RenateLollRadboudUniversity,Nijmegen

Corfu,18Sep2019

La@ceGravity,DiffeomorphismsandCurvature

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Thecontextofmytalkisthesearchforatheoryofquantumgravitybeyondperturba4ontheoryandtheongoingresearchprogramofCausalDynamicalTriangulaJons(CDT)addressingtheproblem.SinceJmeismuchtooshortforacomprehensiveoverview,Iwillmerelysummarizetheapproachandthendescribesomenewinsightsandstructuralaspectsithasbroughtintofocus.MypresentaJonwillbeabout!• moJvaJonandcontext!• la@cegravityandCDTinanutshell!• theroleofdiffeomorphisms• makingsenseofRiccicurvatureinaquantumcontext

LifeintheCenturyofGravity• urgent:completeourquantumgravitytheoriestomakereliablepredicJons,minimizingfreeparametersandadhocassumpJons!• myroute:tacklequantumgravityandgeometrydirectlyinanon-perturbaJve,Planckianregime(noappealtoduality/dicJonaries)

!• thebeautyofclassicalGR:“theoryofspaceJme”,capturedbyitscurvatureproperJes!• giventhecentralroleofcurvatureclassically,isitalsotruethat!nonperturb.quantumgravity=theoryofquantumcurvature?!• Sofar,no-onehasbeenabletomakemuchsenseofsuchaproposiJon.WehaverecentlyiniJatedalineofresearchintohowtodefineandmeasurequantumRiccicurvatureinquantumgravity.

(©User:Johnstone,Wikipedia) (©R.Hurt/Caltech-JPL/EPA)

These8ng• followingtheextremelysuccessfulexampleofQCD,weexplorethenonperturbaJveregimequanJtaJvelyby“la8cequantumgravity”!• la@cegaugefieldconfiguraJonsàlaWilson(PRD10(1974)2445)arereplacedbypiecewiseflatgeometries(triangulaJons)àlaRegge(NuovoCim.19(1961)558)

!!!!!!!• modernimplementaJon:CausalDynamicalTriangula>ons(CDT),anonperturbaJve,background-independent,manifestlydiffeo-morphism-invariantpathintegral,regularizedondynamicalla@ces

!• N.B.:nontrivialscalinglimitneeded,no“fundamentaldiscreteness”

(©G.B

ergner,Jen

a)

triangulatedmodelofquantumspace

•Classically,differenJablemanifoldsMprovidepowerfulandextremelyconvenientmodelsofspaceJme.!•geometricproperJesencodedin

theRiemanncurvaturetensorRκλμν(x)differenJablemanifoldMandacoordinatechart

!•However,thisdescripJoncomeswithanenormousredundancy,the“freedomtochoosecoordinates”withoutaffecJngthephysics.!•The“gauge”groupofGRistheinfinite-dim.groupofcoordinate

transformaJons(diffeomorphisms)onM.Thekeychallengesofquantumgravityarehowtoimplementthissymmetryanddescribephysicsintermsofdiffeomorphism-invariantquantumobservables.

Contrarytofolklore,givingupsmoothspace>mesandtensorcal-culusisnotacrazyidea,buthasbeenkeytorecentprogressinquantumgravity(c.f.nonclassical’discretegeometry’inmaths).

TheframeworkofCausalDynamicalTriangula->ons(CDT),anonperturbaJvecandidatetheoryofquantumgravity,hasprovenbothfruikulandwellsuitedtostudyingtheissueofobservables.!Severalquantumobservableshavebeensuccessfullydefinedandimplemented,andtheirexpectaJonvaluesbeenmeasured.!CDTemploysadirectquanJzaJonofclassicalspaceJmegeometry=Metrics(M)/Diff(M),usingala@ceregularizaJon.AsonewouldexpectinstandardQFT,thetheoryhasdivergencesintheconJnuumlimitastheUVregulatorisremoved,whichmustberenormalizedappropriately.!CDTdynamics:nonperturbaJve,background-independent,unitarypathintegral;exactlysolubleinD=2,MonteCarlosimulaJonsinD=4.

partofa(piecewiseflat)causaltriangulaJon

CDTQuantumGravity

CDTisacounterexampletothefolklorethat“pu@nggravityonthela@cebreaksdiffeomorphisminvariance”.

