knorrer,fermionic functional integrals and the renormalization group

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1、FermionicFunctionalIntegralsandtheRenormalizationGroupJoelFeldmanDepartmentofMathematicsUniversityofBritishColumbiaVancouverB.C.CANADAV6T1Z2feldman@math.ubc.cahttp://www.math.ubc.ca/feldmanHorstKn•orrer,EugeneTrubowitzMathematikETH{ZentrumCH-8092Zuric•hSWITZERLANDAbstractThe

2、RenormalizationGroupisthenamegiventoatechniqueforanalyzingthequalitativebehaviourofaclassofphysicalsystemsbyiteratingamaponthevectorspaceofinteractionsfortheclass.Inatypicalnon-rigorousapplicationofthistechniqueoneassumes,basedonone'sphysicalintuition,thatonlyacertainnitedim

3、ensionalsubspace(usuallyofdimensionthreeorless)isimportant.ThesenotesconcernatechniqueforjustifyingthisapproximationinabroadclassofFermionicmodelsusedincondensedmatterandhighenergyphysics.ThesenotesexpandupontheAisenstadtLecturesgivenbyJ.F.attheCentredeRecherchesMathematique

4、s,UniversitedeMontrealinAugust,1999.TableofContentsPrefacepixIFermionicFunctionalIntegralsp1xI.1GrassmannAlgebrasp1xI.2GrassmannIntegralsp7xI.3DierentiationandIntegrationbyPartsp11xI.4GrassmannGaussianIntegralsp14xI.5GrassmannIntegralsandFermionicQuantumFieldTheoriesp18xI.

5、6TheRenormalizationGroupp24xI.7WickOrderingp30xI.8BoundsonGrassmannGaussianIntegralsp34xIIFermionicExpansionsp40xII.1NotationandDenitionsp40xII.2Expansion{Algebrap42xII.3Expansion{Boundsp44xII.4SampleApplicationsp56Gross{Neveu2p57NaiveMany-fermion2p63Many-fermion2{withsector

6、izationp67AppendicesxAInniteDimensionalGrassmannAlgebrasp75xBPfaansp86xCPropagatorBoundsp93xDProblemSolutionsp100Referencesp147PrefaceTheRenormalizationGroupisthenamegiventoatechniqueforanalyzingthequalitativebehaviourofaclassofphysicalsystemsbyiteratingamaponthevectorspace

7、ofinteractionsfortheclass.Inatypicalnon-rigorousapplicationofthistechniqueoneassumes,basedonone'sphysicalintuition,thatonlyacertainnitedimensionalsubspace(usuallyofdimensionthreeorless)isimportant.Thesenotesconcernatechniqueforjustifyingthisapproximationinabroadclassoffermio

8、nicmodelsusedincondensedmatterandhighenergyphysics.Therstchapterpro