JournalofMathematicalPhysics556{572(1975).Observables,operators,andcom - PDF document

JournalofMathematicalPhysics556{572(1975).Observables,operators,andcomplexnumbersintheDiractheoryDavidHestenesThegeometricformulationoftheDiractheorywithspace-timealgebraisshowntobeequivalenttotheusualmatrixformalism.ImaginarynumbersintheDiractheoryareshowntoberelatedtothespintensor.Therelationofobservablestooperatorsandthewavefunc-tionisanalyzedindetailandcomparedwithsomepurportedlygeneralprinciplesofquantummechanics.AnexactformulationofLarmorandThomasprecessionsintheDiractheoryisgivenforthe¯rsttime.Fi-nally,somebasicrelationsamonglocalobservablesinthenonrelativisticlimitaredetermined.IntroductionTheDiracelectrontheoryiswidelyacknowledgedtobethemostpreciseavailablequantumtheoryofasingleparticle.YettheprinciplesandinterpretationofquantumtheorycontinuetobediscussedatgreatlengthintheliteraturewithouttakingtheDiractheoryintoaccount.ThoughthemathematicalformalismoftheDiractheoryunfortunatelydoesnotuniquelydetermineitsinterpretation,oneshouldexpectareasonableinterpretationtoprovidesomeunderstandingofthemathematicalstructureofthetheory.Onthisscoremanywidelyproclaimedgeneralprinciplesofquantumtheoryprovetobeprofoundlyde¯cient,ifnotsubstantiallywrong.Forexample,iftheuncertaintyprincipleisbasictotheinterpretationofquantummechanics,whyhasitneverbeengivenaclearformulationinthefullDiractheory?AndifPlanck'sconstantprovidesanabsolutelimitontheprecisionofmeasurements,whydoesitalsodeterminethemagnitudeofthespin?ThispaperaimstoclarifythegeometricalandphysicalinterpretationoftheDiractheoryanddeterminesomeofitsimplicationsfortheinterpretationofquantumtheoryingeneral.WithaformalismdevelopedinRefs.l,2.3theDiractheorycanbegivenacompletelygeometricalformulation,involvingneithermatricesnorcomplexnumbers.Intheprocess,thefollowingsigni¯cantfactsareestablishedconcerningtheinterpretationofthetheory:(l)ThefourgammamatricesplayacompletelygeometricalroleintheDiractheory.Theyarematrixrepresentationsoffourorthonormalvectors(notfourmatrixcomponentsofasinglevectorasoftensuggested).Thealgebraofgammamatriceshasthesamegeometricalsigni¯canceastensoralgebra.ThegammamatriceshavenofunctionintheDiractheorywhichentitlesthemtoberegardedasoperatorswithanyspecialquantum-mechanicalsigni¯cance.(2)TheunitimaginaryinthecustomaryformulationoftheDirachasade¯nitegeometricalandphysicalsigni¯cance.Itisasuper°uousadjunctofthemathematicalformalisminthesensethatitcanbereplacedbythetensorquantityitactuallyrepresents.Speci¯cally,thequantity ,whereveritappearsinequationsoftheDiractheory,isarepresentationofthespintensorbyoneofitseigenvalues.When hasbeenreplacedintheformalismbythespinbivectoritrepresents,antiparticleconjugationcanbeidenti¯edasageometricalofobservablesofthetheory.(3)TheDiracequationandthewavefunctioncanbeeliminatedfromtheDiractheory,leavingasetofconservationlawsandconstitutiverelationsforobservables.TheabovefactsandfurtherdetailshavealreadybeenestablishedinRefs.2and3.However,thesepapersareopentomisunderstanding,becausetheydonotexplainindetailexactlyhowtheyarerelatedtotheconventionalmatrixformulationoftheDiractheory.Thispaperaimstosupplythemissingdetailsandmakeiteasytotranslateexpressionsofthematrixformulationintothegeometricallanguageandvice-versa.The¯rstfewsectionsaredevotedtothisend,withemphasisonestablishingthe¯rsttwofactscitedabove.Section3derivesthegeometricalformoftheDiracequation,whichinvolvesnocomplexnumbers.Sections4and5analyzetherelationofthefunctiontoobservablesindetail.InSec.5somecommonnotionsaboutacorrespondencebetweenobservablesandoperatorsarecriticizedinthelightoftheDiractheory.Correspondencesofobservableswiththewavefunctionareheldtobefundamental.Itissuggestedthatthesigni¯canceofeigenvaluesinquantumtheoryistobefoundmoreintheirassociationwithconstantandhomogeneouslocalobservablesthanwithoperators. Section6introducesthefundamentalrelativeobservablesandanalyzesthephysicalinterpretationoftheenergydensityinconsiderabledetail.Forthe¯rsttimeexpressionsfortheLarmorandThomasprecessionenergiesarederivedfromtheDiractheorywithoutanyapproximations.Section7provesthatinthenonrelativisticlimittheGordoncurrentisequaltothelocalmomentumdensitybutdi®ersfromthechargecurrentbythemagnetizationcurrentassociatedwiththespindensity.2.Geometricsigni¯canceoftheDiracmatricesRecallsomeofthefundamentalpropertiesoftheDiracmatrices(asdeveloped,forexampleinRef.4).TheDiracmatricesarede¯nedasasetofirreduciblematrices,whichsatisfytheanticommutationruleswherethe=0,l,2,3)arecomponentsofthespacetimemetrictensorandistheunitmatrix.Sinceitcanbeprovedthatthemustbe44matrices,theirirreducibilitycanbeexpressedbytheconditionthatthetraceof(2.1a)gives Conditions(2,1)canbesatis¯edonlybytracelessmatrices,thatis, =0(2Conditions(2.1)donotdeterminetheDiracmatricesuniquely.However,itcanbeshownthatanytwosetsofDiracmatricesarerelatedbyasimilaritytransformation,thatis,isanonsingularmatrix.Theoverthecomplexnumbersgeneratethecompletealgebraof4matrices,Butcertainelementsofthealgebraaredistinguishedbytheirassociation(2.1)withthespacetimealgebra,andthisgivestheentirealgebraageometricsigni¯cance.The44matrixalgebrawiththegeometricinterpretationinducedbytheconditions(2.1)iscalledtheDiracAlgebraClearly,(2.1)indirectlyassignssomegeometricsigni¯cancetotheDiracmatricesthemselves.Butintheusualapproachthefullgeometricsigni¯canceoftheisnotdetermineduntiltheirrelationtoaDiracspinorhasbeenspeci¯ed.Equivalentresultscanbeachievedmoree±cientlybyachangeinviewpoint,Toseehow,itisonlynecessarytorealizethatthekeyrelation(2.1a)doesnotdependinanyessentialwayontheassumptionthatthearematrices.A11thatisrequiredisthatthebelongtoanassociativenoncommutativealgebra.Theappropriatechangeinviewpointisachievedsimplybyinterpretingtheasvectorsofaspacetimeframeinsteadofasmatrices.Byde¯nitionthescalarproductsofthesevectorsarejustthecomponentsofthemetrictensor.Sothetwoequations(2.1a)and(2.1b)formatricescorrespondtothesingleequation forvectors.ThevectorsgenerateanassociativealgebraovertherealswhichhasbeendubbedthespacetimealgebrainRef.1,becauseitprovidesadirectandcompletealgebraiccharacterizationofthegeometricpropertiesofMinkowskispacetime.Byprovidingageometricalinterpretationofthewhichdoesnotdependonthenotionofspinor,geometricalgebrareleasesthefromtheircon¯nesinrelativisticquantumtheoryandelevatesthemtoacentralpositioninthemathematicaldescriptionofallphysicalsystemsinspacetime.TheadvantagesofusingspacetimealgebrainplaceoftheusualtensormethodshavebeenadequatelydemonstratedinRefs.1,5,and6,towhichthereaderisreferredfordetails.OfcourseanyequationinthespacetimealgebracanbereexpressedasanequationintheDiracalgebra,butbesideshavingamoredirectgeometricinterpretation,thespacetimealgebraismathematicallymoree±cient,asshouldalreadybeevidentfromacomparisonof(2.1)with(2.4),Thisbecomesclearerasmanipulationswiththetwoalgebrasarecomparedinmoredetail. SincetheDiracmatricesin(2.1)canobviouslyberegardedasmatrixrepresentationsofthevectorsin(2.3).itisconvenienttopersistinusingthesamesymbolsforbothtofacilitatecomparisonofthealgebrasandtranslationofequationsfromonesystemtotheother,Forthesamereason,itisconvenienttoestablishacorrespondencebetweenthenomenclaturesofthetwosystems,Theelementsofthespacetimealgebraarecalledmultivectorsandeverysuchelementcanbeexpressedasapolynomialoftheoverthereals,whichhasatmost16linearlyindependentelements(seeRefs.1and5formoredetails).Amultivectorissaidtobeodd)ifitdoes(doesnot)changesignonreplacementofinitspolynomialrepresentation.Thesameterminologywillbeappliedheretomatrixrepresentationsofmultivectors.Ofcourse,theDiracalgebraadmitspolynomialsoftheoverthecomplexnumbers,but,incontrasttootherelementsofthealgebra,theunitimaginaryofthecomplex¯eldhasnogeometricsigni¯canceexcept,aswillbeshowninthenextsection,inconnectionwithspinors.Itmaybeworthpointingoutthata(iscommonlyintroducedasa`scalar'intheDiractheoryintwodistinctways,¯rstasarootofnegativetermsinandsecondasanessentialelementoftheDiracequation.Geometricalgebrashowsthat(hasadi®erentgeometricalsigni¯canceineachcase,anditprovidesthemachinerytokeepthedistinctionclear.Thus,Eq.