MagneticMonopoleandelectricchargequantizationconditionreviewMohamedElashri[email protected]UniversityofscienceandtechnologyUniversity-ZewailCity.(Dated:August15,2019)Inthispaper,variousmethodsforderivingtheelectricchargequantizationconditionareinves-tigated.FirsttheDiracoriginalargumentisrevisited.Thenitisderivedalsothroughangularmomentumquantizationtogetthesameresult.Aphysicalinterpretationforthemonopoleexis-tencenegativeexperimentalresultsisdiscussed.Thenthemagneticmonopolemassisdiscussed.I.INTRODUCTIONThemagneticmonopoleconceptisoldastheorigintomagnetism,whichfirstaccountsin1269.Thenwiththebeginningofthe19thcenturythereweremanyexperi-mentsanddiscussionsabouttheexistenceofthemag-neticisolatedpoles.Diracproposedhisfamouspaperin1931sayingthatexistenceofmonopolewillexplaintheobservedphenomenaofelectricchargequantization.HeobservedthatmagneticmonopolecanbeIncorpo-ratedintoquantummechanicsifbothofthemagneticandelectricarequantizedcharge[1].Diracworkresultsinarelationbetweenthebasicele-mentaryelectricchargeeandthebasicmagneticchargegasfollows:ge=n~c2(1)wherenisinteger,thenthequantitygD=~c2eiscalledtheDiracunitcharge.Inthefollowingsectionsthequan-tizationconditionisderivedusingdifferentways,afterthatthequestionofexperimentalnegativeresultsforsearchesformonopoleisinvestigated.ThenconsequencesoftheDiracrelationaresummarized.AndfinallytheDiracmonopolemassiscalculated.II.ELECTRICQUANTIZATIONCONDITIONAfterDiracmonopole[2]papertherearemanyotherpaperswithdifferentmethodsderivedthesameDiracquntizationrelationeq(1).ThroughthissectionsomeofthismethodsarediscussedandfirsttheoriginalDiracderivationispresentedthenthequntizationofangularmomentumisusedtoderivethesameresult.A.DiracoriginalderivationDiracintroducesaderivationbasedonthemagneticvectorpotentialargumentandintroduceanewconceptcalled”Diracstring”todealwiththesingularitiesthatarises.Diracusestwosingularvectorpotentialsforthemagneticmonopoleattheoriginofcoordinatesystem:A±=g±1cosθrsinθˆφ(2)Diracdividedthespaceoutsidethemonopoleintotwooverlappingregionsandwrotethelastequationforvec-torpotentialineach.Thisvectorpotentialsaregauge-related,ThenorthernregionA+spanslatitudes0θπ!andSouthernregionspans!θπThetwopotentialsaregauge-equivalentA+A=2grsinθˆφ=2gφ(3)Asthetwopotentialsaregauge-equivalenttheyleadtothesamemagneticfieldBB=∇×A+or=∇×(g(±1cosθ)φ)=g((±1cosθ))×∇φ=gsinθrˆφ×1rsinθˆφ=g1r2ˆr(4)itisknownthatinquantummechanicsagaugetrans-formationofvectorpotentialmustbecombinedwithaphasetransformationofthewavefunctionofchargedpar-ticle.Hence,usingthetwogauge-relatedvectorpoten-tialsineq(2)forcingtousetwowavefunctionψ+for0θπ!andψfor!θπ.Intheoverlapregion!θπ!thetwowavefunctionrelatedbyagaugetransformationψ+=ψe2igeφ~c(5)Thisequationinordertomakesensephysically,awavefunctionsψ±mustbesingle-valuedandcontinu-ous.Basedonthatthishappensonlyif(4πieg/~c)=1.thisresultsinelectricchargequantizationconditionas(4πieg/~c)=1meansthat(4πieg/~c)=2πinandthisgiveseq(1).B.UsingAngularMomentumQuantizationTheDiracquantizationconditioncanderivedbyquan-tizingtheangularmomentumcarriedbyelectromagnetic
