Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(8)

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