Hans Bethe, Quantum Mechanics, and the Lamb Shift

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GENERAL  ARTICLE

A N Mitra

A n e le m e n ta ry d e riv a tio n o f th e L a m b S h ift, a s o rig in a lly g iv e n b y H a n s B e th e , is a tte m p te d , w ith a sh o rt h isto ric a l b a ck g ro u n d o n th e e v o lu - tio n o f Q E D sin c e th e b e g in n in g o f th e la st c e n - tu ry . T h e tre a tm e n t is fo r a n o n -re la tiv istic e le c - tro n , in k e e p in g w ith B e th e 's p ro v e rb ia l in sig h t in to th e p ro b le m , w h ich re n d e rs irre le v a n t th e A N Mitra has had a long e ® e c t o f th e e le c tro n sp in o n th e p ro b le m . T h e teaching service from K ra m e r id e a o f re n o rm a liza tio n is im p le m e n te d 1955 to 1994 when he th ro u g h a sim p le su b tra c tio n o f th e se lf-e n e rg y retired from Delhi University as INSA o f a fre e e le c tro n fro m th a t o f th e e le c tro n b o u n d Einstein Professor of in a h y d ro g e n a to m . physics. He spent a year (1995) as visiting professor 1 . In tro d u c tio n : B e th e 's Im p a c t o n P h y sic s at NIAS (Bangalore). Since then he has been H an s A lb rech t B eth e, th e last lin k am on g th e fou n d in g working as a free-lance fath ers of p hy sics of th e last cen tu ry, p assed aw ay on writer-cum-researcher in M arch 6, 2005 at th e age of 98. In th e w orld of p h y sics,it physics, with periodic is d i± cu lt to d escrib e B eth e's im p act in ad eq u ate term s, publications on scientific matters of current sin ce it ex ten d ed over a w h ole sp ectru m , from atom ic interests. p ro cesses resp on sib le for th e p rop erties of m atter, to th e n u clear forces govern in g th e stru ctu re of atom ic n u clei, w ith in a 75 year tim e sp an ran gin g from th e m id -th irties to th e last d ecad e of th e tw entieth cen tu ry. N o less w as h is con cern for th e so cial resp on sib ility of scien ce, a su b ject on w h ich h e h ad w ritten n u m erou s articles in in ° u en tial jou rn als. W h ich ever sector of p h ysics h e h ad set h is eyes on , B eth e in variab ly left an in d elib le m ark of h is m asterly grasp w ith d eep in sight b orn ou t of h is tw in ch aracteristics of sim p licity an d th orou g h n ess, b e it in solid state p h y sics (B eth e A n satz), or nu clear p h ysics Keywords (B eth e S econ d P rin cip le T h eory ), or even q u an tu m ¯ eld Quantum mechanics, Lamb th eory (B eth e{S alp eter E q u ation ). H is fan tastic p ow ers shift.

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Bethe’s Nobel Prize of fast yet accu rate calcu lation (in th e d ays lon g b efore (in 1967) was a th e com p u ter revolu tion ) h ad m ad e h im a legen d of h is rather belated tim e, th an ks to h is sin gle-h an d ed su ccess in d eterm in - recognition of his in g th e en ergy p ro d u ction in th e S u n an d th e stars b y a genius. correct ch oice of a sequ en ce of n u clear reaction s resu ltin g in th e fu sion of fou r h yd rogen atom s in to on e h eliu m atom , th rou gh fou r step s of th e `carb on cycle', leav in g th e origin al carb on to com e ou t u n scath ed in th e en d . (T h e story of h ow B eth e ach ieved th is feat w h ile travel- lin g in a train is v ivid ly d escrib ed by G eorge G am ow in h is b ook B irth and D eath of the Sun). H is N ob el P rize (in 1967), th ree d ecad es after th is great d iscovery, w as a rath er b elated recogn ition of h is gen iu s. S om e of B eth e's early w ork s h ave b ecom e h ou seh old w ord s in p h ysics textb o ok s. T h e B eth e{W eiszacker m ass form u la is a classic exam p le of p h ysicalin sight, in asm u ch as each term in th e form u la rep resen ts a d istin ct p h y sical e® ect. A n oth er h ou seh old w ord is th e B eth e{H eitler form u la for B rem sstrah lu n g w h ose agreem en t w ith d ata th en availab le p rovid ed on e of th e earliest exp erim en tal su p p orts for th e th en n ascen t stru ctu re of Q E D . B eth e w as eq u ally d ex terou s in ap p lyin g cla ssica l m ath em atical m eth o d s to i) th e calcu lation of electron d en sities in cry stals; ii) th e ord er-d isord er tran sition in alloy s (th e B eth e{P eierls A p p rox im ation ); iii) th e ion ization p ro cesses in sh ock w aves, to n am e on ly a few . N o w on - d er h e h ead ed th e grou p of th eoretician s for th e M an h at- tan P ro ject in L os A lam os, sin ce h is d eep an d versatile k n ow led ge of p h y sics h ad few eq u als. A t th e n ex t h igh er level of sop h istication , th ere h as b een rich ev id en ce of B eth e's tran sp arent ap p roach , all b earin g th e u n m istakab le m ark s of a \y in -yan g" T ao of sim p licity w ith th orou gh n ess. S om e ex am p les are h is fam ou s p ap ers on (i) L am b S h ift in Q E D ; (ii) B eth e{ S alp eter E q u ation in th e stron g interaction sector of q u an tu m ¯ eld th eory (Q F T ); an d (iii) e® ective ran ge th eory, an d B eth e{B ru eck n er th eory of nu clear m atter.

