General Principles of Quantum Field Theory ||

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 530.143--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 self-adjoint and unitary operators (29).

1.5. Algebras with Involution. C-Algebras . . . . . . . . . . . . . . . 31 A. Definition and elementary properties (31). B. The spectrum (33). C. Pos-itivefunctionals (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 se-quences (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 gen-eralized 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. Operator-valued 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 func-tions on canonically closed regular subsets (84). A.3. Application: general-ized 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 "mathe-matics" of dispersion relations (103). B.6. Restriction of the Laplace trans-form (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 cov-ering of the group L~ (121). D. Finite-dimensional representations of the group SL(2, C) (125). E. Simply reducible finite-dimensional representa-tions 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 gen-eralized functions with support at a point (132). C. Odd invariant general-ized 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 rep-resentations. 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. De-parture 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. Representa-tions 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 ana-lytic 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 repre-sentations 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. Ir-reducibility of fields (329). D. Separating property of the vacuwn vector (331).

8.3. Wightman FUnctions . . . . . . . . . . . . . . . . . . . . . . . 332 A. Characteristic properties of Wightman functions (332). B. Kiillen-Lehmann 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 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . 353

Appendix G. General Form of the Covariant Two-Point Function ...... 355 G.1. Covariant decompositions compatible with locality (355). G.2. De-composition 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. Prop-erties 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 pseudo-Hilbert 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. Gold-stone'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 represen-tation (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. Phys-ical 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. Construc-tion 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. Pas-sage to the complex domain with respect to the momentap2,P4 (557). C. Dis-persion 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. Stein-mann 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. Pas-sage 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 inter-action 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 prop-erties 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 "phe-nomenological" 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 Insti-tute 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 Nu-clear 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 mod-ern 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 struc-tures 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 recom-mend 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 quan-tum 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 pos-sible 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 some-thing better; this is a formal infinite series for the solutions of quantum equations in the class of physically interesting renormalizable Lagrangians. The axiomatic ap-proach 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 quan-tities, 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 func-tions 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 Wight-man 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.

Part I Elements of Functional Analysis and the Theory of Functions

Synopsis

The reader will undoubtedly have encountered in his study of quantum mechanics, the idea of a Hilbert space and the spectral analysis of self-adjoint operators in it. Further information on this topic appears in 1.1,1.4. From the remaining material of Chapter 1, we single out the concept of a locally convex space and the Gel'fand-Naimark-Segal (GNS) representation of a CO-algebra.

A non-negative function p( u) on a vector space n is called a seminorm if it is positive homogeneous: p(.\u) = 1.\lp(u) (for any scalar .\) and satisfies the triangle inequality p(u + v) ~ p(u) + p(v). Each system of seminorms {Pal in a space n that satisfies the separation property (Pa(u) = 0 for all a in the given index set A implies that u = 0) defines a (separated) locally convex topology on n. In 1.2.A we introduce a natural notion of equiValence (mutual subordination) of two systems of seminorms; two locally convex topologies on a given linear space n are the same if and only if the seminorms inducing them are equivalent. A complete (separated) locally convex space with a countable system of seminorms is called a Frechet space. Every Frechet space is metrizable and hence the Baire category theorem (Theorem 1.3 of 1.2.B) is applicable to it.

An involutive Banach algebra 11 is called a CO-algebra if IIA" All == IIAII2 (1.5.A). This notion is abstracted from algebras of bounded operators in Hilbert space. A linear functional F over an involutive algebrall is said to be positive if F(A* A) ~ 0 for all A in 11 (1.5.C). The GNS construction enables one to construct from a given positive functional F over a C* -algebra 11 a (cyclic) representation 7rF of 11 in a Hilbert space JiF with cyclic vector ~F such that F(A) = (~F' 7rF(A)~ F) for all A Ell (Theorem 1.25 of 1.5.D).

An important example of a Frechet space is the space S(Rn) of (complex) infinitely smooth (COO for short) rapidly decreasing functions (1.2.C); a topology can be defined on it by means of the increasing system of Hilbert norms

(where Ixl2 == xi + '" + x~, A = ~ + ... + f.r) (which is equivalent to the system (1.42)). 3: 1 :t'n

Generalized functions are defined (2.1.A) as continuous linear functionals over S(Rn). (Sometimes the terminology "tempered generalized functions" (or tempered distributions) is used for these objects; but we shall reserve the terminology "distribution" of Schwartz for the more general notion of a linear functional over the space 1)(0) of COO-functions with compact support in a given open subset 0 of Rn.) Generalized functions (or distributions) can be differentiated and multiplied by smooth functions of polynomial growth without going outside the space S'(Rn) (or 1)'(0)). A remarkable property of the space S(Rn) and its dual is stability under the Fourier transformation; that is, if u(x) E S(Rn), then

u(p) = J eiPxu(x)d"x E S(Rn), where S(Rn) is the space of functions of P with measure dnP = dnpj(27r)n (Proposition 2.6). The Parseval identity

J f(x)u(x)d"x = J7(P)u(p)dn p

1

2 SYNOPSIS OF PART I

(which holds for arbitrary test functions u and f) is used to define the Fourier transform of a generalized function f. Within the framework of the theory of Fourier transforms of generalized functions (2.5.B) one can obtain a justification of a formula of the type 6(z) = f e-ipz: dnp.

In 2.7 we introduce the notions of vector- and operator-valued generalized functions. The first of these enables us to define generalized eigenvectors (corresponding to the continuous spectrum of a self-adjoint operator, see 2.7.C); the second is needed for the statement of the field axioms in Part III.

Chapter 3 is devoted to a study of the Lorentz-invariant and covariant generalized functions. By way of introduction, we give a classification of the finite-dimensional representations of the group SL(2, C) (the Lorentz group of quantum mechanics). Proposition 3.2 (3.2.B) is a precise statement of the intuitive idea that any even Lorentz-invariant function of a 4-vector P is a function of p2. Also related to this are the invariant generalized functions concentrated at a point, which have the form P(D)6(p) (see (3.89. An arbitrary odd invariant function is defined, on its part, by a functional of type

(f,u) = J l(p)u(p)d4P= J dT1/ij(T) J d4pf(po)6(T-p2)u(p) (u E S(M, where ,pJ is a generalized function in S'(R) that vanishes when T < (3.2.C).

