern Primer, 2nd ed., Addison Wesley Loriman, 1988.

[2] K.G. Wilson and J. Kogut, Phys. Rept. 12C (1974) 75. [3] J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, 1983. [4] C. Itzykson, J.B. Zuber, Quantum Field Theory, McGraw-Hill, 1980. [5] L.H. Ryder, Quantum Field Theory, 2nd ed., Cambridge University Press, Cambridge, 1996. [6] J. Zinn-Justin, Quantum Field theory and Critical Phenomena, 3rd ed., Oxford University Press, Oxford 1996. [7] H.M. Nussenzveig, Causality and Dispersion Relations, Academic Press, London 1972.

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Chapter 8 Renormalization Group 8.1

Renormalization Group

Renormalization Group Equation Because the effective action is observable its value must not depend on the way it is calculated. In this chapter, we shall study an important consequence of this obvious statement. The effective action is invariant with respect to the renormalization point, that is, the value of external momenta at which the theory parameters are defined. In the previous chapter, the value of such points were k2 = -m2 in (7.64), k2 = 0 in (7.67) and kl = k r , . . . k4 = k? in the definition (7.68). In chapter 5, we used a definite value of the field cp rather than momentum when renormalising the effective potential. These were particular cases of the general situation: the effective action depends on dimensional quantity (or quantities), that is on a renormalization point (or points). Let us denote the set of such points by p. While l? has an explicit dependence on p, there should not be an overall dependence on it. The explicit dependence must be compensated by an implicit dependence on p through the theory parameters: the mass, the coupling constant and the field renormalization constant. Examples of such dependences are given by (5.67), (7.67), (7.69). The fact that l? has no overall dependence on p can be written as a differential equation:

where, by convention, the differentiation is with respect to ln p rather than p because, as we have already seen, this is a typical l? dependence on the renormalization point.

190

CHAPTER 8. RENORMALIZATION GROUP

To obtain a more explicit form of equation (8.1) we assume p to be the unique point at which all of parameters are defined: the mass squared m2 according to (7.64), the wave function renormalization constant Z, according to (7.75) and the coupling constant X according to (7.68) and (7.77). The renormalization procedure makes it necessary to clarify our notation. The bare field cpo does not depend on p and the renormalised field cpphys = (PO does (here, we extend definition (7.77) to the total field including the background as well as fluctuations). In this chapter cp,h, is called the field and the quantities are defined as the coefficients of the functional expansion in this field (7.78). Below we shall omit the subscript phys for the sake of compactness. Equation (8.1) takes the form

.

.

where l n p is considered as the variable. The dependence of theory parameters on the renormalization point is most transparently manifested in the case of the coupling constant X. Here, we use the calculation results and the definition of X (7.68) from the previous chapter. First, let us make the definition (7.68) more specific. For this purpose let us fix the renormalization point of I'(4)(k1. . . k4) by choosing a set of momenta kl = kp, . . . k4 = kp (see expression (7.68) and the discussion thereafter). We shall call the change of the renormalization point the change of the overall scale of momenta. That is, the renormalization point at an arbitrary p is defined as kl = . . k4 = p i r , where @" = kY/Ikr l . . . k r = kf/IkfI are unit vectors. Because of (7.68) and (7.78) the renormalised expression for the vertex function is

&F.

We emphasise that the function f introduced here depends only on the ratio of the momenta to the mass. The dependence of the mass and the constant Z, on p is not as straightforward. It depends on the context of the problem. If the subject of interest is the propagation and the interaction of quanta of the cp field, then the most reasonable definition of m2 is as the pole of propagator (7.64) and Z, is defined as the constant normalising the residue at this

8.1. RENORMALIZATION GROUP

191

pole to unity (7.75). In this case the renormalization point is fixed: k2 = -m2. Let us give an example in which the definition of m2 and 2, is not so obvious. Let the field cp describes the local magnetisation or another order parameter of a sample near a second-order phase transition. The effective action in this case is the free energy of the system. Strictly speaking, the system is three-dimensional and one has to reproduce the treatment of the previous chapter for this case. One should have four dimensions only in the case of very low temperature when quantum effects are essential but we can hardly imagine the system which exhibits a second-order phase transition at such a temperature. Thus let us return to equation (8.2) and discuss the meaning of m2 and Z, depending on p despite the incorrect dimension. For this case, the experimentally observable quantity is the correlator of the order parameter fluctuations (cp(x)cp(y)) in k-space (this can be measured, for example, by neutron scattering). At the tree level it is nothing but k2 m2, where m-' is the correlation length. A more exact expression taking account of fluctuations is G(k2) (7.64) or G(k2)/Z, if the field normalisation is controlled. It describes the observable shape of G(k2) better if m2 and 2, are properly chosen. They can be specified, for example, as follows. The propagator is required to take the classical form k2 m2 near a point p:

+

+

G-'(p2) = p2

+ m2,

and

d -dk2 ~-'(k~)l

k2=$

=

1 .

(8.4)

One can regard these conditions as a set of equations with respect to m2 and 2 . The explicit form of the propagator (7.64) with the calculated value of C(k2)as well as the experimentally measured values of G(k2) and dG(k2)/k2should be substituted therein. Thus the parameter definition does not differ from that of X (7.68). It should be noted here that the same conditions can be also applied in particle physics in the region far from the propagator pole. Such an application of equation (8.2) will be discussed below. Let us consider equation (8.2) for the vertex functions l?(") appearing in the expansion of I' (see (6.75)) in powers of the field xphys (7.77). This expansion substituted into (8.2) results in

This is the renormalixation group (RG) equation. It is a cp4-analogue of the Callan-Symanzilc equation in QED (see, e.g. [l]).

