; TeX output 2012.07.07:2215 e̍6XIcolor push Blackf cmcsc8DocumentuUa#Math. :K`y cmr10401Un color pop+4# o- cmcsc10OntheGroundSt32ateEnergy?oftheTransla32tionInvariantP3auli-FierzModel.IYI.ٵeG- cmcsc10Jean-MarieBarbarouxandSemjonA.VugalUTterٴ +B|{Y cmr8ReceivÎed: FJebruaryX17,2012ۍ ǨCommÎunicatedXbyHeinzSiedentop"P!JXIAbstract.vmJXIW*edeterminethegroundstateenergyofthetranslationinvqariant JXIPauli-Fierz}moGdel}foranelectronwithspin,tosubleadingorder !", cmsy10OG( b> cmmi10 z^ٓR cmr72)JXIwith_respGect_topowers_of_thenestructureconstantiandprove_rig-JXIorous|errorbGoundsoforderO( z^3).F)A[mainob 8jectiveofourargumentJXIisUUitsbrevity*.JXI2000MathematicsSub 8jectClassication: :81Q10,|35P15,46N50,JXI47N50JXIKeywordsWandPhrases:T*ranslationinvqariantPauli-FierzHamilton-JXIian,UUspGectraltheory*,groundstateenergy.U x1. wIntroduction 6XIW*e?continue>thestudyofthetranslationinvqariant>Pauli-FierzmoGdel>[2 ],yde-6XIscribinganonrelativisticfreeelectroninteractingwiththequantizedelectro-6XImagneticeld.Incontrastwith[2 ],Wwestudynowelectronwithspin.W*eare6XIinterested8in8quantitativepropGerties8ofthegroundstateenergy(Theorem2.1)6XIand{itszassoGciatedeigenfunctions(Theorem2.2).8Inparticular,wedetermine6XItheZJsubleadingtermsofthegroundstateenergyuptoZIorder z^2,wherecIJdenotes6XIthenestructureconstant,andrigorouslybGoundtheerrorbyatermoforder6XI z^3.Incomparisonwith[2 ],thegroundstateenergyisanorderofmagnitude6XIlargerUUinpGowersUUof z,duetothepresenceofelectronspin.6XIF*ollowingdthedtechniquedevelopGeddin[2 ](seealso[4 ]),ourmethoGdisbased6XIonpGerturbationsaroundthetruegroundstateofthetranslationinvqariant6XIopGerator,togetherZ_withZ`aboundonZ`theexpectedphotonZ`numberZ_forZ`thisground6XIstate,obtainedhbyhChenandF*rohlich[8 ]."Inparticular,anhimpGortantingredient6XIoftheproGofistheimprovementofphotonnumbGerestimatesfordierentparts6XIofUUthegroundstate.6XIAwell-knowndicultyconnectedtothisproblemarisesfromthefactthatthe6XIground-Lstateenergy-KisnotanisolatedeigenvqalueoftheHamiltonian,5Mandthat 6XIcolor push BlackODocumentuUa#Mathematica17(2012)401{415Un color pop *e̍6XIcolor push Black4020IJean-MarieBarbarouxandSemjonA.VugalUTterUn color pop+4e̍6XItheµform´factorintheinteractiontermoftheHamiltoniancontainsacritical 6XIfrequencyspacesingularity(theinfraredproblemofQuantumElectroGdynamics6XI(QED)).6XIEstimatesonthegroundstateenergyplayanimpGortantrole,forinstance,in6XIbindingUUproblems,e.g.,thedeterminationoftheHydrogenbindingenergy[3 ].6XIThe8systematic7studyofPauli-FierzHamiltonianwasinitiatedin[1 ].ZTherst6XIestimateHforHthetranslationinvqariantHopGeratorforspinlesselectronwasobtained6XIby[12 ].I#Lateronin[6 ],themoGdelforelectronwithspinwasconsidered,and6XIthe%0bGoundwasobtaineduptotheorder z^2withanerrortermoftheorder{6XI<Z cmr55<x W g P2log