The Contextual Character of Modal Interpretations

arXiv:0705.1660v1[quant-ph]11May2007TheContextualCharacterofModalInterpretationsofQuantumMechanicsGracielaDomenech∗1,3,HectorFreytes2andChristiandeRonde3,41.InstitutodeAstronom´ıayF´ısicadelEspacio(IAFE)CasilladeCorreo67,Sucursal28,1428BuenosAires,Argentina2.DipartimentodiScienzeePedagogicheeFilosofiche-UniversitadegliStudidiCagliariViaIsMirrionis1,09123,Cagliari-Italia3.CenterLeoApostel(CLEA)4.FoundationsoftheExactSciences(FUND)BrusselsFreeUniversity-Krijgskundestraat33,1160Brussels-BelgiumAbstractInthisarticlewediscussthecontextualcharacterofquantummechanicsintheframeworkofmodalinterpretations.WeinvestigateitshistoricaloriginandrelatecontemporarymodalinterpretationstothoseproposedbyM.BornandW.Heisenberg.Wepresentthenageneralcharacterizationofwhatweconsidertobeamodalinterpretation.FollowingpreviouspapersinwhichwehaveintroducedmodalitiesintheKochen-Speckertheorem,weinvestigatetheconsequencesofthesetheoremsinrelationtothemodalinterpretationsofquantummechanics.IntroductionModalpropositionsrefertopossibilityandnecessity,aboutwhat‘mustbe’andwhat‘maybe’thecase.Ifonethinksabouthowthingsareintheactualworld,thenonemayalsothinkofhowthingsmighthavebeeninanalternative,nonactualbutpossible,stateofaffairs.Modalitieswereintroducedinphysicsbystatisticalmechanicsinordertotakeintoaccountpossibleconfigurationsofphysicalsystemsregardlessofwhatisactuallythecase.Incontemporaryphysics,thedevelopmentofquantumtheorieshasthrownnewlighttotheproblemoftherelationbetweenpossibleandactual.Weintendtoanalyzetherˆoleofmodalitiesinquantummechanics,preciselyinwhatsensewemaysaythatasystempossiblypossessesaproperty.Wealsostudywhethertheadditionofmodalpropositionstothelatticeofactualpropositionswouldcircumventthecontextualcharacterofthequantumdiscourseaboutproperties.Thepaperisorganizedinfivesections.Insection1,westudythemodalcharacterofearlyinterpreta-tionsofquantummechanics.Insection2,wediscussthecontextualcharacterofthetheory.Insection3,weoutlineabriefreviewofcontemporarymodalinterpretations.Wethenpresent,insection4,ageneralcharacterizationoftheseinterpretations.Insection5weanalyzetheircontextualcharacter.Finally,wepresentabriefdiscussion.1EarlyModalInterpretationsofQuantumMechanicsThefirstconsistentformalizationofthemethodsappliedtothestudyofquantumphenomenawasdevel-opedintheformof‘matrixmechanics’firstundertheideasofW.Heisenberg,andthenincollaborationwithM.BornandP.Jordan,duringthesecondhalfof1925(Heisenberg,1925;BornandJordan,1925;Born,HeisenbergandJordan,1926):∗FellowoftheConsejoNacionaldeInvestigacionesCient´ıficasyT´ecnicas(CONICET)1“[...]inthesummerof1925itledtoamathematicalformalismcalledmatrixmechanicsor,moregenerally,quantummechanics.TheequationsofmotionofNewtonianmechanicswerereplacedbysimilarequationsbetweenmatrices;itwasastrangeexperiencetofindthatmanyoftheoldresultsofNewtonianmechanics,likeconservationofenergy,etc,couldbederivedalsointhenewscheme.LatertheinvestigationsofBorn,JordanandDiracshowedthatthematricesrepresentingpositionandmomentumoftheelectrondidnotcommute.Thislatterfactdemonstratedclearlytheessentialdifferencebetweenquantummechanicsandclassicalmechanics.”W.Heisenber(1958,p.41)InJanuary1926E.Schr¨odingercompletedthefirstpartofhisapparentlyquitedifferentapproachtothesameproblemintermsofeigenfunctionsofadifferentialequation,whichisknownas‘wavemechanics’(Schr¨odinger,1926a).Schr¨odingerhadreadEinstein’spaperonthequantumtheoryoftheidealgas.Herecognizedlateritwastheseideas,duetoEinsteinanddeBroglie,whichallowedhimtodevelophisowntheory(Jammer,1966,p.257).ByMarch1926,Schr¨odingerinterpretedthewavefunctionψasa‘mechanicalfieldscalar’(mechanischerFeldskalar)inconnectionwiththeclassicaltheoryofelectro-magneticradiation.Followingthisidea,theeigenvalueswheretakenasfrequencyvalues(Schr¨odinger,1926c).Thequantumpostulatewasthenconceivedinconnectionwithresonancephenomena,makingpossibletorejecttheexistenceofdiscreteenergylevelsandquantumjumps.Schr¨odinger’swavetheorywaswellreceivedbytheorthodoxphysicscommunitywhichwantedtoputanendtothedeviationsoftheatomictheoryfromtheclassicalconceptionofnaturalsciencesdevelopedfromNewtontoEinstein.AtthetimeHeisenbergwasquiteunhappyaboutthisturn:“InJuly[1926]IvisitedmyparentsinM¨unichandonthisoccasionIheardalecturegivenbySchr¨odingerforthephysicistsinM¨unichabouthisworkonwavemechanics.ItwasthusthatIfirstbecameacquaintedwiththeinterpretationSchr¨odingerwantedtogivehismathematicalformalismofwavemechanics,andIwasdisturbedabouttheconfusionwithwhichIbelievedthiswouldburdenatomictheory.Unfortunately,nothingcamefrommyattemptduringthediscussiontoputthingsinorder.MyargumentthatonecouldnotevenunderstandPlank’sradiationlawonthebasisofSchr¨odinger’sinterpretationconvincednoone.WilhemWien,whoheldthechairofexperimentalphysicsattheUniversityofM¨unich,answeredrathersharplythatonemustputanendtoquantumjumpsandthewholeatomicmysticism,andthedifficultiesIhadmentionedwouldcertainlysoonbesolvedbySchr¨odinger.”W.Heisenberg(quotedfromWheelerandZurek,1983,p.56)Schr¨odingerrelatedthematterfieldandtheelectromagneticemissionorabsorptionofatomics