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1、MULTIVALUEDFIELDSinCondensedMatter,Electromagnetism,andGravitationMultivaluedFieldsinCondensedMatter,Electromagnetism,andGravitationHagenKleinertProfessorofPhysicsFreieUniversit¨atBerlin(Germany)ICRANet,Pescara(Italy)andNice(France)ToAnnemarieandHagenIIThemodernphysicaldevelopmentshaverequiredam
2、athematicsthatcontinuallyshiftsitsfoundations.P.A.M.Dirac(1902–1984)PrefaceThetheorypresentedinthisbookhasfourroots.ThefirstliesinDirac’sseminalpaperof1931[1]inwhichhepointedoutthatMaxwell’sequationscanaccommo-datemagneticmonopoles,inspiteofthevanishingdivergenceofthemagneticfield,thankstoquantumm
3、echanicsItisalwayspossibletocreateamagneticfieldemerg-ingfromapointbyimportingthefieldfromfardistancetothepointthroughaninfinitelythinmagneticfluxtube.Butitisonlyduetoquantummechanics,thatsuchafluxtubecanbemadephysicallyundetectable.ThisistrueprovidedthefamousDiracchargequantizationconditionisfulfille
4、dwhichstatesthatallelectricchargesareintegermultiplesof2πhc/g¯,wheregisthetotalmagneticfluxthroughthetube.TheundetectablefluxtubeiscalledtheDiracstring.Fromtheendpointofthestring,magneticfieldlinesemergeradiallyoutwardsinthesamewayaselec-tricfieldlinesemergefromanelectricpointcharge,sothattheendpoin
5、tactsasamagneticmonopole.Theshapeoftheundetectablestringiscompletelyirrelevant.Itisamathematicalartifact.Forthisstunningobservation,PauligaveDiracthenicknameMonopoleon.TheDiracquantizationconditionwassubsequentlysharp-enedbySchwinger[2]whoshowedthatthedouble-valuednessofthespin-1wave2functionsof
6、electronrestrictedtheintegermultiplestoevenmultiples.Experimen-tally,nomagneticmonopolewasfoundinspiteofintensivesearch,andtheDiractheorywasputadactaforalongtime.Itresurfaced,however,inthelast35years,intheattempttoexplainthephenomenonofquarkconfinement.Thesecondrootinthisbookliesinthetheoryofthes
7、uperfluidphasetransition.HerethecrucialpaperswerewrittenbyBerezinski[3]andbyKosterlitzandThouless[4].Theyshowedthatthephasetransitioninafilmofsuperfluidheliumcanbeunderstoodbythestatisticalmechanicsofvorticesofsuperflow.Theirdes
