Quantum Gravity Quantumgravityisperhapsthemostimportantopenprobleminfundamentalphysics. It is the problem of merging quantum mechanics and general relativity, the two great conceptual revolutions in the physics of the twentieth century. This book discusses the many aspects of the problem, and presents technical and conceptual advances towards a background-independent quantum theory of gravity, obtained in the last two decades. The first part of the book is an exploration on how tore-thinkbasicphysicsfromscratchinthelightofthegeneral-relativisticconceptual revolution.Thesecondpartisadetailedintroductiontoloop quantum gravityandthe spinfoamformalism.Itprovidesanoverviewofthecurrentstateofthefield,including resultsonareaandvolumespectra,dynamics,extensionofthetheorytomatter,appli- cationstoearlycosmologyandblack-holephysics,andtheperspectivesforcomputing scatteringamplitudes.Thebookiscompletedbyahistoricalappendixwhichoverviews the evolution of the research in quantum gravity, from the 1930s to the present day. Carlo Rovelli was born in Verona, Italy, in 1956 and obtained his Ph.D. in Physics in Padua in 1986. In 1996, he was awarded the Xanthopoulos International Prize, for the development of the loop approach to quantum gravity and for research on the foundation of the physics of space and time. Over the years he has taught and worked in the University of Pittsburgh, Universit´e de la M´editerran´ee, Marseille, and Universit`aLaSapienza,Rome.ProfessorRovelli’smainresearchinterestslieingeneral relativity,gravitationalphysics,andthephilosophyofspaceandtime.Hehashadover 100 publications in international journals in physics and has written contributions for major encyclopedias. He is senior member of the Institut Universitaire de France. CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS General editors: P. V. Landshoff, D. R. Nelson, S. Weinberg S.J.AarsethGravitationalN-BodySimulations J.Ambj/orn,B.DurhuusandT.JonssonQuantumGeometry:AStatisticalFieldTheory Approach A.M.AnileRelativisticFluidsandMagneto-Fluids J.A.deAzc´arrageandJ.M.IzquierdoLieGroups,LieAlgebras,CohomologyandSome ApplicationsinPhysics† O.Babelon,D.BernardandM.TalonIntroductiontoClassicalIntegrableSystems V.BelinkskiandE.VerdaguerGravitationalSolitons J.BernsteinKineticTheoryintheExpandingUniverse G.F.BertschandR.A.BrogliaOscillationsinFiniteQuantumSystems N.D.BirrellandP.C.W.DaviesQuantumFieldsinCurvedspace† M.BurgessClassicalCovariantFields S.CarlipQuantumGravityin2+1Dimensions J.C.CollinsRenormalization† M.CreutzQuarks,GluonsandLattices† P.D.D’EathSupersymmetricQuantumCosmology F.deFeliceandC.J.S.ClarkeRelativityonCurvedManifolds† B.S.DeWittSupermanifolds,2ndedition† P.G.O.FreundIntroductiontoSupersymmetry† J.FuchsAffineLieAlgebrasandQuantumGroups† J.FuchsandC.SchweigertSymmetries,LieAlgebrasandRepresentations:AGraduateCourse forPhysicists† Y.FujiiandK.MaedaTheScalar–TensorTheoryofGravitation A.S.Galperin,E.A.Ivanov,V.I.OrievetskyandE.S.SokatchevHarmonicSuperspace R.GambiniandJ.PullinLoops,Knots,GaugeTheoriesandQuantumGravity† M.Go¨ckelerandT.Schu¨ckerDifferentialGeometry,GaugeTheoriesandGravity† C.G´omez,M.RuizAltabaandG.SierraQuantumGroupsinTwo-dimensionalPhysics M.B.Green,J.H.SchwarzandE.WittenSuperstringTheory,volume1:Introduction† M.B.Green,J.H.SchwarzandE.WittenSuperstringTheory,volume2:LoopAmplitudes, AnomaliesandPhenomenology† V.N.GribovTheTheoryofComplexAngularMomenta S.W.HawkingandG.F.R.EllisTheLarge-ScaleStructureofSpace-Time† F.IachelloandA.ArimaTheInteractingBosonModel F.IachelloandP.vanIsackerTheInteractingBoson–FermionModel C.ItzyksonandJ.