Selected Papers on Quantum Electrodynamics

H t t p r - d n , rt d ' r , p u ' . ' d * , r " ,

durl,, Hnrur6r"rr" ''' drrv,r"N,+

identisch.

" 1- dnrtrr0rrp,,drrp. "'LfnN, ur\,

(63)

Wir schreiben zur AbkiirzunE

(64) K

, , l r (p ',,...p :" *,) ! r r , 1 t 2 ,, " ,

pY :

I

Aus diesem,lt(P't,P'',..., 0'N')bildenwir - genauwie in (62)ein 4r (-lr'(A), ..., -lI'(pp) durch

W ( I { ' ( F r ) , . .f .r ,' $ ' * ) ) : r t , ( p '.' ., . , p ' ^' ) + ' lln't

(6Za)

Wir behaupten nun, dafi

p (ur'(pi),. . ,, I' (0'd): ,taP (lr' @\),. . ., N' (F'd), (66) K

wo der Operator ,fl f,J :

)

Ht|a!,a1

z, 1': 1

ist, mit den a aus (31). * Wir sstzen voraus, da8 I

wieder iiberall verschwindet, rvo nicht

N, (p\)+ ... + N, (f,r): N,

(66 a)

55 tlber das Paulische Aquivalenzverbot.

Dz+i)

Gleichung (65) ist zuniichst sicher richtig an allen Stellen (fiir alle Wertsysteme der Argumente), wo die Anzahl der 1 im Argumentsystem ootr ,l nicht eben gleich -lI' ist. Dann verschwindet nii,mlich die jinke Seite wegen (62), (62a),

und auch auf der rechten Seite stehen lauter

Nullen, da a!,.a1 die Anzahl der 1 nicht iindert. An den Stellen aber, wo etwa die

N' (pi,),N' (pi,),. . ., x'(p;,",) (also genauN) gleich 1 sind, ist /ln {r gleich .. ., \'ry,\, i (P'ur, also gleich K 2 H r r . . . d N , ; l t t . . . v y ,l b ( F r r , . . . , p ' 1 , 7 0 , )

P1 .'. plvf : r KK

p2:7

PL:7

K

... + > Hry,yp,rt,(pi,,p;r,..., p'p,*) l.lv, :

1

K

r:l

tt:t

das zweite ersieht man aus (63). Unsere Absicht ist nun, die rechte Seite von (67) durch die T auszudriicken. Zu diesemZweckebemerken wir, da8 in (67) rechts die pi schon in der richtigen Reihenfolge stehen, nur das jeweils auftretendeBi ist an der falschenStelle. Ist etwa+

FI,_,{ F',r3 Fto,, so ktjnnen wir die Reihenlolge der p' zur richtigen machen,indem wir pj, tiber clie zwischenliegentlen Fi oott der Stelle zwischen 01,_, ,od Fi, +, an die Stel-lezwischen0i;_, u"a 0r1 verschieben. Dabei multipliziert sich die antisymmetrischeFunktion { mit (-1)",,rro a clie Anzahl der zwischen den beiden angegebenen stelle' stehentlenp; ist. Wir kiinnen (67) also auch schreiben tfK

,lt(P;', . . ., Fi.t,) * Das Gleichheitszeichen kommt nicht in Frage, fu dann das entsprechende r,a doch verschwindet.

56 P. Jordan und E. Wigner'

646 u-o natiirlich

noch das Vorzeichen *

von r und p abhiingt,

aber schon

P:r,<"'{F't,<"'
Wenn wir (62) beachten,kiinnen rvir dies auch*

V (*r...rr) :

)

)

xi:l

It:o

* $ t V ( n r , . . . , f r i _t , 0 ,n i 4 r t . . . t f r t r. ,_l , q , . . . , u 6 )

+ 2 H . i i T(t n r ,. . . . n x )

(68)

,j: r

schreiben, u'ie man leicht iiberlegt. In den Summantlen nii,mlich, in (67a)' wo d, f g, ist [erstes Glied in (68)], sind dieselben Argumente vorhanden rvie an der linhen Seite, es fehlt nur pi,., was links vorhanclen war (ri : 7), clagegen ist p|, hinzugekommen, und wir k0nnen annehmen, 0), da sonst rp doch verschwinclen wiirtle. da8 es nicht da war (r7 p fzweites Glied in (68)], so si:rd rechts in rl.r dieseJ.benArgumente wie links.

fst i, :

Das Vorzeichen in (68) bestimmt sich offenbar daclurch, da.8 man 0 und 1 f oder - setzt, je nachdem zwischen der ausgeschriebenen einb gerade oder ungerade Anzahl von 1 steht. (Uber ebenso,viete p,: mu-Bte man das entsprechende Bi, hiniiberschieben.) Dies ist aber die Anzahl der 1, die links von 17 stehen, vermindert um die Anzahl iler 1, die links von f,; stehen. Obrn'ohl jetzt die Richtigkeit

der vorangehenden

Formeln

l
schon

Wir wissen,

dafi der Wertbereich der Argumente von T ingesamt 2r Stellen umta$t, inilem liir jedes der r1,: .lI'(B;) entweder f 1 oder 0 gesetzt kann. Ein darauf wirkender linearer Operator ist also eine werden. Matrix mit 2r Zeilen uncl ebensoviel Spalten. Wir bezeichnen jede Zeile oder Spalte mit -K Indizes (den or, ..., nK entsprechend), die jeweils 1 oder 0 sein kiinnen. Der Operator a1 ist entsprechend $ 3 und $ 6 zu definieren (wir :chreiben der bessoren Ubersicht halber die Indizes als Argumente) durch ay(rr, fist . . ., tKi

U , t ,U s , . . . , Ur )

:(-f)or*12+"'1'rt-r6r.n,0*rn,...drr-ry,t-rdrr.odr^rdr,*

!t).*1-..drrsr, (69)

und entsprechend ist dann at -

a f ,( n u . . . , f r K i A r , . . . , , ! r ) (- 1)"* "' * a,-r6rrar... 6*r*rur-r0",

1d'go r ,. . . 3 r r ' r .

* Wir setzenaus Bequemlichkeitsgriinden rr fiir N' (p).

(69a)

ijber das paulische Aquivalenzverbot.

647

Mit Hilfe dieser Formeln kann (68) auch so geschrieben werden ; W (nr, . . ., *x) K

.2 u;^

t, /-:t

lt..,Ag:9,1 zl. '.2K:n'7

wie man sich mit einiger Miihe iiberregenkann, was aber schwer hingeschriebenwerden kann. Damit ist (66) gewonnen. S 9. Wir haben also in $ B folgendesgesehen:Jeder antisymme_ trischen Funktion, vrelche definiert ist in den Koordinatenr?i,umen mit allen AnzaLlen ,}tr'( K 'on Dimensionen,entspricht durch (62) eine tr'unktion im neuen Raume. Es entspricht dann dem operator (60) .] V - V, ) Ys * ... t Vr fmit der Matrix jn (64\ FI_ . Koordinatenraum der f)perator ,fJ von (68) ;=;;.'; iiil;",*'# Operator ,fl ist dabej .{}:

)

H,;alo4

(66 a)

xr7:1

mit den a|, at von (69), (6ga). Es folgt hieraus, dafi einer Eigenrunktion von z eine Eigenfunktion von ,fJ entspricht' wenn wir noch zeigen kiinioen, dafi das innere Produkt zweier Funktionen im Koordinatenraum denselbenwert hat wie dasjenige der entsprechendenFunktionen im neuen -r[,-Raum, so sind lrir mit dem Beweis fertig. Jm Koordinatenraum ist PK

(tp) :

.)

r t ' $ ' ' . . .p ' * ' ) q $ ' , . . . l l ' n ' ) ,

F '1 ,. . . ,F 'N ' : F ' r v/as wegen der Antisymmetrie

(7r)

IJK

(4)E) : ergibt.

N'l

p'r,...,F,N':Fr F'I<
(73) f,l ,..r,X-:0,1

was mit Riicksicht auf (62) eben (22) ist. Wir mdchten noch bemerken,daf die q_Zabbelationen (86), (40) natiirlich sofor.taus der Irormel (6g) hervorgehen. s 10. wir miissen schiieflich operatoren betrachten, die keine Zerlegungmehr in die Gestart (60) gestatten. v.n dieser Forrn ist crie Zoitschdft fnr ?hysik.

Bd.4z.

58 P. Jordan und E. Wigner,

648

Energie eines nichtidealen Gases. Wir beschriinken rrns dabei zun?ichst auf solche, die in Teile zerlegbar sind, die jeweils nut zwei, stets verschiedene, Teilchen enthalten. Die Erledigung clieser Aulgabe ergibt Formeln fiir di"e Bosesich durch Analogisierung der entsprechenden Einsteinsche Statistih*. Der Onerator T IaBt sich dann schreiben: y-

)

Tit,

(60 b)

i, k:r j
wobei Z die Matrix Hrr,..

:

r 1 1 .; , p 1 - . . . t L N l

Erirlrruirr6rtpr'.'6,i-rt'i-'r6ri1rr.7 2 i, k:r j
Y l t( F ' ' ,. . . , F ' ' ' ) :

d17611t76+r..'d"lr,p,r', (63b) 11'..dr1-1rr7a-1

4 ,( p ' t ,. . . , 0 '* ' )

(64b)

mit Hilfe von (63b) [ebensowie (65)] zu berechnen. Dann ist wieder die ,,richtige Reihenfolge" der p auf der rechten Seite herzustellen. Dabei ktjnnen die pi stehengelassenbleiben, 01, und pj., miissen iiber eine Anzahl von Bi hiniibergeschoben werden, $'obei sich wieder das Vorzeichen ilndern kann. Beachten wir wieder (62), so kdnnen wir wiecler links und rechts fiir I bzw. r/' einsetzen {r bzw. ep. Unter Beacht,ng von (69) findet man nun, ila8 dem Operator (60 b) nunmehr der Operator 1K

t-

'

>

Hrr.r, 1.r1,a!,.ral,ra7.ra7,

(66 b)

)1, tr'2:1 zt, z2:l

entspricht. Es ist befriedigend, da8 die Riickwirkung der 'Ieilchen auf sich solbst wieder durch die nichtkornmutativen Multiplikationseigenschaften der Wellenamplituden im dreidimensionalenRaume automatisch ausgeschlossen wird. Im Bose-Einsteinschen Falle wurde clieserUmstancl deutlich gemacht durch Formel (40) der Arbeit von Jordan und Klein. DaB dieselbe Fornrel auch hier gilt, folgt aus cler leicht beweisbaren Tormel af,a1,a! a1- afra!a,*1,: 6t ta!,arr. Wir wenden uns endlich zu dem Fall von Operatoren, die aus Summandenbestehen,welchejeweils in ro) 2 feilchen symmetrischsind, * P. Jordan

und O. Klein.

a. a. O.

59 Uber das Paulische Aquivalenzverbot.

649

w?ihrend in der Summe alre diejenigen summanden auszurassensind, welche dasselbeTeilchen zweimal enthalte' (was eine wechselwirkung des Teilchensmit sich serbst bedeutenwiirde). Eine verallgemeinerung der obigen Betrachtungen fiihrt daurr zu den folgenden Fo"Jero, welche analog sind der in B angegebenenVerallgemeinerung* der trormel von Jordan und Klein: 1KK

il ^,4:r

tt"'i' nf,'"'aI"nt,"'a7'' (66c)

,,..4:r'""'f;t'i

s 11' wir kdnnen endrich crieerhaltenenErgebnissein einer elwas anderen Form so aussprechen: Es gibt bei einem paurischen Mehrkdrperproblem eine wahrscheinlichkeitsamplitude,.welche die warrrscheinlichkeit dafiir bestimmt, da6, nachdem fiir die meBbarenGrd,rjen lf (pi), N(P;), ..., N((d ein gewissesWertsystem N, (p,r),N,(p;), ..., N' (A'd gemessenworde' ist, fiir die anderen, entsprechenddefinierten G r i i f e nN ( q r ) , N ( , 1 ) , . . . , N ( q ' r ) d i e W e r t eN , ( q r ) , Ul(S;),...,li,(q,r) ge_ funden werden. Eine solche Amplitude wird erhalten durch N (q') 'b '6\'i: mN s'' (F'),

6po) :

0 fiir

> rr'(B) * F'

und

o!$iif,l!"ory,r- &'! wo,Xtw

)

f,fZ) Y'

Ear,fp)

F' Y, wohei Q(fl, W!"X aie in $ 4 besprochenenFunktionen sind. Die in A in der Form

{)a'"ulu,_w\a:o angegebeneno"r.tioo"tgreichung fiir die zum Gesamtsystem gehiirige Amplitude @ lautet nunmehr bei unserer vorzeichenbestirnmung ie. i,, @ enthaltenen Determinanten:

l) A, aiau- w] o :

o;

dieseAbanderung ist niitig, weil in A die vorzeichenzwei.eutigkeitdieser Determinantennicht ausreichendberiicksichtigt wurde; die Matrizen a, unterscheidensich, wie wir wissen, von den b, nur beziiglich der Vor_ zeichenihrer verschiedenenEremente. Es scheint sehr befriedigend,daB die fiir die Bildung des Energieausdrucks notwendige U*rri"r""_ U* * Ygl. B, tr'ormel(34).

60 P. Jordan uncl E. Wigner,

650

Grij8en a, at gleichzeitig auch zu einfachenMultiplikationsgesetzen fiihrtet wie wir in $ 6 gesehenhaben. Es sei endlich hervorgehoben,da8 die in $ 6 eriirterten Multiplikationsgesetze der gequantelten Amplituden a, (S) it Analogie zu den von Jordan untl Pauli entwickeltenrelativistischinvariantenMultiplikationsregeln des laclungsfreienelektromagnetischen Teldes leicht relativistisch verallgemeinertwerden kiinnen, so daf man die dem Paulischen AquivalenzverbotentsprechendeQuantelungtler tle Broglieschen WelIen in relati'r'istisch invarianter Form erh?ilt. Eine genauere Darlegung soll jedoch vorlaulig zuriickgeslellt werden. Z:usal,zbei der Korrekt'ur: Zwischen den 2K Operatorena' ...t ar; al,,aL ..., a! bestehendie Relationen a,21 &zaz -f atat : 't *

+

0

+

^(

(36)

\

a,;ai+a)a):uJ und

-- 6/r. ol,.o^ + qdI

(40)

Wir wollen nun zeigen,daf diese{-Zahbelationen die Operatorena, ot schoneindeutig bestimmen,wenn man sich auf irreduzible Matrizensysteme beschriinkt uncl Matrizensysteme, clie auseinander durch Ahnlichkeitstransformation hervorgehen, als nicht verschiedenvoneinander ansieht*. Um dies einzusehen,bilden wir zunii,ohstfolgende Grdfien a, aK +,

:

a,. I

a!,-,

1.

T \a'x -

)

*,1 a';).

(r)

)

Dte 2 K Matrizen a bestimmen umgek€hrt die a eindeutig' Nun gilt fiir die a, allgemein GI) a * N z * d 4 d , , 1: 2 6 1 7 . Man iiberzeugt sich z. 8., wenn % {K, a , a t l - & ] . \ t x-

1' < K ist, da8

a I )+ @ 1 + * b . @ , * a I ) -

\o, * d).@iI

26,t.

IIan kann (II) auch schreiben a?'': u.a&1:

'I -

L axa,

fiir

] ,t + 1, I

(II a)

* Dies ist tlie Transformation aller Matrizen a durch dieselbe Matrir s zu S-1aS, also natrh dem Sprachgebrauch der allgemeinen Quantenmechanik tlie kanonische Transformation der Matrizendarstellung'

6l Uber das Paulische Aquivalenzverbot.

651

was aber mit anderenworten so vier bedeutet, dag die 2 K MaLrizenu zusammenmit der }fatrix _ I eine Gruppe aufspannen. Wenn z. B. K : 2 ist, so hat diese Gruppe folgende Elemente 1;

Nt,

dz,

&8,

Nti

d1d9t

dldgt

&1d4t

-7; -w1, -d2t _.,tst-a4; -d.tot2,-&r*3t -af(',

dzd,,

d,2a44t a.a4;

I

-d,2a;, _oron, _"i"rrl " otrOl,2dgt ara,3a41t d,rd2clat a20hd4' d,lct2c(3'a4. j(m)

-

C 4 , t C l 2 d s t-

1qABA4t -

A1&gd4t

-

&gd3Ci4t -

ArA,ZABC:,4.

J

Das sind 32 Elemente, im allgemeinen 22r*1 Elemente. Das irreduzible }latrizensystem, cras (II) geniigt, ist sicher eine irreduzible Darstellung dieser Gr.ppe (umgekehrt braucht es nicht der FaIr zu sein, die rsomorphiekann ja mehrstufigsein). wir werdennur die irretluziblen Darstellungenbestimmen. Unsere Gruppe hat den Normalteiler 1, _ 1 (das Zentrum), ihre Faktorgruppe vom Grade 22K is!, abelsch. sie hat also 22r dieser Abelschen Faktorgruppe entsprechendeirreduzible DarstellunEen vom Grade 1, die auch Darstellungen der ganzen Gruppe sind. fod.*..o kommen sie fiir uns nicht in Betracht, da sie den Gleichungen rr nicht gentigen(da sie ja kommutativ sind). wie viele Klassen hat unsere Gruppe? Die beiden Eremente I und - 1 bilden je eine Klasse fiir sich, sonst ist aber iedes Element /i mit - 1 .l? in einer Klasse. Besteht n?imlich-E in (rrr) aus einer ungeradenAnzahl von laktoren, so ist aBa- t - _ 1 .-8, wenn a jn R nicht enthalten ist, rvenn -B aus einer geraden Anzahl von Faktoren besteht,ist aRa-, - - 1.J?, n,enn a in R enthalten ist. Die Aruahl der Klassen ist also 22K+r; dies ist auch die Anzahl der vdneiaander verschiedenenirreduziblen Darstelrungen. Da wir 22r Darstellungen schon kennen und diese nicht fiir uns in Betracht kommen, kann es nur eine einzige, die letzte sein, die die Gleichungen(II) befriedigt, alle anderenLtisungenvon (rr) gehendarausdurch Ahnlichkeitstransformation hervor. wir bestimmen noch die Anzahl von Zeilen und spalfen, die Dimension dieser Darsteilung. (Es mug dabei natiirli ch 2K herauskommen.) In der Tat ist der Grad der Gruppe ZzK+t gleich der summe der Quadrateder Dimensionenihrer Darsteilungen. Es haben r 22 die Dimension 1, die letzte muB die Dimension 2K haben, damit (2R)' + 22K . Lz : NzK | 1. Sie stimmt arso tatsiichrich mit unserern Matrizensystem(69) oder $ B uncl 6 iiberein. G o t tin g en, fnslitut iiir theoretischephvsik.

