APPROACHTOQUANTUMMECHANICS
J.J.Halliwell
TheoryGroup,BlackettLaboratory,ImperialCollege
London,SW72BZ,UnitedKingdom
ABSTRACT:Ireviewthedecoherent(orconsistent)historiesapproachtoquantummechanics,duetoGriffiths,toGell-MannandHartle,andtoOmn`es.Thisisanapproachtostandardquantumtheoryspecificallydesignedtoapplytogenuinelyclosedsystems,uptoandincludingtheentireuniverse.Itdoesnotde-pendonanassumedseparationofclassicalandquantumdomains,onnotionsofmeasurement,oroncollapseofthewavefunction.Itsprimaryaimistofindsetsofhistoriesforclosedsystemsexhibitingnegligbleinterference,andtherefore,towhichprobabilitiesmaybeassigned.Suchsetsofhistoriesarecalledconsistentordecoherent,andmaybemanipulatedaccordingtotherulesofordinary(Boolean)logic.Theapproachprovidesaframeworkfromwhichonemaydiscusstheemer-genceofanapproximatelyclassicaldomainformacroscopicsystems,togetherwiththeconventionalCopenhagenquantummechanicsformicroscropicsubsystems.Inthespecialcaseinwhichthetotalclosedsystemnaturallyseparatesintoadistin-guishedsubsystemcoupledtoanenvironment,thedecoherenthistoriesapproachisclosedrelatedtothequantumstatediffusionapproachofGisinandPercival.
(Toappearinproceedingsoftheconference,FundamentalProblemsinQuantumTheory,
Baltimore,June18-22,1994,editedbyD.Greenberger)ImperialCollegepreprintIC/93-94/52.July1994
1.Introduction
Quantummechanicswasoriginallydevelopedtoaccountforanumberofun-explainedphenomenaontheatomicscale.Thetheorywasnotthoughttobeapplicabletophysicsatlargerscales,norwastheirfeltanyneedtodoso.Indeed,itwasonlybyreferencetoanexternal,classical,macroscopicworldthatthetheorycouldbeproperlyunderstood.Thisviewofquantummechanics,theCopenhageninterpretation,haspersistedforaverylongtimewithnotoneshredofexperimentalevidenceagainstit[1].
arXiv:gr-qc/9407040v1 27 Jul 1994Today,however,moreambitiousviewsofquantummechanicsareentertained.Experimentshavebeencontemplated(e.g.,involvingSQUIDS)thatmayprobedomainstraditionallythoughtofasmacroscopic[2].Evenintheabsenceofsuchexperiments,theCopenhageninterpretationrestsonunsatisfactoryfoundations.Macrosopicclassicalobjectsaremadefrommicroscopicquantumones.ThedualistviewoftheCopenhageninterpretationmaythereforebeinternallyinconsistent,andisatbestapproximate.Mostsignificantly,therehasbeenaconsiderableamountofrecentinterestinthesubjectofquantumcosmologyinwhichthenotionofanexter-nalclassicaldomainiscompletelyinappropriate[3].Generalizationsofconventionalquantumtheoryarerequiredtomeetthesenewchallenges.
JohnWheelerwasoneoftheveryfirstpeopletobesoboldastoeventalkabout“thewavefunctionoftheuniverse”[4].Hehascontributedextensivelytoourunderstandingofquantummechanicsandquantumcosmology,boththroughhisownwork,andthroughhisinspirationofmanyothersinthefield.Itisagreatpleasuretocontributetothismeetingorganizedinhishonour.1.1TheHistoriesApproach
Theobjectofthispaperistoreviewoneparticularapproachtoquantumme-chanicsthatwasspecificallydesignedtoovercomesomeoftheproblemsoftheorthodoxapproach.Thisisthedecoherent(or“consistent”)historiesapprochduetoGriffiths[5,6,7,8,9],Gell-MannandHartle[10,11,12,13,14,15,16,17,18,19]andOmn`es[20,21,22,23,24,25,26].Itis,inparticular,apredictiveformulationofquan-tummechanicsforgenuninelyclosedquantumsystemsthatissufficientlygeneraltocopewiththeneedsofquantumcosmology.Inbrief,itsaimsareasfollows:1.Tounderstandtheemergenceofanapproximatelyclassicaluniversefromanun-derlyingquantumone,withoutbecomingembroiledinthedetailsofobservers,measuringdevicesorcollapseofthewavefunction.Predictionofaclassicaldomainsimilartotheoneinwhichwelivewillgenerallydependontheini-tialconditionoftheuniverse,andmoreover,couldbeoneofmanypossibilitiespredictedbyquantummechanics.Accommodation,ratherthanabsolutepre-diction,ofourparticularclassicaluniversemaybeasmuchascanbeexpected.2.Tosupplyaquantum-mechanicalframeworkforreasoningaboutthepropertiesofclosedphysicalsystems.Suchaframeworkisnecessaryiftheprocessofpre-2
dictioninquantummechanicsistobegenuinelyquantum-mechanicalateverysinglestep.Thatprocessconsistsoffirstlogicallyreconstructingthepasthis-toryoftheuniversefromrecordsexistingintheclassicaldomainatthepresent,andthenusingthepresentrecordstogetherwiththededucedpasthistorytomakepredictionsaboutthefuture(strictlyspeaking,aboutcorrelationsbetweenrecordsatafixedmomentoftimeinthefuture).Aframeworkforreasoningmayalsoleadtoclarificationofmanyoftheconceptuallytroublesomeaspectsofquantummechanics,suchastheEPRparadox.
