Generalized Quantum Theory of Recollapsing Homogeneous Cosmologies

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

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agr-qc/0309117GeneralizedQuantumTheoryofRecollapsingHomogeneousCosmologiesDavidCraig DepartmentofPhysics,HamiltonCollege,Clinton,NY13323JamesB.Hartle DepartmentofPhysics,UniversityofCaliforniaSantaBarbara,CA93106-9530(Dated:February7,2008)AbstractAsum-over-historiesgeneralizedquantumtheoryisdevelopedforhomogeneousminisuperspacetypeABianchicosmologicalmodels,focussingontheparticularexampleoftheclassicallyrecol-lapsingBianchiIXuniverse.Thedecoherencefunctionalforsuchuniversesisexhibited.Weshowhowtheprobabilitiesofdecoherentsetsofalternative,coarse-grainedhistoriesofthesemodeluni-versescanbecalculated.Weconsiderinparticulartheprobabilitiesforclassicalevolutionde nedbyasuitablecoarse-graining.Forarestrictedclassofinitialconditionsandcoarsegrainingsweexhibittheapproximatedecoherenceofalternativehistoriesinwhichtheuniversebehavesclas-sicallyandthoseinwhichitdoesnot.Forthesesituationsweshowthattheprobabilityisnearunityfortheuniversetorecontractclassicallyifitexpandsclassically.Wealsodeterminetherelativeprobabilitiesofquasi-classicaltrajectoriesforinitialstatesofWKBform,recoveringforsuchstatesapreciseformofthefamiliarheuristic“J·dΣ”ruleofquantumcosmology,aswellasageneralizationofthisruletogenericinitialstates.PACSnumbers:03.65.Ca,03.65.Yz,04.60.Gw,04.60.Kz,04.60.-m,98.80.Qc

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

I.INTRODUCTION

Hamiltonianquantummechanicswithitsrequisitepreferredtime(s)mayneedtobegen-eralizedtoprovideaquantumframeworkforcosmologywherespacetimegeometry uctuatesquantummechanicallyanddoesnotspecifya xednotionoftime.Oneapproachtosuchageneralizationisthesum-over-historiesgeneralizedquantumtheoryofspacetimegeometry,describedmostcompletelyin[1]wherereferencestotheearlierliteraturemaybefound.Thisisaformulationofquantumtheoryinfullyfour-dimensionalspacetimeform.Theessentialelementsofthissum-overhistoriesformulationare:

1.Fine-GrainedHistoriesThesearetheelementsofthesetoffour-dimensionalhisto-riesofspacetimemetricandmatter eldcon gurations.Theyarethemostre neddescriptionsoftheuniverseitispossibletogive.

2.Coarse-GrainedHistoriesPartitionsofthe ne-grainedhistoriesintofour-dimensionaldi eomorphisminvariantclassesarecalledcoarse-grainedhistories.Suchsetsofcoarse-grainedhistoriesarethemostgeneralnotionofalternativedescribableinspace-timetermsforwhichquantumtheorypredictsprobabilities.

3.DecoherenceFunctionalThisisameasureofthequantummechanicalinterferencebetweenmembersofasetofalternativecoarse-grainedhistories.Itisconstructedaccordingtosum-over-historiesprinciplesandincorporatesatheoryoftheuniverse’sinitialand nalcondition.Thedecoherencefunctionalisanaturalgeneralizationtoclosedquantumsystemsofthealgebraicnotionofquantumstate[2,3].Setsofhistorieswithnegligibleinterferencebetweenallpairsofmembers,asmeasuredbythedecoherencefunctional,aresaidtodecohere,ortobeconsistent.Itislogicallyconsistenttoassignprobabilitiesinanexhaustivesetofalternativehistorieswhen,andonlywhen,thatsetisdecoherent.Itisthecriterionofdecoherence,ratherthananynotionof“measurement”,whichdeterminestheconsistencyofthequantitativepredictionsofthetheory.

Thispaperappliessum-over-historiesgeneralizedquantumtheorytoaclassofhomoge-neousminisuperspacecosmologicalmodels.1OtherformulationsofthequantummechanicsofhomogeneouscosmologicalmodelshavebeenproposedbyAshtekar,Tate,andUgla[8],byWald[9](seefurther[10]),andbyMarolf[11].Animplementationofgeneralizedquantumtheoryfortheseminisuperspacemodelsisthereforeusefulforcomparisonwiththeseotherapproaches.ThatisespeciallythecasesincetheformulationofWaldmakesthesurprisingpredictionthataquantumuniversewhoseexpansionisaccuratelypredictedbyclassicalEin-steindynamicscannothaveanearlyclassicalrecontractingphase.(Instead,timeappearsto“freeze”astheuniverseapproachesitsclassicalmaximumsize.)Inthispaperwewillshowthat,insuitablecircumstances,sum-over-historiesgeneralizedquantumtheorypredictsthatauniversemayremainclassicalinbothitsexpandingandrecontractingphases.

SectionIIrevisitstheclassicalhomogeneouscosmologicalmodelsasanintroductiontotheirlaterquantization.SectionIIIsetsoutthesum-over-historiesgeneralizedquantumme-chanicsfortheclassicalmodelsofSectionIIfortheespeciallysimplecaseofaclosedBianchi

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

IXcosmologywithasinglehomogeneousscalar eldandvanishingcosmologicalconstant.(CurrentobservationssuggestthatΛissmallbutnotzero.AssumingΛvanishes,whileev-identlynotrealistic,simpli essomeelementsoftheanalysisbyallowingquantumevolutiontoberestrictedtouniverseswhichalwaysrecontract.WhilethequantizationframeworkdescribedinSectionIIIisinprincipleapplicabletoalltypeABianchicosmologies,thecasewheretheuniverseisallowedtoexpandforever–eitherclassicallyorquantummechanically–willnotbeconsideredhere.)SectionIVexplicitlyconstructsthedecoherencefunctionalforthesemodelsforaspeci cclassofboundaryconditions:a“pure”initialstateand“in-di erent” nalconditions.InSectionVweexamineitssemiclassicalpredictionsforinitialconditionsthatcorrespondtoasingleclassicaltrajectoryandshowhowclassicalevolutioncanbeanapproximationtoquantummechanicalevolutioninauniversewithexpandingandcontractingphases,aswellasstudymoregeneralchoicesofinitialstate.