62

P o p e r5

SITZUNGVOM 23.JULI 1934.

iJber die mit der Entstehung von Materie aus Strahlung verkniipften Ladungsschwankungen Von W. Heisenberg

Die Diracsche Theorier) des Positrons hat gezeigt,daB Materie aus Strahlung entstehen kann, inilem z. B. ein Lichtquant sich in ein negatives urrd ein positives Elektron verwanclelt. Dieses auch experimentell bestii,tigte Ergebnis hat zur X'olge, daB iiberall dort, wo zur Messung eines physikalischen Sachverhalts gro8e elektromagnetische X'elder beniitigt werd.en, mit einer bisher nicht beachteten Stiirung des Beobachtungsobjel:tesclurch das Beobachtungsmittel gerechaet werden muB, niimlich mit der Erzeugung von Materie clurch den MeBapparat. So gering diese Stiirung fiir die iiblichen Experimente auch sein mag, ihre Beriicksichtigung ist fiir das Verstdnalnis der Theorie tles Positrons von prinzipieller Becleutung. Zu ilrer nbheren Untersuchung in einem sehr einJachenX'all betrachten wir ein quanten-mechauischesSystem von foeien negativen Elektronen, die sich gegenseitignicht merklich beeinflussen; ihre Anzahl pro ccm sei f;. In einem Volumen o(nt4V\

wird dann im Mittel die Ladung

e:-il+

(1)

zu finden sein (e ist der Absolutbetrag cler Elementarladung). X'iir clasmittlere Schwankungsquatlrat der Ladung im VqtrFmeno wiirile man nach der klassischen Statistik den Wert

n':"'n j,

p)

erwarten. Es soll nun gezeigt werden, claB sich nach tler Diracschen Theorie im allgemeinen ein griiBeres Schwankungsquailrat ergibt. Der Uberschu,E gegeniiber Gl. (2) ist auf ilie mtigliche Entstehung von Materie bei der Messung der Ladung im Volumen o zuriickzufiihren. The principles of Quantum mechanids,. p. 265. Oxforcl 1930. Proc. 1) P. A. Dirac, Cambr. Phil. Soc. 30, f50, f934.

63 318

W. Heisenberg:

Die Eigenfunktionender Elektronen hiingen vom ort t, der zeit f und der Spinvariable o ab, sie sollen ot, (t., t, o) hei8en und in einem Volumen Z (v ) a) normiert sein. Die allgemeinewellenfunktion der Materie wird dann ry("t.,t,o):1o,u^(r,t,oy,

(B)

wobei wegendes PaulischenAusschlieBungsprinzips die V.R.

ala*{ a*fi:6n*

(4)

gelten' Darausfolgt don:Nn;

a*io:r-^r,.

(b)

n'iir die Zustiindenegativer Energie (E* < o) soll noch eingefilhrt werden:

NL:I-N*.

(6)

Es bedeutetdann 1[, die Anzahl der Elektronen im Zustand n, Ni die Anzahl der Positronen im Zustand zi,. n'flr die Ladungsdichte ergibt sich nach der Diracschen Theorie der folgenile Ausdruckl) :

-'4]r|{"d",-uZn;")"*+

)aia*uiu*)'

F)

Fiir die Ladung e im Volumen o erhdlt man daher

e:-e["3J" i,Z{#

(s) +-Zr -"Iw 7 {awiQto)u*('dl'

Aus Griinclen,die von Bohr und Rosenfeldz) ausfiilrlich diskutiert wurden, soll der zeitliche Mittelwert von e iiber ein endliches Intervall ? betrachtet werden, wobei auch die Grenzendes Intervalls eventuell noch unscharf gelassen werden. Wir fiilren daher eine X'unl:tion l(t) en, die nur im Bereich O
I l(t)dt:r . Auch die Grenzen des volumens o sollen eventuell unscharf bleiben; es soll daher g(t) eine X'unktion von r beileuten, die in o bis auf die Umgebung der Grenzen l ist, au8en verschwindet und fiir dte Der entsprefd,rg(r):a. chenclezeitliche und riiumlicho Mittelwert der Laduns e wiril dann L) Ygl. z. B. W. H. X'urry u. J. R. Oppenheimer, phys. Rev. 4b,245, lgg4. 2) N.3ohr und L. R,osenfeld, Verh,der Kgl, Dbn. Gesellsch. d. IViss.XII, 8, f 9$.

64 tlber

die mit der Entstehung

319

von Materie usw.

a: - e1,7, " i -,2 ; + +p*,i, *| | at dnZ t(t)s(t)u)(rto)u* (rt'1]'1oI { ! Bildet man nun den Erwartungswert z von e fiir denjenigenZustand, bei dem die Anzahlen y'f, bekannte c-Zahlen'sind, so erhiilt man

(ro)

i:_,18,*"_e,*4u.

in Ubereinstimmungmit Gl. (1). X'iir den Erwartungswert des Schwankungsquadrats ergibt sich jedoch:

(Tf:7-t:ezZ

ZT"fr;^/

n+m n'+n'

d6'

fat[atlar

o')u'*'(r' t' o') J'dr'l(t) l(t') s O) c U')"*(tt') u*(rto) u,\(t't'

: e Z N"(1-n ) -> [ d'tI dt'I d'r ao' n+m

(11)

o) I dr' t(t)| (t') c Q)s Q'),|Ato) un(t' t' o')ufi(r't' o')u* (r t :ezpN"(l-N*)J*^. Diesen Ausdruck kann man in drei Teile zerlegen. Der erste w6re bereits vorhanden, wenn die Erwartungswerte der If* filt Er) 0 untt N" lidrtEn < 0 alle verschwinden,d. h. im Vakuum (man beachte Jn*: J**): e2

(12)

E">OtEn
Ein zweiter Teil wird, wenn nur negative Ele}:bronenvorhanden sind (N' : 0) :

*F-,*^(F,;F^)t"*' SchlieBlich bleibt noch als dritter Teil -a2 N^N*J,* /

(13)

(14)

E',)>Oi E6>O

iibrig. Dieser dritte Teil enthii,lt jedoch, wie man aus (11) berechnet, den n'aktor (1)" t u, kann daher gegeniiberclen beid.enerstenvernachliissigtwerden, wenn - wie angenommenwurde - t:
" t \ ,

/ > - 2 1 " ! - , ( t ' t "" 'o')u*(t.to) \aSo n-*
(15)

-(r + *#-) = +t(t-"'o'+:*") iVn-'''-o"('-', "*[na-n+'*a''\7. "B

65 320

W' Ileisenbers:

Es wiril also

s$)se)I #41 ('-a,-,ero*:AlilfMJarlar'to tQ) . u | , 1 r t o 1 u " ! r ' t ' 4[:*

>1{/r -drq! +y)lj(po-pt)l' J k"# z r\-

- (r + L!+!:) Po wobei

| | @o-t eu l, | | c $ - p")l, bL@)b*(o,),

'ao'

f -

4p"ot

l(Pr): J itt f(q eh' rn s $) :J dt g 1r)eie' h" " - oT') un(r t o) : b^(o) 'Wellengleichung "* gesetztist. Aus der folgt dann weiter y\./ / t: ! [ a p l t ( . , - p -T)lf@o-fr),' Tpi+(mc\z\ ,t,)dnn:r J-p ti\t + \"2^-, 'wenn

- (r-"';!*i*'' rrro, _F*) tz. ) + fi)r|tcw

(16)

,r'

die zum Zustancl ro gehdrige wellenliinge k]ein ist gegen die rii,umliche Ausdehnung des Gebiets o und wenn ferner d.ie zeit, iber die gemittelt werden soll, so Hein ist, daB die Elektronen in dieser zeit n'w strecken durchlaufen, die ebeufalls klein sind im verhii,ltnis zur rii,umlichen Ausdehnu:rg von o, so ist g(p-pft) als sehr schnell veriinderlich gegeniiberdem Bruch

4+#v

anzusehen,ferner kannindieserA:rnii,herunSll{1po--pt)12-l gesetztwerden. Dann wircl

("3;#,)r*** +l*tc(F-F")t': | | a's@* ?, (u) und fiir den zweiten Teil des Schwankungsquadratserhblt man:

*(A,'4'+,

\/ in Ubereinsti-mung mit Gl. (2). Hierzu kommt nun noch der erste Teil, der auch im Vakuum auftritt:

'-*Als' Ahnncnwie in Gl. (tb) findet man:

LIBRARY JARROTLCOLLEGE HHSNA, ilt0NTAl'lA 5$ffil

(re)

66 Uber die mit der Entstehuns von Materie usw.

32\

- "'o'r-u* ", drId,r Id,r' ) |e)|Q')g(r)se')l'#i H'01, 2 ! "y=I^drf

En>>oi En<.o

d

(r *y2* \-

'

f--!\ olun-n<'-,t -@"+pbQ-t',1

p'o

u*-#ot! : l dtldtldldr'l (t)l (t')s(r)s(tI# l#

)-

- o'tr'-''t- @'+ p6)G-t')l "|xv

,,," fdb fdb , ,, --JPJPtt\?o-r?o)t"'.lq(b_b,) ppo?o-PP'_-m2cz . " popo scharfumgrenzten Setztman9(t) : 1 in einemrechteckigen, Yolumenmit den SeitenlL,I,z,l,s u. au8erhalb Null, sowird /i

\/l

n L L r, d to'n* onu+pzz)- ( "t; 1-l e$): I d,s(r)ei-: k. Ih,io,""n l( +!-l

\/t

\

er)

\f4! hP" t\ /\ eA*6**zg:"ffi!)# "',,,,0, \r/ I

prrpr;pZ"---

2h

-'*

2h ""'

Zh'

Begrenzt man dag Zeitinterwall scharf, so wiril lI

d^.-

f(pi:*[ur,;,""'-#i

-p"cT

"m. 1(po\lz:a

p2)

man ai"w*tl 1zr serzt das rntegrar ,,"u rrri'i,l,.,r0, "r, ." u,Jrf,"J

auf der rechten Seite von Gl. (20). Man erkennt dies am einfachsten,inclem : S einfii}rt; es man als neue Integrationsvariabeln .p-.p' : f und t+{ -// tz \t tz wird dann: -($I1z- Pa4 l -mzcz + nx'a'l yl,' \ l' S' + + " 4 4 |

o* P')t''Ic$)t' ' -4 {;:; J#"[#r|@

.

tl@T.;q4''+

Bei eirrem vorgegebenengroBen W'ert von f (fr > rnn) eryibt die fntegration zu I g(f )2| ; iiber $, clie konvergiert, einen X'akborder ungefrihren GrABe Vf Ty dann wesentlichen iiber fiihrt im auf das fntegral die Integration f

dtc" ++ I q#,rq dh,dku "i"' "*'

fu'lz u' !z!,

das divergiert. Das Schwankungsquadratvon 7 wiril also unendlich gro8, werul man das Raum-Zeitgebiet, iiber das man die Mittelung der Laclung ausf.iiihrt, scharf begrenzt. Dieses Ergebnis entspricht den Resultaten, die sich bei der Untersuchu:rg der Energieschwankuagenin einem Strahlungsfelil herausgestellt habenl). Auch dort erwies es sich als notwendig, das Raum1) Vgl. W. Heisonborg, Verh. d. S6ohe.Ak. 83, 3, 1931,

(23)

67 322

W. Heisenberg:Uber dio mit der Entstehungvon Materieusw.

q:bt:t,_- dem_dieBnergieau{gesuchtwerd.ensoll, unscharf zu begrenzen. X,iir $1 ladungsschwankungen {er Gl. (20) geniigt es, entweder die Grenzen des .Weise Zeitintervalls oder die des Ra,uminf"r-ril]* ir geeigneter unscharf zu wiihlen. Nehmen wir z.B. an, d.agdie Griige g(r) in einem Gebietd.erBreite b umgebung des Randes von o etwa nach de, Art einer Gaugschen T 1* Fehlerfunktion von 1 auf 0 abnimmt. Dann verschwindet g(p) fiir Werte von 1o) ebenfalls wie die Fehrerfunktion; nimmt u man weiter an, daB fie Zeit T klem sui gegen-!, , so kann man in geniigender Approximation den Bruch in (23) entwickeln ftir /c2q *r"". Es + ergibt F, sich dann

|cG)t' ,3,, {:z; I # f# |t QP)12

2(P2+mzc2)'

Bis auf unwesentliche konstante X'aktoren wird daraus I rd.r'

F ;4 J ;l s (t)l 't'

7

f .Lt

4;19:61fi1t61't'

(24)

tu,T<.fr, TfuT}J-.

nz c"

n'iir die GrtiBenordmrng des erstenTeilesdes schwankungsquadrats erhdrt / man daher(tii" tr- tr-h:li),

Ag__L I,t:,,4 @d,-

-2

rurr
" <',i##;t

*," r> ,=- I ftt {*

P5)

Der x'aktor tfr zeigthier deutlich, dag es sich bei diesen schwankungenum . 'wi,nden, einen oberfliicheneffekt handert, der davon herriihrt, daB an den die das vorgegebenevolumen o abgrenzen,Materie entstehen kann. Er wird um so grd8er, je kleiner die zeit gewiihlt wird, in der d.ie Messungder Ladung vorgenommenwerden soll'nd je schri,rferdie Begrenzung des iolumens ist. Diese Ergebnissediirften eng zusammenhdngenmit dem umstand, daB die fiir die Erzeugung von Materie maBgebendeInhomogenitiit der Diracschen Gleichung die zweiten Abreitungen der elektromagnetischen n'eldsttirken ent_ hiilt, daB also eine unstetigkeit im ergten Differentiarquotienten der x,eldstii,rken bereits zur Entstehung von unendlich viel niaterie ,toUn g"m' kiinnte. Zusammenfassend sei festgestelt, daB bei der Messungder Ladung in einem vorgegebenen Raum-Zeitgebietschwankungenauftretuo, ai. n d.erliiassischen Theorie kein Analogon besitzen; sie werden verursacht durch die Materie, die an der oberflii,chedes betrachteten Raumgebietsbei der Mess'ng entsteht.

P o p e r6

68

On the Self-Energy and the Electromagnetic Field of the Electron V. F. Werssxoen Uniaersity of Rochester, Rochester, New York (Received April 12, 1939) The charge distribution, the electromagnetic field and the self-energy of an electron are investigated. It is found that, as a result of Dirac's positron theory, the charge and the magnetic dipole of the electron are extended over a finite region; the contributions of the spin and of the fluctuations of the radiation field to the self-energy are analyzed, and the reasons that the self-energy is only

logarithmically infinite in positron theory are given. It is proved that the latter result holds to every approximation in an expansion of the self-energy in powers of ez/hc. The self-energy of charged particles obeying Bose statistics is found to be quadratically divergent. Some evidence is given that the "critical length" of positron theory is as (-hc/e2). smallask/(mc).exp

I. IN:rnoouctroN AND DrscussroNs or' REsur-ts -I-HE self-energyof the electron is its total I energy in free space when isolated from other particles or light quanta. It is given by the expression

(b) The quantum theory of the relativistic electron attributes a magnetic moment to the electron, so that an elbctron at rest is surrounded by a magnetic field. The energy

f

W:T-l(1/8") J

| (H'zlE'z)d.r.

u^,s:(r/81)r r?dt

(1)

Here I is the kinetic energy of the electron; .FI and E are the magnetic and electric field strengths. In classical electrodynamics the selfenergy of an electron of radius @ at rest and without spin is given by W-mczle2f a and consists solely of the energy of the rest mass and of its electrostatic field, This expression diverges linearly for an infinitely small radius. If the electron is in motion, other terms appear representing the energy produced by the magnetic field of the mcving electron. These terms, of course,can be obtained by aLorentz transformation of the former expression. The quantum theory of the electron has put the problem of the self-energy in a criticalstate. There are three reasonsfor this: (a) Quantum kinematics shows that the radius of the electron must be assumedto be zero. It is easily proved that the product of the charge densities at two difierent points, p(r-{/2) XpG*t/2), is a delta-function d6(t). In other words: if ooe electron alone is present, the probability of finding a charge density simultaneously at two different points is zero for every finite distance between the points. Thus the energy of the electrostatic field is infinite as

of this field is computed in Section III and the result is '

. U^^e: a2h2 f (6rm2c2a3)

This correspondsto the field energy of a magnetic dipole of the moment eh/2mc which is spread over a volume of 'the dimensions a. The spin, however, does not only produce a magnetic field, it also gives rise to an altern4ting electric field. The closer analysis of the Dirac wave equation has shownl that the magnetic moment of the spin is produced by an irregular circular fluctuation movement (Zitterbewegung) of the electron which is superimposed"tothe translatory motion. The instantaneous value of the velocity is always found to be c. It must be expected that this motion will also create an alternating electric field. The existence of this field is demonstrated in Section III by the computation of the expression g"r:(t/8r)

f E"rdr.

There E, is the solenoidalpart (div. Er: 0) of the etectric field strength created by the electron. The fact that the above expression does not vanish for an electron at rest proves the existence r E. Schroedinger,Berl. Ber. 1930'418 (1930).

