Quantum Field Theory: Perspective and Prospective NATO Science Series

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Series C: Mathematical and Physical Sciences - Vol. 530 LES HOUCHES

CENTRE DE PHYSIQUE

Quantum Field Theory: Perspective and Prospective edited by

Cecile DeWitt-Morette Department of Physics and Center for Relativity, University of Texas, Austin, U.S.A. and Jean-Bernard Zuber Service de Physique Theorique, CEASaclay, Gif-sur-Yvette Cedex, France ....

"Springer Science+Business Media, B.V. Proceedings of the NATO Advanced Study Institute on Quantum Field Theory: Perspective and Prospective Les Houches, France 15-26 June 1998

A C.I.P. Catalogue record for thls book Is avallable from the Llbrary of Congress

ISBN 978-0-7923-5673-8 ISBN 978-94-011-4542-8 (eBook) DOI 10.1007/978-94-011-4542-8

Printed an acid-free paper

AII Rights Reserved @ 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 Softcover reprint of the hardcover 1st edition 1999 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 photo• copying, recorcling or by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS

Preface VII

List of Participants Xlll

Correlation Functions through Variational Methods R. Balian 1

Reader's Guide to String Duality and Matrix Theory T. Banks 19

An Introduction to Equivariant Cohomology R.Bott 35

Hopf Algebras, Renormalization and Noncommutative Geometry A. Connes and D. Kreimer 59

The Peierls Bracket B. DeWitt 111

Gauge Theory without Ghosts and Quantum Gravity without Ghosts B. DeWitt and C. Molina-Paris 137

Quantized Branes and Symmetries of String Theory L. Dolan 139

Instructive History of the Quantum Inverse Scattering Method L. Faddeev 161

The Electron is Inexhaustible J. Frohlich 179

Exact Solution of Random Tiling Models B.lVienhu~ 217

Dimensionally Reduced Chern-Simons Terms and their Solutions R. Jackiw and So-Young Pi 245

BPS Branes in Supergravity K. Stelle 257 vi

Selected Topics in Perturbative Field Theory: Renormalization, Differential Algebras R. Stora 353

Lectures on Random Matrices and Hall's Fluids A.Zee 369

Renormalization and Renormalization Group: From the Discovery of UV Divergences to the Concept of Effective Field Theories J. Zinn-Justin 375

Les Houches 1970: List of Participants 389

Les Houches 1975: List of Participants 393

Index 399 Preface

The 1998 NATO AS! on Quantum Field Theory: Perspective and Prospec• tive, is a legacy of two NATO ASI's organized at the Ecole de Physique (Les Houches).

1970 Statistical Mechanics and Quantum Field Theory organized. by C. DeWitt-Morette and R. Stora.

1975 Methods in Field Theory organized by R. Balian and J. Zinn-Justin.

Quantum Field Theory experienced a renaissance in the late sixties. We asked participants in the 1970/1975 sessions, now key players in Quantum Field Theory and its multifaceted developments, two questions: * How do you assess the developments since 1970/1975 in your field of interest ? (legacy) * What advice and guidance do you give to your students and young collaborators? (exploiting the legacy)

At the end of the volume, the reader will find the lists of the 1970 and 1975 participants. For permission to reproduce the cover page, list of lecturers, and list of participants, we thank The Gordon and Breach Publishing Group for the Les Houches 1970 proceedings, and Elsevier Science for the Les Houches 1975 proceedings. After several months of consultation, we identified the following objectives: * An interdisciplinary AS! for physicists and mathematicians who started their careers in Quantum Field Theory, and have since made important contributions not only in its original domain (particle physics) but also in areas of statistical mechanics, condensed matter physics, cosmology, gravitation, algebra, differential geometry or topology. * Guidance for young researchers challenged by the far reaching concepts of Quantum Field Theory, but overwhelmed by its rich and complex details. vii viii

With this goal in mind, we planned the courses listed in the table of contents. On average the courses consisted of lectures totaling four and a half hours.

