Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(2)

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