Perturbation theory for bound states and resonances where potentials and propagators have a

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

Perturbationtheoryforboundstatesandresonanceswherepotentialsandpropagatorshavearbitraryenergydependence

A.N.Kvinikhidze1,2, andB.Blankleider2

1DepartmentofPhysicsandAstronomy,UniversityofManchester,ManchesterM139PL,United

Kingdom

2DepartmentofPhysics,TheFlindersUniversityofSouthAustralia,BedfordPark,SA5042,3

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aAustralia(February1,2008)AbstractStandardderivationsof“time-independentperturbationtheory”ofquan-tummechanicscannotbeappliedtothegeneralcasewherepotentialsareenergydependentorwheretheinversefreeGreenfunctionisanon-linearfunctionofenergy.Suchderivationscannotbeused,forexample,inthecon-textofrelativisticquantum eldtheory.Herewesolvethisproblembypro-vidinganew,generalformulationofperturbationtheoryforcalculatingthechangesintheenergyspectrumandwavefunctionofboundstatesandreso-nancesinducedbyperturbationstotheHamiltonian.Althoughourderivationisvalidforenergy-dependentpotentialsandisnotrestrictedtoinversefreeGreenfunctionsthatarelinearintheenergy,theexpressionsobtainedfortheenergyandwavefunctioncorrectionsarecompact,practical,andmaximallysimilartotheonesofquantummechanics.Forthecaseofrelativisticquan-tum eldtheory,ourapproachprovidesadirectcovariantwayofobtainingcorrectionstoboundandresonancestatemasses,aswellastowavefunctionsthatarenotinthecentreofmassframe.

11.10.St,11.80.Fv,13.40.Ks,31.15.Md

TypesetusingREVTEX

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

I.INTRODUCTION

Thereisagrowinginterestincalculations,withinacovariantquantum eldtheoryframework,ofchangesinthepropertiesofboundstatesandresonancesinducedbysmallperturbationsintheinteractionHamiltonian.Thefour-dimensionalBethe-Salpeterequationanditsvariousthree-dimensionalreductions(so-calledquasi-potentialequations)arethemostpopulartoolsinthisrespect.AcurrentexampleistheNambuJona-Lasinio(NJL)modelwherethenucleonisdescribedintermsofthreerelativisticquarksinteractingviacontactpotentials,andwheremesonexchangeprovidesanimportantperturbativecorrection

[1].AnotherexampleisprovidedbyrelativisticcalculationsofhadronicatomswherethestronginteractionperturbstheCoulombboundstate[2,3],andyetanotherbyvariousothercorrectionstorelativisticcalculationsofelectromagneticboundstates[4].

Theperturbationprobleminvolvedinsuchcovariantcalculationscanbeformulatedasfollows.Denotingthetotalfour-momentumofthesystembyP,onewouldliketodeterminetheboundstatesolutionoftheequation

1G 0(P) K0(P) K1(P)Ψ=0

whereK1(P)isaperturbationtotheunperturbedkernelK0(P),andwhereitisassumedthattheunperturbedGreenfunctionGu(P),de nedasthesolutiontotheequation

Gu(P)=G0(P)+G0(P)K0(P)Gu(P),(2) (1)

isknowncompletely.1ThusweseekthemassMandwavefunctionΨsuchthatEq.(1)withP2=M2issatis ed.AconsequenceofthecompleteknowledgeofGu(P)isthatthemass

nspectrumMu(n=1,2,3,...)andcorrespondingwavefunctionsΦnoftheunperturbed

equation

1G 0(P) K0(P)Φn=0

n2whereP2=(Mu),areknown.

ThetaskofsolvingEq.(1)byexpressingthemassMandwavefunctionΨasaperturba-tionserieswithrespecttoK1isaproblemwhosesolutioniswell-knowninthecorrespondingcontextofnon-relativisticquantummechanics(givenbyso-calledtime-independentpertur-bationtheory).Unfortunatelythe(textbook)derivationusedtoobtainthequantumme-1chanicalresultisrestrictedtothecasewheretheinversefreeGreenfunctionG 0(P)islin-earlydependentonenergyP0andwheretheunperturbedkernelK0isanenergy-independentHermitianoperator.Althoughtheserestrictionsleadtotheclosureandorthonormalitycon-ditions (3)

