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Geometric Formulation of Classical and Quantum Mechanics Geometric Formulation of Classical and Quantum Mechanics 7816 tp.indd 1 8/19/10 2:57 PM This page intentionally left blank Geometric Formulation of Classical and Quantum Mechanics Giovanni Giachetta University of Camerino, Italy Luigi Mangiarotti University of Camerino, Italy Gennadi Sardanashvily Moscow State University, Russia World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 7816 tp.indd 2 8/19/10 2:57 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 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. GEOMETRIC FORMULATION OF CLASSICAL AND QUANTUM MECHANICS Copyright © 2011 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-13 978-981-4313-72-8 ISBN-10 981-4313-72-6 Printed in Singapore. RokTing - Geometric Formulaiton of Classical.pmd1 8/13/2010, 10:25 AM July 29, 2010 11:11 World Scientific Book - 9in x 6in book10 Preface Geometry of symplectic and Poisson manifolds is well known to provide the adequate mathematical formulation of autonomous Hamiltonian mechanics. The literature on this subject is extensive. This book presents the advanced geometric formulation of classical and quantum non-relativistic mechanics in a general setting of time-dependent coordinate and reference frame transformations. This formulation of me- chanics as like as that of classical field theory lies in the framework of general theory of dynamic systems, Lagrangian and Hamiltonian formalism on fibre bundles. Non-autonomous dynamic systems, Newtonian systems, Lagrangian and Hamiltonian non-relativistic mechanics, relativistic mechanics, quantum non-autonomous mechanics are considered. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over the time axis R. Quantum non- relativistic mechanics is phrased in the geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization. Relativistic mechan- ics is adequately formulated as particular classical string theory of one- dimensional submanifolds. The concept of a connection is the central link throughout the book. Connections on a configuration space of non-relativistic mechanics describe reference frames. Holonomic connections on a velocity space define non- relativistic dynamic equations. Hamiltonian connections in Hamiltonian non-relativistic mechanics define the Hamilton equations. Evolution of quantum systems is described in terms of algebraic connections. A con- nection on a prequantization bundle is the main ingredient in geometric quantization. v July 29, 2010 11:11 World Scientific Book - 9in x 6in book10 vi Preface The book provides a detailed exposition of theory of partially integrable and superintegrable systems and their quantization, classical and quantum non-autonomous constraint systems, Lagrangian and Hamiltonian theory of Jacobi fields, classical and quantum mechanics with time-dependent pa- rameters, the technique of non-adiabatic holonomy operators, formalism of instantwise quantization and quantization with respect to different refer- ence frames. Our book addresses to a wide audience of theoreticians and mathemati- cians of undergraduate, post-graduate and researcher levels. It aims to be a guide to advanced geometric methods in classical and quantum mechanics. For the convenience of the reader, a few relevant mathematical topics are compiled in Appendixes, thus making our exposition self-contained. July 29, 2010 11:11 World Scientific Book - 9in x 6in book10 Contents Preface v Introduction 1 1. Dynamic equations 7 1.1 Preliminary. Fibre bundles over R ............. 7 1.2 Autonomous dynamic equations . 13 1.3 Dynamic equations . 16 1.4 Dynamic connections . 18 1.5 Non-relativistic geodesic equations . 22 1.6 Reference frames . 27 1.7 Free motion equations . 30 1.8 Relative acceleration . 33 1.9 Newtonian systems . 36 1.10 Integrals of motion . 38 2. Lagrangian mechanics 43 2.1 Lagrangian formalism on Q R . 43 ! 2.2 Cartan and Hamilton{De Donder equations . 49 2.3 Quadratic Lagrangians . 51 2.4 Lagrangian and Newtonian systems . 56 2.5 Lagrangian conservation laws . 58 2.5.1 Generalized vector fields . 58 2.5.2 First Noether theorem . 60 2.5.3 Noether conservation laws . 64 2.5.4 Energy conservation laws . 66 2.6 Gauge symmetries . 68 vii July 29, 2010 11:11 World Scientific Book - 9in x 6in book10 viii Contents 3. Hamiltonian mechanics 73 3.1 Geometry of Poisson manifolds . 73 3.1.1 Symplectic manifolds . 74 3.1.2 Presymplectic manifolds . 76 3.1.3 Poisson manifolds . 77 3.1.4 Lichnerowicz{Poisson cohomology . 82 3.1.5 Symplectic foliations . 