J. F. van Huele Department of Physics, Oakland University, Rochester, Michigan (Received 18 March 1988)

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1 PHYSICAL REVIE% A VOLUME 38, NUMBER 9 NOVEMBER, 988 Quntum lctrodynmics bsd on slf-filds, without scond quntiztion: A nonrltivistic clcultion of g 2 A. O. Brut nd Jonthn P. Dowling Dprtmnt ofphysics, Cmpus Box 390, Univrsity of Colordo, Bouldr, Colordo J. F. vn Hul Dprtmnt of Physics, Oklnd Univrsity, Rochstr, Michign (Rcivd 8 Mrch 988) Using formultion of quntum lctrodynmics tht is not scond quntizd, but rthr bsd on slf-filds, w comput th nomlous mgntic momnt of th lctron to first ordr in th finstructur constnt. In th nonrltivistic (NR) cs nd in th dipol pproximtion, our rsult is, = (g 2)/2=(4A/3m)(n/), whr A is positiv photon nrgy cutoff nd m th lctron mss. A rsonbl choic of cutoff, A/m = ', yilds th corrct sign nd mgnitud for g 2 nmly,, =+/.. In our formultion th sign of 3 is corrctly positiv, indpndnt of cutoff, nd th dmnd tht, =+/ implis uniqu vlu for A. This is in contrdistinction to prvious NR clcultions of, tht mploy lctromgntic vcuum fluctutions instd of slf-filds; in th vcuum fluctution cs th sign of, is cutoff dpndnt nd th qution,, =/2 dos not hv uniqu solution in A. Bth' first clcultd I. INTRODUCTION th Lmb shift for th hydrogn tom in 947, using mthod tht ws ssntilly nonrltivistic (NR), but nvrthlss pproximtly corrct. In 948, Wlton gv n intuitiv drivtion of th Bth rsult by considring th coupling of th lctron to scond-quntizd lctromgntic vcuum fluctutions lding to th gnrlly hld folklor tht vcuum fluctutions r th physicl cus of th Lmb shift. Howvr, whn th Wlton pproch ws usd to comput th nomlous mgntic momnt of th lctron, ' th incorrct sign for g 2 ws obtind. Complmntry to th vcuum fluctution pictur is th slf-fild pictur, in which on viws rditiv corrctions s rising from rdition rction ffcts du to th intrction of prticl with its own slf-fild. Th slffild pictur is thus in lin with th clssicl point of viw whr thr r no infinit nrgy dnsity zro-point fluctutions nd th vcuum fild is idnticlly qul to zro. Th 95 ppr of Clln nd Wlton on th Auctution dissiption thorm showd tht thr is n intimt connction btwn vcuum fluctutions nd th procss of rdition rction. Th xistnc of on implis th xistnc of th othr. In th 970s svrl workrs in th fild formultd stndrd QED in trms of th Hisnbrg qutions of motion nd wr bl to show tht th phnomnon of spontnous mission could b intrprtd s bing cusd by vcuum fluctutions or by rdition rction or indd by ny linr combintion of th two ffcts. Ths intrprttions r dpndnt upon whthr on uss symmtric ordring or norml ordring, or som linr combintion of ths two ordrings, rspctivly, whn writing down th fild oprtors. This dos not llow on to do wy with th vcuum fild oprtors, howvr. In stndrd QED th vcuum fild must b mintind in th qutions of motion of th tomic oprtors, which othrwis would dcy to zro s th tom rdits. This would violt unitrity. Dlibrd t l. hv rgud tht only th symmtric ordring of th fild oprtors cn b usd if on dmnds tht th slf-fild nd vcuum oprtors b sprtly Hrmitin, which would sm to forc upon us th vcuum fluctution intrprttion. It is intrsting in this contxt to not tht