Scattering in Quantum Field Theories: The Axiomatic and Constructive Approaches

312 Pages·1993·18.736 MB·English

by  Daniel Iagolnitzer

Scattering in Quantum Field Theories
PRINCETON SERIES IN PHYSICS
Edited by Phillip W. Anderson, Arthur S. Wightman,
and Sam B. Treiman (published since 1976)
Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann
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Works on the Foundations of Statistical Physics N. S. Krylov
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The Large-Scale Structure of the Universe by P. J. E. Peebles
Statistical Physics and the Atomic Theory of Matter, From Boyle and
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Quantum Theory and Measurement edited by John Archibald Wheeler and
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Current Algebra and Anomalies by Sam B. Treiman, Roman Jackiw, Bruno
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Weak Interactions in Nuclei by Barry R. Holstein
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Instabilities and Fronts in Extended Systems by Pierre Collet and
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Introduction to Algebraic and Constructive Quantum Field Theory
by John C. Baez, Irving E. Segal, and Zhengfang Zhou
Principles of Physical Cosmology by P. J. E. Peebles
Scattering in Quantum Field Theories: The Axiomatic and Constructive
Approaches by Daniel Iagolnitzer
Scattering in Quantum Field Theories
The Axiomatic and Constructive Approaches
Daniel Iagolnitzer
Princeton Series in Physics
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
Copyright ©1993 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
Chichester, West Sussex
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Iagolnitzer, Daniel.
Scattering in quantum field theories : the axiomatic and constructive
approaches / Daniel Iagolnitzer.
p. cm. — (Princeton series in physics)
Includes bibliographical references and index.
ISBN 0-691-08589-7
1. Scattering (Physics) 2. Quantum field theory. I. Title.
II. Series.
QC174.52.S32I24 1992
539.7'58—dc20 92-15633
This book has been composed in Computer Modern using TgX
Princeton University Press books are printed on acid-free paper and
meet the guidelines for permanence and durability of the Committee
on Production Guidelines for Book Longevity of the Council on Library
Resources
Printed in the United States of America
10 9 8 7 6 5 4 3 21
Contents
PREFACE
INTRODUCTION xiii
1. Organization of the book xiii
2. Description of contents xiv
3. Technical remarks xx
THE MULTIPARTICLE S MATRIX 3
1. Introduction 3
2. General 5-matrix formalism 6
2.1 Free-particle states and S matrix 6
2.2 Cluster property, connected S matrix, unitarity equations 11
3. Multiple scattering and Landau surfaces 14
3.1 Definitions and examples 15
3.2 +α-Landau surfaces, causal directions, plus ίε rules 20
4. The physical region macrocausal S matrix 24
4.1 Macrocausality and physical region analyticity 24
4.2 Macrocausal factorization and local discontinuity formulae 28
4.3 The 3 —» 3 S matrix below the 4-particle threshold 35
5. The analytic S matrix 36
5.1 Hermitean analyticity, crossing, and all that 37
5.2 Analyticity in unphysical sheets 39
5.3 Basic discontinuity formulae for 3 —* 3 processes 42
6. Analysis of Landau singularities 44
6.1 Graphs with single and double lines: holonomic cases 44
6.2 Simplified theory of the m-particle threshold and expansions
in terms of holonomic contributions 47
6.3 The nonsimplified theory: outlook and conjectures 51
Appendix: The multiparticle S matrix in two-dimensional
space-time 52
vi CONTENTS
II. SCATTERING THEORY IN AXIOMATIC FIELD THEORY 55
Introduction and General Formalism 55
1. Introduction 55
1.1 Axiomatic field theory: general preliminaries 55
1.2 Scattering theory: historical survey 58
1.3 Description of contents 66
2. General formalism 67
2.1 The axiomatic framework 67
2.2 Fields and particles: preliminary discussion 71
2.3 Asymptotic states and S matrix (Haag-Ruelle theory) 77
Causality and Analyticity in the Linear Program 81
3. Causality and local analyticity 81
3.1 Asymptotic causality properties of chronological iV-point
functions 81
3.2 Momentum-space analyticity and local decompositions 89
3.3 Reduction formulae and results on the S matrix 91
4. The analytic TV-point functions 96
4.1 General primitive results of the linear program 96
4.2 2-point and 4-point functions 104
Particle Analysis in the Nonlinear Program 108
5. The nonlinear program-direct methods 108
5.1 Discontinuity formulae for absorptive parts and a class of
generalized optical theorems 108
5.2 4-point function and 2-particle threshold 112
5.3 The 5-point function in the 3-particle region 115
5.4 The 6-point function in the 3-particle region: Landau
singularities and structure equation 119
6. The nonlinear program based on irreducible kernels 125
6.1 Some general preliminary theorems 125