¯nΦm=δnm,Φ n¯n=1,ΦnΦ(4)

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

whicharecrucialforthederivationoftime-independentperturbationtheory,theyarenotvalidintheBethe-Salpetercase.Indeednoneoftheserestrictionsarerequiredinthecontextofacovariant eldtheoreticapproach.Inthispaperwethereforepresentanewsolution1totheperturbationproblemwhichisvalidforanyformofG 0(P)andK0(P);inparticu-1lar,oursolutionisvalidforthecaseofcovariant eldtheoreticapproacheswhereG 0(P)dependsnonlinearlyonP0andwhereK0(P)canbeenergy(P0)dependent.Oursolution,giveninEq.(24)andEq.(25)forthenondegeneratecase,andinEq.(43)andEq.(46)forthedegeneratecase,expressesthemassMoftheboundstateorresonanceandthecorrespondingwavefunctionΨintermsofcompactexpressionsthattakeintoaccounttheperturbationtermK1toanyorder.Atthesametime,ourformulationallowsustowritetheperturbationseriesforbothMandΨ,uptoanyorder,inastraightforwardwaywhichismaximallyclosetotheanalogousquantummechanicalformulation.Afurtherimportantaspectofourapproachisthatitismanifestlycovariant.ThisfeatureenablesthedirectuseoftheperturbationseriesforΨalsoincaseswheretheboundstateorresonanceisnotatrest.InthiswaythemoreinvolvedapproachofLorentzboostingwavefunctionscalculatedperturbativelyintherestframe,canbeavoided.Assuch,ourapproachtotheperturbationproblemwherenorestrictionisputontheenergydependenceofkernelsandinversefreeGreenfunctions,mayprovidesomeimportantadvantagesoverpreviousformulations[5,6,2].

II.PERTURBATIONTHEORY

A.Basicequations

Inthispaperweusetheframeworkofrelativisticquantum eldtheorytoillustrateourapproachtoperturbationtheory.Althoughthisisdonepartlyforpresentationalpurposes–itisaparticularcasewherethekernelisenergydependentandwheretheinverseGreenfunc-tionisnon-linearlydependentonenergy,itisalsoaparticularlytopicalcase,asdiscussedintheIntroduction.Ontheotherhand,weemphasizethatourapproachtoperturbationtheorydoesnotdependontheparticulartheoreticalframeworkinwhichtheboundstateproblemisset–itcanbethatofnon-relativisticquantummechanics,relativisticquantum eldtheory,three-dimensionalrelativisticquasi-potentialequations,etc.Similarly,ourap-proachdoesnotdependonthefunctionalformtakenbytheenergydependenceofeitherthekernelortheinversefreeGreenfunction.Allweneedtoassumeistheusualoverallstructureofthedynamicalequationsinvolved,asexempli edbyEq.(1)andEq.(2).WethusconsidertheGreenfunction

G(P)=G0(P)+G0(P)K(P)G(P),(5)

wherePisthetotalfour-momentum,G0isthefullydisconnectedpartofG,andwherethekernelKconsistsofapartK0forwhichthecorrespondingGreenfunctionisknown,andasmallpartK1whichcanbetreatedasaperturbation.Thus

K(P)=K0(P)+K1(P),(6)

anditisassumedthattheunperturbedGreenfunctionGu(P)hasbeenpreviouslydeter-minedbysolvingEq.(2).WeareinterestedinthecasewhereGu(P)hasapolecorrespondingtoaboundorresonancestate.Thuswecanwrite

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

Gu(P)=¯P)iΦ(P)Φ(

¯P)arelikewiseassumedtobecovariantfunctionswhichThewavefunctionsΨ(P)andΨ(¯,whereP¯2=M2,totherespectivesolutionsoftheboundstatereduceinthelimitP→P

equations

¯)=G0(P¯)K(P¯)Ψ(P¯),Ψ(Pand¯P¯)=Ψ(¯P¯)K(P¯)G0(P¯).Ψ((10)P2 M2 b(P).+G(9)

TowriteaperturbationseriesforG,weexpressGintermsoftheknownunperturbedGreenfunctionGuthroughtheequation

G(P)=Gu(P)+Gu(P)K1(P)G(P),(11)

