GENERAL PRINCIPLES OF QUANTUM FIELD THEORY

Mathematical Physics and Applied Mathematics

Editors:

M. Flato, Universite de Bourgogne, Dijon, France

R. R1\czka, Institute of Nuclear Research, Warsaw, Poland

With the collaboration of"

M. Guenin, Institut de Physique TMorique, Geneva, Switzerland

D. Stemheimer, College de France, Paris, France

Volume 10

GENERAL PRINCIPLES OF QUANTUM FIELD THEORY

by

N. N. BOGOLUBOV U.S.S.R. Academy o/Sciences and Moscow State University. U.S.S.R.

A. A. LOGUNOV U.S.S.R. Academy o/Sciences and Moscow State University. U.S.S.R.

A. I. OKSAK Institute/or High Energy Physics.

Moscow. U.S.S.R.

and

I. T. TODOROV Bulgarian Academy 0/ Sciences and

Bulgarian Institute/or Nuclear Research and Nuclear Energy. Sofia. Bulgaria

Translatedfrom the Russian by G. G. Gould

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

Library of Congress Cataloging in Publication Data

Ollshchle printsipy kvantovol teori i poi fa. Engl ish. General princlples of quantum field theory / b~ N.N. Bogolubov

ret al.l ; translated from the Russian by G.G. Gould. p. cm. -- (Mathematical physics and applied mathematics v.

10 ) Translation of: Obshchie printsipy kvantovol teorii polia. ISBN-13: 978-94-0 1 0-6707 -2 e-ISBN-13 :978-94-009-0491-0 DOl: 10.1007/978-94-009-0491-0

1. Quantum field theory. Nikolaevich), 1909- II. QC174.45.02613 1990 530.1·43--dc20

ISB N-13: 978-94-0 1 0-6707 -2

I. BogolfUbov, N. N. (Nikolal Title. III. Series.

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . .. XUl

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . xv The place of the axiomatic approach in quantum field theory (xv). The layout of this book (xviii).

Part I ELEMENTS OF FUNCTIONAL ANALYSIS AND THE THEORY OF FUNCTIONS 1

Synopsis. . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 1. Preliminaries on Functional Analysis . . . . . . . 3 1.1. Normed Spaces . . . . . . . . . . . . . . . . . . . . . . 3

A. Linear spaces (3). B. Direct sum and tensor product of linear spaces (5). C. Normed spaces (7). D. Hilbert spaces (8). E. Direct sum and tensor product of Hilbert spaces (12). F. Linear functionals and dual spaces (14).

1.2. Locally Convex Spaces . . . . . . . . . . . . . . . . . . . . . . 16 A. Equivalent systems of seminorms. Structure of LCS's (16). B. Frechet Spaces (17). C. Examples (18).

1.3. Linear Operators and Linear Functionals in Frechet Spaces . . . . . . 20 A. Continuous maps of LCS's (20). B. The uniform boundedness principle. The weak and weak* topologies (22). C. The closed graph and open mapping theorems (23).

1.4. Operators in Hilbert Space . . . . . . . . . . . . . . . . . . . . 25 A. The notion of an (unbounded) self-adjoint operator (25). B. Isometric, unitary and anti-unitary operators (28). C. The spectral theory of self­adjoint and unitary operators (29).

1.5. Algebras with Involution. C·-Algebras . . . . . . . . . . . . . . . 31 A. Definition and elementary properties (31). B. The spectrum (33). C. Pos­itivefunctionals (34). D. Representations (36). E. Trace class operators (41). F. Von Neumann algebras (43).

CHAPTER 2. The Technique of Generalized Functions . . . . . . . . 46 2.1. The Concept of a Generalized Function . . . . . . . . . . . . . . . 46

A. Functional definition (46). B. Definition in terms of fundamental se­quences (49). C. Local properties of generalized functions (51).

2.2. Transformation of Arguments and Differentiation ......... 53 A. Change of variables in a generalized function (53). B. Differentiation of generalized functions. Examples (54).

2.3. Multiplication of a Generalized Function by a Smooth Function . . 56 A. The problem underlying multiplication of generalized functions. The concept of a multiplicator (56). B. The division problem (58).

2.4. The Kernel Theorem. Tensor Products of Generalized Functions. . 61 A. Bilinear functionals on spaces of type S (61). B. Tensor products (62).

