Lagrangian description of the radiation damping

P.M.V.B.BaroneandA.C.R.Mendes∗

DepartamentodeF´ısica,UniversidadeFederaldeJuizdeFora,

36036-330,JuizdeFora,MG,Brasil

February6,2008

Abstract

WepresentaLagrangianformalismtothedissipativesystemofachargeinteractingwithitsownradiationfield,whichgivesrisetotheradiationdamping[11],bytheindirectrepresentationdoublingthephase-spacedimensions.

1Introduction

Thestudyofdissipativesystemsinquantumtheoryisofstronginterestandrelevanceeitherforfundamentalreasons[5]andforitspracticalapplications[2,6].TheexplicittimedependenceoftheLagrangianandHamiltonianoperatorsintroducesamajordifficultytothisstudysincethecanonicalcommutationrelationsarenotpreservedbytimeevolution.Thendifferentapproacheshavebeenusedinordertoapplythecanonicalquantizationschemetodissipativesystems(see,forinstance,[8,7]).

Oneoftheseapproachesistofocusonanisolatedsystemcomposedbytheoriginaldissipativesystemplusareservoir.OnestartfromthebeginningwithaHamiltonianwhichdescribesthesystem,thebathandthesystem-bathinteraction.Subsequently,oneeliminatesthebathvariableswhichgiverisetobothdampingandfluctuations,thusobtainingthereduceddensitymatrix[1,2,3,4,7].

Anotherwaytohandletheproblemofquantumdissipativesystemsistodoublethephase-spacedimensions,soastodealwithaneffectiveisolatedsystemcomposedbytheoriginalsystemplusitstime-reversedcopy(indirectrepresentation)[9,10].Thenewdegreesoffreedomthusintroducedmayberepresentedbyasingleequivalent(collective)degreeoffreedomforthebath,whichabsorbstheenergydissipatedbythesystem.

Thestudyofthequantumdynamicsofanacceleratedchargeisappropriatedtousetheindirectrepresentationsinceitlosestheenergy,thelinearmomentum,andtheangularmomentumcarriedbytheradiationfield[11].Theeffectoftheselossestothemotionofchargeisknowasradiationdamping[11].

ThereactionofaclassicalpointchargetoitsownradiationwasfirstdiscussedbyLorentzandAbrahammorethanonehundredyearsago,andneverstoppedbeingasourceofcontroversyandfascination[12,13].Nowadays,itisprobablyfairtosaythatthemostdisputableaspectsoftheAbraham-Lorentztheory,suchasself-accelerationandpreacceleration,havebeenadequatelyunderstood.Self-accelerationreferstoclassicalsolutionswherethechargeisunderacceleration

evenintheabsenceofanexternalfield.Preaccelerationmeansthatthechargebeginstoacceleratebeforetheforceisactuallyapplied.

Theprocessofradiationdampingisimportantinmanyareasofelectronacceleratoroperation[14],asinrecentexperimentswithintense-laserrelativistic-electronscatteringatlaserfrequenciesandfieldstrengthswhereradiationreactionforcesbegintobecomesignificant[15,16].

ThepurposeofthisletteristopresentaLagrangianformalismtothestudyofquantumdynam-icsofacceleratedcharge,yieldinganeffectiveisolatedsystem,wherethecanonicalcommutationrelationsarepreservedbytimeevolution.InSection2webrieflyreviewtheequationofmotionoftheradiationdampingandaspectsofthesolutionstotheequationofmotion.Insection3wepresentaLagrangiandescriptiontotheradiationdampingbytheindirectrepresentation,doublingthephase-spacedimensions.Section4containstheconcludingremarks.

2Theequationofmotion

Thederivationofanexactexpressionfortheradiationdampingforcehaslongbeenanoutstantingproblemofclassicalelectrodynamics[8,11,12,13,15,17,18,19].IntheclassicderivationgivenbyLorentzandAbraham[12,13],whichreliesonenergy-momentumconservation,theself-electro-magnetic-energyandmomentumofachargedrigidspherearederivedforanacceleratedmotion.Inthisfirst-orderapproximation,thisderivationyieldsthewell-knownAbraham-Lorentzforcewhichdependsonthesecondtimederivativeoftheparticlevelocityofmassmandchargee:

m

󰀉

d󰀛v

dt2

󰀁

=F

󰀛,(1)

