Generalized Quantum Theory of Recollapsing Homogeneous Cosmo(11)

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

3.TheDecoherenceFunctional

wherethesumisoverpathsinCextthatlieintheclasschwhichbeginatq′andendatq′′.Sisthe(Lagrangian)actionforpaths—afunctionalofq(t)andN(t).Inordertogiveade nitemeaningtothefunctionalintegralin(3.3),and,inparticular,to xthe“measure”onthespaceofpaths,itisconvenienttoconsiderthecorrespondingintegraloverpathsinphase-space,

q′′A q′′ Ch q′ =δN≥0δpδqq(3.4)′ G[q,N]δG[q,N]expiSpA,q,N

chThedecoherencefunctionalisconstructedintwostepsfollowingtheanalogyoftherela-tivisticparticlediscussedin[1];the nalresultisexpressedin(3.9).Itwillturnouttobeanaturalgeneralizationtoareparametrizationinvarianttheoryofthecanonicaldecoherencefunctionalofordinaryquantummechanics.Firstwede nematrixelementsofclassoperatorscorrespondingtoindividualcoarse-grainedhistorieshbyasum-over-historiesintheclass.Schematically,wede ne ′′′expiS[path](3.3) q Ch q = ′′ paths∈q′hq

Theingredientsinthisexpressionareasfollows:Sistheaction(2.8).GisafunctionsuchthattheconditionG=0 xesauniquerepresentativefromeachreparametrizationinvariantequivalenceclassof ne-grainedhistories; GistheassociatedFaddeev-Popovdeterminant.TheintegralisoverallpathsinCextthatlieintheclasschandpassbetweenthecon gurationspacepoints(qA)′and(qA)′′.Allpossiblemomentumpathsareintegratedover.WetakethemultiplierfunctionalintegraltobeoverpositivevaluesN≥0.Otherchoices,forexample,bothpositiveandnegativevalues,wouldleadtodi erentgeneralizedquantumtheories.(Someoftheissuesthatariseinchoosingtheallowedrangeforthelapsearediscussedin[38–41]and[1,sectionVII].)ThemeasurefortheqandpintegrationsistheusualLiouville“dqdp/2π”measureonphase-spacepaths.TheintegrationovermomentamaybeseenasadevicetoinducethemeasureonpathsinCextfromtheLiouvillemeasuresincetherangeofintegrationoverthemomentaareconstrainedneitherbytheclassch,theendpoints(q′,q′′),orthegauge xingdelta-function.Theskeletonizedpathintegralsweenvisionarequitestandard[30,39],andhavealreadybeenbrie ydescribedin[1].

Thedecoherencefunctionalisconstructedfromtheclassoperators(3.4)whichchar-acterizethephysicalhistoriesinquestion,butalsoincorporatesatheoryofcosmologicalboundaryconditionsimposedattheendsofthehistoriesthataretheanalogsofinitialand nalconditionsinaquantumsystemwitha xednotionoftime.

Aninitialor nalconditionisrepresentedbyasetofcon guration-spacewavefunctionstogetherwithaprobabilityforeachwavefunction.Thatisthesamekindofinformationneededtospecifyaninitialor naldensitymatrixinordinaryquantummechanics.Thewavefunctionsarerequiredtosatisfyanoperatorimplementationoftheconstraints.Thus,forhomogeneouscosmologicalmodelsaninitialconditionisspeci edbyaset{Ψi(qA),p′i},whereeachΨi(qA)satis esanoperatorformof(2.11).Wetaketheobviousoperatorordering10