Geometrical Properties of Cumulant Expansions累积量膨胀地几何性质.pdf

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1、GeometricalPropertiesofCumulantExpansionsK.KladkoandP.FuldeMax-Planck-InstituteforPhysicsofComplexSystems,BayreutherStr.40H.16,D-01187Dresden,GermanyAbstractCumulantsrepresentanaturallanguageforexpressingmacroscopicproper-tiesofasolid.Weshowthatcumulantsaresubjecttoanontrivialgeometr

2、y.Thisgeometryprovidesanintuitiveunderstandingofanumberofcumulantrelationswhichhadbeenobtainedsofarbyusingalgebraicconsiderations.WegivegeneralexpressionsfortheirinfinitesimalandfinitetransformationsandrepresentacumulantwaveoperatorthroughanintegrationoverapathintheHilbertspace.Casesar

3、einvestigatedwherethisintegrationcanbedoneexactly.Anexpressionoftheground-statewavefunctionintermsofthecumulantwaveoperatorisderived.InthesecondpartofthearticlewederivethecumulantcounterpartofFaddeev‘sequationsandshowitsconnectiontothemethodofincrements.arXiv:cond-mat/9709044v14Sep19

4、97I.INTRODUCTIONInstatisticalphysics,quantummechanics,quantumfieldtheoryandmanyotherfieldsofmoderntheoreticalphysicsweconstantlyfaceobjectsofthetypeλ1A1+λ2A2+...F=lnhei.(1)HereA1,A2,...areabstractelements,λ1,λ2,...arenumbers,h...imeanssomekindofaveragingande(...)denotesageneralizedexpo

5、nentialfunction.1Assumingtheaverageof1tobenonzero,weconcludethatFisananalyticfunctionoftheλiinavicinityofzero.Therefore,ithasaTaylorexpansionwithrespecttotheλi.Thisexpansioniscalledcumulantexpansion,thecoefficientsoftheexpansionarecalledcumulants.Cumulants(semiinvariants)havebeenknownl

6、ongbeforeinmathematicalstatisticsandprobabilitytheory.Kubo[1]pioneeredindemonstratingtheirusefulnessindiversebranchesofphysics.Consideredasabstractobjectscumulantsformaspecialkindofnoncommutingalgebra.InhisoriginalpaperKuboappliedrelationsfoundinthisalgebratoadiversesetofphysicalprob

7、lems.Inparticular,hegaveanelegantderivationoftheUrsell-Mayerexpansionforclassicalandquantumgasesthatisusuallyobtainedbymuchlongerdiagramconsiderations,herederivedGoldstoneperturbationtheoryfortheground-stateofaHamil-tonoperator,andinvestigatedthemotionofanoscillatorwitharandomfrequen

8、cy.Kubostres