Inmoredetail,theprimarymathematicalaimofthehistoriesapproachistoassignprobabilitiestohistoriesofaclosedsystem.Theapproachisamodestgener-alizationofordinaryquantummechanics,butreliesonafarsmallerlistofaxioms.TheseaxiomsarebasicallythestatementsthattheclosedsystemisdescribedbytheusualmathematicalmachineryofHilberttogetherwithaformulafortheprob-abilitiesofhistoriesandaruleofinterpretation.Itmakesnodistinctionbetweenmicroscopicandmacroscopic,nordoesitassumea“system-environment”split;inparticular,aseparateclassicaldomainisnotassumed.Itmakesnoessentialuseofmeasurement,orcollapseofthewavefunction,althoughthesenotionsmaybediscussedwithintheframeworkoftheapproach.Whatreplacesmeasurementisthemoregeneralandobjectivenotionofconsistency(orthestrongernotionofdeco-herence),determiningwhichhistoriesmaybeassignedprobabilities.Theapproachalsostressesclassical(i.e.Boolean)logic,theconditionsunderwhichitmaybeapplied,andthus,theconditionsunderwhichordinaryreasoningmaybeappliedtophysicalsystem.
Thedecoherenthistoriesapproachisnotdesignedtoanswerthequestionheldbysometobethemostimportantproblemofquantummeasurementtheory:whyoneparticularhistoryfortheuniverse“actuallyhappens”whilsttheotherpotentialhistoriesallowedbyquantummechanicsfadeaway.Althoughsomeaspectsofthisproblemareclarifiedbythedecoherenthistoriesapproach,asatisfactorysolutiondoesnotappeartobepossibleunlesssomethingexternalisadded(seeRef.[27],forexample).Noristheapproachintendedtomeetsomephilosophicalprejudiceaboutthewaytheworldappearstobe.Itsaimsareforthelargepartofaratherpragmaticnature,namelyansweringtheveryphysicalquestionofwhytheworldisdescribedsowellbyclassicalmechanicsandordinarylogic,whenitsatomicconstituentsare
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describedbyquantummechanics.1.2Whyhistories?
Thebasicbuildingblocksinthedecoherenthistoriesapproacharethehistoriesofaclosedsystem–sequencesofalternativesatasuccessionoftimes.Whyaretheseobjectsofparticularinterest?
(a)Historiesarethemostgeneralclassofsituationsonemightbeinterestedin.In
atypicalexperiment,forexample,aparticleisemittedfromadecayingnucleusattimet1,thenitpassesthroughamagneticfieldattimet2,thenitisabsorbedbyadetectorattimet3.
(b)Wewouldliketounderstandhowclassicalbehaviourcanemergefromthequan-tummechanicsofclosedsystems.Thisinvolvesshowing,amongstotherthings,thatsuccessivepositionsintimeofaparticle,say,areapproximatelycorrelatedaccordingtoclassicallaws.Thisinvolvestheprobabilitiesforapproximatepo-sitionsatdifferenttimes.