192

CHAPTER 8. RENORMALIZATION GROUP

Let us comment on the origin of this name. It is not difficult to understand that the set of the parameters transformations corresponding to all possible changes of p makes up a group. The transformation from p1 to p2 and then from p2 to p3 leads to the same value of parameters as the direct transformation from p1 to p3, otherwise, the observables depend on the method of calculation. The unit element is the trivial transformation leaving p unchanged.

General Solution of RG Equation An effective action must satisfy the RG equation (8.2) to be renormalization invariant, that is to be independent of variations of p. The RG equation also makes it possible to extend the region where perturbative calculations are possible. An example of this use of the RG equation was given in chapter 5 (eq. (5.74)). We recall that the effective potential was obtained and renormalised near a point p = M. We could not apply the result far away from this point because the one-loop correction, proportional to ln(4/M), became large. Applying the RG equation enabled us to improve the result as 4 -t 0 and to show that at large 4 the difficulty was not removable. Let us now consider that procedure in a more general form. For this purpose let us fix an arbitrary renormalization point p0 and change variable from p to' t = ln(p/po). We shall write l?(") in the form r(")(kl, . . . k,; X, m, p") where the last argument of l?(") indicates the renormalisation point. As a rule, l? has the form of the classical action plus a correction depending, like (8.3), on the dimensionless ratios ki/pO and ki/m. Considerable deviations of k from p0 may make the correction terms in large and thus violate the applicability conditions of the method. For example, this is the case for (8.3) as (kl k2)2+ 00. The situation can be improved if one includes the large correction terms in the classical action.

+

Problem: Show that it can always be done in a renormalisable theory. Technically this means that one should recalculate the corrections for new m, 2, and X taking a new value of p for the renormalization point. The only requirement imposed upon this new renormalization point is that the corresponding ki have to be close to the values of interest. This extends the scope of the approximate calculations, but there arises one more problem to solve: how can one express the theory parameters at one renormalization point, say p, in terms of their values at another ('old')

8.1. RENORMALIZATION GROUP

193

point, say p o This problem is tackled by solving eq. (8.5). Let us consider the general solution of (8.5) in more detail. First we introduce the following notation for the coefficients appearing in (8.5):

1dln 2, 7(X) = Z a t l , = , The function (8.7) is called the 0-function, and 7 (8.8) is called the anomalous dimension of the field. According to (8.6) - (8.8) all these functions only depend on p0 via m and X. It is clear that one can vary the value of p, in (8.6) (8.8) without changing the functional forms of these relations. Now, let us consider the solution of (8.2). It can be found by the method of characteristics. The value of F(") is known in the hyperplane t = 0 in the 4-dimensional space (t, m, Z,,X). We need F(") all over the space. Let us first consider the case n = 0. From (8.5) with n = 0 we get -

dt

----

at

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

-

=0

We see that the differential dt and the differentials dm2 = adt

and

dX = 0dt

,

(8.10)

which depend on it determine a vector field in the subspace (t, m2,X) along which F(') does not change. Integrating the equations (8.10), the are found: lines of constant value of "'(?l m"

z(t)

and

X

= X(t)

(8.11)

(an example is shown in fig. 35). These functions are called the running mass and the running coupling constant respectively. Any function which is constant along these lines is a solution of equation (8.9). The unique solution we need is fixed by the boundary condition in the plane t = 0. To find F(O)(t,X) at an arbitrary point (t, m2,X) it is necessary to draw a line (8.11) (i.e. the characteristics of eq. (8.9)) up to an intersection with the plane t = 0

CHAPTER 8. RENORMALIZATION GROUP I

Figure 35

(fig. 35). The value of l?(') at this point is the desired solution of (8.9). Let us write it down in the form

(

r(0) m2

,X

7

)

t = r(0)( s ( 0 ) m 0)

7

(8.12)

where s ( 0 ) , X(0) are determined by the conditions s ( t ) = m2 and X(t) = X. To avoid misleading notation, let us recall that X and m2 denote the theory parameters whose numerical values are to be chosen to describe a specific physical system. X(p) and m2(p) denote the same values, but with explicit reference to the point p at which the parameters should be measured. At the same time, m2 and X are the characteristics of equation (8.5) which cross the point m2(p), X(p). The solution of the RG equation, taken in the form (8.12), makes apparent the renorminvariance of The effective action depends on its arguments m2,X, t via the combinations s ( 0 ) and X(0) and these are renorminvariant by construction. Any change of t results in a change of m2(t) and X(t) which still belongs to the same characteristic line. Now let us solve eq. (8.5) at n f 0. It is straightforward to check that

-

Thus the problem is solved in general. Let us proceed further with an explicit example using the results of the previous chapter.

8.1. RENORMALIZATION GROUP

Explicit Example Let us apply the method that has been developed to the vertex function theory (6.36), (6.37). For this purpose we need (8.3) in the expressions (7.64), (7.66) and (8.3). We start rewriting the formula (7.64) in the form

+ +

where Z, is expanded in powers of X : Z, = l 6 . . . and the higher terms in X are neglected. We have also omitted the label ' ( 2 ) ' on C. An application of conditions (8.4) gives

where the prime on C means differentiation with respect to k 2 . The substitution of m2 expressed in terms of m 2 ( p ) according to (8.16) leads to an inverse propagator in the form

By differentiation of expression (8.15), we find the anomalous dimension of the field: y=2 ~ 2 ~ " ( p 2 ) (8.18) up to first-order terms in X. A simple way to obtain expressions for the aand p-functions is to differentiate these equations (7.64) and (8.3) with respect to p taking account of the fact that the left hand sides of them do not depend on p :