34 RESONANCE  October 2005 GENERAL  ARTICLE

1 .1 L a m b S h ift: T h e B e th e C o n n ec tio n The ‘Lamb Shift’ was the frequency T h e L am b S h ift, w h ich is th e su b ject of th e p resen t p a- of a microwave p er, ¯ rst n eed s a little h istorical b ack grou n d w h ich h as field that induced b oth th eoretical an d ex p erim en tal asp ects, of w h ich th e latter w as a b y -p ro d u ct of W orld W ar II in con n ection transitions from w ith th e rad ar d evelop m ent w h ich w as con cern ed w ith one excited state a region of th e ord er of 1 cm , term ed th e m icrow a v e of a hydrogen region , ly in g b etw een th e far in frared an d sh ort rad io atom to another. w aves. T h e tech n iqu es of th e m icrow ave region cam e in very h an d y at th e h an d s of W E L am b an d h is colleagu e R C R eth erford [1], in m ak in g ¯ n e stru ctu re m easu re- m en ts of th e atom ic sp ectra. T h e `L am b S h ift' w as th e freq u en cy of a m icrow ave ¯ eld th at in d u ced tran sition s from on e excited state of a hy d rogen atom to an oth er. In p articu lar, th ey fou n d th at th e 2s 1 = 2 levels of h yd rogen w ere sligh tly ab ove th e 2p 1 = 2 level, b y ab ou t 1050 M H z, con trary to w h at w as p red icted b y D irac's th eory, an d ob eyed by ex p erim ent till th en ! T h is `L am b S h ift' w as ad d ressed by B eth e in h is ow n in im itab le style: H e w as retu rn in g to C orn ell after atten d in g a select w ork sh op in th e su m m er of 1947 in S h elter Islan d (N Y ), w h ere th e an om aly p osed by th e L am b {R eth erford d iscovery w as th e su b ject of several b rain -storm in g session s, w ith top exp erts in th e ¯ eld p articip atin g in th read b are d iscu s- sion s on variou s asp ects of th e `p rob lem ' vis-a-v is th e existin g k n ow led ge on Q E D [2]. O n e of th e p articip an ts w as H K ram ers of H ollan d , w h o u n fold ed h is id ea of ren orm a liz a tion in th is con text (see S ection 2 for a logical exp osition of th e con cep t). A p p arently h e fou n d th e solu tion in th e train itself (!), on h is retu rn jou rn ey { a rep etition of a sim ilar feat a d ecad e earlier on th e m ech an ism of en ergy p ro d u ction in th e S u n , w h ich w as to fetch h im h is N ob el P rize. It w as a tw o-p age p ap er w ith a few crisp sym b ols an d eq u ation s [3], th at said it all.

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The last century T h e story of h ow [2] B eth e ob tain ed th e d esired solu tion started with a bang, is th e su b ject of th is article. S in ce th e in tricacies of Q E D with Max Planck’s w ere n ot in volved in h is d erivation , it is of con sid erab le (1901) revolutionary p h ysical in terest to a w id er read ersh ip , to rep ro d u ce th e quantum proposal d erivation w ith ou t goin g in to th e avoid ab le sop h istica- on the one hand, and tion s of Q E D (in clu d in g th e D irac th eory ). T o keep th e Einstein’s single year article reason ab ly self-con tain ed , h ow ever, som e h istor- (1905) of 3 earth- ical p ersp ective on th e statu s of q u an tu m th eory (Q T ), shaking papers which p reced in g th e grow th of Q E D is o® ered for com p leten ess. changed the face of T o th at en d , th e article is organ ized as follow s. S ection 2 attem p ts an overv iew of th e p rin cip al d evelop m ents in physics, on the other. Q T , lead in g to th e statu s of Q E D u p to th e early for- ties of th e last cen tu ry, w ith sp ecial referen ce to B eth e's in volvem en t in early Q E D ap p lication s. S ection 3 p ro- v id es th e actu al d erivation of th e L am b S h ift as given origin ally by B eth e, w h erein th e electron is treated as a n on -relativ istic p article, an d th e p h oton as a ¯ eld w h ich can b e F ou rier an alysed in term s of w ave n u m b ers an d freq u en cies. S ection 4 con clu d es w ith a sh ort d iscu ssion . 2 . Q u a n tu m T h e o ry to Q E D : A P a n o ra m ic V ie w