(c)Thebasicpragmaticaimoftheoreticalphysicsistofindpatternsinpresently
existingdata.Incosmology,forexample,onetriestoexplaintheconnectionsbetweenobserveddataaboutthemicrowavebackground,theexpansionoftheuniverse,thedistributionofmatterintheuniverse,thespectrumofgravita-tionalwaves,etc.Why,then,shouldwenotattempttoformulateourtheoriesinthetermsofthedensitymatrixoftheentireuniverseatthepresentmoment?Thereareatleasttworeasonswhynot.First,presentrecordsarestoredinawidevarietyofdifferentways–incomputermemories,onphotographicplates,onpaper,inourownpersonalmemories,inmeasuringdevices.Thedynamicalvariablesdescribingthoserecordscouldbeveryhardtoidentify.Thecorrela-tionsbetweenpresentrecordsarefareasiertounderstandintermsofhistories.Thepatternsincurrentcosmologicaldata,forexample,areexplainedmosteco-nomicallybyappealingtothebigbangmodelofthehistoryoftheuniverse.Second,thecorrelationbetweenpresentrecordsandpasteventscanneverbeperfect.Inordertodiscusstheapproximatenatureofcorrelationsbetweenthepastandthepresentitbecomesnecessarytotalkaboutthehistoriesofa
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system.
2.TheFormalismofDecoherentHistories
Inowbrieflyoutlinethemathematicalformalismofthedecoherenthistoriesapproach.Furtherdetailsmaybefoundintheoriginalpaperscitedabove.2.1ProbabilitiesforHistories
Inquantummechanics,propositionsabouttheattributesofasystematafixedmomentoftimearerepresentedbysetsofprojectionsoperators.TheprojectionoperatorsPαeffectapartitionofthepossiblealternativesαasystemmayexhibitateachmomentoftime.Theyareexhaustiveandexclusive,
α
Pα=1,PαPβ=δαβPα
(2.1)
Aprojectorissaidtobefine-grainedifitisoftheform|αα|,where{|α}areacompletesetofstates;otherwiseitiscoarse-grained.Aquantum-mechanicalhis-1(t),···Pn(t),toryischaracterizedbyastringoftime-dependentprojections,Pααnn11
togetherwithaninitialstateρ.Thetime-dependentprojectionsarerelatedtothe
time-independentonesby
k(t)=eiH(tk−t0)Pke−iH(tk−t0)Pααkkk
(2.2)
whereHistheHamiltonian.Thecandidateprobabilityforsuchhistoriesis
n11np(α1,α2,···αn)=TrPαn(tn)···Pα1(t1)ρPα1(t1)···Pαn(tn)(2.3)Thisexpressionisafamiliaronefromquantummeasurementtheory,butthein-terpretationisdifferent.Hereitistheprobabilityforasequenceofalternativesforaclosedsystem.Thealternativesateachmomentoftimearecharacterizedbyprojectors.Theprojectorsaregenerallynotassociatedwithmeasurements,astheywouldbeintheCopenhagenviewoftheformula(2.3).Theycannotbebecausethesystemisclosed.
Itisstraightforwardtoshowthat(2.3)isbothnon-negativeandnormalizedtounitywhensummedoverα1,···αn.However,(2.3)doesnotsatisfyalltheaxioms
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ofprobabilitytheory,andforthatreasonitisreferredtoasacandidateprobability.Itdoesnotsatisfytherequirementofadditivityondisjointregionsofsamplespace.Moreprecisely,foreachsetofhistories,onemayconstructcoarser-grainedhistoriesbygroupingthehistoriestogether.Thismaybeachieved,forexample,bysummingovertheprojectionsateachmomentoftime,
¯αP¯=
Pα
(2.4)
α∈α¯
(althoughthisisnotthemostgeneraltypeofcoarsegraining).Theadditivityrequirementisthenthattheprobabilitiesforeachcoarser-grainedhistoryshouldbethesumoftheprobabilitiesofthefiner-grainedhistoriesofwhichitiscomprised.Quantum-mechanicalinterferencegenerallypreventsthisrequirementfrombeingsatisfied;thushistoriesofclosedquantumsystemscannotingeneralbeassignedprobabilities.
Thestandardillustrativeexampleisthedoubleslitexperiment.Thehistoriesconsistofprojectionsattwomomentsoftime:projectionsdeterminingwhichslittheparticlewentthroughattimet1,andprojectionsdetermingthepointatwhichthe
particlehitthescreenattimet2.Asiswell-known,theprobabilitydistributionfortheinterferencepatternonthescreencannotbewrittenasasumoftheprobabilitiesforgoingthrougheachslit;hencethecandidateprobabilitiesdonotsatisfytheadditivityrequirement.