The next step should be to solve the set of equations (8.10). However, the complicated form of C (7.66) and of the unknown function f means there is no real hope of analytic solution. Let us simplify the problem by considering only large momenta k2 >> m 2 , the so called deeply Euclidean region. At such large k2 the expression (7.66) for C reduces to

196

CHAPTER 8. RENORMALIZATION GROUP

That leads to and in the limit k 2 -+m. Of course, such a simple result is not obtained by chance. Let us show that as k 2 -+ CO the functions a , p and y are determined by the contributions of the divergent diagrams only. To this end, we examine fig. 29 and expression (7.19). All the external momenta appear in the denominator of the corresponding matrix elements. Therefore, all the convergent diagrams (7.19) tends to zero as k 2 t cm.The only contribution to the propagator and the vertex functions turns out to come from the divergent 'fish' diagram A:) (7.25). Let us consider the propagator as a function of the external momentum k as k 2 + CO. In this limit we can neglect the mass and write

It is clear that the integrand reaches its asymptotic value llp4only for p2 >> k 2 while the presence of k in the denominator eliminates the infrared divergence of the approximating integral S d4plp4. Thus the divergent part of A(2)takes the form

where A is the cutoff parameter which is replaced by p after renormalization. This expression determines completely the effective action for k 2 >> m 2 . In particular, it results in the second term in the right hand side of (8.3) leading to p=-- 3X2 16n2 for k 2 >> m'. Thus the functions a, 3 !, and y at large momenta are determined by the structure of divergences and do not depend on the details of the renormalization procedure. It explains, for example, the vanishing of y: the corresponding diagrams converge. It should also be noted that expression (8.25) coincides with formula (5.74) derived for the massless theory ( p 4 . To find the characteristics of the RG equations we have to find s ( t ) and X(t) as solutions of eqs. (8.10) with the boundary conditions s ( 0 ) =

8.2. SCALE TRANSFORMATIONS

197

m2 and X(0) = X. From the second equation in (8.10) we obtain an implicit expression for X(t):

It gives, by virtue of the boundary conditions,

which is the same as (5.76) except p replaces M. Similarly we can obtain an expression for the running mass:

For k; >> m', this running mass does not have an essential effect on the effective action for the following reasons. It gives a subdominant contribution to the inverse propagator (8.17) as compared with the leading term k 2 . There in no dependence on m2 in 1'(4), and all F(") with n > 4 are negligible. Therefore, the most relevant quantity in the RG equation in the limit being considered are the running coupling constant and the corresponding p-function.

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 d-dimensional space specified by the

198

CHAPTER 8. RENORMALIZATION GROUP

Let us take an arbitrary field configuration cp(x) and consider scaling its space-time 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 m4-7). 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.

Gell-Mann - 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) = s4-nr(n). . . , X

,

)

.

(8.32)

S

Here, the factor s4-ntakes 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 Gell-Mann - 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 S-YX 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 Gell-Mann - Low equation (8.33) obtained in the limit kg/m2 -+ oo are exact. Nevertheless, the key quantities P- and y-functions 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 one-loop calculations: the vanishing of y and the expression (8.25) for the P-function 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 Gell-Mann - Low equation) enables one to perform a summation in In S. However, the coefficients in (8.36) remain first-order 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 p-function in this region is not known. Let us just discuss possible scenarios for the p-function 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 p-function 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 p-function 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 P-function 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 P-function behaviour seem to be very interesting because the perturbation theory never fails if XI is small;

3. The P-function 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 QCD-charge 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 p-function 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 ,B-function 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) = S4-n(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. Gell-Mann and F.E. Low, Phys. Rev. 95 (1954) 1300. [3] Z. Horvath and L. Palla, Conformal Field Theories and Integrable Models, Springer, Berlin-New 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. Zinn-Justin, Quantum Field theory and Critical Phenomena, 3rd ed., Oxford University Press, Oxford 1996. C. Itzykson, J.B. Zuber, Quantum Field Theory, McGraw-Hill, 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 cp4-model 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 one-particle 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 3-point 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)= G-l 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 3-point vertex function of the expression G-l = 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 4-point 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 higher-order 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 one-particle irreducible diagrams - l? can be obtained from W by cutting all one-particle 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/m4-d. 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 four-line vertex gives 1 / X and the

CHAPTER 9. CONCLUDING REMARKS

212

three lines gives
where l is the number of lines. It can be expressed in terms of the number of vertices if one counts all lines at all vertices and divides this by two to account only one time for each line. This gives l = (3v3 4v4)/2. This number is integer since v3 is even. Finally we get

+

Problem: Perform a similar analysis of the definite integral

<

Let us consider different possible scales. We take as the first example the theory (9.11) with 7 = 0 and m2 > 0. Switching on a spatially homogeneous current leads to a shift of the potential minimum to a point cpo(J). A typical current value (without any special smallness or greateness) is that which causes cp m. In this case, 7 = Uttt(cp) Am, 6 and &(v3,v4) = Xv4+"3,that is, each vertex adds one power of the coupling constant. One faces a more complicated situation at large values of the background field cp,. One can reach such values either by increasing J or even, in the case of spontaneous symmetry breaking by keeping it zero. Let us consider the latter example. Let the potential be -p2Q,2/2 XQ4/4! with p2 > 0. Expanding the field Q> = Q,o cp near the minimum 4 = p m one comes to an action of type (9.11) with the following W

