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1、QuaternionicAnalysis,RepresentationTheoryandPhysicsIgorFrenkelandMatveiLibineMay25,2008AbstractWedevelopquaternionicanalysisusingasaguidingprinciplerepresentationtheoryofvariousrealformsoftheconformalgroup.WefirstreviewtheCauchy-FueterandPoissonformulasandexpl
2、aintheirrepresentationtheoreticmeaning.Therequirementofunitar-ityofrepresentationsleadsustotheextensionsoftheseformulasintheMinkowskispace,whichcanbeviewedasanotherrealformofquaternions.Representationtheoryalsosug-gestsaquaternionicversionoftheCauchyformulafo
3、rthesecondorderpole.Remarkably,thederivativeappearinginthecomplexcaseisreplacedbytheMaxwellequationsinthequaternioniccounterpart.Wealsouncovertheconnectionbetweenquaternionicanalysisandvariousstructuresinquantummechanicsandquantumfieldtheory,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⊕RkandcarvingthedefiningrelationsonastoneofDublin’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