Thereare,however,certaintypesofhistoriesforwhichinterferenceisnegligible,andthecandidateprobabilitiesforhistoriesdosatisfythesumrules.Thesehistoriesmaybefoundusingthedecoherencefunctional:
′n11n
D(α)=TrPαn(tn)···Pα1(t1)ρPα′(t1)···Pα′(tn)
1
n
(2.5)
Hereα
,α,α
thanthatoriginallyintroducedbyGriffiths[5].SeeRef.[12]foradiscussionofthispoint).
2.2ConsistencyandClassicalLogic
Whyaresetsofconsistenthistoriesareofinterest?Asstated,propositionsabouttheattributesofaquantumsystemmayberepresentedbyprojectionoper-ators.Thesetofallprojectionshavethemathematicalstructureofalattice.Thislatticeisnon-distributive,andthismeansthatthecorrespondingpropositionsmaynotbesubmittedtoBooleanlogic.Similarremarksholdforthemorecomplexpropositionsexpressedbygeneralsetsofquantum-mechanicalhistories.
ThereasonwhyconsistentsetsofhistoriesareofinterestisthattheycanbesubmittedtoBooleanlogic.Indeed,atheoremofOmn`esstatesthatasetofhistoriesformsaconsistentrepresentationofBooleanlogicifandonlyifitisaconsistentset[20,25,26].Thatis,inaconsistentsetofhistories,eachhistorycorrespondstoapropositionaboutthepropertiesofaphysicalsystemandwecanmeaningfullymanipulatethesepropositionswithoutcontradictionusingordinaryclassicallogic.Itisinthissensethatthedecoherenthistoriesapproachsuppliesafoundationforreasoningaboutclosedphysicalsystems.
Animportantexampleisthecaseofretrodictionofthepastfrompresentdata.Supposewehaveaconsistentsetofhistories.Wewouldsaythatthealternativeαn(presentdata)impliesthealternativesαn−1···α1(pastevents)if
p(α1,···αn)
p(α1,···αn−1|αn)≡
consistentsetofhistoriesarelabeled“true”[28](seealsoRef.[29]).Theexistenceoftheseso-calledmultiplelogicsmeansthatonecannotsaythatpastpropertiescorrespondingtoreliablepropositions“actuallyhappened”,becausetheydependonaparticularchoiceofconsistenthistories.Inthehistoriesapproach,therecon-structionofhistoryfrompresentrecordsisthereforenotunique.Thismeansthattheapproachdoesnotingeneralallowonetotalkaboutthepasthistoryoftheuniverse“thewayitreallyis”.
Isthisaproblem?Somefeelthatitis[30].Fortheimmediatepracticalpur-posesofquantumcosmology,however,itdoesnotappeartobeadifficulty.Recallthatwhatquantummechanicsmustultimatelyexplainisthecorrelationbetweenrecordsatafixedmomentoftime.Asstatedearlier,itiseasiesttounderstandthosecorrelationsintermsofhistories,buthistoriesenterasanintermediatestep.Thecorrelationsbetweentworecordsatafixedmomentoftimepredictedbyquan-tummechanicsareunambiguous,eventhoughthehistoriescorrespondingtotheserecordsmaynotbeunique.
3.Decoherence,CorrelationandRecords
Howmaytheconsistencycondition(2.6)cometobesatisfied?Firstofall,itisstraightforwardtoshowthat,withsomeexceptions,historiesofcompletelyfine-grainedprojectionoperatorswillgenerallynotleadtoconsistency.Theconsistencyconditionisgenerallysatisfiedonlybysetsofhistoriesthatarecoarse-grained.Whensetsofhistoriessatisfytheconsistencycondition(2.6)asaresultofcoarse-graining,theytypicallysatisfy,inaddition,thestrongerconditionthatboththerealandimaginarypartsoftheoff-diagonaltermsofthedecoherencefunctionalvanish,
D(α
′)
=0,forα
′
(3.1)
ThisIshallrefertoquitesimplyasdecoherence.(Itissometimesreferredtomorespecificallyasmediumdecoherence[12]butweshallnotdosohere).
Physically,decoherenceisintimatelyrelatedtheexistenceofrecordsaboutthesystemsomewhereintheuniverse.Inthissensedecoherencereplacesandgeneral-izesthenotionofmeasurementinordinaryquantummechanics.Setsofhistoriesde-cohere,andhencethesystem“acquiresdefiniteproperties”,notnecessarilythrough
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measurement,butthroughtheinteractionsandcorrelationsofthevariablesthatarefollowedwiththevariablesthatareignoredasaresultofthecoarse-graining.
