Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(16)

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

α

0α0FIG.1:Theminisuperspaceofhomogeneouscosmologicalmodels.Thetimelikecoordinateq0=αandonespacelikecoordinate(e.g.q1=β+)areplotted.Finegrainedhistoriesarepathsinthiscon gurationspacewhichbeginandendonasurfaceσ0thatisspacelikeintheWheeler-DeWittminisuperspacemetricGABandlocatedatlargenegativeα(=α0,say),correspondingtoasmallspatialvolume.Therestrictiontopathswhichbeginandendonσ0correspondstotheimpositionofaboundaryconditionthatsolutionstotheWheeler-DeWittequationvanishasα→∞.Thepathsallpossessα˙(0)>0andα˙(1)<0onσ0,i.e.“expanding”initialand“contracting” nalconditions.Theuniverseexpandsfromasmalltoamaximumvolumeandthenrecontracts.

B.BranchWaveFunctions

First,itisusefultode nethe“branchwavefunctions”

Ψh(q)= q Ch|Ψ ≡ q Ch q′ Ψ(q′).(4.1)

Thesemayberegardedasthewavefunctionscorrespondingtotheinitialstate,propa-gatedbytherestrictedpropagator(classoperator)correspondingtothephysicalhistoryinquestion.12

Thespace{Ψh(q)}ofbranchwavefunctionswillbedenotedB.ItdependsinanessentialwaybothontheallowedspaceofinitialwavefunctionsΨ(q),andontheallowedcoarsegrainings.