6.2 The 2-particle structure of 4-point and TV-point functions 128
6.3 The 6-point function in the 3-particle region (even theories) 132
7. Macrocausal properties: further results and conjectures 138
7.1 Macrocausal factorization: some general results 138
7.2 General conjectures 141
7.3 Macrocausal analysis of the 2-particle threshold 145
III. EUCLIDEAN CONSTRUCTIVE FIELD THEORY 149
1. Introduction 149
1.1 Historical survey and description of contents 149
1.2 The Euclidean axioms 154
1.3 The models 156
CONTENTS vii
2. The perturbative approach 160
2.1 Perturbative series 160
2.2 Perturbative renormalization: phase-space analysis and
effective expansions 164
3. The Ρ(φ) models 170
2
3.1 Preliminaries 170
3.2 Cluster expansions and related expansions of ΛΓ-point
functions 173
3.3 Infinite-volume limit, Euclidean axioms, Borel summability 180
4. The massive Gross-Neveu model in dimension two 186
4.1 Preliminary results 186
4.2 Phase-space analysis and renormalization 192
4.3 Large-momentum and short-distance properties—Wilson
short-distance expansion 198
5. Bosonic models: complements 203
5.1 Ultraviolet limit in (massive) Ρ{ψ)2 models 203
5.2 Phase-space analysis in more general models: massive φ%
and "infrared ψ%" (outline) 205
IV. PARTICLE ANALYSIS IN CONSTRUCTIVE FIELD THEORY 208
1. Introduction 208
1.1 The perturbative approach 208
1.2 Constructive and semi-axiomatic approaches 212
2. Irreducible kernels in super-renormalizable models 217
2.1 Cluster expansions of order ν > 1 217
2.2 The 2-particle irreducible Bethe-Salpeter kernel and the BS
equation in Ρ(φ)ί models 221
3. Irreducible kernels in nonsuper-renormalizable theories 225
3.1 Irreducible kernels satisfying regularized equations 225
3.2 The Bethe-Salpeter and renormalized BS kernels in the
Gross-Neveu model 229
4. Two-particle structure in weakly coupled field theories 237
4.1 Two-particle bound states and asymptotic completeness
in even theories 237
4.2 Noneven theories 244
4.3 Gross-Neveu model and semi-axiomatic approaches 247
5. Many-particle structure analysis: general results and conjectures 249
5.1 Structure equations 249
5.2 Discontinuity formulae of Feynman-type integrals 256
5.3 Asymptotic completeness relations, S-matrix discontinuity
formulae, and all that 264
viii CONTENTS
MATHEMATICAL APPENDIX: DISTRIBUTIONS, ANALYTIC
FUNCTIONS, AND MICROLOCAL ANALYSIS 271
1. Microsupport of distributions 271
2. Local analyticity properties, general decomposition theorems,
generalized edge-of-the-wedge theorems 273
3. Products and integrals of distributions, restrictions to
submanifolds 274
4. Holonomicity (introduction) 275
5. Phase-space decompositions 276
BIBLIOGRAPHY 279
REFERENCES 281
INDEX 287
Preface
Quantum Field Theory has been recognized for a long time as a funda
mental theory in High Energy Particle physics. Its main developments
have been carried out in the perturbative approach in which quantities of
interest are expressed for each given model (corresponding to a specific
type of interaction at the microscopic space-time level), as series with
respect to a possibly renormalized coupling constant. Although pertur
bative renormalization provides well-defined quantities at each order,
these series are in general divergent, however small the coupling is. A
number of works have been devoted to establishing field theory on more
rigorous, nonperturbative bases and to getting a better and deeper un
derstanding of its general properties. To that purpose "axiomatic" and
"constructive" approaches have been developed, with the first aim of
setting a precise formalism to start from: general (model-independent)
axioms, possibly completed by further conditions intended to charac
terize subclasses of theories of interest, and nonperturbative definition
of models respectively. From the viewpoint of particle physics, a fur
ther ambition is to develop a satisfactory relativistic scattering theory
including the analysis of the particle content of the theories and the de
termination of general properties of multiparticle collision amplitudes.
In spite of more general investigations, the best results so far in this
domain apply to theories describing systems of massive particles with
short-range interactions, to which we shall mainly restrict our attention.
This is a strong physical limitation and important aspects of modern
particle physics developed in the 1970s and 1980s are absent from these
theories. Moreover, the very mathematical existence of such theories, es
tablished in space-time dimension 2 or 3, remains doubtful in dimension
4 if we leave aside trivial theories of free particles without interactions.
However, many basic features of field theory are present and the deeper
analysis that can be carried out for these theories already exhibits a large
number of properties of physical and mathematical interest which justify,
in our opinion, their study as a first approach to relativistic quantum
physics.
IX