Decoherenceistypicallyonlyapproximatesomeasuresofapproximatedeco-herencearerequired.First,notethatthedecoherencefunctionalobeysthesimpleinequality[31],
|D(α
′)|2
≤D(α)D(α
′)
(3.2)
Intuitively,thisresultindicatesthattherecanbenointerferencewithahistorywhichhascandidateprobabilityzero.Italsosuggestsapossiblemeasureofap-proximatedecoherence:wesaythatasystemdecoherestoorderǫifthedecoher-encefunctionalsatisfies(3.2)withafactorofǫ2multiplyingright-handside.Thisconditionmaybeshowntoimplythatmost(butnotall)probabilitysumruleswillthenbesatisfiedtoorderǫ[31].
Approximatedecoherencetoorderǫmeansthattheprobabilitiesaredefinedonlyuptothatorder.Intypicalcases,ǫissubstantiallysmallerthananyothereffectthatcouldconceivablymodifytheprobabilities,andhencetheymaybethoughtofaspreciselydefinedforallpracticalpurposes.Alternatively,ithasbeenconjecturedthatagenericapproximatelydecoherentsetofhistoriesmaybeturnedintoanexactlydecoherentsetbymodifyingtoorderǫtheoperatorsprojectedontoateachmomentoftime[30].
3.1RecordsImplyDecoherence
Inowexemplifytheconnectionbetweenrecordsanddecoherence.ConsideraclosedsystemSwhichconsistsoftwoweaklyinteractingsubsystemsAandB.TheHilbertspaceHofSisthereforeoftheformHA⊗HB.ForsimplicityletHAandHBhavethesamedimension.SupposeweareinterestedinthehistoriescharacterizedsolelybypropertiesofsystemA,thusBisregardedastheenvironment.Thesystemisanalyzedusingthedecoherencefunctional(2.5),wherewetakethePαtodenoteaprojectiononHA(hencetheprojectionsinthedecoherencefunctionalareoftheformPα⊗IB,whereIBdenotestheidentityonHB).IalsointroduceprojectionsRβontheHilbertspaceHB.
IshallshowthathistoriesofAsatisfythedecoherencecondition(3.1)ifthesequencesofalternativesthehistoriesconsistofexhibitexactandpersistentcorre-9
lationswithsequencesofalternativesofB.Tobeprecise,supposethatthealter-kateachmomentoftimetareperfectlyrecordednativesofAcharacterizedbyPαkk
inBasaresultoftheirinteraction.SupposealsothatthisrecordinBisperfectly
persistent(i.e.,permanent).ThismeansthatatanytimetfafterthetimetnofthelastprojectiononAthereexistasequenceofalternativesofB,β1,···βn,thatareinperfectcorrelationwiththealternativesofA,α1···αnattimest1···tn.
Foreachmomentoftimetk,thedecoherencefunctional(2.5)maybewritten,
BAkkBk′(3.4)TrI⊗Rβk···Pαk⊗I···ρ···Pα′⊗I···D(α)=
k
βk
k,wherethedotsdenotetheprojectionsusingtheexhaustivityoftheprojectionsRβ
k
attimesotherthantkandtheunitaryevolutionoperatorsbetweenthem.Now,
kaprojector,itmaybereplacedby(Rk)2.Furthermore,theassumptionsinceRββ
kthroughalltheunitaryofpersistencethenallowsustomovetheprojectorRβ
k
evolutionoperatorsoccuringaftertimetkoneachsideofthedecoherencefunctional,
k
k
withtheresult,
D(α
′)
=
βk
k⊗Rk···ρ···Pk⊗Rk···Tr···Pαβkβkα′k
k
(3.5)
Finally,theassumedcorrelatedbetweenthealternativeαkinAandβkinBmeans
k⊗RkoneachsidewillyieldzerowhenoperatingonthatthetermsoftheformPαkβ
k
everythingthatcameearlierinthechain,unlessαk=βk.Eq.(3.5)willtherefore
