Quantum Fields and Quantum Space Time PDF
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Quantum Fields and Quantum Space Time NATO ASI Series Advanced Science Institutes Series A series presenting the results of activities sponsored by the NATO 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 Kluwer Academic Publishers and Physical Sciences Dordrecht, Boston, and London D Behavioral and Social Sciences E Applied Sciences F Computer and Systems SCiences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York, London, H Cell Biology Paris, Tokyo, Hong Kong, and Barcelona I Global Environmental Change PARTNERSHIP SUB·SERIES 1. Disarmament Technologies Kluwer Academic Publishers 2. Environment Springer-Verlag 3. High Technology Kluwer Academic Publishers 4. Science and Technology Polley Kluwer Academic Publishers 5. Computer Networking Kluwer Academic Publishers The Partnership Sub-Series incorporates activities undertaken in collaboration with NA TO's Cooperation Partners, the countries of the CIS and Central and Eastern Europe, in Priority Areas of concern to those countries. Recent Volumes In this Series: Volume 362 - Low-Dimensional Applications of Quantum Field Theory edited by Laurent Baulleu, Vladimir Kazakov, Marco Picco, and Paul Windey Volume 363 - Masses of Fundamental Particles: Cargese 1996 edited by Maurice Levy, Jean lIiopoulos, Raymond Gastmans, and Jean-Marc Gerard Volume 364 - Quantum Fields and Quantum Space Time edited by Gerard 't Hooft, Arthur Jaffe, Gerhard Mack, Pronob K. Mitter, and Raymond Stora Series B: Physics Quantum Fields and Quantum Space Time Edited by Gerard 't Hooft University of Utrecht Utrecht. The Netherlands Arthur Jaffe Harvard University Cambridge. Massachusetts Gerhard Mack University of Hamburg Hamburg. Germany Pronob K. Mitter CNRS - University of Montpellier 2 Montpellier. France and Raymond Stora Laboratory of Particle Physics. Annecy-Ie-Vieux Annecy-Ie-Vieux. France Springer Science+Business Media, LLC Proceedings of a NATO Advanced Study Institute on Quantum Fields and Quantum Space Time, held July 22-August 3, 1996, in Cargese, France NATO-PCO-DATA BASE The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to about 50,000 contributions from international scientists published in all sections of the NATO ASI Series. Access to the NATO-PCO-DATA BASE is possible in two ways: —via online FILE 128 (NATO-PCO-DATA BASE) hosted by ESRIN, Via Galileo Galilei, I-00044 Frascati, Italy —via CD-ROM "NATO Science and Technology Disk" with user-friendly retrieval software in English, French, and German (©WTV GmbH and DATAWARE Technologies, Inc. 1989). The CD-ROM contains the AGARD Aerospace Database. The CD-ROM can be ordered through any member of the Board of Publishers or through NATO-PCO, Overijse, Belgium. Library of Congress Cataloging-in-Publication Data On file ISBN 978-1-4899-1803-1 ISBN 978-1-4899-1801-7 (eBook) DOI 10.1007/978-1-4899-1801-7 © Springer Science+Business Media New York 1997 Originally published by Plenum Press, New York in 1997 Softcover reprint of the hardcover 1st edition 1997 http://www.plenum.com 10987654321 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 PREFACE The 1996 NATO Advanced Study Institute (ASI) followed the international tradi tion of the schools held in Cargese in 1976, 1979, 1983, 1987 and 1991. Impressive progress in quantum field theory had been made since the last school in 1991. Much of it is connected with the interplay of quantum theory and the structure of space time, including canonical gravity, black holes, string theory, application of noncommutative differential geometry, and quantum symmetries. In addition there had recently been important advances in quantum field theory which exploited the electromagnetic duality in certain supersymmetric gauge theories. The school reviewed these developments. Lectures were included to explain how the "monopole equations" of Seiberg and Witten can be exploited. They were presented by E. Rabinovici, and supplemented by = = an extra 2 hours of lectures by A. Bilal. Both the N 1 and N 2 supersymmetric Yang Mills theory and resulting equivalences between field theories with different gauge group were discussed in detail. There are several roads to quantum space time and a unification of quantum theory and gravity. There is increasing evidence that canonical gravity might be a consistent theory after all when treated in. a nonperturbative fashion. H. Nicolai presented a series of introductory lectures. He dealt in detail with an integrable model which is obtained by dimensional reduction in the presence of a symmetry. The lectures by G. Mack started from the hypothesis that a truly fundamental theory should start from an absolute minimum of a priori structure which merely reflects the belief that the human mind thinks about relations between things. It was shown how the fundamental equations of motion of physics, including a discretized version of canonical general relativity in Ashtekar variables can be accommodated. In black holes there is a particularly strong interplay between geometry and quan tum effects. Black holes were a subject of the lectures by G. 