Quantum Ricci Curvature - mphys9.ipb.ac.rsmphys9.ipb.ac.rs/slides/Loll.pdf · The case for quantum...
Transcript of Quantum Ricci Curvature - mphys9.ipb.ac.rsmphys9.ipb.ac.rs/slides/Loll.pdf · The case for quantum...
RenateLollIns%tuteforMathema%cs,AstrophysicsandPar%clePhysics,
RadboudUniversity
QuantumRicciCurvature
M∩𝚽,Belgrade20Sep2017
ExploringspaceAmeatthePlanckscale
OurmaingoalistoconstructatheoryofQuantumGravity,afundamentalquantumtheoryunderlyingGeneralRela%vity.!Mytalktodayisaboutthenewconceptofquantumcurvature,a
quasilocalobservableinanonperturba%ve,Planckianregime,wherespace%meisnolongerdescribedbyasmoothmetricgμν(x).!Iwillexplaintheideainthecon%nuumandthenimplementiton
variouspiecewiselinear(PL)spaces,includingtheequilateralconfigura%onsof(Causal)DynamicalTriangula;ons.CDTisacandidatetheoryofquantumgravity,basedonanon-perturba%vepathintegral,whereour“quantumRiccicurvature”canbecalculatedinastraighOorwardway.However,theconceptisapplicablemuchmorewidely,andyouneednotknowaboutCDT.! ! (jointworkwithNilasKlitgaard,tobepublished)
ThecaseforquantumobservablesEvenassumingwehadresolvedall“technicaldifficul%es”inspecificquantumgravitytheories,wes%llfacethekeyissueofhavingtoconstructmeaningfulobservablestoquan%fythephysicalquantumproper%esofgravityandspace%me(orwhateverremainsofthem)atthePlanckscale.ThesearepriortothedevelopmentofanytrueQGphenomenologyandshould!1.bepurelygeometrical(coordinate-invariant,background-indept.),2.havefinite,nonzeroexpecta%onvaluesintheensemble,3.bemeasurablereliablyinthewindowaccessibletoquan%ta%veevalua%on(simula%onorothernumericalmethods),
4.havea(semi-)classicallimit.!Wehavesuchobservables,e.g.inCDT,buttheyarerathercoarse(dynamicaldimensions,volumeprofiles)andshouldbecomple-mentedbyquan%%escarryingmorelocalgeometricinforma%on.
•CDTquantumgravityisaperfectse^ngtostudysuchnonperturba%vequantumobservables.Itsregularizedpathintegral(“sumoverspace%mes”)isdefinedpurelygeometricallyintermsofsimplicialmanifoldsthataregluingsofiden%calD-dimensionalflatsimplices(➔“RandomGeometry”).Observablesareevaluatedquan%ta%velybyMonteCarlosimula%on.!•Sincelengthsandvolumescomeindiscreteunits,measuringis
oaenreducedtosimplecoun%ng.(C)DTgeometriesareof“Reggetype”,i.e.con%nuous,butwithcurvaturesingulari%es.!•Thislookslikeaconserva%vese^ng,butallowsfornonclassical
behaviourandnoncanonicalscaling.Infact,wehavelearnedthatsuchbehaviourisgenericandoaenpreventstheexistenceofaclassicallimitwhentheUV-cutoffaisremoved.
ThecaseforCDT
pieceofacausaltriangula%onin3D
aflat
a
abuildingblockin2D:
•Curvatureisacrucialconceptindescribingclassicalspace%megeometry.Itisacomplex,derivedobject.Computa%onoftheRiemanntensorRκλμν[g,∂g,∂2g;x)requiresasmoothmetricg.!•Findingameaningfulno%onof“quantumcurvature”,applicableina
moregeneralcontext,hassofarreceivedlifleafen%on.(Notethatdefini%onsintermsofdeficitanglesareoflimitedusefulness.)!•Isthereaclassicalcharacteriza%onofcurvaturethatcanbeusedto
obtainacoarse-grained,robustandcomputableno%onofquantumcurvatureinnonperturba%vequantumgravity?!
