Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(7)

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