(2.4)gives(nosumon).Moreparticularly,=1hasthetimelikevectorasa\rootofunity,"while1hasthespacelikevectorasa\rootofminusone."Thisshowsthatinthe¯rstoftheabovecasesthe\scalar("servestodistinguishbetweenspacelikeandtimelikevectors,andthatsuchaquantityisrenderedsuper°uousbytheuseofspacetimealgebra.Itwillbeshownlaterthatinthesecondcase(representsaspacelikebivector.InaccordancewithconventionsadoptedinRef.5,thescalarterminapolynomialrepresentationofamultivectorisdenotedbyandcalledthescalaror(0-vectorpart.From(2.2)itiseasytoprovethatingeneral whichshowsthat\scalarpart"spacetimealgebracorrespondsto\trace"intheDiracalgebra.Inparticular,from(2.1b)and(2.4) Tr(Moreover,Eq.(2.2)becomes(=0inspacetimealgebraandsimplysaysthatavectorisnotascalar.Thefactor in(2.5)and(2.6)hasnogeometricalimport.ItsappearanceisanotherindicationthatmatrixalgebraisnotideallysuitedtothegeometricalroleitplaysintheDiractheory.Hermitianconjugationplaysanimportantroleinmatrixtheory,butithasnogeometricalsigni¯canceintheDiracalgebraexceptinconnectionwithaspeci¯cmatrixrepresentation.Inspacetimealgebraasimilarroleisplayedbyanoperationcalledreversion.Thereverseofamultivectorisobtainedfrombyreversingtheorderof'sinitspolynomialrepresentation.TherelationofreversiontoHermitianconjugationwillbegiveninthenextsection.Equation(2.3)waspresentedasachangeinrepresentationoftheDiracmatrices,butthefactisthesameequationappearsinspacetimealgebrawherethenotionofrepresentationhasnosigni¯cance.Thegeometricalrequirementofspacetimealgebrathatthein(2.3)mustbevectorsentailsthattheycanbewrittenasalinearcombinationoftheso(2.3)mustbeinvariantunderreversion,fromwhichitfollowsthatcanbechosensothat.Thus(2.3)takesonthespecialformThisequationdescribesaLorentztransformationofaframeofvectorsintoaframe.Moreover,Eq.(2.7)canbesolvedforasafunctiontheandthealonewhichprovesthatisindeedamultivectorandthateveryLorentztransformationcanbeexpressedinthatform.ProperLorentztransformations(i.e.,transformationscontinuouslyconnectedtotheidentity)willbeofspecialinterestintheanalysisoftheDiractheory.Itcanbeshownthat(2.7)isaproperLorentztransformationifandonlyifisanmultivectorFromthisconditionitisonlyanalgebraicexercisetoshowthatcanbeputintheform isabivector.Withoutelaboratingontheimportantgeometricnotionofabivector,itissu±cientforpresentpurposestoremarkthatanybivectorcanbewritteninthe\polynomialform" wherethe 2[°¹º]´1 )(2provideacompletebasisforthespaceofbivectors,andthearealternativeexpressionsforthesixindependentscalarcoe±cients.FormoredetailsandproofsoftheassertionsinthisparagraphseeRef.1,Chap.IVandAppendixB.3.TheDiracEquationwithoutComplexNumbersAt¯rstsighttheDiracalgebraappearstobemoregeneralthanspacetimealgebra,becauseits\scalar¯eld"consistsofcomplexnumbersratherthanrealnumbersonly.But,itwillbeshownthattheimaginaryunitoftheDiracalgebraissuper°uousinthephysicaltheory,anditsuseservesonlytoobscurethegeometricalandphysicalinterpretation.Thissectionshowsthatcanbereplacedbyaspacelikebivectorand¯ndstheappropriateformulationsofthewavefunctionandDiracequationintermsofthespacetimealgebra.Thenextsectionshowsthatissuper°uousintheexpressionforobservablesoftheDiractheory.ADiracspinorªisacolumnmatrixwithfourcomplexcomponents,thatis,wherethe'sand'sarerealnumbersandisthe(ofthematrixalgebra.Therepresentation(3.1)intermsofcomponentspresumesaspeci¯crepresentationoftheDiracmatrices.Itisconvenienttoworkwiththesocalledstandardrepresentation:isthe22unitmatrixandwherethearetheusual22Paulimatrices,thatis,aretracelessHermitianmatricessatisfyingByvirtueofthefactthatthearerelatedtothespacetimemetricby(2.1),ªanditscomponentshaveade¯nitegeometricalsigni¯cance,asigni¯cancetobeascertainedbelowby¯ndinganexpressionforªintermsofspacetimealgebrawhichisindependentofthematrixrepresentation.AsexplainedinRefs.1and5,inthespacetimealgebrathequantitiesde¯nedby=1,2,3)aretobeinterpretedasvectorsrelativetotheinertialsystemspeci¯edbythetimelikevector.ThegenerateanalgebraovertherealswhichisisomorphictothePaulialgebra.Toemphasizethisfactitisconvenienttowriteinanalogyto(3.2b).Ontheotherhand,.Thisfactcanbeexpressedonlyina44matrixrepresentationofthe.The44matrixrepresentationsofthearecommonlydenotedbyintheliterature(e.g.,p.69ofRef.4),buttohelpkeepgeometricsigni¯cancetotheforethesymbolsareusedhere.Fromthestandardrepresentation(3.2), whichimplyNowintroduceabasisinspinorspacesuchthatSupposing(3.1)referstothisrepresentation,byusing(3.6c)toeliminate,and(3.6b)toeliminatetheimaginaryunit,ªcanbewrittenThusanyDiracspinorªcanbewrittenintheformcanbewrittendowndirectlyfromthecolumnmatrixform(3.1)byusingThesigni¯canceof(3.7)and(3.8)isthattheunitimaginaryhasbeeneliminatedtoexpressªuniquelyasafunctionofanmultivector.Butcanbeexpressedasanelementofthespacetimealgebraatoncesimplybyinterpretingthe'sasvectorsinsteadofmatrices.Aswillbeseen,thishelpstomakethegeometricalsigni¯canceofspinorsexplicit.Dirac'sequationforanelectronwithchargeandmassinanexternalelectromagnetic¯eldcanbe .With =°¹@¹¹=@ theDiracequationassumestheform ¡e cA¢ª:(3:11)5 ItiscrucialtonotethattheunitimaginaryintheDiracequationcannotbeabsorbedintothede¯nitionofthewhilebotharekeptreal.SincetheDiracequationdescribesaphysicalpropertyoftheelectron,thenontrivialexplicitappearanceofintheequationimpliesthathasaphysicalsigni¯cance.Thatsigni¯canceremainstobedetermined.intheDiracequationcanbereplacedbyamultivectorbyusing(3.7)alongwith(3.6b)toput(3.11)intheform 2°1¡e mcÃuwasinsertedintherightsideof(3.12)byusing(3.6a)tomakethecoe±cientsofevenmultivectors.doesnothaveaninverse,thecoe±cientsofin(3.12)canbeequated,because,as(3.6c)shows,evenmultivectorsoperatingongenerateacompletebasisforDiracspinors.Therefore,(3.12)yields 2°1¡e mcÃ;orequivalently, 2°1¡e mcðConversely,bymultiplying(3.13)ontherightbyandusing(3.6)and(3.7)theDiracequationisrecovered.So(3.13)isfullyequivalenttotheDiracequation(3.11),and,byusing(3.7),asolutionoftheoneequationcanbeeasilyexpressedasasolutionoftheother.Byinterpretingthe'sasvectorsinsteadofmatrices,Eq.(3.13)becomesanequationinspacetimealgebrawhichmayfairlybecalledtheDiracequationinthatlanguagebecauseofitsequivalencetotheusualmatrixequation.Likewise,itisappropriatetorefertoasainthespacetimealgebra.Inthespacetimealgebra(3.10)simplyexpressesthepotentialasalinearcombinationofbasisvectorsandthesocalled\Diracoperator"introducedby(3.9)canbedirectlyinterpretedasthederivativewithrespecttoaspacetimepointThemostsigni¯cantfeatureof(3.13)isthatthewhichappearsin(3.11)hasbeenreplacedbythebivector,becausebivectorshaveastraightforwardgeometricalinterpretationinspacetimealgebra.SotranslationoftheDiractheoryintothelanguageofspacetimealgebrapromisestorevealahiddensigni¯canceofimaginarynumbersintheDiractheory.Thispromisewillbecompletelyful¯lledinthenextsection.Thederivationof(3.13)madeuseofaspeci¯cmatrixrepresentationofthe.TheconsequenceofusinganyotherrepresentationcanbeascertainedbyconsideringanarbitrarychangeofbasisinDirac\spinspace"whichtakesNowthetransformationmatrixin(3.14)canbetakentobeanevenmultivector;forhasanoddpart,becauseof(3.6a)thatpartcanbemadeevenwithouta®ecting(3.14)bymultiplyingitontherightby;similarily,any\complexcoe±cient"inapolynomialrepresentationofcanbereplacedbyanevenmultivectorbyusing(3.6b).Byanargumenttobegivenlaterinconnectionwithamoreimportantissue,thefactthatisevenimpliesthatitcanbeputintheform=exp(satis¯es(2.8),isascalar,anditwillberecalled,satis¯es(3.4b).Theinverseof=exp(Hence(3.14)gives=exp(Equations(3.6a,b)canbetransformedintothenewrepresentationbymultiplyingthembyandusing(3.15)and(3.16)togetherwiththefactthatanticommuteswiththe;onegets