W"r:limt"-or-z /a,

72

69 ELECTROMAGNETIC

FIELD

OF

THE

ELECTRON

73

:: a solenoidal field2 apart from the irrotational =.ectric field of the charge. The energies of the .-ectric and magnetic fields of the spin are found :t be equal. The spin movement does not, of : rurse', give rise to a radiation. The time average : the Poynting vector is zero. The electromagnetic figld of the spin does not :tntribute to the self-energy of the electron. It is :iown in Section IV, that the charge dependent :art of the self-energy to a fi_rst approximation is iven by

has been proveds only for the first approximation of the self-energy expanded in powers of. e2/hc. However it will be shown in Section VI that the divergenceis logarithmic in every approximation. The maln purpose of this paper is to show the physical significance of the logarithmic divergence and to demonstrate the reasons of its occurrence. Let us .consider the case of one electron embedded in the vacuum as described by the positron theory. The vacuum is represented by the state in which all negative energy states are filled with electrons. The charge density of these "vacuum electrons" is not observable in the un1 perturbed state of a fi_eld-freevacuum. Ho*tver, iJere p and i are the charge and current densities, the differences between the actual density and c and A are the scalar and vector potential, rethe unperturbed density are observable. spectively. If the self-energy is expressed in terms The presence of an electron in the vacuum ,'f the field energres, the electric and magnetic causesa considerablechangein the distribution :arts have opposite signs,s so that the contribuof the vacuum electrons becauseof a oeculiar :ions of the electric and of the magnetic fields of effect of the Pauli exclusion principle. According :he spin cancel one another. to this principle it is impossibleto find two or (c) The quantum theory of the electromagnetic more electronsin a single cell of a volume lr3in :eld postulates the existence of field strength the phase space. If two electrons of equal spin luctuations in empty space. These give rise to an are brought together to a small distance d, their additional energy, which diverges more strongly momentum differencemust be at least h/d,.This rhan the electrostatic self-energy. The following effect is similar to a repulsive force which causes rmde calculation may demonstrate how this partwo particles with equal spin not to be found :icular part of [he self-energy arises: L.i us closertogether than approximately onede Broglie consider an electron with radius a. The field waveJength. fluctuations in a volume aB are of the order As a consequence of this we find at the position B-hc/aa.a The mean frequency of the fluctuaof the electrona "hole" in the distribution of the vacuum electrons which completely compensates tions is v-c/a. This field induces the electron to its charge.But we also find around the electron perform vibrations with an amplitude x-eE/mv2 a cloud of higher charge density coming from the and an energy Wno"t-ezEz/my2-e2h/mca2. This displaced electrons, which must be found one energy diverges quddratically for infinitely small radius. The exact value is calculated in Section wave-length from the original electron. The total effect is a broadening of the charge of the electron IV and is tl4r"ci:lim( rntca.2. ":srezhf over a region of the order hfmc as it is indicated A new situation is created by Dirac's theory of schematicallyin Fig. 1. The product p(t-t/Z) the positron: The self-energy diverges only logaXp(rIE/2) is no longerzero for a finite distance rithmically with infinitely small radius. This fact f, and is given by the function

w,: +f (,0-j' ^)',:*/

(E,- rr,)dr

2 A solenoidal electric field is.neces*rily an atternating field for its time average vanishes in a stationary state] whereas.the time average of a magnetic field d6es not vansn lI slatlonary currents are present. 3 This at first sight unfamiliar result is connected with the well-known_ fact_t_hat a system of steady currents increases its magnetic field energy if it performi mechanical work. whe.reas.a systm oJ ciarges decreases its field energy by perrormlng mecnanlcal worl, a The fluctuations are of the order of maqnitude of the field-strength of one light quantum with waieJength.

,i mcl} -l{oct>(imcl/h) G(0 : e"= . h l0l2r (SectionII). Hels lJgrrr(*)is the Hankel function of first kind. G({) has still a quadratic singularity zeits'f' Phvsik8e,27 (1934);s0,817 (risYl.w"b"kopt,

70 74

V.

F.

WEISSKOPF

for {: g. It is shown quantitatively in SectionII, that this broadening of the charge distribution is just sufficient to reduce the electrostatic selfenergy to a logarithmically divergent expression. The broadening effect also changes the magnetic field distribution of the spin moment. In positron theory the magnetic field energy is given by Frc. la. Schematic charge distribution of the electron,

(2rmca2) U-,*: litn 1,-o1le2h/ - ezmc/(4rh)'ls @/mca)f.

(2)

This is equal to the field energy of a momentum distribution spread over a finite region, which is proportional to the spread of charge described above. The divergence, which is less strong than in the one-electron theory,6 comes from the quadratic singularity of the distribution. The electric field energy of the spin, however, is not equal to the magnetic field energy becauseof the following effect, which is again based upon the exclusion principle. The vacuum electrons which are found in the neighborhood of the original electron, fluctuate with a phaseopposite to the phase of the fluctuations of the original electron. This phase relation, applied to the circular fluctuation of the spin, decreasesits total electric field by means of interference, but does not change the magnetic field of the spins since the latter is due to circular currents and is not dependent on the phase of the circular motion. Thus the total solenoidal electric field energy is reduced by interference if an electron is added to the vacuum. The electric field energy U"r of an electron in positron theory is therefore negati,aesince it is the difference between the field energy of the vacuum plus one electron, and the energy of the vacuum alone. The exact calculations of Section III give Uer:- [/*,". Thus the contribution of the spin to the self-energy does not vanish in positron theory and is by Eq. (2)

Frc.

1b. Schematic charge distribution of the vacuurn electrons in the neighborhood of an electron.

the action of the electromagnetic field fluctuations upon the electron. The efiect of an external field upon an electron in positron theory is to a first approximation the same as one expects for an electron with infinitely small radius, since the effect of the field upon the displaced vacuum electrons can be neglected. For instance, no destructive interference effect would occur in the interaction with a light wave whose wave-length is smallerthan h/mc.The exclusionprinciple does not alter the interaction of an electron with the field as long as one considers that action to a first approximation to be the sum of independent actions at every point; it has only an efiect on the probability of finding one particle in the neighborhood of another. The energy Wno"tin positron theory is therefore not difierent from the same quantity in one electron theory as shown in Section IV. In the former theory, however, it is balanced by the spin energy lV"n the most strongly divergent terms of which are just oppositely equal 1e fi/n""1. The sum of W"o and l/1,,"t is only logarithmically di(rmcaz) W : - 2 U-,"- - lim
71 ELECTROMAGNDTIC

FIELD

r::c in the one-electrontheory, is negative and ::adratically divergent in the positron theory. l-:.rs is because of the negative contribution of .-,: magnetic field and the interference efiect of '---:electric field of the vacuum electrons. c) The energy l71ru"1of forced vibrations -:der the influence of the zero-point fluctuations :i the radiation field. The energies (b) and (c) r-mpensateeach other to a logarithmic term. It is interesting to apply similar considerations :: the scalartheory of particlesobeying the Bose ':atistics, as has been developed by Pauli and .:e author.? Here the probability of finding two :qual particles closer than their waveJengths is :rger than at longer distances. The effect on the ':lf-energy is therefore just the opposite. The .:.fluenceof the particle on the vacuum causesa ::gher singularity in the charge distribution :.stead of the hole which balanced the original ::large in the previous considerations.It is shown .: SectionV that this giv.esrise to a quadratically :rvergent energy of the Coulomb field of the :article. Thus the situation here is even worse ,:ran in the classical theory. The spin term :rviously does not appear and the energy lVlo"t ,s exactly equal to its value for a Fermi particle. A few remarks might be added about the :ossiblesignificanceof the logarithmic divergence f the self-energy for the theory of the electron. k is proved in Section VI that every term in the of the self-energy in powers of e2/hc "xpansion

w:Ew(")

(3)

liverges logarithmically with infinitely small electron radius and is approximately given by

OF

THE

ELECTRON

75

which is about 10-58 times smaller than the classical electron radius. The "critical length" of the positron theory is thus infinitely smaller than usually assumed. The situation is, however, entirely different for a particle with Bose statistics. Even the Coulombian part of the self-energydiverges to a first approximation as W"t-e2h/(mca2) and requires a much larger critical length that is a:(hc/e2)-t'ft/(mc), to keep it of the order of magnitude of rncz. This may indicate that a theory of particles obeying Bose statistics must involve new features at this critical length, or at energiescorresponding to this length; whereas a theory of particles obeying the exclusion principle is probably consistent down to much smaller lengths or up to much higher energies. II. TUB Cnencp DrsrnrsurroN on rgo ElpcrnoN The charge distribution in the neighborhood of an electron can be determined from the expression f

G({): I p(r-E/z)p(ttt/2)dr; J

(4)

here p(r) is the charge density at the point r. G({) is the probability of finding chargesimultaneously at two points in a distance {. If ap. plied to a situation in which one electron alone is present, direct information can be drawn from this expressionconcerningthe chargedistribution in the electron itself. The charge density is given by

W o) - z &nc2(e2/ hc)"llg (h/ mca)ft, ta n.

p(r):el**(r)'l'(r)l -",

Here the a" are dimensionless constants which cannot easily be computed. It is therefore not sure, whether the series (3) converges even for nnite o, but it is highly probable that it converges if 6:e'z/(hc).le (h/mca)(1. One then would get W:mc2O(6)where 0(6):1 for a value of 6<1. We then can define an electron radius in the same way as the classical radits ezfmc2is defined, by putting the self-energyequal to mc2.One obtains then roughly a value a-h/(mc).exp (-hc/e,) -l W. Fifi and V. Weisskopf, Helv. Phys. Acta7, 709 (1934).

where ry'(r), the wave function, is a spinor with four componentsrltr,p:1,2,3,4. We write

(5)

1

l,l'*tl:D{,*,1, for the scalar product of two spinors. a is the charge density of the unperturbed electrons in the negative energy states which is to be subtracted in the positron theory. In the oneelectron theory o is zero. The wave function ry' can be expanded in wave functions 9o of the

72 76

V.

F.

WEISSKOPF

momenta p of the states,the following result:

stationarystatesg of a freeelectron: r!(r):la,e

nQ).

relation holds for the 9o

The following

{e*oQ)eo$)l:11Y'

r e x Di ( {-' o- ) l h :e26(t). (11) GG):r,lap-' grzPt J

(6)

Thus in the one-electrontheory, G(4) is equal to

( 7 ) the 6-function. We now apply (10) to the vacuum of the

where 7 is the total volume of the system.We positron theory, that is, we set

denote functions with positive energy values by gao and with negative energy values by q-o. We apply the method of quantized waves and consider the ry''sas operators acting on eigenfunctions c(..'No" ') whose variablesare the numbers .lIo of electrons in different states q. If the ry''sare written in the form (6), the o's are operatorswhich fulfill the well-known relations: an*ao:Nq,'anao*:1-19o,

a o*&o'a o'*6o , :Nq(1

(g)

-Nq')'

A1l other combinationsdo not contribute to the expectationvalue G(!) of G(t) becausethey have no diagonal elements.We obtain then G(0 : t'L X,N oNn,* ezL trrq(l - Nq,) qq'qq'

X | { p n * ( r ' ) p n , ( r 'I)l , p o , x ( rpr )o f t z ) l d r -2oelN o!o2V.

f

G,*(0=e'L I

ls -s' J

f { p - 0 , * ( t ' ) p r o ( "|' ) X\p+o*?)

p-o'(dlat.

The fact that this expression is different from zero and even infinite in the vacuum is closely connected with the charge fluctuations of the empty space which have been investigated by Heisenberg and Oppenheimer.s Heisenberg has shown that the charge fluctuations are infinite if the region in which they are measured is sharply limited. This result is due to the electron pairs produced when the charge is measured in a sharply defined region. We are at present interested in the expression G({) corresponding to the charge distribution of one electron. This can be obtained by calculating G*+r(€) for the state in which one electron in the state f qois present (1/"" : 1 all other N+o : 0, and by subtracting the effect of the ff-q:l),

^ t

"> [

t:a*".

( 1 0 ) vacuum G"*(€):

Here and in the following formulas we put fr: r- 1/2, rz:r*{/2. We first apply this expression to a single electron. We then put o:0, and Nn":1 1ot q:qs, Nr:O forg*qo: C
N-o:1,

It is easily seen,that the first, the third and the fourth term cancel each other' The terms \ remaining.give

(8)

We now insert (6) into (5) and (5) into (4) and keep only terms which contain the products of two a's of the form (8) or the following combinations of four o's: a q+0.na o'*6 n': N nN o' ,

ff+o:0,

| * q (r1),pq(rt) \ { p o*Q,) p o"(r,) \ i].r. "*

This expression can be evaluated by inserting the wave functions of a free electron. If qo is the state of an electron at rest, one obtains after replacing the sum over q by an integral over the

" , : t'( El

uvacr

r \ 1.,

-

u soc\ l/ :

e qlt)\ l,p,*(rz),pq"(r,r)\d'r. \,pa,x(rt) I "-L) If one inserts the actual solutions 9o of Dirac's wave equation of the free electron, this expression can be readily evaluated. One obtains after replacing the sum by an integral as before: ,'(L*

exPi(E'P)/h f (12) G ( t ) : e ' m c ' l o -p * t J 8n3h3E(?) -l Akad'86,317(1934); W. HJr"nb"re,Verh.d. Siichs.

J . R . O p p e n h e i m F i ,e l y s ' R e v . 4 7 , 1 4 4 ( 1 9 3 4 ) .

73 ELECTROMAGNETIC

FIELD

here E(.p1:6,1f*m"c'1t. This integral can be evaluatedand gives: mcl6 i limc \ G(il:e'z= . Ilot')l -'il. ht|tZn \ll / Hyor(tc)is the Hankel function of first kind; this function has a logarithmic singularity for r:0 and falls off exponentially for r))1. We obtain thus

i*t

G t):1 " (\s/

mc t lez -. f or {41hfmc lh Ez l4r2 I ..-.". 2 / m""c\' le'( | (h/zrLrnce)r' e-ncEttt \h/ | |

OF

'

forwhich

77

w * : i JfG\t) ^ot Tfi- quadratic ^,,.n,.ri^ .i--,,r..i+., The singularity of G(€) at €:0 gives rise to a logarithmic divergenceof 17"1.By substituting (12) and by performing the integration over f first' we obtain the result ) ,'n

4n2J

This expression replaces the delta-function of the one-electron theory and indicates a spread of charge over a finite region of the order of h/mc. It is of interest to construct a charge density i(z)

ELECTRON

and gives rise to transitions from any occupied state q' to any unoccupied state q. These transitions take place quite independently of the ratio of h/k to the linear dimensions h/mc of. the spread of charge. The energy W* of the electrostatic field can be calculaleddirectly from G(6;:

for l>)h/mc. W*: -l

L

THE

dP-

mc2

-

hrE
-li^*-.rL*r"WPjg'+y2 rhc

(13) lnc

**''* ii"oxoo,l,",*Lf";3/,.'h:Hn:J,f""?*":n

f a{,,)it)or--"{t).

ezh Wu1- lim 1a:o1-tm2c2lg -, TnC tnco

This density is given by _

f _ / mcz1l expi(l.p) /h

o:'J oP\E@))

u*

IIL

'

and for lllc : fi-s-2-stzr-312r-6t2 IL

r<
'

tor r))hf mc, ! falls off exponentially. In order to show that this "spread of charge" does not reduce the effect of a periodical field rvith a short wave-length, let us consider the operator f

J

I P(r) e*P ik't d'r,

rvhich represents an interaction energy between the chargeand a field of wave number t. By inserting (5), this operator can be written in the form elao*aq,, q

Po,:Po*k

TnB Er,pcrnoMAcNErrc FrBr,o or rue ELecrnoN

We calculate in this section the solenoidalpart, E" and H, of the electromagnetic field produced by the electron. It is given by 1 AA"', E": __ __, H: curl Ar. cOt Here A"' is the solenoidal part of the vector potential A which is given by I r i ( r ' , l - l r - r ' l ' '/ c'\U t ' . A ' ( r ,l ) : - , cJ lr_r,l .4 is primed to indicate that this field is produced by the electron. The current density i is defined by i,: ecV*o"9\ ,

etc_

Here cu;are the well-known Dirac matrices. We consider in our approximation the wave func-

74 78

F.

V.

WEISSKOPF

tions ry'to be the solutions of the wave equations of the field-free electron. If we expand the wave functions according to (6) we obtain E La o*ao'\9o*(r't')a,9o,(r't')l aq' A,'(r,t):, I dr'

tt:t-' .c

lr-r'I

lr-r'l

The wave functions

9o of the free electron are

1l eo:--tuoexp-(ipo-r-iEot). vth

(14)

This divergent expressioncorrespondsto the field ehergy of a magnetic dipole density concentrated in a sphereof infinitely small radius a. If one now calculated expression (16) for the vacuum of the positron theory (ff*o:O, N-a:1) one obtains a highly divergent expressionwhich representsthe magnetic field energy U"**(Vac.) produced by the current fluctuations of the vacuum. We are interested in the 6eld energy of one electron at rest which is obtained by calculating I/-,*(Vac.*1) for the state (No.:1, Po,:Q, Eq":mc2 all other i[+o:0, ff-c: 1) and by subtracting the effect of the vacuum: f-*":

po and Eo are momentum and energy of the state q, I/ is the volume of the space considered, ao is a normalized spinor. We obtain then for .C,' 2rehzc2

A,(r,t):-22 V

X e x p i [ ( p o ,- p o ) ' r -

(Eo,- E

:2re'zhz(Z-I)hq +q

-q

{ u q " * a" u , l \ u o * a " u q o l mz(E o-m2ca)z

Averaging over the spin directions gives:

lu o*a"u o'\a o*a o, a a'EoP4-ry'cn-c'Po

-I/-,*(Vac.*1) - U*,"(Vac.)

