Generalized Quantum Theory of Recollapsing Homogeneous Cosmo

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