Renormalized Quantum Field Theory

536 Pages·1989·15.283 MB·English

by  O. I. Zavialov (auth.)

Renonnalized Quantum Field Theory
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M.HAZEWINKEL
Cenlre for Mathematics and Computer Science, Amsterdam, TM Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, U.S.s.R.
Yu.1. MANIN. Stelclov Institute of MatMmalics, Moscow, U.s.s.R.
N. N. MOISEEV. Computing Cenlre, Academy ofSc~nces, Moscow, u.s.s.R.
S. P. NOVIKOV. Landau Institute ofTMoretical Physics, Moscow, U .s.S.R.
M. C. POL YV ANOV, Stelclov Institute ofM athematics, Moscow, u.s.s.R.
Yu. A. ROZANOV, Stelclov Institute ofM atMmalics, Moscow, U.S.s.R.
Volume 21
O. I. ZAV IALOV
Steklov Institute, Moscow, U.S.S.R.
Renormalized
Quantum Field
Theory
KLUWER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LANCASTER
Library of CODeress Catalogina·ID.PubUcation Data
Zav'i~lov, Oleg Ivanovich.
Renonnalized quantum field theory.
(Mathematics and its applications. Soviet series)
Translation of: Perenormirovannye diagranny
Fcl'nmana.
Bibliography: p.
Includes index.
1. Quantum field theory. 2. Feymaan diagrams.
3. Green's functions. I. Title. II. Series:
Mathematics and its applications (Kluwer Academic
Publishers). Soviet series.
QC175.45.Z3813 1989 530.1'43 88-12650
ISBN·13: 978·94·010·7668-5 e-ISBN-13:978-94-009-2585-4
DOl: 10.1007/978-94-009-2585-4
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other counlries, sold and dislributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
This is a revised and eruarged translation of the book
Perenorrnirovannye diagrammy Fetnmana.
published by Nauka, Moscow, 1979.
prillfed 01/ acid Fcc paper
All Rights Reserved
This English edition @ 1990 by Kluwer Academic Publishers
Softcover reprint of the hardcover I st edition 1990
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
SERIES EDITOR'S PREFACE
'Et moi. ...• Ii j'avait su CClIIIIIIaIt CD 1'CVCDir, ODe scmcc matbcmatK:s bas I'CIIdcRd !be
je D', semis paiDt ~. humaD mcc. It bas put common sease bact
Jules Vcmc 'WIIcR it bdoDp, 011 !be topmost sbdl JlCXt
10 !be dully c:uista' t.bdlcd 'cIiIc:arded DOlI-
The series is diverpt; therefore we may be sense'.
able 10 do sometbiD& with it Eric T. BcII
O. Heavilide
Mathematics is a tool for thought. A highly ncceuary tool in a world where both feedback and non-
1inearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the l'Iison d'etre of this series.
1bis series, Mothmralics and lu AppllcatlotU, started in 1977. Now that over one bundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more efi'ort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and morc)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much morc. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vi
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no 1V; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- in1luences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
There is no doubt that renormalization theory it a most important topic and becoming more
important also in other fields than those of theoretical physics, where it is already an established
and often-used technique. Examples are certain parts of probability and a number of topics where
scaling phenomena play an important role.
At the same time the phrase 'renormalization theory' carries for many a certain tlavour of mys
tery: one has to be an initiate in one sense or another to make proper use of it. Indeed a fairly
recent article on the topic has as its title: "'Renormalization: from magic to mathematics".
This book is a mathematically rigorous book on the topic and it should do much to remove that
air of mystery and inaccessability; it should thus enable many more to learn the art of balancing
infinities.
In addition to a thorough and mathematically rigorous treatment of the main results and tech
niques of (the many different kinds of) renormalization theory it contains original so far unpub
lished work by this well-known author in this field, and as such is a welcome addition to this series.
