Quantum Gravity and Quantum Cosmology

402 Pages·2012·3.728 MB·English

by  Gianluca Calcagni, Lefteris Papantonopoulos, George Siopsis, Nikos Tsamis

Lecture Notes in Physics
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Gianluca Calcagni (cid:2) Lefteris Papantonopoulos (cid:2)
George Siopsis (cid:2) Nikos Tsamis
Editors
Quantum Gravity
and Quantum
Cosmology
Editors
GianlucaCalcagni GeorgeSiopsis
InstitutodeEstructuradelaMateria DepartmentofPhysicsandAstronomy
CSIC TheUniversityofTennessee
Madrid,Spain Knoxville,TN,USA
NikosTsamis
LefterisPapantonopoulos CreteCenterforTheoreticalPhysics
DepartmentofPhysics DepartmentofPhysics
NationalTechnicalUniversityofAthens UniversityofCrete
Athens,Greece Heraklion,Greece
ISSN0075-8450 ISSN1616-6361(electronic)
LectureNotesinPhysics
ISBN978-3-642-33035-3 ISBN978-3-642-33036-0(eBook)
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Preface
ThisbookisaneditedversionofthereviewtalksgivenintheSixthAegeanSchool
on Quantum Gravity and Quantum Cosmology, held in Chora on Naxos Island,
Greece, from 12th to 17th of September 2011. The aim is to present an advanced
multiauthoredtextbookmeetingtheneedsofbothpostgraduatestudentsandyoung
researchers,inthefieldsofgravity,relativity,cosmologyandquantumfieldtheory.
Quantum gravity in a broad sense is a fast-growing subject in physics and its
studyisexpectedtogiveanswersontheshort-distancebehaviourofthegravitational
interaction.Probingthehigh-energyandhigh-curvatureregimesofgravitatingsys-
tems can shed some light on the ways to achieve an ultraviolet complete quantum
theory of gravity, giving us information about fundamental problems of classical
gravitysuchas theinitialbig-bangsingularity,thecosmologicalconstantproblem
andthephysicsatandbeyondthePlanckscale.Ontheotherhand,itcangivevital
informationontheearly-timeinflationaryevolutionofourUniverse.
The selected contributions to this volume discuss quantum gravity theories in
connection with cosmological models and observations, and explore what type of
signaturemodernandmathematicallyrigorousframeworkscanbedetectedbyex-
periments.
In the first part of the book, the idea of quantum gravity is introduced and ap-
proachedfromdifferentangles.InthearticlebyKellyStelle,anoverviewisgiven
ofthewayinwhichtheunificationprogramofparticlephysicshasevolvedintothe
proposal of superstring theory as a prime candidate for unifying gravity with the
otherforcesandparticlesofnature.Akeyconcernwithquantumgravityhasbeen
the problem of ultraviolet divergences, which is naturally solved in string theory
byreplacingparticleswithspatiallyextendedstatesasthefundamentalexcitations.
Next, Abhay Ashtekar is presenting a broad perspective on loop quantum gravity
and cosmology,while the article by Carlo Rovellisummarizes the present state of
the covariant formulation of the loop quantum gravity dynamics. A lattice spinor
gravity is formulated in the next article by Christof Wetterich, explaining why the
keyingredientforlatticeregularizedquantumgravityisdiffeomorphismsymmetry.
Andrzei Görlich describes the method of causal dynamical triangulations, a non-
perturbative and background independent approach to quantum theory of gravity.
v
vi Preface
The first part of the book ends with the article by E. Bergshoeff, M. Kovacevic,
J.RosseelandY.Yinwhoreviewtherecentdevelopmentsinmassivegravity.
The second part of the book deals with quantum cosmology. Martin Bojoward
presentsloopquantumcosmologyasanattempttounderstandthedynamicsofloop
quantumgravitybyrealizingcrucialeffectsinsimpler,usuallysymmetricsettings.
ThenextarticlebyMartinReuterandFrankSaueressig,afterintroducingthebasic
ideasoftheasymptoticsafetyapproachtoquantumEinsteingravity,discussesthe
implicationsofasymptoticsafetyforthecosmologyoftheearlyUniverse.Thelast
articleisbyPaulMcFadden,abouttherecentdevelopmentsinholographiccosmol-
ogywhichenablesfour-dimensionalinflationaryuniversestobedescribedinterms
ofthree-dimensionaldualquantumfieldtheories.