VI CONTENTS

2.5. Fourier Transform and Convolution ................ 63 A. Fourier transform of test functions (63). B. Fourier transform of gen­eralized functions (65). C. Convolutes (66). D. Generalized functions of integrable type (67). E. Convolution of generalized functions (70).

2.6. Generalized Functions Dependent on a Parameter . . . . . . . . . . 72 A. General information (72). B. Restriction of generalized functions (74). C. More on the multiplication of generalized functions (76).

2.7. Vector- and Operator-Valued Generalized Functions . . . . . . . . . 78 A. Generalized functions with values in Hilbert space (78). B. Operator­valued generalized functions (80). C. The notion of a generalized eigenvector (82).

Appendix A. Generalized Functions on Subsets of Rn . . . . . . . . . . . 83 A.l. Generalized functions on an open subset (83). A.2. Generalized func­tions on canonically closed regular subsets (84). A.3. Application: general­ized functions on the compactified sets [A, (0), Roo, [-00, +(0) (86).

Appendix B. The Laplace Transform of Generalized Functions . . . . . . . 89 B.l. The Laplace transform as an analytic function in the complex plane (89). B.2. The case of a generalized function with support in a pointed cone (96). B.3. Example: generalized functions of retarded type (98). BA. Boundary values of the Laplace transform (99). B.5. Example: the "mathe­matics" of dispersion relations (103). B.6. Restriction of the Laplace trans­form (105).

Appendix C. Homogeneous Generalized Functions ............. 106 o

C.l. Homogeneous generalized functions in Rn (106). C.2. The single real variable case (109). C.3. Extension of homogeneous generalized functions (110). CA. Application to covariant homogeneous generalized functions (113). C.5. Homogeneous generalized functions in the complex plane (114).

CHAPTER 3. Lorentz-Covariant Generalized Functions . . . . . . . . 118 3.1. The Lorentz Group ........................ 118

A. The geometry of Minkowski space (118). B. Definition of the general Lorentz group and its connected components (119). C. The universal cov­ering of the group L~ (121). D. Finite-dimensional representations of the group SL(2, C) (125). E. Simply reducible finite-dimensional representa­tions of SL(2, C). Spatial reflection (128).

3.2. Lorentz-Invariant Generalized Functions in Minkowski Space .. . . . 131 A. Definition (131). B. Even invariant generalized functions. Invariant gen­eralized functions with support at a point (132). C. Odd invariant general­ized functions (136).

3.3. Lorentz-Covariant Generalized Functions in Minkowski Space . . . . . 138 A. Definition (138). B. Structure of covariant generalized functions (139).

304. The Case of Several Vector Variables . . . . . . . . . . . . . . . . 143 A. Generalized functions that are invariant with respect to a compact group (143). B. Generalized functions that are covariant with respect to a compact group (149). C. Applications to Lorentz-invariant and Lorentz-covariant generalized functions (155).

Appendix D. Vocabulary of Lie Groups and their Representations ...... 159

CONTENTS

D.l. Abstract groups. Algebraic properties (159). D.2. Lie groups (160). D.3. Lie algebras (162). D.4. Relation between Lie groups and Lie algebras (163). D.5. Local Lie groups. Canonical parametrization. Lie's theorems (164). D.6. Linear representations (166). D.7. Adjoint and co-adjoint rep­resentations. Killing forms (167).

Vll

CHAPTER 4. The Jost-Lehmann-Dyson Representation . . . . . . . . 170 4.1. Relation between the JLD Representation and the Wave Equation . . . 170

A. Preliminary remarks (170). B. Outline of the derivation (171). C. De­parture into six-dimensional space (173).

4.2. Properties of Solutions of the d'Alembert Equation in S' ....... 175 A. Notation (175). B. F\mdamental Solution of the Cauchy Problem (176). C. Cauchy problem on a spacelike hypersunace; Huygens' principle (179). D. The Asgeirsson formula and its applications (183).

4.3. Derivation of the Jost-Lehmann-Dyson Formula ........... 185 A. Construction of the spectral function (185). B. Further properties of the support of the spectral function (188). C. Examples (192). D. Representa­tions for generalized functions of retarded and advanced types (193).

CHAPTER 5. Analytic Functions of Several Complex Variables .... 197 5.1. Properties of Holomorphic Functions. Plurisubhannonic Functions . . . 197

A. Space of holomorphic functions (197). B. Holomorphy and analyticity (199). C. Analytic continuation (200). D. Generalized principle of ana­lytic continuation; "edge of the wedge" theorem (204). E. Holomorphic distributions (207). F. Invariant and covariant analytic functions (209). G. Plurisubhannonic functions (211).