whereτ0=2e2/3mc3,cisthevelocityoflight,󰀛v=d󰀛r/dtdenotesthevelocityofF

󰀛thecharge,and

istheexternalforce.Afullyrelativisticformulationoftheequationofmotionwasonlyachievedin1938byDiracinhisclassicpaper[20],wheretheLorentz-Diracequationreads

maµ=

e

µλ

3c3

󰀅

a˙−aaλ

uµ

freeparticle[20].Inthiscase,thesolutionofEq.(1),withthehelpoftheintegratingfactoret/τ0,

forarathergeneraltime-dependentforceF

󰀛(t)readsmd󰀛v

d2󰀛

v¯dt

+τ0

dt󰀛r˙−τ...0󰀛r+

∂Vdt

󰀛r˙...

¯+τ0󰀛r¯+∂V∂󰀎r

=−F

󰀛¯and∂V

2

󰀃

󰀛r

˙.󰀛¨r¯−¨󰀛r.󰀛r˙¯󰀄

−V(󰀛r,󰀛r¯)󰀊

,whereγ=mτ0=2e2/3c3.Itisthenpossibletoidentify

L=m󰀛r

˙.󰀛r˙¯+γ(8)

astheappropriateLagrangianintheindirectrepresentation.So,thesystemmadeoftheradiationdampingandofitstime-reversedimagegloballybehavesasaclosedsystem.TheLagrangian(9)canbewritteninasuggestiveformbysubtitutionofthehyperboliccoordinates󰀛r1and󰀛r2[25]definedby

󰀄󰀄1󰀃1󰀃

󰀛¯=(10)󰀛r=󰀛r(1)+󰀛r(2);r󰀛r(1)−󰀛r(2)

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WefindthattheLagrangianLbecomes

m˙(i).󰀛¨ǫij󰀛rr(j)−V[󰀛r(1),󰀛r(2)](11)L=

2

wherethepseudo-euclidianmetricgijisgivenbyg11=−g22=1,g12=0andǫ12=−ǫ21=1.ThisLagrangianissimilartotheonediscussedbyLukierskietal[26](thatisaspecialnonrelativisticlimitofrelativisticmodeloftheparticlewithtorsioninvestigatedin[27]),butinthiscasewehaveapseudo-euclidianmetric.TheequationsofmotioncorrespondingtotheLagrangian(11)are

...∂V¨m󰀛r(1)−γ󰀛r(2)=−

∂󰀛r(1)

.(12)

Onthehyperbolicplane,theequations(12)showsthatthedissipativetermactullyactsasa

couplingbetweenthesystems󰀛r(1)and󰀛r(2).

Recently,oneofus,in[28]havestudiedthecanonicalquantizationoftheradiationdamping.AHamiltoniananalysisisdoneincommutativeandnoncommutativescenarios,whatleadstothequantizationofthesystem,wherethedynamicalgroupstructureassociatedwithoursystemisthatofSU(1,1).In[29],asupersymmetrizedversionofthemodeltotheradiationdamping,Eq.(11),wasdeveloped.ItssymmetriesandthecorrespondingconservedNoetherchargeswerediscused.ItisshownthatthissupersymmetricversionprovidesasupersymmetricgeneralizationoftheGalileialgebraofthemodel[28],wherethesupersymmetricactioncanbesplitintodynamicallyindependentexternalandinternalsectors.

4Concludingremarks

Wehaveshownthatinthepseudo-Euclideanmetricsthesystemmadeofachargeinteractingwithitsownradiationanditstime-reversedimage,introducedbydoublingthedegreesoffreedomasrequiredbythecanonicalformalism,actuallybehavesasaclosedsystemdescribedbytheLagrangian(11).Thisformalismrepresentsanewscenariointhestudyofthisveryinterestingsystem.TheLagrangian(11)describes,inthehyperbolicplane,thedissipativesystemofachargeinteractingwithitsownradiationfield,wherethe2-labeledsystemrepresentsthereservoirorheatbathcoupledtothe1-labedsystem.NotethatthisLagrangianissimilartotheonediscussedin[26](whichisaspecialnonrelativisticlimitofrelativisticmodeloftheparticlewithtorsioninvestigatedin[27]),butinthiscasewehaveapseudo-Euclideanmetricandtheradiation-dampingconstant,γ,isthecouplingconstantofaChern-Simons-liketerm.Thisformalismisimportantbecauseitallowsustostudythecanonicalquantizationofthemodel(seeRef.[28]),andtostudythesymmetriesofthemodelandtheirsupersymmetricversion(seeRef.[29]).Infutureworks,wewillstudytheintroductionofgaugeinteractionsintothemodel.

