Recent Developments in Gauge Theories

437 Pages·1980·9.21 MB·English

by  M. F. Atiyah (auth.)

Recent Developments in
Gauge Theories
NATO ADVANCED STUDY INSTITUTES SERIES
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Recent Developments in
Gauge Theories
Edited by
G. 't Hooft
InstitUte for Theoretical Physics
Utrecht, The Netherlands
c. Itzykson
CENSaclay
Gif-sur-Yvette, France
A. Jaffe
Harvard University
Cambridge, Massachusetts
H. Lehmann
University of Hamburg
Hamburg, Federal Republic of Germany
P. K. Mitter
University ofP aris
Paris, France
I. M. Singer
University of California
Berkeley, California
and
R. Stora
Center of Theoretical Physics, CNRS
Marseille, France
PLENUM PRESS. NEW YORK AND LONDON
Published in cooperation with NATO Scientific Affairs Division
Library of Congress Cataloging in Publication Data
Main entry under title:
Recent developments in gauge theories.
(NATO advanced study institutes series: Series B, Physics; v. 59)
"Published in cooperation with NATO Scientific Affairs Division."
Includes index.
1. Gauge fields (Physics)- Addresses, essays, lectures. I. 't Hooft, G. II. Series.
QC793.3F5R42 530.1'43 80-18528
ISBN 978-1-4684-7573-9 ISBN 978-1-4684-7571-5 (eBook)
DOl 10.1007/978-1-4684-7571-5
Proceedings of the NATO Advanced Study Institute on
Recent Developments in Gauge Theories, held in Cargese, Corsica,
August 26-September 8, 1979.
© 1980 Plenum Press, New York
Softcover reprint of the hardcover 1st edition 1980
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N.Y. 10011
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted,
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PREFACE
Almost all theories of fundamental interactions are nowadays
based on the gauge concept. Starting with the historical example
of quantum electrodynamics, we have been led to the successful
unified gauge theory of weak and electromagnetic interactions, and
finally to a non abelian gauge theory of strong interactions with
the notion of permanently confined quarks. The.early theoretical
work on gauge theories was devoted to proofs of renormalizability,
investigation of short distance behaviour, the discovery of
asymptotic freedom, etc .. , aspects which were accessible to tools
extrapolated from renormalised perturbation theory. The second
phase of the subject is concerned with the problem of quark
confinement which necessitates a non-perturbative understanding of
gauge theories. This phase has so far been marked by the introduc
tion of ideas from geometry, topology and statistical mechanics
in particular the theory of phase transitions. The 1979 Cargese
Institute on "Recent Developments on Gauge Theories" was devoted
to a thorough discussion of these non-perturbative, global aspects
of non-abelian gauge theories. In the lectures and seminars
reproduced in this volume the reader wilf find detailed reports
on most of the important developments of recent times on non
perturbative gauge fields by some of the leading experts and
innovators in this field. Aside from lectures on gauge fields
proper, there were lectures on gauge field concepts in condensed
matter physics and lectures by mathematicians on global aspects
of the calculus of variations, its relation to geometry and
topology, and related topics. The presence of mathematicians
as enthusiastic participants and masterful lecturers in this
school deserves special mention. We hope this trend will continue
in the future and that, in the last quarter of this century,
common concerns about the fundamental interactions will bring ever
closer the physical and mathematical communities as in the
days of yore.
We wish to express our gratitude to NATO whose generous
financial contribution made it possible to organise this school.
We also thank the Centre National de la Recherche Scientifique,
a
the Delegation la Recherche Scientifique et Technique, the
v
vi PREFACE
C.E.N. de Saclay, as well as the University of Hamburg for
financial help. We thank the University of Nice for making
available to us the facilities of the Institut d'Etudes Scientifiques
de Cargese. Grateful thanks are due to Marie-France Hanseler for
much help with the material aspects of the organisation. Last but
not least we thank the lecturers and participants for their
enthusiastic involvement which contributed much to the scientific
atmosphere of the school.
G. 't Hooft
C. Itzykson
A. Jaffe
H. Lehmann
P.K. Mitter
I.M. Singer
R. Stora
CONTENTS
Remarks on Morse Theory . . . . . . . . . . . . . . . . . 1
M.F. Atiyah
Morse Theoretic Aspects of Yang-Mills Theory . . • . . . 7
R. Bott
A Semiclassical Approach to the Strong Coupling
Physics of QCD . . . . . . . . 29
C.G. Callan, Jr.
Lattice Gauge Theories 45
J. Glimm
Some Results and Comments on Quantized Gauge
Fields .............. . S3
J. Frohlich
String States in Q.C.D. 83
J.-L. Gervais and A. Neveu
Why Do We Need Local Gauge Invariance in
Theories with Vector Particles?
An Introduction. (Lecture I) 101
G. 't Hooft
Which Topological Features of a Gauge Theory