QuantumGravityfromCDTThe(formal,ill-defined)conJnuumgravitaJonalpathintegral!!!!

isturnedintoafiniteregularizedsumovertriangulatedspaceJmes,

Z(GN ,⇤) =

Z

spacetimesg2G

Dg eiSEHGN,⇤[g]

Z(GN ,⇤) := lima!0N!1

X

inequiv.triangul.sT2Ga,N

1

C(T )eiS

ReggeGN,⇤ [T ]

|Aut(T)|

Newton’sconstant

cosmologicalconstant

#buildingblocks

UVcutoff

Einstein-HilbertacJon

whoseconJnuumlimitsareinvesJgatedarerananalyJcconJnuaJon.(N.B.:theinclusionofmaserisstraighkorward)

SEH =1

GN

�d4x

⇥�det g(R[g, ⇥g, ⇥2g]� 2�)gravityacJon:

(“sumoverhistories”)

bare,discreJzedEHacJon

WhatistheoveralloutlookofCDTQG?!•CDTquantumgravitydependsonaminimalistsetofingredients—metricd.o.f.andjusttwofreeparameters—andisconceptuallysimple.!•ItbuildsonasignificantbodyofanalyJcalandnumericalresultson“dynamicaltriangulaJons”(a.k.a.“randomgeometry”)sincethe1980s,whichgiveusanewviewongeometryandtheroleofdiffeomorphisms.(2DDTquantumgravityreproducesresultsofconJn.Liouvillegravity.)

!•OnehasbeenabletoextractnewanduniqueresultsfromevaluaJngahandfulofnonperturba>vequantumobservables.TheseresultsarerobustandquanJtaJve,andpotenJallyfalsifiable(veryrareinQG!).!•causalstructureplaysacrucialrole(EuclideanQG‘notgoodenough’)

!•quanJtaJve4Dresultshavesofarbeenobtainedinahighlyquantum-fluctuaJngregime,farawayfrom(semi-)classicality.

REVIEWS:J.Ambjørn,A.Görlich,J.Jurkiewicz&RL,Phys.Rep.519(2012)127[arXiv:1203.3591]);NEW:RL,arXiv:1905.08669

TheEmergenceofClassicalityfromCausalDynamicalTriangula>ons(CDT)

FrompurequantumexcitaJons,CDTgeneratesaspaceJmewithsemiclassicalproperJesdyna-mically,withoutusingabackgroundmetric.

!•crucialroleofcausalstructure•noJonofdiscreteproperJme(notcoordinateJme)•existenceofawell-defined“WickrotaJon”(unique)•amenabletocomputersimulaJons•nontrivialphasestructure,with“classical”phases•second-orderphasetransiJons(unique)•scale-dependentspaceJmedimension(2→4)•applicabilityofrenormalizaJongroupmethods

Otherkeyresults/proper>es:

howtoobtainamacroscopicuniversewithadeSi:ershape:

fromasuperposiJonof“wild”pathintegralhistories:

WhatwehavelearnedsofarinCDTquantumgravityabout

!(i) thephasestructureandcriJcalproperJesoftheunderlying

staJsJcalsystemof‘randomgeometry’,(ii) thesystem’sbehaviouralongRGtrajectories,and(iii) theproperJesofthedynamicallygenerated“quantum

spaceJme”!

comesfrommeasuringafewquantumobservables.

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

-40 -30 -20 -10 0 10 20 30 40

<V(t)

>

t

K0 = 2.200000, 6 = 0.600000, K4 = 0.925000, Vol = 160k

a = 8253 b = 14.371 w = 7.5353

Monte CarloFit, a sin3(t/b)

MC, <V(t)>cos

red: size of typical quantum fluctuations

almostperfectfittocos3(t/b)

!20 !10 10 20

!0.3

!0.2

!0.1

0.1

0.2

(dataforN4=80k)

first order

Cb CdS

second order

new (h.o.) phase transition

time t

V3(t)

(bareinverseNewtonconstant)

(asymmetryparam

eter)

✔

✖✔

✖

PhasediagramofCDTQG TheuniverseisdeSiUer-shaped

Volumefluctua>onsarounddeSiUerSpectraldimensionoftheuniverse

(diffusionJme)

green: error bars

(spe

ctraldim

ensio

n)(spaJa

lvolum

e)

(properJme)

(low-lyingeigenmodematcheswithsemiclassicalexpectaJon)

“dynamical dimensionalreduction” at Planckian

scalesRESULT

S

Crucial:QGwithoutdiffeomorphisms!Strategy:ataregularizedlevel,representcurvedspaceJmesbysim-plicialmanifolds,followingtheprofound,butunderappreciatedideaof“GeneralRelaJvitywithoutCoordinates”(T.Regge,1961).!•Use‘piecewiseflat’gluingsof4Dtriangularbuildingblocks(four-simplices)todescribeintrinsicallycurvedspaceJmes.!•Geometryisspecifieduniquelybytheedgelengthsℓofthese

simplicesandhowsimplicesare‘glued’together.NocoordinatesareneededandtheCDTpathintegralhasnocoordinateredundancies.