wherethearerelatedtotheby(2.7).With(3.16),(3.7)canbeputintheformexp(Substituting(3.18a)intotheDiracequation(3.11),andusing(3.17a,b)togetanequationforthesamewaythat(3.6a,b)wereusedtogetanequationfor,onenoticesthatthefactorexp()beeliminatedyieldingtheequation Ã0°02°01¡e mcÃThishasthesameformas(3.13).Indeed(3.19)canbeobtaineddirectlyfrom(3.13)bymultiplyingitontherightbytheconstantfactor.Fornotethat,by(2.8)and(2.7),ÃRRwhichshowshowthetermsontheleftof(3.13)and(3.19)arerelated.Thusithasbeenprovedthattheformof(3.13)isuniquelydetermined,independentlyofthechoiceofarepresentationfortheDiracmatrices.Thespeci¯cationofvectorsin(3.13)isdeterminedonlytowithinaproperLorentztransformation;thisarbitrarinesscorrespondstothefreedomtochooseamatrixrepresentationfortheDiractheory.TheformoftheusualDiracequation(3.11)isrepresentationindependent.However,nosolutiontothatequationcanbeexhibitedwithoutchoosingaspeci¯crepresen-tation.Thesigni¯canceofsuchachoiceishiddeninthematrixformulationoftheDiractheory.Ithasbeenuncoveredabove.Achoiceofrepresentationamountstoaspeci¯ccorrespondencebetween44matricesandvectors.Thoughthechoiceistosomeextentarbitrary,ade¯nitechoicemustneverthelessbemade.Theexplicitappearanceof'sin(3.13)and(3.19)isaconsequenceofsuchachoice.Ithasalreadybeenpointedoutthatthisimpliesthatmustbeinterpretedgeometricallyasaspacelikebivector.Amorecompleteinterpretationwillbeprovidedinthenextsection.ItshouldbenotedthatinconventionaltreatmentsoftheDiractheory,itisprovedthattheDiracequationretainsitsformunderLorentztransformations.Thatproofiseasilytranslatedintothepresentlanguage,butitwouldbesuper°uous,forhavingeliminatedmatricesandinterpretedthey,,asvectors,wehaveputtheDiracequationintheform(3.13)whichismanifestlyindependentofcoordinates.Itcannotbeoveremphasizedthatthevectorsappearingin(3.13)neednotbeassociatedwithanycoordinateframe;theyaresimplyasetofarbitrarilychosenorthonormalvectors.AdoptionofacoordinateframewithasthetimecomponentisequivalentintheconventionaltheorytoadoptingamatrixrepresentationforwhichisHermitianandtheareanti-Hermitian.Adoptionofthestandardrepresentation(3.2)associatesby(3.6b)theofthematrixrepresentationwiththebivectorsin(3.13).Thus,thestandardrepresentationisdistinguishedbyrelatingHermitianconjugationandcomplexnumbersofthematrixalgebratointrinsicfeaturesoftheDiracequation.ThereisanalternativeformulationoftheDiracequationintermsofspacetimealgebrawhichshouldbementioned.De¯nethequantity©bytheequation (1+)(1+Multiplying(3.13)ontherightbyandnotingthat,onegets ©i¡e cA©:(3:21)7 Exceptforthechoiceofunitsandadi®erenceinsignwhichismerelyamatterofconvention,(3.21)isidenticaltoEq.(13.2)ofRef.1.Itisthe¯rstformulationoftheDiracequationintermsofspacetimeComparisonof(3.21)withthematrixDiracequation(3.11)suggeststhatthepseudoscalar(3.22)isthegeometricalquantitywhichcorrespondstotheimaginaryin(3.11).Butthisismisleading,becausecertainessentialgeometricalfeaturesoftheDiractheoryarehiddeninthestructureofasde¯nedby(3.20),muchastheyarehiddeninthepropertiesofthebasespinorin(3.7).Thesefeatureswere¯rstuncoveredinRef.2,whereitwasconcludedthat(3.13)isamuchmoresigni¯cantequationthan(3.21).TonaildowntheinterpretationofintheDiractheory,itisnecessarytoexaminethede¯nitionsofobservables.Thisisundertakeninthenextsection.4.ObservablesandtheWavefunctionThegeometricalsigni¯canceofthewavefunctionªinDirac'stheoryisdeterminedbyrequiringthatcertainbilinearfunctionsofªbetensors.Interpretationofthesetensorsasobservablesdeterminesthephysicalsigni¯canceofª.Thissectionexplainshowthesocalled\bilinearcovariants"oftheDiractheorycanbeexpressedintermsofspacetimealgebra.ThismakesitpossibletoprovideadirectinterpretationoftheDiracwavefunction,showingthattheinterpretationoftheasvectorsisincompleteaccordwiththeDiractheoryandsojusti¯edbyitssimplicity.Moreover,thegeometricalandphysicalinterpretationoftheunitimaginarytheDiractheoryisascertainedbyprovingthat isarepresentationofthespinbivectorbyoneofitseigenvalues.Equation(3.8)explicitlyshowsthatisasumofscalar,bivector,andpseudoscalarparts.Ofcourse,everyevenmultivectorhasthisproperty.Notethat,thatis,theunitpseudoscalarinvariantunderreversion.However,everybivectorchangessignunderreversionasisshownby.Hencecanbeobtainedfromsimplybychangingthesignofitsbivectorpart.Sinceisanevenmultivector,so,is.Butisclearlyinvariantunderreversion,soitsbivectorpartmustvanish.Moreover,sincecanbeputinthe\polarform"exp(arescalars.Onecanthende¯nebytheequationequation½exp( ,orjustwrite exp(Becauseof(3.9),Theexpression(4.2a)isthe\canonicalform"foraspinorinthespacetimealgebra¯rstfoundinRef.2.Thequantities,andhavedistinctivegeometricalandphysicalinterpretationswhichareindependentofanymatrixrepresentation.Soitisbesttousetheminsteadofthe'sand'sin(3.8).Itissimplesttosetforththeinterpretationofcategoricallyandafterthatexplainhowitisre-latedtotheusualformulationoftheDiractheory.Thequantityin(4.2)determinesaproperLorentztransformationofaframeintoaframeaccordingtotheequationThisequationhasexactlytheformoftheLorentztransformation(2.7),sincecomparisonof(4.2b)with(2.8)showsthathasthesamealgebraicstructureas.However)isagenerallydi®erentiablefunctionofthespacetimepoint,whileisconstant.Thus,(4.3)speci¯esa(generallydi®erentiable)setoffourvector¯eldswithvalues)ateachpointdeterminedbyaproperLorentztransformationofa¯xedframe.Thiscompletelydescribesthegeometricalsigni¯canceof.Byvirtueof(4.3),thespinormayberegardedasarepresentationofaLorentztransformation. Sincethepseudoscalaranticommuteswiththevector,(4.2a)and(4.3)implyIf(3.12)isregardedasageneralizationofthetransformation(4.3),thenmultiplicationofthemustbeinterpretedasadilatation.ThegeometricalinterpretationofaDiracspinorgivenhereismoredirectanddetailedthanthecon-ventionalone.Ordinarily,DiracspinorsaresaidtoberepresentationsoftheLorentzgroupbecausetheytransforminacertainwayunderLorentztransformation(seeRef.4)Incontrast,wesaythatrepresentsaLorentztransformationbecause,byEq.(4.4),ita\rotation-dilatation"oftheframeintotheframe.Actually,)determinesacontinuousin¯nityofLorentztransformations,oneateachspacetimepoint.Perhapsitisworthaddingforemphasisthatdoesnotoperateinsome\abstractspinspace"detachedfromspacetime,ittransformsspacetimevectorsintospacetimevectors.Thephysicalinterpretationsofare¯xedbyspecifyinginterpretationsforthe.Thequantityistobeidenti¯edastheprobabilitycurrentoftheDiractheory.Itfollowsthatthetimelikevectorcanbeinterpretedthe(local)\worldvelocity"ofaDiracparticle,whileistheprobabilitydensityinthelocalrestframedeterminedby.Thetensorcomponentsoftheprobabilitycurrentrelativetotheframeandthelocalconservationofprobabilityisexpressedby Thespacelikevector 2¹3=1 hR°canbeidenti¯edasthe(local)spinvectoroftheDiractheory.Thecorresponding\current" hðhascomponents Ofcourse,cannotbeinterpreteddirectlyastheelectronspinbecauseangularmomentumisabivector.The\properspindensity"oftheelectronis,whereisthe(local)spinbivectorgivenby 2¹2e1=1 