Alain Connes (les Houches 1970) fascinated the 1998 participants with the powerful and far reaching Noncommutative Geometry which he has devel• oped in the intervening years; for some participants it was a first and heady exposure, for others it was an opportunity to learn recent developments in a field they are exploring. In 1998, Strings, Branes, and M-theory are dominant themes; they were cov• ered by T. Banks, L. Dolan, and K. Stelle (all Les Houches 1975 students).

It may seem that these two fields of investigation, Noncommutative Geome• tryon the one hand, and Strings, Branes, and M-Theory on the other hand tackle the fundamental issues of space-time-matter from totally different per• spectives. Nevertheless in a challenging lecture, J. Frohlich (1970) explored possible bridges between Noncommutative Geometry and Supersymmetric Quantum Theory. This program is described in more detail in his 1995 Les Houches lectures (Quantum Symmetries, eds. A. Connes, K. Gawedzki and J. Zinn-Justin). His more recent work is included here in "The Electron is Inexhaustible.'l Physicists hope one day to have an honest understanding of the quantum mechanical nature of spacetime.

Selected important issues of Quantum Field Theory and of its impact on Condensed Matter Physics can be found in this volume. Needless to say, in the limited time of a two week session and the limited space of a book, equally important issues could not be included. But the following selection offers already a view of Quantum Physics from high power mathematics to novel table top. physics: * Equivariant Cohomology by R. Bott (1970). * Some features of the Chern-Simons action which has no explicit depen• dence on the spacetime metric (a topological field theory) by S.-Y. Pi (1975). * New, or little known, basic topics in field theory, Variational Methods by R. Balian, ix

The Peierls Bracket by B.S. DeWitt (1975). * Two contributions on renormalization one by R. Stora, which relates it to differential algebras, one by J. Zinn-Justin in the context of an historical survey of Quantum Field Theory as reflected in a number of Les Houches sessions. * Gauge Theory without Ghosts by B.S. DeWitt, to be found in the volume only by references and abstracts. * An Instructive History of the Quantum Inverse Scattering Method by L. Faddeev (1975), who presented also some new results (not included in this volume) on solitons and the Hopf invariant, and on the duality of the sine-Gordon theory. * Examples of the impact of Quantum Field Theory on Condensed Mat• ter Physics were given by

J. Frohlich and A. Zee in the analysis of the Quantum Hall Effect, B. Nienhuis (1975) in Quasicrystals and Random Lattices, A. Zee (1975) in Random Matrix Theory.

Not included in this volume are the short presentations by the participants, a vital part of the session but difficult to capture in a written document. They are listed by titles at the end of the preface.

We have mixed feelings on the value of proceedings. Fortunately the NATO Science Committee policy discourages the publication of "ordinary" proceed• ings which are simply collections oflecture notes, together with a few research papers. The output of the meetings, required by NATO, is a book which ex• tends the benefits of the ASI to scientists who did not have the opportunity of participating in the session. In order to fulfill this goal we have included three papers which have been or will be shortly published elsewhere; we are grateful to the publishers who have given us the necessary authorizations: x * The American Mathematical Society for permission to include a modi• fied version of the article by J. Frohlich "The electron is inexhaustible" in Prospects in Mathematics Hugo Rossi, Ed. pp 17-44, 1999. * Elsevier Science for permission to reproduce the article of B. Nienhuis which appeared in Physics Reports. * World Scientific for permission to reproduce the notes of K. Stelle based on lectures given at the 1996 and 1997 ICTP Summer Schools in High Energy Physics and Cosmology. * Kluwer Academic Publishers for leaving the copyright of A. Cannes and D. Kreimer in their own name. * Kluwer Academic Publishers for permission to reproduce the article of L. Faddeev which had appeared in Acta Applicandae Mathematicae. It has been said "String theorists talk to String theorists and everyone else wonders what they are saying". We hope this book will help researchers challenged by modern Quantum Field Theory.

The preparation of this volume was made onerous by the publisher's request for a camera-ready copy. To produce a unified camera-ready copy complete with photographs from the electronic versions of the authors turned out to be a time consuming painstaking chore. We would not have been able to do it without the competence and the superb help of Alexander Wurm who fixed up countless details and never lost his friendly composure.