1 1 1whichfollowsfromthefactthatG 1=G 0 KandGu=G0 K0.ByiteratingEq.(11)

weobtainaperturbationseriesforG(P)withrespecttotheperturbationK1(P).What

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

appearsmoredi cultisto ndacorrespondingperturbationseriesforthemassMandwavefunctionΨ.Yetifonecloselyexaminesthestructureoftheaboveequations,itcanbediscoveredthatamathematicallysimilarproblemwassolvedlongagobyFeshbach[7]albeitintheratherdi erentcontextofnuclearreactiontheory.Indeedthereareanumberofothercontextswhereanalogousproblemshavebeensolved,thecaseofmassandvertexrenormalizationinpion-nucleonscatteringbeingparticularlynoteworthy[8].InthenextsectionweshallthereforeusethemethodofFeshbachtoderivethesolutionofourcovariantperturbationtheoryproblem.

B.Solution

¯andtheInthissubsectionwederiveexpressionsfortheboundstatewavefunctionsΨ,Ψ,

boundstatemassMcorrespondingtothefullkernelKofEq.(6).Althoughourgoalistoformulatethecovariantperturbationtheoryforthisproblem,weinfactderiveexpressions¯andM,thatareexactwithallordersofK1beingtakenintoaccount.StartingforΨ,Ψ,

fromtheseexactexpressionsitisthentrivialtogeneratealltermsoftheperturbationseries.Topresentoursolutionitwillbeconvenienttodiscussthecasesofnondegenerateanddegeneratestates,separately.

1.Nondegeneratecase

Inthenondegeneratecase,toeachunperturbedboundstatemassMutherecorrespondsauniqueboundstatewavefunctionΦ.TheunperturbedGreenfunctionGu(P)thenhasthe“poleplusbackground”structure,asgiveninEq.(7).HavinginmindthatthefullGreenfunctionG(P)hasasimilarstructureasgiveninEq.(9),andthatourgoalistorelatethequantitiesinthesetwoexpression,webeginbyintroducinga“background”GreenfunctionGb(P)de nedasthesolutionoftheequation

bbGb(P)=Gbu(P)+Gu(P)K1(P)G(P).(12)

b(P)whereG b(P)wasde nedinEq.(9).FromEq.(12)itfollowsNotethatGb(P)=G

that

(1+GbK1) 1Gb=Gbu,(13)

wherewehavedroppedthemomentumargumentsforconvenience.SimilarlyEq.(11)implies

G(1+K1G) 1=Gu.

Subtractingthelasttwoequations,weobtain

G(1+K1G) 1(14) (1+GK1)G=b 1b¯iΦΦ

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

(1+GK1)G G(1+K1G)=(1+GK1)bbb¯iΦΦ

¯+K1G)bywritingwhichcanbesolvedforΦ(1

¯K1(1+GbK1)iΦΦ(1¯+K1G)Φb¯¯Φ(1+K1G)=Φ(1+K1G)+

12P2 Mu,(17)

UsingthisresultinEq.(17)weobtaintheresultweareseeking:

¯(P)iψ(P)ψG(P)=2P2 Mu¯+K1Gb).Φ(1(19)

¯),andΨ(¯P¯)=Zψ(P

′ ¯P)[K1(P)+K1(P)Gb(P)K1(P)]Φ(P) 1 iΦ(

u√,¯2=M2P2=P(23)withtheprimeindicatingaderivativewithrespecttoP2,and 22b¯P¯)K1(P¯)+K1(P¯)G(P¯)K1(P¯)Φ(P¯).M=M+iΦ((24)

InthisrespectitisworthnotingthatbecauseallourwavefunctionsandGreenfunctionsareLorentzcovariant,thequantityinthecurlybracketsofEq.(23)[whichalsoappearsinEq.(24)],isaLorentzscalardependingonlyonP2.