5.2. Domains of Holomorphy ..................... 215 A. Holomorphic convexity (215). B. Pseudo-convexity (217). C. Modified principle of continuity (219). D. Single-sheeted envelopes of holomorphy (221). E. Invariant domains (223). F. An example of hoi om orphic extension (226).

Part II RELATIVISTIC QUANTUM SYSTEMS 231

Synopsis . . . . . . . . . . . . 231

CHAPTER 6. Algebra of Observables and State Space 233 6.1. Algebraic Formulation of Quantum Theory . . . . . 233

A. Algebra of observables. States (233). B. Transition probability (235). C. Relationship to representations (236).

6.2. Superselection Rules . . . . . . . . . . . . . . . . . . . . . . . 239

A. The role of pure vector states (239). B. Standard superselection rules (243). C. Connection with gauge groups (245). D. Example of non-abelian gauge groups (247).

6.3. Symmetries in the Algebraic Approach ............... 249 A. The concept of symmetry (249). B. Proof and discussion of Wigner's theorem (252). C. Symmetry groups (256).

viii CONTENTS

6.4. Canonical Commutation Relations . . . . . . . . . . . . . . . . . 260 A. The role of the Schrodinger representation (260). B. Infinite number of degrees of freedom (263). C. Proof of von Neumann's uniqueness theorem (267).

CHAPTER 7. Relativistic Invariance in Quantum Theory ....... 270 7.1. The Poincare Group ....................... 270

A. Definition (270). B. Reflections (271). C. The Lie algebra of the Poincare group (272).

7.2. Unitary Representations of the Proper Poincare Group ........ 274 A. Poincare invariance condition (274). B. Classification of irreducible repre­sentations of Po. Spectral principle (275). C. Description of representations corresponding to particles with positive mass (280). D. Manifestly covariant realization of "physical" irreducible representations (284).

7.3. Fock Space of Relativistic Particles. . . . . . . . . . . . . . . . . 288 A. Second quantization space (288). B. Connection with (anti-)commutation relations (292). C. Covariant creation and annihilation operators (296). D. Symmetries of the general Poincare group (299). E. Relativistic scattering matrix (302). F. Kinematics of two-particle processes (307).

Appendix E. Four-Component Spinors and the Dirac Equation ....... 310 E.1. Clifford algebra over Minkowski space (310). E.2. Spinor representation of the Lorentz group; various realizations of the -y-matrices (312). E.3. Dirac equation; representations of the Poincare group with spin 1/2 (314).

Part III LOCAL QUANTUM FIELDS AND WIGHTMAN FUNCTIONS . 318

Synopsis . . . . . . . . . . . . . . . . 318

CHAPTER 8. The Wightman Formalism 321 8.1. Quantwn Field Systems ...... 321

A. Concept of localization (321). B. Principle of local commutativity (322). C. "FUndamental" fields and "physical" fields (323).

8.2. Definition and Properties of a Local Quantum Field . . . . . . . . . 324 A. Wightman's axioms (324). B. Discussion of the axioms (325). C. Ir­reducibility of fields (329). D. Separating property of the vacuwn vector (331).

8.3. Wightman FUnctions . . . . . . . . . . . . . . . . . . . . . . . 332 A. Characteristic properties of Wightman functions (332). B. Kiillen­Lehmann representation for a scalar field (335). C. Reconstruction of the theory from the Wightman functional (337).

8.4. Examples: Free and Generalized Free Fields . . . . . . . . . . . . . 340 A. Free scalar neutral field (340). B. Free scalar charged field (345). C. Free Dirac field (348). D. Generalized free fields (351).

Appendix F. Swnmary of Invariant Solutions and Green's Functions of the Klein­Gordon Equation . . . . . . . . . . . . . . . . . . . . . 353

Appendix G. General Form of the Covariant Two-Point Function ...... 355 G.1. Covariant decompositions compatible with locality (355). G.2. De­composition with respect to spin (356).

CONTENTS ix

CHAPTER 9. Analytic Properties of Wightman Functions in Coordinate Space .................. 359

9.1. Bargmann-Hall-Wightman Theorem and its Corollaries ........ 359 A. Complex Lorentz transformations (359). B. Lorentz-covariant analytic functions in the past tube (362). C. Real points of the extended tube (366). D. Analyticity of Wightman functions in a symmetrized tube (368). E. Global nature of locality (371).

