Constraints Algebra and Equations of Motion in Boh

arXiv:gr-qc/0311076v122Nov2003AEI–2002–071ConstraintsAlgebraandEquationsofMotioninBohmianInterpretationofQuantumGravityALISHOJAI∗PhysicsDepartment,TehranUniversity,Tehran,IranandInstituteforStudiesinTheoreticalPhysicsandMathematics,P.O.Box19395-5531,Tehran,IranandMPIf.Gravitationsphysik,Albert–Einstein–Institut,AmM¨uhlenberg1,14476GolmnearPotsdam,GermanyFATIMAHSHOJAI†InstituteforStudiesinTheoreticalPhysicsandMathematics,P.O.Box19395-5531,Tehran,IranandMPIf.Gravitationsphysik,Albert–Einstein–Institut,AmM¨uhlenberg1,14476GolmnearPotsdam,GermanyAbstractItisshownthatintroducingthequantumeffectsusingdeBroglie–Bohmthe-oryinthecanonicalformulationofgravitywouldchangetheconstraintsal-∗Email:[e-mail]†Email:[e-mail]dshownthatitistheclearprojectionofgeneralcoordinatetransformationstothespatialandtemporaldiffeomor-phisms.ThequantumEinstein’sequationsarederivedanditisshownthattheyaremanifestlycovariantundertheabovediffeomorphisms,asitwouldbe.PACSNo.:04.60.Ds,98.80.Hw,03.65.TaTypesetusingREVTEX2I.INTRODUCTIONTheissueofconstraintsalgebraisanimportantproblemincanonicalquantumgravity1.UnfortunatelytheconstraintsalgebraisnotatrueLiealgebrabecause,dynamicalvariablesappearinthestructureconstants.MoreimportantisthatthepoissonbracketofHamiltonianconstraintwithitselfisnotzero.Thereforethealgebraisnotaclearprojectionof4–diffeomorphisminfoliatedspace–time[1,2].Usingafixedembeddingrestrictsthesymmetryofthetheorytoasubgroupofgeneralcoordinatetransformations,invarianceunderspatialandtemporalreparametrizations.Spatialreparametrizationinvarianceisexplicitlypresentinthealgebra,buttemporalreparametrizationinvarianceisnotpresentexplicitly.Recently[1–3]somemodificationsaredonetohavetheappropriatealgebra.Ontheotherhand,Bohm’scausalinterpretationofquantummechanics(seee.g.[4])mayprovideabetterunderstandingofsomeaspectsofquantumgravityincomparisontotheCopenhageninterpretation[5].WeshalldiscusssomeaspectsofBohmianinterpretationinthenextsection.InBohm’stheoryanysystemhasadefinitetrajectoryandanyquantityisac-numbernotanoperator.Onemustmodifytheclassicalhamiltonianbyaddingthequantumpotential.Inthiswaytheconstraintsalgebracouldbeevaluatedandmaydifferfromtheclassicalone.ThisiswhatweshalldoinsectionIII.InadditionsinceinBohm’sinterpretationofquantummechanicsthesystemhasawell–definedtrajectory,theevolutionequationsofthemetric(quantumEinstein’sequations)can1InthispaperwearedealingwithWDWapproachtocanonicalquantumgravityandAshtekar’snewvariablesarenotconsidered.3beobtained.ThisisdoneinsectionIV.II.WHYBOHMIANQUANTUMGRAVITY?InthisworkweusetheBohmianinterpretationofcanonicalquantumgravity,becauseithassomeusefulaspects[4,6].Someofthemare:•Itleadstotimeevolutionofthedynamicalvariableswhetherthewavefunctiondependsontimeornot.ThereforewehavenotthetimeprobleminBohmianquantumgravity.•Bohm’stheorydescribesasinglesystem,unliketheCopenhageninterpretationofquantumtheory,whichdoesnottellanythingaboutasinglesystem.Aboutanen-sembleofthesystembothinterpretationsareequivalent.ThisisbecauseofthespecificformofBohm’sequationsofmotion.TheyaretheBohm-Hamilton-Jacobiequationandtheconservationequationofprobability.TheseequationscanbetransformedtotheSchr¨odingerequationbysomecanonicaltransformation.Thisaspectisusefulinquantumcosmologywherethesystemistheuniverse,andanensembleofitdoesnotexist.Thereforewehavenotheretheconceptualproblemofthemeaningoftheuniverse’swavefunctioninquantumcosmology.•Normalizationofthewavefunctionisneededonlyfortheprobabilisticdescription.Herethereisnoneedtonormalizethewavefunctionforasinglesystem.•Theclassicallimithasawell-definedmeaning.Whenthequantumpotentialislessthantheclassicalpotentialandthequantumforceislessthantheclassicalforceweareintheclassicaldomain.4•Thereisnoneedtoseparatetheclassicalobserverandthequantumsysteminthemeasurementproblem.IntheBohmianpictureofthemeasurementprocesswehavetwointeractingsystems,thesystemandtheobserver.Aftertheinteractiontakesplace,thewavefunctionofthesystemisreducedinacausalway.ItmustbenotedthatthesamestatisticalresultsfortheCopenhagenandBohmianinter-pretationsdoesn’tmeanthatthetwotheoriesareequivalent.Theyaredifferentinphysicalconcepts.ThemostimportantdifferenceisthatinBohmianinterpretationonedealswithtrajectories.Thiscanleadtonewconcepts.E.g.,onecanevaluatethetunnelingtimeoftheparticlethroughapotentialbarrierinthenon–relativisticquantummechanics.ThisisaconceptthathasnotaclearmeaningintheCopenhageninterpretation[4,7].TillnowBohmianinterpretationofWDWquantumgravityandcosmologyhavegivensomephysicalresultsthatcouldbefoundintheliterature:•InBohmianquantumcosmologythequantumforcecanremovethebigbangsingu-larity,becauseitcanbehaveasarepulsiveforce.Thishasbeenshownforaradiationdominateduniversein[8,9],foradustfilledflatFRWuniversein[8]andforFRWmodelwithaminimalmasslessscalarfieldin[10].•Thequantumforcemaybepresentinlargescalesbecausethequantumeffectsofquantumpotentialareindependentofthescale[11].ThisbehaviourcanbeseenforflatFRWuniversewithdustandaconformalinvariantscalarfieldin[12].•Onecanfindthegracefule