'Po'

(15)

")t7/h. The field strengths can immediately be computed from this expression. The time average of the magnetic field energy, U-,* is found to be

- *TT vmes-'

:-

e'h' f |

dp

* J aoth"E(b)

I e2 -tim(p__r 2rmch

mzcz P*Pol | ls l. I PPo-2 mc J L

(18)

Here Po is definedby Pq: (P2fmzcz)t.l\is quadratically divergent expression is just what one would expect for the field energy of the magnetic 1f density of the electron if this density is dipole :-e'h'caLL(Po - Po,)','No(1-ffq,) equal to the charge distribution calculated in qc' 2 the previous section. In order to show directly, that this magnetic lu o*a"u o'l lu o'*o"uol (16) X field is equal to the "static" field of the magnetic lE oE o' mzca czpo. p 4f' moment of the spin, we consider the magnetic if one usesthe relations (9). a, is the component polarization (dipole density) M, of c'which is perpendicularto pq- pq,.We apply eh this expression first to a single electron at rest M":_[l/*poro"9l , 2mc ( / f o : 1 , g : q o ; N o : O , q + q o )|

u*,"=rl/Gu rt A'126,

o [uoo*a"uolluo*a"uqol tl^"t:-e2h2Lf o'-' ?-m2(Eo'm2ca)2 2 One obtains after averaging over the spin directionsand replacing the sum by an integral:

and calculate the function

ilr) : I wt, - t/ 2).N{(r* E/ z)d'r,

which correspondsto G(f) (see (4)) and which provides information about the "sprn distribu1 e2h2 e2h2rdp in the electron. If one evaluatesthis intetion" ( 1 7 ) u^"":"*nJ g " w : r " l i m 1 ' : o r3 * r r r o " gral by the method which is used in Section II,

( I

75 ELECTROMAGNETIC

FIELD

::1e obtains

OF

THE

79

IV. Tus SsLr-ENBncy oF THE Er,pcrnor 3h2

r(f):--G(s), 4 tn2c2

o'hich shows0 that the spin is distributed in :ractly the same v/ay as the charge. The mag:etic field energy of this distribution is given by div, M(rr) div. M(rz) U^ru

trl

::rd can be evaluated by the methods used for the :iher calcnlations in this section and leads to the :spressions (17) for the one-electron theory and :c (18) for the positron theory. Hence we are :.llowed to consider them as the field energy of :he magnetic moment. The energy of the solenoidal electric 6eld itrength is given by

The self-energy is calculated in this section by means of a method which is different from the usual perturbation method, in order to outline the physical significance of the difierent terms. It is similar to the method applied to this problem previously by the author The Hamiltonian of a system of charged particles and their electromagnetic field can be written in the form

fi:(#)'.,i,(#):

iC:+

4Gi)'-*r.ao),]ar 1 r / o--i'Aldr + f (" p J\

c

I r /6A"\2 :'"r:-,1-ldr 8rc2J \ 0t / ::

ELECTRON

t

f

*cI iJ

r e2h2 ^ tI(Eo,_E)' 2Coqq'

{u o*a"uo'lluo'*a"uol -ry26e - c2p p lE oE o, o. 4fz

ffr,).

I f /;*(".pi*9nc){';ld.r.

The summation in tbe last term is performed over all particles. The solutions of this Hamiltonian are restricted by the condition

a0/at+c div. A:0. -\pplied to a single electron at rest, this expression By introducing the field strengths instead of the rives potentials we obtain the expression e2h2 r

Ua:r-----rt

do

-:

:U^"s.

fnzczJ gT8h8

(le)

*:!-[

-{pplied to an electron at rest in positron theory, one obtains, however, Y"r: 6[1(Vac.*1) - U"r(Vac.) :

Ed.r
[ (,0-'-t.t)a'

2ne2h2 (E-t)(Eo-mc2)2 " +q -q t1uqr*a

*cIJ u a q "u rl I o* "u ul

m2(Eo-m2ct)2

| 9,*(". p* flmc),! ;ld.r.

This is equal to (1) if one uses the relations div. E:4rp and

and then e'h' f dp : -[J^^". Uet: -r-f ^J m 8r|hsE(f)

r :'> (20)

The interpretation of this result is given in Section I. lThillito, i is s(s*1) for s:i.

[

ft,-ta. (p- (e/ c)A)* pmcftld.t.

The interaction

energy

f Go-0/c)i'L):eH'

76 80

V.

F.

WEISSKOPF

between matter and field contains the electronic charge a explicitly as a linear factor, so that in the above notation f/'is explicitly independent of e; dHt/6e:0. Let us consider the energy 17" of a stationary state s of this Hamiltonian. If the electronic charge e is increased by de the Hamiltonian gets the additionaltermde f1'. According to perturbation theory the increase of W, is d.e(H')a",where (fl')r is the time average of 11' in the state s, assuming that the electronic charge has its original value a. We therefore get

w":w"to)+ f"
the relations 1.. 66'-.;Q'=

-A'rP,

1.. alt'--A':

-Ari,

and we obtain

- L ^ar. *, : L <
We consider now the state s to be the state of an electron at rest. The charge dependent part W' (21) of the self-energycan then be written in the form W':W**W.rlWrn"t,

where l4l"(oris the value of the energy ior e:0, We now expand H'(e) in a power series of e :

Here

(t) H' (e): g'
w",:! f 8rJ

lf

upon the sign of a, and we obtain by neglecting all terms containing e in a higher power than the second W':W"-W"$' e2ef1

:-(It"(r))A": -(H')*:L 22Jc

6: Qo*Q'.

Ao and do are the potentiats of the field when no electron is present. In empty space,.40is the potential of the zero-point oscillations and 6o:0. A' and 6' is the field produced by the electron. We thelr get

w': +[ (oo'r"-l1i'r,')^,)d,

((E"'z)e'- (H'z)^')dx:

'"":

Ua-

U^"u' (23)

uJ

It contains the contribution of the field produced by the spin and is calculated in the previous section; Wtu"t ls the energy produced by the fluctuations of the radiation field,

wrwt: _![i.xar.

| ((pO)^'--(i'A)n)dr.

The potentials can be split into two parts A:Ao*A',

"",,0,

is the static field energy of the irrotational field E"r,i W"p is defined by

2

The second term is zero since W" cannotdepend

Q2)

In order to calculate l/luct, w€ divide i into two parts i:ief i', whereio is the current density for the field-free case. The term Jfio'A&t vanishes when averaged over the time because of the absenceof phase relations between io and Ao. The remaining term

wrrct: -!

[t

. t a,

can be evaluated as follows: i' is to a first approximation given by

-|[ <''o'>^'o''

i ' : [ 9 0 * c / r ]* { 9 t * c r l o } .

of ffi:J;r:*[:ti:,Ll?,Yil;H::Hil;Xbymeans Thefirsttermcanbetransformed

(24)

77 ELECTROMAGNETIC

FIELD

tion of the perturbed wave function:

OF

THE

ELECTRON

81

We have

,h:Laoeq'. go' canbe calculated by means of the ordinary perturbation method. The interaction energy with an arbitrary field A is given by

(Af).:

7 rch

(2s)

,J lildk

and we get finally, on replacing lkl by p/h,

- esA.:- aaU,lAk+) expi(k. r{c t t) e2 rcdo e2 1 i I -lim1p--1P2 :. trVn,,"t:7---tf .' i ( k . 2 n 2 m c 2 hJ l ? l n h cm * A *e) exp r * c l t Ir ) 1 . The sum is taken over all wave numbersk of We now collect the resultsobtained for the other

an orthogonal system of plane waves in the volume V. lf we write the wave functions 9" of lhe free electron in the form (14), we get for 9o'

parts of the self-energyof an electron at rest. The one-electron theory gives LEqs.(11),

(23),(1e)l

f {u 4*auolI o{-t I q' : eLp q,l:----:---- --t-e+ icl,tl f l/"r:limr,:or l, W,p: (Ja- [/-"*:0. t LEo-Eo,lclkl a -1 I u 4*auol/ rt- ) ktt - pq. The positron theory gives (Eqs. (13), (23), (20)) f --e-i't l, hk:gq, Eo-Eo,-clkl I e2 P*Po ,. --, ThisexpresSionis introduced into (24) and gives l/"t:limrp:-r . mc' lg Tttc 'nc for the current i"t : EEN aIuq*a,u,,1lu o,*au ol

W"p: U6- U^,":

. _ f-.._-..........-.-A , * , e x pi ( k . r f c l i I r ) /\t L Eo-Eo,lclhl

-,

-

A(-)exp-i(k.t*c'klt)1

ar-u"-u,

,

ltconJ'

We haveretained only terms containingproducts (8) or (9). It is seen from this expression,that i'is the sarnein the one-electrontheory and in the positron theory. In the latter casewe have to consider d:it (Vac.+l)-l'(Vac.). The actual value can easily be evaluated for an electron at rest. One obtains le2

y:_-4,

which is immediately understood as the forced vibrations of a point charge under the action of an oscillating field A. W1u"1is directly obtained if one replacesA by the field fluctuations Ao of the vacuum:

wtttct:-!

[

t.xar:firo,r^,.

e2 1 frn2c2P+P^1 -limcp:-rlPPo-Igjl. Thcm L 2 mc J

W"o is partly balanced by Wrru.r. The total self-energy in positron theory is then given by 3 e2 W':--mc2 zT kc

P-lP, lim1r:'lB j*finite rtc

(26) terms,

The self-energy of a free electron in motion can be obtained by a Lorentz transformation from (26). The direct calculation from the above methods is ambiguous because it leads to a difierence of terms, each of which diverges quadratically. The factor of the logarithmielly divergent difference of these terms depends essentially on the way in which the infinite terms are subtracted. The calculation of the self-energy of an electron at rest is not so much exposed to these ambiguities because of the spherical symmetry of the problem, which suggestsonly one natural way of subtracting two divergent integrals over the momentum space, namely, the subtraction of the contributions of concentric spherical shells around the center. It must be expected, that the value of the self-energy of a moving electron can only be covariant to the value (26) if one performs the subtraction appropriately. This is why the expressions for the self-energy obtained in reference 5 are apparently not relativistically covariant.

78 V. F. WEISSKOPF

82

V. TnB SeI,r-ETERGYoF e Penrrcr.e OgBvrwc

and obtain for

BOSB-S:TETISTICS

tUE

lr

It has been shown that the quantization of the scalar wave equation of Klein and Gordon leads to a theory of elementary particles with Bose statistics and charges of both signs. The theory includes a description of pair creation and of all related phenomena. The quantitative results are not very different from the results of Dirac's posilron theory. The formalism has been recently applied to particles with intrinsic angular momentum. It will be shown here that the calculation of the self-energy, however, gives results quite difierent from the positron theory. The energy of the electrostatic field of the electron is found to be more strongly divergent than in the classical theory; the energy of the radiation field diverges quadratically and is equal to the corresponding energy of a single electron in the one-electron theory. The qualitative arguments for this behavior are given in I. The following calculation is based on the formulas derived elsewhere.T The operator of the charge density is given by p:ie(9*o* - {r) '

o@:nJ P(r)exP( -is' r)dr the following

pG):

expression :

E(k)+E(t) e -b*(k)b(1)] [o*(k)o(l) zv? f"(D"(Df- E(t) +E(k) [o(k)6(l)-a*(k)b*(l)], [E(e)E(')]t-

where l:kfs. The o(k) and D(k) fulfill the .elations a*(k)a(k) : l{(k),

a*(k)b(k) : M(k) '

o(k)o*(k) : 1+N(k),

b(k)b*(k): 1+M(k).

The N(k)'s are the numbers of positrons, the M(k)'s the number of negatrons in the states with the wave vector k. The electrostatic selfenergy is given by: tt ( r - E5/ / 2-/-r\' ) p ( r *'r !rr-/drd{ F\' f p w"r:+ t r' ltl p( - s) p(s)

where ry'is the wave function and n its conjugale operator: A,t* ":h

1 t:-E s ( k )e x p i k . r , v^ h e7) P(k) exP-ik'r'

LZ\-

;s2

x (n(k)tN(l)+ 1l+ M(k)[M0)+ 1]) t(E(k)-E0))' E(k)E(t)

vrh

Here l/Vt.expzkr form a set of orthogonal functions in the volume Z. We further introduce rE(l')t,

p@):l ___:_: | (o*(k)+D(k)), \'2/

E(k): c(h'k2*m2c2)+,

t-

1r(E(fr)*E(l))' re2 rr,-.:-- tt-l s "L E(k)E(t) 2y'i7

We introduce new variables by means of

| 1| / o&): _il -I (o*(k) _D(k)), \28&)/

-

We obtain by introducing (29) and retaining only the diagonal terms:ro

*,

|Q)r(r') - t(r'){(r) : 6(r r') '

1 r:-.8

(2s)

(28)

I

+ uf(k)+ 1l[M(l)+ 1])l x (N(k)M(l) This expressiondoes not vanish for the vacuum for every k). We calculate the (li/(k):M(k):0 difference iV

",:

W (Y ac.* 1) - ilz"t(Vac.) "t

r0 The term s=0 is omitted. It can easily be shown that this term does not contribute for tr/*e.

79 ELECTROMAGNETIC

FIELD

and assume that the particle is at rest:

plpol 7 p 2 . m z c 2 - ----lg l. " Lh' h2 mc J

' \'' \mcal

We now show that the particle does not produce a solenoidal electric or magnetic field in the ap_ proximation considered here. The current density is given by Eq. (43) of reference 7: i:ir*ig,

wrn"t:€2 [ Aorg*gdr. J

By introducing the new variables and retainins only diagonalelementswe find

e2 nr(k)+M(k)+l Wn"t:=(Ao')^">_--.--'T 2 E(h\ and finally f,Vru"t: tr;Zl,c(Vac.* 1) - I4lrr"*(Vac.)

grad. '1,),

the new variables

This is identical with the correspondingexpression for the Dirac electron. (21) into iy

VI. THn Hrcnon AppnoxruerroNs oF TrrESELFENoncv IN tne TnBony oF THE posrrnor

exp ir . s,

i ( s ) : i 'h c e-il -

We obtain then

e2

it:i.hce(t

ir: Ii(s)

83

:;(Ao')^" grad. ,!,*-{*

iz: -2e2L{*{. We introduce and get

ELECTRON

iz': -2e2Aot*{.

This is an expression which diverges quadratically. By putting P:h/a one obtains '"'=\n"( 4

THE

vanishes. The second part i2l is in the required approximation directly given by

e2h r l*2m2c2 /h2s2 W.t:_ | ds _- (s21m262/ft211 16ncJ h r : e2- - limrp:*l mc4

OF

k+l

( o * ( k ) (al )* b ( k ) b * ( l )

la1n1a14y

- a ( k ) b ( 1 )- b ( k ) a ( l ) ) ,

l:k*s,

i1(s) is proportional to k*l and ii is therefore irrotational for all transitions which start or end with a particle at rest. (kfl is parallel to s if k:0 or l:0). Thus i1 does not produce a solenoidalfield. is is proportional to e2so that its field does not come into consideration.The particle does not give rise to a solenoidal field as -long as it is at rest since it has no magnetic spin moment. The remaining term of the self-energy is

wru.,: _![

r. ua,.

(22)

i'is defined as the current density produced by the field A6. Here the first part i1/ can be shown to be again irrotational. The integral over the product of i1f and the solenoidal vector Ao

It will be proved in this section that the successiveapproximations of the self-energy of the electron vanish in the limit zr+0. Furthermore it is shown that the divergence of the selfenergyis logarithmicin every approximation.rr We consider the total system containing the electrons and the radiation field and calculate the energy W(s) oI the state s of this system. W(s) can be expanded in a series of approximations W(s):2"14t<">1t1 corresponding to an expansion in powers of the parameter ezfhc, Since this proceduredoes not give zero for the vacuum in Dirac's positron theory, the selfenergy of the electron must be defined as the difference between the energy I,Iz(Vac.f 1) of the state in which one electron is present and the energy lV(Vac.) of the vacuum alone. We confine ourselvesto the calculation of the electrodynamic rrRecently_A. Mercier (Helv. Phys. Acta tZ, 55 (193g)) . has treated the same problem and has obtained a hisher d.lvergence.As he dces not compute the numerical factois of the drvergent expressions,he cannot exclude a factor zero for the highest divergent terms. The following considerations, howeve-r,show that.the highest nonvanGhing terms qlverge only loganthmlcally.

80 84

V.

F.

WEISSKOPF

self-energy. The calculation of the electrostatic The only differencebetween the pair (a), (b) and energy and the mixed terms in higher approxima- the pair (c), (d) consists in the fact that the specified transitions start from a positive or tion can be made along the same lines. We now consider the detailed form of the ath from a negative state, respectively. This fact approximation Ws@\of the energy of the state s. does not affect the energy differences in the W,@)is a sum of terms containing a product of denominators in the limit m--+0, as in this limit 2n matrix elements of the interaction energy the energiesof s"'6and s-s are eeuol. lt.remains whicq correspond to consecutive transitions of to show that the numerators are also unchanged. the total system from one state to another This can be seen in the following way: the starting from the state s and returning to it. transition matrix elements appearing in the deThe terms have denominators that are products nominator belong to a chain of consecutive of energy differencesbetween the original state s transitions starting from and qeturning to the initial state of the system. Thrls the transition and intermediate states. The self-energyof the electron I7(") is given elements from or to s+0or s-o belong to a chain of transitions of an electron starting from the by the difference state s1o(or from s-s, respectively) and returning Ii/@:W@(Vac.f 1) - il/r")(Vac.). (30) to this state. The transition elements form the We now prove that ltr/@:0 for m:0 by com- product: paring it with the self-energy of a positron 2 a : 'fu'<"> I i l " ( * 0 ) H ' { ( f ' ) \ l , t * @ t ) H r V @ ) | .. . in the same state:

x \t*@"-,)H"/(+o) ). (33)

fi '
Here -Fleis the interaction energy with the light quantum which is emitted or absorbed with the ith transitlon and '!(p;) is the wave function of the electrlrn with the momentum fr which performs the transition, ry'(+O) is the wave function of the state sae or s-6, respectively' After averaging over the two spin states for every momentum pr, this product can be written as trace of the following matrix:

W (") (Y ac.I l) t W <">(Y ac.- l)

:2w(n')(Vac.) for m:0. (sz) r*:r'"t"ll/, ttt4 7<"r(Vac.-1) Comparing ryt"r(!ac.f1) w'th l4l(r)(Vac.)we notice: (a) Wr"t(Yac.-l) lacks all terms containing transitions of the electron in the state s-0. (b) t/(,)(Vac.f 1) contains additional terms from transitions of the additional electron in s+0. (c) I4zt"r(Vac.-1) contains additional terms from transitions of one of the vacuum electrons into the empty state s-0. (Vac. { 1) lacks terms containing tran(d) I4z<"t sitions of the vacuum electrons into the state s+o of the additional electron We now prove that the missing terms of (a) and (d) are in the limit m+0 identical with the additional terms of (b) and (c), respectively.

