Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(3)

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.