T h e last cen tu ry started w ith a b an g, w ith M ax P lan ck 's (1901) revolu tion ary qu an tu m p rop osalon th e on e h an d , an d E in stein 's sin gle year (1905) of 3 earth -sh ak in g p a- p ers w h ich ch an ged th e face of p h y sics, on th e oth er. T h e S p ecial T h eory of R elativ ity, w h ich w as in sp ired b y th e invarian ce of M axw ell's equ ation s u n d er L orentz tran sform ation s, as w ell as b y th e n u ll resu lt of th e M ich elson {M orley ex p erim en t, u n i¯ ed sp ace an d tim e (h ith erto regard ed as tw o sep arate en tities) in to a sin g le sp ace-tim e in varian t (x 2 c2 t2 ) u n d er L orentz tran sfor- m ation s, alb eit at th e co¡st of a n ew u n iversal con stan t c rep resentin g th e velocity of ligh t in vacu u m . (E in stein later gen eralized th e S p ecial T h eory into th e G en eral T h eory of R elativ ity, b u t su ch gen eralization is n ot rel- evan t in th e p resen t con tex t). A s a logical con seq u en ce of th e S p ecial T h eory, E in stein d ed u ced th e d y n am i- cal resu lt E = m c2 w h ose im p ortan ce in th e w orld of

36 RESONANCE  October 2005 GENERAL  ARTICLE p h ysics h ard ly n eed s tellin g. T h e In tern ation al Y ear of P h y sics (2005) in cid entally m arks th e cen ten ary of th is great d iscovery. 2 .1 O ld Q u a n tu m T h eo ry

T h e ram i¯ cation s of th e qu an tu m con cep t to ok m u ch lon ger to sin k in , alth ou gh E in stein q u ick ly u p h eld P lan - ck 's hy p oth esis w ith h is celeb rated p h oto electric e® ect w h ich gave th e p h oton a p article p rop erty over an d ab ove its alread y k n ow n w ave ch aracter. W h at w as n ow n eed ed w as a v iab le atom ic m o d el to accou n t for som e cru - cial em p irical d ata like R itz's classi¯ cation of sp ectral lin es, th e F ran ck{H ertz ex p erim en t on th e d iscrete en - ergy losses of electron s on collision w ith atom s, etc. T h e n ecessary b o ost w as p rov id ed b y R u th erford 's d iscov - ery (1911) of atom ic stru ctu re w h ich led N iels B oh r (1913) to p rop ose h is atom ic m o d el in term s of tw o p ostu lates: i) an atom ic sy stem can stay in p articu lar q u an tized states each of w h ich corresp on d s to a d e¯ - n ite en ergy of th e sy stem ; ii) a rad iation qu an tu m h as a freq u en cy equ al to th e d i® eren ce of tw o atom ic en - ergy levels d ivid ed b y h . T h ese p ostu lates, forti¯ ed by W ilson {S om m erfeld (W S ) qu an tization con d ition s, su f- ¯ ced to give a q u an titative d escrip tion of large n u m b er of sp ectroscop ic d ata for atom s an d d iatom ic m olecu les. T h e W S con d ition s in tu rn w ere b ased on th e H am ilton { Jacob i form u lation of classical m ech an ics w h erein th e en ergy of th e sy stem w as ex p ressib le en tirely in term s of th e so-called àction ' in tegrals (after elim in ation of th e cy clic àn gle' variab les). In th is rep resentation , th e W S con d ition s con sisted sim p ly in d em an d in g th at th e action in tegrals b e `qu antized ', i.e., rep resented as in tegral m u ltip les of h . T h is w as th e so-called òld Q u antu m T h eory ' w h ich w as to ru le th e w orld of atom ic p h ysics for ab ou t a d ecad e at th e en d of w h ich it gave w ay to th e new Q u antu m M ech an ics (Q M ) of H eisen b erg, S ch rÄod in ger an d D irac.

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T h e n eed for ch an ge w as occasion ed b y som e m a jor in - ad eq u acies in th e old Q T , su ch as its in ap p licab ility to ap erio d ic sy stem s, an d in com p lete treatm en t of th e in - ten sities of sp ectral lin es, as w ell as som e in n er con - trad iction s resu ltin g in its failu re to give a con cep tu ally satisfactory accou n t of fu n d am en tal p h en om en a. F or ex - am p le, it w as d i± cu lt to u n d erstan d w h y th e C ou lom b in teraction b etw een th e electron an d th e p roton w as so e® ective for th e sp ectra, w h ile th e ab ility of an electron to em it em rad iation from a sta tion a ry state w as zero! A n d th e assu m p tion of a d u alch aracter of ligh t ap p eared self-contrad ictory [4]. 2 .2 N e w Q u a n tu m M ech a n ic s