<

N

+

+

values of the parameters: n2= 2p2, 7 = m m . We see that large values of the background field Q, -- m / f i lead to the large parameter 7 that gives -- 1 in (9.16). The consequence is that the diagrams of the same order in the coupling constant really have different orders of magnitude. For example, a few diagrams of the order X and X2 shown in fig. 42 belong to different orders of the perturbation theory (from the first to the fourth order). Let us show that the order of any diagram (including those with external lines) is determined by the number, L, of independent closed loops of the graph: E(%, v4) = XL-'. It is obviously so for the diagrams in fig. 42. In the general case, note that L is equal to the number of

c

9.2. ON THE PROPERTIES OF PERTURBATION SERIES

213

Figure 42

independent integrations over momenta. To count these, it is sufficient to recollect that each propagator (the total number of which is I) gives just one integration (see (6.21), (6.23)). Each vertex (the total number is v = v3 v4) contributes a S-function of all momenta entering it (a detailed analysis of this fact was done for a particular case in section 7.1). All the S-functions result in one common &function expressing the energy-momentum conservation for the external lines (7.13). The remaining v - 1 &functions decrease the number of integrations to

+

which is the number of diagram loops. Comparing this expression with (9.15), we obtain the desired relation in the form

In theories with spontaneous symmetry breaking C 1. Otherwise, < 1 at not very large J, that causes further dependence of the diagram order on the type of vertices it contains. To find the formal loop expansion parameter which does not depend on we turn to dimensional units, i.e. we introduce llfi into (9.11). Then, as can easily be seen from (9.13), one should replace X by fiX while remains unchanged. That leads to

Thus the Planck constant is the loop expansion parameter. We already used this fact, which was checked to the first order in fi, when the effective action was introduced in chapter 6 (see expression (6.65)). We saw also

CHAPTER 9. CONCLUDING REMARKS

214

that the one-loop correction to the effective action (6.72) contained all orders of the coupling constant. It should be stressed that both the expansion in the coupling constant and the loop expansion are variants of the same perturbation theory approach. They differ from each other only by the way diagrams are arranged according to their order of smallness. The origin of perturbation theory is the expansion of the exponential function containing the third and the fourth powers of the field in the Taylor series near a properly chosen point. The point can be taken in different ways. In the next section we shall discuss some principle restrictions on the accuracy of the perturbation theory itself.

On the Asymptotic Nature of Perturbation Series In this section we shall not obtain any exact statements about the convergence of the perturbative expansion. Instead we model it with the definite integral

and present some evidence that the petrurbation series of this integral is asymptotic.

Let us expand the expression ecXX4in (9.20) in a series and integrate each term individually. This gives

The series obtained evidently diverges because of the r(2n),which grows faster than n!. At the same time, the first term of this series, I ( X ) z 1 - X/2 3A/8 . . ., improves considerably the accuracy of the zeroth 1for X << 1. That can be verified by numerorder approximation I ( X ) ical calculation. Thus it would not be wise to disregard the possibility of making use of diverging series such as this one. In fact, all the impressively precise calculations in QED are done using this method. Let us look closely at the structure of the series (9.21). The reason for its divergence is clear. The exponential factor ePXx4in (9.20) is very small as X + CO. However, this smallness is a result of the cancellation of very large terms in the corresponding Taylor expansion.

+

+

9.2. ON THE PROPERTIES OF PERTURBATION SERIES

215

We destroyed this subtle compensation by expanding the exponential function in (9.20) and taking the limit X + CO as is required by the integral. As a result, the n-th term of the series in the second part of (9.21) has a maximum at r = @ which is absent in the original function. The value of the maximum becomes of the order unity at n % 4 e ~ ' / ~ / XThus . one should expect that the series does not approximate the initial integral beyond n 1/X. Let us discuss this problem in more detail. Let us find the common term C,(X) of the series as n >> 1. To estimate it, we use the Stirling formula for the F-function and factorial

-

We see that for X << 1 the terms of the series at first decrease as n grows, but then, beginning from some n = no, they increase again. Let us find no under the assumtion that it is large. For this purpose we differentiate the second factor in (9.23) with respect to n:

Equating that to zero, we find no = 1/(4X). Substituting n = no into (9.23), we get an estimate for the smallest term of the series (9.21):

Thus the smaller X is, the later the divergence of the series manifests itself and the smaller the minimal term is. It is natural to expect that the quantity (9.25) is an estimate of the maximal accuracy which can be achieved when summing the series (9.21). Let us state here without a proof that this is precisely the case [14]. The best approximation for the exact expression can be obtained by summing series (9.21) up to the terms with numbers of order no. Besides that, the exact expression contains the term of type e-const/x which, of course, cannot be expanded in Taylor series near X = 0 (all derivatives of this function vanish at X = 0 ) . In quantum field theory such contributions are called nonperturbative as opposed to the perturbative ones which are terms of the perturbation series.

CHAPTER 9. CONCLUDING REMARKS

Figure 43

Figure 44

Let us show that nonperturbative terms do indeed exist. This will, at the same time, be a proof of the statement that the radius of convergence of series (9.21) is zero. It follows from the fact that I ( X ) (9.20) has a cut along the negative part of the real axis on the complex plane of the variable X. This cut begins at the point X = 0. I ( X ) has an imaginary part on the edges of the cut, which in the case X -t 0, takes the form

on the upper and the lower edge respectively. To show this, let us consider the analytic continuation of I ( X ) , (9.20), on the X complex plane to values X < 0. We define the change to negative X as a rotation X = IXleia, with the phase a varying from 0 to T . It is evident that integral (9.20) diverges for A < 0 but this divergence no longer has anything to do with the function I(X). The correct integral representation of the analytic continuation I(X) can be derived from the integral (9.20) by rotation of the integration contour in the X complex plane so that the integral remains convergent for all values of the phase cu including a = T . This can easily be shown. Let us consider the integral (9.20) in the complex plane of X. By convention, we will refer to the sets of directions in which the integrand vanishes (goes to infinity) for 1x1 + oo on sectors of convergence (divergence). The divergence sectors of the integral are depicted in fig. 43 by the bold solid lines, and the integration

