GRAVITATION AND GAUGE SYMMETRIES Series in High Energy Physics, Cosmology and Gravitation Other books in the series Electron–Positron Physics at the Z M G Green, S L Lloyd, P N Ratoff and D R Ward Non-accelerator Particle Physics Paperback edition H V Klapdor-Kleingrothaus and A Staudt Ideas and Methods of Supersymmetry and Supergravity or A Walk Through Superspace Revised edition I L Buchbinder and S M Kuzenko Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics F Weber Classical and Quantum Black Holes Edited by P Fr´e, V Gorini, G Magli and U Moschella Particle Astrophysics Revised paperback edition H V Klapdor-Kleingrothaus and K Zuber The World in Eleven Dimensions Supergravity, Supermembranes and M-Theory Edited by M J Duff Gravitational Waves Edited by I Ciufolini, V Gorini, U Moschella and P Fr´e SERIES IN HIGH ENERGY PHYSICS, COSMOLOGY AND GRAVITATION Series Editors B Foster, L Grishchuk, E W Kolb, M A H MacCallum, D H Perkins and B F Schutz GRAVITATION AND GAUGE SYMMETRIES Milutin Blagojevi ´ c Institute of Physics, Belgrade, Yugoslavia INSTITUTE OF PHYSICS PUBLISHING BRISTOL AND PHILADELPHIA c IOP Publishing Ltd 2002 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with the Committee of Vice-Chancellors and Principals. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0767 6 Library of Congress Cataloging-in-Publication Data are available Cover image: A burst of outgoing gravitational radiation originating from a grazing merger of two black holes. (Courtesy Konrad-Zuse- Zentrum fuer Informationstechnik Berlin (ZIB) and Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut, AEI). Simulation: AEI Numerical Relativity Group in cooperation with WashU and NCSA (Ed Seidel, Wai-Mo Suen et al). Visualization: Werner Benger.) Commissioning Editor: James Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing Executive: Laura Serratrice Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in L A T E X2 by Text 2 Text, Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall Contents Preface xi 1 Space, time and gravitation 1 1.1 Relativity of space and time 1 Historical introduction 1 Relativity of motion and the speed of light 3 From space and time to spacetime 6 1.2 Gravitation and geometry 9 The principle of equivalence 9 Physics and geometry 11 Relativity, covariance and Mach’s ideas 14 Perspectives of further developments 18 2 Spacetime symmetries 20 2.1 Poincar´e symmetry 21 Poincar´e transformations 21 Lie algebra and its representations 22 Invariance of the action and conservation laws 24 2.2 Conformal symmetry 27 Conformal transformations and Weyl rescaling 27 Conformal algebra and finite transformations 29 Conformal symmetry and conserved currents 32 Conformal transformations in D = 235 Spontaneously broken scale invariance 37 Exercises 39 3 Poincar ´ e gauge theory 42 3.1 Poincar´e gauge invariance 43 Localization of Poincar´e symmetry 43 Conservation laws and field equations 47 On the equivalence of different approaches 50 3.2 Geometric interpretation 51 Riemann–Cartan space U 4 51 Geometric and gauge structure of PGT 61 vi Contents The principle of equivalence in PGT 62 3.3 Gravitational dynamics 65 Einstein–Cartan theory 65 Teleparallel theory 68 General remarks 72 Exercises 75 4 Weyl gauge theory 78 4.1 Weyl gauge invariance 79 Localization of Weyl symmetry 79 Conservation laws and field equations 83 Conformal versus Weyl gauge symmetry 85 4.2 Weyl–Cartan geometry 86 Conformal transformations in Riemann space 86 Weyl space W 4 89 Weyl–Cartan space Y 4 92 4.3 Dynamics 94 Weyl’s theory of gravity and electrodynamics 95 Scalar fields and the improved energy–momentum tensor 96 Goldstone bosons as compensators 100 General remarks 102 Exercises 105 5 Hamiltonian dynamics 107 5.1 Constrained Hamiltonian dynamics 108 Introduction to Dirac’s theory 108 Generators of gauge symmetries 119 Electrodynamics 123 5.2 The gravitational Hamiltonian 125 Covariance and Hamiltonian dynamics 125 Primary constraints 128 The (3 + 1) decomposition of spacetime 129 Construction of the Hamiltonian 131 Consistency of the theory and gauge conditions 134 5.3 Specific models 136 Einstein–Cartan theory 136 The teleparallel theory 141 