T h e ad vent of th e new Q M w as p reced ed by tw o con cep - tu al d iscoveries. T h e ¯ rst w as th e p rop osalb y d e B roglie ab ou t th e w ave n atu re of m aterial p articles w ith w ave len gth given b y ¸ = h = p, w h ich su ggested th at w ave- p article d u ality w as a u n iv ersa l p h en om en on , n ot con - ¯ n ed to th e electrom agn etic ¯ eld alon e, a resu lt w h ich fou n d overw h elm in g su p p ort from th e D av isson {G erm er (1927) an d G P T h om p son (1928) exp erim en ts. T h e secon d w as S N B ose's (1924) p rop osal for a n ew m o d e of cou n tin g b ased on tru e in d istin g u ish a bility w h ich em p h asized th e classi¯ cation of sta tes accord in g to th e n u m b er of p h oton s in th eir k itty, in stead of th e `classical' M ax w ell{B oltzm an n m o d e of cou n tin g in w h ich th e in d i- v id u al `n am es' of th e p h oton s cou ld n ot b e fu lly erased , (d esp ite d iv ision b y n ! !). B ose's p rop osal fou n d a n at- u ral ech o in H eisen b erg's M atrix M ech an ics in w h ich a d y n am ical variab le (like p osition or m om en tu m ) w as n o lon ger con sid ered asso ciated w ith in d iv id u al p article m otion s, b u t w as regard ed as an op erator (m atrix), op eratin g on states th at w ere su p p osed to b e lab elled b y th e n u m b er of p articles w ith sp eci¯ ed ch aracteristics. In th is form u lation , th e corresp on d en ce w ith th e classical p ictu re w as estab lish ed b y rein terp retin g th e H am ilton ian eq u ation H (q;p) = E as an op erator (m a-

38 RESONANCE  October 2005 GENERAL  ARTICLE trix ) eq u ation w h ich m ad e sen se on ly after sp ecify in g Heisenberg’s matrix th e state (vector) on w h ich it w as su p p osed to op erate! formulation succeeded in H eisenb erg's m atrix form u lation su cceed ed in rem ov in g th e con cep tu al ob stacles associated w ith th e old Q u an - removing the tu m T h eory, w h ile im p rov in g if any th in g th e agreem en t conceptual obstacles w ith th e d ata. A n altern ative form u lation b y S ch rÄod in ger, associated with the w h ich p roved m ore e® ective in p ractice, con sisted in u s- old Quantum Theory, in g a loca l `d i® eren tial' rep resen tation for th e op erators while improving if in q sp ace: anything the agreement with the p = i¹h @ q ; E = + i¹h @ t; H = H (q; ih¹ @ q ); (1) data. ¡ ¡ an d w ritin g th e eq u ation of m otion as

H Ã (q;t) = ih¹ @ tÃ (q;t); (2) w h ere th e w ave-fu n ction Ã stan d s for th e state u n d er stu d y. In th is d escrip tion , th e w ave fu n ction is th e rep os- itory for th e fu ll d yn am ical in form ation of th e q u antu m state u n d er stu d y. T h e H am ilton ian is ad ap ted from th e corresp on d in g `classical' stru ctu re, w ith d u e regard to its op erator ch aracter in th e qu an tu m con tex t. (F or its p recise form u n d er th e electrom agn etic in teraction , see b elow ). 2 .3 R ela tiv istic Q M : Q u a n tu m F ie ld

T h e n ext stage of th e d evelop m ent, v iz., relativistic for- m u lation of th e q u antu m eq u ation s, p roved cru cial n ot on ly in con crete m ath em atical term s, b u t con ceptu a lly in term s of m ath em atical self-con sisten cy of th e gen - eralization en visaged . N ow th e gen eralization from th e (n on -relativ istic) N ew ton ian eq u ation s of m otion to th ose in con form ity w ith th e sp ecial th eory of relativ ity h ad p resented n o p rob lem . In d eed for a free p article, it on ly m ean s th e rep lacem ent

p 2 = 2m = E ; E 2 = p 2 c2 + m 2 c4 : (3) ) F or a p article interactin g w ith th e electrom agn etic ¯ eld

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(A ;Á ), th e form given ab ove stays relativ istically in varian t, sin ce it m ean s th e rep lacem ents

E E eÁ ; p p eA =c: (4) ) ¡ ) ¡ T h u s th e classical relativistic eq u ation in th e p resen ce of an em ¯ eld read s as

(E eÁ )2 = (cp eA )2 + m 2 c4 ; (5) ¡ ¡ w h ich also h ap p en s to b e con sisten t w ith g a u g e in varian ce! T h e n on -relativ istic form of th e last eq u ation n ow follow s in th e lim it of sm all p= m c as

