towards a generalized distribution formalism for gauge quantum fieldsnew

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1、P.N.LebedevInstitutePreprintFIAN/TD/2–94I.E.TammTheoryDepartmentMarch1994TowardsaGeneralizedDistributionFormalismforGaugeQuantumFields1M.A.SolovievDepartmentofTheoreticalPhysicsP.N.LebedevPhysicalInstituteLeninskyprospect,53,117924,Moscow,Russia†ABSTRACTWeprovethatthedistributionsdefinedontheGelf

2、and-ShilovspacesSβwithαβ<1,andhencemoresingularthanhyperfunctions,retaintheangularlocaliz-abilityproperty.Specifically,theyhaveuniquelydeterminedsupportcones.Thisresultenablesonetodevelopadistribution-theoretictechniquessuitableforthearXiv:hep-th/9403083v115Mar1994consistenttreatmentofquantumfield

3、switharbitrarilysingularultravioletandinfraredbehavior.Theproofcoveringthemostgeneralanddifficultcaseβ=0isbasedontheuseofthetheoryofplurisubharmonicfunctionsandH¨ormander’sL2–estimates.†E-mailaddress:soloviev@td.fian.free.net1ThisworkwassupportedinpartbyaSorosHumanitarianFoundationGrantawardedbythe

4、AmericanPhysicalSociety.21IntroductionRecentlytheproblemofanadequatechoiceoftestfunctionsinquantumfieldtheory[1]cameonceagainintonoticewithemphasisoninfraredsingularitieswhichoccuringaugeQFTinconnectionwiththenecessarylackofpositivity[2].Namely,analysisofsomeexplicitlysolublemodelshasshown[3–5]th

5、atthesesingularitiesareingeneralmoreseverethanthoseofultradistributionsorevenhyperfunctionsandthatappropriatetestfunctionspacesareSβbyGelfandandShilov[6].Theindicesαandβcontrolrespectivelytheαultravioletandinfraredsingularitieswhoseorderincreaseswithdecreasingtheirvalues,whiletheSchwartzspaceuse

6、dintheoriginalWightmanformalismcanbeidentifiedwithS∞.Of∞greatimportanceisthefactthattheusualdefinitionofsupporthassenseonlyforgeneralizedfunctionsonSβwithβ>1,forotherwisetheconfiguration-spacetestfunctionsareanalytic.αThesameistruereferringtothemomentum-spacetestfunctionsandα>1.Justthenoneusesthete

7、rmultradistributionsandtheformerinequalityselectsinfacttheJaffestrictlylocalizablefields[7].ThespaceS1,whichwasputforwardin[8]asmostadequateto1Meiman’slocalityprinciple[9],admitsadirectgeneralizationofthenotionofsupportandunde