bediagonalinαk.Repeatingtheargumentforallothervaluesofk,wethusfindthat,asadvertized,aperfectandpersistentcorrelationofalternativesofAwiththoseofBleadstoexactdecoherenceofthehistoriesofA.Itisnotjusttheconsistencycondition(2.6)thatissatisfiedthroughpersistentcorrelationwithanothersubsystem,butthestrongerconditionofdecoherence,(3.1).ThisargumentwasinspiredbyanargumentgivenbyHartle[14]inhisdiscussionoftherecoveryoftheCopenhageninterpretationfromthedecoherenthistoriesapproach.AmoredetailedversionofitisgiveninRef.[32].3.2DecoherenceImpliesGeneralizedRecords
Thereisaconversetotheaboveresult,namelythatEq.(3.1),inacertainsense,impliestheexistenceofrecords[12].Considerthedecoherencefunctional(2.5),for
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anysystem(notjustthespecialonediscussedabove).Introducetheconvenientnotation
Cα
|Ψarean
orthogonal(butingeneralincomplete)set.TherethereforeexistsasetofprojectionoperatorsRβ
Cα
NotethattheCα
atanytimeafterthefinaltime.Thedecoherencefunctionalforsuchhistoriesis
†′′D(α|α)=TrRβ|ΨΨ|Cα(3.11)Theseextendedhistoriesdecohereexactlybyvirtueof(3.10)and(3.1),andthusthediagonalelementsof(3.11),whichwedenotep(αEq.(3.10)impliesthatp(α
)=δαp(α
),aretrueprobabilities.Theandβ
correlationscontainedintheseprobabilitiesmaythereforebediscussed.Indeed,
β
|Ψ(3.10)
arereferredtoasgeneralizedrecords:informationaboutthehistoriescharac-terizedbyalternativesα1···αnisrecordedsomewhere.Itis,however,notpossible
tosaythattheinformationresidesinaparticularsubsystem,sincewehavenotspecifiedtheformofthesystemS;indeed,itisgenerallynotpossibletodivideitintosubsystems.
4.TowardsaQuasiclassicalDomain
Giventheframeworksketchedabove,oneoftheprincipleaimsofthedecoherenthistoriesapproachistodemonstratetheemergenceofanapproximatelyclassicalworldfromanunderlyingquantumone,togetherwiththequantumfluctuations
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aboutitdescribedbythefamiliarCopenhagenquantummechanicsofmeasuredsubsystems.Suchastateofaffairsisreferredtoasaquasiclassicaldomain[11,12,13].Inmoretechnicalterms,aquasiclassicaldomainconsistsofadecoherentsetofhistories,characterizedlargelybythesametypesofvariablesatdifferenttimes,andwhoseprobabilitiesarepeakedaboutdeterministicevolutionequationsforthevariablescharacterizingthehistories.
Thehistoriesshould,moreover,bemaximallyrefinedwithrespecttoaspecifieddegreeofapproximatedecoherence.Thatis,onespecifiesadecoherencefactorǫintheapproximatedecoherenceconditiondiscussedabove.Thisshould,forexample,bechosensothattheprobabilitiesaredefinedtoaprecisionfarbeyondanyconceiv-abletest.Then,thehistoriesshouldbefine-grained(e.g.,byreducingthewidthsoftheprojections)tothepointthatfurtherfine-grainingwouldleadtoviolationofthespecifieddegreeofapproximatedecoherence.Theresultingsetofhistoriesarethencalledmaximallyrefined.Thereasonformaximallyrefiningthehistoriesistoreduceasmuchaspossibleanyapparentsubjectiveelementinthechoiceofcoarse-graining.