E eÁ = (p eA = c)2 = 2m ; (6) ¡ ¡ w h ere th e en ergy E on th e left-h an d -sid e exclu d es th e rest en ergy m c2 . N ex t, th e qu an tu m form u lation , ¶a la S ch rÄod in ger, from th e n on -relativ istic form of classical m ech an ics go es th - rou gh b y regard in g (2) or (6) as op erator eq u ation s an d ap p ly in g th em to Ã on th e right. B u t th e sam e p rescrip - tion d o es n ot w ork w ith th e relativ istic form (5), w ith ou t sacri¯ cin g m ath em atical self-con sisten cy ! T o see th e n a- tu re of th e self-con sisten cy th at is in volved , con sid er th e q u an tu m form of th e secon d p art of (3):

2 2 2 2 2 2 4 h¹ @ t Ã = [ h¹ c + m c ]Ã ; (7) ¡ ¡ r w h ere th e d i® eren tial form s of E an d p , ¶a la (1), h ave b een in serted . N ow th e p resen ce of th e secon d d erivative w .r.t. tim e im p lies th at th e p article d en sity

¤ ¤ ½ = i(Ã @ tÃ @ tÃ Ã ); ¡ th at is n eed ed to con serve th e cu rrent, is n o lon ger p ositive d e¯ n ite, w h ich is th e sou rce of th e in con sisten cy. T h is p rob lem w as resolved b y P au li{W eisskop f [5] by rein terp retin g th e qu an tity ½ as th e average ch arge d en - sity of th e Ã f ield , (w h ich can h ave eith er sign lo cally),

40 RESONANCE  October 2005 GENERAL  ARTICLE in stead of as a sin g le p article d en sity ! T h is is th e p rice The marriage of of qu an tization in th e relativistic regim e: th e m in im u m Relativity with d .o.f. is th at of a f ield (in ¯ n ite n u m b er of p articles), Quantum theory an d n ot of a sin gle p article, as stip u lated by D yson [6]. resulted in the T h e sam e story go es for th e D irac form u lation for an prediction of anti- electron w ith sp in [5], d esp ite th e ap p earan ce of E in matter. a lin ear form , sin ce th e in con sisten cy n ow sh ow s u p in th e form of n eg a tiv e en ergy states! H ere again , th e solu tion lies in d em an d in g th at th e n egative en ergy states b e n orm ally fu ll, th u s p reventin g (µa la ex clu sion p rin - cip le) th e electron from en terin g th em , u n less en ou gh en ergy (> 2m c2 ) is p u m p ed in from ou tsid e to `lift' on e of su ch electron s to a p ositive en ergy state, so as to accom m o d ate th e ou tsid e electron in th e `h ole' so cre- ated . T h u s th e m essage is th e sam e again , v iz., on e m u st d eal w ith an in ¯ n ite n u m b er of electron s { a q u antu m (ferm ion ) f ield ! T h e `h ole' is a sign atu re for an an ti- electron (p ositron ) w ith ex actly op p osite p rop erties to th e electron 's. In th e case of th e b oson ¯ eld Ã of (7) to o, th e corresp on d in g an ti-p article is a b oson of op - p osite ch arge. T h u s th e m essage is th e sam e for b oth [5]: T h e m arriage of R elativ ity w ith Q u an tu m th eory resu lted in th e p red iction of anti-m atter. In th e n on -relativ istic d om ain on th e oth er h an d , th e ¯ eld con cep t is op tion al for qu an tization , sin ce th e corresp on d in g S ch rÄod in ger equ ation is n ow h¹ 2 2 i¹h @ tÃ = Ã ; 2mr ¡ for w h ich ½ = ¹h Ã ¤Ã is p ositive d e¯ n ite even for a sin gle p article. T h e sam e logic goes th rou gh in th e p resen ce of an em ¯ eld to o, for w h ich th e q u an tu m equ ation can b e read o® from th e corresp on d in g classical eq u ation (6): [E eÁ ]Ã = [(p eA =c)2 =2m ]Ã (8) ¡ ¡ in volvin g a sin gle tim e d erivative w h en th e d i® eren tial form s are in serted from (1). O n th e oth er h an d , for th e