in th thory of stochstic or rndom lctrodynmics th n-point corrltion functions of th clssicl, stochstic bckground fild gr with thos of th QED vcuum fild only if th QED fild oprtors r symmtriclly ordrd. Th rdition rction pictur, howvr, lso hs its mny dvocts. To quot Jyns: "This complt intrchngbility of sourc-fild ffcts nd vcuum fluctution ffcts dos not show tht vcuum fluctutions r "rl." It shows tht th sourc-fild ffcts r th sm s if vcuum fluctutions r prsnt. " H hs shown tht th nrgy dnsity of th rdition fild, ovr th spctrl intrvl of th nturl linwidth, is xctly th sm s tht of th vcuum fild. Finlly, w quot Milonni: "It sms... tht th gnrliztion of ths ids... my ld us to viw th vcuum fild mor s forml " rtific or subtrfug thn "rl" physicl thing. As mntiond bov, nonrltivistic ndvors to driv th nomlous mgntic momnt of th lctron by coupling it to th vcuum fluctutions yild th wrong sign for, :=(g 2)/2. (Hr :=b is symbolic logic nottion for " is bing dfind s qul to b;" th quntity on th sid of th colon is th quntity bing dfind. ) Grotch nd Kzs' hv mngd to obtin th corrct sign of, in such NR vcuum fluctution pproch by Th Amricn Physicl Socity

2 A. O. BARUT, JONATHAN P. DO%'I.ING, AND J. P. vn HUEI.E including th ffcts of mss rnormliztion trm 5m. Howvr, th sign of, in this pproch is itslf cutoff dpndnt for cutoff A in th rng AE(0, 4m) on gts, &0, but for AE(4m, oo ) w hv th incorrct sign,, &0. Evn whn, &0 th choic of A which solvs, =+/ is not uniqu. Ths difficultis do not occur with th slf-fild pproch which is usd hrwith; th sign of, is corrct, indpndnt of cutoff, nd, = + /2n. lds to uniqu choic of th prmtr A. Bfor mss rnormliztion (MR), w obtin, in th slf-fild pproch, 4A = (l) 277 3m to th first ordr in th fin-structur constnt. Aftr mss rnormliztion is mployd, th corrct sign rsults nd w hv 4A, =+ 2' 3m (2) W s tht choic of cutoff A=3m /4 in Eq. (2) yilds th corrct vlu for,. Th sign of, is positiv for ll AC(0, ) nd th function is linr in th prmtr x:=a/m, lding to uniqu vlu of x for th solution th rsult of th vcu- of, =/2nIn con. t. rdistinction, um fluctution (VF) clcultion' is qudrtic function of x with ngtiv concvity; hnc th sign of th VG formul for, (x) dpnds on x, nd th solution of, (x)=+/2 is not uniqu, s, (x) is not singl vlud. Th fct tht th slf-fild pproch yilds in strightforwrd nd unmbiguous mnnr th corrct sign nd mgnitud for nd tht th rsult divrgs only linrly rthr thn qudrticlly with th cutoff prmtr x, might b viwd s vidnc for th intrprttion tht it is th slf-fild of th prticl itslf (rthr thn th hypothticl fluctuting vcuum) which is th physicl origin of th nonzro vlu of g 2 in fr spc. II. METHOD W brifly rviw th slf-fild pproch to QED which ws usd rcntly to vlut numbr of rditiv procsss." ' Th bsic id of th pproch is quit simpl: on includs th slf-fild of th prticl from th bginning, rthr thn introducing scond-quntizd rdition fild. Thus vcuum fluctutions, dirct consqunc of th scond quntiztion procdur, do not ppr. Th vcuum fild is idnticlly zro. Th clssicl fild A (x}surrounding th chrgd prticl is concptully sprtd into slf-fild contribution 3 ' nd n xtrnl fild A ' if rquird. Th A r usd to construct th fild tnsor F vi th usul dfinition F Th F oby th inhomognous