Generalizedquantumtheory[12,13]isabroadframeworkfordescribingandcompar-ingdi erentformulationsofquantummechanics.Areductiontoessentialsofthegeneralprinciplesofthequantummechanicsofclosedsystems[14–16],theframeworkprovidesanaturallanguagewithwhichtoframequestionsincosmologyconcerningwhetherproba-bilitiesareconsistentlyassignedbyquantumtheorytoasetofalternativehistoriesoftheuniverse.Thespeci csum-over-historiesimplementationofitsprinciplessketchedaboveisbutoneofseveralapproachestoaconceptuallycoherentandmanageablequantumtheoryofspacetime;forlucidreviewsofsomeofthemandthedi cultiestheyencountersee[17].II.HOMOGENEOUSCOSMOLOGICALMODELS

Heretheσiaret-independentspatialone-formspreservedbytheisometrieswhosedualvector eldsσiobey

[σi,σj]=ck(2.2)ijσk,

wheretheckijarethecomponentsofthestructuretensoroftheLiealgebraoftheisometrygroupintheσbasis.3ThequantitiesL(t)andα(t)arefunctionsoftalone;β(t)isa3×3tracelesssymmetricmatrixthatmeasuresthe√deviationsfromisotropy.ThecoordinatevolumeelementofthespatialslicescaleslikeInthissectionwereviewtheessentialfeaturesofhomogeneouscosmologicalmodelsnec-essaryforthesubsequentdiscussionoftheirquantization.2Aspatiallyhomogeneouscosmo-logicalgeometryisaspacetimepossessingagroupofisometrieswhoseorbitsareafamilyofspacelikesurfacesthatfoliatethespacetime[20].Usingacoordinatetthatlabelsthesespacelikesurfaces,themetricofaspatiallyhomogeneousspacetimemaybeputinthestan-dardform[20,21] ds2= L2(t)dt2+e2α(t)e2β(t)ijσiσj.(2.1)

2

3Aclassicalgeneralreferenceis[18].Extendeddiscussionscanbefoundin[5,19]andinChap.7of[20].SeeWald[20,section7.2]forexample.Toavoidpossibleconfusion,notethatwhileWald’sσ’scoincidewiththeω’sofMacCallum[5],MacCallum’sstructuretensorcisde nedwiththesignoppositetothatofWald.

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

“BianchiI”through“BianchiIX”models.The(simply-connectedcovering)groupmanifoldwithitsnaturalmetricisthemanifoldforspatialgeometry.IntheBianchiImodels,forexample,thegroupisgeneratedbythetranslationsofthree-dimensional atspaceandthemanifoldisR3.IntheBianchiIXmodelsthegroupisSU(2),ckij=εijk(inappropriatecoordinates),andthemanifoldisthethree-sphereS3.TheclassicalFriedmann-Robertson-Walker(FRW)modelsarethemostfamiliarexamplesofBianchiuniverses:theopenFRWuniverseisofBianchitypeI,the atuniversetypeV,andtheclosedFRWuniverseisoftypeIX.

The“typeA”Bianchimodelsarethoseforwhichciij=0;therestarecalled“typeB”.WerestrictattentiontotypeAmodelsbecausetheactionprincipleforthetypeBBianchimodelsdeducedfromthatforgeneralrelativitybysubstitutionofthehomogeneityansatz(2.1)doesnotleadtothecorrectequationsofmotion[24];inthesecasesthehomogeneityofthespatialmetricobstructstheeliminationofboundarytermsproportionaltothetraceofthestructuretensor.TheBianchitypesI,II,VI0,VII0,VIII,andIXarealltypeA.Avarietyofmattercontentsareconsistentwithhomogeneity.Asanillustrativeexampleweshallrestrictattentiontoasingle,minimallycoupled,homogeneousscalar eldφ(t)togetherwithapositivecosmologicalconstantΛ.Fortheactionofthescalar eldwetake

SM[g,φ]= 1 g( φ)+Vφ(φ) 2

forsomepotentialVφ(φ).[Wefollow,asfaraspossible,theconventionsof[20]withrespecttosignature( ,+,+,+),de nitionsofthecurvaturetensors,theextrinsiccurvatureofahypersurface,etc..Weemployunitswhere =c=G=1.]

AcanonicalactionforthetypeABianchicosmologieswithscalarmattermaybearrivedatinthefollowingmanner.FirstnotethatforallthetypeAmodelswith“diagonal”matter(T0i=0),asinourexample,itispossibleclassicallytochoosetheone-formsσiin(2.1)sothatthematrixβijisdiagonal,andtheckijtaketheircanonicalvalues[18,19];theEinsteinequationsguarantee thatβ√remainsdiagonalastime passes.Itisthentraditionaltoparametrizeβasβ=diagβ++3β , 2β+.

Assumingthatthemetricremainsdiagonalquantummechanicallyisequivalenttosolvingtheclassicalmomentumconstraintsbeforequantization[10](cf.[20,equationE.2.34]);theonlyremaininggaugefreedomthenliesinthetime-reparametrizationsofthesurfacesofhomogeneity.Weshallassumethatthemetricremainsdiagonalinthesequel.However,thequantumtheorythusobtainedisnotobviouslyequivalenttoatheoryinwhichthemomentumconstraintisinsteadimposedasanoperatorcondition.4 Insertingadiagonalhomogeneousmetric(2.1)intotheactionS=(1/16π)(R 2Λ)+SManddoingthespatialintegrationoverastandardcoordinatevolumeof(4π)2(thecoordinatevolumeofBianchiIX’sSU(2)closedspatialmanifold;cf.[25,Box30.1])yieldsthereducedminisuperspaceaction

3α ˙++p β˙ +pφφ˙ LeS=dtpαα˙+p+β

4 (2.3)Theissueisthatwhilethemomentumconstraintimpliesthatthespatialmetricmayalwaysbediagonal-izedatanyonemomentoftime,theclassicalequationsofmotionarerequiredtoshowthatitremainssothereafter.TheseissuesarediscussedwithcharacteristiclucidityinsectionIVof[10].

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

whereHisthesuper-Hamiltonian

2224α6α6αH= p2α+p++p +pφ+eVβ(β+,β )+eVφ(φ)+eΛ

afterrescalingφ,Vφ,andΛbypositiveconstants((2.5)

2β4βtr1 2e+e,becauseVais3positivede nitewithaglobalminimumof0atβ=0,andistriangularlysymmetricabout

β=0.)ThehomogeneousisotropicFRWuniversemayberecoveredbysettingβ=0intheequationsforBianchiIX.Forthem(3)R=(3/2)e 2αandVβ= (6π)2.(NotetheusualFRWscalefactora=2eαafterchangingfromtheEuler-angleσibasistothestandardpolarcoordinatesonS3.)

Thecon gurationspacefortheseminisuperspacemodelsisspannedbythevariables(α,β+,β ,φ).Thatis,itisthesuperspaceofspatialgeometriesplusthespaceofscalar eldvalues.Theaction(2.4)maybeexpressedinamorecompactformbyintroducingthenotationqA,A=0,1,2,3forthefourvariables(α,β+,β ,φ),the atLorentzianDeWittmetricGAB=diag( 1,1,1,1),andarescaledlapsefunctionN(t)=e 3α(t)L(t)/24π.Then

1 AASpA,q,N=dtpAq˙ NH(2.8)

where

H=GABpApB+e4αVβ(β+,β )+e6α[Vφ(φ)+Λ]

=GABpApB+V(α,β+,β ,φ,Λ),(2.9)

andwehavetakenadvantageofthearbitrarinessintasacoordinatelabeltoassignthevalues0and1totheendsoftherangeofintegration,achoiceweshallmakewhereverconvenient.Inthisformtheanalogywitharelativisticparticlemovinginapotentialisclear[1,28]andweshallexploitthisinwhatfollows.

