資源描述：
《Quaternionic_Analysis__Representation_Theory_and_Physics外文》由會(huì)員上傳分享，免費(fèi)在線閱讀，更多相關(guān)內(nèi)容在學(xué)術(shù)論文-天天文庫。
1、QuaternionicAnalysis,RepresentationTheoryandPhysicsIgorFrenkelandMatveiLibineMay25,2008AbstractWedevelopquaternionicanalysisusingasaguidingprinciplerepresentationtheoryofvariousrealformsoftheconformalgroup.We?rstreviewtheCauchy-FueterandPoissonformulasandexpl
2、aintheirrepresentationtheoreticmeaning.Therequirementofunitar-ityofrepresentationsleadsustotheextensionsoftheseformulasintheMinkowskispace,whichcanbeviewedasanotherrealformofquaternions.Representationtheoryalsosug-gestsaquaternionicversionoftheCauchyformulafo
3、rthesecondorderpole.Remarkably,thederivativeappearinginthecomplexcaseisreplacedbytheMaxwellequationsinthequaternioniccounterpart.Wealsouncovertheconnectionbetweenquaternionicanalysisandvariousstructuresinquantummechanicsandquantum?eldtheory,suchasthespec-trum
4、ofthehydrogenatom,polarizationofvacuum,one-loopFeynmanintegrals.Wealsomakesomefurtherconjectures.Themaingoalofthisandoursubsequentpaperistore-vivequaternionicanalysisandtoshowprofoundrelationsbetweenquaternionicanalysis,representationtheoryandfour-dimensional
5、physics.Keywords:Cauchy-Fueterformula,Feynmanintegrals,Maxwellequations,conformalgroup,Minkowskispace,Cayleytransform.1IntroductionItiswellknownthatafterdiscoveringthealgebraofquaternionsH=R1⊕Ri⊕Rj⊕Rkandcarvingthede?ningrelationsonastoneofDublin’sBroughamBrid
6、geonthe16October1843,theIrishphysicistandmathematicianWilliamRowanHamilton(1805-1865)devotedtheremainingyearsofhislifedevelopingthenewtheorywhichhebelievedwouldhaveprofoundarXiv:0711.2699v4[math.RT]25May2008applicationsinphysics.Butonehadtowaitanother90yearsb
7、eforevonRudolfFueterproducedakeyresultofquaternionicanalysis,anexactquaternioniccounterpartoftheCauchyintegralformulaI1f(z)dzf(w)=.(1)2πiz?wBecauseofthenoncommutativityofquaternions,thisformulacomesintwoversions,oneforeachanalogueofthecomplexholomorphicfuncti
8、ons–left-andright-regularquaternionicfunc-tions:Z1(Z?W)?1f(W)=·?dZ·f(Z),(2)2π2?Udet(Z?W)Z1(Z?W)?1g(W)=g(Z)·?dZ·,?W∈U,(3)2π2?Udet(Z?W)1whereU?Hisaboundedopenset,thedeterminantistakeninthes