Thecaseforcurvature
•Curvatureisacrucialconceptindescribingclassicalspace%megeometry.Itisacomplex,derivedobject.Computa%onoftheRiemanntensorRκλμν[g,∂g,∂2g;x)requiresasmoothmetricg.!•Findingameaningfulno%onof“quantumcurvature”,applicableina
moregeneralcontext,hassofarreceivedlifleafen%on.(Notethatdefini%onsintermsofdeficitanglesareoflimitedusefulness.)!•Isthereaclassicalcharacteriza%onofcurvaturethatcanbeusedto
obtainacoarse-grained,robustandcomputableno%onofquantumcurvatureinnonperturba%vequantumgravity?YES.!•N.B.:weareworkinginaRiemanniancontext(“aaerWickrota%on”)
Thecaseforcurvature
Thekeyidea(classical):
δvεw εw’p
qq’
p’
v,wunitvectorsatp,v⊥wε,δ>0w’istheparalleltransportofwalongvp’=expp(δv),q=expp(εw),⇒thereisauniquepointq’=expp’(εw')
OnaD-dimensionalmanifold(M,gμν),comparethedistanceoftwosmall(D-1)-sphereswiththedistanceδoftheircentres.!Step1:
thenwehave
whereK(v,w)isthesec8onalcurvaturecorrespondingto(v,w),
K(v, w) =<R(v, w)w, v>
<v, v><w,w> � <v,w>2
d(q, q0) = �⇣1� ✏2
2K(v, w) +O(✏3 + �✏2)
⌘, ✏, � ! 0
εwε
εδv
M
εw’p
SεpSεp’
qq’
p’
d(S✏p, S
✏p0) :=
1
vol(S✏p)
Z
S✏p
d
D�1q
ph d(q, q0),
Step2:!LetSεpdenotetheε-sphereofallpointsatgeodesicdistanceεfromitscentrep.Thenthespheredistancetoanotherε-sphereSεp’whosecentrep’liesatadistanced(p,p’)=δ,andwhosepointsareobtainedbyparalleltransportisdefinedby
whichforsmallδ,εisgivenby
d(S✏p, S
✏p0) = �
✓1� ✏2
2DRic(v, v) +O(✏3 + �✏2)
◆,
whereRic(v,v)istheRiccicurvatureofthevectorv,i.e.theaverageofthesec%onalcurvatureK(v,w)overalltwo-planescontainingv.
(noteuniqueassocia%onq↔q’)
Thisformula,
d(S✏p, S
✏p0) = �
✓1� ✏2
2DRic(v, v) +O(✏3 + �✏2)
◆,
capturesourkeystatement:“OnaD-dim.manifoldwithposi%ve(nega%ve)Riccicurvature,thedistancebetweentwonearby(D-1)-spheresissmaller(larger)thanthedistancebetweentheircentres.”!Thisobserva%onisthestar%ngpointforOllivier’s“coarseRiccicurvature”(Y.Ollivier,J.Funct.Anal.256(2009)810-864).[→c.f.workbyC.Trugenberger,G.Bianconi(“QGfromgraphs”)]!However,hisdefini%oninvolves“transportdistance”(inabsenceofparalleltransport),whichisverydifficulttocomputeinprac%ce.!Instead,wearelookingforano%onof“quantumRiccicurvature”whichiscomputable,scalable(alsoworksfornon-infinitesimalscales)androbust.
DefiningquantumRiccicurvature
d̄(S✏p, S
✏p0) :=
1
vol(S✏p)
1
vol(S✏p0)
Z
S✏p
d
D�1q
ph
Z
S✏p0
d
D�1q
0ph
0d(q, q0),
Ourclassicalstar%ngpointistheaveragesphere“distance”
whichusesonlyvolumeanddistancemeasurements,andthereforecanbeimplementedstraighOorwardlyongeneralmetricspaces.!Wethensetε=δ,correspondingtooverlappingspheres,anddefinethe“quantumRiccicurvatureKqatscaleδ”by
d̄(S�p , S
�p0)
�= cq(1�Kq(p, p
0)), � = d(p, p0), cq > 0,
wherecqappearstobeanon-universalconstantdependingonthespaceunderconsidera%on.WehaveevaluatedKqonclassicalmodelspaces,andtesteditonavarietyof(mainly2D)PLspaces(flatregular%lings,“nice”and“not-so-nice”triangula%ons).