The shortest path between two truths in the Never lend books, for no one ever returns
rcal domain passes througb the complex them; the only books 1 have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique DC nous donne pas seulement The function of an expert is not to be more
l'occuion de resoudre des probIemes ... eIle right than other people, but to be wrong for
nous fait pressentir la solution. more sophisticated reasons.
H. poincaJt David Buder
Bussum, June 1989 Michiel Hazewinkel
CONTENTS
Preface xi
Chapter I. Elements of Quantum Field Theory 1
1. Quantum Free Fields 1
1.1. Fock Space 1
1.2. Free Real Scalar Field 6
1.3. Other Free Fields 13
2. The Chronological Products of Local Monomials
of the Free Field 20
2.1. Wick Theorem 20
2.2. Wick Theorem for Chronological Products
of Free Fields 24
2.3. Regularized T-Products 26
2.4. Ambiguity in the Choice of Chronological
Products 29
3. Interacting Fields 38
3.1. Interpolating Heisenberg Field 38
3.2. Connection Between Two Systems of Axioms 44
3.3. T-Exponential, Lagrangian, Renormalization
Constants 50
3.4. Green Functions, Functional Integral,
Euclidean Quantum Field Theory 57
3.5. Interaction Lagrangians 65
Chapter II. Parametric Representations for Feynman
Diagrams. R-Operation 69
1. Regularized Feynman Diagrams 69
1.1. Intermediate Regularization. Divergency
Index 69
1.2. Parametric Representation for Regularized
Diagrams 75
1.3. The Proof of Statements (16)-(21) 80
1.4. Parametric Representations in Other
Dimensions and in Euclidean Theory.
Coordinate Representation 90
2. Bogoliubov-Parasiuk R-0peration 93
2.1. Subtraction Operators M and Finite
Renormalization Operators P. Definition of
R-Operation 93
2.2. The Structure of the R-0peration 103
2.3. R-0peration with Non-Zero Subtraction Points
or Other Subtraction Operators 107
viii CONTENTS
3. Parametric Representations for Renormalized
Diagrams 112
3.1. Renormalization over Forests 112
3.2. Non-Zero Subtraction Points 119
3.3. Renormalization over Nests 121
3.4. Renormalization by Means of Integral
Operators 128
Chapter III. Bogoliubov-Parasiuk Theorem. Other
Renormalization Schemes 133
1. Existence of Renormalized Feynman Amplitudes 133
1.1. Division of the Integration Domain into
Sectors. The Equivalence Classes of Nests 133
1.2. The Ultraviolet Convergence of Parametric
Integrals 143
1.3. The Limit £ ~ 0 147
2. Infrared Divergencies and Renormalization in
Massless Theories 151
2.1. Infrared Convergence of Regularized
Amplitudes 151
2.2. Illustrations and Heuristic Arguments 156
2.3. Classification of Theories 160
2.4. Ultraviolet Renormalization 163
2.5. More Refined Arguments 179
3. The Proof of Theorems 1 and 2 180
3.1. Preliminaries 180
3.2. Basic Lemma 182
3.3. Theorem 1. The Case of a Diagram without
Massive Lines 186
3.4. Theorem 1. The Case of a Diagram with
Massive Lines 191
3.5. The Scheme of the Proof for Theorem 2 195
3.6. The Structure of the Forms D, A, Bl , K .. 196
. . 4v 1)
