CPT Invariance and the Spin-Statistics Connection PDF
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CPT INVARIANCE AND THE SPIN–STATISTICS CONNECTION CPT Invariance and the Spin–Statistics Connection Jonathan Bain DepartmentofTechnology,CultureandSociety,TandonSchool ofEngineering,NewYorkUniversity 3 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©JonathanBain2016 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2016 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2015951929 ISBN978–0–19–872880–1 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. Acknowledgments Thisworksprangoutofseveralinitiallydisparatetrainsofthoughtandcoalesced in the guise of various workshop and conference presentations, and a smatter- ing of journal articles. I would first like to thank the Department of Technology, Culture and Society at NYU-Tandon for providing me with a reduced teaching loadforthesemesterduringwhichthebulkofthebookwaswritten.Iwouldalso liketothankaudiencesatvariousmanifestationsoftheInternationalConference on the Ontology of Spacetime, the Philosophy of Science Association Biennial Meeting, the British Society for Philosophy of Science Annual Conference, the European Philosophy of Science Association Biennial Conference, the UK and EuropeanMeetingontheFoundationsofPhysics,andtheAnnualMeetingofthe SocietyforExactPhilosophy.The2011BoulderConferenceontheHistoryand PhilosophyofScienceallowedmetopresentincoherentthoughtsontheCPTand spin–statisticstheoremsandnon-relativisticfieldtheories,asdidWesternUniver- sity’s 2009 Workshop on the Philosophy of Quantum Field Theory. I’d also like tothanktheorganizersandparticipantsintheNewYork/NewJerseyPhilosophy of Science Group for providing a forum for foundational issues in the Big City. Finally,I’dliketothankMonaforallthefigbars,amongotherthings. Material in Chapters 1, 2, and 3 has appeared in the articles “Relativity and quantum field theory,” in V. Petkov (ed.) Space, Time, and Spacetime, Berlin: Springer,129–146,(2010);“CPTinvariance,thespin–statisticsconnection,and the ontology of relativistic quantum field theories,” Erkenntnis 78, 797–821, (2013); and “Pragmatists and purists on CPT invariance in relativistic quantum field theories,” forthcoming in European Philosophy of Science Association 2013 Conference Proceedings. I would like to thank Springer Verlag for permission to reproducethismaterial.MaterialinChapter3hasappearedinthearticle“Quan- tum field theories in classical spacetimes and particles,” Studies in History and PhilosophyofModernPhysics42,98–106,(2011),<http://www.sciencedirect.com/ science/journal/13552198>. Thanks to Elsevier for permission to include this materialinthecurrentwork. Introduction This is a book on the philosophy of quantum field theory that seeks to answer the question “What explains CPT invariance and the spin–statistics connection (SSC)?” CPT invariance is the property of being invariant under the combined transformationsofchargeconjugation(C),spaceinversion(Pfor“parity”),and time reflection (T). It forges a link between matter and antimatter states on the one hand (which are related by charge conjugation), and spacetime symmetries ontheother.TheSSCisthepropertythatholdsofastatejustwhen,ifthestate ischaracterizedbyFermi–Diracstatistics,thenitpossesseshalf-integerspin,and if the state is characterized by Bose–Einstein statistics, then it possesses integer spin. It thus forges a link between the spin property of a physical system and the statistics it obeys; in particular, it provides a foundation for Pauli’s exclusion principle.PhysicistsappealtotheCPTandspin–statisticstheoremsinrelativistic quantumfieldtheories(RQFTs)foranexplanationoftheseproperties,insofaras these theorems entail that any state of a physical system described by an RQFT must possess them. This appeal, however, is made problematic by the fact that there are multiple, and in some cases, mutually incompatible, ways of deriving thesetheoremswithinRQFTs.Moreover,theSSCplaysanessentialroleinexpla- nations of phenomena described by both non-relativistic quantum field theories (NQFTs) and non-relativistic, non-field-theoretic quantum mechanics (NQM). Inthesetheories,thespin–statisticstheoremfails,asdoestheCPTtheorem.Iwill arguethattheseconsiderationsentailthatnoneofthestandardaccountsofscien- tificexplanation succeedsinprovidinganunderstandingofCPTinvarianceand theSSC.Thegoalofthisbookistoworktowardsuchanunderstandingbyfirst providing an analysis of the necessary and sufficient conditions for these prop- erties, and second by advocating an account of explanation appropriate for this context.Underthisaccount,anexplanationisgivenofageneralregularitythatis expressed in one type of theory (NQFTs and NQM), and that can only be ad- equately understoodby an appeal toamorefundamentaltheory (RQFTs). The explanatory work is done in part by means of an appeal to intertheoretic rela- tions,andinpartbymeansofaderivation,withinthemorefundamentaltheory, based on a set of non-fundamental principles, i.e., principles that do not form an agreed-uponfoundationforallformulationsofthefundamentaltheory. CPTInvarianceandtheSpin–StatisticsConnection.JonathanBain. ©JonathanBain2016.Published2016byOxfordUniversityPress. 