9.2. TCP-Theorem ......................... 375 A. TCP-invariance (375). B. Weak locality (378). C. Borchers classes; the notion of a local composite field (378).

9.3. Connection between Spin and Statistics . . . . . . . . . . . . . . . 381 A. Statement of the results (381). B. Necessary conditions for anomalous commutation relations (383). C. Reduction of w to canonical form (385). D. Construction of the Klein transformation (387).

904. Equal-Time Commutation Relations. Haag's Theorem . . . . . . . . 388 A. Three-dimensional version of Haag's theorem (388). B. Haag's theorem in the relativistic theory (391). Comments on Haag's theorem (392).

9.5. Euclidean Green's Functions .................... 394 A. Group of rotations of four-dimensional Euclidean space (394). B. Prop­erties of the Schwinger functions (396). C. Reconstruction theorem in terms of Schwinger functions (400).

Appendix H. Parastatistics . . . . . . . . . . . . . . . . . . . . . . . 403 H.l. Free parafields and paracommutation relations (403). H.2. Comment on the TCP-theorem and the connection between spin and parastatistics for local parafields (406).

Appendix I. Infinite-Component Fields ................. 407 1.1. Elementary representation of SL(2, C) (407). 1.2. Concept of a quantum IFC (408). 1.3. Covariant structure of the two-point function. Infinite degeneracy of mass with respect to spin (410). 104. Absence of I.P+ -covariance and connection between spin and statistics in ICF models (413).

CHAPTER 10. Fields in an Indefinite Metric . . . . . . . . . . . . . 417 10.1. Pseudo-Wightman Formalism ................... 417

A. Pseudo-Hilbert space (417). B. Axioms of pseudo-Wightman type (420). C. Vacuum sector and charged states (423). D. Physical subspace of pseudo­Hilbert space (427).

10.2. Abelian Models with Gauge Invariance of the 2nd Kind . . . . . . . . 428 A. The field of the dipole ghost and the gradient model (428). B. Local formulation of quantum electrodynamics (434).

10.3. Internal Symmetries . . . . . . . . . . . . . . . . . . . . . . . 440 A. Symmetries and currents in the Wightman formalism (440). B. Gold­stone's theorem (443). C. Spontaneous symmetry breaking in abelian gauge theories (446).

CHAPTER 11. Examples: Explicitly Soluble Two-Dimensional Models 450 11.1. Free Scalar Massless Field in Two-Dimensional Space-Time . . . . . . 450

A. One-dimensional non-canonical scalar field (450). B. Physical represen­tation (454). C. Free "quark" fields; bosonization offermions (461). D. Free scalar massless "ghost" field (467).

x CONTENTS

11.2. The Thirring Model . . . .. . ............ 469 A. Solution of the field equation (469). B. Currents and charges; vacuum representation (473).

11.3. The Schwinger Model ...................... 474 A. Solution in the Lorentz gauge (474). B. Vacuum functional (480). C. Phys­ical fields; observables (481).

Part IV COLLISION THEORY. AXIOMATIC THEORY OF THE S-MATRIX 484

Synopsis . . . . . . . . . . . . . . . . . . . . . . 484

CHAPTER 12. Haag-Ruelle Scattering Theory 486 12.1. Scheme of the Quantum Field Theory of Scattering 486

A. The one-particle problem in quantum field theory (486). B. Construc­tion of in- and out-states (488). C. S-matrix and TCP-operators in the asymptotically complete theory (489).

12.2. Existence of Asymptotic States . . . . . . . . . . . . . . . . . . 491 A. Truncated vacuum expectation values (491). B. Strengthened cluster property (495). C. Spread of relativistic wave packets (497). D. Proof of the main result (501).

CHAPTER 13. Lehmann-Symanzik-Zimmermann Formalism. . . . . . 503 13.1. Basic Concepts ... . . . . . . . . . . . . . . . . . . . . . . 503

A. T-products of fields (503). B. Retarded products (509). C. LSZ axioms (512).

13.2. Asymptotic Conditions and Reduction Formulae . . . . . . . . . . . 515 A. LSZ asymptotic conditions (515). B. Yang-Feldman equations (520). C. Partial reduction formulae (522). D. Reduction formulae for the scatter-ing matrix (526).

