Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(18)

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

whereψ±∈B±.±SuchanidentityImaybeexplicitlyconstructedbychoosingbases{Φ±i}forBwhichareorthogonalintheinnerproduct(3.6),

±Φ±i Φj=±δij(4.5)

(4.6)and

Settingallthep′′i=1,withsuchabasiswemayconstructprojectionoperators

±′′′′′′± ′′I(q2,q1)=Φ±i(q2)Φi(q1)

i ΦΦ±j=0.i(4.7)

forwhich

Iisthengivenby

sothat′′′′′′′′I±(q2,q1) ψ(q1)=±ψ±(q2).(4.8)(4.9)

(4.10)I=I++I I ψ=ψ+ ψ .

ExplicitcalculationsrequireanexplicitchoiceoftheΦ±i.SincewearerestrictingourattentiontocoarsegrainingswhosebranchwavefunctionsΨh(q)lieinF(σ0),wecanim-plementthesplitofthespaceofbranchwavefunctionsintoB±bychoosingthe{Φ±i}tobeofde nitefrequencyonσ0withrespecttoα.Thatis,relabellingtheindexi→p ,

α,Hf(α,β±,φ)=4e4αVβ(β±)f(α,β±,φ)+6e6αVφ(φ)f(α,β±,φ), (4.12)sothat / αandHapproximatelycommuteeverywhereonsurfacesoflargenegativeαsolongastheyactinthespaceoffunctionsF(σ0)forwhichVβfandVφfremainbounded.SinceVβandVφarebothboundedbelow,thisisequivalenttotheconditionthatVf≈0onσ0–ormoreprecisely,thate 4αVfisbounded–whenceourchoiceofboundaryconditionsontheallowedsolutionstotheconstraintinSectionIIIA3.±TheexplicitformoftheΦp isnotdi cultto nd.Itisclearfrom(2.9)thatonσ0therearelargeregionsneartheoriginin q-spaceforwhichthepotentialVistotallynegligiblesolongasVφremainsbounded.Infact,itisastandardpartoftheloreofmixmastercosmologies15thatVa,theanisotropypotentialforBianchiIX–seebelow(2.7)√–iswell-approximatedasα→ ∞byatriangularsetofpotentialwallsatα= (β+±15[47]areusefuladditionstotheliteraturealreadycited.ThediscussioninsectionIVof[10]providesanextremelyusefulformulationforgeneraltypeABianchimodels;seealso[29]forBianchiIX.