+(r+1J4)

U'

,(+Y#)...a.+rr*or]. is the energy of the Here E6: *(b?lm'c2)t wave function V@t. fi we now go to the limit v7+0, all terms with B disappear except those containing the p in the last pbrenthesis. (fli does not contain the operator B.)12Since the trace of every expression containing a's but only one B is zero, we are allowed to omit the last also, and it becomesevident that P*:p-. From (30), (31) rr This conclusion does not hold if one or more of the p;'s As l7r"l is an integral over all possible inter' are Mc, mediate momenta pi in the volume I/, the terms with one contain one or more factors (nc;l'"o ,nu, or more bi-nc these contributions should be neglected.

8l ELECTROMAGNETIC

FIELD

and (32) it follows directly that =0 1fi7<",

for

OF

THE

ELECTRON

85

tional to xc-ft, x-n-L...rc-2". We get, therefore,

m:O.

2-5n-l

F(kr'...k"'):

We now prove that this result infers a losarithmical divergenceof.,Th, for finite zz. The terms of W@ contain transitions in lnhich n lisht quanta are emitted and absorbed.We write ll/(n) 2s an integral over the wave vectors kr. . .k, of theselight quanta:

E

c,x-" for

p>)mc.

z:Zn_1

Not all of tbese c, need to be difierent from zero. If cs is the first coefficient different from zero, we can write F(b'' . ' .k,') :6ryt

=s-r p(kr. . .k")

(35)

u . .k^) (34) w*, :,[o'' atr.. . uo^r(kt. fo"'u

because we certainly can neglect the terms with z ) {, which are smaller by the rati,o - (mc /.pt) "-8. We are now interested in the function fr<"t(p\. T h e r e l a r i o n ( 3 5 , ;i s n o t v a l i d i n t h e e n t i r e r e e i o n and integrate to a finite lirnlt p/h)mc/h l,n o f i n t e g r a t i o n i n ( 3 a ) . T h e r e g i o n s i n w h i c h the order to r,nake ltrr<">finite. The properties of conditions PD)mc are not fulfilled are restricted F(kt'.k") are very simple for all p1)mc where to certain small areas in the 3z-dimensional P; are as above the momenta of the electrons s p a c e o f t h e w a v e v e c t o r s kr...k". The conwhich change their states in the transitions to tributions of these areas can be neelected for interrnediate states (except of course the mo_ P))mc. We therefore get

mentum Po:O of.the electron under considera_ tion in its initial state). It can be shown that _ fPlh nPlh W(aQP)=l dk,,...I dk,,Flhr,...h^,) replacing every k 1 (l: I, . . n) by k / : xk 1 gives vo

F ( k t , .. . k " , ) : x N F ( k r . . k , )

if

=rB"_r. W, (n.)(p) + smaller terms.

pn>>*r,

where .|y'is an integer. This result may be under_ stood in the following way. By means of the substitution kt':xkt the momenta /i of the excitedelectronsin rhe intermediate.tut", ur" also multiplied by a since the pd are sums or differencesof the momenta of the absorbed or emitted light quanta. lf now k1))mc,all momenta pi involved are large compared to mc, and, the correspondingenergiescan be replacedby clp;1. This neglect of mc compared to p; has as consequencethat all energy differences.8"-Er be_ tween the initial state and the intermediate statesare multiplied by_nif one replacesk fty k i . Since all terms of lT@ have 2n-! energy differences in the denominator, the latter is proportional 1o *2r-1. The numerators consist of n matrix elements which form expressionslike (33). From the fact that every }r; is proportional to &t-i un4 fror,- ihe structure of (33) it follows that the numerators are sums of lerms propor-

Ju

The additional smaller terms come from the regions of integration in which these considerations are not applicable and from the neglected terms. From this relation follows, that

,i,,, =,(;)" #

(.

:)

.mc2tsmater ".

Here N:3nf, c is a numerical factor and 0=t=n becauseany of the z integrationsmight give rise to a logarithm. The factors nxc are appiied in order to make the dimensionsfit. It was proved that :g. liml^-o;'f4t<"> This is only possibleif .^/=0. This is equivalent to the fact that litr"t does not diverge stronger than

*,",-(;)"^.(*#)

T[IE0RIE DU POSITRON P,ln II.

La d6couverte r6cente de l'6lectron positif ou positron a tamen6 l'attention vers une th6orie d6jA ancienne sur les 6tats d'6nergie n6gative de l'6lectron, les r6sultats exp6rimentaux obtenus jus' qu'ici se trouvant d'accord avec les pr6visions de cette th6orie. La question des 6nergiesn6gatives se pose dds que l'on 6tudie le rnouvement d'une particule conform6ment au principe de relati'r,it6 resbreinte. Dans la m6canique non relativiste, l'6nergie W d'une particule est donn6e en fonction de sa vitesse p ou de sa quantit6 de mouvement p par : \V:

IItttv!: tr.

F., 2ill'

ce qui correspond h un W toujours positif; mais en m6canique relativiste ces formules doivent €tre remplac6es par : w):

n L 2 c L - +c- 7 p ) ,

ou w:c

lrrl\'-+p',

,re qui permet h W d'6tre positif ou n6gatif. On fait d'ordinaire l'hypothise suppl6mentaire que l'6nergie W {oit toujours 6tre positive. Cela est admissible en th6orie classique of les grandeurs varient toujours de manidre continue et oir W ne peut par cons6quent jamais passer d'une de ses valeurs positives, qui doit €tre ) mcz, d une de ses valeurs n6gatives qui doit 6tre S - mcz. Dans Ia th6orie quantique, au contraire, une variable peut subir des changem'ents discontinus, de sorte que W peut passer d'une valeur positive h une valeur n6gative'

83 20

STRUCTURE ET PROPRIETfS DES NOYAUX ATOMIQUES.

Il n'a pas 6t6 possible de d6velopper une th6orie quantique relativiste de l'6lectron dans laquelle les transitions d'une valeur. positive i une valeur n6gative de l'6nergie soient exclues. Il n'est donc plus possible d'admettre que l'6nergie est toujours positive sans qu'il en r6sulte des inr:ons6quences dans la th6orie. Dans ces conditions, deux possibilit6s nous restent ouvertes. Ou bien nous devons trouver une signi{ication physique pour les 6tats d'6nergie n6gative, ou bien nous devons admettre que la th6orie quantiq.e relativiste est inexacte dans la mesurp or'relle pr6voit des transitions entre les 6tats d'6nergie positive.'et ceux d'6nergie n6Eative. Or de sernblablestransitions sont en g6n6ral pr6vues pour tous les processus rnettant en jeu des 6changes d'6nergie de l'ordre de rnczet il ne semble y avoir aucune raison de principe contre I'applicabilit6 de la m6canique quantique actuelle h de semblables6changesd'6nergie. Il est vrai que cette m6canique ne semble pas pouvoir s'appliquer aux ph6nomines dans lesquels interviennent des distances de I'ordre du ravorr classique de l'6lectron Irirque la th6orie actuelle ne peut fi, en aucune fagon rendre compte de la structure de l'6lectron, mais de telles distances, consid6r6escomme longueurs d'onde 6lectroniques, comespondent d des 6nergies de I'or.dre ryo;, rncz, beaucoupplus grandesque les changementsen question. Il semble donc que la solution la plus raisonnable est de chercher un sens physique pour les 6tats d'6nergie n6gative. Un 6lectron dans un 6tat d'6nergie n6gative est un objet tout i fait 6tranger d notre exp6rience,mais que nous pouvons cepend a n t 6 t u d i e r a u l r o i n t d e v u e l h 6 o r i q r r e n; o u s l r o u v o n s e , n p a r ti culier, pr6voir son molrvement dans un charnp 6lectromagn6tique quelconque donn6. Le r6sultat du calcul, e{Iectu6soit en m6canique classique, soit en th6orie guantique, est qu'un 6lectron d'6nergie n6gative est d6vi6 par le champ exactement comme le serait un 6lectron d'6nergie positive s'il avait une charge 6lectrique positive * e au lieu de la charge n6gative habituelle - e. Ce 16sultat suggdre imm6diatement une assimilation entrt l'6lectron d'6nergie n6gative et le positron. On serait tent6 d'admettre qu'un 6lectron dans un 6tat d'6nergie n6gative constitue pr6cisr4mentun positron, mais cela n'est pas acceptable.

84 T H E o R I ED U P o s I T R o N .

:2o5

parce que Ie positron observ6 n'a certainement pas une 6nergie cin6tique n6gative. Nous pouvons obtenir urr meilleur r6sultat en utilisant le principe cl'exclusionde Pauli, en vertu duquel un 6tat quantique donn6 ne peut 6tre occup6 par plus d'un 6lectron. Admettons que dans l'Univers tel que nous le connaissons,les 6tats d'6nergie n6gative soient presque tous occup6s par des 6lectrons, et que la distribution ainsi obtenue ne soit pas accessibleh notre observation h causede son uniformit6 dans toute l'6tendue de I'espace. Dans ces conditions, tout 6tat d'6nergie n6gative non occuir6 repr6sentant une rupture de cette uniformit6, doit se r6v6ler h I'observation comme une sorte de laoune. Il est possible d'admettre que ces lacunes constituent les positrons. Cette hypothdse r6sout les dilficult6s principales de l'interpr6tation des 6tats d'6nergie n6gative. Une lacune dans la distribution des 6lectrons d'6nergie n6gative repr6sente une 6nergie positive, puisqu'elle correspondh un d6faut local d'6nergie n6gative. De plus, le molrvement de cette. lacuue dans un chamJr 6lectromagn6tique guelconque est exactement 1e mdme que celui de l'6lectron n6cessairepour combler la lacune. Nous pouvons tirer de lA deux conclusions : d'abord que le nrouvement de la lacune peut 6tre repr6sent6 par une fonction d'onde de Schrridinger analogue h celle qui repr6sentele nrouvement d'urr 6lectron, et ensuite, que la lacune se comporte dans un cham;r de la m6me manidre qu'un 6lectron positif d'6nergie positive. Ainsi la lacune prend exactement l'aspect d'une particule ordinaire 6lectris6epositivement et son identification avec le positrorr se pr6sente comme tout h fait plausible. Si notre hvpothdse est correcte,nous devons pouvoir en d6duire un certain nombre de cons6quencesexp6rimentalement r'6ri{iables. Tout d'abord, la masse du positron doit 6tre exacternent 6gale d celle de l'6lectron et sa chargedoit 6tre exactement6galeet oppos6eir cellede l'6lectron. De plus nous pouvons pr6voir certains r6sultats concer,nantla cr6ation et la disparition des positrons. Un 6lectron ordinaire d'6nergie positive ne peut pas sauter dans l'un des 6tats occup6sd'6nergie n6gative, en raison du prirrcipe de Pauli; il peut, au contraire, sauter dans trne lacune pour la combler. Ainsi un 6lectron et un positron peu\.ent se d6ttuire

85 206

S T R U C T U R E T P R O P R I E T ED SE S N O Y A U XA T O M I Q U E S .

r6ciproquement. Leur 6nergie doit se retrouver sous forme de photons et il r6sulte des principes de conservation de l'6nergie et de la quantit6 de mouvement que deux photolls au moins doivent 6tre produits. On peut calculer la probabilit6 pour qu,un tel processusait lieu et obtenir ainsi la vie probable d'un positron se mouvant d travers une distribution donn6e d'6lectrons. Le r6sultat est une vie moyenne de 3 X ro-z sec pour un positron en mouvement lent dans l'air h la pression atmosph6rique, cette dur6e moyenne augmentant avec la vitesse. La valeur ainsi obtenuel est d'un ordre de grandeur compatible avec l'exp6rience, puis-t qu'elle est su{Iisante pour permettre h un positron rapide de traverser une chambre de condensation cle Wilson sans y 6tre en g6n6ral d6truit, et assez petite cependant pour que les positrons ne soient pas des objets commun6ment pr6sents au laboratoire. Un 6lectron et un positron peuvent s'annihiler r6ciproquement en donnant naissance i un seul photon si un noyau atomique est pr6sent pour absorber la quantit6 de mouvement lib6r6e. Le processus inverse consiste dans la production d'un positron et d'un 6lectron par Ia rencontre d'un seul photon d'6nergie su{Iisante avec un noyau atomique. On peut se le repr6senter comme un e{Tet photo-6lectrique sur un des 6lectrons d'6nergie n6gative tl6crivant des orbites hyperboliques au voisinage du noyau; cet 6lectron 6tant 6lev6 vers un 6tat d'6nergie positive et apparaissant ainsi comme 6lectron ordinaire, tandis qu'il laisse deni.dre lui une Jacune se comportant comme un positron. La probabilit6 d'apparition d'un tel proeessus a 6t6 calcul6e approximativement par Oppenheimer et ind6pendarnment par Peierls, et le r6sultat est d'un ordre de grandeur eui concorde avec les observations relatives ir la production de positrons par des rayons y durs tombant sur des noyaux lourds. Pour que Ia conception que nous proposons des 6tats d'6nergie n6gative se d6veloppe en th6orie compldte, nous devons consid6rer non seulement le mouvement des 6lectrons et des lacunes dans un champ, mais aussi la manidre dont un.champ 6lectromagn6tique est produit par les 6lectrons et les lacunes. Pour cela, il est n6cessaired'introduire une hypothdse nouv€lle puisque Ia conception ordinaire que chaque cha"ge - e sur un 6lectron

86 THEORIE DU POSITRON.

207

contribue h produire la densit6 6lectrique p qui d6termine le champ 6lectrique E conforrn6ment h l'6quation de Maxwell : 4t )

, l i vi : 4 r o

conduirait 6videmment d un champ infini en tout point. Faisons l'hypothdse que la distribution d'6lectrons dans laquelle aucun 6tat d'6nergie positive n'est occup6, tandis que tous les 6tats d'6nergie n6gative le sont ne produit aucun champ, et que/ ce sont les 6carts h partir de cette distribution qui d6terminent les champs conform6ment h l'6quation (r). Dans cette hypothdse, un 6tat d'6nergie positive occup6 produirait un champ correspondant d une charge n6gative - e etr un 6tat d'6nergie n6gative non occup6 produirait un champ correspondant h une charge positive f e. Nous obtenons ainsi une nouvelle propri6t6 des lacunes qui contribue ir rendre vraisemblable notre assimilation de ces lacunes avec les positrons. La nouvelle hypothdse est tout i fait satisfaisante lorsqu'il s'agit d'une r6gion de I'espace orl n'existe aucun champ, et oir la distinction entre les 6tats d'6nergie positive et ceux d'6nergie n6gative est nettement d6finie; mais elle doit 6tre pr6cis6elorsqu'il s'agit d'une r6gion de I'espace oi le champ 6lectromagn6tique n'est pas nul pour pouvoir conduire i des r6sultats libres de toute ambiguit6. Il faut sp6cifier math6matiquement quelle distribution d'6lectrons est suppos6ene produire aucun chamlr et donner aussi une rbgle pour soustraire cette distribution de celle qui existe e{Iectivement dans chaque probldme particulier, de manidre ir obtenir une di{I6rence finie qui peut figurer dans (r ), puisque, en g6n6ral, I'op6ration math6matique de soustraction entre deux in{init6s est ambigu6. Cette question n'a pas encore 6t6 examin6e ni r6solue pour le cas g6n6ral d'un champ 6lectromagn6tique arbitraire. Il y a, cependant, un cas particulier dans lequel les hypothdses n6cessaires semblent assez6videntes : celui d'un champ 6lectrostatique permanent. Nous allons traiter ici ce cas, en supposant le champ su{fisamment faible pour qu'une m6thode de perturbation puisse 6tre utilis6e. Nous constaterons que la distribution qui ne produit aucun champ ne satisfait pas aux 6quations du mouvement. En

87 1)O8

S T R U C T U R E T P R O P R I E T ED S E S N O Y A U XA T O M I Q U E S .

retranchant cette distribution de celle qui satisfait h ces 6quations et qui correspond ir un 6tat or) ne sont pr6sents ni 6lectrons ni positrons, nous obtiendrons une di{f6rence qui pourra 6tre interpr6t6e physiquement comme un effet de polarisation par le champ 6lectrique de la distribution normale des 6lectrons d'6nergie n6gative. Nous emploierons la m6thode d'approximation de HartreeFock qui attribue ir chaque 6lectron sa propre fonction d'onde individuelle ,1,(C), et nous introduirons la matrice densit6 R d6finie par ( E ' I t \ l q "t : 2 " V \ q ' ) ' ! ,( q ") , Ia sommation s'6tendant ir tous les 6lectrons, c'est-h-dire h tous les 6tats occup6s. Toute distribution d'6lectrons peut €tre oaract6ris6e par une semblable matrice, au degr6 de pr6cision que comporte la m6thode de Hartree-Fock. Cette repr6sentation n'est pas relativiste puisque les valeurs g', q" des variables associ6es ir un 6l6ment de la matrice R correspondent ir deux points diIT6rents de I'espace,mais ir un mdme instant. N6anmoins elle convient pour notre probldme actuel. L'6guation de mouvement pour R est (1) : i i t r \ : r { R- R I I ,

(z)

ori H est l'hamiltonien pour un 6lectron mobile dans le champ : II -

a e , ( o , p ) - + p t n l . c 2-

eY ,

les p et o 6tant les matrices de spin habituelles et V le potentiel 6lectrostatique. V doit contenir une partie repr6sentant la contribution au ohamp des autres 6lectrons pr6sents. La condition Irour que la distribution satisfasse au principe d'exclusion est (3)

ll2 :

lt.

Repr6sentonspar R6la distribution qui est suppos6ene produire (r) Cl. Drnec, Proc. Cam.b, Phil, Soc., t. 25, rgzg, p. 6z; et 26, r93o, p.376.

88 THEoRIE DU PosITRoN.

:IOII

aucun charnp. I-'hypothdse la plus irnm6diate pour Ro est

',,-i('-+)

i

ori W est l'6nergie cin6tique d'un 6lectron : \\r :

cgr(O, p)r-

93lzc,t.