CHAPTER 14. The S-Matrix Method . . . . . . . . . . . . . . . . . 530 14.1. S-Matrix Formulation of the Basic Requirements of the Local Theory . 530

A. The concept of extending the S-matrix beyond the mass shell (530). B. Choice of the class of test functions (534). C. Axioms of the S-matrix approach (535). D. Radiation operators; current (537).

14.2. Fields in the Asymptotic Representation . . . . . . . . . . . . . . 540 A. Construction of quantum fields and their T-products (540). B. Fulfillment of the LSZ axioms (544).

Part V CAUSALITY AND THE SPECTRAL PROPEIITY: THE ORIGINS OF THE ANALYTIC PROPERTIES OF THE SCATTERING AMPLITUDE . . . . . . . . . . . . . . 546

Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

CHAPTER 15. Analyticity with respect to Momentum Transfer and Dis-persion Relations ................... 548

15.1. The Lehmann Small Ellipse .................... 548 A. Introductory remarks (548). B. JLD representation for retarded and advanced (anti)commutators (551). C. Analyticity with respect to t (553).

CONTENTS Xl

15.2. Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . 556 A. The main steps for the derivation of the dispersion relations (556). B. Pas­sage to the complex domain with respect to the momentap2,P4 (557). C. Dis­persion relation for non-physical "masses" (560). D. Analytic properties of the absorptive part of the amplitude (563). E. Dispersion relation on the mass shell (570).

CHAPTER 16. Analytic Properties of the Four-Point Green's Function 574 16.1. Generalized Retarded Functions . . . . . . . . . . . . . . . .. 574

A. Generalized retarded products (574). B. Supports in x-space (576). 16.2. Four-Point Green's Functions . . . . . . . . . . . . . . . 579

A. Notation (579). B. Domains of coincidence in p-space (581). C. Stein­mann identities (583). D. Analyticity near physical points (586).

16.3. Crossing Relation . . . . . . . . . . . . . . . . . . . . . . . . 589 A. Statement of the result (589). B. The case of imaginary "masses" (590). C. Analytic continuation with respect to the mass variables (591). D. Pas­sage to the mass shell (596).

Appendix J. The Role of Unitarity . . . . . . . . . . . . . . . . . . . 598 J.1. Partial wave decomposition of the scattering amplitude of a two-particle process (598). J.2. Analytic continuation of the dispersion relation with respect to t (603).

CHAPTER 17. Consequences for High-Energy Elementary Processes . 608 17.1. Restrictions on the Behaviour of Cross Sections at High Energies . 608

A. Froissart bound (608). B. Comparison of the cross sections of the inter­action of a particle and an antiparticle with the same target (613).

17.2. Inclusive Processes . . . . . . . . . . . . . . . . . . . . . . . . 617 A. Physical characteristics of inclusive processes (617). B. Analytic prop­erties of differential cross sections with respect to angular variables (621). C. Asymptotic estimates (624).

Commentary on the Bibliography and References 626 Bibliography . . 644 References . . . 649 Index of Notation 683 Index . . . . . 687

Preface

The majority of the "memorable" results of relativistic quantum theory were obtained within the framework of the local quantum field approach. The explanation of the basic principles of the local theory and its mathematical structure has left its mark on all modern activity in this area.

Originally, the axiomatic approach arose from attempts to give a mathematical meaning to the quantum field theory of strong interactions (of Yukawa type). The fields in such a theory are realized by operators in Hilbert space with a positive Poincare-invariant scalar product. This "classical" part of the axiomatic approach attained its modern form as far back as the sixties. *

It has retained its importance even to this day, in spite of the fact that nowadays the main prospects for the description of the electro-weak and strong interactions are in connection with the theory of gauge fields. In fact, from the point of view of the quark model, the theory of strong interactions of Wightman type was obtained by restricting attention to just the "physical" local operators (such as hadronic fields consisting of ''fundamental'' quark fields) acting in a Hilbert space of physical states. In principle, there are enough such "physical" fields for a description of hadronic physics, although this means that one must reject the traditional local Lagrangian formalism. (The connection is restored in the approximation of low-energy "phe­nomenological" Lagrangians.) Therefore our desire to include in our discussion such "unobservable" (that is, gauge dependent) fundamental fields as 4-vector potentials and the local field of an electron or quark, which are used in practical calculations in perturbation theory, necessitates a certain broadening and modification of the Wightman scheme.

This monograph is devoted to a logical account of the principles of local quantum field theory, including the gauge theories with indefinite metric. Although the amount of space allotted to the gauge theories proper is relatively small, the entire make-up of this book gives due attention to their inclusion. We have laid great emphasis on the algebraic approach (by comparison with [BBl).