Theformofthemetric(2.1)isleftunchangedbyreparametrizationsofthetimet→f(t).Asalreadynoted,thisinvarianceistheremnantofthefour-dimensionaldi eomorphisminvarianceofthefulltheoryofgeneralrelativityoncethediagonalform(2.1)hasbeen xed.Correspondinglytheaction(2.8)isinvariantunderreparametrizationtransformationsoftheform

qA(t)→q A(t)=qA(f(t)),

pA(t)→p A(t)=pA(f(t)),

(t)=N(f(t))f˙(t),N(t)→N

(2.10a)(2.10b)(2.10c)

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

betweencoordinatesandtheirconjugatemomenta.VariationwithrespecttothepAandqAgiveEinstein’sequationsofmotion.Thecharacterofthesolutionstotheseequations—thepossibleclassicalhistories—dependsontheisometrygroupandthevalueofthecosmologicalconstantΛ.Forexample,inthecaseofΛ=0BianchiImodelsweexpectcosmologiestoexpandforeverfromaninitialsingularity.Moreprecisely,weexpecttheextrinsiccurvatureK=(3/L)dα/dt(traceoftheextrinsiccurvaturetensor)ofthet=constantsurfacesofhomogeneitytoremainpositivetothefutureofaninitiallysingularsurface.BianchiIclassicalsolutionswithΛ>0alsoexpandforever,whilemodelswithΛ<0alwaysrecollapse.BianchiIXmodelshaveclosedspatialsectionswiththree-spheretopology.TheΛ=0BianchiIXuniversesarejusttheanisotropicgeneralizationsoftheclosedFRWuniverse.Moreprecisely,LinandWald[29]haveshownthatwhenthedominantenergyconditionissatis edandthetraceofthespatialprojectionofthestress-energytensor(i.e.thesumoftheprincipalpressures)ispositive–thusexcludingacosmologicalconstant–therearenoclassicalsolutionswhichexpandforeverfromaninitialsingularityinthesensethattheextrinsiccurvatureofconstanttsurfacesremainspositive.

Thestress-energytensorarisingfrom(2.3)satis esthedominantenergyconditionsolongasVφ≥0,thoughthepressuresmaybenegativeifthepotentialenergyinthescalar eldexceedsthekineticenergy.(Indeed,itispreciselythisfeaturewhichallowsascalar eldtomimicacosmologicalconstantinin ationarymodels.)Thus,scalar eldsonlysatisfytheconditionsoftheLin-Waldrecollapsetheorematlatetimesforcertainchoicesofscalarpotential–afree,massless,minimallycoupledscalar eld,forexample.ThegeneralconditionsonVφforwhicharecollapsetheoremholdsareasfarasweareawarenotcurrentlyknown.

WithacosmologicalconstanttheconditionsoftheLin-Waldtheoremarenotsatisi ed.TheexampleofdeSitterspaceisenoughtoshowthattherewillbeBianchiIXsolutionswithvanishingscalar eldwhichevolvenon-singularlyfromaninitialcontractingphase(K<0)inthein nitepasttoanexpandingphase(K>0)inthefarfuture.Inbetweenthevolumeoftheuniversereachesanon-zerominimumvalue.Theinclusionofasmallamountofhomogeneousscalar eldwouldnotbeexpectedtodisturbthisbehavior.However,thehomogeneous,isotropicBianchiIX(Friedmann)modelsshowthatapositivecosmologicalconstantdoesnotremovethesingularityineverycase.Therearealsomodelsinwhichthestress-energyofthescalar elddominatesthatofthecosmologicalconstantthroughoutthemodel’shistory.SuchmodelsdisplaythequalitativefeaturesoftheΛ=0case—aninitialsingularityleadingtoa niteexpansionfollowedbyrecontraction,formodelssatisfyingtheconditionsoftheLin-Waldtheorem.See[7]forawide-rangingsurveyofthedynamicsofthevariouscosmologicalmodels.solongasf(0)=0andf(1)=1.Variationof(2.8)or(2.4)withrespecttothemultiplierNgivestheconstraint HpA,qA=0(2.11)

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

III.GENERALIZEDQUANTUMMECHANICSOFΛ=0BIANCHIIXUNI-VERSES

A.GeneralizedQuantumTheory

InthissectionwedescribeageneralizedquantumtheoryforBianchiIXminisuperspacecosmologicalmodelswithΛ=0.Whiletheconstructionisinprinciplevalidevenformodelswhichmayexpandforever,certaintechnicalaspectsoftheanalysisaresimpli edbyrestrictingattentiontoquantumhistorieswhichalwaysrecontract,andthemostgeneralcasewillnotbeconsideredhere.

Weworkwithinthegeneralprinciplesofthequantummechanicsofaclosedsystem[14–16].Theclosedsystemismostgenerallyandaccuratelytheuniverseasawhole.Themostgeneralpredictionsofquantummechanicsaretheprobabilitiesofindividualmembersofsetsofalternativecoarse-grainedhistoriesoftheclosedsystem.Probabilitiesarenotpredictedforeverysetofalternativehistories,butonlythoseforwhichthequantummechanicalinterferencebetweentheindividualhistoriesinthesetisnegligibleasaconsequenceofthesystem’sboundaryconditionsanddynamics.Suchsetsofhistoriesaresaidtodecohere,orbeconsistent.

Generalizedquantumtheory[1,12,13]isacomprehensiveframeworkforimplementingtheprinciplesofthequantummechanicsofclosedsystems.Asnotedintheintroduction,thefollowingelementsspecifyageneralizedquantummechanics:(1)Thesetsof ne-grainedhistorieswhicharethemostre neddescriptionofthesystempossible.(2)Theallowedcoarse-grainingswhichgenerallyarepartitionsofasetof ne-grainedhistoriesintoanex-haustivesetofmutuallyexclusiveclasses{ch},h=1,2,3···calledcoarse-grainedhistories.(3)Adecoherencefunctional,D(h,h′)thatmeasurestheinterferencebetweenpairsofhis-toriesinacoarse-grainedset.Thedecoherencefunctionalisacomplex-valuedfunctionalonpairsofclassesthatsatis escertaingeneralrequirements:Itis(i)Hermitian,(ii)nor-malized,(iii)positive,and(iv)consistentwiththeprincipleofsuperpositioninsensesmadeprecisein[1,12,13].Thedecoherencefunctionalincorporatesaspeci cationofthebound-aryconditionsfortheclosedsystem—typically“initial”and“ nal”conditions.Itisanaturalgeneralizationtoclosedsystemsoftheideaofquantumstate,asthetermisusedinquantumlogicandinalgebraicquantummechanics,tomeasurethequantuminterferencebetweenhistoriesinadditiontotheirprobabilities[2,3].