9.2. ON THE PROPERTIES OF PERTURBATION SERIES

217

contour coincides with the real axis. Formula (9.20) defines the analytical function I(X) in the region -7r/2 < argX < 7r/2. Let us rotate the contour by the angle -P so that it occupies the position pictured in fig. 43 by the slanted dashed line. The value of the integral will not change after that. However, it now defines the function I(X) in the region -7~12 p < arg X < 7r/2 ,B. Thus two analytical functions defined by the integrals over both the contours in fig. 43 coincide with each other ,f3 < argX < n/2. Hence, by in the intersection of the regions -7r/2 virtue of the theorem about the uniqueness of the analytic continuation, two integrals define the same analytic function I(X). The statement for an arbitrary phase a can be proved in an analogous way by repeating the above arguments several times.

+

+

+

After the continuation of X to negative values of X = IXlem",he integration contour in (9.20) takes the position shown in fig. 44 by the dashed line. The points z = 0 and z = fl/mare saddle points of the integrand. They are marked off by circles in the figure. The integrand equals to unity at z = 0, falls monotonously as it approaches these points along the real axis and then grows to infinity. Its value a t z = f1 1 0 is exp {-1/(4X)) and determines the imaginary part of I(IXlei"). To extract the expression ImI(lXlei") from the total integral, we take the integration contour as shown in fig. 44 by the solid line. Then, the integrand is real in the interval from z = -l/& to z = l/&. The vicinity of the peak at X = 0 gives the main contribution to ReI(lXlei") = 1. At the same time, the imaginary part of I(IX(ei") is due to the parts of the contour lying beyond the real axis. The main contribution to ImI(lXlei*) is given because the contour comes out of by the regions close to r = f these points in the direction of the steepest descent which is parallel to the imaginary axis. Taking account for this contribution in the Gaussian approximation we come to formula (9.26).

l/m

Now we consider the change of sign of X when going to the lower In this case the rotation of the integration half-plane: X = contour in (9.20) takes place in the anti-clockwise direction opposite to that shown in fig. 44. Therefore, the sign of the imaginary part of (9.26) is reversed. Thus we have shown that X = 0 is the branching point of I(X). Consequently, the series in powers of X cannot converge near this point. It should also be noted that (9.26) contains the same exponential factor as the accuracy estimate (9.25). The example considered here serves as a good model for the situation in QFT. To demonstrate the divergence of the perturbation series and the existence of the nonperturbative part In Z let us make analytically

CHAPTER 9. CONCLUDING REMARKS

Figure 45

continue (9.11) to X < 0 keeping m2 > 0. After that, the potential in (9.11) takes the form shown in fig. 45. The alternation of the sign of X does not affect the properties of the perturbation series near the point cp = 0. At X sufficiently small one can use the vacuum state cp = 0 and study cross-sections 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 first-order 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 -ie-const~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: Second-Order 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 second-order 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 field-theoretical models discussed in chapters 6-9. 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 second-order 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 large-energy 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 two-point 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 tree-level two-point 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 p-field (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 (p4-theorydoes not affect the content of this book. Our aim was to present the method of loop expansion and to discuss the basic properties of the field systems. We used the (p4 model as a simple example. If the theory is trivial, this only shows how nontrivial the nonperturbative properties of field systems can be.

>

>

9.4

<

<

Conclusion

This chapter completes the part of the book in which we have attempted to study the general features inherent in any quantum field theory using the example of the (p4 model. In the next chapters we shall consider in more details a quantization procedure, which leads logically to fermionic fields. Both these and gauge fields will be quantized. In addition, we shall discuss interesting classical objects which appear in non-linear field theory.

2Note that the experiments sensitive to the presence of the Higgs particle suggest that its mass might be not much higher that 80 GeV [13].

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Bibliography [l]J.S. Langer, Ann. of Phys. 41 (1967) 108.

[2] S. Coleman, Phys. Rev. D15 (1977) 2929; Erratum: ibid D16 (1977) 1248. [3] K.G. Wilson and J . Kogut, Phys. Rep. 12C (1974) 75. [4] S.-K. Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, MA 1976. [5] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover Publications, 1975. [6] A.Z. Patashinskij and V.L. Pokrovskij, Fluctuation Theory of Phase Transitions, Oxford, Pergamon Press 1979. [7] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 3rd ed. Oxford, University Press, 1996. [8] F.R.S. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982. [g] L.D. Landau, E.M. Lifshits and L.P. Pitaevskii, Course of Theoretical Physics. v.5: Statistical Physics. Pt.1, 3rd ed., London, Pergamon, 1980. [l01 D.J.E. Callaway, Phys. Rep. 167C (1988) 241. [l11 S. de Jong, Higgs Search at LEP, talk at the 23rd Rencontres de Moriond (Electroweak Interactions and Unified Theories), Les Arcs, France, March 1998; C. Caso et al, The European Physical Journal C3 (1998) 1.

[l21 R. Dashen and H. Neuberger, Phys. Rev. Lett. 50 (1983) 1897.

224

BIBLIOGRAPHY

[l31 D. Treille, Searching for New Particles at Existing Accelerators, talk a t the ICHEP'98, Vancouver, B.C., Canada, July 23-29 1998, Canada (http://ichep98.triumf.ca). [l41 A. Erdelyi, Asymptotic Expansions, Dover, 1956.