2fields[3][4].Ifanelectricchargeeatoriginandmag-neticchargegareseparatedbydistancedthisassumedtoresultinelectricandmagneticfieldsasfollows:E=14π!0er2ˆrB=14πgr2bˆrb(6)Nowplacemagneticchargegonz-axisondistancedfromelectricchargee,thenitslocationisdˆz,rb=rdˆz.Then:rb=p(x+d)2+y2+z2=pr22xd+d2=rr12xdr2+d2r2r(1xdr2)=rxdr(7)Thenthetotalangularmomentumofthesystemis:L=1c2Zr×(E×H)d3x=1c2Zr×(14π!0er2×H)d3x=e4π!0Z1rrr.H)Hr.ˆr))d3x=e4πZ1rrr.B)B)d3x(8)Aftersomemanipulationeq(8)givesL=e4πZˆr(.B)d3x=eg4πZˆ3(rdˆz)d3x=eg4πˆz(9)Nowiftheangularmomentumisquantizedintheunitsof~itgivestheDiracquantizationconditioneq(1)III.MAGNETICCOUPLINGSTRENGTHFordiracquantizationconditioneq(1),theminimalmagneticchargeisforn=1whichcanbedefinedasgD=~c2e=e2αe=68.5e(10)whereαeisthefinestructureconstantαe=e2/~c=1/137andeistheminimalelectriccharge.Andifthereexistafreequarkwithe/3,thenitwouldbeexpectedthattheminimalmagneticchargetobethreetimeslarger.Thisminimalmagneticchargeisverylargecomparedwithminimalelectriccharge,andalthoughtheconceptofmagneticmonopoledefinesasymmetricMaxwell’sequations,itleastonumericalasymmetryincontextofmagneticandelectriceffects.Themagneticlikefinestructureconstantisαm=g2/~c=34.25.Thisisverylargecouplingconstant,andratherthanelec-tromagneticinteractionforelectriccharges,magneticmonopoleselectromagneticinteractionisverystronganddoesnotallowforusingperturbativetechniques.Asanexample,theproblemofelasticscatteringofelectronbyamagneticmonopolecannotbedescribedbytheexchangeofonephoton.butcanbedescribedifconsideringtheexchangeofmanyphotons.Thisisduetothefactthatαmα=O(1),whileinordinaryelectronelectronscat-teringithasα2O(104).TheThismeansthatthecouplingofamagneticmonopoleisabout4690timesstrongerthanthecouplingofaelectriccharge.αmαe=14α2e=4692(11)Thusthemagneticmonopolesareattractedtoeachotherbylargeforce,soitisnoteasytoisolatethepolestogeteachseparately.Andinparticleacceleratorstheparame-terspaceoftheexpectedtheoreticalmagneticmonopoleparametersprobedwithoutanypositiveresults.Thisputsanexperimentallimitsonmagneticmonopolespa-rametersspeciallymonopolemasswhichwillbediscussedinthenextsection.IV.CONSEQUENCESOFTHEDIRACRELATIONTheconsequencesoftheDiracrelationcanbesumma-rizedasfollows[1]-Magneticchargeforn=1andthebasicelectricchargeisjusttheelectroncharge,thenthebasicmagneticchargeisgD=~c/2e=137e/2=3.29×108CGS.-Couplingconstantinanalogywiththefinestructureconstantαe=e2/~c=1/137,thedimensionlessmag-neticcouplingconstantisαm=g@D/~c=34.25-Energyacquiredinamagneticfield:E=ngDBl=20.5nkeV/Gcm.-trappingofmagneticmonopolesinferromagneticmaterials:magneticmonopolesmaybetrappedininferromagneticmaterialsbyanimageforce,whichreachthevalueof-Energylossesinmatter.AnyfastmagneticmonopolewithmagneticchargegDandwithvelocityv=βc,itbehaveslikeequivalentelectriccharge(Ze)eq=gDβV.MAGNETICMONOPOLEMASSDiracdidnotgiveanypredictionforthemassofthemagneticmonopole,butitcanbeestablishedinaclassi-
3caltheorybyassumingthattheclassicalelectronradiuswillbeequaltoclassicalmonopoleradius[5]re=e2mec2=rM=g2mMc2(12)FrompreviousequationmM=mec2/e2=4700me=2.4GeVwhichisasexpectedverylargecomparedwiththeelectronmassbecausebasicmagneticchargeismuchlargerthanbasicelectricchargeACKNOWLEDGMENTSIwouldthankDr.Ahmedabdealimforintroducingmeforthefirsttimeforthemagneticmonopoleidea,alsoIwouldtothankDr.AliNassarandMs.NadaKhaledfortheirhelpfuldiscussionduringwritingthispaper.[1]G.GiacomelliandL.Patrizii,[hep-ex/0112009].[2]P.A.M.Dirac,Proc.Roy.Soc.Lond.A133,60(1931).doi:10.1098/rspa.1931.0130[3]SahaMN1936Ind.J.Phys.10141[4]WilsonHA1949Phys.Rev.75309[5]G.GiacomelliandL.Patrizii,ICTPLect.NotesSer.14,121(2003)[hep-ex/0302011].