-M.DrouffeStatisticalFieldTheory,volume1:FromBrownianMotionto RenormalizationandLatticeGaugeTheory† C.ItzyksonandJ.-M.DrouffeStatisticalFieldTheory,volume2:StrongCoupling,Monte CarloMethods,ConformalFieldTheory,andRandomSystems† C.JohnsonD-Branes J.I.KapustaFinite-TemperatureFieldTheory† V.E.Korepin,A.G.IzerginandN.M.BoguliubovTheQuantumInverseScatteringMethod andCorrelationFunctions† M.LeBellacThermalFieldTheory† Y.MakeenkoMethodsofContemporaryGaugeTheory N.MantonandP.SutcliffeTopologicalSolitons N.H.MarchLiquidMetals:ConceptsandTheory I.M.MontvayandG.Mu¨nsterQuantumFieldsonaLattice† L.O’RaifeartaighGroupStructureofGaugeTheories† T.Ort´ınGravityandStrings A.OzoriodeAlmeidaHamiltonianSystems:ChaosandQuantization† R.PenroseandW.RindlerSpinorsandSpace-Time,volume1:Two-SpinorCalculusand RelativisticFields† R.PenroseandW.RindlerSpinorsandSpace-Time,volume2:SpinorandTwistorMethodsin Space-TimeGeometry† S.PokorskiGaugeFieldTheories,2ndedition J.PolchinskiStringTheory,volume1:AnIntroductiontotheBosonicString J.PolchinskiStringTheory,volume2:SuperstringTheoryandBeyond V.N.PopovFunctionalIntegralsandCollectiveExcitations† R.J.RiversPathIntegralMethodsinQuantumFieldTheory† R.G.RobertsTheStructureoftheProton† C.RovelliQuantumGravity W.C.SaslawGravitationalPhysicsofStellarandGalacticSystems† H.Stephani,D.Kramer,M.A.H.MacCallum,C.HoenselaersandE.HerltExactSolutions ofEinstein’sFieldEquations,2ndedition J.M.StewartAdvancedGeneralRelativity† A.VilenkinandE.P.S.ShellardCosmicStringsandOtherTopologicalDefects† R.S.WardandR.O.WellsJrTwistorGeometryandFieldTheories† J.R.WilsonandG.J.MathewsRelativisticNumericalHydrodynamics †Issuedasapaperback Quantum Gravity CARLO ROVELLI Centre de Physique Th´eorique de Luminy Universit´e de la M´editerran´ee, Marseille cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521715966 © Cambridge University Press 2004 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2004 Reprinted 2005 First paperback edition published with correction 2008 Reprinted 2010 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data isbn 978-0-521-83733-0 Hardback isbn 978-0-521-71596-6 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. 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Contents Foreword, by James Bjorken page xi Preface xiii Preface to the paperback edition xvii Acknowledgements xix Terminology and notation xxi Part 1 Relativistic foundations 1 1 General ideas and heuristic picture 3 1.1 The problem of quantum gravity 3 1.1.1 Unfinished revolution 3 1.1.2 How to search for quantum gravity? 4 1.1.3 The physical meaning of general relativity 9 1.1.4 Background-independent quantum field theory 10 1.2 Loop quantum gravity 13 1.2.1 Why loops? 14 1.2.2 Quantum space: spin networks 17 1.2.3 Dynamics in background-independent QFT 22 1.2.4 Quantum spacetime: spinfoam 26 1.3 Conceptual issues 28 1.3.1 Physics without time 29 v vi Contents 2 General Relativity 33 2.1 Formalism 33 2.1.1 Gravitational field 33 2.1.2 “Matter” 37 2.1.3 Gauge invariance 40 2.1.4 Physical geometry 42 2.1.5 Holonomy and metric 44 2.2 The conceptual path to the theory 48 2.2.1 Einstein’s first problem: a field theory for the newtonian interaction 48 2.2.2 Einstein’s second problem: relativity of motion 52 2.2.3 The key idea 56 2.2.4 Active and passive diffeomorphisms 62 2.2.5 General covariance 65 2.3 Interpretation 71 2.3.1 Observables, predictions and coordinates 71 2.3.2 The disappearance of spacetime 73 2.4 *Complements 75 2.4.1 Mach principles 75 2.4.2 Relationalism versus substantivalism 76 2.4.3 Has general covariance any physical content? 