Thus,inthenondegeneratecase,theproperlynormalizedwavefunctionsforthefullperturbationtheoryare

¯)=1 iΦ(¯P¯)K1(P¯)+K1(P¯)Gb(P¯)K1(P¯)Φ(P¯)Ψ(P

b ¯P¯)=Φ(¯P¯)[1+K1(P¯)G(P¯)]1 iΦ(¯P¯)K1(P¯)+K1(P¯)G(P¯)K1(P¯)Φ(P¯)Ψ(b

Wenotethatthesewavefunctionssatisfythenormalizationcondition

¯P)iΨ( G 1(P) ′ 1/2¯)K1(P¯)]Φ(P¯),(25)[1+Gb(P ′ 1/2.(26)

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

2.Referenceframedependenceofthewavefunctions

Asfarasweknow,allpreviousattemptsatdevelopingperturbationtheoryforrelativis-ticsystemshaveconsideredboundstatesonlyatrest(seee.g.[5]).Ontheotherhand,forobservablesinvolvingscatteringo theboundstate(e.g.electromagneticform-factors)takingintoaccountthetotalmomentumdependenceoftheboundstatewavefunctionisimportant.Intherelativisticcasetherearesomesubtletiesinthedeterminationofthisde-pendenceperturbativelyandatthesametimeinamanifestlycovariantway.Onepossiblewaytodothisistoderivethewavefunctiontotheneededorderintherestreferenceframe,andthentoboostitinordertogiveitthedesiredmomentum.Therearetwodisadvantagestothisapproach:oneisthatitinvolvestwoseparatesteps-theperturbationexpansion¯/Mwhichdeter-andtheboosting.Theseconddisadvantageisthattheunitvectorn=P

minestheboost[14],itselfmayneedtobecalculatedperturbatively.Toillustratethis,we¯)to rstorderintheconsiderthedeterminationofascalarboundstatewavefunctionΨ(P

perturbation.Showingexplicitlyonerelativemomentumpinadditiontothetotalon-shell¯,we rstwritetheperturbedwavefunctionasaboostedwavefunctionatrest:momentumP

¯,Lnp)=SLnΨ0(Lnp)¯,p)=SLnΨ(LnPΨ(P(28)

¯=(M,0),SLnistheassociatedtrans-whereLnistheboostLorentztransformation,LnP

formationmatrixactingonthespinindicesoftheconstituents,andΨ0(q)istheboundstatewavefunctionatrest.NextstepistocalculateΨ0(q)to rstorderintheperturba-tion:Ψ0(q)=(1+η1)Φ0(q),wherethe rst-ordercorrectionfactorη1isgivenexplicitlyinEq.(57).Thus

¯,p)=SLn(1+η1)Φ0(Lnp).Ψ(P

√¯AsLnisafunctionoftheunitvectorn=P/M=((29)

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

P¯u,p)allthesolutionsoftheboundstatewavefunctions,sotothisendwedenotebyΦ(¯uhasthepropertyboundstateequation[ rstofEqs.(8)]forwhichthetotalmomentumP¯2=M2.Wethennotethatonecannotsimplyde neΦ(P,p)=Φ( P¯u,p)whereP=(P0,P)Puu ¯u=(andP

P2wherePisarbitrary.Thus,ifwede newavefunctionΦ(P,p)as

MuP,p,Φ(P,p)=ΦP2

itimmediatelyfollowsthat

Φ(P,p)=SLΦ(LP,Lp),(32) (31)

whichisthestatementthatwavefunctionΦ(P,p)isLorentzcovariantinthewayweneed.InthiswaywehaveconstructedawavefunctionΦ(P,p)thatsatis esthesought-afterLorentz P¯u,p)ascovariance,whileatthesametimereducingtotheboundstatewavefunctionΦ(¯u(infactΦ(P,p),asde nedbyEq.(31),istheboundstatewavefunctionwithtotalP→P√momentumMuP/

Asforthenon-degeneratecase,weshallassumeourwavefunctionstobecovariantbutnotdependentonP2.ThewavefunctionsΦjare,bytheassumptionofr-folddegeneracy,

2P2 Mu+Gbu(P).(33)

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

1linearlyindependent.ApplyingthisfacttothepolestructureoftheidentityGuG uGu=Gu,

weobtainthenormalizationconditionforthesewavefunctions:

1 G¯iu(P)iΦ

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

¯S(P).SincedetD(P)= jDj(P)=0,thewithsimilarde nitionsholdingforΦS(P)andΦ

GreenfunctionG(P)willhavepolesatP2=Mj2,j=1,2,3,...,r,whereMjisthesolutionoftheequation

Mj2=2MubS¯S+iΦj(Pj)K1(Pj)+K1(Pj)G(Pj)K1(Pj)Φj(Pj),

¯S(P)andΦS(P)beingPjbeinganymomentumsatisfyingPj2=Mj2,andthefunctionsΦjjSS¯thej’thelementsofΦ(P)andΦ(P),respectively.