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The quantum field em ¯ eld , w h ich is alw ay s relativ istic, th e q u antization (QFT) concept is is n ecessarily for th e en tire ¯ eld , an d n ot for a sin gle essentially one of p h oton . extending the 2 .4 Q F T M eth od s fo r Q E D : P ertu rba tio n T h e- methods of o ry quantum mechanics to an infinite number T h e qu an tu m ¯ eld (Q F T ) con cep t is essentially on e of of harmonic exten d in g th e m eth od s of q u antu m m ech an ics to an in - oscillators. ¯ n ite n u m b er of h arm on ic oscillators. H ere's h ow . T h e q u an tization for a free ¯ eld is ach ieved b y ¯ rst F ou rier an aly zin g th e ¯ eld variab le say A ¹ in term s of a d iscrete b u t in ¯ n ite set of qk ;p k variab les (u n d er b ox n orm alization ). In a (q;p) rep resen tation , th e free ¯ eld H am ilton ian b eh aves like an in ¯ n ite set of h arm on ic oscillators, so th at th e p rob lem of qu an tization gets red u ced to solv in g for an in ¯ n ite set of n on -in teractin g h arm on ic oscillators, in th e stan d ard m an n er. T h e treatm ent gets greatly sim p li¯ ed b y an altern ative com p lex rep resentation : (a ;a y) = (q ip)= 2 p w h erein th e a an d a y b eh ave§ like step -d ow n (an n ih ila- tion ) an d step -u p (creation ) op erators resp ectively in a n u m b er (N ) rep resen tation for th e q u anta of th e free ¯ eld u n d er stu d y. T h e sam e p ro ced u re is ap p licab le for th e in teraction of tw o ¯ eld s, say th e electron an d em ¯ eld s, w ith sep arate tracks of th eir resp ective N rep resen tation s. A n d sin ce in p ractice, on ly on e or tw o q u an ta of th e resp ective ¯ eld s are in volved , th e N -d egrees of freed om u su ally get elim in ated in a triv ial m an n er, leav in g th e rem ain in g (n on -triv ial) p arts of th e m atrix elem en ts to th e ru les of ord in ary q u antu m m ech an ics. T h e ¯ rst ap p lication of th e Q F T m eth o d s w as to th e p rob lem of in teraction of th e em ¯ eld w ith th e D irac electron , term ed q u an tu m -electro d y n am ics, Q E D for sh ort. T o th at en d , th e u n p ertu rb ed p art H 0 of th e

42 RESONANCE  October 2005 GENERAL  ARTICLE

H am ilton ian in clu d es th e resu ltant of th e lon gitu d in al Bethe realized an d tem p oral p arts of th e em ¯ eld , togeth er w ith an from intuition that extern al ¯ eld if an y (e.g., th e ¯ eld of a p roton in a h y - it was most d rogen atom ), as an e® ective C ou lom b in teraction , w h ile practicable to treat th e tra n sv erse p art of th e em ¯ eld , term ed th e rad ia- the electron as a tion ¯ eld , is treated b y stan d ard p ertu rb ation th eory particle, and the w ith th e interaction term : photon as a field.

eÃ¹A ¹ ° ¹ Ã ; e = 4¼ =137:

(T h e sm alln ess of th e electron 's ch arge, a m easu re of its cou p lin g w ith th e em ¯ eld , jqu sti¯ es th e p ertu rb ation ap p roach ). 2 .5 B e th e 's In v o lve m e n t in E a rly Q E D

S om e of th e earliest ap p lication s of Q E D w ere to several secon d ord er p rocesses like C om p ton scatterin g (K lein { N ish in a form u la [7]), an d th e related p ro cess k n ow n as B rem sstrah lu n g w h ich w as given b y B eth e in collab oration w ith H eitler [8], b oth of w h ich tu rn ed ou t to give excellen t agreem ent w ith exp erim en t. T hu s B eth e w as from th e b egin n in g of Q E D closely asso ciated w ith its ap p lication s to atom ic p h en om en a. H e seem ed to h ave realized from in tu ition th at it w as m ost p racticab le to treat th e electron as a p article, an d th e p h oton as a ¯ eld . S o h e w en t on to collab orate w ith F erm i, w h o h ad d evelop ed th e Q E D on th ese lin es [2], to calcu late th e retard ed in teraction of tw o electron s v ia th e ex ch an ge of a p h oton [9]. A fter th e F erm i collab oration , h e w rote a b ig rev iew article [10] on th e q u an tu m th eory of on e- electron an d tw o-electron sy stem s, w h ich is a stan d in g testim on y to th e th orou gh n ess of h is ap p roach to atom ic p rob lem s w ith th e to ols of Q E D . 2 .6 In ¯ n itie s in Q E D : R e n o rm a liza tio n ?

T h e Q E D ap p lication s in th e th irties, w ere n ot on ly su ccessfu l in th e secon d ord er, b u t also for h igh er or- d er p ro cesses, p rov id ed th e calcu lation w as m ad e in th e