't Hooft, and by L. Susskind. String theory was not a main topic at the school, but it was discussed in the context of black hole physics in the lectures of Susskind. Recent evidence for the necessity of going beyond strings as basic constituents was also addressed. The question of how we can extract geometry from the quantum mechanics of extended objects was also discussed from a more general point of view in the lectures of J. Frohlich. He started from a Pauli-electron on a spin manifold and ended with a model of quantum space time which has a quantum symmetry as an essential feature, v and is based on noncommutative differential geometry. A comprehensive introduction to noncommutative differential geometry was pre sented by A. Connes, and quantum field theories with quantum symmetry were exten sively discussed in lectures by Alekseev, Faddeev, Pressley and Zumino. In this way the connection was made with the topics which had been of main interest at the last school. Alekseev showed that quantum symmetries could also appear as local gauge symme tries. Faddeev constructed lattice models of current algebra by formulating them as integrable models. Pressley gave an introduction to quantum groups and quantum affine algebras with application to affine Toda theories. The lessons from quantum field theory are important in the study of other complex systems. This was demonstrated in K. Gawedzki's lectures on turbulence, which applied renormalization group techniques borrowed from field theory. They are also important for mathematics. A. Jaffe discussed elliptic genus and Stora lectured on equivariant cohomology, both in a quantum field theory context. This volume presents the lecture notes. We regret that the contributions of A. Jaffe, E. Rabinovici and L. Susskind were not available in time for inclusion. We wish to express our gratitude to NATO whose generous financial contribution made it possible to organize the school. We thank Professor Elisabeth Dubois-Violette, the director of the Institut d'Etudes Scientifique de Cargese, as well as the Universite de Nice and Universite de Corte for making available to us the facilities of the Institute. Grateful thanks are due to Annie Touchant for much help with the material aspects of the organization, to the staff of the institute, and to Max GrieBl, York Xylander and Volker Schomerus, who helped in Hamburg. for the editors, G. Mack, director of the ASI vi CONTENTS Lectures Non-commutative gauge fields from quantum groups A. Yu. Alekseev and V. Schomerus .......... . 1 Duality in N = 2 SUSY SU(2) Yang-Mills theory: a pedagogical introduction A. Bilal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Noncommutative differential geometry and the structure of space time A. Connes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45 Quantum integrable models on 1 + 1 discrete space time L. Faddeev and A. Volkov . .................. . 73 Supersymmetry and non-commutative geometry J. Frohlich, O. Grandjean, and A. Recknagel 93 Turbulence under a magnifying glass K. Gaw~dzki . . . . . . . . . . . . . . . 123 Quantization of space and time in 3 and in 4 space-time dimensions G. 't Hooft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Pushing Einstein's principles to the extreme G. Mack . ................. , ................ 165 Integrable classical and quantum gravity H. Nicolai, D. Korotkin, and H. Samtleben ........ 203 Quantum affine algebras and integrable quantum systems V. Chari and A. Pressley . ........................... 245 Exercises in equivariant cohomology R. Stora ............... . . ......... 265 Some complex quantum manifolds and their geometry C. S. Chu, P. M. Ho, and B. Zumino .......... . 281 vii Seminars T-duality and the moment map C. KlimiHk and P. Severa. . . . ...... 323 Symmetries of dimensionally reduced string effective action J. Maharana ................................... 331 Non local observables and confinement in BF formulation of Yang-Mills theory F. Fucito, M. Martellini, and M. Zeni . . . . . . . . . . . . . . . . . . . . . 339 Disorder operators, quantum doubles, and Haag duality in 1 + 1 dimensions M. Miiger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 The cohomology and homology of quantum field theory J. E. Roberts. 357 Index. . . . . . . 369 viii NON-COMMUTATIVE GAUGE FIELDS FROM QUANTUM GROUPS Anton Yu. Alekseev 1, Volker Schomerus 2 1 Institute of Theoretical Physics, Uppsala University, Box 803 S-75108, Uppsala, Sweden. e-mail: [email protected] 2 II. Institut fur Theoretische Physik, Universitiit Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany e-mail: [email protected] ABSTRACT We give a non-technical introduction into the theory of non-commutative lattice gauge fields with quantum gauge group. The general construction is illustrated at the example of the Hamiltonian Chern-Simons theory. We also review the counterpart of the Noether's theorem for quantum groups. INTRODUCTION The aim of this contribution is to give a non-technical introduction into the theory of R-matrix algebras and its applications to non-commutative lattice gauge theories developed during last 5 years. We start in Section 1 with the idea of introducing a quantum group as a gauge group in some gauge theory. We treat quantum groups in the physicist-oriented for malism due to Faddeev, Reshetikhin and Takhtajan (FRT) [1]. This idea can be im plemented in the most simple way for lattice gauge theories [2]. The first example of such a construction was given in [3]. We show that gauge invariance with respect to a quantum gauge group leads to non-commutative gauge fields. This was realized in [4]. The resulting theory is closely related to non-ultralocal integrable systems [5]. Lectures presented by A.Yu. Alekseev 1
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