TakingintoaccountthediagonalnatureofD(P),Eq.(40)canbewrittenas

G(P)=i jS 1¯S(P)+Gb(P).ψj(P)Dj(P)ψj (43)(44)

Thus,assumingthattheperturbedboundstatemassMjisitselfnondegenerate, wecan nditscorrespondingwavefunctionΨjasinthenondegeneratecaseabove:Ψj=

Sb¯S1 iΦj(Pj)[K1(Pj)+K1(Pj)G(Pj)K1(Pj)]Φj(Pj)

Thus,inthedegeneratecaseoftheunperturbedtheory,theproperlynormalizedwavefunc-tionscorrespondingtothe(nondegenerate)boundstatemassMjofthefullperturbationtheory,are

Ψj= ′.(45)¯S(Pj)[1+K1(Pj)Gb(Pj)].ZjΦj(47)

ments

ThemainresultsofthissubsectionaretheexpressionsforM2andΨgiveninthenondegeneratecasebyEq.(24)andEq.(25),andinthedegeneratecasebyEq.(43)andEq.(46),respectively.Notonlyaretheseexpressionsexactandcompact,buttheycanalsobeeasilyusedtowritedowntheexplicitperturbationseriesforthesequantities.Forthis

2purposeitismost√convenienttotreatallfunctionsofPasfunctionsofPandtheunitfour-

vectorn=P/

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

1=K1+K′δK1+δ2

2!

whereGbu+...′′(50)

2δ≡M2 Mu(51)

2andeachtermwithoutatildeisevaluatedatP2=Mu,wecanimmediatelywriteM2asa

perturbationserieswithrespecttoordersofK1≡K1(Mu):

2M2=Mu+δ1+δ2+δ3+...(52)

where

¯K1Φδ1=iΦ

¯δ1K′+K1GbK1Φδ2=iΦ1u

¯δ3=iΦ ′δ2K1(53) (54)+2δ1

2 1+GbuK1

2

81

4(57) η2=η3=

etc.1+δ1GbuK1 ′+η1GbuK1δ221 (58)GbuK1 ′′1 1 2+15+η2GbuK1(59)

where iisderivedfromδibyputtinganextraderivativeoneachK1andGbu;thatis,

′¯K1 1=iΦΦ

′′′¯δ1K1 2=iΦ+(K1GbuK1)Φ(60) (61)

¯ 3=iΦ ′′δ2K1+2δ1

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

Asimilarprocedurecanbeusedtogeneratetheperturbationseriesforthedegeneratecase.Itisworthnotingthattheperturbativecorrectionstotheboundstatewavefunction,asderivedhere,areparticularlyimportanttotakeintoaccountwhencalculatingcorrectionstovertices(electromagnetic,axial,etc.)withinconstituentmodels.Itisonlybytakingintoaccounttheappropriateorderofwavefunctionperturbationexactly,willsymmetryproper-ties,likeforexamplegaugeinvariance,bepreservedateachorderinthevertexcorrection–foraconcreteexample,seeRef.[10]whereEq.(57)wasusedtodeterminethefulllowestorderpioniccorrectiontothenucleonvertexfunctionintheNJLmodel.

ItisalsoworthpointingoutthatinthecasewheretheperturbationK1istoolargeforaperturbativetreatment,ourexpressionsofEq.(24),Eq.(25)Eq.(43),andEq.(46)maystillbeusefulforperformingpracticalnonperturbativecalculationsofM2andΨ.Indeed,inboththedegenerateandnondegeneratecases,themaincalculationale ortwouldbeinsolvingEq.(12)forthe“background”GreenfunctionGb.Yetthisisanespeciallysimpleequation,ofstandardLippmann-Schwingerform,whereGbhasnopoleatP2=M2and

22GbuhasnopoleatP=Mu(sincetheyhavebeensubtracted),andwherethereareno

singularitiesintheintegrationovermomenta.EvenintheunlikelyeventthatGbuhappens

22tohaveanunsubtractedpoleclosetoP=M,thiscasecanbeeasilyhandlednumerically.

Finally,itisusefultonotethatGbuhasalreadybeenconstructedfortheimportantcaseofthenonrelativisticCoulombproblembySchwinger[11]–aresultthatcanbeeasilyadaptedtotherelativisticCoulombcase[2].