In the study of the Lorentz-covariant (generalized) functions (3.3) we shall use the formalism of homogeneous polynomials of the spinor variables w (E C 2) and w (instead of working with spin-tensors). The function I(p;w,w) is covariant if I(A(!J..)p; ~w, /),w) = I(p;w,w). A non-trivial covariant function 1 exists provided that the degree of homogeneity in w is the same as that in w. Denoting this common degree by n, we have In(P;w,w) = 1o (p)(w,{;'W)n ,where 10 is a Lorentz-invariant (generalized) function (see (3.127.

A generalized function I+(z) in S'(M) is called retarded if its support lies in the future cone V+; similarly, I_(z) is an advanced function if supp 1- c V-. It is proved in Appendix B that the Fourier transform of a retarded (advanced) function is the limiting value of an analytic function that is holomorphic in the tube domain T = R4 + iV.

Let h+(p) and h_(p) be the Fourier transforms of retarded and advanced functions and let their difference g(p) = h+(p) - h_(p) be concentrated in the domain -m,O) + Vt) U m,O) + VM), where Vii = {p E R4 : po ;::: JM2 + p2}. Then we have the Jost-Lehmann representation

where ~o, ~l are generalized functions in S'(Ra x R+) concentrated on the set

Chapter 4 is devoted to a derivation of the Jost-Lehmann-Dyson (JLD) representation, which gen-eralizes the formula to the case of a more general (asymmetric) region in which the function g(p) vanishes.

The JLD representation enables one to extend by analytic continuation the single function h(k) originally defined on the union of the domains T+ U T- U 0 (where 0 is the (real) region where h+ and h_ coincide):

to a larger region of 4-dimensional complex space. This is the simplest example of the "edge of the wedge theorem" (Theorem 5.12). It illustrates an important fact which distinguishes the theory of analytic functions of several complex variables from the corresponding single-variable theory. Whereas any domain D of the complex plane C is the domain of holomorphy of an analytic function that does not admit an analytic continuation outside D, for the case of several variables there exist domains that are not domains of holomorphy of any analytic function. This enables one to introduce the notion of analytic continuation, domains of holomorphy and envelopes of holomorphy; Chapter 5 is devoted to the study of these.

CHAPTER 1

Preliminaries on Functional Analysis 1.1. Normed Spaces

A. LINEAR SPACES

In the study of infinite-dimensional linear spaces by the methods of functional analy-sis, emphasis is placed on topological properties. Leaving aside the topological details for the moment, we give a brief run-through here of the basic concepts relating to linear spaces and familiar from linear algebra.

A linear space is a set of elements of any kind for which the operations of addition and multiplication by a real (or complex) number are defined, the usual laws being satisfied for these operations. Examples of a linear space are vectors in n-dimensional Euclidean space, or the set of continuous (or integrable) functions defined on some set of points in finite-dimensional space, or functionals defined on some class of functions. The concrete nature of the elements is irrelevant for the abstract theory of linear spaces.

We now give the precise definitions. A set 0 is called a linear space (over the field R of real numbers or the field C of

complex numbers) if Conditions I-III hold. I. A commutative and associative law of addition is defined on O. This means

that for any two elements U and v of 0 a third element U + v is defined such that la. U + v = v + Uj lb. U + (v + w) = (u + v) + Wj Ie. For any u and v there exists an element x, depending on u and v, such that

u + x = v (this element being denoted by v - u). II. Multiplication by numbers A, p., ... is defined on 0, where lIa. 1 u = Uj lib. A(P.U) = (AP.)U for any U E O. III. The operations of addition and multiplication by a number are related by the

distributive laws: IlIa. (A + p.)u = AU + I-IUj I1Ib. A(U + v) = AU + AV. The elements of a linear space are also referred to as points or vectors, and the

linear space itself as a vector space. (It is called a real or complex (linear or vector) space depending on the choice of R or C as the field of scalarsj in this book we shall be dealing, in the main, with complex spaces.)

Condition Ie is equivalent to the requirement that both the following conditions hold: Ie'. There exists an element 0 E n such that u + 0 = u for all Uj

3

4 CHAPTER 1

Ie". There exists for each u E n a negative element -u such that u + (-u) = O. Exercise 1.1. Prove that Condition Ie is equivalent to the pair of conditions le' and le".

Exercise 1.2. Deduce from the axioms that 0 . u = 0 and (-I)u = -u.

A subset X of 0 is called a linear subspace if it is itself a linear space with respect to the linear operations (addition and multiplication by scalars) inherited from X. IT X is a linear subspace of 0, then we can introduce an equivalence relation in 0 by defining two vectors u, v E 0 to be equivalent if u - v EX. Thus, the equivalence class of an element U E 0 is the subset V. = u + X == {u + v: v E X} of O. The set of equivalence classes is called the quotient space of 0 modulo the subspace X and is denoted by 01 X. It is not difficult to see that this is a linear space if we define the linear operations thus: (u + v)" = V. + ii, (AU)" = AV..

For a given set X in 0 there is a smallest subspace of 0 containing X; it is called the linear span of X (or the linear space spanned by X) and consists of all possible linear combinations of elements of X, that is, vectors of the form Ej=1 AjUj, where the Aj are arbitrary scalars and Uj E X. IT the equality Ej=1 AjUj = 0 holds only when all the Aj are equal to zero, then the vectors of X are said to be linearly independent. A space 0 in which any set of linearly independent vectors is finite is called finite-dimensional, and each such maximal set is called a basis in O. IT a basis consists of n elements, then 0 is said to be n-dimensional.