can be Responsible for Permanent
Confinement? (Lecture II) ...... . 117
G.'t Hooft
Naturalness, Chiral Symmetry, and Spontaneous
Chiral Symmetry Breaking. (Lecture III) 135
G. 't Hooft
Introduction to Lattice Gauge Theories 159
C. Itzykson
vii
viii CONTENTS
Classical Gauge Theories and Their Quantum
Role . • • . . • . . • • • • • • 189
A. Jaffe
The Coupling Constant in a ~4 Field Theory 201
J. GI~ and A. Jaffe
Exact Instanton Gases • • 205
M. Luscher
Properties of Lattice Gauge Theory Models at
Low Temperatures . . . . . . . 217
G. Mack
Geometry of the Space of Gauge Orbits and the
Yang-Mills Dynamical System . . 265
P .K. Mitter
On the Str~cture of the Phases in Lattice
Gauge Theories •........ 293
G. Parisi
Superalgebras and Confinement in Condensed
Matter Physics . . .. •....• 303
v.
Poenaru
Cutoff Dependence in Lattice ~4 Theory 313
K. Symanzik 4
Gauge Concepts in Condensed Matter Physics 331
G. Toulouse
Monte-Carlo Calculations for the Lattice
Gauge Theory . . . • . . . . . . . 363
K.G. Wilson
The lIN Expansion in Atomic and Particle
Physics ..•.. .• . . . 403
E. Witten
Generalized Non-Linear a-Models with. Gauge
Invariance 421
J. Zinn-Justin
Index 437
REMARKS ON MORSE THEORY
M. F. Atiyah
University of Oxford
'Mathematical Institute
24-29 St. Giles, oxford
§l. INTRODUCTION
Morse theory is a topological approach to the Calculus of
Variations. It aims to relate the critical points of a functional
to the topology of the function space on which the functional is
defined. It is only directly applicable in special rather res
trictive conditions, notably for problems involving one independent
variable. However I will discuss a number of special examples, in
some of which the Morse theory really works, and others in which
it clearly fails but where nevertheless some aspects appear still
to survive. These examples include those of physical interest and
it would be interesting to investigate these further. One can
make a number of speculations in this direction.
§2. GEODESICS ON A GROUP
The classical example for which Morse developed his theory is
that of geodesics. This can be formulated as follows'. Given a
compact Riemannian manifold M we consider closed paths in M
described by a periodic function f(t) with values in M or more
formally a map
where Sl is the unit circle. As our functional we take
E(f)
where f' (t) is a tangent vector to M and If' (t) I denotes its
length in the Riemannian metric of M. The critical points of
2 M. F. ATIYAH
E , that is the solutions of the corresponding Euler-Lagrange
equatioris, correspond to closed geodesics on M parametrized by
arc length.
As a special example we may take M = G a compact Lie group
with bi-invariant metric. The closed geodesics through the
identity of G are just the closed I-parameter subgroups, and so
are completely known in terms of the group structure of G.
Using the Morse theory one can then derive information about the
topology of the loop space n (G) , i. e. the space of closed paths
based at the· identity. This programme was carried through many
years ago by Bott [3J. As a simple example if G = SU(2) , the
3-sphere, the closed geodesics are the great circles (including
their n-fold iterates and for n = 0 the degenerate point-map) .
From this one finds that the homology of nG is infinite cyclic
in every even dimension and zero in odd<dimensions.
§3. YANG-MILLS IN TWO DIMENSIONS
We consider Yang-Mills theory over a compact two-dimensional
surface, for example the 2-sphere S2. The Yang-Mills equations
are rather trivial in 2-dimensions since they assert that the
field Fuv (or curvature) is covariant constant. From this one
can easily describe all solutions and one finds that they are
precisely the homogeneous connections, i.e. connections which
admit the action of SU(2) (the double cover of the rotation
group SO(3)). By a general principle homogeneous connections
with group G are given by homomorphisms of the isotropy group,
in this case the circle subgroup of SU(2), into G. Thus we
see that the critical points of the Yang-Mills action correspond
to the critical.points of §2.
If we look at the function space situation, the space of all
connections is contractible but since our action is invariant
under gauge transformations it is appropriate to factor out by
these (actually for technical reasons it is best to use only
gauge transformations which are fixed to be the identity at some
base point of S2). The resulting space turns out to have the
same homotopy type as nG. Thus we see that our example is, in
its essential features, like that of §2. This should not be too
surprising to physiCists since, if we put the Yang-Mills theory
in Hamiltonian form (treating I-dimensional space as compact) we
get the motion of a free particle on G , which is precisely the
Hamiltonian version of §2.
Mathematically there is interest in replacing S2 by a
general surface of genus g and this case has been extensively
studied by Bott and myself. For a preliminary account see [lJ.
It is perhaps worth pointing out at this stage that solutions