Withthesethreeelementsspeci ed,theprocessofpredictionproceedsasfollows:Asetofalternativecoarse-grainedhistories(approximately,medium)decohereswhenD(h,h′)isnegligibleforallh=h′.Theprobabilitiesp(h)oftheindividualhistoriesinadecoherentsetarethediagonalelementsofD.Therulesfordecoherenceandprobabilitiesarethussummarizedbythefundamentalformula

D(h′,h)≈p(h)δh′h(3.1)

obeyedbyhistoriesindecoheringsets.

Thefour-dimensionaldi eomorphisminvariancecharacteristicofageometrictheoryofgravityismosteasilyaccomodatedbyemployingasum-over-historiesformulationofquan-tummechanics.Asum-over-historiesquantummechanicspositsauniquesetof ne-grainedhistorieswhichinthecaseofgravityarefour-dimensionalspacetimemetricsandmatter eldcon gurations.Asum-over-historiesgeneralizedquantummechanicsforcosmologywasde-scribedin[1].Generalizedquantummechanicsformodelswithasinglereparametrization

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

invariancewasdescribedaswell.5Wenowapplythosediscussionstohomogeneous,minisu-perspacecosmologicalmodelswhich,asformulatedhere,possessasingletimereparametriza-tioninvariance.Todothatwespecifyexplicitlythethreeelementsoftheirgeneralizedquantummechanics.

1.Fine-grainedHistories

Wetakeforthesetof ne-grainedhistoriesthepathszM(t)=(qA(t),N(t)),M=0,...,4intheextendedcon gurationspaceCextofqA=(α,β+,β ,φ)andmultiplierz4=N.Wearethusconsideringasum-over-historiesquantummechanicsinwhichthereisaunique ne-grainedsetofhistories.Weputnorestrictionofsingle-valuednessonthepaths.Thus,forexample,thetotalspatialvolume(4π)2exp(3α(t))mayincreaseanddecreaseoverthecourseofthehistoryand,indeed,gothroughsuchcyclesanarbitrarilylargenumberoftimes.Inparticular,theclassicalhistoriesofaBianchiIXuniversewhichexpandandrecontractareincludedamongthepossiblequantummechanicalhistories.

Weputnorestrictiononthedi erentiabilityofthepaths,butdorequirethemtobecontinuous.

The ne-grainedhistorieshaveendsatwhichthecosmologicalboundaryconditionsanal-ogoustoinitialand nalconditionsareimposed,andtheseendsmustbeprescribedtocompletethespeci cationofthesetof ne-grainedhistories.Anaturalprinciplerestrictingthischoiceisthatthesetof ne-grainedhistoriesshouldincludealltheclassicalhistories.OtherwisethereisnohopeofrecoveringEinstein’sclassicaltheoryasasuitablelimitofquantumtheory.(ThiswillbethesubjectofSectionV.)TheworkofLinandWald[29]discussedattheendofsectionIIshowsthatallclassicalBianchiIXcosmologieswithscalar eldandΛ=0expandfromaninitialsingularityofvanishingthree-volumeandeventuallyrecontracttoa nalsingularityofvanishingthree-volume,assumingthatthepotentialen-ergyinthescalar elddoesnotdominatethekineticatlatetimes.Weshallcon neourattentiontopotentialsforwhichthisisso.Classicalhistoriesmaythereforebethoughtofasbeginningandendingonasurfaceσ0oflargeconstantnegativeα(=α0)inthecon gurationspaceofpaths.Theclassofallpathsthatbeginandendonsuchasurfaceisthereforethenatural,minimalsetof ne-grainedhistoriesforageneralizedquantumtheoryofBianchiIXminisuperspacecosmologicalmodelswithscalar eldandΛ=0.ThiswasthechoiceadvocatedbyTeitelboiminhistheoryofquantumcosmological“scattering”betweeninitialand nalsingularities[30].Itisthechoiceweshalladopthere.(Restrictingthepathsinthiswaycorrespondstotheimpositionofaboundaryconditionthatwavefunctionsvanish

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

asα→∞.)Finally,weaddtherestrictionthatallpathspossessα˙(0)>0andα˙(1)<0onσ0,i.e.“expanding”initialand“contracting” nalconditions.

2.Coarse-grainedHistories

Coarse-grainedhistoriescorrespondtothephysicalquestionsthatmaybeaskedofasys-tem.Wethereforeallowascoarse-grainedsetsofalternativehistoriesanypartitionofthe ne-grainedhistoriesintoreparametrizationinvariantclasses{ch},h=1,2,···,becausereparametrizationinvarianceiswhatremainsinminisuperspaceofthedi eomorphismin-varianceofgeneralrelativity.TheclassesgenerallymaybethoughtofaspartitionsbyvaluesofreparametrizationinvariantfunctionalsF[qA,N]ofthepathsinCext.Explicitly,foranexclusivesetofranges{ h},h=1,2,···oftherealline,andasinglefunctionalFwede ne A Ach=(q(t),N(t))|F[q,N]∈ h.(3.2)Anypartitionmaybethoughtofasofthisformbecausewemayalwaysconsiderthefunc-tionalwhichhasthevaluehforpathsinchandasetofrangesthatbrackettheintegers.Simpleexamplesofinterestingpartitionsintodi eomorphisminvariantclasses–herereducedtoreparameterizationinvariantclasses–arereadilygiven:

Onecouldpartitionthehistoriesbyrangesofvaluesofthevolumeofthelargestvolumethreesurfaceofhomogeneity.Theresultingprobabilitiesareforthevaluesofthevolumeofmaximumexpansionoftheuniversereachedinthecourseofitsexpansionandcontraction.

Onecouldpartitionthehistoriesofthesehomogeneousspacetimesintotheclasseswhichhaveasurfaceofhomogeneitywithavolumelessthan andtheclassofthosewithnosuchsurface.Thevalueoftheprobabilitythatthereisasurfacewithvolumelessthan consideredas →0wouldbeonewayofassigningaprobabilitytotheuniversebecomingsingular.6

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

OnecouldpartitionthehistoriesintotheclasswhichremainclosetoasolutionoftheclassicalEinsteinequationbysomestandard(cf.(5.3))andtheclasswhichexhibitasigni cantexcursionawayfromclassicalbehavior.Theprobabilityofthe rstclassistheprobabilitythattheuniversebehavesclassicallyaccordingtothegivenstandard.WeshallemploysuchcoarsegrainingsinSectionV.

Onecouldpartitionthe ne-grainedhistoriesbyrangesofvaluesofthevolumeand anisotropyβ±theyassumeagiven“propertime”7Ldtaftertheinitialcondition.Theresultingprobabilitieswouldbetheprobabilitiesforthevolumesandanisotropiestheuniversecouldhaveatagivenpropertimefromtheinitialsurface.Thesearenotunliketheprobabilitiesthatwouldbeofinterestincomparingthepredictionsofarealisticquantumcosmologywithobservation.

Allofthecoarsegrainingsmentionedaboveareintomanifestlyreparametrizationinvari-antclassesof ne-grainedhistories.Mostarepartitionsthatarenotde nedbyalternativesthatareinanysensealternatives“atonemomentoftime”,andhencearenotde nedsimplybyobservablesonsuperspace.Rathertheyarespacetimealternativesreferringtopropertiesofhistoriesextendedovertime.Forexample,theprobabilitiesforclassicalbehaviorrefertowhetherasuitablycoarse-grainedhistoryapproximatelyobeystheEinsteinequationoveracourseoftime.