Ceci signifie que, dans une repr6sentation matricielle ori W est. diagonale, Ro sera 6galement diagonale et aura pour 6l6ments diagonaux o ou r suivant que l'6nergie W est positive ou n6gative. C'est l'6nergie cin6tique W qui doit figurer dans (,{) et non l'6nergie totale H parce que, dans ce dernier cas, I'expression({) serait modifi6e seulement par I'addition d'une constante au protentiel 6lectrique V et ne pourrait par cons6quent avoir aucune signification phvsique. Consid6rons un 6tat permanent pour lequel l'6quation de rnouvement (z) se r6duit h : o:IIR-ltll

(5)

Cette 6quation n'est pas satisfaite par R : Ro ir moins que V lle soit une constante. Supposonsque V soit une petite quantit6 du premier ordre et cherchons une solutiorr de (3) et (5) de la forme R : Ru * R, oir R, soit une quantit6 du premier ordre. En n6gligeantles petites quantit6s du secondordre, l'6quation (5) donne: ((j)

.r:(W-eV)(Ro+-R,)-(Ilo-fllr)(W-eV) : W l t ,- R r W - e ( V R 6 - R n V ) .

L'op6rateur I W I peut 6tre d6fini comme Ia racine carr6e positive de W2 ou m2 c' * cz p2. Ainsi : Wl:c{r.r":4p:;i. S i r r o u sp o s o n s

w l\Y1

-. i'

llous avoDs I

W :

ci(,elc't4

et aussi F, o : lxsrr?uT soLvAY {PHy-*IQUEJ,

I' -(I-1).

ptl'.t

89 2]O

S T R U C T U R E T P R O P R I E T ED SE S N O Y A U XA T O M I Q U E S .

L'6quation (6) peut, par cons6quent, sn6crire: i1

y ( m 2 c 2 + - p r ) - R 1 - R 1 1 ( n r l 6 1 1 p: : ; i ii,l.\'

0)

-\'t).

L'6quation (3) donne : (Ro-+Rr)r:Ro-fRr, RoRr-+RlRs:

qui se r6duit h

R1,

1 R 1- + R 1 1: o .

En utilisant cette relation et l'6quation y2: r, nous d6duisons d" (y), aprds multiplication des deux membres h gauche par y : tl

1z z : c 1- r p r ) i R , - + R r ( t t z 1 c 2 -p+.-) T :

ii,\'

- 1\ 1 r.

La quantit6 qui nous int6resse est la densit6 6lectrique correspondant d la distribution \. Pour l'obtenir, nous devons former la somme diagonale d" R , par rapport attx variables de spin et prendre ensuite l'6l6ment diagonal g6n6ral, multipli6 par - e, de la matrice r6sultante par rapport aux variables de position o. Si D repr6sente la somme diagonale par rappor[ aux variables de spin, nous avons, aprds un caleul simple : -!r ( t r t ! c ) - r p : l E D r R r l - 1 -I l ( l l 1 ) ( z r 1 c 1 - + - p r r T

: 129C o < v - y v T ) : r:11''

|

---l----(lr:c:-;

lrp. \ p ! -+ t,trc2\ ----l------r l. 1 7 1 1 : 3 1 4P : ; ' : )

P:)E

Si nous supposon$maintenant I'emploi d'une repr6sentation dans laquelle la matrice quantit6 de mouvement p est diagonale et si (p' ] D (Rt)l p") d6signe,dans ces conditions, l'6l6ment g6n6ral de la matrice D (Rl), nous avons lr

( nztcz-+ p't 7t (p' I D ( R, ) | p',) -+-(p' I D ( It, ) | 1t") ( mz ct a p"t)z PI

: r l ( p ' , V 7 , , r! r - - - - - l - ! _ , "

|

rp'. p, )+- ntlclf ----l-rl

( nzt cz -+-p't 17

ce qui donne ( s ) ( r r 'l D ( R r ) I p " ) :

z 1 @ ' ) l ' tr r " , '

(nt2 c2 +- lttt':f I

(9" P") -+ ntct

I m ' c 2 ' +p - ' 1) 2( n t zc 2- + p " t ) t . ( 1azf -4-P'z 17I (nt2 ct -+-P"t 11

90 THEORTE DU POSTTRON.

2.Ir

Nous pouvons maintenant transformer D (Rr) dans une repr6sentation pour laquelle les variables de position c sont diagonales et en calculer l'6l6ment diagonal. En utiiisant les lois habjtuelles de transformation, on ohtient t, ( r r ) ( . i r lD 1 1 1)' i r ) : [ [ " - ' , r , n - p ' ) t t ' ( p ' l D ( R)1l p " ) d l t 'r Q : : t " . n' .J .,1

Maintenant, puisque V n'est fonction que des variables de position , et pas des quantit6s de mouvement p, (p' I V i p'1) ne doit d6pendre que de la diff6rencep'-p//. Par suite, si nous substituons I'expression donn6e par le second membre de (8) dans l'int6grale (9), et si nous prenons pour nouvelles variables d'int6gration p' + p' et p' -ptt, nous pouvons effectuerI'int6gration par rappor[':, p'+ p" en laissant V quelconque.Le r6sultat contient un infini logarithmique. On pourrait croire, d premidre vue, que la pr6sence de cet infini rend la th6orie inacceptable. Cependant, nous ne pouvons pas supposer que la th6orie s'applique lorsqu'il s'agit d'6nergies supr6rieuresi I'ordre de r37 tncz, et la manidre de proc6der la plus raisonnable semble €tre de limiter arllitrairement le domaine d'int6gration i une valeur de Ia quantit6 de mouvement I (p' f p") correspondant h des 6nergies 6lectroniques de I'ordre indiqu6. Cela revient, physiquement, h admettre que la distribution concernant les 6lectrons d'6nergie n6gative inf6rieure h un niveau d'environ - r37 mcz ne donne pas lieu A une polarisation par le champ 6lectrique de la manidre indiqu6e par notre th6orie. La place exacte que nous attribuons h ce niveau d'6nergie limite n'a pas grande importance puisque la valeur de ce niveau figure seulement dans un logarithme. Si P est la grandeur du vecteur quantit6 de mouveI ment 1(p'*

p")

h laquelle nous limitons le domaine d'int6-

gration, le r6sultat final, obtenu aprds une int6gration compliqu6e, est :

(r,,)

-e(riD(R,)t.r):-"O*$r#-:)t

- ' *4 n "" lf, \*)

\r-

r"*

of p est la densit6 6lectrique produisant le potentiel V, de sorte

9I S T R U C T U R E T P R O P R I E T ED SE S N O Y A U XA T O M I Q U E S .

2t2

que VrV: - 4:ip, et oir les terrnes contenant les d6riv6es de p d'ordre sup6rieur au second ont 6t6 n6glig6s. Le secondmembre de (ro) donne la densit6 6lectriqueprovenant de la polarisation produite par l'action du champ sur la distributiorr des 6lectrons d'6nergie n6gative. Le terme important est le premier eui, pour 3 o" -*p.

t^

:

r3,J, est sensiblemefi -#?

Ceci signifie qu'il n'y a de densit6 produite par

polarisation que dans les endroits oir se trouve situ6e la densit6 p productrice du champ et que la densit6 induite y neutralise une fractron d'environ

I

d" la densit6 productrice du champ. Le ;7 second terme dans le second membre de (ro) repr6sente une correction importante seulement lorsque la densit6 p varie rapidement avec Ia position et change de maniire appr6ciable sur une d i s l a n c ed e l ' o r d r e d u a . Comme .o.rr6q.,..r.JT,, .ul",rl pr6c6dent, il semblerait que les charges 6lectriques normalement observ6es sur les 6lectrons, protons ou autres particules 6lectris6esne sont pas les charges v6ritables port6es par ces particules et figurant dans les 6quations fondamentales, mais sont l6gdrement plus petites dans le rapport doenviron r36 d r37. Pour des processuscomportani des 6changes d'6nergie de I'ordre d.e mcz, il n'y aurait probablement pas le temps, cependant, pour que la polarisation des 6lectronsd'6nergie n6gative s'6tablisse de manidre compldte, de sorle qu'on doit s'atbendre h ce que les charges observ6essoient plug voisines des charges r6elles. Il en r6sulterait des d6viations de I'ordre de r pour roo dans des expressionstelles que la forrnule de KleinNishina ou la formule de diffusion de Rutherford lorsque des 6nergies de l'ordre d.e mcz sont en jeu. Lorsque la v6rification exp6rir'entale de ces formules pourra 6tre rendue su{fisarnment pr6cise, on y trouvera un contr6le de I'exactitude de nos hvpothdses sur le champ produit par la distribution des 6lectrons d'6nergie n6gative.

P o p e r8

92

UggN DIE ELEKTRODYNAMIKDES VAKUUMSAUF GRUND DER QUANTENTHEORIE DES ELEKTRONS voN

V. VEISSKOPF ines der wichtigsten F'rgebnisse in der neueren Entp '/ I wicklung der Elektronentheorie ist die Moglichkeit, elektromagnetische Feldenergie in Materie zu verwandeln' Ein Lichtquant

z. B. kann bei Vorhandensein von andern

elektromagnetischen Feldern irn leeren Raum absorbiert und in Materie verwandelt .rNerden,wobei ein Paar von Elektronen mit entgegengesetzterLadung entsteht. Die Erhaltung der Energie erfordert, falls das Feld, in welchem die Absorption vor sich geht, statisch ist, dass das absorbierte Lichtquant

die ganze z:ur Erzeagung des Elek-

tronenpaares notwendige Energie aufbringt. Die Frequenz e2 d e r s e l b e nm u s s s o m i t d e r B e z i e h u n g h v : 2 m c 2 * e 1 i geniigen, wobei mcz die Ruhenergie eines Elektrons und e1 und esdie iibrige Energie der beiden Elektronen ist. Diesen Fall haben wir z. B. bei der Erzeugung eittes Blektronenpaares durch ein 7-Quant im Coulombfeld eines Atomkerns vor uns. Die Absorption kann auch in Feldern stattfinden, die von andern Lichtquanten stammen, wobei die letzteren zur Energie des Elektronenpaares beitragen kdnnen, sodass in diesem Falle die Energie 2mcz* e1* e2 der heiden Elektronen gleich der Summe aller bei diesem Prozess absorbierten Lichtquanten sein muss. Das Ph5nomen der Absorption von Licht im Vakuum

93 4

pr. 6. V. wsrssxopn:

stellt eine '\''vesentliche Abweichung vor der M.c.xw'rr,'schen Elektrodynamik dar. Das Vakuum sollte ndmlich unabhAngig von den dort he'rschenden Feldern fiir eine Lichtrveile frei durchdringlich sein, da sich verschiede'e Felder nach den Mlxwrll'schen Gleichungen infolge der Linearitit derselben unabhingig iiberlagern kdnnen. Es ist bereits ohne niheres Eingehen auf die spezielle Theorie verstdndlich, dass auch in solchen Ferdern, die nicht die notige Energie besitzen, um ein Elektronenpaar zu erzeugen, Abrveichungen von der Mlxwnr,r,'schen Elektro_ dynamik auftreten miissen: wenn hochfrequentes Licht in elektromagnetischen Feldern absorbiert werden kann, so man fiir Lichtstrahlen, deren FreqLtenz zar paarer_

wird

zeugung nicht ausreicht, eine Streuung oder Ablenkung er_ warten, analog zur Streuung des Lichts an einem Atom. dessen kleinste Absorptionsfrequenz gr6sser als die des Lichts ist. Das Licht wird sich also beim Durchgang durch elektromagnetische Felder so verhalten, als ob das Vakuum unter der Einwirkung der Felder eine von der Einheit verschiedene DielektrizitAtskonstante erhalten wiirde. Um diese Erscheinungen darstellen zu krinnen, muss die Theorie dem leeren Raum gewisseEigenschaften zuschreiben, die die ern'iihnten Abrn'eichungen von der Maxwrr,r,'schen hervorrufen. Tats:ichlich frihrt die relativistische Wellengleichung des Elektrons auch zu derartigen Folgerungen, \ renn man die aus der Dlnac'schen Elektrodynamik

wellengleichung folgenden Zustfinde mit negativer kinetischer Energie zur Beschreibung des Vakuums heranzieht. Die Grundannahme der Drn.qc'schenTheorie des positrons besteht darin, dass das physikalisc.he Verhalten des Vakuums im gewissen Sinne beschrieben werden kann durch das Verhalten einer unendlichen Menge von Elektronen, die

94 Uber die Elektrodynamik'

des Vakuums.

Vakuumelektronen -

die sich in den ZustAnden negativer kinetischer Energie befi.nden und simtliche dieser Zustiinde besetzthalten. Die ubereinstimmung kann selbstverstandlich nicht vollkommen sein, da die Vakuumelektronen eine unendliche Ladungs- und Stromdichte besitzen,die sicher keine physikalische Bedeutung haben darf. Es zeigt sich aber, dass z. B. die Paarerzeugung(und ihr Umkehrprozess) gut wieder=' gegebenwird als ein Sprung eiues Vakuumelektrons in einen Zustand positiver Energie unter dem Einfluss elektromages als ein reales Elektron in Ercheinung tritt, wiihrend das Vakuum um ein negatives Elektron irmer geworden ist, was sich durch das Auftreten

netischer Felder, wodurch

eines positiven Elektrons dussern muss' Die von diesem -VerBilde ausgehende Berechnung der Paarerzeugung und nichtung zeigt eine gute Ubereinstimmung mit der Erfahrung' Die Berechnung der meisten andern Effekte, die aus der Positronentheorie folgen, stossen immer auf das Problem' in welchem Ausmass das Verhalten der Vakuumelektronen tatsrichlich als das des Vakuums anzusehen ist' Dieses noch durch den Umstand erschwert' dass Ladungs-, Strom- und Energiedichte der Vakuutnelektronen unendlich sind, sodass es sich meistens darum handelt' von

Problern wird

einer unendlichen Summe in eindeutiger Weise einen endlichen Teil abzutrennen und diesem Realitiit zuzuschreiben' Die Losung dieses Problems wurde von DInlc und HnrsENBERGdadurch duichgefiihrt, dass sie eine wiederspruchsfreie Methode angaben, den physikalisch bedeutungsvollen Teil der Wirkungen der Vakuumelektronen zu bestimmen' Es q,ird im folgenden gezeigt,dass diese Bestimmung weitgehend frei von jeder Willkiir ist, da sie in konsequenter Weise nur folgende Eigenschaften der Vakumelektronen als physikalisch bedeutungslos annimmt:

95 Nr.6. V. Wprssxopr: Die Energie der Vakuumelektronen im feldfreien Raum.

l;l

Die Ladungs- und Stromdichte der Vakuumelek-

(r)

tronen im feldfreien Raum.

3) Eine rdumlich und zeitlich konstante feldunabhiingige elektrische und magnetische polarisierbarkeit des Yakuunrs.

Diese Grdssenr beziehen sich nur auf das feldfreie Vakuum, und es darf als selbstr-erstdndlich angesehenwerden, dass diese keine phrsikalische Bedeutung haben k6nnen. Alle drei Gr6ssen er\\'eisen sich nach Sumnrierung der Beitriige aller Takuumselektronen als divergierende Summen. Es sei noch hinzugefigt, dass eine konstante polarisierbarkeit in keiner \\'eise feststellbar r'dre, sondern nur sdmtliche Ladungs- und Feldstdrkens'erte mit einem konstanten Fak_ tor multiplizieren q-rirde. q'erden im nichsten Abschnitt auf Grund dieser Annahmen die physikalischen Eigenschaften des Vakuums bei Anrn'esenheit ron Feldern berechnen, die zeitlich und riumlich langsam rerinderlich sind. Wir verstehen darunter Wir

solche Felder F, die sich auf Strecken der Liinge ft urrd in Zeiten der Linge a" .ro. wenig verindern, ,r#"ro-it den Bedingungen

( t.)

l l e . " o r ' t . . t rr-'rt, . 4 l q 4 l mc,---r mczlEtl..

lFl

genigen. Bei Anwesenheit solcher Felder werden im all_ gemeinen keine Paare erzeugt, da die auftretenden Licht_ quanten zu geringe Energie haben. Die Extremfiille, in denen die Strahlungsdichte so hoch ist, um das Zusammenwirken t Die Annahmen, l.) oder 2.) oder 3.) als bedeutungslos anzusehen, werden im folgenden mit 11, 12 bzw.I3 zitiert. 3 ft ist die durch 2z geteilte pr,lNcr'sche Konstante.

96 Uber die ElektrodynamikdesVakuums.

7

von sehr vielen Quanten zu gestatten, oder in denen elektrostatische Felder mit Potentialdifl'erenzen von iber 2mcz vorhanden sind (in diesem Falle wffrden auf Grund. des Kr,nrn'schen Paradoxons Paare entstehen) r*'ollen n'ir von der Betrachtung ausschliessen. Unter diesen Umstinden lassen sich die elektromagnetischen Eigenschaften des Va=-/ kuums durch eine feldabhiingige elektrische und magnetische Polarisierbarkeit des leeren Raums darstellen, die z. B. za einer Lichtbrechung in elektrischen Feldern oder zu einer Streuung von Licht an Licht fiihrt. Der Dielektrizitfrts- und Permeabilithtstensor des Vakuums hat dann fiir schu'ichere Feldstdrken niherungsweise folgende Form , (i, i, i,i sind die vier elektromagnetischen Feldgrdssenr.)

Dr: (2) oix: dir,*

-j,

I

tikDk,Hr: Z p*Bo 1n, - n ldik+ 7BiBft]

"knlz

tnr*:dix.#fu

i: A' o" t x : \f0l ,, tI k

- 7E,E*]. lz6, - n 1d,*

Die Berechnung dieser Erscheinungen wurde von Eurrn

u.

Kocxer,e und von HnrsnNsrnc u. Eur,en3 bereits durchgeftihrt. Im nflchsten Abschnitt sollen jedoch bedeutend einfachere Methoden angewendet werden. Ausserdem sollen die Eigenschaften des Vakuunrs auf Grund der skalaren relativistischen Wellengleichung des Elektrons von Krrrl

u. Gonoox

berechnet werden. Diese Wellengleichung liefert nach Prurr u. Wnlssropra

die Existenz positiver und negativer Par-

tikel, und ihre Erzeugung und Vernichtung durch elektro1 Es werden im folgenden nur dort Pfeile iiber Vektorgr6ssen gesetzt, wo Verwechslungen mdglich sind. t H. Eur,on u . B . K o c x r r - , N a t u r w i s s . 2 8 , 2 . 1 6 ,1 9 3 b ; H . E u r - n n , A n n . d. Phys. v.26, 398. u W. HnrsoNranc

u. H. Eur-an, ZS. f. Phys. iXl, 214, 1936. a W. Peur,r u. V. Wussxonr, Helv. Phys. Acta. ?, Zl0, 1934.

