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arXiv:gr-qc/0105050v22Apr2002p-ADICANDADELICMINISUPERSPACEQUANTUMCOSMOLOGYGORANS.DJORDJEVI´CSektionPhysik,Universit¨atM¨unchen,Theresienstr.37,D-80333M¨unchen,GermanyDepartmentofPhysics,UniversityofNiˇs,P.O.Box224,18001Niˇs,YugoslaviaBRANKODRAGOVICHSteklovMathematicalInstitute,GubkinSt.8,GSP-1,117966,Moscow,RussiaInstituteofPhysics,P.O.Box57,11001Belgrade,YugoslaviaLJUBIˇSAD.NEˇSI´CDepartmentofPhysics,UniversityofNiˇs,P.O.Box224,18001Niˇs,YugoslaviaIGORV.VOLOVICHSteklovMathematicalInstitute,GubkinSt.8,GSP-1,117966,Moscow,RussiaWeconsidertheformulationandsomeelaborationofp-adicandadelicquantumcosmol-ogy.TheadelicgeneralizationoftheHartle-Hawkingproposaldoesnotworkinmodelswithmatterfields.p-adicandadelicminisuperspacequantumcosmologyiswelldefinedasanordinaryapplicationofp-adicandadelicquantummechanics.Itisillustratedbyafewofcosmologicalmodelsinone,twoandthreeminisuperspacedimensions.Asaresultofp-adicquantumeffectsandtheadelicapproach,thesemodelsexhibitsomediscretenessoftheminisuperspaceandcosmologicalconstant.Inparticular,discretenessofthedeSitterspaceanditscosmologicalconstantisemphasized.1.IntroductionThemaintaskofquantumcosmology1istodescribetheevolutionoftheuniverseinatheveryearlystage.Atthisstage,theuniverseisinaquantumstate,whichisdescribedbyawavefunction.Usuallyonetakesitthatthiswavefunctioniscomplex-valuedanddependsonsomerealparameters.SincequantumcosmologyisrelatedtothePlanckscalephenomenaitislogicaltoreconsideritsfoundations.Wewillheremaintainthestandardpointofviewthatthewavefunctiontakescomplexvalues,butwewilltreatitsargumentsinamorecompleteway.Namely,wewillregardspace-timecoordinatesandmatterfieldstobeadelic,i.e.theyhaverealaswellasp-adicpropertiessimultaneously.Thisapproachismotivated1bythefollowingreasons:(i)thefieldofrationalnumbersQ,whichcontainsallobservationalandexperimentalnumericaldata,isadensesubfieldnotonlyinthefieldofrealnumbersRbutalsointhefieldsofp-adicnumbersQp(pisanyprimenumber),(ii)thereisaplausibleanalysis2withinandoverQpaswellasthatonerelatedtoR,(iii)generalmathematicalmethodsandfundamentalphysicallawsshouldbeinvariant3underaninterchangeofthenumberfieldsRandQp,(iv)thereisaquantumgravityuncertainty3,4ΔxwhilemeasuringdistancesaroundthePlancklengthℓ0,Δx≥ℓ0=r¯hGc3∼10−33cm,(1)whichrestrictsthepriorityofarchimedeangeometrybasedonrealnumbersandgivesrisetoemploymentofnon-archimedeangeometryrelatedtop-adicnumbers,3(v)itseemstobequitereasonabletoextendcompactarchimedeangeometriesbythenonarchimedeanonesinthepathintegralmethod,and(vi)adelicquantummechanics5appliedtoquantumcosmologyprovidesrealizationofalltheabovestatements.Thesuccessfulapplicationofp-adicnumbersandadelesinmoderntheoreticalandmathematicalphysicsstartedin1987,inthecontextofstringamplitudes6,7(forareview,seeRefs.2,8and9).Forasystematicresearchinthisfielditwasformulatedp-adicquantummechanics10,11andadelicquantummechanics.5,12Theyarequantummechanicswithcomplex-valuedwavefunctionsofp-adicandadelicarguments,respectively.Intheunifiedform,adelicquantummechanicscontainsordinaryandallp-adicquantummechanics.Asthereisnotanappropriatep-adicSchr¨odingerequation,thereisalsonop-adicgeneralizationoftheWheeler-DeWittequation.Insteadofthedifferentialapproach,Feynman’spathintegralmethodwillbeexploited.p-adicgravityandthewavefunctionoftheuniversewereconsideredinthepaper13publishedin1991.AnideaofthefluctuatingnumberfieldsatthePlanckscalewasintroducedanditwassuggestedthatwerestricttheHartle-Hawking14proposaltothesummationonlyoveralgebraicmanifolds.ItwasshownthatthewavefunctionforthedeSitterminisuperspacemodelcanbetreatedintheformofaninfiniteproductofp-adiccounterparts.Anotherapproachtoquantumcosmology,whichtakesintoaccountp-adicef-fects,wasproposedin1995.15Likeinadelicquantummechanics,theadeliceigen-functionoftheuniverseisaproductofthecorrespondingeigenfunctionsofrealandallp-adiccases.p-adicwavefunctionsaredefinedbyp-adicgeneralizationoftheHartle-Hawking14pathintegralproposal.ItwasshownthatintheframeworkofthisprocedureoneobtainsanadelicwavefunctionforthedeSitterminisuper-spacemodel.However,theadelicgeneralizationwiththeHartle-Hawkingp-adicprescriptiondoesnotworkwellwhenminisuperspacehasmorethanonedimension,inparticular,whenmatterfieldsaretakenintoconsideration.Thesolutionofthisproblemwasfound16bytreatingminisuperspacecosmologicalmodelsasmodelsofadelicquantummechanics.2Inthispaperweconsideradelicquantumcosmologyasanapplicationofadelicquantummechanicstotheminisuperspacemodels.Itwillbeillustratedbyone-,two-andthree-dimensionalminisuperspacemodels.Asaresultofp-adiceffectsandtheadelicapproach,inthesemodelsthereissomediscretenessofminisuperspaceandcosmologicalconstant.ThiskindofdiscretenesswasobtainedforthefirsttimeinthecontextofadelicthedeSitterquantummodel.15Inthenextsectionwegivesomebasicfactsonp-adicandadelicmathematics.Section3isdevotedtoabriefreviewofp-adicandadelicquantummechanics.p-adicandadelicquantumcosmologyareformulatedinSec.4.Sections5and6containsomeconcreteminisuperspacemodels.Attheend,wegivesomeconcludingremarks.2.p-AdicNumbersandAdelesWegivehereabriefsurvey