z._Such42estimatesarenot43sucienttocomputethe43correctiontothe6XIbindingK,energyduetotheinteractionwiththeradiationeld.SLIn[2 ]anew6XIeectiveimethoGdjwasdevelopGedtojobtaintheselfenergyinthespinlesscase6XIup]totheorder z^3㔲withanerror]OG( z^4).Thisresultwaslaterimprovedin6XI[5 ]i4withcomputingi3thetermOG( z^4)witherrortermOG( z^5).cTheselasttwo6XIresults G[2 , F5 G]werecrucialfor Fprovingthatthebinding Fenergyinthecaseofthe6XIHydrogen\atom[withspinlesselectroncontainedan z^50logղtermandthatthis6XIterm`comesfromthegroundstateenergy`oftheHydrogenatomanddoGesnot6XIexistUUinthetranslationinvqariantUUcase[3 ].6XIIntheworkathand,wearestartingtoimplementthesameprogramforthe6XImoGdelZhofZiaHydrogenatomwithspin1/2electroninteractingwiththequantized6XIradiationueld.|Therstustepofthisprogramis,~ asin[2 ],~computingtheself-6XIenergy*,UUfortheelectronwithspin,uptotheorderOG( z^3).6XITheJcPauli-FierzJdHamiltonianHbforafreeelectroncoupledtothequantized6XIelectromagneticUUeldisdenedbyˍ . e#H=UU:b u cmex10 \oir 0er cmmi7xAĸ 8Ifh Op 7O fe o㍵TA(x)bb֟߱2ia:+OpUWO fe o㍵t8B q(x)+Iel Hf/ : 6XI(1)ˍ6XIwhere4:p:denotes4normalordering,lcorrespGondingtothesubtractionof6XIaXnormalorderingconstantpropGortionalto z.{TheoperatorH(actsonthe6XIHilbGertespacedH:=hܸHel V yF 9,ΩwhereHel E=L^2|s(& msbm10R^3;C^2),ΩisdtheeHilbGertspaceof6XIone:non-relativistic:electron,?R^3isthecongurationspaceoftheelectron,?and6XIC^2ȲaccomoGdatesUUitsspin.6XIW*e0describGe/thequantizedelectromagneticeldbyuseoftheCoulombgauge6XIcondition."=Accordingly*,the'one-photonHilbGertspaceisgivenbyL^2|s(R^3) 6XIC^2|s,:whererR^3 (denotessthephotonmomentumandC^2 (accountsforthetwo6XIindepGendenttransversalpGolarizationsofthephoton.T6ThephotonF*oGckspaceis6XIthenUUdenedbyc sFQ=M+nO! cmsy72'qy msbm7NF 9(n)፴s; t6XIwhereVRthen-photonsVSspaceF: 9(n)]"sxw=ȾNލ 2n% 2sXb L^2|s(R^3)8 C^2|sbX7isthesymmetrictensorݍ6XIproGductUUofncopiesofL^2|s(R^3)8 C^2|s.6XIW*euseunitssuchthat~TL=c=1,=andwherethemassoftheelectronequals6XIm=1=2.X1Theelectronchargeisthengivenbye=po fe o3荵,withВ1=137denot-6XIingUUthenestructureconstant.qAsusual,wewillconsider^ϲasaparameter. 6XIcolor push BlackODocumentuUa#Mathematica17(2012)401{415Un color pop e̍6XIcolor push BlackG[Self-energyofanElectroninNRUQED4>403Un color pop+4e̍6XITheopGeratorthatcouplesanelectrontothequantizedvectorpGotentialisgiven 6XIbyS?A(x)=+X=1;2pcZ: yR 3<$0N (jkPj)+ڟw fe "y V2[ٸjkPjr1=2Ou">:(kP)`heik+Bx 8a(k)8+e ik+Bx a፴>:(k)`idk?;6XIwhereybyxtheCoulombxgaugecondition,div A=0.=)TheopGeratorsya>:,a^vsatisfy6XItheUUusualcommutationrelations_bp[aɲ(kP);a፴>:(k0в)]=`(kw 8kP0);;[aɲ(kP);a>:(k0)]=0;6XIandԣthereexistsauniqueunitray fc2DF 9,vtheF*oGckvqacuum,vwhichsatises I6XIa>:(kP) f H='0,forallk22R^3{and(2f1;2g.oLThevectors">:(kP)(2R^3{arethe 6XIfollowingUUtwoorthonormalpGolarizationvectorspGerpendiculartokP,n""1|s(kP)=<$K(k2; k1;0)Kw fe 0+ ōKݟN4p KޟN4 fe