An Advanced Study Institute is more than a series of lectures, it is a commu• nity of participants who live together for a couple of weeks. A book cannot convey the unique flavor of a session, the atmosphere of a "standing room only" classroom, the questions and challenges exchanged between the senior participants, to the delight of the junior ones who often had thought their seniors to be on a pedestal.

The quality of life during this period contributed immensly to its success: the delicious meals prepared by Claude Coneau and his team, the magnificent setup of the E~ole de Physique, its new or remodelled facilities, the impec• cable and friendly management by the School Secretariat: Brigitte Rousset, xi

Isabel Lelievre, and Ghislaine d'Henry, the services of the staff responsible for the buildings and grounds: Thierry Rousset, Christiane Rocher, and Martine Balbo.

All this was made possible by the financial support from the NATO Sci• entific Affairs Division, the Universite Joseph Fourier, and the Continuing Education program of the CNRS. A number of participants received support from their institutions. NSF-NATO travel grants were awarded to Alexander Wurm and Edwin Karat, a Tubitak-NATO travel grant to Ayfer Kandemir. To Michele Leduc, the Director of the Centre de Physique who offered us the resources of the Centre and to everyone who contributed to this session, we express our gratitude.

Cecile DeWitt-Morette Jean-Bernard Zuber. Seminars given by participants

A. Mostafazadeh: Lattice Topological Field Theories and *-Algebras

M. Berg and C. DeWitt-Morette: Massive Neutrinos

A. Accardi: Topological Sectors in Pure Yang-Mills Theory

F. Gonzalez-Rey : Self-dual (& exact?) Low Energy Effective Action of N=4SYM v. Brindejonc: 3d Yang-Mills Theory as a Gravity

A. Ouarab : Supermulticonformal Structure on Me. -Bosonic Action-

V. Terras - N ~ Kitanine : One-Point Functions for the XXZ Model. Quan• tum Inverse Scattering Method Approach

P. Zinn-Justin : Solution of a New Two-Matrix Model

xii Les Houches, June 15-26 1998 Participants

1. DIRECTORS Raoul BOTT, Dept. of Mathematics, Cecile DEWITT-MORETTE, Harvard University, Department of Physics, Oxford Street, University of Texas, Cambridge, MA 02138, Austin, TX 78712, USA, USA, email: [email protected] email:[email protected] Alain CONNES, Jean-Bernard ZUBER, IHES, Service de Physique Thoorique , 35 route de Chartres, CEA Saclay, F-91440 Bures sur Yvette, 91191 Gif-Sur-Yvette Cedex, France, France, email: connes@ihesJr email:[email protected] Bryce DEWITT, Department of Physics, The University of Texas at Austin, 2. LECTURERS Austin, TX 78712, USA, Roger BALIAN, email: [email protected] Service de Physique Thoorique, CEA Saclay, Louise DOLAN, F-91191 Gif sur Yvette Cedex, Dept. of Physics, France, University of North Carolina, email: [email protected] Chapel Hill, NC 27599-3255, USA, Tom BANKS, email: [email protected] Dept. of Physics, Rutgers University, Ludwig FADDEEV, P.O. Box 849, V.A. Steklov Mathematical Institute, Piscataway, NJ 08855-0849, Fontanka 27, USA, St Petersburg, 191011, email: [email protected] Russia, email: [email protected] xiii xiv

