Suc an equation ust tak in to accoun oth spin states The efunction ust spinor rather than scalar and the spin states are ouple th us simple scalar equation lik the KleinGordon equation just ont do So in the 1920s Dirac set out to 57356nd equation th ID: 56723
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1AntiparticlesTheKlein-Gordonequation @2 @t2+r2 =m2 (1)thatwederivedinthepreviouslectureisnotsatisfactoryfordealingwithmassiveparticlesthathavespin.Suchanequationmusttakeintoaccountbothspinstates.Thewavefunctionmustbeaspinorratherthanascalar,andthespinstatesarecoupled|thusasimplescalarequationliketheKlein-Gordonequationjustwon'tdo.So,inthe1920s,Diracsetoutto ndawaveequationthatwoulddescribetheelectron.1Hemathematicallyshowedthatinordertoavoidnegativeprobabilitydensitysolutions,hisequationwouldhavetobelinearin@@t,unliketheK-Gequation,whichhasa@2@2t.ThushisequationwouldhavetobeoftheformH =(~ ~p+ m) (2)butmustalsosatisfytherelativisticenergy-momentumrelationH2 =(~p2+m2) (3)Thequestionis,whatare~ and ?Lookingatthis,youcanseethattheycan'tjustberealnumbers.Itturnsoutthatthesimplestrepresentationof~ and isasetof4x4matrices.IfonetakesthePaulimatrices:1= 0110!;2= 0 ii0!;3= 100 1!;(4)andthende nesthesetof4x4matrices:~ = 0~~0!; = I00 I!;(5)whereIisthe2x2identitymatrix,onesatis esbothequations(2)and(3).ThuswehavetheDiracequation|itisjustequation(2)with~ and describedby(5).Buthowcanweinterpretthe4elementsoftheresulting1Dirac'smotivationwasactuallynottodescribespin,buttoavoidnegativeprobabilitydensitysolutionsoftheKlein-Gordonequation.Buthisworkendedupnotonlydescribingspin,but ndinganexplanationoftheKlein-Gordonnegativeprobabilitydensitysolutions|aswewillsee,antiparticles.1 wavefunctioneigen-spinor?Ifitwere2elements,thatwouldbesimple{itwouldjustbethe+1/2and-1/2spincomponents.But4?Let'sconsiderafreespaceeigen-spinorsolutionwithmomentum~p,whichwillhavetheform =u(~p)e i~p~x.TheDiracequationbecomes m~~p~~p m! u1;2u3;4!=E u1;2u3;4!(6)Thereare4solutionstothis.Ifwetake 1= 1! 2= 0!(7)thenthesolutionsareu1=N 1~~pE+m 1!;u2=N 2~~pE+m 2!;u3=N ~~pjEj+m 1 1!;u4=N ~~pjEj+m 2 2!:(8)The rst2haveassociatedenergyeigenvalueE=p~p2+m2.Butthesecond2haveenergyeigenvalueE= p~p2+m2|anegativeenergy!Whatistheinterpretationofthesesolutions?ConsideranelectronofenergyEand3-momentum~p.Itselectromag-netic4-vectorcurrentisj(e )= 2ejN2j0Epxpypz1:(9)Nowconsideranantiparticle,apositron,withthesameE,~p.Sinceitschargeis+e,j(e+)=+2ejN2j0BEpxpypz1= 2ejN2j0B E px py pz1;(10)whichisthesameasthecurrentjforanelectronwith E, ~p.Thus,anemissionofapositronwithenergyEisthesameastheabsorptionofanelectronofenergy E.2 Thusthenegative-energystatesintheDiracequationcanbeinterpretedaspositive-energyantiparticles.Orvice-versa.OneespeciallyinterestingthingabouttheDiracequationisthatitputsparticlesandantiparticlesonthesamefooting.ThereisacompletesymmetrybetweenthemasfarastheDiracequationisconcerned.Thefactthatwehadparticlesbethe rst2solutionsandantiparticlesthelasttwowasjustanexampleofourpersonalprejudice|wecouldjustaseasilyhavede ned~ and sothatantiparticleswerethe