5Acknowledgement

ThisworkissupportedbyCNPqBrazilianResearchAgency.Inparticular,ACRMwouldliketoacknowledgetheCNPq.

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References

[1]R.P.FeynmanandF.L.vernon,Jr.,Ann.Phys.24(1963)118.[2]A.OCaldeiraandA.J.Leggett,Phys.Rev.Lett.46,(1981)211.[3]A.OCaldeiraandA.J.Leggett,Ann.Phys.121,(1983)587.[4]A.OCaldeiraandA.J.Leggett,PhysicaA149,(1983)374.[5]W.G.UnruhandW.H.Zurek,Phys.Rev.D40,(1989)1071.

[6]W.S.Warren,S.L.HammesandJ.L.Bates,J.Chem.Phys.91,(1989)5895.[7]P.M.V.B.BaroneandA.O.Caldeira,Phys.Rev.A43,(1991)57.[8]A.C.R.MendesandF.I.Takakura,Phys.Rev.E64,(2001)056501.[9]R.BanerjeeandP.Mukherjee,quant-ph/0205143;quant-ph/0108055.[10]H.FeshbachandY.tikochinsky,Trans.N.Y.Acad.Sci.,SerII38(1977)44.

[11]W.Heitler,TheQuantumTheoryofRadiation(Dover,1970),3nded.;J.D.Jackson,Classical

Electrodynamics(Wiley,NewYork,1975),2nded.,Chaps.14,17;GilbertN.Plass,ReviewsofModernPhysics33,(1961)37.[12]R.Becker,ElectromagneticFieldsandInteractions(BlaidsellPublishingCo.,NewYork,1965)

Vol.1,Chaps.80,84,85,andVol.2,Chap.4.[13]H.A.Lorentz,TheTheoryofElectrons(Dover,NewYork,1952),2nded.,Secs.26-37,178-183.[14]R.P.Walker,“RadiationDamping”,Proceeding,GeneralAcceleratorPhysics,CERNGenebra

-CERN91-04,116-135(1990).[15]F.V.HartemannandA.K.Kerman,Phys.Rev.Lett.76,(1996)624.

[16]C.Bula,K.T.McDonald,E.J.Prebys,C.Bamber,S.Boege,T.Kotseroglou,A.C.Melissinos,

D.D.Meyerhofer,W.Ragg,D.L.Burke,R.C.Field,G.Horton-Smith,A.C.Odian,J.E.Spencer,D.Walz,S.C.Berridge,W.M.Bugg,K.ShmakovandA.W.Weidemann,Phys.Rev.Lett.76,(1996)3116.[17]F.Rohrlich,ClassicalChargedParticles(Addison-Wesley,RedwoodCity,1990).

[18]L.D.LandauandE.M.Lifshitz,TheClassicalTheoryofFields(Pergamon,Oxford,1975),

4thed.[19]F.Rohrlich,Am.J.Phys.65,(1997)1051.

[20]P.A.M.Dirac,Proc.R.Soc.London,Ser.A167,(1938)148.

[21]C.Teitelboim,D.VillaroelandCh.G.vanWeert,Riv.NuovoComento3,(1980)1903.[22]D.Villaroel,Phys.Rev.A55,(1997)3333.

5

[23]R.M.Santilli,FoundationofTheoreticalMechanicsI(Springer-Verlag,NewYork,1984)pp

119-137.[24]H.Bateman,Phys.Rev.38,(1931)815.

[25]M.Blasone,E.Graziano.O.K.PashaevandG.Vitiello,Ann.Phys.252,(1996)115.[26]J.Lukierski,P.C.StichelandW.J.Zakrzewski,Ann.Phys.260,(1997)224.[27]M.S.Plyushchay,Nucl.Phys.B362,(1991)54;Phys.Lett.B262,(1991)71.

[28]A.C.R.Mendes,C.Neves,W.OliveiraandF.I.Takakura,Eur.Phys.J.C45(2006)257.[29]A.C.R.Mendes,C.Neves,W.OliveiraandF.I.Takakura,J.Phys.A:Math.Gen.38,(2005)

9387.

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