Part I11

MORE COMPLEX FIELDS AND OBJECTS

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Chapter 10 Second Quantization: from Particles to Fields It was shown in the chapter 4 that fluctuations of quantum fields are particles. The phonons in the solid state physics are a classical example: they are quanta of the crystal lattice vibrations. However, electrons or other elementary particles seem to be logically more primary than any field. Formally, one may introduce a field that upon quantization gives rise to the particles which really exist in our World. The field that possess the desired properties can be used as a theoretical object justified by the comparison of its predictions with experiment. This is possible in principal, but it is not strightforward. In this chapter, we would like to describe another method of second quantisation in which the elementary particles are considered as primary objects and the quantum field as a result. This approach allows us not only to complete logically the second quantisation presented in chapter 6, but also to introduce fermionic fields in a very natural way.

10.1

Identical Particles and Symmetry of Wave Functions

Elementary particles of the same type are identical. This is a standard principle one can find in any textbook on quantum mechanics. Actually, its meaning is related to the definition of the notion elementary particle. In order to distinguish different interacting particles one has to label them. This is only possible if these particles have an internal structure and if this is the case they are not elementary. Let us stop the discussion of this fundamental property here and consider its main consequences for

228

CHAPTER 10. SECOND QUANTIZATION

the quantum mechanics of many-particle systems. Let us formulate the property of indistinguishability as a condition on the two-particle wave function. The interchange of two particles does not change any observables. Thus, as a result of the interchange, the wave function of a two-particle system can only get a phase factor em". Two consecutive interchanges is the same as none because two interchanges bring the particles back to their initial states. Thus eZi"" = 1, i.e. ei"" = 1 or ei"" = -1. The first case corresponds to bosons and the second to fermions. Let us denote the two-particle wave function as 9 ( x l , x2) where the coordinate xl corresponds to the first particle and xa to the second one. Then 9(x2,xl) = *(g1, x2) for the boson system and 9(x2,xl) = -Q(xl, x2) in the case of two fermions. Consider N identical non-relativistic particles. The Hamiltonian is

where

and so on. Here all the summation indices take the values 1. . . N , and the terms with two or more coinciding indices in multiple sums are excluded. For example, the term v(~)(x,,,sal, xaz) that could appear in we ~ , , xal) in Formally we can include in V(') and the term v ( ~ ) ( x xal, require, as a condition, that the potential is equal to zero if any two of its arguments coincide:

Further, the potentials v('")(x,, . . . , X,,) tations of their arguments.

are symmetric under all permu-

IHowever, in two space dimensions new possibilities arise. This is a special case because any rotation of two particles around each other (which is the way to interchange them) is characterized by only one angle. This allows arbitrary values of or, for example, for particles moving in a plane orthogonal to an external magnetic field. The objects with non-integer or are called anyons [l,21.

10.1. IDENTICAL PARTICLES

229

The operator <(l) is the so-called one-particle Hamiltonian. It is given by the sum of the conventional SchrGdinger operators H:') of noninteracting particles of mass M in an external potential v(')(x). As one example one can consider the motion of electrons in metal. In this case v ( ~ ) ( x is ) a potential of the ionic crystal lattice. The next term <(2) describes the two-particle interaction with the potential energy V('). For example, it can be the Coulomb repulsion of two electrons. The combinatorial factor 1/2! appears to avoid the double counting of each couple of identical interacting particles. This operator is the so-called two-particle Hamiltonian. defined by (10.4) is the three-particle HamiltoThe next term nian. The potential V(" describes the modification of the two-particle interaction energy due to the presence of the third particle. This situation is a bit more complicated. As a relevant example, one can consider the correction to the interaction of two atoms due to the presence of the third one. As the third atom approaches, the electron shells of the initial two atoms get perturbed and this changes the interaction between them. Of course, the separation of these three atom into a two-body system and an external perturbing atom is fairly symbolic when all three atoms are identical. The factor 1/3!, corresponding to the number of permutations of three elements, is introduced in (10.4) to count the contribution from each three particle configurations only once. Let us recall the example of electrons in metal. The presense of an electron affects the lattice polarization. This effect is, in general, nonlinear. Thus, in this simple case, an electron changes the dielectic permittivity of the matter between two other interacting electrons. As a result the energy of their interaction changes and the corresponding correction is V("). The meaning of the other terms in (10.1) could be considered in the same manner. There are many physical situations where particles are less coupled to each other than to a background potential and the character of interaction between them is mainly one-to-one. Then the expansion (10.1) is fast converging and we can restrict ourself to consideration of only the first a few terms. In this chapter we consider a very convenient formalism which takes into account the symmetry of the wave function under interchange of particles. Let Q(xl, x2 . . . ,xN) be an N-particle wave function and let the system be described by the Hamiltonian (10.1). One can introduce the two particle interchange operator by defining

This operator commutes with

6 because its action is just

a renaming of

CHAPTER 10. SECOND QUANTIZATION

230

the summation indices in expressions (10.3) - (10.4). Thus 'Paband the Hamiltonian should have a common set of eigenfunctions. Let us find them. To do this, we choose a complete set of orthogonal normalizable functions &(X), i = 0,1,2 . . .. A common choice is to consider the plane-wave basis qbi(z)= eikz where the momenta take all possible values allowed by the boundary conditions in, say, a potential box. We shall use the expansion

&

Each of the multiparticle basis functions &l (xl)giz(x2). . . qiN(xN) describes a state where the first particle is in the state il, the second one is in the state i2 and so on. It is obvious these functions are not symmetric under particle interchange. More precisely, the interchange of two particles, indicated by the numbers a and b in the functions (10.7), means that the arguments interchange as X, + xb, while the state labels i, and ibremain unaffected. Of course this operation is equivalent to the interchange of indices ia + ib by the unchanged arguments. Thus we have to construct basis functions which are symmetric with respect to such operations. We do this first for bosonic systems.