Along with the predominantly purely theoretical material, we have included in the last part applications of the techniques developed to the derivation of dispersion relations and an examination of the behaviour of the cross sections of the interaction of elementary particles at high energies.

* Even at that time it was the subject of specialist monographs [816], [J3], [B8]. The present book

was originally planned as a second edition of [B8]. However, the overall project grew considerably as

it evolved, and this current new book is the result. (The content of the book is broader than that

suggested by the title: the mathematical methods of quantum field theory are set forth alongside the

principles.)

xiii

xiv PREFACE

This book is intended for theoretical physicists and mathematicians interested in the problems of quantum field theory and mathematical physics. Although we have tried as far as possible to make the exposition independent of other sources, this book cannot be recommended for a first acquaintance with quantum theory. Apart from a familiarity with a regular course in quantum mechanics and the general notions of elementary particles and their interactions, it is helpful to have some ideas about the fundamentals of quantum field theory, for example, within the framework of [BIl] (or the first four chapters of [B10]; see also [H3], [S5]). The ancillary mathematical apparatus that falls outside the scope of the compulsory courses in physics faculties (an account of functional analysis, the theory of generalized functions, the theory of analytic functions of several variables and the vocabulary of the theory of Lie groups and their applications) is set out in the text.

It is our pleasant duty to express our thanks to our colleagues of the Steklov Mathematical Institute, the Joint Institute of Nuclear Research (Dubna), the Insti­tute for High Energy Physics (Serpukhov), the Institute of Nuclear Research of the Academy of Sciences of the USSR, and the Institute of Nuclear Research and Nu­clear Energy of the Bulgarian Academy of Sciences for numerous helpful discussions. One of the authors (I.T. Todorov) expresses his gratitude to M.K. Polivanov for his hospitality at the Steklov Mathematical Institute during the final stages of the work on this book.

January 1984.

Introduction

THE PLACE OF THE AXIOMATIC APPROACH IN QUANTUM FIELD THEORY

In physics as well as in mathematics, the role of the axiomatic method is twofold. On the one hand it clarifies the logical foundations of a given topic by showing what the independent premises are (as , for example, in Euclidean geometry or Newtonian mechanics), and so opening up new possibilities. On the other hand, by isolating their fundamental structures, it enables one to find relationships between branches of science that at first glance appear different. Such an approach is typical of mod­ern mathematics where it has given rise to a number of new areas. The role of the axiomatic method is less in evidence in theoretical physics; but even here, it gained wide dissemination as far back as Newton's time, both as a method of systematizing known results and as a method of describing new phenomena by means of formal schemes developed earlier. At the basis of every axiomatic physical discipline there lie deep physical ideas that are expressible in a mathematically consistent form. It is remarkable that the formal schemes sometimes contain more than was originally invested in them (examples: the action principle, the canonical formalism, the Gibbs ensemble). As a result of this there is a reverse influence of the mathematical struc­tures of theoretical physics on the formation of physical ideas.

In the thirties, the axiomatic method was successfully applied (in the work of Jordan, von Neumann and Wigner) to quantum mechanics (see [V6); we can recom­mend the books [MI], [K5] as examples of the later development in this direction). The structural analysis of quantum mechanics has led to a remarkable synthesis of physical and mathematical ideas, which has become part of the generally accepted formalism of quantum theory and has influenced the development of mathematics. (Under the stimulus of quantum theory, new branches of functional analysis have come into being: the theory of operators in Hilbert space, operator algebras, unitary representations of groups, harmonic analysis.) The mathematical problems of quan­tum theory have in large measure determined the interests of modern mathematical physics.

In quantum field theory, the axiomatic approach relates to the activities of fifty years in connection with the successes and difficulties of the method of perturbations in Lagrangian quantum field theory. The apparatus of renormalizations developed in the perturbation method led to brilliant success in quantum electrodynamics where the parameter of the expansion (the coupling constant) is small, so that it was pos­sible to restrict attention to the first terms of the perturbation-theory series for a comparison with experiment. This method, however, has proved to be unsuitable for a description of the strong interactions of elementary particles (where the effective

xv

xvi INTRODUCTION

coupling constant is greater than unity). The theory of renormalizations yields some­thing better; this is a formal infinite series for the solutions of quantum equations in the class of physically interesting renormalizable Lagrangians. The axiomatic ap­proach was called upon in the first instance to answer the question: what is hidden behind these formal infinite series? For this purpose, the creation of new principles of quantum theory were required that were different from the Lagrangian method with its perturbation theory. *

