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1、MAGNETICMONOPOLES,FIBERBUNDLES,ANDGAUGEFIELDSmenCommentaryChenNingYangbeginspage78InstituteforTheoreticalPhysicsStateUniversityofNewYorkatStonyBrookStonyBrook,NewYork1I794Thereportsinthismonographhaveshowngreatenthusiasmandexuberancefortheunificationofvariousinteractionsthroug
2、htheconceptofgaugefields.Iwouldliketoemphasizeapointthathasnotyetbeenexplicitlystatedbyanyoftheotherauthors:gaugefieldsaredeeplyrelatedtosomeprofoundlybeautifulideasofcontemporarymathematics,ideasthatarethedrivingforcesofpartofthemathematicsofthelast40years.Recallingtherelatio
3、nshipbetweenphysicsandmathematicsinearlierperiods,generalrelativityandRiemanniangeometry,quantummechanicsandHilbertspace,itisalltooobviousthatphysicistsmayagainbezeroinginonafundamentalnewsecretofnature.Themathematicaldevelopmentreferredtoaboveisthetheoryoffiberbundles.Itmayap
4、pear,apriori,thatthistheoryisquiteabstractandisunrelatedtothestructureofthephysicalworld.Toshowthatthisisnottrue,wewillstartwithasimpledemonstrationthatelectromagnetismandquantummechanicstogetherleadnaturallyto“nontrivialfiberbundles.”Wewillthentracetheearlyhistoryofthegaugefi
5、eldconceptanditsgeneralization,emphasizingthreerelatedbutdifferentconceptualmotivations,eachofwhichleadstoageneralformulationofgaugefields.MAGNETICMONOPOLESANDNONTRIVIALBUNDLESThemagneticmonopoleisthemagneticcharge.Thoughtheideaofmagneticmonopolesprobablywasdiscussedinclassicp
6、hysicsearlyinthehistoryofelec-tricityandmagnetism,moderndiscussionsofthisconceptdatebackonlyto1931,whentheimportantpaperofDirac’pointedoutthatmagneticmonopolesinquantummechanicsexhibitsomeextraandsubtlefeatures.Inparticular,withtheexistenceofamagneticmonopoleofstrengthg,electr
7、icchargesandmagneticchargesmustnecessarilybequantized,inquantummechanics.Wewillgiveanewderivationofthisresultbelow.Ifonewantstodescribethewavefunctionofanelectroninthefieldofamagneticmonopole,itisnecessarytofindthevectorpotentialAaroundthemonopole.Diracchoseavectorpotentialtha
8、thasastringofsingularities.Thenecessityofsuchastringofsingula
