Lace Gravity, Diffeomorphisms and Curvaturescience_ru_nl_01.pdf · physics in terms of...
Transcript of Lace Gravity, Diffeomorphisms and Curvaturescience_ru_nl_01.pdf · physics in terms of...
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),[email protected],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