Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(20)

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

D.TheWheeler-DeWittProductandtheDecoherenceFunctional

WiththeresolutionoftheidentityIinhand,wede nethepositivede niteWheeler-DeWittproduct by

ψ ψ=ψ I ψ

=ψ+ ψ+ ψ ψ .(4.21)

Intermsofthispositiveproduct,17thedecoherencefunctionalmaybewrittenverysimplyas

(4.22)D(h,h′)=NΨh′ Ψh,

whereNis(Ψu Ψu) 1.(4.22)istheformofthedecoherencefunctionalwewillusefortheremainderofthediscussion.Thegeneralizationofthisformulatothecaseofanimpuresetofinitialwavefunctions{Ψi,p′i}isobvious.

E.ClassicalInitialConditions

Becausewewillbeinterestedininitialconditionsandcoarsegrainingswhichdistin-guishclassicalbehaviour,itisusefultodiscusstheconditionsnecessarytospecifyclassicalsolutions.

Theclassicalequationsofmotion–thereducedEinsteinequations–resultingfromthevariationδS/δq=0oftheLagrangianactionappearingin(3.17a)are,inthepropertimegauge(3.13),

1= N (4.23)AV,dt2

alongwiththeLagrangianversionoftheconstraintδS/δN=0:

1

dtdqA

17Indeed,uptotechnicaldetails,weexpectthechoiceswehavemadeareintheire ectequivalenttoemployingthe“Rie elinduced”innerproducttoconstructthedecoherencefunctional;see[28,43,44]andreferencesthereinforfurtherdiscussion.(4.22),employingtheRie elinducedproduct,isthusanaturalalternativede nitionofthedecoherencefunctionalthatautomaticallyincorporatesanotionof“ nalindi erence.”Moregenerally,notethat(4.3)de nesapositiveproductonthespaceofbranchwavefunctions.Whenthe{Φi}constituteacompletesetandallthep′′i=0thisisagenuinenon-degenerateinnerproduct.Onemaythusregardthespeci cationof“ nalindi erence”asequivalenttotheproblemofde ninganinnerproductonthespaceofbranchwavefunctions.