ε✂

d=2

α1α2

…

GluingfiveequilateraltrianglesaroundavertexgeneratesasurfacewithGaussiancurvature(deficitangleε)atthevertex.

•disJnctfromReggecalculus:alledgeshaveidenJcallengthℓ=a(uptoglobalJmevs.spacescaling)!•studysuperposiJonsofsuchgeo-

metriesinconJnuumlimita→0(removalofUVcut-off)

ℓs

ℓs

ℓt

Thechallengeof“quantumcurvature”

,

IndividualspaceJmegeometries(=pathintegralhistories)inCDTareconJnuous,butnotsmooth,andfarfrom(semi-)classical.! • WhichproperJesconJnuetoholdonsuchspaces?• Howcanwemakesenseofcurvatureandcurvaturetensors?

!• Howcanweaverage/coarse-grainthem?WehavesuccessfullydefinedandtestedquantumRiccicurvature.(N.Klitgaard&RL,PRD97(2018)no.4,0460008andno.10,106017,workinprogresswithJ.BrunekreefandN.Klitgaard)

fromclassical

toquantum?

IntroducingquantumRiccicurvatureInDdimensions,thekeyideaistocomparethedistancedbetweentwo(D-1)-sphereswiththedistanceδbetweentheircentres.

δp

SpSp’

p’

d_

_

Ourvariantusestheaveragespheredistancedoftwospheresofradiusδwhosecentresareadistanceδapart, δ δ

δp p’

qq’

Thesphere-distancecriterion:“OnametricspacewithposiPve(negaPve)Riccicurvature,thedistancedoftwonearbyspheresSpandSp’issmaller(bigger)thanthedistanceδoftheircentres.”

_

(c.f.Y.Ollivier,J.Funct.Anal.256(2009)810)

ε ε

_

D=2

‣ involvesonlydistanceandvolumemeasurements!‣ thedirecJonal/tensorialcharacteriscapturedbythe“doublesphere”!‣ coarse-grainingiscapturedbythevariablescaleδ

Wemeasurethe“quantumRiccicurvatureKqatscaleδ”,

onthequantumensembleandcompareitwiththebehaviouronsimpleconJnuum“referencespaces”(constantlycurved;ellipsoids;cones).Remarkably,forthehighlyfractalquantumgeometryof2Dquantum

Implemen>ngquantumRiccicurvatureδp p’

d_

d̄(S�p , S�

p�)

�= cq(1 � Kq(p, p�)), � = d(p, p�), 0 < cq < 3,

non-univ.constant

sphere:Kq>0

flatspace:Kq=0

hyperbolicspace:Kq<0

0 1 2 3 4 5 6 �0.0

0.5

1.0

1.5

2.0

2.5

d

�

Kqonclassical,constantlycurvedspacesinD=2(curvatureradius1)

gravity,quantumRiccicurvaturedisplaysarobust,sphere-likescalingbehaviour:

hd̄/�i

#trianglesN∈[20k,240k];errorbarstoosmalltobeshown

DTla@ceartefactsforδ<5

sphere

Summary

NonperturbaJvequantumgravitycanbestudiedinala8cese@ng,incloseanalogywithla@ceQCD,buttakingintoaccountthedynamicalnatureofgeometry,asexemplifiedbyCDT.!TheCDTapproachhasbeenmakingsignificantstridestowardsafull-fledgedquantumtheory.Itswell-definedcomputaJonalla@ceframeworkallowsforquanJtaJveevaluaJonand“realitychecks”.!ThefullpowerofRegge’sideaofdescribinggeometrywithoutcoordinatesunfoldsinnonperturbaJveQGintermsofCDT,yieldingamanifestlydiffeomorphism-invariantformulaJon.!Despitetheabsenceofsmoothness,onecandefineanoJonofcurvaturethatappearstobewell-defined,includinginaPlanckianregime,andgivesusanewtooltounderstandtheproperJesofquantumgravityandthequantumgeometryemergentfromit.

Thank you!

Corfu,18Sep2019

La@ceGravity,DiffeomorphismsandCurvature