The fundamental concepts of the theory of linear spaces are the linear functionals and linear operators. More generally, by a functional on a linear space 0 one means a (usually scalar-valued) function defined on o. A functional F is said to be linear if it takes values in the field of scalars and satisfies the linearity condition:

F(AU + Jlv) = AF(u) + JlF(v), (1.1) where A, Jl are arbitrary scalars and u, v E O. (The value F( u) of the linear functional at the element U is sometimes written in the form (F, u) or (F, u).)

Let 0 1 and O2 be two linear spaces over the same field of scalars; then by a linear operator from 0 1 to O2 we mean a function, say, T : 0 1 -+ O2 , defined on 0 1 with values in O2 and satisfying the linearity conditions of type (1.1). The value T( u) of the linear operator at the element U is usually written as Tu. IT 0 1 = O2 = 0, then T is called a linear operator in O. IT T(O.) = O2 (that is, if the image of 0 1 is the whole of O2 ), then T is said to be an operator from 0 1 onto O2 A one-to-one linear map from 0 1 onto O2 is called an isomorphism of the linear spaces. (An isomorphism from 0 onto itself is called a linear transformation of 0.)

A linear functional is a special case of an operator when O2 is the field of scalars. Another important example of a linear operator is the natural projection J : 0 -+ -+ 0IX that associates with the vector U E 0 its equivalence class V. = U + X.

In the case when 0 1 and O2 are complex linear spaces, the notion of anti-linear operator is also commonly used; this is a function T : 0 1 -+ O2 satisfying the anti-linearity condition

T(AU + Jlv) = XT(u) + ji.T(v) (1.2) for all A, Jl E e, u, v E 0 1 . An example of an antilinear operator is the operation of complex conjugation in en : (TU)j = Uj.

PRELIMINARIES ON FUNCTIONAL ANALYSIS 5

There is an extension theorem for linear functionals according to which, each linear functional Fo defined on a linear subspace X of n can be extended to a linear functional F on the whole of n.

The theorem is proved by means of Zorn's lemma, which is a special version of "transfinite induction". With later references to this in view, we give the lemma here in a form suitable for application. (The various formulations and a discussion of Zorn's lemma can be found in 1.5 of [KlO]). Lemma 1.1 (Zorn). Let A be a family of subsets of some set A with the property that the union of any chain of subsets of the family A is contained in some subset of A. Then the family A has a maximal subset.

Here, a subfamily of A is called a chain if either U C V or V C U for any two elements U, V of it. A subset U of A is said to be maximal if for any V in A, the condition U C V implies U = V.

Exercise 1.3.(a) Prove the extension theorem for linear functionals. [Hint: Choose as the set A in Zorn's lemma the set 0 x K, where K is the field of scalars R or C; for the family A of subsets of A, choose the set of graphs of linear functionals F defined on linear subspaces of 0 containing X and coinciding with Fo on X.]

(b) Let UI, ... , Un be linearly independent vectors in O. Prove that there exist n linear function-als 1I, ... .fn on 0 such that fi(uj) = Oij (where Oij is the Kronecker delta). [Hint: Use the extension theorem.]

The null space (or kerne0 is defined by ker T = {u E nI : Tu = O} ; clearly this is a linear subspace of nl .

Exercise 1.4. Suppose that the null space of the linear operator T : 0 1 -+ O2 contains the linear subspace X COl . Prove that there exists a unique linear operator S : 01/ X -+ O2 such that T = Sp, where p : 0 1 -+ 01/ X is the natural projection. (In this case we say that the operator T is lowered by S from 0 1 onto 01/ X.)

B. DIRECT SUM AND TENSOR PRODUCT OF LINEAR SPACES

U sing the linear spaces at our disposal, one can construct new ones. Let {nv LEN be a family of linear spaces indexed by II (running through an index set N). We consider the set n consisting of all possible families {u v} vEN == u, where the Uv run through the elements of nv and for each such family u, only a finite number of the U v are non-zero. The element U v is called the 11th projection (or component) of u. In particular n has a zero element all of whose projections are zero. It is easy to see that n becomes a linear space if we define the sum u + v and product ,\ . u by a number'\ in terms of the projections: (u +v)v = Uv +VV , ('\u)v = '\uv' It is called the (algebraic)*direct sum of the spaces nv.

Another method of constructing new spaces is tensor multiplication. Let nI , ... . . . , nn be a finite family of linear spaces. We consider the set :F consisting of all possible functions f of n variables VI E nI , . , Vn E nn taking values in the scalar field and such that f( VI, ... , V n ) = 0 everywhere except at a finite number of points (V}, ... ,Vn ) E nt x ... x nn. Clearly:F becomes a linear space if we define the linear operations as follows:

(f + g)(VI,"" vn) = f(vt, ... , vn) + g(Vt, ... , vn), * The algebraic direct sum (or algebraic tensor product) must be distinguished from the topological

direct sum (or topological tensor product); see, for example, l.l.E.

6 CHAPTER 1

(>.f)(V1, , Vn ) = >.. J(V1, , Vn ).

We associate with each fixed point (11."

, un) E n 1 X X nn the element JU1, ... ,un E E :F by setting

f ( ) { a, if(v1, ... ,Vn)=!(u1, ... ,un), Ul,,Un V1, ,Vn = 1, f ( ) ( ) 1 V1, , Vn = 11.1 , , Un Then :F is the linear span of all the JUI, ... ,un (these elements form a basis in :F). We consider the linear subspace X of :F spanned by vectors of the form

fUl, ... ,UV+lIv, ... ,un - !Ul, .. o,UJI, .. "un - !Ul, ... ,VIl I ... ,un,

/U1 1 ... ,"\UJl, ... ,un - >--/Ul, ... ,u", ... ,Un' where the v can take any values from 1 to nj 11.1, ... , ui, Vj, , Un are arbitrary vectors in the appropriate n 1 , , nn, and >. is an arbitrary scalar. The quotient space :FIX is called the (algebraic) tensor product of n 1 ... ,nn. In particular, the equivalence class iUlI ... ,u,. of JUI, ... ,un is called the tensor product of the elements 11.1, , Un and is denoted by 11.1 ... Un.