hR° hRiisv:Theseassortedequivalentexpressionsareeasilyrelatedbyusing=1andtheappropriatede¯nitions.Thetensorcomponentsof¹º®¯whichwegetbyapplying(2.12)to(4.8a)andintroducing,the\alternatingtensor"de¯nedby¹º®¯Therightsideof(4.8a)or(4.8b)showsthatisthedualofthebivector.Thus,given,andareequivalentdescriptionsofthespininthesensethateitheronedeterminestheotherby(4.8). In(4.5),(4.7),(4.8)thede¯nedby(4.3)havebeengivenaphysicalinterpretationbyrelatingthemtotheelectronspinandvelocity.Itisimportanttorealizethattheindexin(4.3)isa\freeindex,"thatis,itneednotberelatedtoanycoordinatesystem.However,thephysicalinterpretationrequiresthattheindexedin(4.3)beidenticaltothesetspeci¯edintheDiracequation(3.13).Itwillbenotedthatthe\changeofrepresentation"transforming(3.13)to(3.10)doesnotaltertheSomethingcannowbesaidaboutthephysicalinterpretationoftheDiracwavefunctioninitscanonicalform(4.2a).Thequantityhasbeenidenti¯edastheproperprobabilitydensity.TheunimodularspinordeterminingtheLorentztransformation(4.3)canbespeci¯edbysixscalarparameters.Fiveofthoseparametersdeterminethevelocityandspindirectionsoftheelectron,which,ofcourse,alsodeterminesthe\spinplane"containingthevectorsorthogonalto.Theremainingparameteristheofthewavefunction.Geometrically,thephasedeterminesthedirectionsofinthespinplane.Physically,thephaseisrelatedindirectlytotheelectronenergy-momentumbyderivativesofthewavefunction;thiswillbeconsideredinthenextsection.Tosumup,exceptforthephaseandtheparameterin(4.2a),theDiracwavefunctiondetermines(orisdeterminedby)theelectronprobabilitycurrentandthespindirection.Togetaninterpretationof,additionalphysicalassumptionsareneeded.Thebivector 2mc2°1eÃ=e exp(isusuallyinterpretedasthemagnetizationormagneticmomentdensityoftheelectron.Therightsideof(4.10)wasobtainedbyusing(4.2a)with(4.8a)andshowsthattheratioofmagneticmomenttospindensitydi®ersfromtheusualFerromagneticratioattributedtotheelectronbythefactorexp().Equation(4.10)showsthatcanbeinterpretedgeometricallyastheangleofa\dualityrotation"ofinto.Italsolendsaphysicalsigni¯canceto,but,asexplainedinRef.3,otherfeaturesoftheDiractheorymakeafullysatisfactoryphysicalinterpretationdi±culttocomeby.Proofthattheexpressions(4.5),(4.7),and(4.10)areequivalenttoconventionalexpressionsforprob-abilitycurrent,spinandmagneticmomentintheDiractheoryissimplyamatterofcomputationusingtheuniquecorrespondencebetweenandthecolumnspinorªestablishedinSec.3.ThecomputationshavebeendiscussedinAppendixAofRef.3,soitsu±cestodisplaytheresultsinTableIintheformofexpressionsfortheso-calledbilinearcovariantsusingbothmathematicalsystems.TableIusesthenotation whereªdenotesHermitianadjoint,inadditiontoconventionsalreadyexplained,especiallyinconnectionwith(2.5)and(2.10).Theterm\observable"isusedheretorefertotensorquantitiessuchaswhich(inprinciple,atleast)areamenabletoexperimentalobservation.Therelationsofwavefunctiontoobservablesgivenby(4.1),(4.5),(4.7),(4.10)aremuchsimplerandeasiertousethanthoseinTableI.Sothetableisusefulonlyforcomparisonwiththeconventionalformulation.TableIhelpsrevealtheroleoftheunitimaginaryinthematrixformulationoftheDiractheory.Observablesarealwaysreducibletorealnumbers.SomeofthebilinearcovariantsinTableIareformedfromantiHermitianproductsofthe,soafactorisneededtomakethequantitiesreal.Thisapparentlytrivialmathematicalfacthasaphysicalsigni¯cance.Todiscoverthatsigni¯cance,itshouldbenoted¯rstthatneverentersanyexpressionforobservablesoftheDiractheory(suchasthoseinTableI)exceptasamultipleofthewavefunctionª,andsecond,thatentersonlyinthecombinationwithPlanck'sconstant.Also,notethat hðiseasilyprovedfrom(4.2)and(4.8).Interpreting(4.11)asamatrixequationandusing(3.7)and(3.6b),oneshowsimmediatelythat thatis, isaneigenvalueofthebivectordescribingthespin.Equation(4.12)canbeusedtoeliminatetheexplicitappearanceofinequationsoftheDiractheory.Forexample,if¡issomematrixoperator, .(4.12)yields ¡ª=ª ¡°¹°ºªS¹º;(4:13)10 showingthefactortobeequivalenttoacontractionofthespintensorwithsomeothertensor.AsTableIillustrates,thefactorappearsexplicitlyonlyinthoseexpressionsforobservablesinvolvingtheelectronspin.Equation(4.12)showsusthatthefactor intheDiractheoryisinfactarepresentationoftheelectronspinbyaneigenvalueofthespintensor.Theeigenvalueisimaginary,becausethespintensorisskew-symmetric.Theisarepresentationofthedirectionofthespintensor,becausetheconventionsofDiractheorycorrespond,inadeviousway,tothegeneratorofrotationsinthephysical\spinplane"withthegeneratorofrotationsintheabstractcomplexplane.Ofcourse,thefactor isjustthemagnitudeofthespintensor. TABLEI.Bilinearcovariantsasobservables. ª ÃeÃ)=½cos¯ª °¹°¹0eÃ)=°¹¢(¹e i0¹h 2ª 1 2°¹^°ºe¹ 2(°¹°º2°1eÃ)°¹^°º)¢M=Mi0¹ 2ª °¹°5Ã=¹h 2(°¹3eÃ)=°¹¢(½s¹ª °5eÃ)(0)=¡½sin¯ Thespintensor,whichiscrucialtotheunderstandingof,isnotmentionedinstandardaccountsoftheDiractheory,sosomeexplanationisinorder.Standardaccounts(e.g.,p.59ofRef.7)eitherimplicitlyorexplicitlyintroducethespin(density)tensorº®¯ 2i0¹hª °º^°®^°¯ª=1 2i0¹hª ¹º®¯¹º®¯whereusehasbeenmadeoftheidentity¹º®¯andtheexpressionforinTableI.Contractionof(4.14)withandcomparisonwith(4.8b)givesthedesiredrelationº®¯¹º®¯mustbeinterpretedasthecomponentsoftheelectronspinangularmomentumdensityinthelocalrestframedeterminedbytheparticlevelocityHavingascertainedthattheimaginaryunitfunctionsintheDiractheoryasarepresentationoftheelectronspindirection,itisevidentthatchargeconjugationmusthaveageometricalinterpretationaswellastheusualphysicalone.Indeed,ithasalreadybeendiscoveredinSec.5ofRef.8thatchargeconjugationentailsarotationofthe\observables"de¯nedby(4.3)aboutanaxisinthespinplane.Reference9pointedoutthatthistransformationin\thespaceofobservables"oftheDiractheoryisisomorphictotheoperationofchargeconjugationinisospacesobyregardingthemasoneandthesame,onegetsaspacetimeinterpretationofisospacerelatingspintoisospinandindicatingaconnectionbetweenthepseudoscalarandisospinpropertiesofthepion.Thisspeculativeideahasnotthusfarbeenpushedbytheauthor.Still,inviewoftheuncertaintiesofstronginteractiontheory.Itseemssu±cientlypromisingtomeritmentiononce5.ObservablesandoperatorsAnumberofstatementsaboutobservables,operators,andeigenvaluesarefrequentlyputforwardinoneformoranotherasgeneralprinciplesofquantumtheory.Thedi±cultiestheseprinciplesfacewhenappliedtotheDiractheoryshowsthattheyarenotsogeneralafterall.Surely,sincetheDiractheoryisthemost ¯rmlyestablishedversionofquantumtheory,onlythoseprincipleswhicharerequiredbyorareatleastconsistentwiththeDiractheorycanberegardedasbothfundamentalandgeneral. TABLEII.Observableswiththekineticenergy-momentumoperator. GordoncurrentEnergy-momentumProperEnergy-momentumdensityKineticEnergydensity(Relative)KineticmomentumdensityTotalAngularmomentumtensorProperTotalAngularmomentumdensity(Relative)TotalAngularmomentumdensityi;j;k)=(1 ^p¹ª´i0¹1 2fª @¹ª¡(@¹ª )ªe cA¹ª ª´(eÃ^p¹Ã)´³eÃn¹h@¹2°1¡e cA¹Ã(0)T¹º [(^pRú¹®¯ f°¹p®x¯¡^p¯x®)¡(i0¹2)°¹^°®^°¯gª°0eÃf°¹p®x¯¡^p¯x®)᡹2)°¹^°®^°¯2°1g¢v¹J¹®¯=½(p^x+isv)¢(°¯^°®)J0ijyf^pixj¡^pjxi¡(i0¹2)°i^°jgª´ªy^JkÃy^JkÃ)´¡Ãyfpixj¡^pjxi)áh=2)°i^°j2°1g¢ Consider,forexample,theEHOO-principlethat\theigenvaluesofperatorscorrespondbservedvalues."Intherelation =1,2,3)fromTableI,thematricesareHermitian.Since(=1,theeigenvaluesof1.Soifªisaneigenstateofwehaveªorª;whence,.Butthissaysthatthevelocityoftheelectronequalsthespeedoflight,whichisinconsistentwiththeDiracequation,nottomentionexperimentalevidence.Notwithstandingtheabsurdityoftheresult,applicationoftheEHOO-principletothesocalled\velocityoperators"hasbeenperpetuatedinaccountsoftheDiractheoryforfortyyears,alongwithfruitlessattemptstoexplainawaytheproblem.Thisandmanyothersimilardi±cultiesareeasilyresolvedbyadmittingthatEHOOisnotageneralprincipleofquantummechanicsand,inparticular,doesnotapplytooperatorscomposedofthe.TheroleoftheintheDiractheoryisclearinthemultivectorformulation;aresimplyanorthonormalsetofvectors.IntheexpressionappearinginTableItheinterpretationoftheasoperatorsistrivialandhasnothingtodowithquantummechanics;the