97 Nr. 6. V. Wnrssxopr:

magnetische Felder ohne jede besondere Zusatzannahme. Jedoch besitzen diese Partikel keinen Spin und befolgen die Bosestatistik, weshalb diese Theorie nicht auf die realen E l e k t o n e n a n w a n d b a r i s t . E s i s t i e d o c h b e m e r k e n s w e r t ,d a s s auch diese Theorie auf Eigenschaften des vakuums fiihrt, denen keine physikalische Bedeutung zukonrmen kann. So erhiilt man z. B. ebenfalls eine unendliche rdumlich u'd zeitlich konstante feldunabhringige polarisierbarkeit des vakuums. Nach Weglassung der entsprechenden Glieder ge_ langt ntan zu ihnlichen Resultaten, wie clie der positronen_ theorie Drnec's. Die physikalischen Eigenschaften des Va_ k u u r n s r u h r e n i n d i e s e r . T h e o r i ev o n d e r > N u l l p u n k t s e n e r g i e < der Materie her, die auch bei nichtvorhandenen Teilchen von den iiusseren Feldstiirken abhiingt und somit ein zusatzglied zu der reinen Maxwelr,'schen Feldenergie liefert. Im 3. Abschnitt behandeln wir die F-olgerungenaus der Drna,c'schen Positronentheorie fiir den F all allgemeiner dus_ serer Felder und wir zeigen, dass mari auf Grund der ge_ nannten drei Annahmmen uber die Wirkungen der Vakuum_ elektronen stets zu endlichen und eindeutige' Resultaten kommt. Die HnrsBNerRc'schen Subtraktionsvorschriften er_ weisen sich als identisch rnit diesen drei Annahmen und erscheinen somit bedeutend weniger willkiirlich als es in der Literatur bisher angenommen wurde. Alle folgende Rechnungen beriicksichtigen nicht explizit die gegenseitigenWechsehvirhungen der Vakuumelektronen sondern betrachten ausschliesslichjedes einzelneVakuumelek_ tron allein unter der Ei'wirku'g ei'es vorhandenen Ferdes. Bei diesem Verfahren sind aber die gegenseitigen Einwirkungen nicht vollkommen vernachlfrssigt,da man das [ussere Feld gar nicht von dem Feld trennen kann, das von den vakuumselektronen selbst erzeugt ist, sodass das in die Rech-

98 Uber die Elektrodynamik

des Vakuums.

nung eingehende Feld die Wirkungen der andern Vakuumelektronen zum Teil implizit enthelt. Dieses Vorgehen ist analog zur Ha,nrnrn'schen Berechnung der Elektronenbahnen eines Atoms in dem Feld, das von den Elektronen selbst verdndert wird. Zur expliziten Berechnung der Wechseli \firkungen

mtisste man

die Quanlenelektrodynamik anwenden, d. h. die Quantelung der Wellenfelder vornehmen. Dies fiihrt bekanntlich auclr bereits ohne die Annahme unendlich vieler Vakuumselektronen zu Divergenzen und soll im folgenden nicht nflher berrihrt werden.

II. In diesemAbschnittsoll die ElektrodynamikdesVakuums ftir Felder behandeltwerden, die den Bedingungen(l) geniigen. Die Feldgleichungensind durch die Angabeder Energiedichte U als Funktion der Feldstirken festgelegt-Wir bestimmen diese aus der Energiedichtef a"r Yakuumelektronen, die ftir das Verhalten des Vakuums massgebendsein sollen. Es ist vorteilhaft, auf die Lagrangefunktion t des etrektromagnetischenFeldes zurrickzugreifen,da diese drrch
u: Zu,#,-t.

In der Maxwer,l'schen Elektrodl'namik gilt:

L : -*(En- B,), L': #(ru+F). Die Zusdlze zu dieser Lagrangefunklion mi-rssenebenso wie diese selbst relativistische Invarianten sein. Solange rvir uns

99 10

Nr. 6. v. wnrssropr.:

nur auf langsam verinderliche Felder beschrinken, (Bedingung (1)), werden diese Zusf,tze nur von den Werten der Feldstiirken abhingen und nicht von deren Ableitungen. Sie krinnen daher nur Funktionen ddr Invarianten (E2 - 82) und (EB)z sein. Entwickeln wir die Zasdtze nach Potenzen der FeldstArken bis zur 6. Ordnung, so erhalten lvir: 1

L : +(Er_R )+ L, 8n' L , : a ( E z _ B z 1 zp+( E B ) r + + t (E',- Br),+ e(8, - B\ (nn;z1 . . . und daher nach (3) (1

l| U :

(t\ t r^z

---(82+82)+U'

67t

)I u ' : o ( E ' - B z ) G E 2 + B ' ) + p ( E B ) ' + + E ( 8 ,- B r ) r ( 5 8 2 + B \ + e @ B ) r ( } B , - 8 2 ) + . . . [

Der Zusatz zur Energiedichte ist somit durch die Invarianzeigenschaften weitgehend festgelegt; es wird

also im fol-

genden nur notwendig sein die vorkommenden Konstanten a(,P,8,E,... zu bestimmen. Diesen Ansitzen liegt schon die spezielle Annahme zugrunde, dass U' keine Glieder 2. Ordnung in den Feldstirken

enthfllt, sondern nur htihere. Dies

ist gleichbedeutend damit, dass das Vakuum keine von den Feldern unabhiingige Polarisierbarkeit besitzt. Die Rechnungen von Eur,nn u. Kocrnr, und von Helsrr.rBERG u. Eur,nn liefern fflr die vier Konstanten die Werte:

": ffi{}u,

f :7o,

I s

e6hT

ffio"/ -t"*'

,

13.

s:2s'

Die in (2) angegebenenDielektrizitiits- und Permeabilitiitstensorenergebensich aus den Beziehungen:

too , Uber die Elektrodynamih des Vakuums.

D,:

AI. 4- .n-f r , dE;

11

AL H - - ' - - 4 -f 't "AB;

Wir werden im folgenden diese Resultate auf eine wesentlich einfachere Weise herleiten. ) Der Zasa\z U' zur MAxwer,I,'schenEnergiedichte des Vakuums soll durch d,en Zusatz D' bestimmt sein, den die Vakuurnelektronen beitragen,Die Energiedichtebei Anwesenheit von Elektronen in den ZustAnden l)t,,Vz. . . r/, . . . ist gegebendurch

U_

1

s , (82+B\+ fr',

fiu,: f {v:, l ( " ' f*,uu*,i)+n^elvl L\

wobei i, P ai" Drnac'schenMatrizen ora i das vektorielle Potential ist. Der Zusatz t' ,o, Mlxwrr,L'schen Dichte ist somit nicht gleich der ganzenmateriellen EnergiedichteU*", (5)

s--f U*", : ih > .lrtt,., A I *Vr|

sondern (6)

fr' : (J^^r-Z{rp,*, eV!)i}

wobei V das skalare Potential ist. Man kann U' als die kinetischeEnergiedichtebezeichnen.Die gesamtematerielle Energiedichte U-r* lisst sich, wie wir sehen werden, leicht berechnen;der zweiteTerm von (6) - die potentielleEnergiedichte- ergibt sich aus U-,, in folgenderWeise: S'enn x Zwei Eigenfunktionen ry' und g in geschwungenenKlammern , {rp, r} bedeutet hier und im folgendendas innere Produkt der beiden Spinozen q tr. p: : rlroi*, wobei ft der Spinindex ist. {V, ,} / k

rol Nr.6. v. wntssxoPr':

12

konman sich das skalare Potential proportional zu dem stanten Faktor 2' denkt, so gilt:1

(i)

;\:rv,* ,

eVtPr)dr

: ^#\(J^u,dc

rrobei die Integrationen sich iiber den ganzen Raum erstrecken.Im Grenzfall konstanter Felder, den wir hier wegen der Bedingungen (1) betrachten wollen, kdnnen wir die Feldstarke ,E selbst als den konstanten Faktor l' ansehen' und kdnnen ausserdem die Beziehung (7) auch auf die Energiedichten fibertragen. wir erhalten dann fiir die kinetische

Energiedichte (i a)

i i , _u

ATI r at _E--mar. -r m AE

Vergleicht man dies mit (S) so sieht man, dass zwischen der materiellen und der kinetischen Energiedichte dieselbe - L und U' U-u, kann also Beziehung besteht, wie zwischen hier dem durch die Vakuumelektronen hervorgerufenen Zusalz zur Lagrangefunktion

(8)

1- 1 mar

gleichgesetzt rverden: -f)

:

ir

Da die Form von U' weitgehend durch die relativistischen Invarianzforderungen festgelegt ist, so gentigt es, U' fflr ein 1 Der Beweis liiuft folgendermassen: Wenn der Energieoperator 't[ von II,,'des einem Parameter I abhiilgig ist, so lndert sich das Diagonalelement dl von Anderung adiabatischen infinitisemalen einer Energieoperators bei it um:

l0 H\

| * | aA, \oA /ii

dHii:

Wenn wir Dun setzen:

H :

Ho* heV

so gilt dann:

l . ( e v ^) , :

0 H,' X ^oL

102 Uber die El.:ktrodynamikdesVakuums.

13

spezielles Feld zu hestimmen. Wir wAhlen ein homogenes (B*,0,0) uud ein dazu paralleles magnetisches Feld n: riumlich

periodisches elektrostatisches Feld, dessen Poten-

tial durch ige

(e)

V:

-_!9e

v o e h+ V o * e n

gegeben ist. Wir vergleichen dann dieses Resultat mit dem allgemeinen Form (4) und x'erden daraus die Koeffizienten dieser Form bestilnmen. HetsBNsnnc u. Eulpn

rn'iihlen im Gegensatz hierzu ein

konstantes elektrisches Feld, wodurch Schwierigkeiten infolge des Kr,nrx'schen Paradoxons entstehen: Jedes noch so schwache homogene elektrische Feld erzeugt Elektronenpaare, wenn es sich iiber den ganzen Raum erstreckt. Die Elektronenbesetzung der Energiezustinde ist dann nicht exakt stationir. In der vorliegenden Rechnung kann durch die Periodizitit vermieden werden, dass Potentialdifferenzen ffber 2 mcz vorkommen, sodass keine Paarerzeugungenstattfinden. Die materielle Energiedichte ist bei voller Besetzung aller negativen Energiezustiinde gegeben durch

(10)

U-ut :

Zw,{rp,*, rlt}

W, ist die zur Eigenfunktion r/, gehorigeEnergie,summiert *'ird rlber alle negativeZustflnde.Die Summe ist selbstverstindlich unendlich.WelcherendlicheTeil dieserSummevon physikalischerBedeutungist, rvird sich eindeutig aus dem explizitenAusdruck fiir U-., ergeben. Die Q, befolgendie Wellengleichung: ( 11 )

-0 a,ih"*.4r *,-+* {+

r03

ru

Nr.6. V. Wnrssropr:

(t2)

K : auih.f *,1,^*-ilnlo)- u^". u*

Wir folgen vorl5ufig der Rechnung HnrseunERGSu. Eur,ans l. c., wobei wir nur unwesentliche Anderungen in der Bedeutung der Variabeln anbringen. Als Ldsung setzen wir an:

o"' 'u (c)

: =J:ro ,p, '

(13)

X(p).

V2nh

Der Operator K ergibt, zweimal auf tp angewendet:

K'rl,:

\2

|

I

^32 e o'u - i aoo e h |, r^ , |+ L(r. + i I n I a + *' "' . ) )r/ "= o, l-

Wir setzennun

1 . . - 2 P " ht\/ ,

,^t -:

\u+

b ,,

2or,

^ 1-: ? !- !t IrJot r. 1

b ist das Mass des Magnetfeldes.Durch Einfrihrung von ? erreichen wir, dassff die Form einer Oszillator-Hamiltonfunktion erhilt. Wir setzendaher

u(D: fi^(n)(+o),,, wobei fi,Q) die n-te aui 1 normierte Oszillator-Eigenfunkund rion ist. Dann gilt \1" ful l'ac :1

(14)

t(tp :

( (

+ b lmz c2"

(n* \

*))* '.

/l

,

6,:

i ,ou,.

Es liisst sich nun eine Darstellung der 4-komponentigenp wihlen, in der o, diagonal ist:

104 Uber die Elektrodynamik

""-\3 ,_(;

00 10 0 -1 00

des Vakuums.

15

3\ i/

I)en ersten beiden Komponenten von r/ entspricht dann ein positiver, den beiden andern ein negativer Spin in der c-Richtung. Bei dieser Wahl ze?fiillt die Wellengleichung (11) in zrvei getrennte Gleichungssysteme fiir die beiden Komponentenpaare mit

gleichem Spin, sodass wir zv,ei Wellengleichungen mit zweireihigen Matrizen gewinnen. Der Operator K l:isst sich dann in der Form K : flK| schreiben, rvobei I eine zweireihige Matrix ist, die die Bedingung /2 : L erfrillt und l1{l die gewdhnliche Zahl

^,",+b(,+=)2 t lKl: lf y \ bedeutet, die vom Wert o, des Spins abhiingt. Ebenso ist die in der Wellengleichung auftretende Matrix a, zweireihig und ist mit 7 antikommunativ i a, T * f o, : 0, da a, nach (12) auch mit K antikommutativ ist. Die beiden Wellengleichungen lassen sich dann in der Form

(16)

q,: o tKt) ff f,. *,ih,3o-?r*/

schreiben,wobei a, und 7, zweireihige Matrizen sind, die sich nur auf ein KornponentenpaargleichenSpins beziehen. Der Unterschied in der Wellengleichung fiir die beiden Spinrichtungenliegt nur in dem verschiedenen Wert von lKl. Nachdemdie Abhnngigkeitvon p von den Variabelny and z durch (13) bereits festgelegtwurde, stellt (16) eine Wellengleichung ftir die Funktion X(r) allein dar. Bisher ist der

105

16

Nr. 6. V. Wrrssropr:

Rechnungsgang im wesentlichen identisch mit dem Hnrsrn_ B E R G su n d E u r , e n s . Nun behandeln wir vorerst den Fall V: 0. Die Eigen_ werte und die normierten Eigenfunktionen fiir (16) lauten

(r7) xf+)(r,,):a(+) @) ^ * ,J:"t{.PP' l/2nh (18)rvj(p"): + clg+lxP: rc rt.@ / Der obere Index (f) oder (-) unterscheidetdie Zustiinde positiver und negativerEnergie. dr (p) ist ein normierter 2-komponentiger>Spinor<. Die Gleichung (tO) und ihre L6sungen(17), (18) stellenein eindimensionales Analogon zu Drnrc'gleichungdar, in dem ylff lrp statt des Massengliedesf mc'4t steht. Zu einem Impuls p, gehdrenein positiver und ein negativerEnergiewert.(Die beiden andern Energiewerteliefert die Wellengleichung mit entgegengesetztem Spin). Setzenwir nun diese Grdssenin die Energiedichte(10) ein, so erhalten wir _+t,

.'",", 1{J

_@

!#orp,) |n@p(#)+r .4'(p,) r'. 55 rtl

riber P" liefert infolge dp": I,Er, :-:._,t::,-"tion, 1x"'\P)1": Zrn +1

(1e)

o

a-@

u-ut: *-!,* Z Z \ apw;rr). 1+€ n:0 6:-l

Von hier ab schreibenwir p statt p,. Um die Summation durchzufrihren.bilden n'ir

trru

r06 0ber die Elektrodvnamik

z

\6:

-|

w;:

n:0

des Vakuums"

\_-r

w ; + 2 .1-J)

t7

w;

n:1

Wir verwenden nun die Eulnn'sche Summenformel fiir eine Funktion F(r):

: *''D+rr (a* Nb) tr',"r+:= F(n :

r ffo*"u

- 1 e r e)^ B b':'r\FQ"'-"(o dr + Nb)2 ; L):,") ffi>, _

F\2m

t,(o))

, I|

B,,,,istdie m-te BeRrvour,r,r'sche Zahl. F"" (*) ist die m-te Ableitung von F(r). Wenn wir dies auf (19) anwenden,erhalten wir:

: ^,t*\*lllo, a*+i u-o, ai'

(20)

"

Q)

LtJo

f(r):

*-"-t

-rVp'117f

u'"';#.1-;"'nt2"'-rt 1o;l s2ls.

In dem Spezialfall eines reinen Magnetfeldes, kann man nach (7 a) U-u. und fr' gleich setzen. Dieser Ausdruck stellt bereits die Energiedichte dar, in einer Entwicklung

nach

Potenzen der magnetischen Feldstiirke b. Nun ist es sehr leicht, jenen Teil des Beitrag". fr' d", Vakuumelektronen zu bestimmen, der fiir das wirkliche Vakuum massgebend sein soll: Das von b unabhdngige Glied stellt die Energiedichte des feldfreien Vakuums dar und ist ein divergentes Integral; da die Energiedichte frir das feldfreie Vakuum verschwinden muss, kann dieser Ausdruck keine reale Bedeutung haben. Weiter miissen die (iibrigens auch divergierenden) Glieder mit

bz weggelassen werden, da die Energiedichte keine

Vldensk. Selsk. Math.-fys. Medd. XIV,6.

t07 18

Nr. 6. v. welssropr:

Glieder zweiter Ordnung in den Feldstiirken besitzen soll. Das Weglassen dieser Glieder ist durch die Annahme begriindet, dass die Polarisierbarkeit des Vakuums mit verschwindenden Feldern gegen Null strebt. Es sei hervorgehoben, dass die hier vorgenommenen Subtraktionen ausschliesslich auf triviale Annahmen iiber das t'eldlose Vakuum beruhen. So erhalten wir ftir denZtsalz zur Mlxwnr,r,'schen Energiedichte:

(21) u': - +#n +;]' 6"'rs . ( a m - b ) z

+@

dp

7

,2nr

4 m-3'

(pz l mz cz) z

-a

Diese Potenzreihe l6sst sich leicht durch die Potenzreihenentwicklung des hyperbolischen Ctg darstellen. Man erhilt:

U,:

- r-f *') ctszab # ^r (T)'ffr* {za5

wobei !5 die magnetische Feldstiirke gemessen in Einheiten m'c" der kritischen Feldstirke - - , ISt l en

a

:

eh *D'

m'c"

Das erste und zweite Glied der Entwicklung liefert:

[J':-

!!.Bn+= 360n2 m 4 C 7 "

'

- 1" 4 q , B 6 + . . .