Inthethirdpartof thebook,theobservationalstatusofdarkmatter(thearticle
byJoeSilk)andtheobservationalstatusofdarkenergy(overviewedbyShinjiTsu-
jikawa)arepresented.ThecontributionbyRobertBrandenbergerdescribestwoal-
ternativestothecurrentcosmologicalscenario,thematterbounceandthestringgas
cosmologyscenarios.Thelastarticle,byM.Romania,N.TsamisandR.Woodard,
presents a class of non-local, gravitational models obtained in quantum gravity in
anacceleratingcosmologicalbackground.
The Sixth Aegean School and the present book became possible with the kind
support of many people and organizations. The School was organized and spon-
sored by the Albert Einstein Institute in Potsdam, the Physics Department of the
UniversityofCrete,thePhysicsDepartmentoftheUniversityofTennesseeandthe
PhysicsDepartmentofNationalTechnicalUniversityofAthens,anditwascospon-
soredbytheMunicipalityofNaxosandtheGeneralSecretariatofAegeanandIsland
Policy.WespeciallythanktheMunicipalityofNaxosformakingavailabletousall
theexcellentfacilitiesoftheCulturalCenterintheformerUrsulineSchoolandall
thestaffofthecenterforhelpingustorunsmoothlytheschool.WealsothankKa-
terinaChiou-LahanasforhervaluablehelpinorganizingtheschoolinNaxos.The
administrativesupportoftheSixthAegeanSchoolwastakenupwithgreatcareby
Fani Siatra and Katerina Papantonopoulou. We acknowledge the help of Vassilis
Zamarias who designed and maintained the webside of the School. We also thank
PetrosSkamagoulisforhelpingusineditingthisbook.
Last, but not least, we are grateful to the staff of Springer-Verlag, responsible
fortheLectureNotesinPhysics,whoseabilitiesandhelpcontributedgreatlytothe
appearanceofthisbook.
GianlucaCalcagni
LefterisPapantonopoulos
GeorgeSiopsis
NikosTsamis
Contents
PartI QuantumGravity
1 StringTheory,UnificationandQuantumGravity . . . . . . . . . . . 3
K.S.Stelle
1.1 Introduction:TheUltravioletProblemsofGravity . . . . . . . . . 3
1.2 StringTheoryBasics . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 ReparametrizationInvariance . . . . . . . . . . . . . . . . 6
1.2.2 TheStringAction . . . . . . . . . . . . . . . . . . . . . . 7
1.3 EffectiveFieldEquations . . . . . . . . . . . . . . . . . . . . . . 10
1.4 DimensionalReductionandT-Duality . . . . . . . . . . . . . . . 12
1.4.1 DimensionalReductionofStringsandT-Duality . . . . . . 13
1.5 M-TheoryandtheWebofDualities . . . . . . . . . . . . . . . . . 15
1.6 BranesandDuality. . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 TheOnsetofSupergravityDivergences . . . . . . . . . . . . . . . 20
1.7.1 SupergravityCountertermAnalysis . . . . . . . . . . . . . 22
1.7.2 SupergravityDivergencesfromSuperstrings . . . . . . . . 25
1.8 OtherAspectsofStringTheory . . . . . . . . . . . . . . . . . . . 26
1.8.1 TheStringScale . . . . . . . . . . . . . . . . . . . . . . . 26
1.8.2 BoundariesofModuliSpace . . . . . . . . . . . . . . . . 27
1.8.3 StringandGravityThermodynamics . . . . . . . . . . . . 28
1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 IntroductiontoLoopQuantumGravityandCosmology . . . . . . . 31
AbhayAshtekar
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.1 DevelopmentofQuantumGravity:ABird’sEyeView . . . 31
2.1.2 PhysicalQuestionsofQuantumGravity . . . . . . . . . . 36
2.2 LoopQuantumGravityandCosmology . . . . . . . . . . . . . . . 38
2.2.1 Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.2 Advances. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
vii
viii Contents
2.2.3 ChallengesandOpportunities . . . . . . . . . . . . . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 CovariantLoopGravity . . . . . . . . . . . . . . . . . . . . . . . . . 57
CarloRovelli
3.1 TheDefinitionoftheTheory . . . . . . . . . . . . . . . . . . . . 57