Coarse-grainedhistoriescorrespondtothephysicalquestionsthatmaybeaskedofasys-teminthefollowingsense.Askfortheprobabilitythattheuniversehasanyreparametriza-tioninvariantpropertyexpressibleinspacetimeterms.8Toanswerthisquestiononecon-sidersthepartitionofthe ne-grainedhistoriesintotheclasswhichhavethepropertyandtheclasswhichdonot.Ifthissetofcoarse-grainedhistoriesdecoheres,thenthequantummechanicspredictstheprobabilitythattheuniversehasthepropertyinquestion.Ifonecannottellwhetheragiven ne-grainedhistoryhasthepropertyornottheneitheritdoesnotmakesenseoritisnotexpressibleintermsofmetricand/ormatter eldcon gurationsalone.

Whileitiseasytoexhibitphysicallyinterestingsetsofalternativereparametrizationin-variantcoarse-grainedhistoriesforthesehomogeneousminisuperspacecosmologicalmodelsitismuchharderto ndsetsofsuchhistoriesthatdecohere.Thatisbecauseofthesmallnumberofdegreesoffreedomofthemodel.Coarse-grainingisessentialfordecoherence.Realisticmechanismsofdecoherencethataree ectiveinavarietyofinitialconditionsqual-itativelyinvolvethedissipationofphasesfromvariablesfollowedbythecoarse-grainingintovariablesthatareignored.9However,thepresentminisuperspacemodelsdonotpresentmanydegreesoffreedomtobeignored!

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

3.TheDecoherenceFunctional

wherethesumisoverpathsinCextthatlieintheclasschwhichbeginatq′andendatq′′.Sisthe(Lagrangian)actionforpaths—afunctionalofq(t)andN(t).Inordertogiveade nitemeaningtothefunctionalintegralin(3.3),and,inparticular,to xthe“measure”onthespaceofpaths,itisconvenienttoconsiderthecorrespondingintegraloverpathsinphase-space,

q′′A q′′ Ch q′ =δN≥0δpδqq(3.4)′ G[q,N]δG[q,N]expiSpA,q,N

chThedecoherencefunctionalisconstructedintwostepsfollowingtheanalogyoftherela-tivisticparticlediscussedin[1];the nalresultisexpressedin(3.9).Itwillturnouttobeanaturalgeneralizationtoareparametrizationinvarianttheoryofthecanonicaldecoherencefunctionalofordinaryquantummechanics.Firstwede nematrixelementsofclassoperatorscorrespondingtoindividualcoarse-grainedhistorieshbyasum-over-historiesintheclass.Schematically,wede ne ′′′expiS[path](3.3) q Ch q = ′′ paths∈q′hq

Theingredientsinthisexpressionareasfollows:Sistheaction(2.8).GisafunctionsuchthattheconditionG=0 xesauniquerepresentativefromeachreparametrizationinvariantequivalenceclassof ne-grainedhistories; GistheassociatedFaddeev-Popovdeterminant.TheintegralisoverallpathsinCextthatlieintheclasschandpassbetweenthecon gurationspacepoints(qA)′and(qA)′′.Allpossiblemomentumpathsareintegratedover.WetakethemultiplierfunctionalintegraltobeoverpositivevaluesN≥0.Otherchoices,forexample,bothpositiveandnegativevalues,wouldleadtodi erentgeneralizedquantumtheories.(Someoftheissuesthatariseinchoosingtheallowedrangeforthelapsearediscussedin[38–41]and[1,sectionVII].)ThemeasurefortheqandpintegrationsistheusualLiouville“dqdp/2π”measureonphase-spacepaths.TheintegrationovermomentamaybeseenasadevicetoinducethemeasureonpathsinCextfromtheLiouvillemeasuresincetherangeofintegrationoverthemomentaareconstrainedneitherbytheclassch,theendpoints(q′,q′′),orthegauge xingdelta-function.Theskeletonizedpathintegralsweenvisionarequitestandard[30,39],andhavealreadybeenbrie ydescribedin[1].

Thedecoherencefunctionalisconstructedfromtheclassoperators(3.4)whichchar-acterizethephysicalhistoriesinquestion,butalsoincorporatesatheoryofcosmologicalboundaryconditionsimposedattheendsofthehistoriesthataretheanalogsofinitialand nalconditionsinaquantumsystemwitha xednotionoftime.

Aninitialor nalconditionisrepresentedbyasetofcon guration-spacewavefunctionstogetherwithaprobabilityforeachwavefunction.Thatisthesamekindofinformationneededtospecifyaninitialor naldensitymatrixinordinaryquantummechanics.Thewavefunctionsarerequiredtosatisfyanoperatorimplementationoftheconstraints.Thus,forhomogeneouscosmologicalmodelsaninitialconditionisspeci edbyaset{Ψi(qA),p′i},whereeachΨi(qA)satis esanoperatorformof(2.11).Wetaketheobviousoperatorordering10

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

andwrite HΨi(q)= GAB 2

α

Havingintroducedtheproduct wenowde neΨ(α, q).(3.7)

Φi|Ch|Ψj =Φi(q′′) q′′ Ch q′ Ψj(q′).(3.8)

Thisde nitionappearsat rstsighttodependonthechoiceofsurfaceonwhich isde ned,butinfactitdoesnot,solongasthecoarsegrainingchdoesnotrestrictthepathsonthosesurfaces.Thisisbecausetheclassoperators(3.4)forsuchcoarsegrainingsgenerallysolve(3.5)aswellasΨandΦ,afactwedemonstrateintheappendix.First,however,wecompletethede nitionofthegeneralizedquantumtheoryofhomogeneousminisuperspacecosmologies.

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

ThedecoherencefunctionalD(h′,h)isde nedthrough

′′′pi Φi|Ch′|Ψj p′j Φi|Ch|Ψj D(h,h)=N

i,j(3.9)

whereNisanormalizingfactordeterminedsothatΣh′hD(h′,h)=1.Speci cally,ifuistheclassofall ne-grainedhistories

′′ 1N=pi| Φi|Cu|Ψj |2p′j.(3.10)ij

Thedecoherencefunctionalde nedby(3.9)isthedirectanalogueofthe“canonical”decoherencefunctionalofordinaryHamiltonianquantummechanicswithinitialand nalboundaryconditions[1,3],writteninfunctionalintegralform,withaccomodationsappro-priatetothereparametrizationinvarianceofthepresenttheory.Itsatis esthegeneralcon-ditionsrequiredofageneralizedquantumtheory.Itis(i)Hermitian,D(h′,h)=D(h,h′) ,(ii)normalized,Σhh′D(h′,h)=1,(iii)positiveonthediagonalelements,D(h,h)≥0,and,(iv)consistentwiththeprincipleofsuperpositioninthesensethatif{c¯h¯}isapartitionoftheclasses{ch}intocoarserclasses,then ′¯¯D(h,h)=D(h′,h)(3.11)

¯′h∈h¯h′∈h

Thesefourconditionsareenoughtoensurethatforsetsofhistoriesthatdecohereaccordingto(3.1),thenumbersp(h)de nedby(3.1)areprobabilitiessatisfyingthemostgeneralformoftheprobabilitysumrules.Byusingin(3.1)thespeci cform(3.9)wecanassesstheprobabilitiesofalternative,coarse-grained,decoheringhistoriesofthemodelhomogeneouscosmologiesunderdiscussion.

