Time-Symmetrized Counterfactuals in Quantum Theory

Recently, several authors have criticized the time-symmetrized quantum theory originated by the work of Aharonov et al. (1964). The core of this criticism was a proof, appearing in various forms, which showed that the counterfactual interpretation of time-

8

9

9

1

l

Ju

7

2

1

v

5

7

7

8

9

/h

p

-

t

n

a

u

:qv

i

X

r

aTIME-SYMMETRIZEDCOUNTERFACTUALSINQUANTUMTHEORYLevVaidmanSchoolofPhysicsandAstronomyRaymondandBeverlySacklerFacultyofExactSciencesTelAvivUniversity,Tel-Aviv69978,Israel.AbstractRecently,severalauthorshavecriticizedthetime-symmetrizedquantumtheoryoriginatedbytheworkofAharonovetal.(1964).Thecoreofthiscriticismwasaproof,appearinginvariousforms,whichshowedthatthecounterfactualinter-pretationoftime-symmetrizedquantumtheorycannotbereconciledwithstandardquantumtheory.Iargueherethattheapparentcontradictionisduetoalogicalerror.Ianalyzetheconceptofcounterfactualsinquantumtheoryandintroducetime-symmetrizedcounterfactuals.Thesecounterfactualsdonotleadtoanycon-tradictionwiththepredictionsofquantumtheory.Idiscussapplicationsoftime-symmetrizedcounterfactualstoseveralsurprisingexamplesandshowtheusefulnessofthetime-symmetrizedquantumformalism.

Recently, several authors have criticized the time-symmetrized quantum theory originated by the work of Aharonov et al. (1964). The core of this criticism was a proof, appearing in various forms, which showed that the counterfactual interpretation of time-

1.Introduction.Ishalldiscussmeasurementsperformedonapre-andpost-selectedquantumsystem,i.e.atatimebetweentwoothermeasurements.Thetime-symmetrizedformalismdescribingsuchsystemswasproposedbyAharonov,Bergmann,andLebowitz(ABL)(1964)andhasbeendevelopedinrecentyears.ApartiallistofreferencesincludesAharonovetal.(1985),AharonovandVaidman(1990,1991).Severalauthorscriticizedthetime-symmetrizedapproachtoquantumtheoryingeneralandsomeofitsparticularapplications.ThemostrepresentativeexampleistheworkofSharpandShanks(1993).Theypresentedaproof,whichwaslaterrepeatedandusedbyothers,thatthecounter-factualinterpretationoftheABLprobabilityrule(Eq.2below)cannotbereconciledwiththestandardquantumtheory.Ishallclaimherethattheproofcontainsalogicalerror.Ishallanalyzetheconceptofcounterfactualsinquantumtheoryandintroducetime-symmetrizedcounterfactuals.TheABLrulecanbeappliedtosuchcounterfactualsandIshallpresentsituationsforwhichitisuseful.

Theplanofthisworkisasfollows.InSection2Ipresentabriefreviewofthetime-symmetrizedformalism.Insection3Ianalyzetheconceptofcounterfactualsinquantumtheoryandintroducetime-symmetrizedcounterfactuals.Section4isdevotedtotheanalysisoftheinconsistencyproofofSharpandShanksanditsvariations.Insection5Idiscussrelatedtimeasymmetrypreconceptionsinquantumtheory.Insection6thetimesymmetry(andasymmetry)oftheprocessofquantummeasurementisanalyzedinordertogivearigorouscontexttothepreviousdiscussionofthetimesymmetryoftheABLrule.Applicationsoftimeasymmetricandtimesymmetriccounterfactualsinquantumtheoryareconsideredinsections7-9.Section10summarizetheargumentsofthepaper.

2.Time-SymmetrizedFormalism.Instandardquantumtheoryacompletedescrip-tionofasystematagiventimeisgivenbyaquantumstate|Ψ .ItyieldstheprobabilitiesforalloutcomesaiofameasurementofanyobservableAaccordingtotheequation

Prob(ai)=| Ψ|PA=ai|Ψ |,(1)

wherePA=aiistheprojectionoperatoronthesubspacede nedbyA=ai.Eq.1isintrinsicallyasymmetricintime:thestate|Ψ isdeterminedbysomemeasurementsinthepastanditevolvestowardthefuture.Thetimeevolutionbetweenthemeasurements,however,isconsideredtimesymmetricsinceitisgovernedbytheSchr¨odingerequationforwhicheachforwardevolvingsolutionhasitscounterpart(itscomplexconjugatewithsomeotherwellunderstoodsimplechanges)evolvingbackwardintime.Theasymmetryintimeofthestandardquantumformalismismanifestedintheabsenceofthequantumstateevolvingbackwardintimefromfuturemeasurements(relativetothetimeinquestion).Time-symmetrizedquantumtheorydescribesasystematagiventimebyatwo-statevector Ψ2||Ψ1 .Ityieldsthe(conditional)probabilitiesforalloutcomesaiofameasurementofanyobservableAaccordingtothegeneralizationoftheABLformula(AharonovandVaidman,1991):

Prob(ai)=| Ψ2|PA=ai|Ψ1 |2

Recently, several authors have criticized the time-symmetrized quantum theory originated by the work of Aharonov et al. (1964). The core of this criticism was a proof, appearing in various forms, which showed that the counterfactual interpretation of time-

Thetimesymmetrymeansthat Ψ2|and|Ψ1 entertheequations,andthusgoverntheobservableresults,onequalfootings.Moreover,thetimesymmetrymeansthat,inregardtotimesymmetricmeasurements,asystemdescribedbythetwo-statevector Ψ2||Ψ1 isidenticaltoasystemdescribedbythetwo-statevector Ψ1||Ψ2 .Iwillanalyzethetimesymmetryoftheprocessofmeasurementinsection6;hereIonlypointoutthatidealmeasurementsaretimesymmetric.Indeed,thesymmetryundertheinterchangeof Ψ2|and|Ψ1 isexplicitinEq.2whichreferstoidealmeasurements.Anotherbasicconceptoftime-symmetrizedtwo-statevectorformalismisweakvalue.An(almost)standardmeasurementprocedureformeasuringobservableAwithweakenedcoupling(whichwecallweakmeasurement,AharonovandVaidman1990)yieldstheweakvalueofA: Ψ2|A|Ψ1 Aw≡

Recently, several authors have criticized the time-symmetrized quantum theory originated by the work of Aharonov et al. (1964). The core of this criticism was a proof, appearing in various forms, which showed that the counterfactual interpretation of time-

amiracle(aviolationofthefundamentallawofnature)forA?DoesAcomebyitself,oritisaccompaniedbyotherchanges?However,thesequestionsarenotrelevanttotheproblemofcounterfactualsinquantumtheory.ThequestionsaboutAarenotrelevantbecauseAdependssolelyonanexternalsystemwhichisnotunderdiscussionbythede nitionoftheproblem.Indeed,inquantumtheorythecounterfactualshav …… 此处隐藏：31742字，全部文档内容请下载后查看。喜欢就下载吧 ……