Noncommutative Conformally Coupled Scalar Field Cosmology an(3)

2 2

(37)+4a χ2Ψ(a,χ)=0, 2 χwhichistheWheeler-DeWittequationfortheconformallycoupledscalar eldmodel.3Thisequationcanbe

solvedbyseparatingtheaandχvariables,asithasbeendoneintheliterature(see[13,25]andreferencestherein).However,bymakingachangeofvariablesthereisanotherroutetotackletheproblemthatisinterestingbytheensembleofsolutionsitgenerates.Weshallpresentithere,andshow,lateron,thatsucharouteitisparticularlysuitableforapplicationinthenoncommutativequantumcase.Bymakingthecoordinatechange

a=ξcoshη,

wecanrewrite(37)as

2

ξ

ξ2

2

χ=ξsinhη,

(38)

ξ2

+

1 ξ

+

α2

3

Aparticularfactororderingisbeingassumed.

We study the implications of a noncommutative geometry of the minisuperspace variables for the FRW universe with a conformally coupled scalar field. The investigation is carried out by means of a comparative study of the universe evolution in four differen

thatarealsoadmissibleininterpretationsotherthantheBohmianone.WeshallthereforediscardtheIν(x)contributionandwritethesolutionof(39)as4

Ψ(ξ,

η

)

=

(43)AαKiα/2ξ2eiαη.

α

A.QuantumTrajectoryFormalism

Inorderestablishaframeworkwhereallversionsoftheuniversemodelcanbecompared,itisinterestingto

appealtoacommonlanguage.ThisisprovidedbytheBohmianquantumtrajectoryformalism,whichwebrie ydescribeinthissection.Foramoredetailedaccountofthesubject,seethereferencesgivenintheintroduction.Intheformulationpresentedhere,weshallbene tfromideasproposedin[20].Thewavefunctionwillbeassumedasnotaconstituentofthephysicalsystem,asoriginallyassumedbyBohm[14],butasarepresentationofit.Quantuminformationtheorytellsusthatthewavefunctionhasanon-physicalcharacter[32].Bohmianquantumphysicsshouldinsomewaybeinaccordancewiththisfact.Actually,thecomprehensionofthemeaningofthewavefunctionasarepresentationofaquantumsystemiscrucialtoachievingtheunderstandingofquantummechanicsfromanyperspective[20].Thesamepointofviewseemstobesuitableforadoptioninquantumcosmology.Weshallbrie ycommentthisaspectwhenapplyingtheBohmianformalisminthedescriptionofthenoncommutativeversionofouruniversemodel.

Whendealingwithaquantummodelonemusthaveaclearpictureoftheelementsofontologyofthetheory.Bytheelementsofontology,wemeanwhatthetheoryisessentiallyabout.5TheorthodoxquantumtheorybasedontheCopenhageninterpretation,e.g.,isaboutobserversthatrealizemeasurements.IntheBohmianinterpretation,ontheotherhand,quantumtheoryisconcernedwiththephysicalsystems,whichcanbeparticles,waves,strings,etc.Inthisworktheobjectofattentionistheprimordialquantumuniverse,characterized,intheminisuperspaceformalism,bythecon gurationvariablesaandχ.Having xedtheobjectsofontologyofthetheory,wemustdeterminehowtheyevolveintime.Thisisdonewiththeaidofthewavefunction,whoseroleistoprovideustheevolutionlaw.Theprocedureisbestillustratedinthecontextofnon-relativisticquantummechanics.

Bohmiannon-relativisticquantummechanicsisconcernedwiththebehaviorofpointparticlesthatmoveinspacedescribingquantumtrajectories.Anevolutionlawisascribedtothemaccordingtotherule

1

Ψ Ψ

S

x˙=Re

i

=

t

=

2

45

Sinceαisacontinuousindex,inthemostgeneralcasethesummationcanbereplacedbyanintegral.

Everyphysicaltheorymustnecessarilybeessentiallyaboutsomething,theprimitiveontologyofthetheory[16].Inthissense,allthetheoriesareontological.

We study the implications of a noncommutative geometry of the minisuperspace variables for the FRW universe with a conformally coupled scalar field. The investigation is carried out by means of a comparative study of the universe evolution in four differen

“beable”

i ΨAx , i iΨ

)=ReB(A

m

.(48)

Asaconsequenceofbeinganobjectivetheoryofpointparticlesdescribingtrajectoriesinspace,Bohmian

quantummechanicsdoesnotgivetoprobabilityaprivilegedrole.Instead,asdiscussedin[19],suchaformulationprobabilityisaderivedconcept,adecurrentofthelawofmotionofthepointparticles.TheBohmianformulationisthuseminentlysuitableforthestudyofindividualsystems,astheprimordialquantumuniverse.Intheremainingofthissection,wewillbeconcernedwiththeapplicationofthetheorytothecommutativequantumuniversediscussedinthebeginningofthissection.Inthenextsectionsasimilarstudywillbecarriedoutforthenoncommutativequantumcase.

B.

ApplicationtoQuantumCosmology

InthissubsectionweshallapplytheBohmianformalismtoobtaininformationabouttheevolutionofthequantumFRWuniversewithaconformallycoupledscalar eld.Inthedescriptionofquantumcosmologyemployingquantumtrajectoriesweshallextendtheevolutionlaw(48)totheminisuperspacevariables.InthecommutativecasetheresultingBohmianminisuperspaceformalismmatcheswiththeminisuperspaceversionoftheBohmianquantumgravityproposedin[18],andemployedtostudytheconformallycoupledscalarmodelin[13].From(48)we nd,inthegaugeN=a,

S[Ψ( i a/2)Ψ]

a˙=Re

2

1

η=1 χ.

(50)

Ψ Ψ

dξ

2 S(ξ,η)

dt

Bychanging(49)and(50)intothe(ξ,η)coordinatesde nedby(38)weobtain

=

.

(51)

Inwhatfollowsweshallsolvethesystem(51)intwoexampleswheretheauniverseischaracterizedbyawavefunctionofthetype(43).

We study the implications of a noncommutative geometry of the minisuperspace variables for the FRW universe with a conformally coupled scalar field. The investigation is carried out by means of a comparative study of the universe evolution in four differen

1.

Case1

whereAisaconstant.SincetheBesselfunctionKiν(x)isrealforνrealandx>0,7thephasecanbereaddirectlyfromtheexponentialin(52):S=αη.Theequationsofmotion(51)inthisstatearethereforereducedto

dξ

dt=

α

Letusconsider rsttheexamplewherethereisasingleBesselfunctionin(43). …… 此处隐藏：5372字，全部文档内容请下载后查看。喜欢就下载吧 ……