10.2

Bosons

Let the N particles under consideration be bosons, i.e.

for any a and b. Our goal is to symmetrize the set of the basis functions (10.7) and calculate the matrix elements of the Hamiltonian in the new basis. To symmetrize the functions &l (xl)qbi2(x2). . . +iN (XN) we have to perform all permutations of indices, sum up the results and introduce a normalization factor. The correct expression for the wave function takes the form

Here Zn1..,ni...n3is a normalization factor. Since in the sum over permutations only interchanges of different states are included, all terms in

10.2. BOSONS

231

(10.9) are orthogonal to each other. Thus Z is just the number of configurations in the sum. Let us find it. One can introduce the so-called i-th state occupation number ni;this is the number of times the term Gi appears in the product +i,( x ~ ) (+ x 2~). .~. +iN ( x N ) (Each time, of course, with a new argument X,). This number is the number of particles with a given state number i. Its possible values are ni = 0 , 1 , . . . N . It is apparent that 00

xni=N . (10.10) i=O Let us expresss 2,,,.,, i,.,n,,., in terms of the occupation numbers. This is the number of all particle permutations N ! divided by the number of permutations of the particles in identical states:

Formula (10.9) is rather cumbersome. It would be more convenient not to write all the arguments X , because only the occupation numbers are really meaningful. Let us use representation independent notation

for the state described by the wave function (10.9). This notation contains only information about the occupation numbers of each states. Knowing these numbers allows the whole wave function (10.9) to be restored. The states (10.12) are orthogonal if the occupation numbers are different for at least one single-particle state. We shall use, for a while, the representation (10.9) in order to derive the rules of operation on the many-particle states. Our goal is to undestand how the Hamiltonian (10.1) acts on the functions (10.9). Let us start with the one-particle Hamiltonian (10.2).

One-Particle Hamiltonian Let us denote by H$) the matrix elements of the one-particle Hamiltonian (10.2):

The operator H(') acts on the basis function

232

CHAPTER 10. SECOND QUANTIZATION

Consider the action of the one-particle Hamiltonian (10.2) on the function (10.9). This operator transforms the first term in (10.9) to

and so on. Obviously the number of terms in the wave function increases, but the number of factors gi in each term is still the same. The only changes to the occupation numbers come from the change of one lltj to qhi and this change gives rise to a factor H,. Thus

Here the Bii are as yet unknown coefficients. This form should be supplemented with a convention that any term with at least one negative occupation number is zero. Indeed, if in the initial state ni = 0 then the coefficient Hij does not appear by the action of the operator f i ( l ) . To calculate the coefficients Bij we track down the contributions to a matrix element Hi,. Let us consider an arbitrary term in (10.9). It contains nj factors $ j which differ by their argument. Only these functions contribute to Hi,under the action of the '$l). Each term gives rise to nj new terms proportional to Hij in which $ j are replaced by $i with unchanged argument. Thus the total number of terms proportional to the H,. in the whole of the wave function is Znj where Z is defined by (10.11). As before these terms differ by all possible permutations of states, but a part of them is now superfluous if ni > 0. Indeed, (xl, 2 2 . . . xN) inherit all the terms proportional to H,(:) in ";I(')@ the interchanges between the initial state i in (10.9) and identical states generated by the Hamiltonian action. Such interchanges are forbidden because we permutate only different states. Let us find the number of these obsolete transmutations. The total number of terms differing by permutation of the states i and j in the initial function is ,,,,,,,,,,i,,.

As we already noted, this number is multiply by nj by the action of the Hamiltonian. At the same time the correct number of permutations

10.2. BOSONS

233

between ni + 1 particles in the state i and n j - 1 particles in the state j (generated as a result of the action of E(') in (10.16)) is (ni

(ni + n j ) ! + l)!(nj l ) ! -

Multiplying (10.17) by nj and then dividing by (10.18) we find the number of the obsolete transmutations to be ni+l. This is the additional in front of the matrix element coefficient to the factor l/,/=

H::'. Let us write this coefficient in the form

where Z,l...712+1...7L3-~... is the correct normalization factor for the wave function after the action of 'Id1). Thus we have found the coefficients Bi, and (10.16) now takes the form

Although this notation is already compact, it is only the first step towards a very convenient formalism specially created to describe many-particle wave functions.

Creation and Annihilation Operators To develop this formalism, let us introduce operators ai and a! according to the definitions

(note the square root is taken of the bigger of the new and the old state i occupation numbers). Traditionally the operators ai and a! are written without a hat. Operator ai decreases the state i occupation number by one. The occupation numbers of all other states do not change. The action of ai gives zero if ni = 0. This operator is the so-called annihilation operator of a particle in the state i. Operator a! increases the state i occupation number by one. This operator is called the creation operator.

234

CHAPTER 10. SECOND QUANTIZATION

It is easy to see that the operators introduced here are Hermitian conjugates (this has already been reflected in the notation a f ) :

In the last term here the operator af acts to the left on the state i with occupation number mi and 6,, is the Kronecker symbol. It follows from the definitions (10.21), (10.22) and the orthogonality of the states (10.12) that [ai,a;] = . (10.24) Let us introduce one more definition. The product fii alai is called the particle number operator or occupation number operator of the state i because (10.25) a f a i ( .. . n i . . .) = nil.. . ni . . .) .