Exercises 148 6 Symmetries and conservation laws 152 6.1 Gauge symmetries 153 Constraint algebra 153 Gauge generators 154 6.2 Conservation laws—EC theory 157 Asymptotic structure of spacetime 158 Contents vii Improving the Poincar´e generators 160 Asymptotic symmetries and conservation laws 163 6.3 Conservation laws—the teleparallel theory 168 A simple model 168 The Poincar´e gauge generators 170 Asymptotic conditions 171 The improved Poincar´e generators 173 Conserved charges 176 6.4 Chern–Simons gauge theory in D = 3 179 Chern–Simons action 180 Canonical analysis 183 Symmetries at the boundary 187 Exercises 191 7 Gravity in flat spacetime 195 7.1 Theories of long range forces 196 Scalar field 196 Vector field 197 The symmetric tensor field 202 The sign of the static interaction 206 7.2 Attempts to build a realistic theory 207 Scalar gravitational field 207 Symmetric tensor gravitational field 210 Can the graviton have a mass? 214 The consistency problem 219 Exercises 220 8 Nonlinear effects in gravity 222 8.1 Nonlinear effects in Yang–Mills theory 222 Non-Abelian Yang–Mills theory 222 Scalar electrodynamics 226 8.2 Scalar theory of gravity 228 8.3 Tensor theory of gravity 231 The iterative procedure 231 Formulation of a complete theory 232 8.4 The first order formalism 237 Yang–Mills theory 237 Einstein’s theory 239 Exercises 243 9 Supersymmetry and supergravity 245 9.1 Supersymmetry 246 Fermi–Bose symmetry 246 Supersymmetric extension of the Poincar´e algebra 251 The free Wess–Zumino model 255 viii Contents Supersymmetric electrodynamics 257 9.2 Representations of supersymmetry 259 Invariants of the super-Poincar´e algebra 259 Massless states 261 Massive states 264 Supermultiplets of fields 267 Tensor calculus and invariants 269 The interacting Wess–Zumino model 271 9.3 Supergravity 273 The Rarita–Schwinger field 273 Linearized theory 276 Complete supergravity 278 Algebra of local supersymmetries 281 Auxiliary fields 283 General remarks 286 Exercises 290 10 Kaluza–Klein theory 293 10.1 Basic ideas 294 Gravity in five dimensions 294 Ground state and stability 298 10.2 Five-dimensional KK theory 303 Five-dimensional gravity and effective theory 303 Choosing dynamical variables 307 The massless sector of the effective theory 310 Dynamics of matter and the fifth dimension 312 Symmetries and the particle spectrum 315 10.3 Higher-dimensional KK theory 320 General structure of higher-dimensional gravity 320 The massless sector of the effective theory 325 Spontaneous compactification 329 General remarks 331 Exercises 335 11 String theory 338 11.1 Classical bosonic strings 339 The relativistic point particle 339 Action principle for the string 341 Hamiltonian formalism and symmetries 344 11.2 Oscillator formalism 346 Open string 347 Closed strings 349 Classical Virasoro algebra 350 11.3 First quantization 353 Quantum mechanics of the string 353 Contents ix Quantum Virasoro algebra 355 Fock space of states 356 11.4 Covariant field theory 358 Gauge symmetries 358 The action for the free string field 361 Electrodynamics 362 Gravity 364 11.5 General remarks 366 Exercises 369 Appendices 373 A Local internal symmetries 373 B Differentiable manifolds 379 C De Sitter gauge theory 390 D The scalar–tensor theory 396 E Ashtekar’s formulation of GR 402 F Constraint algebra and gauge symmetries 410 G Covariance, spin and interaction of massless particles 415 H Lorentz group and spinors 421 I Poincar ´ e group and massless particles 433 J Dirac matrices and spinors 443 K Symmetry groups and manifolds 451 L Chern–Simons gravity in three dimensions 473 M Fourier expansion 487 Bibliography 489 Notations and conventions 513 Index 517 [...]