Here are the main properties of the tensor product. (a) The tensor product of n 1 , , nn is spanned by the elements 11.1 ... Un. (b) Multilinearity:

= AUI ... U v ... Un + 1-'11.1 ... Vv ... Un

(c) IT V~I), ... , v~mv) is a family of linearly independent vectors of nv , then the vectors v~id ... v~in) (where the iv take all possible values from 1 to mv) are linearly independent.

Exercise 1.5. Prove the property (c) of the tensor product. [Hint: Suppose that

(1.3) i 1 in

To prove, for example, that A1...1 = 0, define for each II the linear functional hv on Ov such that hv (uV) = 1 and hv(uV = 0 for i > 1. Then define the linear functional H on the space:F (defined above) as:

H(!)= L ... L f(Vl, ... ,vn)hl(VI) ... h,.(vn). "leOl "neOn

Verify that this functional is lowered to a linear functional h on the quotient space :F / X = 0 1 ... Onl where h (v~id ... v~n) is equal to 1 for i l = 1, ... I in = 1 and is equal to 0 for all other values of il I ... ,in. FinallYI let h act on both sides of (1.3).]

Exercise 1.6. Suppose that each of the spaces Uv is finite-dimensional and that the {v~)} (i = = 1, ... I mv) forms a basis in Ov. Prove that the vectors v~il) ... v~in) then form a basis in


the tensor product of the 01, ... , On, SO that the dimension of the tensor product is equal to the product of the dimensions of the spaces 0 1, , On.

C. NORMED SPACES We now tum to the topological concepts. We begin with the simplest example of a linear topological space, namely, a normed space. A more general class of linear topological spaces (the class of locally convex spaces) will be introduced in 1.2.

A real function p( u) defined on n is called a norm if it satisfies the following conditions:

(a) p(oXu) = IAlp(u) for any scalar A (positive homogeneity); (b) p( u + v) ~ p( u) + p( v) (triangle inequality, or convexity of the norm); (c) if p(u) = 0, then u = 0 (separation property). It follows from (a) and (b) that the norm is non-negative:

o = p( u - u) ~ p( u) + p( -u) = 2p( u). FUnctions satisfying conditions (a) and (b) only (but not necessarily (c)) are called semlnorms.

A linear space in which a norm is defined is called a normed space. The norm of an element u is sometimes denoted by lIuli.

IT we define the distance between two elements u, v as d( u, v) = p( u - v), then the normed space becomes a metric space [that is, the set of n is endowed with a non-negative function d( u, v) defined on n x n, called a metric, which by definition must satisfy the conditions of symmetry (d( u, v) == d( v, u)), the triangle inequality (d(u,v) ~ d(u,w)+d(v,w)forallu,v,w En), and separation (d(u, v) = O::::} u = v)]. We say that a sequence {un} in n converges to u if the distance from Un to u tends to zero as n -t 00: lim d( Un, u) = O. This type of convergence is called convergence

n-+oo in norm or strong convergence. Exercise 1.7. (a) Prove that the norm is continuous, that is, lIunll ..... lIuli if Un ..... U in norm as n ..... 00.

(b) Prove the continuity of the sum U + v with respect to u, v and also the continuity of multiplication AU with respect to u, A, that is, Un + Vn ..... U + v if Un ..... U, Vn ..... v as n ..... 00, and Anun ..... AU if Un -+ U, An ..... A as n ..... 00.

A sequence {un} of elements of a normed (or metric) space n is called fundamental if for any e > 0, a number N( e) can be found such that d( Un, um) < e for any n > N( e) and m > N(e) (that is, if . lim d(un,um ) = 0). (A fundamental sequence is

mm(m,n)-+oo sometimes called convergent in itself or a Cauchy sequence.) It is not difficult to see that if a sequence {un} converges to some element u E n then it is fundamental. The converse does not always hold. A normed (or metric) space in which every fundamental sequence converges to some element of this space is called complete. A complete normed space is called a Banach space. There is a theorem that states that any normed space can be completed to form a Banach space (see, for example, [Yl], 1.10). A Banach space n is said to be separable if it contains a countable subset that is everywhere dense in n (that is, each element of n is the limit of a sequence of elements of this countable set).

A trivial example of a Banach space is n-dimensional real Euclidean space Rn, in which the norm is defined by the formula

8 CHAPTER 1

(1.4) ;=1

Another example of a (this time complex) Banach space is the space C([a, b]) of continuous complex functions on the interval [a, b] with norm

p(u) = sup lu(x)l. (1.5) ze[a,b]

Exercise 1.8. Prove that C([a, b)) is complete.

Our next example of a Banach space is the space S,\,m(Rn), which plays an important role in the theory of generalized functions. Here A, m are arbitrary non-negative integers. The space S,\,m(Rn) consists of all complex functions of n real variables x == (Xl"'" Xn) E Rn having continuous partial derivatives up to order m and decreasing at infinity at least as rapidly as Ixl-m. In other words, for each function u in S,\,m(Rn) all its derivatives of the form

afJ1 + ... +.Bn u(x) xa DfJu( x) == X~l x:n (h fJn

aXl ... axn (1.6)

are bounded for all multi-indices a and f3 whose (integral) components satisfy (1.7)

The norm in S,\,m(Rn) is defined by the equality

lIulkm = max sup Ixa DfJu(x)l. (1.8) lal~'\ zeRR IfJl$m

. In the next subsection we shall be considering examples of Hilbert spaces, which are an important special case of Banach spaces.

Two normed spaces are said to be isomorphic * if there exists a one-to-one map between them that preserves the linear operations and the norms of the vectors (that is, a linear isometric map from one space onto the other). Exercise 1.9. Let X be a closed linear subspace of the Banach space O.

(a) Prove that X is a Banach space in its own right (with the linear operations and norm inherited from 0).