Behaviouronconstantlycurvedmodelspaces
¯d(S�p , S
�p0)
�=
8<
:
1.5746 flat space
1.5746� 0.1440( �⇢ )2+ h.o. spherical case
1.5746 + 0.1440( �⇢ )2+ h.o. hyperbolic case
d(S✏p, S
✏p0)
�=
8<
:
1 flat space
1� 14 (
✏⇢ )
2+ h.o. spherical case
1 +
14 (
✏⇢ )
2+ h.o. hyperbolic case
Normalizedspheredistancefor2Dmodelspaces,forsmallε,δ:
ThisisconsistentwiththegeneralformulagivenaboveforRic=1/ρ2.!Normalizedaveragespheredistancefor2Dmodelspaces,forε=δandsmallε,δ:
Thisisqualita%velysimilartothespheredistanceresultsforε=δ.
Behaviouronconstantlycurvedmodelspaces
¯d(S�p , S
�p0)
�=
8<
:
1.5746 flat space
1.5746� 0.1440( �⇢ )2+ h.o. spherical case
1.5746 + 0.1440( �⇢ )2+ h.o. hyperbolic case
d(S✏p, S
✏p0)
�=
8<
:
1 flat space
1� 14 (
✏⇢ )
2+ h.o. spherical case
1 +
14 (
✏⇢ )
2+ h.o. hyperbolic case
Normalizedspheredistancefor2Dmodelspaces,forsmallε,δ:
ThisisconsistentwiththegeneralformulagivenaboveforRic=1/ρ2.!Normalizedaveragespheredistancefor2Dmodelspaces,forε=δandsmallε,δ:
Thisisqualita%velysimilartothespheredistanceresultsforε=δ.
new
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
�
�
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
�
�
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 1 2 3 4 5 60
1
2
3
4
5
6
��
� �
0.5
1.0
1.5
2.0
2.5
3.0
0 1 2 3 4 5 60
1
2
3
4
5
6
��
� �
0.5
1.0
1.5
2.0
2.5
3.0
Behaviouronconstantlycurvedmodelspaces
Comparingspheredistanceandaveragespheredistanceforany(δ,ε).Observethequalitativesimilaritiesalongthediagonalsδ=ε.
flatcase sphericalcase
new new
averagespheredistanceinthecon%nuum(2D)
normalizedaveragespheredistanceinthecon%nuum(2D)
Behaviouronconstantlycurvedmodelspaces
d̄/�
d̄
sphere:Kq>0
flatspace:Kq=0
hyperbolicspace:Kq<0
d̄
�= cq(1�Kq)recall
0 1 2 3 4 5 6 �0
5
10
15
d hyperbolicspace
flatspace
sphere
0 1 2 3 4 5 6 �0.0
0.5
1.0
1.5
2.0
2.5
d
�
(aaerse^ngε=δ)
Wemeasurednormalizedaveragespheredistancesforregularflatla^cesin2Dand3D.
Example:spheredistanceona2Dsquarela^ce(δ=3)
Theirtypicalbehaviouris
d̄/�
-correc%ons
(cqnotuniversal).
“Nice”spacesI:regulartriangulaAons
�
�� � � � � � � � � � � � � � � � � �
�
� � � � � � � � � � � � � � � � � � �
0 5 10 15 20�1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
d
�squarela^cein2D!hexagonalla^cein2D
d̄/� = cq +1
�2
Howcanweconstructequilateralrandomgeometriesthatare“close”toagivensmoothclassicalgeometry?
“Nice”spacesII:DelaunaytriangulaAons
ADelaunaytriangula%onisatriangula%onTofafinitepointsetP⊂R2(thever%cesofT)ifthecircumcircleofeverytrianglecontainsnopointsofPinitsinterior.Itmaximizestheminimumangleandavoidsthin,elongatedtriangles
1. generatePusingPoissondiscsamplingona2Dconstantlycurvedspace(plane,sphere,hyperboloid)
2. constructtheDelaunaytriangula%onofP
3. setalledgelengthsto1
Ourprocedure:
HownicearetheresulAngPLspaces?