Mxwll qutions ( &0, li/ =c = I) In th bsnc of xtrnl currnts, j"(x) is just th four-currnt of th lctron. Th slf-fild A' is com- P pltly dtrmind by j" nd by boundry conditions. Eqution (3}cn b solvd formlly for th slf-fild lon s" A '(x)= Jdy D,(x y)j (y), (4) whr D& is n lctromgntic Grn's function. In th Coulomb gug, nd for mpty spc without cvitis, w hv Ik (x y) D,"(x y)= f dk (5J+;tr ), Ik (x y) dk (2n) It [ i (5) D 0(x y)=d O (x y}=0 whr w r using th four-vctor nottion: k x:=k x",dk:=d k, tc. nds. ;:=k;/ k. W procd to clcult ll nrgy shifts from n ction 8', with corrsponding ction dnsity w W= xw x;y;a (Indics p,, v, tc. r supprssd. } For scttring (3) problms th dimnsionlss ction W is rltd to th scttring mplitud pr unit spc-tim 6, nd for bound stts is rltd to th totl invrint nrgy 8 of th systm by W/ (2n) 5 (P f P;)6, (7) Wp (2n)5(Ef E;)8. (Th P's nd E's r th initil or finl momnt nd nrgis in th fr- nd bound-stt css, rspctivly. ) For th purpos of computing g 2 w tk y to b two-componnt Puli spinor fild which w coupl to th lctromgntic fild A vi th Puli Hmiltonin: H = [cr (p A)] +go. (8) Including th slf-fild Lgrngin contribution,, 'F F"',nd using intgrtion by prts to symmtriz H, w hv for th totl ction dnsity [(V+i A) o][o"(v +AO i 0, y+, 'F F" i' A)]- Th Eulr Lgrng qution-s of motion yild th Puli qution upon vrition with rspct to y*: V i i + A. V+ V- A m 2 o.b+ A y=0, nd thy giv th inhomognous Mxwll qutions

3 38 QUANTUM ELECTRODYNAMICS BASED ON SELF-FIELDS,... 5W 5W g y FP 0 /As P /As P V V, P, upon vrition with rspct to A so long s w dfin th four-currnt j"s 68 =. j" 5A ' which implis in turn tht (S, M, nd F lbl th spin, momntum, nd fild contributions, rspctivly) j"=q', '. &+ (VX rrx& ) A q i m lp 3M +3SM +3F (0) With this currnt th intrction of th mttr fild with th lctromgntic fild (EM) fild is of th stndrd form A j". Eqution (0) is th NR vrsion of th Gordon dcomposition of th Dirc currnt +y %. Th vctor portion of j" sprts nturlly into momntum (M), spin momntum (SM), nd fild (F) currnts. In fct, th nonrltivistic pproximtion md by th sprtion of 4' in %y % into uppr nd lowr (lrg nd smll) componnts yilds Eq. (0). In wht follows w ssum tht A'(x) vnishs t infinity. Thus, in clcultions involving th ction dnsity w(x), qulity will b undrstood s qulity with rspct to intgrtion by prts nd possibl surfc intgrls vnishing t infinity long with A'. III. FIELD CONTRIBUTION Th lctromgntic fild Lgrngin F F""in w(x) contins both A' nd A'. Howvr, cross trms my b convrtd to surfc intgrls which vnish t infinity, lving only F F""=F 'g I"'+F 'g,"'. Th xtrnl fild tnsor contrction is th invrint:,'f 'g," =, '(E, B,), which w shll drop from w(x), bcus it is nondynmicl fixd quntity. Th slf-fild tnsor contrction my b writtn,'f 'g,"'=, ' A I } F, "" (+.I (+ (2) whr [, ] implis n ntisymmtriztion with rspct to th two indics. W rcll tht j"givn by Eq. (0) contins A which is th sum of A' nd A'; kping this in mind, w xpnd out th product Ap". Intgrtion by prts is usd to sndwich ll th V's btwn y' nd p, nd th spin lgbr of th o's is usd to simplify vrious trms. Th rsult is 'F, F" = 'Ap" 4 PV whr B':= V g A. 2 4m + o.b' ( A'. A'+ A, ) 4m IV. TOTAL ACTION (3) Th kintic portion of th ction