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1HiddenvariablesandhiddentimeinquantumtheoryKurakinP.V.(KeldyshInstituteofAppliedMathematics,RussianAcademyofSciences)Bell’stheoremprovesonlythathiddenvariablesevolvingintruephysicaltimecan’texist;stillthetheorem’smeaningisusuallyinterpretedintolerablywide.Theconceptofhiddentime(and,ingeneral,hiddenspace-time)isintroduced.Suchconceptprovidesawholenewclassofphysicaltheories,fullycompatiblewithcurrentknowledge,butgivingnewtremendouspossibilities.ThosetheoriesdonotviolateBell’stheorem.Comments:15pagesReport-no:KeldyshInstitutepreprintNo.33,20041.Introduction(a)TheproblemofhiddenvariablesishistoricallyconnectedtoagedankenexperimentproposedbyEinstein,PodolskyandRosen[1],whentryingtorevisesomepositionsofquantummechanics.EPR-experiment(byauthors'lastnames)initssimplestcaselooksasfollows.Somequantumsystemisdividedintotwosubsystems,whilesomesummaryphysicalquantityispreserved.Say,letitbeangularmomentum(spin).Thensubsystemsmovearbitraryfarfromeachother.Ifonemeasuresthepreservedquantityforsomeonesubsystemthen-atappropriatemeasurementbasissupplied-theresultofanalogousmeasurementforpairedsubsystemispre-determined.ProfessorJohnG.CramerofWashingtonUniversity(Seattle,USA)offers,asIseeit,thebestdescriptionofEPRparadoxandcorrespondingexperimentinitsmodernview[2]:Thiskindofnon-localityisdemonstratedbytheFreedman-Clauserexperiment(1972)...Hereexcitedcalciumatomsundergoa0+to1-to0+atomiccascadeandprovideapairofphotons,assumedtobeemittedback-to-back,whichareinarelativeL=0angularmomentumstate.Becauseofangularmomentumconservationthesephotonsarerequiredtohaveidenticalhelicitiesorlinearcombinationofhelicities,i.e.,theymustbeinidenticalstatesofcircularorlinearpolarization.ForthisreasontheSVofthetwophotonsystempermitsthephotonstobeinanypolarizationstate,providedonlythatbothareinthesamestate.Experimentallythismeansthatifthephotonsaretransmittedthroughperfectpolarizingfiltersbeforedetection,theymustbetransmittedwith100%probabilityifthepolarizationsofthefiltersselectmatchingstatesand0%ifthefiltersselectorthogonalstates,nomatterwhatorientationorpolarizationselectivitythefiltershave.TheFreedman-Clauser(FC)experimentemployslinearpolarizingfiltersandmeasuresthecoincidenttransmissionyieldofthetwophotondetectorswhentheprincipalaxesofthetwofiltersaresetatanglesQAandQB,whicharevariedindependently.QuantummechanicspredictsthattheexperimentallyobservedyieldwilldependonlyontherelativeangleQrel=QA-2QBbetweenthetwoprincipalaxes,andfurtherthatforidealfilterstheyieldwillhavethenormalizedangulardependence:R[Qrel]=Cos2(Qrel)(1)(b)Suchcorrelationsinquantummechanicalpredictions,fromviewpointofauthorsof[1],lookverysuspicious.Thisiswhyauthorsof[1]supposedthefollowing:thesepredictionsshouldbewrongorquantummechanicsinincompletetheory.TheFreedman-Clauserexperimentdemonstratesthatsuchcorrelationsdoexistindeed.So,quantummechanicsiscorrect.Thenotionofcompletenessofatheory,asintroducedbyA.Einstein,iscomplexenough,andweshallnotdealit.Truephysicalproblem,inmyopinion,isthatstandardquantummechanicsdoesnotexplaininanywayquantumphenomenainsteaditofferspredictedprobabilityamplitudesforelementaryprocesses.Quantumsystemsdopossesanytrajectories.So,itisunclearhowexactlyaquantumtransitionfrominitialstatetoafinaloneisaccomplished.Thetheorydoesnotexplaininanyway,whyandкакspinsofarbitraryremoteparticlesturnouttobecorrelated.Itisveryimportantheretounderstandthatit'sfundamentallyincorrecttoimagine,asifcorrelatedvaluesdoexistjustimmediatelyafterthepartsofsystemdepartfromeachother.Laterwe'llgiveadetailedexplanationofthisbyProf.J.Cramer.Trueproblemhereisthateachsubsystemgetsitsownvalueofpreservedquantityatthedetectioninstantonly.(c)UnlocalcorrelationslikeinFCexperiment,donotcontradicttoprinciplesofspecialrelativity,but,asRogerPenrosestates,[3],theycontradicttothespiritofrelativity.Whentryingtoexplainsuchunlocalcorrelationinlocalway,andthustryingtodescribeexplicitlyanyquantumsystemtransition,A.Einsteinintroducedahypothesisof«hiddenvariables».Somesmoothanddeterminedevolutionofhiddenvariablesshould,asA.Einsteinsupposed,giveatoutputtheobservedvaluesofphysicalquantities.Asiscommonlyassumed,Bell'stheoremexperiments(seebelow)denythepossibilityofhiddenvariablestoexistintheNature.2.Bell'stheorem(a)Here'stheformulationofBell'stheoremaccordingtoclassicalpaper[4].IsupposethatIshouldretelltheformulationinmanydetails,sinceareviewerofonerespectedRussianphysicalmagazinestatedthatIdon'tcatchtheessenceandformulationofthetheorem.Particularly,thatreviewerthinksthat:3Authors'remarkthatinformationonparticlesandcorrelatedquantityis,accordingto[4],localizedinparticlesthemselves,isnotvalid.Descriptionin[4]isnotinlanguageofparticles,suchanotioncannotevenbestrictlydefined.So,Icite:ConsideranensembleofcorrelatedpairsofparticlesmovingsothatoneentersapparatusIa,andtheotherapparatusIb,whereaandbareadjustableapparatusparameters.IneachВкаждомapparatusaparticlemustselectoneoftwochannelslabeled+1and-1.LettheresultoftheseselectionsberepresentedbyA(a)andB(b),eachofwhichequals±1accordingasthefirstorthesecondchannelisselected