83 3.1.6 Group action on Poisson manifolds . 87 3.2 Autonomous Hamiltonian systems . 89 3.2.1 Poisson Hamiltonian systems . 90 3.2.2 Symplectic Hamiltonian systems . 91 3.2.3 Presymplectic Hamiltonian systems . 91 3.3 Hamiltonian formalism on Q R . 93 ! 3.4 Homogeneous Hamiltonian formalism . 98 3.5 Lagrangian form of Hamiltonian formalism . 99 3.6 Associated Lagrangian and Hamiltonian systems . 100 3.7 Quadratic Lagrangian and Hamiltonian systems . 104 3.8 Hamiltonian conservation laws . 105 3.9 Time-reparametrized mechanics . 110 4. Algebraic quantization 113 4.1 GNS construction . 113 4.1.1 Involutive algebras . 113 4.1.2 Hilbert spaces . 115 4.1.3 Operators in Hilbert spaces . 118 4.1.4 Representations of involutive algebras . 119 4.1.5 GNS representation . 121 4.1.6 Unbounded operators . 124 4.2 Automorphisms of quantum systems . 126 4.3 Banach and Hilbert manifolds . 131 4.3.1 Real Banach spaces . 131 4.3.2 Banach manifolds . 132 4.3.3 Banach vector bundles . 134 4.3.4 Hilbert manifolds . 136 4.3.5 Projective Hilbert space . 143 4.4 Hilbert and C∗-algebra bundles . 144 4.5 Connections on Hilbert and C∗-algebra bundles . 147 4.6 Instantwise quantization . 151 July 29, 2010 11:11 World Scientific Book - 9in x 6in book10 Contents ix 5. Geometric quantization 155 5.1 Geometric quantization of symplectic manifolds . 156 5.2 Geometric quantization of a cotangent bundle . 160 5.3 Leafwise geometric quantization . 162 5.3.1 Prequantization . 163 5.3.2 Polarization . 169 5.3.3 Quantization . 170 5.4 Quantization of non-relativistic mechanics . 174 5.4.1 Prequantization of T ∗Q and V ∗Q . 176 5.4.2 Quantization of T ∗Q and V ∗Q . 178 5.4.3 Instantwise quantization of V ∗Q . 180 5.4.4 Quantization of the evolution equation . 183 5.5 Quantization with respect to different reference frames . 185 6. Constraint Hamiltonian systems 189 6.1 Autonomous Hamiltonian systems with constraints . 189 6.2 Dirac constraints . 193 6.3 Time-dependent constraints . 196 6.4 Lagrangian constraints . 199 6.5 Geometric quantization of constraint systems . 201 7. Integrable Hamiltonian systems 205 7.1 Partially integrable systems with non-compact invariant submanifolds . 206 7.1.1 Partially integrable systems on a Poisson manifold 206 7.1.2 Bi-Hamiltonian partially integrable systems . 210 7.1.3 Partial action-angle coordinates . 214 7.1.4 Partially integrable system on a symplectic manifold . 217 7.1.5 Global partially integrable systems . 221 7.2 KAM theorem for partially integrable systems . 225 7.3 Superintegrable systems with non-compact invariant submanifolds . 228 7.4 Globally superintegrable systems . 232 7.5 Superintegrable Hamiltonian systems . 235 7.6 Example. Global Kepler system . 237 7.7 Non-autonomous integrable systems . 244 7.8 Quantization of superintegrable systems . 250 July 29, 2010 11:11 World Scientific Book - 9in x 6in book10 x Contents 8. Jacobi fields 257 8.1 The vertical extension of Lagrangian mechanics . 257 8.2 The vertical extension of Hamiltonian mechanics . 259 8.3 Jacobi fields of completely integrable systems . 262 9. Mechanics with time-dependent parameters 269 9.1 Lagrangian mechanics with parameters . 270 9.2 Hamiltonian mechanics with parameters . 272 9.3 Quantum mechanics with classical parameters . 275 9.4 Berry geometric factor . 282 9.5 Non-adiabatic holonomy operator . 284 10. Relativistic mechanics 293 10.1 Jets of submanifolds . 293 10.2 Lagrangian relativistic mechanics . 295 10.3 Relativistic geodesic equations . 304 10.4 Hamiltonian relativistic mechanics . 311 10.5 Geometric quantization of relativistic mechanics . 312 11. Appendices 317 11.1 Commutative algebra . 317 11.2 Geometry of fibre bundles . 322 11.2.1 Fibred manifolds . 323 11.2.2 Fibre bundles . 325 11.2.3 Vector bundles . 328 11.2.4 Affine bundles . 331 11.2.5 Vector fields . 333 11.2.6 Multivector fields . 335 11.2.7 Differential forms . 336 11.2.8 Distributions and foliations . 342 11.2.9 Differential geometry of Lie groups . 344 11.3 Jet manifolds . 346 11.3.1 First order jet manifolds . 346 11.3.2 Second order jet manifolds . 347 11.3.3 Higher order jet manifolds . 349 11.3.4 Differential operators and differential equations . 350 11.4 Connections on fibre bundles . 351 11.4.1 Connections . 352 July 29, 2010 11:11 World Scientific Book - 9in x 6in book10 Contents xi 11.4.2 Flat connections . 354 11.4.3 Linear connections . 355 11.4.4 Composite connections . 357 11.5 Differential operators and connections on modules . 359 11.6 Differential calculus over a commutative ring .
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