dnsity, wo, my b xpndd nd intgrtd by prts to giv wo cp* 2 l8 V S l + A' V+ V A' m cr B'+ 2 V2 w=g* +A'+ A' ib + A' V 2 m 2 A' V 8' 8'+ A 4m A (4) kping in mind tht A = As+ A', so tht Vx A=B=B'+B'. Combining th rsults of Eqs. (3) nd (4), w my writ th totl ction dnsity w (x) s 2 =:g w,. + A'A'+ V A'+ V A' 4m Not tht th trm A, drops out in th totl ction. For prticl of chrg moving in uniform xtrnl mgntic fild A'(x}=, 'B'Xx w my st Ao(x)= 0. In th Coulomb gug, V. A'=V A'=0. W drop, for wk filds, th trm proportionl to A, nd tht proportionl to A' A' will turn out to b of O( ) nd so w drop it too. Trms w, +w will giv ris to th stndrd Lndu orbitl solutions, whi w7 is th norml mgntic momnt contribution. (Our choic of ction dos not giv spin-orbit trm. ) Brut nd Vn Hul'i hv nlyzd th contributions from w3 nd w6. Th trm w3 corrsponds to th lctrosttic slf-nrgy nd cn b writtn in such wy s to suggst NR nlogu to th QED vcuum polriztion; it contributs hr only to mss rnormliztion. Th ction dnsity w6 cn b shown to giv ris to th phnomnon of spontnous mission, with th corrct Einstin A cofficint nd lso to th Bth xprssion for th Lmb shift. In ddition, w6 contins mss rnormliztion trm. Of primry intrst for us in this work, howvr, is th trm wz,

4 k so 4'. A. O. BARUT, JONATHAN P. DO%'LING, AND J. F. vn HUELE 38 Writing th four-vctors k"= [co,k], x"= [t,x], w& g ct B (6) y"=[u, 4m y] nd tking th DA, ' '" "' =, w rriv t ] lo( t u) which contins th nomlous mgntic momnt. VXA'= f f dydk m () k V. CALCULATION OF ws Xp*[, 'B'+, '«(«B' )]y. (9) In ordr to vlut m 8 w must comput Exprssion (9) my b simplifid by th following trick: o'b'=cr (V X A'). Th slf-fild A '(x) is spcifid by lt b n rbitrry constnt vctor. Dot into both Eqs. (4) nd (5). W nd not us th ntir xprssion sids of (9) nd crry out th d Qk intgrtion with th (0) for th currnt j". W shll vntully b mking id of th idntity th dipol pproximtion (DA) nd lso b tking th limit of th lctron momntum going to zro. Only th f dqk( «)(b.«)= b. 3 portion of j"which coupls th spin to th fild will thn contribut to th finl rsult: Thn, sinc b= c V b=c, w my xtrct to obtin (dy:=d y) A (7) F 0 m Using Eq. (4) for th slf-fild A' (x), nd Eqs. (5) for th Grn's function D,(x y), w hv A'(x) = 2 ik (x ' y) ffdydk Xy'I A(y) «[» A (y)]lq&. (8) Hr A= A'+ A', but in n itrtiv procdur, to within O(), w my tk A(y)= A'(y) undr th intgrnd. Furthrmor, nticipting us of th DA nd th zro-momntum limit, w cn tk A'(y) = A'(x). 2 I CO( t u) P'X A'= f dy f df", dcoa, Xtp 8m' 3 (20) whr A, := k =co A.. This xprssion (20) is intrsting in its own right; it rlts th mgntic fild producd by circulting lctron B', to th xtrnl mgntic fild B' cusing th circultion. W my now writ w8 s 8 =0' ' 0' 4m = y'(x) 4 n (po B') f f f dydkdcui, ' 3 (2)2 m 0 i(o( f u) p(y) y(x), (2) whr po ij,ocj with po / th Bohr mgnton, nd p(y) =y'(y)p(y) s usul. In our units = /4' VI. THE CALCULATION OF g 2 W now procd with th nlysis of th ction 8' = dx w x in th mnnr st forth by Brut nd whr Ho givs ris to th norml Lndu solutions, nd H is prturbtion contining spin nd slf-fild rt'cts. W now prform Fourir xpnsion of th Puli fild s (s Rfs. nd 2) Krus" or Brut nd Vn Hul. ' From Eq. (5) w y(x) my writ th totl ction dnsity s =g y (x) (23) m =p (Ho+H' ib, )q&, H V i m = o".b 4m H = + A'. V :=H]+H2+H3, cr 8'+ A'. V (22) whr w nticipt tht th y (x) will b, in th lowst ordr of itrtion, th Lndu solutions for Ho. W considr th H'z contribution in dtil (dx:=d x, dy:=d y):