pickoutthethcomponentofthevectorbyinnermultiplication.Noristhenoncommutivemultiplicationruleforthejusti¯edbyanyprincipleconcerningtheimpossibilityofsimultaneouseigenvaluesintheDiractheory;ithasstraightforwardgeometricalsigni¯cancealreadydiscussedinSec.2andinRefs.1and5.ItshouldbeevidentthatthehavethesamegeometricalroleinalltherelationsofTable1.TotheextentthatpropositionsaboutHermitianoperatorsandeigenvaluesaresigni¯cantintheDiractheory,theyarespecialpropertiesofthe\kineticenergy-momentumoperators,"customarilywritten inthematrixformulation.Itisoftensaidthat^=1,2,3)correspondstothekineticmomentumobservable,butthisisinaccurateandleavesroomformisinterpretation.Rather,theoperator^determinesarelationbetweenthewavefunctionandthekineticmomentumobservable.Thoserelationsofthewavefunctiontobasicobservableswhicharedeterminedby^aregiveninTableII.Thetablegivesboththemultivectorandmatrixexpressionsfortheobservables.TherelationsinTableII,likethoseinTableI,canbeprovedbythemethodofAppendixAinRef.3.Theoperator^isde¯nedimplicitlyforboththematrixandmultivectorformalismsbytheexpressionsfortheGordoncurrent.Theexpressionfor^usedinthetabledi®ersfrom(5.1)inthatitis\hermitized"togiverealquantitiesfortheobservables.TablesIandIIgiveacompletesetofrelationsofthewavefunctiontothefundamental(local)observablesintheDiractheory.Actually,TableIIaddsonlyonerelationwhichismathematicallyindependentofthesevenindependentrelationsinTableIItrelatestheenergy-momentumtensortothegradientofthephaseofthewavefunction.Thenatureofthisrelationisbestseenbyexpressingthe(proper)kineticenergy-momentumintheformpRà cA¹=2S¢(@¹e)¡e Thequantity2=2(canberegardedasthelogarithmicderivativeorangularvelocityofthefamilyofLorentztransformationsdeterminedbyaccordingto(4.3).Thequantity2)istheprojectionofthisangularvelocityontothespinplane;itdescribestwoe®ects:therateatwhichthespinplaneprecessesandtherateatwhichthephase(angleofrotationinthespinplane)changes.Onlythelaste®ectappearsintheSchrÄodinger'stheory,wheretheenergyandmomentumarecompletelydeterminedbythephase.Inspiteofthefactthatthe\properenergy-momentumdensity"isafundamentalquantityinrelativisticcontinuummechanics,itisrarelyconsideredindiscussionsoftheDiractheory.Indeed,theexpressionforintermsofthematrixwavefunctionissounwieldythatithasnotbeenincludedinTableII,thoughitisnotdi±culttoobtainfrommatrixexpressionsforinTablesIandII.Thesigni¯canceofEq.(5.2)liesintheinsightitgivesintotheroleoftheoperatorintheDiractheory.Thehasbeenreplacedin(5.2)bythephysicalandgeometricalquantityitrepresents,thespinbivector.Moreover,(5.2)showsthatinrelatingthewavefunctiontoenergyandmomentumthetheoperatorfunctionsasaprojectionoperator,eliminatingfromthederivativesofretainingonlytheangularvelocityofinthespinplane.Thesespeci¯cfactsaboutintheDiractheoryshouldbecomparedcarefullywithgeneralpropositionsaboutHermitianoperatorsandobservablesinquantummechanics.Forthemostpart,suchpropositionshavebeendevelopedtogeneralizeproperties.Itmaybeconcludedthatinsofarastheyrelatethephaseofawavefunctiontotheenergyandmomentumofaparticle,thepropositionsarewellgrounded.Beyondthis,theirvalidityisproblematic.Certainlynoneoftheusualpropositionsrecognizetheessentialrelationoftothespin.And,asshownearlier,theyproducenonsensewhenappliedtotheDiracmatrices.\Hermiticity"isoftensaidtobeanessentialattributeofoperatorscorrespondingtoobservables.ThevalidityofthisviewshouldbejudgedbyexaminingtheroleofhermiticityintheDiractheory.Severaldi®erentbutinterrelatedrolescaninfactbedistinguished.IntheDiracmatrixalgebrahermiticityisgivenageometricalsigni¯cancebyadoptingthestandardrepresentationinwhichthe=1,2,3)representingspacelikevectorsareanti-Hermitian,whilerepresentingatimelikevectorisHermitian.ThusHermitianconjugationservestodistinguishaspeci¯c,thougharbitrarilychosen,restframeinspacetime.Itshouldbe notedthatassignmentofthisroletohermiticityisnotentirelyarbitrary;itisinlargepartdictatedbyEq.(2.1)whichrelatesthematrixrepresentationtotheinde¯nitemetricofspacetime.HermiticityalsoplaysaroleintheassociationifwiththespinthroughtherequirementmetinTableIthatonlyrealcovariantsaretobeinterpretedasobservables.Neitheroftheserolesisrecognizedintheusualdiscussionsofoperatorsinquantummechanics.Moreover,whenthematrixalgebraandcomplexnumbersarereplacedbyspacetimealgebra,theindirectrepresentationofgeometricalandphysicalfeaturesbyhermiticityisThesigni¯canceofHermitianmatricesisquitedi®erentfromthesigni¯canceofHermitiandi®erentialoperatorsintheDiractheory,thoughthesetwokindsofhermiticityarecontinuallyconfoundedintheliterature.Thedistinctionappearsinathirdroleofhermiticity.AstudyoftherelationsinTableIIrevealsthathermiticityinsuresthattheoperatorrelatesenergyandmomentumdirectlytothephaseandnottotheamplitudeofthewavefunction.ThisistheroleofHermiticityintheSchrÄodingertheoryandseemstobethefeatureresponsibleforthesuccessesofabstractoperatorformulationsofquantummechanics.Amisplacedemphasisonoperatorsinquantummechanicshascontinuedtocover-upthemeaningofhermiticityandtherelationofcomplexnumberstospin.TherelationsofobservablestothewavefunctionexpressedinTablesIandIIisfundamentaltotheDiractheory.TheyareonlyindirectlyandimperfectlyexpressibleasrelationsofobservablestoHermitianoperators.ByusingtherelationsinTablesIandIIalongwiththeDiracequation,thewavefunctionandtheoperatorcanbecompletelyeliminated,resultinginaformulationoftheDiractheoryasasetofconservationlawsandconstitutiveequationsforobservables.ThisreformulationhasbeencarriedoutinRef.3andbringstolightotherfeaturesoftheDiractheorywhicharehiddenintheconventionalformulation.HavingdeterminedthatthepurportedlyfundamentalcorrespondencebetweenobservablesandoperatorsisneitheradequatenornecessaryintheDiractheory,thesigni¯canceofeigenvaluesinthetheoryshouldbeexaminedclosely.Tobeginwith,itshouldbeemphasizedthatthefundamentalassumptionsoftheDiractheorydonotrequireanyreferencetoeigenvalues,soacceptableassertionsaboutthephysicalsigni¯canceofeigenvaluesmustbederivedratherthanassumed.Considertheappearanceofenergyeigenvalues.SolutionoftheDiracequationforaboundelectron,togetherwiththecorrespondenceofenergytothewavefunctioninTableII,givesadiscretespectrumofvaluesfortheenergy.Itistruethatthesenumberscanberegardedastheeigenvaluesofanenergyoperator,butthisfactisnotneededeithertoidentifythenumbersasobservablesortoexplaintheexistenceofaspectrum;thefundamentalrelationoftheenergytothephaseofthewavefunctionsu±ces.Thereisanotherpropertyofenergyeigenvalueswhichmaybephysicallysigni¯cantbutgoesunremarkedinstandardexpositionsofquantummechanics,becausetheydealwiththetotalenergywithoutexaminingtheenergydensity.Theglobal(ortotal)energyofanelectroninastationarystateissubjecttofairlydirectexperimentalmeasurement.Theexistenceofastationarystaterequiresonlythatindependentoftime.However,thestationarystatesolutionsoftheDiracequationentailtheadditionalproperty,thatis,theenergydensityhasthestrikingpropertyofbeingeverywhereaconstantmultipleoftheprobabilitydensity.ItmaybenotedthatthisiswhatonewouldexpectifitissurmisedthatthestationarysolutionsoftheDiracequationdescribeanensembleofparticlemotionseachwiththesameenergy.Thissuggeststhatthephysicalsigni¯canceofeigenvaluesinquantumtheoryistobefoundinthefactthattheycorrespondtolocalobservableswhicharehomogeneousinspaceandconstantintime,ratherthanintheirconnectiontoHermitianoperators.AlocalobservablemaywellbehomogeneousandconstantwithoutbeingtheeigenvalueofsomeHer-mitianoperator.Forexample,withoutspecifyinganyHermitianoperator,onemightneverthelesssaythatanelectronisinavelocityeigenstatewitheigenvalueifthelocalvelocityishomogeneousandconstant.Itiseasytoshowthatonlythefreeparticle\planewave"statesareeigenstatesofthevelocityinthissense.Theyarealsoeigenstatesofspinandmomentumintheusualsense,buttheyarequiteunphys-icalbecausetheyrequireuniformdensity.ThephysicalfreeparticlesolutionsoftheDiracequationarewavepackets,forwhichitcanbeshownthatthelocalvelocitycannotbeconstant.Soitmaybethatsomephysicallysigni¯cantlocalobservablescannotbeassociatedwithphysicallysigni¯canteigenvalues.Angularmomentumoperatorshaveanimportantplaceinthesystematicanalysisofquantumtheory.TheangularmomentumoperatorsintheDiractheoryarede¯nedimplicitlybytheexpressionsforthe(relative)angularmomentumdensityinTableII[wherethevaluesof()areunderstoodtobecyclic