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Kramers idea of low est `e-ord er' for th e p ro cess u n d er stu d y, i.e., n o renormalization v irtu a l su b -p ro cess (w ith em ission an d re-ab sorp tion helped remove the of rad iation ) w as in volved . O n th e oth er h an d it w as infinities in the fou n d th at if th e e-ord er of a given p ro cess in volved calculation of som e v irtu a l su b -p rocess, th en invariab ly som e d iver- electron mass. gen ces w ere en cou n tered . A p rim ary v irtu al p rocess is on e in w h ich an electron em its an d reab sorb s a rad iation q u an tu m , thu s con trib u tin g to its `self-en ergy '. If th is is an entire p ro cess, it corresp on d s to th e `self-en ergy ' of a f ree electron . O r it cou ld b e p art of a b igger p ro cess, e.g., an electron in sid e a hy d rogen atom , in w h ich case it is term ed th e self-en ergy of a bou n d electron . F or all su ch p ro cesses, th e ru les of p ertu rb ative Q E D invariab ly give in ¯ n ities. W h y ? In a secon d ord er p ro cess in volv - in g th e em ission of a p h oton , follow ed by reab sorp tion of th e sa m e p h oton , th e p h oton m om en tu m k can h ave an y valu e, so th at all su ch am p litu d es m u st b e su m m ed over all p ossib le valu es of k , w h ich p ro d u ces a d ivergen t exp ression ! U n fortu n ately, th e th eory as d evelop ed u p to th e 1940s, w as n ot eq u ip p ed to d eal w ith su ch h az- ard s. O n th e oth er h an d , any p h y sical q u antity to b e ob servab le, m u st h ave a ¯ n ite valu e, an d th ere seem ed to b e n o ob v iou s w ay to get rid of su ch in ¯ n ities b efore con fron tin g th e corresp on d in g am p litu d es w ith ob serva- tion . 2 .6 .1 K ra m e rs' Id ea : It w as a b rilliant id ea of H K ram ers th at led h im to p rop ose th at th e b u lk of th e in ¯ n ity in su ch calcu lation s w as n ot sign i¯ can t, b u t on ly a sm all (h op efu lly ¯ n ite!) p art th at rem ain s after th e am ou n t corresp on d in g to a free or ba re electron w as iso- lated an d su b tracted ou t, sin ce th e self-en ergy con trib u - tion for th e latter w ou ld n ever b e ob servab le ! T h erefore th e `tru e' self-en ergy of an electron b ou n d in a h yd rogen atom is th e resu lt of su b traction of th e valu e accru - in g from th e free or ba re electron , an d sh ou ld h op efu lly b e ¯ n ite. T h is w as th e id ea of R en orm a liz a tion w h ich am ou n ted to red e¯ n in g th e ob served m ass of th e elec-

44 RESONANCE  October 2005 GENERAL  ARTICLE tron as a su m of its b are m ass m 0 an d th e `self-m ass' ±m (ob tain ed as ab ove, after d iv id in g b y c2 ). It w as h ow ever left to B eth e to sh ow h ow to calcu late th e n et e® ect w ith h is great in sigh t [2] th at led h im to con sid er equ ation (8) in stead of its fu ll-° ed ged relativ istic cou n - terp art, th u s avoid in g th e ¯ eld con cep t for th e electron . T h is d erivation is sketch ed b elow . 3 . B e th e 's D e riv a tio n o f th e L a m b S h ift

W e n ow give a b rief sketch of B eth e's origin al d erivation [3,8] of th e L am b S h ift, on e in w h ich th e sp irit of ren orm alization w as im p lem ented in a p ragm atic fash - ion , m erely by su b tractin g th e `free' electron self en ergy from th e fu ll (b ou n d electron ) am ou nt. T o th at en d w e w rite th e fu ll H am ilton ian from an in sp ection of (8), an d sp ecializin g to a C ou lom b p oten tial, as

2 2 H = p =2m Z e =r + H 1 ; H 1 = ep :A =m c; :A = 0; ¡ ¡ r (9) w h ere th e in teraction H 1 w ith th e (tran sverse) rad iation ¯ eld is ex p licitly sh ow n . T h e rad iation ¯ eld A is F ou rier an aly zed as

3 2 h¹ c d k y A (~x ) = ~²k i(a k i + a ); (10) (2¼ )3 2! k i p k 1 w h ere th e twso tran sverse p olarization s of th e p h oton are in d icated b y th e iZn d ex i. NXow th e secon d ord er self- en ergy of a b ou n d electron in a qu an tu m state of en ergy E m , d u e to em ission an d reab sorp tion of rad iation of en ergy k c, is given b y th e stan d ard form u la:

< H 1 > ; < H 1 > ; ¢ E = m k n k n k m k ; n k E m E n ck ¡ ¡ w h ere th e su m is over all p ossib le in term ed iate states n of th e atom , as wXell as over all p ossib le states k of th e em itted an d reab sorb ed p h oton . S u b stitu tin g from (9) an d (10), an d skip p in g a cou p le of step s of th e triv ial

RESONANCE  45 October 2005 GENERAL  ARTICLE

d .o.f.'s of th e N -rep resentation , th e sim p li¯ ed form of th e self-en ergy b ecom es [11]

2® L p p ¢ E = k d k m n n m : (11) 3¼ m 2 c2 0 E + k E ¡ n n m ¡ H ere p is th e m om entu m op erator for th e electron ; th e su m over th e tw o p olaZrizationX d irection s as w ell as th e an gu lar in tegration over th e p h oton d irection s h ave b een carried ou t; an d k in sid e th e integral h as b een n orm alized to th e d im en sion of en ergy. T h e u p p er lim it L of k -integration is su p p osed to b e in ¯ n ite. T h e cru cial step is n ow to su btra ct th e `free' electron con trib u tion , w h ich m ay b e read o® from (11) b y rep lacin g th e d en om in a- tor on th e r.h .s. w ith k on ly, so th at th e su b tracted d en om in ator read s as: 1 1 1 E E = m n : E n + k E m E n + k E m k k (E n + ¡k E m ) ) ¡ ¡ ¡ ¡ T h en th e `ren orm alized ' form of ¢ E b ecom es