5 po k Vn). n 38 QUANTUM ELECTRODYNAMICS BASED ON SELF-FIELDS,... W'= dxy' cr 8' y 4m f f dx dy y'(x)(po B')q(x) f da, Af d 2 p(y) 4 i co(t u) 3 (2)2 m CO ICO(t u] f f dx dy p *(x)(po B')q (x) f da A f dc@ q', (y)q', (y) (2)' m 0 cg Q) Crrying out th whr k:= [co,k] nd A, := dx:=dt, dy:=du intgrtions givs 5 function: 5(co +co), whr t0 :=E E, tc. Crrying out th dko=dco intgrtion nd using Dirc nottion for simplicity, Eq. (22) givs, with n =s nd m =r, th slfnrgy contribution B' ), W2 f d A. g & n n whr w hv usd orthonormlity of th p. (Th trms with n =m nd r =s, which r vcuum polriztion trms in th rltivistic cs, do not ris in NR clcultions. ) W my nlyz W, = Jdx q'( po B')qr in th sm fshion. W divid 8' by. to obtin n nrgy shift. [S Eq. (7).] Extrcting th contribution to th nth Lndu lvl, th totl nrgy shift proportionl to po'8 is givn by he = (W', + W', ) 4 A = &n p, B'n) 3 2' 77 && xp[i (E E)t +i (E E, ) u ], (24) mliztion in NR clcultions such s this. In Sc. VII, w comput th mss rnormliztion, nd show tht th fctor of ', is ctully + ',. VII. MASS RKNORMALIZATION (25) From Eq. (22) w now considr 03 which givs pic of th ction, 8'3, (26) (Rcll, w hv usd &0 throughout. ) W hv introducd cutoff A in th photon momntum intgrtion, "dx J';dX=: A. Th fr-spc contribution to th mgntic momnt is thn p 277 3&i +0( ) (27) Tking th cutoff of A=3m /4 givs vlu of 5p/p corrct in mgnitud, but incorrct in sign. As mntiond in th Introduction, Grotch nd Kzs' hv pointd out th ncssity of including mss rnor- W,'=, g f da. &n I W3= f dx g)" A'. V W r concrnd hr with chng in th mss s coscint of inrti of th prticl. Thus w will b intrstd in chng in th kintic nrgy oprtor which is proportionl to V. An inspction of Eq. (0) for th currnt j" shows tht only j nd jsm contin th momntum oprtor xplicitly. Dtild clcultions show tht th contribution from jsm is zro; thrfor w shll concntrt on jm(y) =qr'vylmi Th. nlysis procds similrly to tht of 8", nd 8'z in th mgntic momnt clcultion. With j(y)=j(y) w comput A'. ik (x y) A'= f f dy dk [j «(» j)] CO p Vp, (29) whr w hv usd th dipol pproximtion, vlid in th limit B'0 nd & iv)0. Insrting th xprssion (29) into Eq. (28) nd xpnding q&(x)=g y (x)xp( s bfor, w obtin ie t) &m Vm) &n fv/m). &m /V/n) (30) 3 m with co :=E End A.= k, s ws prviously. Brut nd Vn Hul' hv shown tht th first trm in th lrg prnthss vnishs if on dos not mk th dipol pproximtion, nd so w drop it. Th scond trm in th lrg prnthss cn b modifid by noticing tht, du to th symmtry in indics n nd m, w hv th prtil frction xpnsion L nm nm 2 nm + nm nm + n, m (3) whr th scond qulity is, with rspct to th doubl