Santa Barbara, CA 93106 USA, Jurg FROHLICH, email: [email protected] Theoretical Physics, ETH-Honggerberg, Jean ZINN-JUSTIN, CH-8093 Ziirich, Service de Physique Thoorique , Switzerland, CEA Saclay, email: [email protected] F-91191 Gif sur Yvette Cedex, France, Bernard NIENHUIS, email: [email protected] Dept. of Physics, University of Amsterdam, Valckenierstraat 65, NL-1018 XE Amsterdam, 3. ASI STUDENTS The Netherlands, email: [email protected] Alberto ACCARDI, Dipartimento di Fisica, So Young PI, Universita di Milano, Dept. of Physics, Sez. Astrofisica c/o M. Martellini, Boston University, via Celoria, 16, 590 Commonwealth Ave., 20100 Milano, Boston, MA 02215, Italy, USA, email: [email protected] email: [email protected] Marcus BERG, Kellog STELLE, Center for Relativity (C1606), Blackett Laboratory, UT Austin, Imperial College, Austin, TX 78712, Prince Consort Rd., USA, London SW7 2BZ, email: [email protected] United Kingdom, email: [email protected] Gabrielle BONNET, Service de Physique Thoorique , Raymond STORA, CEA Saclay, LAPTH, 91191 Gif-Sur-Yvette Cedex, BP 110, France, 74941 Annecy-Ie -Vieux Cedex, email: [email protected] France, email: [email protected] Vincent BRINDEJONC, Service de Physique Nucleaire, Anthony ZEE, CEA-Saclay, Institute for Theoretical Physics, 91190 Gif sur Yvette cedex, University of California, France, xv email: [email protected] email: [email protected]

Raffaele BUFFA, Dmitry GITMAN, Centre de Physique Theorique, Institute of Physics, 13288 Marseille Cedex 9, University of Sao Paulo, France, CP 66318, email: [email protected] Sao Paulo, 05315-970, S.P., Brazil, email: [email protected] Pierre CARTIER, IRES, 35 route de Chartres, Francisco GONZALEZ-REY, F-91440 Bure sur Yvette, Institute for Theoretical Physics, France, Physics Department, email: [email protected] S.U.N.Y. at Stony Brook, Stony Brook, NY 11794-3840, USA, Francois DA VID, email: [email protected] Service de Physique Theorique , CEA Saclay, Christopher HERZOG, F-91191 Gif sur Yvette Cedex, Laboratoire Kastler Brossel, France, E.N.S., email: [email protected] 75231 Paris Cedex 05, France, Hiroshi EZAWA, email: [email protected] Department of Physics, Gakushuin University, Masao HIROKAWA, Mejiro, Toshima-ku, Tokyo Gakugei University, Tokyo, 171-8588, Department of Mathematics, Japan, Koganel, email: [email protected] Tokyo 184, Japan, Giovanni FEVERATI, email: [email protected] Dipartimento di Fisica, Universita di Bologna, Fumio HIROSHIMA, via Irnerio 42, Institute for Applied Mathematics, 40100 Bologna,Italy, University of Bonn, email: [email protected] Wegelerstr. 6, D-53115 Bonn, Daniela GARAJEU, Germany, Centre de Physique Theorique, email: [email protected] CNRS - Luminy - Case 907, 13288 Marseille Cedex 9, Conor HOUGHTON, France, DAMTP, xvi