L 2® p m n p n m (E n E m ) ¢ E = 2 2 d k : (12) 3¼ m c 0 n E n + k ¡E m ¡ N ote th at th e d egree of d ivergen ce in k n ow red u ces to Z logarith m ic from lin ear, soXth at th e sen sitiv ity to th e u p - p er lim it L is greatly red u ced . T h is is as far as cou ld b e ach ieved w ith B eth e's n on -relativ istic treatm en t. H ow - ever, B eth e correctly su rm ised th at w ith a p rop er relativ istic treatm ent, th e d ivergen ce w ou ld b e fu rth er re- d u ced from logarith m ic to a convergent in tegral [11]. T o treat (12) fu rth er, th e k -integration gives a factor 2 ln L =[E n E m ], w h ere L m ay b e taken as of ord er m c , as b e¯ ts ¡a n on -relativ istic treatm en t. F u rth er, th e rela- tive in sen sitiv ity of th e logarith m to its argu m en t w ar- ran ts an ap p roxim ation w h ich elim in ates its (m ;n ) d e- p en d en ce, b y rep lacin g E n E m w ith an average valu e ¡

46 RESONANCE  October 2005 GENERAL  ARTICLE

< E E m > w h ich can th en b e taken ou t of th e n - The fuller su m m ¡ation ! T h e resu lt of all th ese m an ip u lation s is ramifications of the 2® Kramers ¢ E = 2 2 p m n p n m (E n E m ) renormalization idea 3¼ m c n ¡ were to extend from 2 [ln m c = < E E m > ]: (13) that of mass, to N ow u sin g th e ru les of qXu an t¡u m m ech an ics, on e ¯ n d s every conceivable [11] physical quantity: mass, charge, wave 1 2 (E n E m )p n m p m n = < [p;[ Z e =r;p]] > m m = function, 2 n ¡ ¡ 2 2 2 2¼ ¹h e Z [Ã m (0)] : T oXsee h ow th e last form com es ab ou t, th e d ou b le com - m u tator red u ces to a 2 (1=r) w h ich is a 3D ±-fu n ction ! y 3 T h u s th e in tegral is prop ortion al to < Ã ± (x )Ã > , i.e., to th e sq u are of th e atom ic w ave fu n ction at zero d is- tan ce. T h is m ean s in tu rn th at th e sh if t a® ects on ly th e 2s 1= 2 state, w h ile th e 2p 1 = 2 state rem ain s u n sh ifted , w .r.t. th e D irac th eory p red iction s. B eth e estim ated th is valu e as 1040 M H z in am azin gly goo d accord w ith th e L am b -R eth erford valu e [1]. D iscu ssio n

F or a h istorical p ersp ective on h ow B eth e zo om ed in (like a h om in g p igeon ) on a n on -relativ istic treatm ent, th e interested read er is referred to D y son 's n arrative [2] on h ow B eth e's p reviou s train in g an d exp erien ce led h im to th e righ t an sw er, w ith ou t b ein g led astray b y irrele- van t d etails! A lesser m ortal w ou ld h ave b een overaw ed b y th e Q E D in ¯ n ities, so as n ot to th in k of an y th in g b u t a fu ll-° ed ged relativ istic treatm en t of b oth th e electron an d p h oton ¯ eld s. A n d in d eed , th e fu ller ram i- ¯ cation s of th e K ram ers ren orm alization id ea w ere to exten d from th at of m ass, to every con ceivab le p h y sical q u an tity : m ass, ch arge, w ave fu n ction , etc., n ot m erely in secon d ord er, b u t to every con ceivab le ord er in p ertu rb ation th eory in a closed form . D y son sh ow ed h ow

RESONANCE  47 October 2005 GENERAL  ARTICLE

to d o th is sy stem atically in term s of ap p rop riate classes of F ey n m an d iagram s [12]. A n d of cou rse th e treatm en t h ad to b e en tirely in term s of a relativistically in varian t Q F T , at th e en d of w h ich th e agreem en t b etw een th e- ory an d ex p erim en t w as 1 in 10 1 2 ! B u t th e im p etu s for th is d rive cam e from H an s B eth e's p ilot p ro ject w h ich covered ju st tw o p ages of th e P hysical R eview . I am gratefu l to F reem an D y son for m ak in g availab le to m e h is th ou gh ts [2] on H an s B eth e, p rior to p u b lication , w h ich h as h elp ed p u t several th in gs in th is p ap er in p ersp ective.