... J and M (Dirac spinors, Fourier expansion) are indispensable for the exposition in chapters 9 and 11 Appendices A, H and I (internal local symmetries, Lorentz and Poincar´ e group) are very useful for the exposition in chapters 3 (A) and 9 (H, I) Appendices C, D, E, F, G and L (de Sitter gauge theory, scalar–tensor theory, Ashtekar’s formulation of general relativity, constraint algebra and gauge symmetries, ... exposition of their localization; the structure of the corresponding gauge theories is explored in chapters 3 and 4 Then, in chapters 5 and 6, we present the basic features of the Hamiltonian dynamics of Poincar´ gauge theory, discuss the relation between gauge symmetries and e conservation laws and introduce the concept of gravitational energy and other conserved charges The second part of the book treats... gravitation, which are important for our understanding of gravitation as a gauge theory These aspects include: the development of the principle of relativity from classical mechanics and electrodynamics, and its influence on the structure of space and time; and the formulation of the principle of equivalence, and the introduction of gravitation and the corresponding geometry of curved spacetime The purpose... importance of these spacetime symmetries in particle physics, in this chapter we shall give a review of those properties of Poincar´ e and conformal symmetries that are of interest for their localization and the construction of related gravitational theories Similar ideas can be, and have been, applied to other symmetry groups Understanding gravity as a theory based on local spacetime symmetries represents... my contribution to that text, and include some of his formulations (in sections 5.2 and 8.4) and exercises Finally, it is my pleasure to thank Friedrich Hehl and Yuri Obukhov for their sincere support during this project The exposition of Poincar´ gauge theory and its Hamiltonian structure is e essentially based on the research done in collaboration with Ignjat Nikoli´ and c Milovan Vasili´ The present... April 2001 Chapter 1 Space, time and gravitation Theories of special and general relativity represent a great revolution in our understanding of the structure of space and time, as well as of their role in the formulation of physical laws While special relativity (SR) describes the influence of physical reality on the general properties of and the relation between space and time, the geometry of spacetime... of our understanding of their dynamical structure It has its origins in Maxwell’s unification of electricity and magnetism in the second half of the nineteenth century, matured in Weyl’s and Kaluza’s attempts to unify gravity and electromagnetism at the beginning of the last century, and achieved its full potential in the 1970s, in the process of unifying the weak and electromagnetic and also, to some... gravity and the principle of gauge invariance, which may lead to a consistent unified theory The first part of this book, chapters 1–6, gives a systematic account of the structure of gravity as a theory based on spacetime gauge symmetries Some basic properties of space, time and gravity are reviewed in the first, introductory chapter Chapter 2 deals with elements of the global Poincar´ and conformal symmetries, ... covariance, spin and interaction of massless particles, and Chern–Simons gravity) are supplements to the main exposition, and may be studied according to the reader’s choice The material in appendices B and K (differentiable manifolds, symmetry groups and manifolds) is not necessary for the main exposition It gives a deeper mathematical foundation for the geometric considerations in chapters 3, 4 and 10 The... of gravitational interaction Perhaps the biggest barrier to a full understanding of these remarkable ideas lies in the fact that we are not always ready to suspect the properties of space and time that are built into our consciousness by everyday experience In this chapter we present an overview of some aspects of the structure of space, time and gravitation, which are important for our understanding . Conformal symmetry 27 Conformal transformations and Weyl rescaling 27 Conformal algebra and finite transformations 29 Conformal symmetry and conserved currents 32 Conformal transformations in D. 108 Generators of gauge symmetries 119 Electrodynamics 123 5.2 The gravitational Hamiltonian 125 Covariance and Hamiltonian dynamics 125 Primary constraints 128 The (3 + 1) decomposition of spacetime 129 Construction. ENERGY PHYSICS, COSMOLOGY AND GRAVITATION Series Editors B Foster, L Grishchuk, E W Kolb, M A H MacCallum, D H Perkins and B F Schutz GRAVITATION AND GAUGE SYMMETRIES Milutin Blagojevi ´ c Institute