(b) Prove that the quotient space 0/ X is a Banach space with norm

D. HILBERT SPACES

lIull = inf lIu + vII for all u E O. vEX

(1.9)

A function w that associates with each pair u, v of elements of a complex linear space n a complex number w( u, v) is called a sesquilinear form on n if w( u, v) is linear in v and antilinear in tI, that is, if

* More general is the notion of topological isomorphism of (normed) spaces (see 1.3.A).


W(AtUt + A2V2, v) = XtW(Ut, v) + X2W(U2, v) for all U,Ut,U2,V,Vt,V2 En, At,A2 E C. If, moreover, the Hermitian condition

W(u,v) =w(v, u) holds, then W is called a Hermitian form.

The expression w( u, v) is called a scalar product of the vectors u, v E n if this Hermitian form is non-degenerate, that is, if the condition w( u, v) = 0 for all v E n implies that u = 0, in other words, if there exists for each element u =f 0 an element v E n such that w( u, v) =f 0). The usual notation for the scalar product w( u, v) is (u, v) (or (ulv)).

It is clear that the "scalar square" (u, u) of any u E n is a real number. If it can take all real values, then n is called a space with an indefinite metric.

For the present, we are interested in a different case. A Hermitian form w is said to be non-negative-definite if the scalar square of any vector is non-negative:

w( u, u) :2: 0 for all u E n, (1.10) and positive-definite* if, in addition to (1.10) we have

w(u,u) =0, onlyif u=o. (1.11 ) It follows from the next exercise (more precisely from (1.12)) that a non-negative definite form w is positive-definite if and only if it is non-degenerate, which means that the expression w(u,v) == (u,v) can be called the scalar product of the vectors u, v. (It is not difficult to prove from this (by contradiction) that a space with an indefinite form contains a vector u =f 0 for which (u, u) = 0.) Exercise 1.10. Let w be a non-negative definite Hermitian form on O.

(a) Prove that the following inequality (called the Cauchy-Bunyakovsky-Schwarz inequality) holds for any vectors u, v E 0:

Iw(u,v)12 ::; w(u,u)w(v,v). [Hint: The expression w(Au + v, AU + v) is non-negative for all A E C.)

(b) Prove that the expression p(U) = Jw(u, u)

is a seminorm on 0; it is a norm if and only if w is non-degenerate.

(1.12)

(1.13)

A space n with a positive-definite Hermitian form w(u,v) = (u,v) is called a (complex) pre-Hilbert space.

It follows from (1.10) that every pre-Hilbert space is a normed space with norm

(1.14)

Furthermore, the Cauchy-Bunyakovsky-Schwarz inequality holds:

I(u, v)1 :::; lIulllIvll (1.15) * Sometimes non-negative-definite and positive-definite forms are called positive-definite and strictly

positive-definite respectively.

10 CHAPTER 1

A complete pre-Hilbert space 'H is called a Hilbert .!pace. *

E~erciBe 1.11. Prove that the scalar product is continuous in U,V, that is, (un,Vn ) --+ (u,v) as Un --+ U, Vn --+ V in norm.

The notion of orthogonality can be defined in a Hilbert (and in a pre-Hilbert) space: two vectors ~, 111 are said to be orthogonal if their scalar product (~, III) is zero. A subset X of a Hilbert space 'H is said to be total if the linear span of X is everywhere-dense in 'H, that is, the closure of this linear span is the whole of 'H.

E~erciBe 1.12. (a) Prove the parallelogram law for the Hilbert space norm:

(1.16)

(b) Prove the polarization identity:

(~, 1) = 1 E wllw~ + 1)112; (1.17) w=~

here w runs through the four complex roots of unity (that is, w = 1, i). The polarization identity shows that the scalar product in a Hilbert space is completely deter-

mined by the norm. Of interest is the question, what properties of the norm distinguish Hilbert spaces in the class of all Banach spaces. It turns out that the parallelogram law is such a characteristic prop-erty of Hilbert space (Jordan and von Neumann, 1935). In other words, if the norm in a Banach space 1t satisfies the parallelogram law, then 1t is a Hilbert space with scalar product (1.17). Prove this. Clearly, it suffices to verify the equations

(~, 1)1 + 1)2) = (~, 1)1) + (~, 1)2) for~, 1)11 1)2 E 1t,

(~, ;\1) = ;\(~, 1) for~, 1) E 1t,;\ E C. It follows from (1.17) that

We transform the right hand side by means of the parallelogram law:

= (211~ + 1)1112 + 2111)2112 -II~ + 1)1 -1)2112) -II~ -1)1 -1)2W == == II~ + 1)lW + II~ + 1)1112 + 2111)2112 - (II~ + 1)1 -1)2W + II~ -1)1 -1)2112 =

= II~ + 1)1112 + (-II~ -1)1112 + 211~1I2 + 2111)1112) + 2111)2112-

(1.18)

(1.19)

-(211~ -1)2112 + 2111)1112) = (II~ + 1)1112 -II~ -1)1112) + (211~1I2 + 2111)2W - 211~ -1)2W) = = 4Re(~, 1)1) + 4Re(~, 1)2).

Hence and from the identity Im(~, 1) == Re(i~, 1), (1.18) follows. For;\ = ~, (1.19) follows from (1.17); therefore it suffices to prove (1.19) for ;\ > O. For integral ;\ > 0 and for rational ;\ > 0, (1.19) follows from (1.18); for the remaining ;\ > 0, it follows from the continuity of the scalar product.

E~erciBe 1.13. Let 1t1 be a closed linear subspace of1t, and ~ any vector in 1t. (a) Prove that there exists a vector ~1 E 1t1 such that

* We deal here only with complex Hilbert spaces, these being of primary interest to us.

(1.20)


[Hint: Let d = inf II~ - 1)11 and I),. a sequence in 'HI such that II~ - 1),.11--+ d as n --+ 00. Deduce 9E1t,

from the parallelogram identity (1.16) applied to the vectors ~ - I)m, ~ - I),., that IIl)m - 1),.11--+ 0 as min(m,n) --+ 00, that is, the sequence I),. is fundamental.]

