The Nine Lives of Schroedinger's Cat

arXiv:quant-ph/9501014v527Jan1995UniversityofLondonImperialCollegeofScience,TechnologyandMedicineDepartmentofPhysicsTheNineLivesofSchr¨odinger’sCatOntheinterpretationofnon-relativisticquantummechanicsZviSchreiberRakachInstitu[e-mail]ialfulfilmentoftherequirementsforthedegreeofMasterofScienceoftheUniversityofLondonOctober1994ContentsIIntroduction1I.1Introduction2I.1.1Abriefhistoryofquantummechanics..................2I.1.2Theinterpretationalissues........................4I.1.3Outlineofthesis..............................6IIQuantummechanics7II.1Formulationofquantummechanics8II.1.1Diracnotation...............................8II.1.2Systemsandobservables.........................8II.1.3Formulation................................9II.1.4TheHeisenbergpicture..........................14II.1.5QuantisationandtheSchr¨odingerrepresentation...........15II.1.6Mixedstatesandthedensitymatrix..................17II.1.7TheLogico-algebraicapproach......................21II.1.8Spinhalf..................................24II.2Histories25IIIInterpretation33III.1TheOrthodoxinterpretation34III.2Bohr’sinterpretation40III.3Mindcausescollapse46III.4Hiddenvariables53III.5Many-worldsinterpretation62III.6Many-mindsinterpretation69III.7Bohm’sinterpretation73III.8Decoherenthistories(Ontology)82III.9Decoherenthistories(Epistemology)89IVConclusion94IV.1Conclusion95PartIIntroductionThemostremarkableaspectofquantummechanicsisthatitneedsinterpretation.Neverbeforehastherebeenasituationinwhichaphysicaltheoryprovidescorrectpredictionsforthebehaviourofasystemwithoutprovidingaclearmodelofthesystem.Bywayofintroducingtheinterpretationalissues,PartIbrieflysketchesthehistoricaldevelopmentoftheformalismandconceptsofquantummechanics.(Adetailedhistorymaybefoundin[Jam89].)1ChapterI.1IntroductionI.1.1AbriefhistoryofquantummechanicsInclassicalphysics,someaspectsofnaturearedescribedusingparticles,othersusingwaves.Light,inparticular,wasdescribedbywaves.In1900,MaxPlanckshowedthatthespectrumofblackbodyradiation,whichcouldnotbeexplainedintermsofwaves,couldbeexplainedbytheassumptionthatlightisemittedindiscretequantaofenergy[Pla90,Pla91].Later,in1905,AlbertEinsteinshowedthatthephotoelectriceffect,whichalsodidnotfitthewavemodel,couldbeexplainedbytheassumptionthatlightisalwaysquantised[Ein05].Thiscontrover-sialsuggestionwasnotaccepteduntilafterthefamousexperimentsofArthurHollyComptonin1923[Com23a,Com23b].In1923,LouisdeBrogliesuggestedthat,conversely,‘particles’mightdisplaywave-likeproperties[Bro23b,Bro23c,Bro23a,Bro24].Thiswaslaterconfirmed,moststrikinglybytheDavisson-Germerexperimentof1927[DG27].In1925,WernerHeisenbergintroducedmatrixmechanics[Hei25].Thisformalismwasabletopredicttheenergylevelsofquantumsystems.ItwascastintheformofawavefunctionanddifferentialequationbyErwinSchr¨odingerinthefollowingyear[Sch26].TheeffortsofPaulDirac[Dir26],PascualJordan[Jor27]andothersatsynthesisingthetwoapproaches(thesocalledtransformationtheory)ledtothemoregeneralformalismofquantummechanics[Neu55].Thesedevelopmentsmarkedaturningpointinphysics.Forthefirsttime,physi-calresultswerebeingderived,notfromamodeloftheuniverse,butfromabstractmathematicalconstructssuchastheSchr¨odingerwavefunction.Therewasanun-precedentedneedforinterpretation.In1926,MaxBornsuggestedthatthevaluesinaparticle’swavefunctiongavetheprobabilitiesoffindingtheparticleinagivenplace[Bor26].Thisreallysetthecatamongstthepigeons.Thesuggestionthatthelawsofphysicswerenon-deterministic2CHAPTERI.1.INTRODUCTION3flewinthefaceofeverythingachievedsinceNewton.Inretrospect,theworkofBornprecipitatedanevenmoreradicalrevolution:aswitchinphysicsfromontology(discussionofwhatis)toepistemology(discussionofwhatisknown).AccordingtoBorn,anexperimenttodeterminethepositionofaquantumsystemwouldgiveresultswithprobabilitiesdictatedbythewavefunction.Thisstatementtalksaboutwhatresultsareobtainedinsteadofmodellingtheprocess.Itinvolvesanartificialdivisionoftheworldintosystemandobserver.Thefollowinginterpretationalquestionswereraised.•Acceptingquantummechanicsasanepistemologicaltheory,isthereaspecificplaceatwhichtheobserver-systemcutshouldbelocated?Inmoreconcreteterms:aretherecertainsystemswhichmaybedescribedbyawavefunctionandotherswhichmaynot?Ifthereisnosuchfixedobserver-systemcut,isitconsistenttoallowthiscuttobemadeatdifferentplacesaccordingtoconvenience?Inotherwords,maywealwayschoosehowmuchofasystemtodescribebyawavefunction?•Isthereanontologicaltheoryunderlyingtheepistemologicalformulationofquan-tummechanics?Inotherwords,isthereamodeloftheuniversefromwhichwecanderivethefactthatcertainsystemsmaybetreatedusingquantummechanics?Ifsuchamodelisproposed,doesitexplainthenon-determinismofquantumme-chanicsintermsofanunderlyingdeterministicmechanism?Inresponsetothesequestions,twomainschoolsofthoughtarose.Bohrandhisfollowers(notablyHeisenberg)embracedtheepistemologicalviewpointandarguedthatitcannotbepossibletofindanunderlyingontologicaltheory.Einsteinandhisfollowers(notablySchr¨odinger)