>

In accordance with its meaning, ni 0. Formally, this property is secured by the zero coefficient when ai acts on the state ni = 0. In other words, it is impossible to construct a state with a negative occupation number by any consecutive action of the annihilation operator. Using expression (10.20) let us write the one-particle Hamiltonian 'H(') in terms of the annihilation and creation operators. This yields

This formula turns to be especially simple if the wavefunctions & ( X ) , n = 0 , 1 , 2 . . . are the eigenfunctions of the one-particle Schrodinger o p erator H('). In this case H(;) = Eibij and (10.26) takes the form

The meaning of this expression is quite obvious. The Hamiltonian of a system of non-interacting particles is equal to the sum of the energies of the one-particle states multiplied by the corresponding occupation numbers. Note that the latter are operators and the numbers of particles in any state are not fixed. Let us investigate the question of the completeness of the set of all states with all possible occupation numbers (and variable total number

10.2. BOSONS

235

of particles N). These states form the so-called Fock space. The annihilation and creation operators act on this space. Obviously N is increased by one under the action of the creation operator (10.22) and there is no upper bound on its value. As was mentioned above, there is a restriction on the lowest possible value of N because all occupation numbers ni 2 0. Thus by consecutive action of the annihilation operators, one can obtain a single state without any particles. Note that the Fock space without this state is incomplete with respect to the action of the annihilation operators. Let us denote this state as

This is the vacuum state or vacuum. F'uther action of the annihilation operator on this state gives zero while one can construct all states by acting with appropriate creation operators on the vacuum. It would be impossible to describe a system with variable numbers of particles without including the vacuum state in the complete set. However, it is impossible to write a wave function of the vacuum state in analogy with (10.9).

Total Hamiltonian Let us consider the action of the two-particles Hamiltonian (10.3) on the wave functions (10.9), (10.12). In principle, we have to repeat the combinatorial considerations of section 10.2. However, such an approach is getting rather cumbersome and awkward. Let us apply a more formal approach using the already derived result (10.26). Let us expand the two-particle interaction potential in (10.3) on a complete set of functions wi(x) (this set has nothing to do with functions (10.7)):

Let us consider the functions wi as the one-particles Hamiltonian, then, applying the relation (10.26), we obtain

) ~ (wj),, ~ are the matrix elements of wi(x) and wj(x) sandHere ( w ~ and wiched between the one-particle basis functions. In order to rewrite this

CHAPTER 10. SECOND QUANTIZATION

236

expression in its final form, we have to express it in terms of the matrix of the operator V ( ~ ) ( X x2) ~ , which are elements

vL~~,

A substitution of this expression into (10.30) results in

It should not be forgotten to check whether this expression is unique. One could change the order of aial and aka, in (10.30). This would give rise to two additional sums in (10.32) which included the commutators [al,ak] and [a,, ail. According to (10.24) these commutators are proportional to S-symbols. The additional terms in (10.32) would, therefore, include matrix elements of the two-particle potential with coinciding indexes from the first and the second pairs. One of two such terms would

Note it has the form of a correction to the one-particle Hamiltonian. As follows from the middle expression in (10.31) and because of the completeness of the eigenfunction set $i, the summation over l results in this additional term being proportional to v ( ~ ) ( x , x ) .This is zero according to (10.5). A generalization of the procedure considered above for the interaction potentials of higher number of particles is obvious:

The meaning of these relations is rather clear. An n-particle Hamiltonian annihilates n particles in all ossible states and create n particles in other states with an amplitude v ; ~ ~ ~a.,-.kn., , ~ ~ ~ ~ ~

P

The Field Operator Let us introduce an operator

10.2. BOSONS

237

This operator, when it acts on the vacuum, creates a particle at a point X. Let us check it:

In the coordinate representation the one-particle state wave function which appears in (10.36) is Gi(y). Exploiting the completeness of the set we see that the right-hand side of the expression (10.36) is just S(Y- 4 . An operator 03

+(~)=X$i(x)ai

,

(10.37)

i=o

conjugate to +?(X), is called a field operator. Its action is to annihilate a particle at a point X:

Here we have used the action of +(X) on one-particle states:

It follows immediately from the definitions (10.35), (10.37) and the commutation relations (10.24) that

We are going to re-express the Hamiltonians (10.26), (10.32) and, generally, (10.34) in terms of the field operator. To this end we invert the expressions (10.35) and (10.37) as

and substitute these relations into (lO.26), (10.32) and (10.34). This results in the following form for the one-particle Hamiltonian

CHAPTER 10. SECOND QUANTIZATION

238

It follows from the definition of the matrix element (10.13) that the expression in squared brackets is S(x1 - x ~ ) H ( ~ ) ( xThen ~ ) . we obtain, finally, A(')= d x ~ t ( x ) ~ ( 1 ) ( x ) 8 (.x ) (10.44)

/

In the same way we arrive at the following expression for the twoparticle Hamiltonian

The generalization of these formulae for many-particle Hamiltonians is obvious. Let us now write the equation of motion for the time-dependent field operator in the Hamiltonian representation. We start from the basic equation d (10.46) ih-@(X, t ) = ["ri,$ ( X , t ) ] ,

-

at

substitute in it the expansion (10.1) - (10.4) for f i in the n-particle Hamiltonians and use the results (10.44) (10.45) along with the commutation relations (10.40). The result is the operator equation -

d ih-$(X, t ) = H(')@(x,t )

at

+ J d y ~ t ( yt ,) ~ ( ~ -) (y )x8 ( y ,t ) 8 ( x ,t ) + . . .

+A / dy, . . .dyn-,8t(yl, t ). . . 8t(yn-, ,t ) (n l ) !

(10.47)

-

v ( ~ ). .(. yn)8(y1, Y~ t )