Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(6)

时间:2025-07-12

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

betweencoordinatesandtheirconjugatemomenta.VariationwithrespecttothepAandqAgiveEinstein’sequationsofmotion.Thecharacterofthesolutionstotheseequations—thepossibleclassicalhistories—dependsontheisometrygroupandthevalueofthecosmologicalconstantΛ.Forexample,inthecaseofΛ=0BianchiImodelsweexpectcosmologiestoexpandforeverfromaninitialsingularity.Moreprecisely,weexpecttheextrinsiccurvatureK=(3/L)dα/dt(traceoftheextrinsiccurvaturetensor)ofthet=constantsurfacesofhomogeneitytoremainpositivetothefutureofaninitiallysingularsurface.BianchiIclassicalsolutionswithΛ>0alsoexpandforever,whilemodelswithΛ<0alwaysrecollapse.BianchiIXmodelshaveclosedspatialsectionswiththree-spheretopology.TheΛ=0BianchiIXuniversesarejusttheanisotropicgeneralizationsoftheclosedFRWuniverse.Moreprecisely,LinandWald[29]haveshownthatwhenthedominantenergyconditionissatis edandthetraceofthespatialprojectionofthestress-energytensor(i.e.thesumoftheprincipalpressures)ispositive–thusexcludingacosmologicalconstant–therearenoclassicalsolutionswhichexpandforeverfromaninitialsingularityinthesensethattheextrinsiccurvatureofconstanttsurfacesremainspositive.

Thestress-energytensorarisingfrom(2.3)satis esthedominantenergyconditionsolongasVφ≥0,thoughthepressuresmaybenegativeifthepotentialenergyinthescalar eldexceedsthekineticenergy.(Indeed,itispreciselythisfeaturewhichallowsascalar eldtomimicacosmologicalconstantinin ationarymodels.)Thus,scalar eldsonlysatisfytheconditionsoftheLin-Waldrecollapsetheorematlatetimesforcertainchoicesofscalarpotential–afree,massless,minimallycoupledscalar eld,forexample.ThegeneralconditionsonVφforwhicharecollapsetheoremholdsareasfarasweareawarenotcurrentlyknown.

WithacosmologicalconstanttheconditionsoftheLin-Waldtheoremarenotsatisi ed.TheexampleofdeSitterspaceisenoughtoshowthattherewillbeBianchiIXsolutionswithvanishingscalar eldwhichevolvenon-singularlyfromaninitialcontractingphase(K<0)inthein nitepasttoanexpandingphase(K>0)inthefarfuture.Inbetweenthevolumeoftheuniversereachesanon-zerominimumvalue.Theinclusionofasmallamountofhomogeneousscalar eldwouldnotbeexpectedtodisturbthisbehavior.However,thehomogeneous,isotropicBianchiIX(Friedmann)modelsshowthatapositivecosmologicalconstantdoesnotremovethesingularityineverycase.Therearealsomodelsinwhichthestress-energyofthescalar elddominatesthatofthecosmologicalconstantthroughoutthemodel’shistory.SuchmodelsdisplaythequalitativefeaturesoftheΛ=0case—aninitialsingularityleadingtoa niteexpansionfollowedbyrecontraction,formodelssatisfyingtheconditionsoftheLin-Waldtheorem.See[7]forawide-rangingsurveyofthedynamicsofthevariouscosmologicalmodels.solongasf(0)=0andf(1)=1.Variationof(2.8)or(2.4)withrespecttothemultiplierNgivestheconstraint HpA,qA=0(2.11)

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