3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 TheDiscretizationofParametrizedSystems . . . . . . . . . . . . 59
3.4 TheDiscretizationofClassicalGeneralRelativity . . . . . . . . . 62
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 SpinorGravityandDiffeomorphismInvarianceontheLattice. . . . 67
C.Wetterich
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 SpinorsasFundamentalDegreesofFreedom . . . . . . . . . . . . 68
4.3 ActionandFunctionalIntegral . . . . . . . . . . . . . . . . . . . 71
4.4 GeneralizedLorentzTransformations . . . . . . . . . . . . . . . . 72
4.5 LorentzInvariantSpinorBilinears . . . . . . . . . . . . . . . . . . 74
4.6 ActionwithLocalLorentzSymmetry . . . . . . . . . . . . . . . . 75
4.7 GaugeandDiscreteSymmetries . . . . . . . . . . . . . . . . . . . 77
4.8 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.9 LatticeAction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.10 LatticeDiffeomorphismInvariance . . . . . . . . . . . . . . . . . 81
4.11 LatticeDiffeomorphismInvarianceinTwoDimensions . . . . . . 83
4.12 EffectiveAction . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.13 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.14 EffectiveActionforGravityandGravitationalFieldEquations . . 89
4.15 ConclusionsandDiscussion . . . . . . . . . . . . . . . . . . . . . 90
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 IntroductiontoCausalDynamicalTriangulations . . . . . . . . . . . 93
AndrzejGörlich
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1.1 CausalTriangulations . . . . . . . . . . . . . . . . . . . . 94
5.1.2 TheReggeActionandtheWickRotation . . . . . . . . . . 96
5.2 PhaseDiagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 TheMacroscopicdeSitterUniverse . . . . . . . . . . . . . . . . . 100
5.3.1 TheSpatialVolume . . . . . . . . . . . . . . . . . . . . . 100
5.3.2 TheMini-superspaceModel . . . . . . . . . . . . . . . . . 102
5.3.3 TheFour-DimensionalSpace-Time . . . . . . . . . . . . . 103
5.4 QuantumFluctuations . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4.1 TheEffectiveAction . . . . . . . . . . . . . . . . . . . . . 109
5.4.2 FlowoftheGravitationalConstant . . . . . . . . . . . . . 112
5.5 TheGeometryofSpatialSlices . . . . . . . . . . . . . . . . . . . 113
5.5.1 TheHausdorffDimension . . . . . . . . . . . . . . . . . . 113
Contents ix
5.5.2 SpectralDimension . . . . . . . . . . . . . . . . . . . . . 115
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 MassiveGravity:APrimer . . . . . . . . . . . . . . . . . . . . . . . 119
E.A.Bergshoeff,M.Kovacevic,J.Rosseel,andY.Yin
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 GeneralSpin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2.1 “BoostinguptheDerivatives” . . . . . . . . . . . . . . . . 121
6.2.2 “TakingtheSquareRoot” . . . . . . . . . . . . . . . . . . 125
6.3 Spin1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 Spin2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4.1 3DNewMassiveGravity . . . . . . . . . . . . . . . . . . 131
6.4.2 3DTopologicalMassiveGravity . . . . . . . . . . . . . . 136
6.4.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Appendix Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
PartII QuantumCosmology
7 LoopQuantumCosmology,Space-TimeStructure,andFalsifiability 149
MartinBojowald
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2 CanonicalGravity . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2.1 CosmicSubtleties . . . . . . . . . . . . . . . . . . . . . . 154
7.2.2 DeformationsofSpace . . . . . . . . . . . . . . . . . . . 156
7.2.3 GaugeTheory . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2.4 QuantumCorrections . . . . . . . . . . . . . . . . . . . . 159