3.7. Trans1t1on from the Space S' (R '{q O})
4v ~
to the Space S' (R 'E) 206
4. Analytic Renormalization and Dimensional
Renormalization 207
4.1. Introductory Remarks 207
4.2. The Recipe for Analytic Renormalization 208
4.3. The Equivalence of R-Operation and Analytic
Renormalization 211
4.4. Dimensional Renormalization 216
4.5. The Parametric Representation in the Case
of Dimensional Renormalization 222
4.6. Equivalence of the Dimensional Renormalization
and R-Operation 225
4.7. Modifications. Zero Mass Theories 227
4.8. Examples 229
CONTENTS ix
5. Renormalization 'without Subtraction'.
Renormalization 'over Asymptotes' 234
5.1. Intermediate Regularization and the
Recipe of Renormalization 'without
Subtraction' 234
5.2. The Equivalence of the R-Operation and
the Renormalization 'without Subtractions' 244
5.3. Renormalization 'over Asymptotes' 248
Chapter IV. Composite Fields. Singularities of the
Product of Currents at Short Distances and on the
Light Cone 252
1. Renormalized Composite Fields 252
1.1. Basic Notions and Notations 252
1.2. The Subtraction Operator M 261
1.3. The Structure of Renormalization 274
1.4. Generalized Action principle 282
1.5. Zimmermann Identities 288
2. Products of Fields at Short Distances 294
2.1. A Lowest Order Example 294
2.2. Wilson Expansions 302
2.3. A Massless Case 307
2.4. An Important Particular Case 311
3. Products of Currents at Short Distances 315
3.1. Short-Distance Expansions for Products
of Currents 315
3.2. The Proof of the Lemma 321
3.3. The Structure of Renormalization with
Incomplete Subgraphs. The Short-Distance
Expansion in the Weinberg Renormalization
Scheme 333
4. Products of Currents near the Light Cone 339
4.1. Lower Order Consideration 339
4.2. Subtraction Operator i(a). Light-Ray
Fields 343
4.3. The Light-Cone Theorem 349
4.4. An Example. General Discussion.
A Massless Case 356
5. Equations for Composite Fields 365
5.1. Equations of Motion for the Interpolating
Field 365
5.2. Equations for Higher Composite Fields 369
5.3. The Proof of Relations (273) and (276) 372
5.4. Renorm-Group Equations and Callan-
symanzik Equations 378
6. Equations for Regularized Green Functions 389
6.1. Relation of Renormalization Constants to
Green Functions 389
6.2. Relations of Green Functions to Derivatives
of the Renormalization Constants 395
x CONTENTS
Chapter V. Renormalization of Yang-Mills Theories 401
1. Classical Theory and Quantization 401
1.1. Classical Yang-Mills Fields 401
1.2. Quantization 403
1.3. Fields of Matter. Abelian Theory 408
2. Gauge Invariance and Invariant Renormalizability 412
2.1. Abelian Theories. Ward Identities 412
2.2. Non-Abelian Yang-Mills Theories.
BRST Symmetry. Slavnov Identities 415
2.3. A Linear Condition for the Gauge Invariance
of Non-Abelian Yang-Mills Theories 421
2.4. The Structure of Subtractions 426
2.5. Invariant Renormalizability of the Yang
Mills Theory 433
3. Invariant Regularization and Invariant
Renormalization Schemes 438
3.1. ~reliminary Discussion 438
3.2. Scalar Electrodynamics. Recipes for
Regularization 442
3.3. Scalar Electrodynamics. Arguments in Favour
of the Recipe 447
3.4. Spinor Electrodynamics. Recipes for
Regularization 451
3.5. Spinor Electrodynamics. Argumentation 454
3.6. Examples and Remarks 458
3.7. Non-Abelian Yang-Mills Theories 465
3.8. An Example: Gluon Polarization Operator.
Arguments 468
4. Anomalies 478
4.1. Is It Always Possible to Retain a
Classical Symmetry in a Quantum Field Theory? 478
4.2. Main Statements 482
4.3. Heuristic Check of Ward Identities
(Momentum Representation) 488
4.4. The Triangle Diagram in the a-Representation 490
4.5. Ward Identities 495
Appendix. On Methods of Studying Deep-Inelastic
Scattering 498
A.l. Deep-Inelastic Scattering 498
A.~. The Traditional Approach to Deep-Inelastic
Scattering 501
A.3. The Non-Local Light-Cone Expansion as the
Basic Tool to Study Deep-Inelastic Scattering 508
A Guide to Literature 513
References 516
Index 522