rst2solutionsandparticleswerethelast2.Butwhenwelookaroundus,weseeonlyparticles.AntiparticleswerediscoveredafewyearsafterDiracpublishedhisequation,andwemakethemeverydayinlaboratories,etc.,buttheyarerareinthenaturalworld.Butmathematicallythereisasymmetrybetweenparticlesandantiparticles.So,whydoweseeonlyparticles?2BaryonicfreezeoutThehotbigbangmodeloftheuniverseimpliesthatatanearlyepochoftheuniverse,leptonicandbaryonicpairsexistedinafullymixedstateinequi-libriumwithradiation.Astheuniverseexpandedandcooled,matterandantimatterwascontinuallybeingcreatedandannihilated,2buteventuallyfreezesoutintoasteady-statenumberdensity.Thisphasetransitionoccurswhentheexpansionrateoftheuniverseexceedstheannihilationrate,thelatterofwhichisdeterminedbythematter-antimatterannihilationcross-section.Wecanusethesamemethodweusedtwoweeksagofordarkmatterneutrinosandneutralinostodeterminethefreeze-outdensityofprotonsandantiprotons.Unlikeneutrinosandneutralinos,protonsandantiprotonsannihilateviathestronginteraction,viadiagramssuchas2TherewillbetimetomurderandcreateAndtimeforalltheworksanddaysofhandsThatliftanddropaquestionontoyourplate.|T.S.Eliot(1917)3 Thecarrierofthestrongforceisthegluon,butatlowenergiessuchasthis,thecouplingconstant sislarge,andinsteadoflargenumbersofgluons,theintermediaryparticlecanbethoughtofasthe0.3Followingtheanalysisofp.9-11ofthesecondweek'slecturenotes,thecross-sectionis 2=m20.Thus,analogoustothecaseofneutralinos,wehavenp/e mp=TZ10p2dpe p2=(2mpT)/(mpT)3=2e mp=T(11)andwemustsolvetheequation 2m20m3x 3=2e x=m2mPlx(12)andweobtainxln 2mpmPlm20! 12lnln 2mpmPlm20!50(13)wheremp1GeV,m0100MeV,and s1.Thus,protonsandantiprotonsfreezeoutatatemperatureTmp=5020MeV.Againanalogoustothecaseofneutralinos,weexpectapost-freezeoutabundancenpn\rHf=vn\r=T2f=mPl( 2m2p=m40)mPlT3f510 22(14)whereHfandTfarethevaluesoftheHubbleconstantandtemperatureatfreezoutrespectively.Eachprotonweighsapproximately1GeV,sothiscorrespondstoamassdensity(usingn\r=400cm 3today)of\nbaryon410 13today.Butthisiscompletely,utterlydi erentthanwhatweobserve!Weseethatbaryonsmakeupapproximately4%oftheclosuredensityoftheuni-verse,not410 13ofit.WeknowthiswithexcellentprecisionfrombothBBNandtheCMB.Sowhydowehavesomanymorebaryonsthanweexpect?Andwheredidalltheantibaryonsgo?3Matterandantimatterdomains?Perhapswearejustinaregionoftheuniversethatismetter-dominated,andotherregionsaredominatedbyantimatter.Thispicturecanbecompared3The0'squarkcontentis1p2(uu+dd),soitactssortoflikeacreationandannhiliationoperatorforlightquarks.4 tothespontaneousformationofferromagneticdomainswhenapieceofunmagnetizedironcoolsbelowthecriticaltemperatureintheabsenceofamagnetic eld.[1]Astothemechanismforhowsuchascenariocouldberealized,wewillnotneedtodelveintohere,becausewecanshowthatitisexcludedbyacombinationofinformationfromtheCMBandfromtheCDG(cosmicdi usegammaradiation)spectrum.Weknowthatifmatterandantimatterdomainsexist,theymustbeincontact.Thisisbecauseiftherewerevoidsinbetweenthedomains,suchvoidswouldshowupasanisotropiesintheCMB.Sincethedomainsareincontact,thereshouldbeannihilationbetweenthenucleonsandantinucleonsatthedomainboundary.Themainreactionsareproton-antiprotonandneutron-antineutronannihilationstoseveralpi-onsandphotons,ie.p+pandn+ntomultiple+, ,0,and\r.Thedecayprimarilyto+,andthethendecaytoee.The0decayto\r+\r.Therewillthusbetwopotentiallyobservablee ectsfromtheseinter-actions.TheelectronsfromthechargedpiondecayswillComptonscattero CMBphotons,whichcouldcontributetotheso-calledSunyaev-Zeldoviche ect,andwillalsoheattheinterstellarmedium.Bothofthesee