6 o"b' cr.b' o"b' V 440 A. O. BARUT, JONATHAN P. DO&LING, AND J. F. vn HUELE 38 sum, g. Thus W', =, y f'dx 3m MR m o nm nm + )&& nvm) (m, g(n iv[m) (m [Vin) 2' 3m m SA n n 277 3m n). (32) Th constnt trm in th lrg prnthss of (32) lds to our mss rnormliztion, whil th othr trm givs ris to spontnous mission nd th Lmb shift (s Rf. 2}. Tking only th mss rnorrnliztion trm from (32) nd xtrcting only th contribution to singl nrgy lvl n, w gt (33) Clling th br mss mp, nd th rnormlizd mss m, w cn dfin nw kintic nrgy oprtor for fr lctron s g2 which thn implis g2 p m=mp 27T 27T 8A 3m p th mss rnormliztion 8A 3m p SA =mp + 2' +O( ) 3m p =:mp+5m, (34) whr us of th binomil xpnsion is vlid if w ssum tht A-m. Th totl shift proportionl to "B will b th sum of contributions from m7 nd m8 with m rnormlizd: be7+bes = (n o n ) 4A 2' 3m This is our min rsult. Our first vlu for givn in Eq. (27) ws ngtiv, but ftr including th ffcts of mss rnormliztion, it bcm positiv with vlu of th sm mgnitud s bfor. Th uniqu choic of th vlu A=3m/4 for th cutoff prmtr givs n grmnt with th QED vlu of, =+/. +0 ( ). VIII. SUMMARY AND CONCLUSIONS Thus w s tht thr r two sprt Slf-fild 4x Vcuum r 2x 3 ffcts contributing to th NR clcultion of g 2. On must comput th ctul mgntic momnt chng 5p nd thn includ th ffcts of mss rnormliztion 5m. Blow, w compr th rsults of our slf-fild (SF) pproch to tht of th vcuum fiuctution pproch [th VF rsults r thos of Grotch nd Kzs (GK}, Rf. 0]. Hr x:=a/m x 4x x fluctution 8x X +2 3 whr to O() ithr m or mo cn b usd intrchngbly. Th sm cutoff A f' or th photon momntum is usd in ll formuls. It hs bn climd tht th vcuum fiuctution rsult is positiv, but in fct th GK formul for, " s function of A is only positiv in th rstrictd rng A E(0,4m); thus th sign is ctully cutoff dpndnt (s Fig. ). If w dfin th unitlss prmtr x by x:= A/rn, thn th formul for, "=, "(x ) yilds, "=+/2' for th two vlus x =2+ '& t -x/4 ) 4, := =+ (n n ) + m p (n 2 2K 3m n) 2' 3m (35) (0 42) (4 2) whr, to O(), ithr m or mo my b usd in th lrg prnthss of th lst lin of (35). Thus th mss rnormliztion combins with th originl momnt shift 6p to yild th corrct sign for, Our finl xprssion for th g gyromgntic rtio (g FIG.. 2)/2 Grotch nd Kzs's formul for is, =(g 2)/2 s function of th unitlss cutoff prmtr x:=a/m. Notic tht 2 4A 2 +O( ). (36), (x) &0 only if x is in th intrvl (0,4). Th qution, (x)=+/2n. hs two solutions nd, (x)= /. hs on.

7 QUANTUM ELECTRODYNAMICS BASED ON SELF-FIELDS, =I3.58, 0.42I. (Th vlu of 0.42 is quotd by GK in Rf. 0). Hnc th choic of x hr is not uniqu. In ddition, on cn qully rriv t th wrong nswr of '&78=4.2. Sinc, "= /2n. by choosing x =2 +, non of ths choics of x pprs to b physiclly unrsonbl, it sms to th prsnt uthors tht th problm of th corrct sign for, hs not bn stisfctorily solvd within th contxt of th NR vcuum fluctution pproch, mss rnormliztion notwithstnding. Evn th choic of A which yilds th corrct positiv xprimntl rsult is mbiguous thr. In contrst, our formul for, (x), which ws drivd from th slf-fild pproch, is linr in th vribl x =A/m, nd it is uniformly positiv for ll (physicl) positiv vlus of x. Thus in th domin x E(0, ) th incorrct qution, "(x)= /2 stisfctorily hs no solutions, sinc, "(x)&0. Also,, "(x)=+/2n hs th singl uniqu solution x = '. (S Fig. 2.) Complictions ros in th UF cs du to th fct tht, "(x) in th VF thory dpnds qudrticlly on x. In th SF cs such complictions do not occur, s, "(x) hs linr x dpndnc which lds to th positivity of, nd uniqu solution for, "(x)=+/. Th curious businss of th ltrnting + signs which