(b) Set ~2 = ~-~1. Prove that the vector ~2 is orthogonal to 'HI. [Hint: it follows from (1.20) that 1I~2 + ~l)dl2 ~ 1I~2112 for all 1)1 E 'HI, ~ E C.]

It is clear that every closed linear subspace ?i1 of a Hilbert space ?i is itself a Hilbert space. The set

is called the orthocomplement (or orthogonal complement) of the subspace ?i1. Exercise 1.14. Let 'HI be a closed linear subspace of the Hilbert space 'H.

(a) Prove that 'Hr is a closed linear subspace of'H and that 'HI n 'Hr = {OJ.

(1.21 )

(b) Prove that any vector in 'H can be uniquely represented in the form ~ = ~1 + ~2, where ~1 E 'HI, ~2 E 'Hr. [Hint: Use the preceding exercise and part (a) of this exercise.]

The result of part (b) of Exercise 1.14 can also be stated as follows: the Hilbert space?i can be decomposed into a direct sum of orthogonal subspaces ?i1 and ?it; this is written in the form

?i = ?idB ?it. (The orthocomplement is sometimes called "orthogonal difference" and one uses the notation: ?il. = ?i e?iJ.) In the decomposition q; = q;1 + q;2, the vectors q;1 E ?iI, q;2 E ?it are called the (orthogonal) projections of the vector q; onto the subspaces ?i1 and ?it.

Exercise 1.15. (a) Prove that a set X C 'H is total if and only if the set of vectors in 'H that are orthogonal to X (that is, orthogonal to all the vectors in X) consists solely of the zero vector. [Hint: Apply Exercise 1.14 to the closed linear span of X.]

(b) Prove that a Hilbert space 'H is separable if and only if there exists a finite or countable orthonormal (that is, orthogonal and normalized) family of vectors {e,.}, (ei, ej) = Dij forming a total set in 'H.* [Hint: Use the orthogonalization process for a sequence of vectors.]

A trivial example of a Hilbert space is n-dimensional complex Euclidean space en with scalar product

n

(1.22) ;=1

The complex measurable functions of a real variable x with square-integrable modulus over the interval [a, b) form a Hilbert space. More precisely, it is the space of equivalence classes of functions, where functions differing only on a set of Lebesgue measure zero are deemed to be equivalent. This space is usually denoted by C2([a, b)). The scalar product is defined in it by the formula

(q;, 'II) = lb ~(x)w(x)dx. (1.23) One often obtains Hilbert spaces by means of the following general construction.

Suppose that we are given a non-negative definite Hermitian form w on the complex

* We call such a family of vectors an orthonormal basis of the (finite- or countable-dimensional) Hilbert space.

12 CHAPTER 1

linear space n. Then (1.13) defines a seminorm p(u) on n. Let no be the set of all vectors in n whose seminorms are zero, that is, no = {u En: p(u) = o}. Clearly no is a linear space and it follows from (1.12) that no is orthogonal to all the vectors of n. It is not difficult to see that the formula

(it, v} =w(u,v) for all U,v E n (1.24) well-defines a positive-definite Hermitian form (or scalar product) on the quotient space nino, thus converting it into a pre-Hilbert space. On completing this, we obtain a Hilbert space 'H (in which nino is an everywhere-dense linear subspace).

E. DIRECT SUM AND TENSOR PRODUCT OF HILBERT SPACES

In 1.1.B we introduced the notion of algebraic direct sum and algebraic tensor product of linear spaces. In the case when the original spaces are Hilbert spaces, the resultant spaces are in general (incomplete) pre-Hilbert spaces which need to be completed so as to obtain Hilbert spaces. We consider these constructions in greater detail.

Suppose that we are given a family of Hilbert spaces {'H"}"EN' We consider the family . -+ Ea" 'H" which is an isomorphism of the Hilbert space 'H>. onto a closed linear subspace 'H~ c Ea 'H" (consisting of all the vectors of the direct sum whose components


Exercise 1.17. Let {?ill} be a family of closed linear subspaces of the Hilbert space ?i. Prove that?i is the direct sum of these subspaces ?ill if and only if they are pairwise orthogonal and there does not exist a non-zero vector in ?i orthogonal to all of the ?ill.

By choosing an orthonormal basis in a Hilbert space 'H, we obtain an example of a direct-sum decomposition of'H into one-dimensional subspaces. By an orthonormal basis we mean any family {ell }IIEN of vectors in 'H with the properties: 1) (e.>., el') = = 6.>.1" 2) the set {ell }IIEN is total in 'H. It is easy to conclude from this that any vector u E 'H can be uniquely represented in the form

(1.27 a)

where (1.27b)

The sum of the series (1.27a) is to be understood in the following sense: for any f > 0 there exists a finite index subset Mo C N such that 111> - LIIEM Allellil < f for every finite index subset M :J Mo. The scalar product of two vectors 1> = LII Allell and IJI = LII p.llell is given by the formula

(1), IJI) = L ),IIP.II. lieN

Every Hilbert space has an orthonormal basis {ell }IIEN and the cardinality of this basis (that is, the cardinality of the index set N) is called the dimension of the Hilbert space (see, for example, (Yl] Ch. III, 4).

We now turn to the tensor product. Let n be the algebraic tensor product of a finite family of Hilbert spaces 'HI, ... , 'Hn. An arbitrary pair of vectors 1>, IJI in n can be represented in the form

M N

1> = L u~;) @ . @ u~) , IJI = L v~j) @ ... @ v~) , (1.28) ;=1 j=1

where uS;), vV) E 'HII . We define the scalar product of this pair of vectors by the formula

M N

(1), Ill) = L L(u~i), v~j)) ... (u~), v~). (1.29) ;=1 j=1

However, since the representation (1.28) is not unique, it is not clear that the right hand side of (1.29) depends only on the vectors u, v and not on the specific represen-tation (1.28). The following exercise shows that the scalar product is well defined by (1.29), that is, the right hand side does not depend on the choice of the representation (1.28). Exercise 1.18. (a) We define a Hermitian form on the space :F of 1.1.B by setting

w(f,g) = (1.30)

14 CHAPTER 1

Prove that the form w is non-negative definite. (b) Prove that if 1 e X or 9 e X, then w(/,g) = O. Deduce that w can be lowered onto

:FIX = n, that is, there exists a form won n such that w(i,g) = w(/,g) for all/,g e:F. Verify that w(u,v) is the same as the expression (u,v) defined by (1.29).

