A Review of the Decoherent Histories Approach to Quantum Mechanics

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|α󰀐󰀏α|,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

|Ψ󰀐arean

orthogonal(butingeneralincomplete)set.TherethereforeexistsasetofprojectionoperatorsRβ

Cα

NotethattheCα

atanytimeafterthefinaltime.Thedecoherencefunctionalforsuchhistoriesis

󰀅󰀆

†′′D(α|α)=TrRβ|Ψ󰀐󰀏Ψ|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

11

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

15

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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