Silver Str., Edwin KARAT, Cambridge CB3 9EW, MIT Center for Theoretical Physics, United Kingdom, 77 Massachusetts Ave, email: [email protected] Room 6-409A, Cambridge, MA 02139, Stephen HOWES, USA, Department of Physics, email: [email protected] University of Wales Swansea, Singleton Park, Robert L. KARP, Swansea SA2 8PP, EOTVOS University, United Kingdom, Theoretical Physics Department, email: [email protected] Puskin 5-7, Budapest, 1088, Takashi ICHINOSE, Hungary, Department of Mathematics, email: [email protected] Kanazawa University, Kanazawa,920-1192, Daniel KASTLER, Japan, Centre de Physique Theorique, email: [email protected] CNRS - Luminy, case 907, F-13288 Marseille cedex 9, Patrick IRWIN, France DAMTP, University of Cambridge, Achim KEMPF, Silver St., DAMTP, Cambridge CB3 9EW, Silver St., United Kingdom, Cambridge CB3 9EW, email: [email protected] United Kingdom, email: [email protected] Arthur JAFFE, Department of Physics, Annamaria KISS, Harvard University, Centre de Physique Theorique, Cambridge, MA 02138, Luminy - case 907, USA, 13288 Marseille Cedex 9, email: [email protected] France, email: [email protected] Ayler KANDEMIR, Ankara University, Nikolai KITANINE, Faculty of Sciences, Laboratoire de Physique, Physics Dept. 06100, Groupe de theorie, Tandogan-Ankara, ENS - Lyon, Turkey, 46 allee d'Italie, email: [email protected] 69364 Lyon Cedex 07, France, xvii email: [email protected] Universita. di Trieste, P.le Europa, 1, John KLA UDER, 1-34127 Trieste, Dept. of Physics, Italy, Univ. Florida, email: [email protected] P.O. Box 118440, Gainesville FL 32611-8440, Serge LAZZARINI, USA, Centre de Physique Th&lrique, email: [email protected] CNRS - Luminy, case 907, F-13288 Marseille cedex 9, Christian KORFF, France, Institut fiir theoretische Physik, email: [email protected] Freie Universitat Berlin, Arnimalle 14, Thierry MASSON, D-14195 Berlin, LPTHE Bat 210, Germany, UniversiteParis XI, email: [email protected] 91405 Orsay Cedex, France, Thomas KRAAN, email: [email protected] Instituut-Lorentz for Theoretical Physics, Leiden University, Hyunsoo MIN, P.O. Box 9506, Department of Physics, NL 2300 RA Leiden, University of Seoul, The Netherlands, Seoul, 130-743, email: [email protected] Korea, email: [email protected] Thomas KRAJEWSKI, Centre de Physique Theorique, Ali MOSTAFAZADEH, Campus de Luminy, Math Dept., 13288 Marseille cedex 9, Ko~ Uni., France, Instinye 80860 Istanbul, email: [email protected] Thrkey, email: [email protected] Dirk KREIMER, Dept. of Physics (ThEP), Aziza OUARAB, Mainz Univ., Faculte des Sciences, D-55099 Mainz, Laboratoire de Physique theorique, Germany, Av. Ibn Battouta, email: [email protected] B.P. 1014, Rabat, Morocco, Giovanni LANDI, [email protected] Dipartimento di Scienze Matematiche, xviii

Jean-Claude PISSONDES, Hebrew University, Dept. d' Astrophysique Extragalactique, Givat Ram, et de Cosmologie (DAEC), Jerusalem - 91904, 5, place J. Janssen, Israel, 92195 Meudon Cedex, email: [email protected] France, email:[email protected] Vironique TERRAS, Laboratoire de Physique, Henning SAMTLEBEN, Groupe de theorie, Max-Planck-Institut , ENS de Lyon, fiir Gravitationsphysik, 46 allee d'Italie, Schlaatzweg 1, 69364 LYON Cedex 07, D-14473 Postdam, France, Germany, email: [email protected] email: [email protected] Koumarane VALA VANE, Emanuel SCHEIDEGGER, Centre de Physique Theorique, Ludwig-Maximilian-Universitat, Case 907,163 avenue de Luminy, Sektion Theoretische Physik, 13288 Marseille cedex 09, Lehrstuhl Wess, France, Theresienstr. 37, email: [email protected] D-80333 Miinchen, Germany, email: [email protected] Marcel VENERONI, Institut de Physique Nuclerure, Jorn SCHIMMEL, 91406 Orsay, Institut fiir theoretische Physik, France Bunsenstr. 9, D-37073 Gottingen, Raimar WULKENHAAR, Germany, Centre de Physique Theorique, email: CNRS-Luminy, Case 907, [email protected] F-13288 MARSEILLE Cedex 9, France, Andrzej SITARZ, email: [email protected] Institute of Physics, Jagiellonian University, Alexander WURM, Reymonta 4, Department of Physics, Krakow, 30-059, Center for Relativity, Poland, The University of Texas at Austin, email: [email protected] Austin TX 78712, USA, Lakshmanan SRIRAMKUMAR, email: [email protected] Racah Institute of Physics, xix

Paul ZINN-JUSTIN, Laboratoire de Physique Theorique, de l'Ecole Normale Superieure, 75231 Paris Cedex 05, France, email: [email protected]

4. VISITORS

Klaus HEPP, Theoretical Physics, ETH-Honggerberg, CH-B093 Ziirich, Switzerland, email: [email protected]

Ivan KOSTO V, Service de Physique Theorique , CEA Saclay, 91191 Gif-Sur-Yvette Cedex, France, email: [email protected]