(c) Let the vector ~ be as in (1.28) and let {e~)}j=l, ... ,m" be an orthonormal basis in the subspace of?i" spanned by the vectors u~l), ... , u!f'l). Then ~ can be expressed in the form

~ _ '"' \ (id"" "" (in) 'I!- LJ J\iL ... inel '


where 4>(x) is a function of bounded variation (see [K2j,Ch.VI,3) (Riesz represent-ation theorem). The norm of the functional is equal to the total variation of 4>:

10-1

p'(F) = V:4>(x) == sup sup L 14>(xj+I) - 4>(Xj)l 10 a'u) = >'lIulD and apply the extension theorem to extend Fo (as a continuous linear functional) onto the whole of n (where F can be chosen so that IWII = 1).

In fact, far-reaching conclusions can be drawn from this argument. We say that a given set S of linear functionals on a linear space n separates the points of n if the condition F(U1) = F(U2) for all F E S, implies that U1 = U2 Exercise 1.22. Prove that the dual of a normed space n separates the points of n.

* In Dirac's notation, "ket vectori' (that is, elements of 7f.) are converted into "brn vectori' (elements of 7f.').

16 CHAPTER 1

The space 0" == (0')' dual to 0' is called the second dual of the normed space O. It is a Banach space with norm p". The space 0 can always be regarded as a linear subspace of Oil (with norm induced by Oil). Exercise 1.23. We associate with each element U E 0 the functional u(u) on 0' according to the formula u(u)(F) = F(u), FE 0'. Prove that the map u: 0 -+ 0" is a linear isometric map.

Thus we can identify 0 with a subspace of Oil by means of the map (J' (of Exercise 1.23). IT n = nil, that is, if (J' is an isomorphism, then n is said to be reflexive.

The space C([a, b]) of Example 1) is non-reflexive, whereas every Hilbert space is reflexive.

1.2. Locally Convex Spaces

A. EQUIvALENT SYSTEMS OF SEMINORMS. STRUCTURE OF LCS's.

In addition to the normed spaces, an important role is played by the more general classes of linear spaces with convergence. Of these, the Frechet spaces, or F-spaces, are the closest generalizations of normed spaceSj the aim of this section is to acquaint the reader with such spaces.

Let n be a linear space (say for definiteness, complex). To estimate the degree of closeness of an arbitrary vector u E n, we take some fixed family of seminorms {P"'}"'E.A on nj here a is an index distinguishing the seminorms and taking values in some (finite or infinite) set A. We say that the system of seminorms {P"'}"'E.A on n is subordinated to the system of seminorms {qp}PE8 on n if there exists for any a E A a finite set of indices (31, ,(3k in B and a number c ~ 0 such that P'" :::; C SUPj=1, ... ,k qPi'

According to the next exercise, the quantity on the right hand side of this last inequality is a seminorm on n.

Exercise 1.24. Let {Pa}aEA be a system of semi norms on n such that the quantity p(u) = sUPaEAPa(u) is finite for all u E O. Prove that P is a seminorm on O.

Two systems of seminorms on n are said to be equivalent if each system is subor-dinated to the other. A linear space n endowed with a system of seminorms {Per} is called a locally convex space (LCS for short) where by definition, two different systems of seminorms define the same LCS structure on n if and only if they are equivalent. * A given system of seminorms on n defining an LCS structure is called a defining system of seminormsj we also use the more detailed notation (n, {P"'}"'E.A) for an LCSj this notation contains a reference to the determining system of seminorms.

A determining system of seminorms on an LCS can, without any harm, be re-placed by an equivalent system. Because of this, we can (and shall) henceforth assume that the determining system {Per} of seminorms satisfies the condition: any finite subsystem is subordinated to some seminorm of the given system. (IT this con-dition is not satisfied, then we replace the original system by the equivalent system of seminorms P"'l''''''k = SUPj=1 ..... k P"'i j here k runs through all the natural numbers and a}, ... ,ak run through all the indices in A.) The aim of the above understanding, * It will be clear from what follows that LCS structures are the same if and only if the topologies

defined by them are the same.


which does not sacrifice any generality, is to simplify a number of the statements. (For instance, relations of type (1.37), (1.43) (see below) would have to be replaced by something much more cumbersome.)

An LCS (0, {Pc>}), is called separated if the condition Pc>( U - v) = 0 for all a E A implies that u = v, that is, if for any non-zero u E 0 there exists an index a E A such that Pc> ( u) > o. (In particular, an LCS (0, {Pc>}) is separated if at least one of the seminorms Pc> is a norm.) We can define the notion of a limit in a separated LCS in similar fashion to that for normed spaces. Namely, a sequence Uk in 0 converges to an element u E 0 (Uk -t U or u = limk-+oouk) if limk-+oopc>(uk - u) = 0 for any a E Aj in view of the above separatedness condition, the limit, if it exists, is unique.

From now on we shall suppose (without special mention) that all our LCS's are separated.

IT X is a linear subspace of the LCS (0, {Pc>}), then the restrictions of the semi-norms Pc> to X define an LCS structure on X called the induced structure (induced from 0).

