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Connections in Classical and Quantum Field Theory HOE • iT.LH inP Connections in Classical and Quantum Field Theory .World Scientific Connections in Classical and Quantum Field Theory This page is intentionally left blank L. Mangiarotti University of Camerino, Italy G. Sardanashvily Moscow State University, Russia Connections in Classical and Quantum Field Theory World Scientific Singapore •NewJersey •London • Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. CONNECTIONS IN CLASSICAL AND QUANTUM FIELD THEORY Copyright C 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-02-2013-8 Printed in Singapore by Uto-Print Preface The present book is based on the graduate and post graduate courses of lectures given at the Department of Theoretical Physics of Moscow State University and the Department of Mathematics and Physics of Camerino University. It is addressed to a wide audience of physicists and mathematicians, and aims at showing in a unified way the role that the concept of a connection plays both in classical and quantum field theory. To our knowledge, this is the first attempt to present connections as a main geometrical object which underlies many relevant physical ideas. The concept of a connection in quantum field theory. is rather new. It is phrased in algebraic terms, in comparison with the purely geometric one used in classical field theory. This concept is based on modern development of quantum mechanics, SUSY models, BRST formalism, and non-commutative field theory. In our opinion, connections provide a new link between classical and quantum physics. In classical field theory, we follow the general notion of a connection on fibre bundles, formu- lated in terms of jet manifolds. We emphasize the role of connections as the main ingredient in dynamic equation theory, and Lagrangian and Hamiltonian formalisms. As is well-known, connections on principal bundles have a role of gauge potentials of fundamental interactions while, in gravitation theory, connections characterize the space-time geometry. Classical mechanics is also presented as a particular field theory, involving the concept of a connection in many aspects. The reader will find connections in quantum mechanics, too. In fact, we collect together the basic meth- ods concerning different types of connections in quantum field theory. These are superconnections, connections in BRST formalism, topological field theory, theory of anomalies, and non-commutative geometry. Mathematics is not the primary scope of our book, but provides the powerful methods of studying contemporary field models. v This page is intentionally left blank Contents Preface ..................................... v Introduction . 1 1 Geometric interlude 3 1.1 Fibre bundles . 3 1.2 Differential forms and multivector fields . 12 1.3 Jet manifolds . 24 2 Connections 35 2.1 Connections as tangent-valued forms . 35 2.2 Connections as jet bundle sections . 38 2.3 Curvature and torsion . 40 2.4 Linear connections . 42 2.5 Affine connections . 45 2.6 Flat connections . 47 2.7 Composite connections . 48 3 Connections in Lagrangian field theory 55 3.1 Connections and dynamic equations . 56 3.2 The first variational formula . 58 3.3 Quadratic degenerate Lagrangians . 62 3.4 Connections and Lagrangian conservation laws . 65 4 Connections in Hamiltonian field theory 71 4.1 Hamiltonian connections and Hamiltonian forms . 72 4.2 Lagrangian and Hamiltonian degenerate systems . 77 4.3 Quadratic and affine degenerate systems . 80 vii viii CONTENTS 4.4 Connections and Hamiltonian conservation laws . 83 4.5 The vertical extension of Hamiltonian formalism . 86 5 Connections in classical mechanics 91 5.1 Fibre bundles over R . 92 5.2 Connections in conservative mechanics . 96 5.3 Dynamic connections in time-dependent mechanics . 98 5.4 Non-relativistic geodesic equations . 103 5.5 Connections and reference frames . 107 5.6 The free motion equation . 111 5.7 The relative acceleration . 113 5.8 Lagrangian and Newtonian systems . 116 5.9 Non-relativistic Jacobi fields . 124 5.10 Hamiltonian time-dependent mechanics . 127 5.11 Connections and energy conservation laws . 136 5.12 Systems with time-dependent parameters . 142 6 Gauge theory of principal connections 149 6.1 Principal connections . 149 6.2 The canonical principal connection . 159 6.3 Gauge conservation laws . 165 6.4 Hamiltonian gauge theory . 175 6.5 Geometry of symmetry breaking . 182 6.6 Effects of flat principal connections . 188 6.7 Characteristic classes . 195 6.8 Appendix. Homotopy, homology and cohomology . 206 6.9 Appendix. Cech cohomology . .. 212 7 Space-time connections 215 7.1 Linear world connections . 215 7.2 Lorentz connections . 220 7.3 Relativistic mechanics . 226 7.4 Metric-affine gravitation theory . 