Quantum Field Theory: Perspective and Prospective NATO Science Series A Series presenting the results of activities sponsored by the NA TO Science Committee. The Series is published by lOS Press and Kluwer Academic Publishers, in conjunction with the NA TO Scientific Affairs Division. General Sub-Series A. Life Sciences lOS Press B. Physics Kluwer Academic Publishers C. Mathematical and Physical Sciences Kluwer Academic Publishers D. Behavioural and Social Sciences Kluwer Academic Publishers E. Applied Sciences Kluwer Academic Publishers F. Computer and Systems Sciences lOS Press Partnership Sub-Series 1. Disarmament Technologies Kluwer Academic Publishers 2. Environmental Security Kluwer Academic Publishers 3. High Technology Kluwer Academic Publishers 4. Science and Technology Policy lOS Press 5. 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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.