The first attempt to go beyond the framework of the Lagrangian approach goes back to Heisenberg (1943). In analysing what, in fact, is measured in the physics of elementary particles, Heisenberg came to the conclusion that the basic observable is the scattering matrix; he suggested that a theory should be constructed directly in terms of the elements of the S-matrix which would do away with the notion of a field, the adiabatic hypothesis of the exclusion of the interaction (which was at the basis of perturbation theory) and so on. It turned out, however, that the Heisenberg approach was too radical. The complete banishment of the local quantities of the theory deprives us of the possibility of considering the evolution of the system in space and time by taking the causality principle into account. Therefore the development of the axiomatic approach proceeded via the study of local quantities; and at the very beginning (in the 50's) at least three lines of approach took shape.

The Wightman formalism singles out as the basic objects the most regular quan­tities, namely, the quantized fields in the Heisenberg representation and the vacuum expectation values of their ordinary products (the Wightman functions are analogous to the correlation functions in statistical physics). In principle, the Wightman func­tions enable one to extract all the physical information contained within the theory. In particular, the asymptotic condition (which was originally stated by Haag as one of the postulates of the theory) and the scattering matrix are derived concepts. (Only the asymptotic completeness condition remains as an independent hypothesis.)

In the Lehmann-Symanzik-Zimmermann (LSZ) approach, the basic concepts are the chronological (or T- ) products of the fields (also their vacuum expectation values - the Green's functions) and the asymptotic condition. In this connection it can be shown that it makes no sense to talk about an independent approach, since the Green's functions and the T-products are formally expressed in terms of the Wight­man functions and the ordinary derivatives of the Heisenberg fields. In fact, this formal definition is not mathematically well defined since it contains the product of an (operator-valued) generalized function with discontinuous 8-functions (which leads to divergences of the same type as in perturbation theory); and this blocks the application of the alternative point of view, that the T-products (or, equivalently, the retarded products) are the primitive objects of the theory along with the Heisenberg fields and are defined only indirectly by means of a certain set of properties. This sort of approach is not the most economical one (since the scattering theory can, in principle, be developed without introducing the notion of T-product in advance; see Ch.12), but it is convenient in practice and brings us close to the traditional Lagrangian method.

The Bogolubov-Medvedev-Polivanov approach, in which the basic object is the

* It is appropriate here to recall the analogous situation that arose in probability theory: the axiom

scheme of Kolmogorov (put forward at the end of the 20's) brought about a decisive restructuring of

this discipline on completely mathematical foundations.

INTRODUCTION xvii

extended S-matrix (beyond the mass shell), is superficially closer to Heisenberg's original programme. More interestingly, it is closely related to the LSZ approach which occurred in parallel, since the extended S-matrix is, in essence, the generating functional of the T-products of the current operators. Whereas the derivation of the reduction formulae (which express the S-matrix elements in terms of Green's functions) are non-trivial in the Wightman or LSZ formalism, in the S-matrix method these formulae are derived automatically by formally taking the variational derivative of the S-operator (or the radiation operators expressed in terms of it) with respect to the asymptotic fields. The practical convenience of the S-matrix method is afforded by the effective calculation of the complicated combinatorics in operations of this kind. It is no accident that the dispersion relations were first proved by this means.

In the 60's quite a considerable number of relationships manifested themselves between the approaches which historically had independent origins. Disregarding certain mathematical niceties and technical differences, we can say that all three approaches are applicable with equal success to the class of quantum field theories with the asymptotic completeness condition. Since this condition is made essential use of only in the last two approaches, the Wightman formalism is somewhat more general.

Alongside this, there arose in the 60's an even more general axiomatic line of development. In the work of Haag, Araki and Kastler, the principles of local quantum theory were stated in the language of the algebraic approach originating from the work of von Neumann and Segal. The significance of this approach for the general statement of the problems of the physics of systems with an infinite number of degrees offreedom is on the increase. In particular, it provides a method of describing gauge theories, spontaneous breakdown of symmetries (and in statistical physics, phase transitions, although the latter are beyond the scope of this book).

Of the various routes along which modern elementary particle theory has moved, the axiomatic approach has occupied a relatively small place (especially if we judge from the number of publications). However, the short life of the phenomenological results of theories based on a number of special assumptions, makes the results arising from the fundamental principles of quantum theory all the more interesting; these principles are: relativistic invariance, the existence of a complete system of states with positive energy, and causality.