ppr in Eqs. (37) wrrnts n ttmpt t n intrprttion. In th UF sitution th lctron is immrsd in stochstic bckground fild nd "fls" drg forc s its cloud of virtul positron-lctron pirs ncountrs th bumps of th vcuum fild. Thus th chrg is bit slow to rspond inrtilly to xtrnl forcs, lding to n incrs of th ffctiv mss by n mount 5mvF. Also, this sm vcuum drg phnomnon tnds to slow th spin ngulr momntum nd thus dcrs th ffctiv intrinsic mgntic momnt by 5pv. For smll portion of th domin of th cutoff prmtr, nmly, x:=a/i E(0,4), th mss shift wins out ovr th spin momntum chng to yild positiv rsult for g 2. In th slf-fild scnrio similr rgumnt holds with th drg bing du to th fild of th chrg itslf. In nlog with clssicl rdition rction thory, th br mss must b rnormlizd by dding on th lctromgntic mss bound in th slf-fild, 5msF. Th spin ngulr momntum of br Dirc lctron is slowd slightly whn on includs slf-fild ffcts, s th chrg is now obligtd to drg its own fild round with it s it spins. Th loss of spin nrgy lowrs th mgntic momnt by 6psF, but th loss is lwys mor thn compnstd by th mss chng, rgrdlss of th choic of cutoff, giving nt positiv sign for g 2. Th simplicity nd strightforwrdnss of th (NR) slf-fild clcultion of g 2 ovr th (NR) vcuum fild 2 K I (0 75) FIG. 2. Our formul for, =(g 2)/2, s function of x:=a/m. Notic tht, {x)0 for ll x in th intrvl (0, ) nd tht th qution, (x) =+/ hs only th uniqu solution x = ;hnc th nonphysicl qution,,(x)= / hs no solutions. computtion is vry ppling. Th slf-fild ffct, with its clssicl corrspondnc to rdition rction thory, givs us n intuitiv physicl cus for th nonzro vlu of g 2. Th slf-fild concpt, s dvlopd by Brut nd Krus, " hs bn usd succssfully to comput NR Lmb shifts nd spontnous mission, ' spontnous mission in cvitis, ' Lmb shift nd Csimir-Poldr vn dr Wls forcs in cvitis, ' nd full rltivistic ccount of th two-body problm, th Lmb shift, nd spontnous mission. "' This pproch to QED touchs indd th vry foundtions of quntum thory nd its intrprttion. If th fild %(x) dscribs n objctiv distribution of lctronic rnttr, s Schrodingr nd d Brogli originlly thought, th lctric currnt ssocitd with this distribution producs fild, nmly, th slf-fild, which must b ddd to th xtrnl fild in th Schrodingr, Puli, or Dirc qutions. Th slfconsistncy of th coupld mttr nd Mxwll fids rquirs this. ' Thus th succssful clssicl clcultions of th rditiv procsss hs ld to possibl rvivl of th Schrodingr intrprttion of quntum thory. ' %ork is in progrss to pply th mthod to g 2 in cvitis, ful- y rltivistic computtion of g 2, nd th Unruh ffct. ACKNQWLKDGMENT On of us (J.P.D.) would lik to cknowldg prtil support from th Itlin Ministry of Forign Affirs during sty t th Intrntionl Cntr for Thorticl Physics in Trist. 'H. A. Bth, Phys. Rv. 72, 339 (947). 2T. A. Wlton, Phys. Rv. 74, 57 {948). Z. Korb, Prog. Thor. Phys. 4, 39 (949). 4H. B. Clln nd T. A. Wlton, Phys. Rv. 83, 34 {95). J. R. Ackrhlt, P. L. Knight, nd J. H. Ebrly, Phys. Rv. Ltt. 30, 955 (973); P. W. Milonni, J. R. Ackrhlt, nd W. A. Smith, ibid. 3, 958 {973). 6J. Dlibrd, J. Dupont-Roc, nd C. Cohn-Tnnoudji, J. Phys. (Pris) 43, 67 (982). 7T. H. Boyr, Phys. Rv. D, 809 (975}.

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