Progress in Gauge Field Theory PDF
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Progress in Gauge Field Theory NATO ASI Series Advanced Science Institutes Series A series presenting the results of activities sponsored by the NA TO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physics New York and London C Mathematical D. Reidel Publishing Company and Physical Sciences Dordrecht , Boston, and Lancaster o Behavioral and Social Sciences Martinus Nijhoff Publishers ,E Engineering and The Hague, Boston, and Lancaster Materials Sciences F Computer and Systems Sciences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York, and Tokyo Recent Volumes in this Series Volume 111-Monopole '83 edited by Jame$ L. Stone Volume 112-Condensed Matter Research Using Neutrons: Today and Tomorrow edited by Stephen W. Lovesey and Reinhard Scherm Volume 113-The Electronic Structure of Complex Systems edited by P. Phariseau and W. M. Temmerman Volume 114-Energy Transfer Processes in Condensed Matter edited by Baldassare Oi Bartolo Volume 115-Progress in Gauge Field Theory edited by G.'t Hooft, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer, and R. Stora Volume 116-Nonequilibrium Cooperative Phenomena in PhysiCS and Related Fields edited by Manuel G. Velarde Volume 117-Moment Formation in Solids edited by W. J. L. Buyers Series B: Physics Progress in Gauge Field Theory Edited by G. 't Hooft Institute for Theoretical Physics Utrecht, The Netherlands A. Jaffe Harvard University Cambridge, Massachusetts H. Lehmann University of Hamburg Hamburg, Federal Republic of Germany P. K. M itter University of Paris VI Paris, France I. M. Singer University of California Berkeley, California and R. Stora Laboratory of Particle Physics Annecy, France Plenum Press New York and London Published in cooperation with NATO Scientific Affairs Division Proceedings of a NATO Advanced Study Institute on Progress in Gauge Field Theory, held September 1-15, 1983, in Cargese, Corsica, France ISBN 978-1-4757-0282-8 ISBN 978-1-4757-0280-4 (eBook) DOI 10.1007/978-1-4757-0280-4 Library of Congress Cataloging in Publication Data NATO Advanced Study Institute on Progress in Gauge Field Theory (1983: Cargese, Corsica) Progress in gauge field theory. (NATO ASI series. Series B, Physics; v. 115) "Published in cooperation with NATO Scientific Affairs Division." "Proceedings of a NATO Advanced Study Institute on Progress in Gauge Field Theory, held September 1-15, 1983, in Cargese, Corsica, France"-Verso t.p. Includes bibliographical references and index. 1. Gauge fields (Physics)-Congresses. I. 't Hooft, G. II. Title. III. Series. QC793.3. F5N37P 1983 530.1'43 84-17869 ISBN 978-1-4757-0282-8 © 1984 Plenum Press, New York Softcover reprint of the hardcover 1s t edition 1984 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher To the memory of Kurt Symanzik Kurt Symanzik (1923-1983) PREFACE The importance of gauge theory for elementary particle physics is by now firmly established. Recent experiments have yielded con vincing evidence for the existence of intermediate bosons, the carriers of the electroweak gauge force, as well as for the presence of gluons, the carriers of the strong gauge force, in hadronic inter actions. For the gauge theory of strong interactions, however, a number of important theoretical problems remain to be definitely resolved. They include the quark confinement problem, the quantita tive study of the hadron mass spectrum as well as the role of topo logy in quantum gauge field theory. These problems require for their solution the development and application of non-perturbative methods in quantum gauge field theory. These problems, and their non-pertur bative analysis, formed the central interest of the 1983 Cargese summer institute on "Progress in Gauge Field Theory." In this sense it was a natural sequel to the 1919 Cargese summer institute on "Recent Developments in Gauge Theories." Lattice gauge theory provides a systematic framework for the investigation of non-perturbative quantum effects. Accordingly, a large number of lectures dealt with lattice gauge theory. Following a systematic introduction to the subject, the renormalization group method was developed both as a rigorous tool for fundamental questions, and in the block-spin formulation, the computations by Monte Carlo programs. A detailed analysis was presented of the problems encountered in computer simulations. Results obtained by this method on the mass spectrum were reviewed. In addition lecture series were devoted to a new approach to the mass spectrum based on a perturbative scheme and asymptotic freedom, and to the mathe matical study of asymptotically free renormalisable field theories in the planar diagram approximation. Recent work in constructive field theory based on random walk methods, the random surface approach to gauge field theory, and random lattice theories were reviewed. Another series of lectures were devoted to monopoles, the calculus of variations and topology, as well as recent wort:"s on anomalies and supersymmetry