78 2.4.4 Meanings of time 82 2.4.5 Nonrelativistic coordinates 87 2.4.6 Physical coordinates and GPS observables 88 3 Mechanics 98 3.1 Nonrelativistic mechanics 98 3.2 Relativistic mechanics 105 3.2.1 Structure of relativistic systems: partial observables, relativistic states 105 3.2.2 Hamiltonian mechanics 107 3.2.3 Nonrelativistic systems as a special case 114 3.2.4 Mechanics is about relations between observables 118 3.2.5 Space of boundary data G and Hamilton function S 120 3.2.6 Evolution parameters 126 3.2.7 * Complex variables and reality conditions 128 3.3 Field theory 129 3.3.1 Partial observables in field theory 129 3.3.2 * Relativistic hamiltonian mechanics 130 3.3.3 The space of boundary data G and the Hamilton function S 133 Contents vii 3.3.4 Hamilton–Jacobi 137 3.4 * Thermal time hypothesis 140 4 Hamiltonian general relativity 145 4.1 Einstein–Hamilton–Jacobi 145 4.1.1 3d fields:“The length of the electric field is the area” 147 4.1.2 Hamilton function of GR and its physical meaning 151 4.2 Euclidean GR and real connection 153 4.2.1 Euclidean GR 153 4.2.2 Lorentzian GR with a real connection 155 4.2.3 Barbero connection and Immirzi parameter 156 4.3 * Hamiltonian GR 157 4.3.1 Version 1: real SO(3,1) connection 157 4.3.2 Version 2: complex SO(3) connection 157 4.3.3 Configuration space and hamiltonian 158 4.3.4 Derivation of the Hamilton–Jacobi formalism 159 4.3.5 Reality conditions 162 5 Quantum mechanics 164 5.1 Nonrelativistic QM 164 5.1.1 Propagator and spacetime states 166 5.1.2 Kinematical state space K and “projector” P 169 5.1.3 Partial observables and probabilities 172 5.1.4 Boundary state space K and covariant vacuum |0(cid:2) 174 5.1.5 * Evolving constants of motion 176 5.2 Relativistic QM 177 5.2.1 General structure 177 5.2.2 Quantization and classical limit 179 5.2.3 Examples: pendulum and timeless double pendulum 180 5.3 Quantum field theory 184 5.3.1 Functional representation 186 5.3.2 Field propagator between parallel boundary surfaces 190 5.3.3 Arbitrary boundary surfaces 193 5.3.4 What is a particle? 195 5.3.5 Boundary state space K and covariant vacuum |0(cid:2) 197 5.3.6 Lattice scalar product, intertwiners and spin network states 198 5.4 Quantum gravity 200 5.4.1 Transition amplitudes in quantum gravity 200 5.4.2 Much ado about nothing: the vacuum 202 viii Contents 5.5 *Complements 204 5.5.1 Thermal time hypothesis and Tomita flow 204 5.5.2 The “choice” of the physical scalar product 206 5.5.3 Reality conditions and scalar product 208 5.6 *Relational interpretation of quantum theory 209 5.6.1 The observer observed 210 5.6.2 Facts are interactions 215 5.6.3 Information 218 5.6.4 Spacetime relationalism versus quantum relationalism 220 Part II Loop quantum gravity 223 6 Quantum space 225 6.1 Structure of quantum gravity 225 6.2 The kinematical state space K 226 6.2.1 Structures in K 230 6.2.2 Invariances of the scalar product 231 6.2.3 Gauge-invariant and diffeomorphism-invariant states 233 6.3 Internal gauge invariance. The space K 234 0 6.3.1 Spin network states 234 6.3.2 * Details about spin networks 236 6.4 Diffeomorphism invariance. The space K 238 diff 6.4.1 Knots and s-knot states 240 6.4.2 The Hilbert space K is separable 241 diff 6.5 Operators 242 6.5.1 The connection A 242 6.5.2 The conjugate momentum E 243 6.6 Operators on K 246 0 6.6.1 The operator A(S) 246 6.6.2 Quanta of area 249 6.6.3 * n-hand operators and recoupling theory 250 6.6.4 * Degenerate sector 253 6.6.5 Quanta of volume 259 6.7 Quantum geometry 262 6.7.1 The texture of space: weaves 268 7 Dynamics and matter 276 7.1 Hamiltonian operator 277 7.1.1 Finiteness 280 7.1.2 Matrix elements 282