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