B.EvaluationoftheClassOperatorsintheProperTimeGauge

Weapply(3.1)topredictionsconcerningthesemiclassicalbehaviourofhomogeneousminisuperspacecosmologiesinthenextsection.Toendthissection,wediscusstheeval-uationoftheclassoperators(3.4)inaparticularlyconvenientgauge–(3.13)–calledthe“propertime”gauge.Forsuitablecoarsegrainings,wealsobrie yarguethatthesematrixelementssatisfytheconstraint(3.5).Becausetheaction(2.8)isessentiallythatofarel-ativisticparticleinapotential,thetreatmentcloselyparallelsthatofthefreerelativisticparticlethathasbeengivenpreviouslyin[1,section7].

The rststepinevaluating(3.4)istochoosea“gauge”that xesthereparametrizationsymmetry(2.10a-2.10c),thein nitesimalformofwhichisinvarianceunderthechanges(setf(t)=t+ /N)

δqA= (t){qA,H}

δpA= (t){pA,H}

δN= ˙(t),(3.12a)(3.12b)(3.12c)

where{,}isthePoissonbracket,and (0)= (1)=0.Aconvenient“gauge” xingfunctionis[38,40](seealso[1,section7])˙G=N.(3.13)

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

TheFadeev-Popovdeterminant G=detδG/δ ～detd2/dt2isinthiscaseindependentoftheintegrationvariables.Thedeltafunctionalin(3.4)thenpermitsonlytheN=constantpathstocontributetotheintegraloverN(t),leaving

1 ′′q q′′ Ch q′ =dN≥0δpδqqidt[pAq˙A NH](3.14)′expch0

afterdroppingtheconstantfactorsthatcancelinthedecoherencefunctional(3.9).Changingvariablesto

ds=Ndt,(3.15)

theGaussianfunctionalintegralsoverpmaybeperformed,leavingsimply

∞

q′′ Ch q′ =dN q′′N Ch q′0

0(3.16)

solongasweassumethecoarsegrainingdoesnotrestrictthevalueofN;otherwisetherangeoftheNintegralmustberestrictedappropriatelyaswell.Herewehavede ned

1 1q′′ q′′N Ch q′0 =δqqidt′exp

ch0 AdqGAB V,(3.17b)4ds

wherethepathintegralmeasureδqhasbeenrenormalizedintheusualmannerinducedbythemomentumintegrations.(Seetheappendixfordetails.)Thenotationonthelefthandsideof(3.17a)isinspiredbytheobservationthatthepathintegralin(3.17b)sharestheformofthatforthepropagatoroveratimeNofarelativisticparticleinapotentialV.(3.16)maybethoughtofasa“restrictedpropagator”fortheclassofpathsch,asshouldbeevidentfromtherestrictedfunctionalintegral(3.4).

Ingeneral,thecomplexityoftheminisuperspacepotentialVprecludesmuchfurtherex-plicitprogressinthenon-perturbativeevaluationof q′′ Ch q′ .Itisstillpossible,however,toshowthattheclassoperatorssatisfytheconstraint(3.5)forcoarsegrainingswhichdonotrestrictthevaluesoftheendpointsq′andq′′orthevalueofN.11Thisisdoneintheappendix.(3.8)isthusasclaimedindependentofthesurfacesonwhichwechoosetoimposetheboundaryconditionsΨandΦ.

IV.DECOHERENCEFUNCTIONALFORRECOLLAPSINGBIANCHICOS-MOLOGIES

ThegeneralformofthedecoherencefunctionalfortypeAhomogeneouscosmologiesisgivenin(3.9).Inthissectionweemploysomespeci cchoicesoftheinitialand nalconditionsapproriatetoΛ=0closed(BianchiIX)cosmologiestocastthedecoherencefunctionalintoasimplerandmorepracticallyusefulform.

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

A.InitialandFinalConditions

Asalreadynoted,itistobeexpectedthattheinitialquantumconditionsoftheuniverseare xedbysometheoryofcosmologicalboundaryconditions.Knowledgeofthistheoryisnot,however,requiredintheconstructionofthedecoherencefunctionalforcosmology.Weshall,therefore,illustratetheprocessofpredictionbasedonthedecoherencefunctional(3.9)withsomesimplechoicesofinitialstate.Thepracticalsigni canceofsuchpredictionsdependsentirelyonwhetherthechoseninitialstatesarerepresentativeoftheboundaryconditionsontheactualphysicaluniverse.

Mostoftheextanttheoriesoftheinitialstateoftheuniverse[6,42]produceabound-arystateconsistinginasingleinitialwavefunction.WewillthereforeinourexamplesconcentrateentirelyonthecaseofapureinitialstateΨ.

RecallfromSectionIIIA1thattheBianchiIXcosmologicalhistorieshave“ends”atwhichweimposeboundaryconditions{Ψi,p′i}and{Φi,p′′i}.Inordertocorrespondtoaconventionalnotionofcosmologicalboundaryconditionsforclosed,Λ=0universes,weimposetheboundaryconditionsonasuitablesurfaceσ0oflarge,negativeα(=α0),i.e.whentheuniverseisverysmall.BecausethewavefunctionsΨiandΦisatisfytheconstraint(3.5),itdoesnotmatteronwhichsurfacetheyareimposedsolongasthecoarsegrainingsunderconsiderationdonotinvolveregionsofminisuperspaceintersectingthosesurfaces,asnotedabove.Inthissense,then,thecosmologicalhistories“begin”and“end”atsmallspatialvolume,justastheclassicalhistoriesdo.(SeeFigure1.)

Maintainingaclosecorrespondencewithordinaryquantummechanics,weshallchoose nalboundaryconditionswhichare“indi erent”tothepaths,inasensetobemadepreciseinSectionIVC.Therewillthene ectivelybeno nalconditionsatallonthecosmologicalhistories.

Asforthechoiceoftheinitialboundaryconditions,wewillexaminetwoinstructiveexamplesinthesequel.Inordertoillustratehowthepresentsum-over-historiesconstructioncanpredictsemiclassicalbehaviourinasuitablelimitasnaturallyassum-over-historiesformulationsalwaysdo,asa rstexampleweconsiderasingleinitiallocalizedWheeler-DeWittwavepacketΨWPwhichisdesignedtopreferaparticularclassicalpathoverallothersinthesemiclassicallimit.Inapartitionoftheminisuperspacehistoriesbyclassicalpaths,asteepestdescentsevaluationofthepathintegralfortheclassoperatorsthenrevealsthattheprimarycontributiontothedecoherencefunctionalcomesfromtheclassoperatorcorrespondingtothecoarse-grainedclassofpathscontainingtheclassicalpathpreferredbytheinitialconditionΨWP.