and index theory. These principal lec tures were supplemented by a seminar program with invited speakers. vii PREFACE A notable absence in the school was Kurt Symanzik who was ailing at the time. Kurt Symanzik was to be a principal lecturer at this school, and his work was of extreme relevance to the central interests of the school. It was this with profound sorrow that we learned of his subsequent death. His death is a great loss for Theoretical Physics and also for the Cargese summer institute with which he had been associated for many years. He lectured successively at Cargese in 1970, 1973, 1976 and 1979 and his lectures had a seminal impact on developments in quantum field theory and statistical mechanics. With deep sorrow we dedicate these proceedings to his memory. We wish to express our gratitude to NATO whose generous financial contribution made it possible to organize this school. We thank Maurice Levy, the Director of the Institut d'Etudes Scientifiques de Cargese, as well as the University of Nice, for making available to us the facilities of the Institut. Grateful thanks are due to Marie-France Hanseler, for much help with the material aspects of the organization. Last but not least we thank the lecturers and the participants for their enthusiastic involvement which contributed much to the success of the school. G. 't Hooft A. Jaffe H. Lehmann P.K. Mitter 1.M. Singer R. Stora CONTENTS An Introduction to Gravitational Anomalies • • • • • • 1 L. Alvarez-Gaume Dirac Monopoles, from d = 2 to d = 5, Lecture I 23 F.A. Bais Charge-Pole Dynamics, Lecture II •••••.••••• 51 F.A. Bais Exact Renormalization Group for Gauge Theories • • • • 79 T. Balaban, J. Imbrie and A. Jaffe The Spectrum in Lattice Gauge Theories 105 B. Berg Lattices, Demons and the Microcanonical Ensemble ••• 155 G. Bhanot Quantum Field Theory in Terms of Random Walks and Random Surfaces • • • • • • • • • • • • • • J. Frohlich Rigorous Renormalization Group and Large N • • • • • • 235 K. Gawedzki and A. Kupiainen On Kahler's Geometric Description of Dirac Fields 247 M. Gockeler and R. Joos Planar Diagram Field Theories 271 G. 't Rooft Fields on a Random Lattice 337 C. Itzykson Defect Mediated Phase Transitions in Superfluids, Solids, and their Relation to Lattice Gauge Theories • • 373 R. Kleinert ix x CONTENTS Lattice Gauge Theory 403 C.P. Korthals Altes Constructive Theory of Critical Phenomena 435 A. Kupiainen and K. Glawedzki On a Relation Between Finite Size Effects and Elastic Scattering Processes • • •• •••• 451 M. Luscher Renormalization Group and Mayer Expansions 473 G. Mack Gauge Invariant Frequency Splitting In Non Abelian Lattice Gauge Theory • • • • • • • • • • • • 497 P.K. Mitter Prolegomena to any Future Computer Evaluation of the QCD Mass Spectrum •••.•••••••••• 531 G. Parisi Algebraic Structure and Topological Origin of Anomalies • 543 R. Stora Morse Theory and Monopoles: Topology in Long Range Forces •.•••••••••••••••• 563 C.H. Taubes Monte Carlo Renormalization Group and the Three Dimensional Ising Model K.G. Wilson . . . . . . . . . . . . . . . . . . . . . . . . Index . AN INTRODUCTION TO GRAVITATIONAL ANOMALIES Luis Alvarez-Gaume Lyman Laboratory of Physics Harvard University Cambridge, MA 02138 INTRODUCTION The study of anomalies in global and gauge currents in Quantum Field Theory has had a remarkable number of important ap~licatjons during the 70's. In the original version of the anomaly one con siders a massless fermion triangle diagram with one axial current and two vector currents. Requiring the vector currents to be conserved, one finds that the axial current is not conserved therefore leading to a breakdown of chiral symmetry in the presence of gauge fields coupled to conserved vector currents. This break down of chiral symmetry led to the understanding of TID decay and to the resolution of the u(l) problem. 2 The anomaly has also been instrumental in posing constraints to insure the mathematical con sistency of gauge theories coupled to chiral currents. If one considers a theory with gauge fields coupled for instance to left handed currents, one must look at a fermion triangle diagram with V-A currents at each vertex. Again, this diagram is anomalous, and unless the anomalies cancel when summing over all the fermion species running around the loop, one finds that the V-A currents are not conserved, implying that gauge invariance is broken and thus the anomaly renders the theory inconsistent. The anomaly cancellation condition has proven to be very useful in constraining the particle content of unified gauge theories.' More recently4, the anomaly has also been shown to be useful in analyzing the spectrum of massless fermions in confining theories. In the context of low energy chiral theories, the Wess-Zumino lagrangian5 has recently played a central role in showing that the soliton solutions of certain models6 can be identified with baryons.' This recent development has in turn shed new light into our understanding of chiral anomalies.
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