Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(13)

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

ThedecoherencefunctionalD(h′,h)isde nedthrough

′′′pi Φi|Ch′|Ψj p′j Φi|Ch|Ψj D(h,h)=N

i,j(3.9)

whereNisanormalizingfactordeterminedsothatΣh′hD(h′,h)=1.Speci cally,ifuistheclassofall ne-grainedhistories

′′ 1N=pi| Φi|Cu|Ψj |2p′j.(3.10)ij

Thedecoherencefunctionalde nedby(3.9)isthedirectanalogueofthe“canonical”decoherencefunctionalofordinaryHamiltonianquantummechanicswithinitialand nalboundaryconditions[1,3],writteninfunctionalintegralform,withaccomodationsappro-priatetothereparametrizationinvarianceofthepresenttheory.Itsatis esthegeneralcon-ditionsrequiredofageneralizedquantumtheory.Itis(i)Hermitian,D(h′,h)=D(h,h′) ,(ii)normalized,Σhh′D(h′,h)=1,(iii)positiveonthediagonalelements,D(h,h)≥0,and,(iv)consistentwiththeprincipleofsuperpositioninthesensethatif{c¯h¯}isapartitionoftheclasses{ch}intocoarserclasses,then ′¯¯D(h,h)=D(h′,h)(3.11)

¯′h∈h¯h′∈h

Thesefourconditionsareenoughtoensurethatforsetsofhistoriesthatdecohereaccordingto(3.1),thenumbersp(h)de nedby(3.1)areprobabilitiessatisfyingthemostgeneralformoftheprobabilitysumrules.Byusingin(3.1)thespeci cform(3.9)wecanassesstheprobabilitiesofalternative,coarse-grained,decoheringhistoriesofthemodelhomogeneouscosmologiesunderdiscussion.

B.EvaluationoftheClassOperatorsintheProperTimeGauge

Weapply(3.1)topredictionsconcerningthesemiclassicalbehaviourofhomogeneousminisuperspacecosmologiesinthenextsection.Toendthissection,wediscusstheeval-uationoftheclassoperators(3.4)inaparticularlyconvenientgauge–(3.13)–calledthe“propertime”gauge.Forsuitablecoarsegrainings,wealsobrie yarguethatthesematrixelementssatisfytheconstraint(3.5).Becausetheaction(2.8)isessentiallythatofarel-ativisticparticleinapotential,thetreatmentcloselyparallelsthatofthefreerelativisticparticlethathasbeengivenpreviouslyin[1,section7].

The rststepinevaluating(3.4)istochoosea“gauge”that xesthereparametrizationsymmetry(2.10a-2.10c),thein nitesimalformofwhichisinvarianceunderthechanges(setf(t)=t+ /N)

δqA= (t){qA,H}

δpA= (t){pA,H}

δN= ˙(t),(3.12a)(3.12b)(3.12c)

where{,}isthePoissonbracket,and (0)= (1)=0.Aconvenient“gauge” xingfunctionis[38,40](seealso[1,section7])˙G=N.(3.13)