Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory

268 Pages·2014·8.93 MB·English

by  Carlo Rovelli

CovariantLoopQuantumGravity
Quantum gravity is among the most fascinating problems in physics. It modifies our
understanding of time, space, and matter. The recent development of the loop approach
hasallowedusto exploredomainsrangingfromblackholethermodynamicstotheearly
universe.
Thisbookprovidesreaderswithasimpleintroductiontoloopquantumgravity,centred
on its covariantapproach.Itfocuseson the physicaland conceptualaspects of the prob-
lemandincludesthebackgroundmaterialneededtoenterthislivelydomainofresearch,
makingitidealforresearchersandgraduatestudents.
Topics covered include quanta of space, classical and quantum physics without time,
tetrad formalism,Holst action, lattice gaugetheory,Regge calculus, ADM and Ashtekar
variables, Ponzano–Regge and Turaev–Viro amplitudes, kinematics and dynamics of
4d Lorentzian quantum gravity, spectrum of area and volume, coherent states, classi-
cal limit, matter couplings, graviton propagator, spinfoam cosmology and black hole
thermodynamics.
CarloRovelliisProfessorofPhysicsatAix-MarseilleUniversite´,wherehedirectsthegravity
researchgroup.Heisoneofthefoundersofloopquantumgravity.
FrancescaVidotto is NWO Veni Fellow at the Radboud UniversiteitNijmegen and initiated
thespinfoamapproachtocosmology.
“The approachto quantumgravity knownas loop quantumgravity has progressedenor-
mouslyinthelastdecadeandthisbookbyCarloRovelliandFrancescaVidottoadmirably
fillstheneedforanup-to-datetextbookinthisarea.Itwillservewelltobringbeginning
studentsandestablishedresearchersalikeup-to-dateondevelopmentsinthisfastmoving
area. It is the only book presenting key results of the theory, including those related to
black holes,quantumcosmologyandthe derivationof generalrelativity fromthe funda-
mentaltheoryofquantumspacetime.Theauthorsachieveagoodbalanceofbigideasand
principleswiththetechnicaldetails.”
LeeSmolin,PerimeterInstituteforTheoreticalPhysics
“Thisisanexcellentintroductiontospinfoams,anareaofloopquantumgravitythatdraws
ideasalsofromReggecalculus,topologicalfieldtheoryandgroupfieldtheory.Itfillsan
important gap in the literature offering both a pedagogical overview and a platform for
furtherdevelopmentsinaforefrontareaofresearchthatisadvancingrapidly.”
AbhayAshtekar,ThePennsylvaniaStateUniversity
Covariant Loop Quantum Gravity
An Elementary Introduction to Quantum
Gravity and Spinfoam Theory
CARLO ROVELLI
Universite’d’Aix-Marseille
FRANCESCA VIDOTTO
RadboudUniversiteitNijmegen
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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Informationonthistitle:www.cambridge.org/9781107069626
(cid:2)c C.RovelliandF.Vidotto2015
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Firstpublished2015
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Contents
Preface page xi
PartI FOUNDATIONS
1 Spacetimeasaquantumobject 3
1.1 Theproblem 3
1.2 Theendofspaceandtime 6
1.3 Geometryquantized 9
1.3.1 Quantaofareaandvolume 14
1.4 PhysicalconsequencesoftheexistenceofthePlanckscale 16
1.4.1 Discreteness:scalingisfinite 16
1.4.2 Fuzziness:disappearanceofclassicalspaceandtime 18
1.5 Graphs,loops,andquantumFaradaylines 18
1.6 Thelandscape 21
1.7 Complements 21
1.7.1 SU(2)representationsandspinors 21