GiventheHamiltonianandinitialstateofthesystem,one’staskistocomputethedecoherencefunctionalforvariousdifferentchoicesofhistories,andseewhichonesleadtoquasiclassicalbehaviour.AssuggestedbythediscussionattheendofSection2,therecouldbe–andprobablyare–manysuchsetsofvariablesleadingtoquasiclassicalbehaviour.Animportantproblemistofindasmanysuchsetsaspossibleanddevelopcriteriatodistinguishbetweenthem.Oneusefulcriterioniswhetheraquasiclassicaldomaincansupporttheexistenceofaninformationgather-ingandutilizingsystem,orIGUS.Thisisacomplexadaptivesystemthatexploitstheregularitiesinitsenvironmentinsuchawayastoensureitsownsurvival.Thisparticularcriterionmayruleoutdomainsdescribedbyparticularlybizarredecoher-entsetsofhistories,suchasonesdescribedbycompletelydifferentvariablesateachmomentoftime,becausetheIGUSmaynothavesufficientinformationprocessingcapabilitiestoassimilateitsenvironment.Also,criteriasuchastheexistenceofIGUSesalleviatetosomedegreethemultiplicityofconsistentsetsofhistoriesdis-12
cussedinSection2.TheseissuesarediscussedfurtherinRefs.[11,12,13,30,33,34,35]4.1HistoriesofHydrodynamicVariables
Whatarethesetsofvariablesthatcanleadtoquasiclassicalbehaviour?Oneparticularsetofvariablesthatarestrongcandidatesforitaretheintegralsoversmallvolumesoflocallyconserveddensities.Agenericsystemwillusuallynothaveanaturalseparationinto“system”and“environment”,anditisoneofthestrengthsofthedecoherenthistoriesapproachthatitdoesnotrelyonsuchaseparation.Certainvariableswill,however,bedistinguishedbytheexistenceconservationlawsfortotalenergy,momentum,charge,particlenumber,etc.Associatedwithsuchconservationlawsarelocalconservationlawsoftheform
∂ρ
concernedquantumBrownianmotionmodels,primarilybecausecalculationscanbecarriedoutwithcomparativeease[12,31].Thesehaveprovedtobequiteinstructive.Verybriefly,suchmodelsconsistofaparticleofmassMinapotentialV(x)linearlycoupledtoanenvironmentconsistingofalargebathofharmonicoscillatorsinathermalstateattemperatureT,andcharacterizedbyadissipationcoefficientγ.ThetypesofhistoriescommonlyconsideredaresequencesofapproximatepositionsoftheBrownianparticle,specifieduptosomewidthσ,whilsttheenvironmentofoscillatorsistracedover.
Theresultsmaybrieflybesummarizedasfollows.Decoherencethroughinter-actionwiththeenvironmentisanextremelyeffectiveprocess.Forexample,foraparticlewhosemacroscopicparameters(mass,timescale,etc.)areoforder1inc.g.s.units,andforanenvironmentatroomtemperature,thedegreeofapproxi-40matedecoherenceisoforderexp−10,averysmallnumber.Theprobabilitiesforhistoriesofpositionsarethenstronglypeakedabouttheclassicalequationsofmotion,butmodifiedbyadissipationterm,
Mx¨+Mγx˙+V′(x)=0
(4.3)
Therearefluctuationsaboutclassicalpredictability,consistingoftheubiquitousquantumfluctuations,adjoinedbythermalfluctuationsfromtheinteractionwiththeenvironment.Thereisagenerallyatensionbetweenthedemandsofdecoherenceandclassicalpredictability,duetothefactthatthedegreeofdecoherenceimproveswithincreasingenvironmenttemperature,butpredictabilitydeteriorates,becausethefluctuationsabout(4.3)grow.However,iftheparticleissufficientlymassive,itcanresistthethermalfluctuationsandacompromiseregimecanbefoundinwhichthereisareasonabledegreeofbothdecoherenceandclassicalpredictability.5.DecoherentHistoriesandQuantumStateDiffusion
Thedecoherenthistoriesapproachiscloselyconnectedtothequantumstatediffusion(QSD)approachtoopensystems.Inthatapproach,themasterequationforthereduceddensityoperatorofanopensystem(essentiallyaclosedsysteminwhichonefocusesonaparticularsubsystem)issolvedbyexploitingapurelymathematicalconnectionwithacertainnon-linearstochasticSchr¨odingerequation
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(Itoequation)[38].SolutionstotheItoequationturnouttocorrespondrathercloselytotheresultsofactuallaboratoryexperiments(e.g.,inquantumoptics),andarethereforeheldtodescribeindividualsystemsandprocesses.Forexample,inaquantumBrownianmotionmodel,thesolutionstotheItoequationbecomelocalizedaboutpointsinphasespacefollowingtheclassicalequationsofmotion.Theconnectionwiththedecoherenthistoriesapproachisthat,looselyspeaking,thesolutionstotheItoequationmaybethoughtofastheindividualhistoriesbelongingtoadecoherentset[39].Moreprecisely,thevariablesthatlocalizeintheQSDap-proachalsodefineadecoherentsetofhistoriesinthedecoherenthistoriesapproach.Thedegreesoflocalizationandofdecoherencearerelated,andtheprobabilitiesassignedtohistoriesineachcaseareessentiallythesame.Thisconnectioncouldbeaveryusefulone,bothconceptuallyandcomputationally,andeffortstoexploititarebeingmade.
6.WhatHaveWeGained?