III.DISCUSSIONANDSUMMARY

InthisworkwehavepresentedageneralformulationofperturbationtheoryapplicabletoboundstatesandresonanceswheretheboundstateequationsinvolvekernelsandinversefreeGreenfunctionsthathaveanarbitraryenergydependence.Ourformulationisthusdirectlyapplicabletotheimportantcaseofrelativisticquantum eldtheory.Onecanconsiderourresultsasextendingthewell-knowntime-independentperturbationtheoryofquantummechanicstothecasewherethekernelsareenergy-dependentandwheretheinversepropagatorsarenon-linearintheenergy.

Inparticular,wehavederivedexpressionsfortheboundstate(orresonance)massMandwavefunctionΨofasystemwhoseinteractionkernelKconsistsofapartK0forwhichthecorrespondingGreenfunctionGuisknown,andapartK1whichplaystheroleofaperturbation.OurresultsforMandΨarecontainedinEq.(24)andEq.(25)forthenondegeneratecase,andinEq.(43)andEq.(46)forthedegeneratecase,andhavethefeaturethattheyareexact,withtheperturbationK1takenintoaccounttoallorders.ThekeyelementintheseexpressionsistheGreenfunctionGbwhichneedstobefoundbysolvingEq.(12).Forsu cientlysmallK1,Eq.(12)canbesolvedsimplybyiteration,inthiswaygeneratingaperturbationexpansioninK1thatistheanalogueofthetime-independentperturbationtheoryofquantummechanics.Ontheotherhand,ifK1isnotsmallenoughtogenerateaconvergentperturbationseries,Eq.(12)couldstillbesolvedbystandardnumericaltechniquesforintegralequations.

Asfarasweknow,ourformulationoftheperturbationtheoryproblemisnew.How-ever,thereareafewalternativeformulationsavailableintheliterature,allpresentedfor

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

theparticularcaseofrelativisticquantum eldtheory.The rstoftheseisamethodwheretheperturbationseriesforM2andΨareexpressedintermsofcontourintegrals.OriginallydevelopedbyKato[9]anddescribedinMessiah’sstandardtext[12]forthecaseofquantummechanics,thecontourmethodwasextendedtothecovariantcasebyLepage[5]andused,forexample,byMurato[13].Anothermethod,duetoBodwinandYennie[6],isclosestinspirittoourapproach,butdoesnothavethefeatureofhavingclosedexpressionsfortheperturbedmassandwavefunction.AthirdapproachistherecentformulationofIvanovetal.[2]whoseperturbativeexpansionisexpressedintermsofacertain“relativisticgeneral-izationofaprojectionoperator”.Inthisapproachthesecondderivativeoftheinversefree12propagator, 2G 0/ E,looksverymuchlikeagenuineandnecessaryrelativisticfeature,yetitdoesnotappearinourformulationatallandisthusjustanartifactoftheparticu-larderivationused.Similarly,theexpressionforthelowest-orderwavefunctioncorrectionderiveddirectlyfromEq.(9)ofRef.[2]containsfourtermsagainstouronlyone.

Ineachoftheabovethreealternativeapproaches,perturbativecorrectionstotheboundstatewavefunctionwerederivedonlyforthespecialcasewheretheboundstateisatrest.Thus,inordertodescribescatteringprocesswheretheboundstatehasnon-zerototalmomentum,suchwavefunctioncorrectionsneedtobemodi edbytheappropriateLorentzboost(thatitselfdependsontheorderofperturbationbeingconsidered).Bycontrast,ourapproachhasenabledustowriteexpressionsfortheboundstatewavefunctioncorrectionsthatareLorentzcovariantateachorderoftheperturbation,thusavoidingthestepofboostingfromtherestframe.Althoughallperturbationexpansionsmustmathematicallybeidentical,itisevidentthattheexpressionsprovidedbyourEq.(24),Eq.(25),Eq.(43),andEq.(46)arethesimplestbothpracticallyandconceptually.

ACKNOWLEDGMENTS

WewouldliketothankA.G.Rusteskyforfruitfuldiscussions.ThisworkispartiallysupportedbytheEngineeringandPhysicalSciencesResearchCouncil(U.K.).

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a non-linear function of energy. Such derivations

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