All the topological concepts which, undoubtedly, are well known to the reader in the context of Euclidean spaces, can easily be carried over to an arbitrary LeS. Thus, the analogues of open balls at the origin (or at the point u E 0) are the sets

v; = {v EO: Pc>( v) < f} (1.37) (or u + V;) associated with each index a E A and number f > O. A subset X of 0 is said to be open (in 0) if for any point u EX, a set of the form u + V; is contained in X (for some a E A, f > 0). The complement of an open set in 0 is called a closed set (in 0). The closure of a subset X of 0 is the smallest closed set containing Xj it is denoted by X. By a neighbourhood of a point u E 0 we mean an open set containing this point. We say that a set Xc 0 is dense in the set YeO if Xc Y and X:) Y. (The latter inclusion is equivalent to the property that any neighbourhood of a point of Y has a non-empty intersection with X.)

We can summarize the above discussion as follows: every LeS has a canonically defined topology.

B. FRECHET SPACES

It should be noted that for the case of normed (and in particular, Euclidean) spaces, the notions of closedness and closure can be restated in terms of sequences (that is, sequential properties). Thus, to say that a set X is closed in a normed space 0 means that X contains the limit of any sequence of points of X that converges in 0, while the relation X = Y means that Y is the set of limits of all possible sequences of points of X that converge in O.

To enable us to go over to such a characterization in an LeS (0, {Pc>}), we have to impose the condition: one can choose from the family V; of neighbourhoods of the origin, a family {V;:} k=l,2, ... that forms a countable basis of neighbourhoods of zero. (By definition, such a basis has the property that each V; contains at least one set V;: of the chosen family.) As in the case of normed spaces, the existence of such a basis is decisive for formulating closedness and closure (also completeness and continuity) in terms of sequences. In particular, this condition turns out to be satisfied for all LeS's with a finite or countable defining system of seminorms. (The

18 CHAPTER 1

case of a finite number of seminorms can be regarded as a special case of a countable number, since the addition of seminorms subordinated to the original system leads to an equivalent system.) Exercise 1.25. (a) In an LCS with a countable determining system of seminorms {Pa}ae,.( there exists a countable basis of neighbourhoods of zero. [Hint: consider a family of sets V:~ , where IX runs through the countable index set A and the E~ are a sequence of positive numbers converging to zero.]

(b) The topology of an LCS n with a countable system of seminorms {p~}r=l is the same as the topology on n defined by the metric

We now single out from the LCS's whose determining systems of seminorms can be chosen to be countable, a class of spaces of the greatest interest. A separated LCS with a countable system of seminorms is called complete if every Cauchy sequence in n converges in n. (By analogy with the case of normed spaces, a sequence u~ in n is, by definition, a Cauchy sequence if for any a E A, Pa(un - um) -+ 0 as min( n, m) -+ 00.) A separated complete LCS with a countable system of seminorms is called an F-space (or Frechet space). We shall, in the main, be concentrating attention on such spaces.

It follows from the metrizability of F-spaces (see Exercise 1.25(b)) that the fol-lowing theorem is applicable.

Theorem 1.3 (Baire category theorem). The intersection of any countable family {M,,}k.:l of open dense subsets M" of a complete metric space n is dense in n.

It suffices to prove that for any point Uo E n and any EO > 0, the open ball Uo = {u En: d(u,uo) < EO} with centre at Uo and radius fO has a non-empty intersection with nMl:. It follows from the density of Ml in n that there exists Ul E Ml n Uo. Since Ml is open, there is a closed ball U1 with centre at Ul and radius fl (0 < fl < fO) contained in Ml n Uo. By continuing this process by induction, we can prove the existence of a sequence of points u~ E n and a monotone decreasing sequence of numbers fl: -+ 0 such that each M~ n U~-l contains a closed ball UI: with centre at UI: and radius f~. By construction, U~ is a Cauchy sequence, that is, d(um,un ) -+ 0 as min(m, n) -+ 00. Consequently it has a limit U co belonging to UI:+1 for each II: ~ O. Since by construction, UHl C MI: n U~, U CO belongs to the intersection of Uo with n~l M~.

C. EXAMPLES

As is clear, Banach spaces belong to the class of Frechet spaces. We now mention a number of other (complex) Frechet spaces that are encountered in applications.

1) The space C(O). Let 0 be an open subset of Rn. Then C(O) denotes the space of all complex continuous functions in (or on) 0 endowed with the system of seminorms

lIull[( = sup lu(x)l, (1.38) xe[(

where K runs through all the compacta * in 0 (or a countable family of compacta * By a compactum in 0 we mean a set K C 0 such that any sequence of points of K has a

subsequence converging to a limit in K. (For the special case under consideration (namely Rn), a compactum K can be defined as a bounded subset of 0 that is closed in R n .)


whose interiors cover 0). In this example, 0 is in fact allowed to be an arbitrary locally compact subset of Rn. *

2) The space (0). Let 0 be an open subset of Rn. We denote by (0) the space of all complex infinitely differentiable (COO for short) functions on O. We use the following notation for the partial derivatives of a function u E (0):

(1.39)

where a == (al, ... , an) is an ordered set of n non-negative integers (called the n-index or multi-index); here DQu(x) == u(x) when a = (0, ... ,0). The order of the derivative DQ is denoted by

(1.40)

Next we endow (0) with the system of seminorms (1.41)

where I can take all possible non-negative values and K runs through all the compacta of 0 (or a countable family of compacta whose interiors cover 0).

In mathematical analysis ([S4],Ch. IV,8) there is a Coo variant of the classical Weierstrass polynomial approximation theorem: for any Coo function u( x) defined on an open set 0, any compactum K c 0 and any I E Z+,** f > 0, there exists a complex polynomial P(x) defined on 0 (or, what is the same, on Rn) such that lIu - PII:'-, < f. In other words, the complex polynomials on 0 form an everywhere-dense linear subspace of ( 0).

3) The space S(Rn). Let S(Rn) be the space of Coo functions on Rn for which the expressions

(1.42)

are finite for all non-negative integers I,m. The space S(Rn) endowed with the system of seminorms IIUIl/,m is called the space of rapidly decreasing test functions on R n. It will play an important role later on.

Exercise 1.26. Prove that the spaces C( 0), t( 0) and S(Rn) are Freehet spaces. Exer

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