Next,weshallturnourattentiontothesemiclassicalpredictionsofthedecoherencefunctionalwithaninitialstateofthemoretraditionalWKBform.Asistobeexpected,undersemiclassicalcoarsegrainingssuchachoiceofinitialstateleadstoadistributionofclassicaltrajectorieswithinitialvalueweightedaccordingtotheWKBprefactor,andinitialmomentumgivenbythegradientofthephase.

Letusnowproceedwiththedetails.

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

α

0α0FIG.1:Theminisuperspaceofhomogeneouscosmologicalmodels.Thetimelikecoordinateq0=αandonespacelikecoordinate(e.g.q1=β+)areplotted.Finegrainedhistoriesarepathsinthiscon gurationspacewhichbeginandendonasurfaceσ0thatisspacelikeintheWheeler-DeWittminisuperspacemetricGABandlocatedatlargenegativeα(=α0,say),correspondingtoasmallspatialvolume.Therestrictiontopathswhichbeginandendonσ0correspondstotheimpositionofaboundaryconditionthatsolutionstotheWheeler-DeWittequationvanishasα→∞.Thepathsallpossessα˙(0)>0andα˙(1)<0onσ0,i.e.“expanding”initialand“contracting” nalconditions.Theuniverseexpandsfromasmalltoamaximumvolumeandthenrecontracts.

B.BranchWaveFunctions

First,itisusefultode nethe“branchwavefunctions”

Ψh(q)= q Ch|Ψ ≡ q Ch q′ Ψ(q′).(4.1)

Thesemayberegardedasthewavefunctionscorrespondingtotheinitialstate,propa-gatedbytherestrictedpropagator(classoperator)correspondingtothephysicalhistoryinquestion.12

Thespace{Ψh(q)}ofbranchwavefunctionswillbedenotedB.ItdependsinanessentialwaybothontheallowedspaceofinitialwavefunctionsΨ(q),andontheallowedcoarsegrainings.

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

Forarbitrarycoarsegrainings,itisnotimmediatelyevidentthatΨh(q)forq∈σ0mustbeinthespaceF(σ0)offunctionswhichfallo rapidlyatin nityonσ0,evenwhentheinitialwavefunctionsΨ(q)arerequiredtobeinF(σ0),aswehavedone.However,wewilltaketheargumentsofHiguchiandWald[10]asstronglysuggestivethatinitialstatesinF(σ0),propagatedbythefullpropagator q′′ Cu q′ o ofσ0andthenbacktoα→ ∞,indeedremaininF(σ0)whenthereisascalar eldpresent.13Forthepresent,wewillassumewhennecessarythatattentionisrestrictedtocoarsegrainingsforwhichallΨh(q)∈F(σ0)whenq∈σ0i.e.forwhichB|σ0 F(σ0).

C.FinalIndi erence

Boundaryconditionsof“ nalindi erence”shouldamount,inessence,tono nalbound-aryconditionatall,inthesensethattheendpointsofallpathsareweightedequally.The nalboundaryconditionthene ectivelydisappearsfromthedecoherencefunctional,justasitdoesinordinaryquantummechanics.

Letusbeslightlymorepreciseabouthowtodothis.

WhatweseekareacompletesetofsolutionstotheWheeler-DeWittequation(3.5)whichmayservetode neapositive“resolutionoftheidentity”inthespaceofbranchwavefunctionsB.Bythiswemeanthefollowing.First,notethatifwede ne

′′′′′′ ′′I(q2,q1)=p′′(4.2)iΦi(q2)Φi(q1)

i

′′′′forq1,q2∈σ0,thedecoherencefunctional(3.9)forthecaseofapureinitialstateΨmaybewrittenverysimplyintermsofthebranchwavefunctions(4.1)as

′D(h,h)=Np′′i(Φi Ψh)(Φi Ψh′)

i

=NΨh′ I Ψh.(4.3)

Inordertocapturethenotionof“ nalindi erence”,wenextsplitBintoorthogonalsectorsB±onwhichtheWheeler-DeWittproduct ispositiveornegativede nite.14Werequirefor nalindi erencethattheΦibechosensothat

I ψ±=±ψ±,(4.4)

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

whereψ±∈B±.±SuchanidentityImaybeexplicitlyconstructedbychoosingbases{Φ±i}forBwhichareorthogonalintheinnerproduct(3.6),

±Φ±i Φj=±δij(4.5)

(4.6)and

Settingallthep′′i=1,withsuchabasiswemayconstructprojectionoperators

±′′′′′′± ′′I(q2,q1)=Φ±i(q2)Φi(q1)

i ΦΦ±j=0.i(4.7)

forwhich

Iisthengivenby

sothat′′′′′′′′I±(q2,q1) ψ(q1)=±ψ±(q2).(4.8)(4.9)

(4.10)I=I++I I ψ=ψ+ ψ .

ExplicitcalculationsrequireanexplicitchoiceoftheΦ±i.SincewearerestrictingourattentiontocoarsegrainingswhosebranchwavefunctionsΨh(q)lieinF(σ0),wecanim-plementthesplitofthespaceofbranchwavefunctionsintoB±bychoosingthe{Φ±i}tobeofde nitefrequencyonσ0withrespecttoα.Thatis,relabellingtheindexi→p ,

α,Hf(α,β±,φ)=4e4αVβ(β±)f(α,β±,φ)+6e6αVφ(φ)f(α,β±,φ), (4.12)sothat / αandHapproximatelycommuteeverywhereonsurfacesoflargenegativeαsolongastheyactinthespaceoffunctionsF(σ0)forwhichVβfandVφfremainbounded.SinceVβandVφarebothboundedbelow,thisisequivalenttotheconditionthatVf≈0onσ0–ormoreprecisely,thate 4αVfisbounded–whenceourchoiceofboundaryconditionsontheallowedsolutionstotheconstraintinSectionIIIA3.±TheexplicitformoftheΦp isnotdi cultto nd.Itisclearfrom(2.9)thatonσ0therearelargeregionsneartheoriginin q-spaceforwhichthepotentialVistotallynegligiblesolongasVφremainsbounded.Infact,itisastandardpartoftheloreofmixmastercosmologies15thatVa,theanisotropypotentialforBianchiIX–seebelow(2.7)√–iswell-approximatedasα→ ∞byatriangularsetofpotentialwallsatα= (β+±15[47]areusefuladditionstotheliteraturealreadycited.ThediscussioninsectionIVof[10]providesanextremelyusefulformulationforgeneraltypeABianchimodels;seealso[29]forBianchiIX.