2 CPTInvarianceandtheSpin–StatisticsConnection Inthecourseofthisdiscussion,otherissuesinthephilosophyofquantumfield theory will arise. These issues include the roles that relativity and locality play in RQFTs, the debate over which formulation of RQFTs should be adopted to address foundational issues, the role of renormalized perturbation theory in in- teracting RQFTs, and the nature of the intertheoretic relations between RQFTs on the one hand, and NQFTs and NQM on the other. I hope to provide clar- ity on these issues by viewing them through the specific lens of the CPT and spin–statisticstheorems.Onereoccurringfeatureofthediscussionwillbeitscom- parison of pragmatist and purist formulations of QFTs. A received view among philosophersofphysicsmaintainsthat,whenitcomestoaddressingfoundational issues associated with RQFTs, we should adopt rigorous mathematical, or as I shall call them, purist formulations of these theories. Little work has been done byphilosophersinemployingthemethodsassociatedwithheuristic,orasIshall callthem,pragmatistformulationsofRQFTs.Thisisunfortunatesinceitisthese latterformulationsthatappearintextbooksandthatareusedbypracticingphysi- ciststoderiveandtestthepredictionsofRQFTs.Whilethisbookisnotintended asadefenseofpragmatismoverpurity,itdoesseektorestoresomebalancetothe debateamongphilosophers.1Asweshallsee,attheendoftheday,bothpositions facesimilarfoundationalissues. The significance of CPT invariance and the SSC extends beyond the philos- ophyofquantumfieldtheorytoimpactbroaderphilosophicaldiscussionsonthe nature of time reversal invariance, and the concept of indistinguishability. Note thatifCPTinvarianceholds,thenaviolationofCPinvarianceentailsaviolation ofTinvariance.2 ThustheCPTtheoremnotonlyentailsthatCPTinvarianceis anessentialpropertyinRQFTs,italsoentailsthatinRQFTsinwhichCPinvar- iance is violated, so too is time reversal invariance. This entailment underwrites an inference found in the physics literature from observations of CP-violation in the decay of neutral kaons to the claim that such decays are evidence for T- violation (such decay processes involve weak interactions; hence evidence that they are T-violating is evidence that the electroweak theory is T-violating). Thus explainingCPTinvarianceinRQFTsprovidesthebasisforanexplanationofwhy asignificantsectoroffundamentalphysics(i.e.,electroweaktheory)distinguishes thepastfromthefuture.Ontheotherhand,itshouldbenotedthatT-violationcan alsobederivedwithouttheassumptionofCPTinvariance.Roberts(2014a)iden- tifiestwosuchinferencestoT-violationthatappearinthephysicsliterature(one involvesobservationsofdifferencesintransitionprobabilitiesforneutralkaonand B-meson oscillations, the other involves potential observations of fundamental electric dipole states), neither of which requires the assumption of CPT invari- ance. Note, finally that there is a large body of recent philosophical literature on the topic of T-violation (for a summary and critique, see Roberts, 2014b), with someauthorspointingoutthathowoneinterpretstimereversalinvariancewillde- terminehowoneinterpretsCPTinvariance(e.g.,ArntzeniusandGreaves,2009; Greaves, 2010). With respect to these claims, one should note that the connec- tion between CPT invariance and T-violation (via CP-violation) is independent ofhoweitherisinterpreted. Introduction 3 The SSC links the statistics of a quantum state with its spin. The statistics of a quantum state underwrites a concept of indistinguishability for the physi- cal system it characterizes. For instance, if the physical system is a collection of particles, then, in some sense, they are indistinguishable. But just how to flesh out this sense is a matter of some debate in the philosophical literature. A sub- stantial body of work attempts to flesh out this sense in terms of the principle of the identity of indiscernibles (PII). A summary and critique of this work is givenbyCaulton(2013),whocharacterizesitinthefollowingway:earlyfolklore claimedthatthePIIholdsforfermions(i.e.,particlesthatobeyFermi–Diracsta- tistics),butnotforbosons(i.e., particlesthatobeyBose–Einsteinstatistics),and thisisbecausebosonscanbeinthesamestate,whereasfermionscannot.Thenew folklore identifies differing degrees of discernibility, and claims that one of these (“weakdiscernibility”)allowsforthePIItoholdforbosonsandfermions(aswell as particles that obey parastatistics). The SSC adds a further dimension to this debate insofar as it connects issues concerning the identity of quantum systems, asencodedintheirstatistics,withthepropertyofspin;althoughitremainstobe seen if this connection has philosophical significance. In particular, if there were reasonstobelievethatthepossessionofspinisrelatedtoidentity,independently of statistics, then explaining the SSC would provide the basis for an explanation ofidentityascriptionstofermionsandbosons. Physicists tend to view the explanatory power of the SSC in more concrete terms. Statistics, we are told, characterizes how a state behaves under permuta- tions.Again,thisiseasiesttothinkofintermsofastatethatrepresentsacollection of particles. If such a multi-particle state is invariant under a permutation of its single-particlesubstates,thelattercanbethoughtofasrepresentingindistinguish- ableparticles.Intuitively,switchingtheorderofanytwoindistinguishableparticles does not affect the entire collection. Such collections of indistinguishable parti- cles may in addition be characterized by an exclusion principle, which prohibits any two particles from being in exactly the same single-particle substate.3 Indis- tinguishable particles that do not obey the exclusion principle are defined to be bosons, whereas indistinguishable particles that obey the exclusion principle are defined to be fermions. A key difference between bosons and fermions is that a collection of bosons can be prepared so that all of them are characterized by the samesingle-particlesubstate.Allempiricalevidencesuggeststhattheparticlesde- scribed by quantum theories are either bosons or fermions.4 Moreover, our best quantum theories of matter (as encoded in the Standard Model) represent fun- damental matter states (leptons and quarks) as possessing half-integer spin and interactingviatheexchangeofintegerspincarriersoffundamentalgaugeforces (spin-0photonsascarriersoftheelectromagneticforce,spin-1W andZ bosons as carriers of the weak force, spin-1 gluons as carriers of the strong force, and thespin-1Higgsbosonthat“carries”mass).TheexplanatorypoweroftheSSC can now be stated: on the basis of our best theories of matter, the SSC entails that fundamental matter states must be fermions that obey the exclusion prin- ciple, whereas the states of carriers of the fundamental forces must be bosons.
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