permutationsof(1,2,3)].Thearecommonlyde¯nedbyrequiring¯rstthattheysatisfythewell-knowncommutationrelationsofangularmomentumoperatorsandsecondthattheycommutewiththeDiracHamiltonianforacentral¯eldandsoareconservedquantities(see,forexample,Ref.10).ThethenfoundtohavetheformgiveninTableII.Suchanapproachgivestheimpressionthatassumptionsabout\angularmomentumalgebra"areessentialtotheinterpretationofDiractheory.Onthecontrary,itwasshowninRef.3thatgiventheDiracequationandtherelationsofthewavefunctiontotheparticlevelocityinTableIandtheenergy-momentumtensorinTableII,expressionsforthespinandthetotalangularmomentumareunambiguouslydeterminedbyintroducingtheconventionalde¯nitionoforbitalangularmomentum.ThentheformofinTableIIcanbeobtainedsimplybywritingtheresultingexpressionsinconventionaloperatorform,andmanifestlywithoutanappealtoassumptionswhichcouldbecalled\quantummechanical."Evidently,thephysicalsigni¯canceoftheisderivedfromtherelationtheyexpressbetweenthewavefunctionandtheangularmomentumtensor.Butnothinginthisrelationimpliesthat,asisusuallysupposed,theeigenvaluesofshouldbeinterpretedasthesquareofthemagnitudeoftheangularmomentum.Thisisnottoquestiontheimportanceofortheangularmomentumalgebraincalculationsortheclassi¯cationofstates;theaimhereisonlytopointoutthatsomeproblemsofinterpretationexist.Furtherquestionsabouttheinterpretationofangularmomentumquantumnumberswillbeconsideredlater.ProbablythemostprofoundproblemposedbytheDiractheoryforconventionalinterpretationsofquantumtheoryliesinthedi±cultyofreconcilingtheusualinterpretationsoftheHeisenberguncertaintyprinciplewiththepropertiesofelectronspin.AsthenatureofthisproblemhasalreadybeenexplainedinRef.3,itwillnotbediscussedhere.6.RelativeObservablesandtheInterpretationofEnergyThemostimportantobservableintheDiractheoryisthetotalenergy,forthisquantityhasbeensubjectedtothemostthoroughexperimentalinvestigationespeciallyinthehydrogenatom,anditisprimarilybyinferencesfromtheexperimentalresultsthattheidenti¯cationofotherDiracobservables,suchasvelocity,spinandmomentumhavebeencon¯rmed.Toidentifydi®erentphysicale®ectscontributingtotheenergyandstudytherelationtootherobservables,theenergymustbedecomposedintoasumoftermsTheusualtreatmentassociatesobservableswithoperatorsratherthandirectlywiththewavefunctionand,strangely,isunabletoseparatedi®erentphysicalcontributionstotheenergyexceptasperturbationsofthenonrelativisticlimit.Incontrast,theapproachhereistoeliminatethewavefunctionalongwiththeoperatorsoftheDiracHamiltoniantogetanexpressionfortheenergydensityofaDiracelectronintermsofthelocalobservablesidenti¯edinTablesIandIl.Theresultsarenotcompleteinthesensethatafullysatisfactoryphysicalunderstandinghasbeenachieved.However,thelinearcontributionofexternal¯eldsisidenti¯edandexactexpressionsfortheThomasandLarmorprecessionsarefound.Also,anumberofpeculiarfeaturesarediscoveredwhicharehiddenbytheoperatorformulation.Comparisonwiththeusualresultswillbemadeinanotherpaper.Energyisarelativeobservable,thatis,itssigni¯canceisrelativetosomeinertialframe.However,anatombindinganelectrondeterminesaninertialframe,andrelativetothatframetheelectron'senergyhasanabsolutesigni¯cance.Thisframeshouldbekeptinmind,thoughthefollowingdiscussionmakesnoassumptionaboutboundstates,andtheresultsholdforanyinertialframedesignatedbyaspeci¯cchoiceofthetimelikevectorAsanaidtophysicalinterpretationandinpreparationforthe\nonrelativisticlimit,"theobservablesinTablesIandIIwillbeexpressedas\relativeobservables"andrelatedbyformulasderivedinRef.3.TheprocedureusedforintroducingrelativevariablesisexplainedindetailinRef.5,sotheresultsandnomenclatureofRef.5arefreelyemployedwithonlythebriefestcomments.The\relative"velocityoftheelectronisrelatedtothe\proper"velocityde¯nedin(4.5)bythe =(1 Thus(1+isthespinorinEq.(6.15)below,whichdeterminesthe\boost"ofinto.The\relative"probabilitydensityintheinertialsystemofAspacetimepointcanbedesignatedbyatimeandapositionandthederivative = canbeexpressedintermsofthederivatives andr=rx=°0^ bytheequation Accordingly,theequationforprobabilityconservation )=0(6canbewritteninthefamiliarrelativeformSimilarly,thepropertimederivative d¿=v¢ isrelatedtothesocalled\hydrodynamicderivative"bytheequation =v0 Nowconsidertheexpressionofmechanicalquantitiesintermsofrelativevariables.Thekineticenergy-momentumvector,whosecomponentsarerelatedtothewavefunctionby(5.2),canbedecomposedintoanenergyandamomentumbyusingthealgebraicrelation (Heretheenergy-momentumhasbeenexpressedin\momentumunits"insteadofthe\energyunits"usedinRef.5.)The(kinetic)momentumdensityintheinertialframeofisthusNowfrom(5.3)and(6.7c) c+p¢(¡x)=²t¡p¢x+ctp¡² c¡p^x: Thescalarpartofthisequationisjusttherelationwhiletheproperbivectorpartis Fromthis,oneimmediatelyseesthattherelativevectorpartof whiletherelativebivectorpartisisp^x](2)=x^p=ix£p´iL;(6:9c)showingthatisanappropriategeneralizationoftheusualorbitalangularmomentumvectororbetter,thecorrespondingangularmomentumbivector.Inaccordancewith(6.8),the\relativeorbitalangularmomentumdensity"isFortheproperspinvector,de¯nedby(4.7),and,since)=0,Using(6.1)and(6.11)in(4.8),onehasfortheproperspinbivector as£v+iv0(s¡s0v=c):Hence,S=s1+S2=s1+is2;(6:12a)wheres1´[S](1)=v0 cs£v=v0 )(6istherelativevectorpartof,andandS](2)=iv0(s¡c¡1s0v)(6istherelativebivectorpart.From(6.9)and(6.12)one¯ndsthattherelativebivectorpartoftheproper\total"angularmomentumatapointtJ](2)=x^p+S2=i(x£p+s2);(6:13)so½0J=½0x£p+½0s2;(6:14)isthetotal(relative)angularmomentumdensity,expressedasasumoforbitalandspinparts. Thetroublewithrepresentingspinbytherelativevectorsistheydonothaveconstantmagnitude,oneoftheprincipalpropertiesoftheproperspin.Thereisanalternativede¯nitionof\relative"spinwhichdoesnotsu®erthisdefect.InthemannerexplainedinRef.5,thespinorintroducedby(4.2)canbefactoredintheformLU;wherethespinordeterminesaspatialrotationanddeterminesaboostof.Nowde¯nearelativespinbivector§andvector Using(6.15)in(4.8)oneeasilyshowsthat§isrelatedtobyaboost:SL;svLWhenexpressedintermsofrelativevelocity,therelationofisfoundtobe[seeEq.