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

α=2β+.Wellinsidethesewalls,wemaytakeVtobeessentiallyzeroforreasonableVφ.16

±Neartheoriginofthespacelikesurfaceσ0,theΦp (q)canthenbetakentohavetheform

±Φp (q)=1p·q e iωpαei ,n(2π)2ωp(4.13)

2whereωp=p 2.IntheasymptoticregionwherethepotentialVisnotnegligiblewemaychoosetomaintaineither(3.5)or(4.11),sinceourbranchwavefunctionsarebydesignessen-tiallyzerothere.Shouldwechoosetoadherestrictlytotherequirement(3.5)everywhere±onσ0,theΦp willofcoursebemorecomplicatedintheasymptoticregion.Practically,however,itisusuallyeasiesttoretaintheform(4.13)everywhere.Solongaswerestrictattentiontoinitialstatesandcoarse-grainingsforwhichB|σ0 F(σ0)thedistinctionwillbepracticallyirrelevantandtheexplicitform(4.13)canbeusedincalculatingthedecoherencefunctional.

Thebasisoffunctions(4.13)areorthogonalintheappropriatesense,

±±Φp Φp ′=±δp p ′,(n)(4.14)

(4.15)and

(nisthenumberof“spatial”minisuperspacedegreesoffreedom–three,inthepresentexample,β±and φ.)TheprojectionsI±of(4.7)maythenbeexplicitlyconstructedas (takingip′′dnp)i→ ′′′′I±(q2,q1)=±′′± ′′dnpΦp (q2)Φp (q1).± Φp Φ p ′=0.(4.16)Finally,wenotethattheorthonormalbases{Φ±i}arealsousefultorepresenttheinitialconditionΨ.Indeed,solutionstotheWheeler-DeWittequationinF(σo)maybewrittenas

++ Ψ=ciΦi+c iΦi

=Ψ+Ψ i+i(4.17)

where

whereFortheparticularchoiceofbasis(4.13)forF(σo),thislookslike + +(p )Φ (q)Ψ(q)=dnpΨ )Φp (q)+Ψ(pp ± ±(pΨ )=±Φp Ψ±c±i=±Φi Ψ.(4.18)(4.19)onσ0.(4.20)

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

D.TheWheeler-DeWittProductandtheDecoherenceFunctional

WiththeresolutionoftheidentityIinhand,wede nethepositivede niteWheeler-DeWittproduct by

ψ ψ=ψ I ψ

=ψ+ ψ+ ψ ψ .(4.21)

Intermsofthispositiveproduct,17thedecoherencefunctionalmaybewrittenverysimplyas

(4.22)D(h,h′)=NΨh′ Ψh,

whereNis(Ψu Ψu) 1.(4.22)istheformofthedecoherencefunctionalwewillusefortheremainderofthediscussion.Thegeneralizationofthisformulatothecaseofanimpuresetofinitialwavefunctions{Ψi,p′i}isobvious.

E.ClassicalInitialConditions

Becausewewillbeinterestedininitialconditionsandcoarsegrainingswhichdistin-guishclassicalbehaviour,itisusefultodiscusstheconditionsnecessarytospecifyclassicalsolutions.

Theclassicalequationsofmotion–thereducedEinsteinequations–resultingfromthevariationδS/δq=0oftheLagrangianactionappearingin(3.17a)are,inthepropertimegauge(3.13),

1= N (4.23)AV,dt2

alongwiththeLagrangianversionoftheconstraintδS/δN=0:

1

dtdqA

17Indeed,uptotechnicaldetails,weexpectthechoiceswehavemadeareintheire ectequivalenttoemployingthe“Rie elinduced”innerproducttoconstructthedecoherencefunctional;see[28,43,44]andreferencesthereinforfurtherdiscussion.(4.22),employingtheRie elinducedproduct,isthusanaturalalternativede nitionofthedecoherencefunctionalthatautomaticallyincorporatesanotionof“ nalindi erence.”Moregenerally,notethat(4.3)de nesapositiveproductonthespaceofbranchwavefunctions.Whenthe{Φi}constituteacompletesetandallthep′′i=0thisisagenuinenon-degenerateinnerproduct.Onemaythusregardthespeci cationof“ nalindi erence”asequivalenttotheproblemofde ninganinnerproductonthespaceofbranchwavefunctions.

A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such uni

F.QuantumInitialStates:Examples

Wechooseour rstexampleinitialstateΨWPtobeapositivefrequencysolutionto(3.5)thatislocalisedwellwithinmixmaster’spotentialwallsontheinitialsurface,andalsoawayfromregionsofverylargeVφ.Suchaninitialstatewillpreferaparticularclassicalpathandpredictapproximatelyclassicalbehaviouralongthecorrespondingclassicalsolution.ΨWPbyassumptionsolves(3.5)withV≈0andislocalizednearsome q0onσ0.Itmaythenberepresentedas

+ WP+(pΨWP(q)=d3pΨ )Φp (q q0)

d3p1p·(q q 0) WP+=e iωp(α α0)ei Ψ(p )(4.25)32ωp(2π)

1/2 WP+ WP+(pforsomeΨ ),whereofcourseωp=|p |.TakingωpΨ(p )tobeaGaussian

centeredaroundsomep 0,forinstance,yieldsaΨWP(q)localizedonσ0about( q0,p 0)tothegreatestextentconsistentwiththeuncertaintyprinciple.

Alternatively,wecouldconsideraninitialstateofWKBform,

ΨWKB(q)=A(q)eiW(q),(4.26)

whereA(q)isinanappropriatesenseslowlyvaryingrelativetoW(q).Approximatecalcula-tionsofthewavefunctionscorrespondingtothevariousproposalsfortheinitialconditionoftheuniversetendtohavecomponentsofthisform.WealsorequirethatA(q)beofcompactsupportonσ0soastoensurethatΨ0∈F(σ0),andthatW(q)be“positivefrequency”inthesensethat W/ α<0onσ0.(Thiswillturnouttocorrespondtoan“expanding”initialconditiononthepathsthiswavefunctionde nes.)

Wewillexaminebothofthesechoicesinthesequel.First,however,weshallmoveontodiscussthecategoryofcoarsegrainingsde ningapproximatelyclassicalbehaviourthatweshallconsider.

V.APPROXIMATECLASSICALITY

Inthissection,weapplythedecoherencefunctional(4.22)forΛ=0,BianchiIXcos-mologiestocoarsegrainingswhichdistinguishbetweenthosepathsinminisuperspacewhichbehave(semi-)classically,andthosewhichdonot.We ndthatforsuitablechoicesoftheinitialconditionΨ,andforasuitableclassofde nitionsofthesemiclassicalcoarsegrain-ings,theuniverseispredictedtobehaveclassicallywithprobabilitynearone.Inparticular,Λ=0BianchiIXcosmologiesarepredictedquantum-mechanicallytorecollapsejustastheydoclassically[29].Thisissatisfying,notleastbecauseofthesurprisingpredictionofWald’s

[10,26]rigorouscanonicalquantizationofBianchiIXthatclosedquantumuniversesdonotrecollapseinaclassicalfashion.18(Foranotherdiscussionofdi cultiesinterpretingwavefunctionsforclassicallyrecontractingcosmologies,see[49].)

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