235 7.5 Spin connections . 242 7.6 Affine world connections . 254 CONTENTS ix 8 Algebraic connections 257 8.1 Jets of modules . 257 8.2 Connections on modules . 269 8.3 Connections on sheaves . 274 9 Superconnections 285 9.1 Graded tensor calculus . 285 9.2 Connections on graded manifolds . 289 9.3 Connections on supervector bundles . 300 9.4 Principal superconnections . 316 9.5 Graded principal bundles . 324 9.6 SUSY-extended field theory . 327 9.7 The Ne'eman-Quillen superconnection . 334 9.8 Appendix. K-Theory . 340 10 Connections in quantum mechanics 343 10.1 Kahler manifolds modelled on Hilbert spaces . 343 10.2 Geometric quantization . .. 350 10.3 Deformation quantization . 360 10.4 Quantum time-dependent evolution . 370 10.5 Berry connections . .. .. 376 11 Connections in BRST formalism 383 11.1 The canonical connection on infinite order jets . 383 11.2 The variational bicomplex . 398 11.3 Jets of ghosts and antifields . .. 404 11.4 The BRST connection . 413 12 Topological field theories 419 12.1 The space of principle connections . 419 12.2 Connections on the space of connections . 424 12.3 Donaldson invariants . 430 13 Anomalies 435 13.1 Gauge anomalies . 435 13.2 Global anomalies . .. ... 440 x CONTENTS 13.3 BRST anomalies . 444 14 Connections in non-commutative geometry 447 14.1 Non-commutative algebraic calculus . 447 14.2 Non-commutative differential calculus . 451 14.3 Universal connections . 456 14.4 The Dubois-Violette connection . 458 14.5 Matrix geometry . 461 14.6 Connes' differential calculus . 463 Bibliography 467 Index 493 Introduction The main reasons why connections play a prominent role in contemporary field theory lie in the fact that they enable us to deal with invariantly defined objects. Gauge theory shows clearly that this is a basic physical principle. In gauge theory, connections on principal bundles are well known to provide the mathematical description of gauge potentials of the fundamental interactions. Furthermore, since the characteristic classes of principal bundles are expressed in terms of gauge strength, one also meets the topological phenomena in classical and quantum gauge models, e.g., anomalies. All gauge gravitation models belong to the class of metric-affine theories where a pseudo-Riemannian metric and a connection on a world manifold are considered on the same footing as dynamic variables. Though gauge theory has made great progress in describing fundamental inter- actions, it is a particular case of field models on fibre bundles. Differential geometry of fibre bundles and formalism of jet manifolds give the adequate mathematical for- mulation of classical field theory, where fields are represented by sections of fibre bundles. This formulation is also applied to classical mechanics seen as a particular field theory on fibre bundles over a time axis. In summary, connections are the main ingredient in describing dynamic systems on fibre bundles, and in Lagrangian and Hamiltonian machineries. Jet manifolds provide the appropriate language for theory of (non-linear) dif- ferential operators and equations, the calculus of variations, and Lagrangian and Hamiltonian formalisms. For this reason, we follow the general definition of connec- tions as sections of jet bundles. This enables us to deal with non-linear connections and to include connections in a natural way in describing field dynamics. In quantum field theory requiring the theory of Hilbert and other linear spaces, one needs another concept of a connection, phrased in algebraic terms as a con- nection on modules and sheaves. This notion is equivalent to the above-mentioned 1 2 INTRODUCTION geometric one in the case of structure modules of smooth vector bundles. Extended to the case of modules over graded and non-commutative rings, it provides the geometric language of supersymmetry and non-commutative geometry. Our book is not a book on differential geometry, but it collects together the basic mathematical facts about various types of connections in Lagrangian and Hamilto- nian formalisms, gauge and gravitation theory, classical and quantum mechanics, topological field theory and anomalies, BRST formalism, and non-commutative ge- ometry. Additionally, it provides the detailed exposition of relevant theoretical methods both in classical and quantum field theory. We have tried throughout to give the necessary mathematical background, thus making the book self-contained. We hope that our book will attract new interest, from theoretical and mathematical physicists, in modern geometrical methods in field theory. Chapter 1 Geometric interlude This Chapter summarizes the basic notions on fibre bundles and jet manifolds which find an application in the sequel [123, 179, 265, 274].
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