Ourconstruc%ongeneratesrandomtriangula%onswithmild(small-scale)localcurvaturefluctua%ons.
Beforese^ngℓ=1:
probabilitydistribu%onofedgelengthsℓinaDelaunaytriangula%onofflatspace,N=6.2k
Beforeandaaer:
distribu%onofvertexorders(flatcase)
volumecofgeodesiccirclesofradiusδ:linearityimpliesflat-spacebehaviour
Aaer:
1.1minDist 1.3minDist 1.5minDist 1.7minDist 1.9minDist 2.1minDist 2.3minDistd0.00
0.02
0.04
0.06
0.08
dmin 2dmin
||
MeasuringthequantumRiccicurvatureKq
comparisonofmeasurementsofthenormalizedaveragespheredistanced̄/�
Weobtainagoodmatchingwithcon%nuumresults,withdiscre%za%onartefactsconfinedtotheregion,andKq≈0elsewhere.� . 5
“flat”regular,hexagonalla^ce
recallthatd̄/� = cq(1 � Kq(�))
_
“flat”randomtriangula%on
flatcon%nuumspace
normalizedaveragespheredistanceforrandomtriangula%onsmodelledonflatspaceandspheresofvarioussizes
d̄/�
Weobservegood“averagingproper%es”.
MeasuringthequantumRiccicurvature,ctd.
d̄/�
“hyperbolic”space(yellow)“flat”space(blue)
normalizedaveragespheredistanceforrandomtriangula%onsmodelledonflatandhyperbolicspace
_
_
AtruequantumapplicaAonofRiccicurvature
• considera2Dtoymodelof(Euclidean)quantumgravity,with
!!!• nonperturba%vepathintegralovergeometrieswithfixedtopologyS2,solublevia“DynamicalTriangula%ons”
Z(Λ) =
!
geom. gDg e−Λ vol(g)
• pathintegralconfigura%onsarearbitrarygluingsof2Dequilateraltriangles;inthecon%nuumlimit,“typical”ensemblemembersarehighlynonclassical,fractalandnowheredifferen%ablegeometrieswithspectraldimension2andHausdorffdimension4!• thequantumdynamicsisgovernedbybranching“babyuniverses”
atypical“universe”inDTquantumgravityin2D
Usingasystemof20.000triangles,wefoundasurprisinglygoodfitofthedatawiththoseofacon%nuumspherein4D(!),withposi%vecurvature.ThispointstoaveryrobustbehaviourofquantumRiccicurvature.(N.B.:theSδpingeneraldonotevenhavesphericaltopologyforδ>1,butaremultiplyconnected.)
QuantumRiccicurvaturein2DDTQGravity
Wehavemeasuredtheexpecta%onvalue!
ofthenormalizedaveragespheredistanceintheensembleof2DEuclideanDTquantumgravity.
hd̄(S�p , S
�p0)/�i
hd̄/�i
�
measureddata:bluecon%nuumsphere:yellow
Summary
Wehavedefinedanovelno%onof“quantumRiccicurvatureatscaleδ”,basedontheaveragespheredistance,andhaveinves%gateditsproper%esinbothaclassicalandquantumcontext,mostlyontwo-dimensionalgeometries.Theresultslookpromising:!•theprescrip%onisstraighOorwardtoimplementonpiecewiseflat
spacesandisfeasiblecomputa%onally;!•onnicepiecewiseflatspaces,la^ceartefactscanbecontrolled
andsmoothresultsarereproducedonsufficientlylargescales;!•“robustness”hasbeenfoundinthecaseofthehighlyquantum-
fluctua%ngquantumensembleof2DEuclideanDTquantumgravity.!Thenextstepisanimplementa%onin4DCausalDynamical
Triangula%ons,anonperturba%vecandidatetheoryofquantumgravity,tounderstandandquan%fyitsquantumgeometry.
d̄(S�p , S
�p0)/�
Thankyou!
M∩𝚽,Belgrade20Sep2017