(4.38)ofRef.5], v2¾¢=¾+v20 +1)Substitutionof(6.18)into(6.12c)yields +1)so(6.14)canbewritten +1)Thisshowsthecorrectwaytocombinewiththeorbitalangularmomentumtogetthetotalrelativeangularmomentumdensity.Especiallyindiscussionsofrelativisticapproximations,itisimportanttobeclearaboutwhichoftheseveraldi®erentrepresentationsofspinisemployed.Fromthekineticenergy-momentumvectoritisconvenienttoformtheanenergy-momentumvectorde¯nedby istheelectromagneticvectorpotentialandtherightsideoftheequationhasbeenobtainedfrom(5.2).Thequantityofchiefinterestinthissection,thedensityofthetotalenergyisrelatedtowhere(6.21)hasbeenused,isthekineticenergyde¯nedby(6.7a),andistheusualelectricpotentialenergy.Thecorrespondingtotallocalmomentumis .Combining(6.22a)and(6.22c),onehasE=cEquation(6.22a)explicitlyexhibitsthedependenceofthelocalenergyonthetimederivativeofthe.Insteadofanalyzingthisexpressiondirectly,itisadvantageous¯rsttostudytheenergyinthe localrestsystemdeterminedbythevelocity.Sinceby(4.5),(4.7),and(4.8)thelocalvelocityandspinarefunctionsonlyofthespinor,equationsofmotionforthese\mechanicalquantities"alonga\streamline"inspacetimewithtangentaredeterminedbyanequationofmotionfor,whichcanbeputintheform or­=2(where­isabivectorandisthepropertimederivativede¯nedby(6.6a).Expressedintermsofthe\angularvelocity"­theequationsofmotionforvelocityandspinare 2¢d¿s=1 2¢d¿S=1 ­;S]:(6.24c)Byusing(6.23)alongwith(6.21),one¯ndsthatthelocalenergyinthelocalrestframeisgivenbytheprojectionoftheangularvelocityontothelocalspinframe,thatis,=(­ Theplannowistoattainaphysicalinterpretationoftheenergydensitybyanalyzingthe\properangularvelocity"­.Toaccomplishthis,somephysicalinputisneededbesidestherelationsofobservablestothewavefunction,whichisallthathasbeenusedsofar.ThatinputcomesfromtheDiracequation,whichwasusedinSec.6ofRef.3togetthefollowingexpressionfor­intermsoflocalobservables: +v¢³cos¯+e cA´S¡1;(6:26)where­ =¡ ^v¡iv^ ¯=e mc2¡ei¯ inwhich istheexternalelectromagnetic¯eld °¹°º;[W¹;Wº]S¡1]g;(6.27b)W¹=(½ei¯)¡1@¹(½ei¯S)=@¹S+S(@¹ln½+i@¹¯):(6.27c)Substituting(6.26)and(6.27)in(6.25)andrecallingtheexpression(4.10)forthemagneticmomentdensity,onegetsthefollowingexpressionforthe\kineticenergydensity"inthelocalrestsystem½mcLestthereaderbelievethatphysicalinterpretationsarebeingarbitrarilyimposedhere,itshouldbepointedoutthattheidenti¯cationofaskineticenergydensityisaconsequenceofadoptingtheconventionalinterpretationoftheoperator(5.1)as\kineticenergy-momentumoperator"andofthe\Diraccurrent" asprobabilitycurrent.Unconventionalasthepresentdiscussionmayappear,itisbasedonconventionalassumptionsoftheDiractheoryandconventionalprinciplesofrelativisticcontinuummechanics.Theultimateaimistodiscoverthefullconsequencesofthoseassumptions.Accordingtoconventionalprinciplesofrelativisticcontinuummechanics,thequantitybeinterpretedastheelectronmassdensity.IftheDiractheoryinfactdescribesastatisticalensembleofparticlemotionsthenmustbeonlyalocalaverage(notanactual)particlevelocity,andadeviationoftherestmassdensityfrom½mcduetostatisticale®ectsistobeexpected.Bethisasitmay,theterm½mchastheappearanceofarestmasstensor,andonemightguessthattheunfamiliarfactorcosisneededtomeettheconstraint=1inastatisticalaverageofparticlevelocities.Theinterpretationofthesecondtermin(6.28)requireslessspeculation,for willberecognizedastheclassicalexpressionfortheincreaseinmassduetotheelectromagneticinteractionofadipole.Interpretationofthelasttermisdi±cult,butmaybecrucialtoacompleteunderstandingoftheDiractheory.Consideringthe expression(6.27b,c)forintermsofthespinandthecorrespondingspindependenceofthemomentum°uxexhibitedinEq.(3.22)ofRef.3,itmaybeguessedthatthislasttermbeinterpretedastheenhancementofmassduetothelocalspin°ux.Torelate(6.28)totheenergydensitynotefrom(6.1)and(6.7)that ):(6hencethetotalenergydensitycanbewrittenc½v Useoftheexpressions(6.26),(6.27)for­togettheenergydensityintheform(6.30),whichisamenabletophysicalinterpretation,isequivalenttotheusualpracticeofexpressingtheenergydensityintermsoftheDiracHamiltonian,andthensystematicallyreplacingtheoperatorsandthewavefunctionbylocalobservables.Togetonwiththeinterpretationof(6.30),itisworthremarkingthatthemagneticmomentoftheDiracelectronwas¯rstidenti¯edtheoreticallyandexperimentallypreciselybyisolatingthecontributionofthetermtothetotalenergy(byadi®erentmethod,ofcourse).Butthereisanothercontributiontotheenergyduetotheinteractionofthespinwiththeexternal¯eldarisingfromtheThomasprecession.OrdinarilythecontributionoftheThomasprecessionisidenti¯edonlyasacorrectiontothenonrelativisticapproximation.ButtheformalismusedheremakesitpossibletodiscusstheroleoftheThomasprecessionexactly.InSec.4ofRef.5thegeneralizedLarmorandThomasprecessionofaclassicalrigidpointparticleisdiscussed.Alltheresultsobtainedthereapplyimmediatelytothepresentproblemifonlythepropervelocityofaparticlethereisidenti¯edwiththevelocityofastreamlineinthepresentformulationoftheDiractheory.Thisparagraphrecallsthoseresultswhicharemostpertinenttothepresentdiscussion.Theangularvelocity­de¯nedbyEq.(6.23)determinestheprecessionofthevelocityandspinthroughEq.(6.24).Itwillbeconvenienttodecompose­intorelativevectorandbivectorparts;thus­=2(=[­]=[­]Introducingthefactorization(6.15),one¯ndsforthespinortheequationofmotion cd¿U=1 !U;forwhichtheangularvelocitycanbeexpressedintheseveralusefulforms (1+ c(v0)(¯£v)£v+v0 =[2(=[2 v0=v20 (1+ (1+ c£¯´:(6:35)20 Theseequationsgivethedecompositionofintoasumtwoterms,thegeneralizedLarmorprecessionandtheThomasprecession,whichisduesolelytotheaccelerationoftheparticle.Onecansolve(6.33)forintermsofL!L+2(whichisthemoreusefulwhenonehastheformula Applying(6.32)and(6.33)totherelativespinvectorde¯nedby(6.16),oneobtainstheequationofmotion cdt¾=!¢¾=!L¢¾+!T¢¾=³¡¯+v0 (1+akeyformuladerivedbyThomas.NowitmustbeemphasizedthattospeakoftheThomasprecessionintheDiractheory,itisessentialtointroducetherelativespinde¯nedby(6.16);thisisthespinobtained,asrequiredbyThomas,fromtheproperspinbya\deboost"intoansystem,asysteminwhichtheaccelerationoftheparticleiszero.ItmakesnosensetospeakoftheThomasprecessionofthespinvectorsde¯nedpreviously.TheformulationoftheThomasprecessionjustgivenadmitsanimmediategeneralization,simplybyreplacingthepropertimederivativebythederivatives .Then(6.31a)isreplacedby=2(fromwhich(6.31a)canberecoveredsince­=.Similarly,(6.32)generalizesto 2!¹:40a)where!¹=Le!¹L¡2Le@¹L=!L¹+!T¹=i³¯¹+v0 (1+=[2( (1+ (1+ c£¯¹´:(6:40c)Also,­¹=L(!¹Le@¹R)Le:41)and@¹¾=!¹¢¾=³¡¯¹+v0 (1+or,equivalently, !¹;§];(6:42b)and,di®erentiating(6.17a)andusing(6.41), 2¹L(@¹§¡1 !T¹;§]) Wearenowpreparedtoseparatetheenergy-momentumduetovariationsinvelocityfromothercon-tributionstothetotalenergy-momentum.Using(6.39)in(6.21)followedby(6.17),(6.41),and(6.42),we=[­=[())§]isde¯nedbyy@¹Ui¾3eU](0)=[!¹§](0)=!¹¢§:(6:44)From(6.43)oneseesthat§(6isjustthegeneralizedLarmorprecessionenergy,whereas,from(6.44),§(6includesalsotheThomasprecessionenergy.Removingthepotentialenergycontributionfrom(6.44)by cA¹h[@¹¾3eU]¡e one¯nds,since Itisconvenienttoreplace­by­ intheformulationofprecessionenergy.Thisisperfectlypermissiblesince,asiseasilyshownbysubstituting(6.26)into(6.24),itdoesnotaltertheprecessionofthevelocityandthespin.Thereplacementdoesnota®ecttheThomasprecession;itmerelychangesthede¯nitionofthegeneralizedLarmorprecessionfrom(6.34)to LLe­ Correspondingchangesinotherquantitieswillalsobeindicatedbyoverbars.Nowswitchingfrom­to­ andsubstituting(6.45)and(6.48)into(6.30a),theenergydensityisputintheform Theterm ¢½00)(! )§)]includesboththegeneralizedLarmorandThomasprecessionenergies.Itisimportanttonotethatthechangeinthede¯nitionofthekineticmomentumfromisessentialtomaketheThomastermexplicit.Theterm§vanishesforstationarystates,sinceby=0if=0.Tostudythedependenceoftheprecessionenergyin(6.50)onrelativeobservables,expressintermsoftheelectricandmagnetic¯eldsrelativetoAlsowritewriteC]1andC2=iC2´[C]2.Thede¯nition(6.27b)(6.42c)showsthatdependson;thesequantitiescanbeexpressedintermsof§and§byusing(6.42a)and(6.17a),butthisstepwillnotbecarriedout,becauseitisnotclearhowtoderiveanyphysicalinsightfromit.Thetermswhichare di±culttounderstandwillbekeptlumpedtogether.Nowusing(6.51)and(6.52)in(6.27b)andthenin(6.33),onegets ¢¡e ³¾¢B+v0 (1+ (1+ m¡e mc³¾¢E¡v0 (1+ (1+ Thetermsin¯rstlineof(6.53)exceptforthefactorcosareexactlythetermsderivedbyThomas(seeRef.5),whoevaluatedtheprecessionenergyundertheassumptionthat­ .Thomaswiselyevaluatedtheenergyonlyinthenonrelativisticapproximationwhere(1+ (and,fortunately,1).Inhigherorderapproximationsthee®ectoftheexternal¯eldthroughtermsinthesecondlineof(6.53)isprobablyimportant,thoughitisnotclearhowtomakethisexplicit.Itmightbeguessedthatbecauseofthesmallnessofsin,theunfamiliartermsinthelasttworowsaregenerallyinsigni¯cant,butitwillbeseenthatthetermmaybeofthesameorderofmagnitudeasthe)terminthe¯rstrow.Foranelectroninacentral¯eld, jxjwithV0=dV