ectscouldaltertheCMBspectrum.Thephotonsfromthe0decayswillcontributetotheCDGspectrum.Ananalysisofthesizeofthesee ectsshowsthatthee ectontheCMBfromtheelectronsisnegligiblecomparedwithexper-imentalsensitivity.However,asshownbyCohen,deRujula,andGlashowin1998,thee ectfromthephotonsontheCDGspectrumplacesquitesig-ni cantlimitsonthesizeofmatter-antimatterdomains[2].Thefollowing gureshowstheobservedCDGspectrum,withcurvesshowingtheexpectedcontributionfrommatter-antimatterannihilationphotonsproducedbydo-mainsofsize20Mpc(uppercurve)and1Gpc(lowercurve).ThedatathusrulesoutdomainsizessmallerthanaGpc,ie.approximatelythesizeoftheuniverse.Thise ectivelyrulesoutmatter-antimatterdomains.Therearestillquibbleswiththisargument,sincee ectssuchasmagnetic eldsatthedomainboundariescane ecttheamountofmatter-antimatterannihilation.ExperimentsarestillbeingdonetobothobtainimprovedsensitivitytotheCDGandtolookforantinuclei,suchasantihelium,thatcoulddriftacrossdomainboundaries.However,themajorityofpeopleinthe eldfeelthatmodelsthatwouldavoidtheCDGlimitstendtoberathercontrived,andpeoplewouldbeextremelysurprisedifantinucleiexperimentssuchasBESSandAMSweretoseeasignal.(Needlesstosay,sofartheyhavenot.)Soweareleftwiththefactthatsomethingintheearlyuniversemusthaveproducedanoverallasymmetrybetweenbaryonsandantibaryons.5 Figure1:Theobservedcosmicdi usegammaradiation(CDG)spectrum,withcurvesshowingtheexpectedcontributionfrommatter-antimatteran-nihilationphotonsproducedbydomainsofsize20Mpc(uppercurve)and1Gpc(lowercurve).ThedatathusrulesoutdomainsizessmallerthanaGpc[2].6 Whatfundamentalphysicscouldcausesuchanasymmetry?Thisques-tionsisknownasthebaryogenesisproblem.Inordertoexplainwhatsortofphysicscouldsolveit,wewillneedtolearnaboutthediscretesymmetriesoftheStandardModel.4DiscretesymmetriesTheStandardModelLagrangiancontainsmanysymmetries.Itofcourseobeysspecialrelativity,thusitissymmetricunderLorentztransformations.Italsoconatinstheso-called\gauge"symmetriesoftheinteractions.Forexample,electromagneticsissymmetricunderarede nitionofthescalarandvectorpotentials~A0=~A+~rV0=V @@t(15)whereisanyscalarfunction.Ingrouptheoryterms,thissymmetryiscalledaU(1)symmetry.TheweakandstronginteractioncontributeSU(2)andSU(3)gaugesymmetriesrespectively,thustheStandardModelgaugesymmetryisSU(3)SU(2)U(1).Inadditiontothesecontinuoussymmetries,therearethreeindependentdiscretetransformationsthatalsopreservetheMinkowskiintervalt2 ~x2.Theyarethechargeconjugationoperator(C),theparityoperator(P),andthetime-reversaloperator(T).TheseformacompletesetofdiscreteMinkowskiinterval-preservingtransformationsoftheHilbertspace.Al-thoughotherdiscreteinterval-preservingtransformationsexistintheStan-dardModel,allcanbeformedfromC,P,T,andthegroupofcontinuousLorentzandgaugerotations.Maxwell'sequations,andthustheelectromagniticinteraction,aresym-metricundereachofthethreediscretetransformations.Detailedstudiesofthemagneticdipolemomentoftheneutronshowthatthestronginteractionissymmetricundereachofthe3transformationsaswell.Generalrelativityisalsosymmetricundereachofthe3interactions.Butwhatabouttheweakinteraction?5TheSakharovconditionsIn1967,Sakharovshowedthat3conditionsarerequiredfor7 6TheweakinteractionandCPviolation7Baryonnumberviolation8SupersymmetryandCPviolation9LeptogenesisReferences[1]F.W.Stecker,hep-ph/0207323.[2]A.G.Cohen,A.deRujula,andS.Glashow,Astrophys.J.495,539(1998).8