InthiscontributionIhavetriedtogiveabriefoverviewofthedecoherenthistoriesapproachtoquantumtheory.Whathasthedecoherenthistoriesapproachtaughtus?
Atthelevelofordinaryquantummechanics,appliedtolaboratorysituations,twothingshavebeengained.Firstofall,aminimalviewofthedecoherenthistoriesapproachisthatitisinasenseamorerefinedversionoftheCopenhageninterpre-tation.Itrestsonaconsiderablesmallernumberofaxioms,andinparticular,itisapredictiveformulationofquantummechanicsthatdoesnotrelyonanykindofassumptionsreferringtomeasurementortoaclassicaldomain.ItisinternallyconsistentandreproducesalltheexperimentalpredictionsoftheCopenhagenap-proach.Secondly,itprovidesaclearsetofcriteriafortheapplicationofordinarylogicinquantummechanics.Sincemanyoftheconceptualdifficultiesofquantummechanicsareessentiallylogicalones,e.g.,theEPRparadox,aclarificationoftheapplicabilityoflogichasbeenarguedtoleadtotheirresolution[7,21,24].SucharesolutionisnotstrictlypossibleinCopenhagenquantummechanics,becauseitdoesnotofferclearguidelinesfortheapplicationofordinarylogic.
Therewill,ofcourse,alwaysbesomewhoclaimthattheycanfinessetheirway
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throughanydifficultyofquantummechanicswithouthavingtoworryaboutthesomewhatcumbersomemachineryofthehistoriesapproachdescribedhere.Inthisconnection,Omn`eshastosaythefollowing[26]:
“Itmaybetrue,assomepeoplesay,thateverythingisinBohr,butthishasbeenamatterforhermeneutics,withtheendlessdisputesanyscripturewillleadto.Itmayalsohappenthatheguessedtherightanswers,butthepedagogicalmeansandthenecessarytechniquedetailswerenotyetavailabletohim.Sci-encecannot,however,proceedbyquotations,howeverelevatedthesource.Itproceedsbyelucidation,sothatfeatsofgeniuscanbecomeordinarylearningforbeginners.”
Intuitionalonemaybesufficienttoseesomethroughthedifficultiesofnon-relativisticquantummechanics,butifwearetoextendquantumtheorytotheentireuniverse,areliablevehiclefortravelbeyondthedomainofourintuitionisrequired.Forquantumcosmology,thedevelopmentofthedecoherenthistoriesapproachhasbeenaconsiderablebonus.Thedecoherenthistoriesapproachsuppliesanunambiguous,workableandpredictiveschemeforactuallyapplyingquantummechanicstogenuinelyclosedsystems.Furthermore,asdiscussedatsomelengthinthispaper,itsuppliesaconceptuallyclearmethodofdiscussingtheemergenceofclassicalityinclosedquantumsystems,andthisisperhapsitsgreatestsuccess.
Stilloutstandingarethelargelytechnicaldifficultiesofquantumcosmologycon-nectedwithquantizinggravity.However,itispossiblethatthehistoriesapproachmightbeofusetherealso.Thefocusonhistoriesmaycircumventthe“problemoftime”encounteredinmostcanonicalapproachestoquantumgravity.Ishamandcollaborators,forexample,arecurrentlyexploringthepossibilityofhistories-basedformulationsofquantumtheorythatdonotrelyontheconventionalHilbertspacestructure,orontheexistenceofapreferredtimecoordinate[40,41,42],buildingonanearliersuggestionofHartle[16].Muchremainstobedone,butonbothconcep-tualandtechnicalgrounds,thehistoriesapproachtoquantumcosmologyappearstobeaparticularlypromisingavenueforfutureresearch.
Furtheraspectsofthedecoherenthistoriesapproacharediscussedin
16
Refs.[43,44,45,46,47,48,49,50,51,52,53,,55,56,57,58,59,60,61,62,63]7.Acknowledgments
Iamgratefultotheorganizersforgivingmetheopportunitytotakepartinsuchaninterestingmeeting.Iwouldalsoliketothanknumerouscolleaguesforusefulconversations,especiallyLajosDi´osi,FayDowker,MurrayGell-Mann,Nico-lasGisin,JimHartle,ChrisIsham,AdrianKent,SethLloyd,RolandOmn`es,IanPercival,TrevorSamols,DieterZehandWojciechZurek.ThisworkwassupportedbyaUniversityResearchFellowshipfromtheRoyalSociety.8.References
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