inwhichcasethesecondtermin(6.53)canbewritten m2c2½ (1+ )cosTheterm(6.54a)hasthewell-knownformofthespin-orbitcoupling;however,di®ersfromtheorbitalangularmomentum,becausetherelationassumedbyThomasdoesnotholdintheDiractheory.Toestimatethemagnitudeofthisdiscrepancythecorrectrelationofwillnowbefound.Thecorrectrelationofthekineticenergy-momentumtotheparticlevelocityintheDiractheorywasfoundinSec.5ofRef.3tobegivenbymc½v ¢( ¢½()¢ Toexpressthisasarelationamongrelativeobserables,multiplybyandrecall(6.7)and(6.1)toget c+p0¡v c¢¡½ ¢()°0¡( But,byvirtueof(6.4)and(6.12a), ¢(°0 =[( ¯)¢(iS°0( =[( Hencethescalarpartof(6.56)canbewrittenwhilethevectorpartcanbewrittenUsing(6.57b)toeliminatein(6.54),oneindeedgetsthedesiredspin-orbittermwith,butthereareseveraladditionaltermsaswell.Someunderstandingoftheadditionaltermscanbeachievedbycomparingwithresultsknownintheliterature,buttodosoitisnecessarytotakethenonrelativisticlimit,sinceitisonlyinconnectionwiththatapproximationthattheThomasprecessionhasbeendiscussedpreviously.Thiswillbedonelater.ThediscussionoftheDiracenergydensityinthissectionhasconcentratedonadetailedinterpretationofafewterms.Asatisfactoryinterpretationofallthetermshasnotbeenfound,butletusreviewthegeneralapproach.Thee®ectivemassdensitygivenby(6.28)di®ersfromwhatappearstobearestenergy½mcbyatermcommonlycalledtheinternalenergydensity.Accordingly,itisnaturaltocallthe\generalizedLarmor"term ¢S= §theinternalenergyofthesystem.Themagnitudeoftheinternalenergydepends,ofcourse,ontheinteractionwithexternal¯elds,which(6.30b)expressestotermslinearintheexternal¯eld .Thekineticenergydensitythesystemisalsoin°uencedbyexternal¯elds,whichwasfoundtotermslinearinthe¯eldbyseparatingtheThomasprecessionenergyfromothercontributionstothekineticenergywithTheLarmorandThomasprecessionswerecombinedin(6.50)togetthetotalin°uenceofexternal¯elds(asidefromthepotentialenergyofcourse).Nevertheless,itisveryimportanttobeabletoseparatecontributionstothemassdensityfromcontributionstothekineticenergy.7.ObservablesintheNonrelativisticLimitTheexactconstitutiverelationsfoundforrelativeobservablesinthelastsectionarerathercomplicatedanddi±culttointerpret.Therelationssimplifygreatlyinthenonrelativisticlimittobedeterminedhere;stilltheremainnontrivial.ItwillbeshownthatthelocalmomentumandtheGordoncurrentareequalinthatlimit,buttheydi®erfromthelocalvelocitybya\magnetizationcurrent."Inasubsequentpaperthisresultwillbeshowntohaveimportantimplicationsfortheinterpretationofspin-orbitcouplingandthePauliandSchrÄodingertheories.Also,forfutureuse,thenonrelativisticformofthespin-electricenergydensityisTheadjective\relativistic"wascriticizedinRef.5,butitwillneverthelessbeemployedhere,becauseitisalmostuniversallyusedinconnectionwiththetopicunderdiscussion.Itmaybewelltorecall,therefore,thatsometimestheword\relativistic"meansthattherelativespeedisoftheorderofthevelocityoflight;sometimesitmeansthataccelerationsaresmall,andsometimesitmeansthatanexpressionorquantityiscompletelyindependentoftherelativevelocity.Theterm\nonrelativistic"willbeusedhereespeciallytoc.Furthermore,itshouldbeemphasizedthattheso-called\nonrelativisticlimit"oftheDiractheoryinvolvesanumberofotherassumptions-reasonableassumptionsaboutthemagnitudeofexternal¯elds,andaboutwhichquantitiesareslowlyvaryingfunctionsofposition,inparticular,aboutthecuriousquantityItwillnotbenecessarytospelloutsuchassumptions,becausetheywillbeimplicitintheapproximateequationswrittendown.From(6.16)onegetsinthenonrelativisticlimit=(1=1+ 2v2 ¢¢¢¼Usingthisintheseveralequationsde¯ningtherelativespinsone¯nds 0jsj;(7.2c)s21s2Sj2=¡S2=1 Also,ofcourse,and(6.20)reducestothefamiliarexpressionFromEq.(2.18)ofRef.3,onegetsHenceforstationaryorslowlyvaryingstatesonehas mc() Tothismaybeaddedthecondition=c;whichseemsreasonableinviewofthenumericalfactorontherightsideof(7.3a).Betterjusti¯cationwillbegiveninasubsequentpaper.Employingtheaboveapproximations,one¯ndsthat(6.57b)reducestotheimportantequationHigherordertermsmustbecarriedtogettheappropriateapproximationto(6.57a),sincethecorrectionstothelarge\restenergy"termareofinterest.Accordingly,recallingespecially(6.12b),one¯nd: 2mv2¡1 2mc2¯2+½¡1(½s£v)+cs¢r¯=1 2mv2+1 22¯2+½¡1[½(sv£s]¼1 2mv2+1 2m³r¢(½s) ½´2+½¡1r¢³¡1 ThisisanappropriateplacetodiscussthephysicalinterpretationandexaminetheN.R.limitoftheGordoncurrent,whosecomponentsarede¯nedinTableII.InSec.5ofRef.3theGordoncurrentwasfoundtoberelatedtothevelocityandmagnetizationorspinbytheexactequation k+ ¢M=e mck+e andtotheproperenergy-momentumdensitybyGiventheconservationlaw(6.5)fortheDiraccurrente½vandtheidentity ¢( )=0,one¯ndsfrom(7.6),theconservationlaw Acceptingtheconventionalinterpretationofasthetotalchargecurrentandidentifying asamagnetizationcurrent,oneisleadby(7.6)tointerpret(asaconvectioncurrent.Inthiswaythe Gordoncurrentisgivenaphysicalinterpretation,however,thesigni¯canceofitscloserelationtotheenergy-momentumexhibitedby(7.7)remainsobscure,thoughitisclearlytiedupwiththesigni¯canceof.Since,ashasalreadybeenmentioned,sinshouldberegardedasasmallquantity,(7.7)showsthattheGordoncurrentisnearlyproportionaltotheenergy-momentumdensity.However,itisnotpossibleintheexactDiractheorytoidentify(asachargecurrentdensity,becauseitdoesnothavevanishingdivergence.Onecanexpress )intermsofotherobservablesbytakingthedivergenceof(6.55),obtainingimmediately mc½v )sin ½iS But,asshowninSec.5ofRef.3theDiracequationalsoimplies i½Smc½v ½q:Hence,onegetstheexactrelations ¢(¢ ¯=½v( ¯)¢ sg=¡½s¢f( ¯)¢ assmall,onegetsimmediatelyfrom(7.7a)½p:By(6.7)and(7.5),the\timecomponent"oftheGordoncurrentis,intheN.R.limit, mc½;whilethe\spacecomponent"isAnd,ifisslowlyvarying,onegetsfrom(7.9) Substituting(7.11b)into(7.4),onegets mk+³½e theN.R.expressionforachargecurrentexpressedasaconductioncurrentplusamagnetizationcurrent.Returningnowtotheexpression(6.53)fortheinteractionenergydensity,weareparticularlyinterestedintheN.R.limitofthetwotermsexplicitlyinvolvingtheelectric¯eld.Letusrefertothesetermscollectivelyasthespin-electricenergydensityanddenotethemby.Recalling(7.3b),we¯ndthat(6.53)givesus 22½s¢(E£v)¡e ½¯:Using(7.4)wecandeterminehowthe¯rsttermin(7.14)couplestothemomentuminsteadofthevelocity.Thus,usingthefactthat )=0,we¯ndthatthatE£(r£½s)]=¡s2E¢r½+E¢ss¢r½+½E¢(s¢rs):(7.15)Byvirtueof(7.3a)thelasttermin(7.14)canbewritten mcs¢E=e m2c2s¢E(½s);:16)26 whichrevealsthatitgivesacontributionofthesameorderofmagnitudeasthenexttolasttermin(7.14).Withthehelpof(7.15)and(7.16)wecanwrite(7.14)intheform 2m2c2f½s¢(E£ps2¡½r¢E¡r¢(½E)¢g¡e Discussionofthisresultwillbedeferreduntilasubsequentpaper,whenwewillbeinapositiontocompleteitwithresultsobtainedbyconventionalmethods.D.Hestenes,Space-TimeAlgebra(GordonandBreach,NewYork,1966).D.Hestenes,J.Math.Phys.,798(1967).D.Hestenes,J.Math.Phys.,893(1973).[Byanerrorinproof,thephysicalinterpretationsoftheDiracandGordoncurrentsin(5.6)ofthisreferencegotinterchangedinthesubsequentsentence.]S.Schweber,AnIntroductiontoRelativisticQuantumFieldTheory(RowPeterson,NewYork,1961),p.70{80.D.Hestenes,J.Math.Phys.,1768(1974).D.Hestenes,J.Math.Phys.,1778(1974).J.JauchandF.Rohrlich,TheTheoryofPhotonsandElectrons(Addison-Wesley,ReadingMass.,1955).D.Hestenes,J.Math.Phys.,809(1967).Equation(2.9b)ofthispapershouldbecorrectedbyaddingthetermtoitsright-handside.(ThecalculationfollowsAppendixBofRef.3.)Thiscorrectiondoesnota®ectthemainideasofthepaper.D.Hestenes,J.Math.Phys.,809(1967).H.BetheandE.Salpeter,QuantumMechanicsofOne-andTwo-ElectronAtoms(Springer-Verlag,Berlin,1957),