1.7.2 Paulimatrices 26
1.7.3 Eigenvaluesofthevolume 27
2 Physicswithouttime 30
2.1 Hamiltonfunction 30
2.1.1 Boundaryterms 35
2.2 Transitionamplitude 36
2.2.1 Transitionamplitudeasanintegraloverpaths 37
2.2.2 Generalpropertiesofthetransitionamplitude 39
2.3 Generalcovariantformofmechanics 41
2.3.1 Hamiltonfunctionofageneralcovariantsystem 44
2.3.2 Partialobservables 45
2.3.3 Classicalphysicswithouttime 46
2.4 Quantumphysicswithouttime 47
2.4.1 Observabilityinquantumgravity 49
2.4.2 Boundaryformalism 50
2.4.3 Relationalquanta,relationalspace 52
2.5 Complements 53
2.5.1 Exampleofatimelesssystem 53
2.5.2 SymplecticstructureandHamiltonfunction 55
viii Contents
3 Gravity 58
3.1 Einstein’sformulation 58
3.2 Tetradsandfermions 59
3.2.1 Animportantsign 62
3.2.2 First-orderformulation 63
3.3 HolstactionandBarbero–Immirzicouplingconstant 64
3.3.1 Linearsimplicityconstraint 65
3.3.2 Boundaryterm 67
3.4 Hamiltoniangeneralrelativity 67
3.4.1 ADMvariables 68
3.4.2 Whatdoesthismean?Dynamics 70
3.4.3 Ashtekarconnectionandtriads 72
3.5 Euclideangeneralrelativityinthreespacetimedimensions 74
3.6 Complements 76
3.6.1 Workingwithgeneralcovariantfieldtheory 76
3.6.2 Problems 79
4 Classicaldiscretization 80
4.1 LatticeQCD 80
4.1.1 Hamiltonianlatticetheory 82
4.2 Discretizationofcovariantsystems 83
4.3 Reggecalculus 85
4.4 Discretizationofgeneralrelativityonatwo-complex 89
4.5 Complements 95
4.5.1 Holonomy 95
4.5.2 Problems 96
PartII THREE-DIMENSIONALTHEORY
5 Three-dimensionaleuclideantheory 99
5.1 Quantizationstrategy 99
5.2 Quantumkinematics:Hilbertspace 100
5.2.1 Lengthquantization 101
5.2.2 Spinnetworks 102
5.3 Quantumdynamics:transitionamplitudes 106
5.3.1 Propertiesoftheamplitude 109
5.3.2 Ponzano–Reggemodel 110
5.4 Complements 113
5.4.1 Elementaryharmonicanalysis 113
5.4.2 Alternativeformofthetransitionamplitude 114
5.4.3 Poissonbrackets 116
5.4.4 Perimeteroftheuniverse 117
ix Contents
6 Bubblesandthecosmologicalconstant 118
6.1 Vertexamplitudeasgauge-invariantidentity 118
6.2 Bubblesandspikes 120
6.3 Turaev–Viroamplitude 123
6.3.1 Cosmologicalconstant 125
6.4 Complements 127
6.4.1 AfewnotesonSU(2) 127
q
6.4.2 Problem 128
PartIII THEREALWORLD
7 Therealworld:four-dimensionallorentziantheory 131
7.1 Classicaldiscretization 131
7.2 Quantumstatesofgravity 134
7.2.1 Yγ map 135
7.2.2 Spinnetworksinthephysicaltheory 137
7.2.3 Quantaofspace 140
7.3 Transitionamplitudes 141
7.3.1 Continuumlimit 143
7.3.2 RelationwithQEDandQCD 145
7.4 Fulltheory 146
7.4.1 Faceamplitude,wedgeamplitude,andthekernelP 147
7.4.2 CosmologicalconstantandIRfiniteness 149
7.4.3 Variants 149
7.5 Complements 151
7.5.1 Summaryofthetheory 151
7.5.2 Computingwithspinnetworks 152
7.5.3 Spectrumofthevolume 155
7.5.4 UnitaryrepresentationoftheLorentzgroupandtheYγ map 159
8 Classicallimit 162
8.1 Intrinsiccoherentstates 162
8.1.1 TetrahedrongeometryandSU(2)coherentstates 163
8.1.2 Livine–Spezialecoherentintertwiners 167
8.1.3 Thinandthickwedgesandtime-orientedtetrahedra 168
8.2 Spinorsandtheirmagic 169
8.2.1 Spinors,vectors,andbivectors 171
8.2.2 Coherentstatesandspinors 172
8.2.3 RepresentationsofSU(2)andSL(2,C)onfunctionsofspinors
andYγ map 173
8.3 Classicallimitofthevertexamplitude 175
8.3.1 Transitionamplitudeintermsofcoherentstates 175
8.3.2 Classicallimitversuscontinuumlimit 180
8.4 Extrinsiccoherentstates 183