7.3 LoopQuantumGravity . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.1 CorrectionsfromLoopQuantumGravity . . . . . . . . . . 162
7.3.2 ConstructionofInverse-TriadCorrections . . . . . . . . . 163
7.3.3 Anomaly-Freedom . . . . . . . . . . . . . . . . . . . . . . 164
7.3.4 Falsifiability . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.3.5 Anomaly-FreeHolonomyCorrections . . . . . . . . . . . 167
7.4 EffectiveTheories . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.4.1 EffectiveCanonicalDynamics . . . . . . . . . . . . . . . 170
7.4.2 MomentDynamics. . . . . . . . . . . . . . . . . . . . . . 172
7.4.3 EffectiveConstraints. . . . . . . . . . . . . . . . . . . . . 174
7.4.4 IsotropicCosmology. . . . . . . . . . . . . . . . . . . . . 177
7.4.5 Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8 AsymptoticSafety,Fractals,andCosmology . . . . . . . . . . . . . . 185
MartinReuterandFrankSaueressig
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
x Contents
8.2 TheorySpaceandItsTruncation . . . . . . . . . . . . . . . . . . 188
8.3 TheEffectiveAverageActionforGravity . . . . . . . . . . . . . . 192
8.4 TheEinstein–HilbertTruncation . . . . . . . . . . . . . . . . . . 194
8.5 TheMulti-fractalPropertiesofQEGSpace-Times . . . . . . . . . 197
8.6 Spectral,Walk,andHausdorffDimension. . . . . . . . . . . . . . 200
8.7 FractalDimensionsWithinQEG . . . . . . . . . . . . . . . . . . 202
8.7.1 DiffusionProcessesonQEGSpace-Times . . . . . . . . . 202
8.7.2 TheSpectralDimensioninQEG . . . . . . . . . . . . . . 204
8.7.3 TheWalkDimensioninQEG . . . . . . . . . . . . . . . . 206
8.7.4 TheHausdorffDimensioninQEG . . . . . . . . . . . . . 206
8.7.5 RelationsBetweenDimensions . . . . . . . . . . . . . . . 207
8.8 TheRGRunningofD andD . . . . . . . . . . . . . . . . . . . 207
s w
8.9 MatchingtheSpectralDimensionsofQEGandCDT . . . . . . . . 210
8.10 AsymptoticSafetyinCosmology . . . . . . . . . . . . . . . . . . 213
8.10.1 RGImprovedEinsteinEquations . . . . . . . . . . . . . . 214
8.10.2 SolvingtheRGImprovedEinsteinEquations . . . . . . . . 214
8.10.3 InflationintheFixed-PointRegime . . . . . . . . . . . . . 215
8.10.4 EntropyandtheRenormalizationGroup . . . . . . . . . . 217
8.10.5 PrimordialEntropyGeneration . . . . . . . . . . . . . . . 219
8.10.6 EntropyProductionforRGTrajectoryRealizedbyNature . 221
8.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9 HolographyforInflationaryCosmology . . . . . . . . . . . . . . . . 227
PaulMcFadden
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9.2 Domain-WallsandCosmologies. . . . . . . . . . . . . . . . . . . 230
9.2.1 DefiningthePerturbations . . . . . . . . . . . . . . . . . . 230
9.2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.2.3 TheDomain-Wall/CosmologyCorrespondence . . . . . . . 234
9.2.4 CosmologicalPowerSpectra . . . . . . . . . . . . . . . . 236
9.3 HolographyforCosmology . . . . . . . . . . . . . . . . . . . . . 237
9.3.1 BackgroundSolutions . . . . . . . . . . . . . . . . . . . . 238
9.3.2 BasicsofHolography . . . . . . . . . . . . . . . . . . . . 238
9.3.3 HamiltonianHolographicRenormalisation . . . . . . . . . 240
9.3.4 TheStressTensor2-PointFunction . . . . . . . . . . . . . 243
9.3.5 HolographicAnalysis . . . . . . . . . . . . . . . . . . . . 244
9.3.6 HolographicFormulaeforthePowerSpectra . . . . . . . . 249
9.4 HolographicPhenomenologyforCosmology . . . . . . . . . . . . 250
9.4.1 APrototypeDualQFT. . . . . . . . . . . . . . . . . . . . 251
9.4.2 CalculatingtheHolographicPowerSpectra . . . . . . . . . 252
9.5 ConfrontingObservations . . . . . . . . . . . . . . . . . . . . . . 257
9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265