There is to date no complete answer to the fundamental question which resulted in the development of the axiomatic approach; namely, are the principles of relativistic local quantum theory (in four-dimensional space-time) compatible with the existence of a non-trivial scattering matrix? Here, considerable advances have been made in the last decade. An essentially new area has emerged, namely, constructive quantum field theory (see 1GB]); as a result of its development, non-trivial models have been constructed in two- and three-dimensional space-time. However, these models are super-renormalizable* and do not require an infinite renormalization of the charge. If one is to consider realistic renormalizable models in four-dimensional space-time, then essentially new methods are required. Even so, the successes obtained along this route, together with the discovery of renormalizability and asymptotic freedom

* That is, they have a finite number of primitively divergent (one-particle irreducible) diagrams

(see, for example, the definition in [S5j,§16).

xviii INTRODUCTION

of the non-abelian gauge fields in the traditional (formal) approach* have opened up further prospects for the development of local quantum field theory.

THE LAYOUT OF THIS BOOK

Part I is of a preliminary character: it contains various topics in functional analysis and the theory of functions required for the subsequent parts. In places the account is somewhat terse and is not, of course, a substitute for a systematic exposition of all the questions touched upon: some proofs are omitted (and replaced by detailed references to the literature), and not all of the definitions and statements are accompanied by covering comments. The systematic account begins in Part II. After an account of the fundamental ideas of the quantum phenomenology (given in algebraic language) we formulate those principles of relativistic quantum theory that do not require the introduction of local quantities: the invariance principle with respect to the Poincare group (that is, the non-homogeneous Lorentz group) and the spectral condition (that is, the existence of a complete system of physical states with non-negative energy).

In Part III we deal, in the main, with the Wightman formulation of the theory of local quantized fields. Examples of free (and generalized free) fields are analysed in detail. A number of general results are given here: the CPT theorem, the theorem on the connection of spin with statistics and the theorems of Haag and Goldstone. One of the chapters is devoted to a generalization of the Wightman formalism for fields with indefinite metric (the importance of this class of theories is that the gauge theories in local covariant gauges come out of it). The chapter on two-dimensional explicitly soluble models serves as an illustration, as it were, of the Wightman formalism and its generalizations.

Part IV contains a survey of the Haag-Ruelle scattering theory, its connection with the LSZ theory, also the S-matrix approach.

In Part V, the apparatus developed in the earlier parts is applied to the analytic properties of the amplitudes of elementary processes. Notwithstanding its simplicity, the idea of analyticity has turned out to have had a very fruitful influence on the development of the theory of strong interactions. (In this connection, we note at least the dual resonance models.) In our account of this, we present the basic results deduced from the principles of quantum field theory; these are, in the first instance, the analyticity with respect to the cosine of the scattering angle, the dispersion relations, and the crossing. An extensive literature is devoted to the applications of the results of local quantum field theory to high-energy elementary particle processes (the reader will find detailed information concerning this in the surveys [G7]). Several typical examples of this sort are given in the final chapter.

Each part is preceded by a brief summary of the contents. This book includes appendices which set out auxiliary material or questions of

independent interest. The numerous exercises form an integral part of the text. They are referred to in the subsequent parts of the text and, as a rule, hints are provided. The appendices, exercises, proofs, also some remarks are printed in a smaller typeface.

The references to the literature are separated into two parts. In the first part are the textbooks and monographs; references to these are given in square brackets (for example, [AI]). The second half contains articles and journals, preprints and lectures at seminars and workshops. In the text, references to this part are given by

* See, for example, the collection [Q1) and the survey article by Crewther (1976).

INTRODUCTION xix

the surnames of the authors (or just the first author plus "et al." if there are more than two authors) and the publication date (for example, Zwanziger, 1979b).

At the back of the book is a glossary of the notation of frequent occurrence in this book. It should be noted that in this book all the coordinates of the 4-vectors of Minkowski space-time M are real and the metric tensor in M is defined by the formulae

gOO = -lie = 1 for k = 1,2,3, (g/Jv = 0 for p. =f v, p.,v = 0,1,2,3).

The three-dimensional spatial part of the 4-vector p is denoted by bold-face type so that p == (pO, p), p2 == (pO)2 _ p2. Throughout, a system of units in which c = Ii = 1 is used.