6}0nz m8c1

Wenn wir dies mit jenen Gliedern von (4) vergleichen, die das Magnetfeld in 4, und 6. Potenz enthalten, bekommen wir: *

I eah 369o2 ^a"7'

!

I

6so;P

e6h3 -m%i5'

r08 iiber die Elektrodynamik des Vakuums.

l9

Es werde nun das elektrischeFeld mitberficksichtigt.Zu diesemZweck l6senwir die Wellengleichung(16) fffr X(c) mit der BonN'schenNflherungsmethode.Wir ers'arten, dass die vom Potential V abhiingigenTeile von L'-., in der zweiten,za V2proportionalen Niiherungerscheinen.Wenn s'ir [.'-r, nach Potenzenvon y entwickeln: U-,t : tljl^+ Cll,,*... so erhalten wir nach (10):

(22) uSl,:

1

,;," ( ,/,1)(o) . * u'a',(,r,,1)(', ?

w;T' ,(l ,/,lt)(*) sind die k-ten Niherungen in der entsprechenden Entwicklung von W, und I r2,12.llan beachte,dass (1)'n W. dem angegebenenelektrischen Feld rerschwindel Es llsst sich leicht zeigen, dass

: i
un: zw,,r,(,/,1';,0,. (l ,/,In)(o) wurde

bereits im Falle des reinen llagnetfeldes

berechnet und wir erhalten somit ganz analog zu (19) +1

ig " m a t.: =]_,_;s g n z h t Lfr L n u f

-

r*r

,,; ,',(p). \oo )1..

(2) Der Wert von W. lisst sich mit der Bons'schenNiherungsmethodeberechnen.Mit den Eigenfunktionen(17) ergibt sich: ,.

109

20

Nr'.6. V. Wnrssxopr:

ot-)(p)} l' * w;Q)(p): ezlvol, il{"t*ilptg), W Q ) w : ( p I 0 L (23)

'@}f <st-q+i' * l{:i=. w"(il_w;(p*s) I* l

* ( d a s s e l b em i t - q ) . Der zlvischen den ( )-ftammern stehende Ausdruck stellt ein skalares Produkt zweier zweikonponentiger Spinoren dar. Bei der Integration von (28) iiber p fallen die zweiten Glieder in den [ ]-Klammern \\,eg, \,enn man die Integration der Glieder mit-g mit der Variabeln p': p_g ausfiihrt:

(p+ g),ot-)(p)]l' : erlv^lrf * ' | '(P) " o, .,u - w'^*) w'; (P+ g) ' J {soo*r"' (l+)*

*(dasselbe mit -.q).

Dieses Vorgehen ist im allgemeinen keineswegs eindeutig, da die Integration des zweiten Gliedes in den [ ]_Klammern von (Z'l) zu einem diver.genten Resultat fiihrt, das aber nach Addition des entsprechend.enGliedes mit - g endlich ge_ macht werden oder, wie in (24), zum Verschwinden gebracht werden kann, je nachdem in welcher weise man die Integrationsvariabeln wiihlt. Diese willkiir bertihrt aber ursere Rechnung nicht, da wir nach Ausfiihrung der Summation tiber n nur die zv b2,b4,etc.proportionalen Glieder verwenden, in denen auf Grund der EurnR'schen Summenformel nur (2)(p) Ableitungen von nach n auftreten. Wie man sich 4 leicht iiberzeugen kann, divergieren diese Ableitungen des zweiten Gliedes in den [ ]-Klammern nicht mehr bei der Integration tiberp, sodassdas Resultat dieser Integration un_ abhAngig von der Wahl der Integrationsvariabeln ist. Es ergibt sich weiter aus (24):

tr0 Uber die Elektrodynamik des Vakuums'

27

s;5,o rotrt" : - e'lvrf ##ht S von I wobei beleits eine Reihenentwicklung nach Potenzen zweiter durchgefiihrt wurde und die Glieder hoherer als wurden. Dies bedeutetdie VernachOrdnung weggelassen der Iiissigung der Ableitungen der Feldstiirke auf Grund ist Bedingungen(1). Ebenso,wie in der vorigen Rechnung' ,ron Ep", durch (20) gegeben,wenn man setzt:

F(r):-r'lvoft "##;rt iiber p

Man erhiilt dann, wenn rnan zuerst

integriert:

: - #* rtrtro' ,Sl, t5,"4. -- f ?,,:,

Bn'(-)"'(o':"-t,--=\ oz,, 1 " m'co*t)*:ol (2m)l \drt--t

Da dieser Ausdruck

quad'ratisch in

den elektrischen

Feldstiirken ist, erhalten wir fiir die kinetische Energiedicfte nach (7a)

- uSl,. fr'(2):

Glieder Aus den friiher diskutierten Griinden kdnnen erst die Vakuum das 4. und hdherer Ordnung in den Feldstiirken fiir divergierende von physikalischer Bedeutung sein, sod'assdas V6 durch die Integlal wegzulassen ist. Wir ersetzen nun elektrische Feldstiirke,E:

E :2fitu,r

ilt 22

t{". O. V. Wnrssxopr.:

\Nobei die Querstriche Raummittelungen bedeuten und erhalten ftr die ersteu beiden Glieder: y 7 r ( 2 )/(25) otr\

Ut\''

5

e{/r

o . 2-R g _ 7

lrOJ; itf cr-

|

"ahs

2 6*0nzffin'Ana...

frir den Grenzfirll schrrach rerinderlichen Felder bei welchem die Raummittelungen rteggelassen rverden k
t-2a:##,

Br--s':*#"

e6hz *s ata

t

und mit den bereits berechneten Werten von d und ,g

i:7o,

r:fq5.

Der in der magnetischen Feldstlrke exakte !.usdruck fflr f"") ergibt sich zu 1r(21

I o / m c \ a *l , f - o n - { ( z a S C t g z E-r), B-r^r\o/ S""\;, do

q'obei g:

irt. *E' m-c" Die hdheren Ndherungen in E lassen sich leicht bis auf einen konstanten Faktor bestimmen. Denken wir die /c-te Ndherung lI; G)(p) der Energie des durch p und n gegebenen Zustandes bestimmt; sie wird. auf Grund. derWellenglei_ chung (16) die folgende Geslalt hatren:

W r o ,( p ) : g o" o| % l o .G( c ,h , l K l , p ) wobei G eine Funktion ist, in welcher nur die angegebenen Grdssen vorkommen. Ulo)-o., infolge der Eichinvarianz min-

112 Uber die Elektrodvnamik

des Vakuums.

23

destens lc-ter Ordnung in g sein. Die hdheren Potenzen von g sind vernachlissigt. Die Energiedichte in k-ter Ordnung wird dann: ,,{K'

U-,t: (26)

-1 -

k

*r%r'tli5!j,. #e^ffi\;j,).J

e t 4 7 1 ; 'S € , \ -^L * r u nr:1

(A'-1) Das Integral iiber G muss d.ie Dirnension (Energie) (Impuls)-(k-1) haben, und darf nur mehr von den Gr6ssen c,h,lKl

abhiingen, was nur in der Form mdglich ist:

P+*

5

t:t : f* -Lroywz: fo;-'C#;i'-:r "t l

wobei fo ein Zahlenfaktor ist. Wenn man dieses in (zO) einsetzt, so lisst sicn U$),, lis auf den Faktor f* vollstiindig angeben. Die Zahlenfaktoren f1 bestimmen sich aber leicht durch die Uberlegung, dass U-*, nach (8) eine relalivistische Invariante sein rnuss. Da deswegen U-o, nur von E2-82 und (EB)' abhiingen darf, rnuss z. B. der Koeffizient von Ek sich von dem Koeffizient von d ttot durch den Faktor k ,.; (-)2 unterscheiden. Der letztere Koeffizient wurde bereits berechnet und ist durch (21) gegeben. Man erhilt dann

fzn ' .:

';::'-" 1: rl(2rn-l)

u' und kann damit d"ieDarstellung berechnen,die HstSeNBERG Eurnn ftrr L'angegeben habenr: 1 In der Frage der Konvergenz dieses Integrals verweisen wir auf die u' Eur'en S' 729' diesbeziiglichen Bemerkungen in derArbeit von Helsrxsrnc

II3

24

L,:

-

Nr. 6. V. \['nrssxopr': aa*

#-" (T)'

\fr

- r+rf<,e' - *,1} ,?{zuctez o.q@cts,;s

g:'t!n, en

E:^',"'B. eh

Dieser Ausdruck ist fiir parallele Felder berechnet. Um ihn auf beliebige Felder zu verallgemeinern, muss man ihn als Funktion der beiden Invarianten Ez - Bz und (EB)z schreiben. Dies ist nach HETsTNBERGund Eur,en in ein_ facher Weise mit der Beziehung

C t g a c t g P : - i g- 9 @

cos1/F-Faznp-conj

moglich und man erhiilt

-a5t+ 2l(qq5)+""4+ rr ,t : - ;I , - \e e' l - - - d,nt . : l r i ? z ( E"Do i ? f , S = . 6n' nc r" cosqlGt-!09+2r'(qs)-conj )0 .

2,.t

,

|

,/;;--

I''

+5;:+trn-'r| wegen der Realit:it des Gesamtausdrucksist diesertatsiichlich nur von Ez - Bz und (EB)z abhiingig. Die Berechnung der Energiedichte und Lagrangefunktion des Vakuums ist in der skalaren Theorie des positrons mit den gleichen mathematischen Hilfsmitteln durchzuffihren. Eine Energiedichte des vakuums entsteht in dieser Theorie durch die Nullpunktsenergie d.er Materiewellen. Die Gesamtenergie ist nach P.q,ur,rund Werssropr L c. (Formel (2g)) durch

E_ut: Zw*(Nf +N;+1) ft

ll4 25

Ober die Elektrodynamik des Vakuumt'

gegeben,wobei l4/ftdie Energie des /c-ten Zustandes ist und Nf, die Anzahl der Positronen, No die Anzahl der Elektronen ist' die diesem Zustand angehdren. lrn leerenvakuum bleibt die Summe iiber alle Energien IVo iibrig, $'obei die Energie Wo des durch den Impuls p und d.erQuantenzahl n charakterisierten Zustandes in einem Magnetfeld B den Wert hat:

wlu"t{p,n1 : "llo'*

b('+;). tttzc2+

Die summation iiber alle Zustinde und Division durch das Gesamtvolumenftihrt zur Energiedichte,die sich leicht analog zu (19) ergibt:

u-,t :

*

r--f*'

\a l -apwlu" (p, B). - -), nr tt 07[ " -]ro a

Der einzige Unterschied gegen die fruhere Rechnung besteht in dem Wegfallen d'er Summierung riber die beiden Spinr.ichtungen. Nun verifiziert rnan leicht die folgende Beziehung in der skalaren und der zwischen der Energie wln"'{l,n) in der DIna.c'schen Elektronentheorie: Energie W^(f,B) +1

N

,)'r*' n:0

+1

N

2N

(p, B) 6:-1

n-g

d:-l

n:0

wir konnen daher die Energiedichte in der skalaren Theorie 0iuu, durch die Energiedichte d'd". Drnac'schen Positronentheorie in folgender Weise ausdriicken:

' 2fr1u", (B) : fr'(D - 2'if(ttlz)' Man sieht daran, dass auch hier der von den Feldstirken unabhiingige und der in ihnen quadratische Anteil unendlich ist. Der letztere liefert also eine unendliche von den Feld-

il5 Nr. 6. V. Wrrssropr:

stirken unabhengige Polarisierbarkeit. Um ftlr das feldlose Vakuum ein brauchbares Resultat zu erhalten, muss man wieder diese beiden Anteile streichen und erhiilt infolge der Beziehung

cteP-zctgf: -r#,

: - i# ** ("';)' lit-'{, * .*--*- 1+'Lr,rl L,.r.u, \ ,n-

a,/(o)

Die Durchfiihrung einer analogen Stdrungsrechnung im elektrischen Feld fiihrt in gleicher Weise zu einem Zusatz zar Lagrangefunktion des Feldes, der mit dem aus der Drnlc'schen Positronentheorie gewonnenen sehr verwandt ist:

f' Lskal

-

-i"';\fr''{

2 i&EB)

cosql/(@2- E') + 2 i(qa5) - conj

- E\\. +^"^.t-Lgz 'l e"h" tt' Fiir die in (4) definierten Koeffizienten a,p erhilt man daher: 7 | eah 1 6 B 6 o n ,^ n { '

u:

^ P:

1 o' 7

Es sei hier noch auf folgende Eigenschaft der Lagrangefunktion des Vakuums hingewiesen. Frir sehr grosse Feldstdrken .E oder B haben die h6chsten Glieder des ZusatzesZ' zur Mlxwnr-l'schen

Lagrangefunktion in der Drnnc'schen

Theorie des Positrons die Form qq

L' .- -

o-

--". - Es lS C 24nzhc-

Bt -lgo q5. -' bzw. L' ..r ^ , :* Z4nohc

Das Verhiiltnis zwischen diesem Z u s a t z l ' u n d d e r M n x 1^ wer-r-'schen Lagrangefunktion Zo : ^ (E'- Bo) ist somit 67r

tl6 Uber die Elektrodvnamihdes vakuums.

27

logarithmisch in den Feldst6rken fiir hohe Werte derselben multipliziert: und ist ausserdem rnit dem Faktor ! hc

"u*-jorrt n,*.fi .-Die Nichtlinearititen

5,i1;.rsu'

der Feldgleichungen stellen somit auch

bei Feldstirken, die wesentlich h
der Feldgleichungen

mit der nichtlinearen Feldtheorie von BonN und Iltpst-l' ist daher nur dusserlich. In der letzteren Theorie sind die Gleichungen bei der kritischen Feldstiirke Mlxwnll'schen -2.4 Fn" e: ' "'"" nam Rande des Elektrons< bereits vollkommen abgedndert, .n'odurch dann die endliche Selbstenergie einer Punktladung erreicht wird. Hier hingegen sind die Abu'eichungen von d.en Mlxwnr,l'schen Feldgleichungen frir Felder d.er Grdsse Fo noch sehr klein und wachsen viel zu langsam an, um eine ihnliche Rolle in den Selbstenergieproblem zu spielen. Die Extrapolation der vorliegenden Rechnungen auf die Felder am ))Rande des Elektrons<
in dieser Hinsicht

ein wesentlich verschiedenes

Resultat liefert. III. In diesem Abschnitt soll der Einfluss beliebiger Felder auf das Vakuurn behandelt werden. Wir beschrinken uns vorerst auf statische Felder. Die stationflren Zustinde des t [{. Bonn u. L. INner,n, Proc. Roy. Soc.143, 410' 1933.

117 Nr. 6. V. Wnrssxopl.:

Elektrons auf Grund der Drnac'schen Wellengleichung und ihre Energieeigenwerte werden sich im allgemeinen in zwei Gruppen einteilen lassen, die bei einer adiabatischen Einschaltung des statischen Feldes aus den positiven bzw. aus den negativen Energieniveaus des freien Elektrons entstanden sind. Dies trifft z. B. im Coulombfeld eines Atomkernes zu und bei allen in der Natur vorkommenden statischen Feldern. Es sind jedoch auch solche statische Felder angebbar, in denen eine derartige Einteilung versagt, da infolge der Felder Ubergiinge von negativen zu positiven Zustinden vorkommen. Ein bekanntes Beispiel hierfiir ist eine Potentialstufe der Hohe )

2mcz. Diese Ausnahmsfiille sind sta-

tionir nicht behandelbar und miissen als ein zeitabhingiges Feld betrachtet werden, das zu einer gegebenenZeit eingeschaltet wird. Dies ist umsomehr schon deswegennotv'endig, da derartige Felder infolge der fortwdhrende Paarentstehung gar nicht stationlr

aufrechterhalten u'erden k6nnten.

Im Falle dass aber das Eigenwertspektrum sich eindeutig in die beiden Gruppen einteilen liisst, kann man die Energiedichte U und die Strom-Ladungsdichte i, g der Vakuumelektronen nach den Formeln ff

_

?t'

(2e)

o-

*' irrl

t ih f lrt,,

" ){'!f

v)

->

;-

" / t {p,* "irp)

berechnen, wobei iiber die Zustinde zu summieren ist, die den negativen Energiezustdnden des freien Elektrons entsprechen. Die angeschriebenenSummen werden divergieren.

tr8 29

Uber die Elektrodynamikdes Vakuums.

Wenn aber die physikalisch bedeutungslosen Teile abgetrennt werden, erhalten wir konvergente Ausdrtcke. Um diese Teile auf Grund der Annahmen (I) festzulegen, entn'ickeln wir die Summanden der Ausdriicke (29) nach Potenzen der iusseren Feldstdrken in der Weise, dass wir uns die letzteren mit einem Faktor l. multipliziert denken und nach Potenzen dieses Faktors entwickeln. Dieses Verfahren ist identisch mit einer successivenStorungsrechnung' die von den freien Elektronen als nullte Niherung ausgeht' Die Annahmen Ir, I, verlangen vor allem das Verschwinden der von l, unabhiingigen Glieder, die aus den Beitriigen der vom Felde unabhiingigen ffeien Vakuumvorldufig nur jene freien p Vakuumelektronen beriicksichtigen, deren Impuls lp I < ist, so erhalten wir hierftir folgende Beitrf,ge:1 elektronen bestehen. Wenn wir

tP uo: -#o'\d;

'tiF+;'.TF'

lPl
(30)

Q o:

"

f t

4#F\dP, lpl
--->

io

e (-

6

+,"';f)anffi lPl
Die Beitriige simtlicher

Elektronen

'-

P-+oo

-

diver-

gieren natiirlich. Durch das Abtrennen der von i,'unabhflngigen Glieder vollstiindig erftrllt' ist aber die Annahme I, noch licht Vakuumfeldfreien Die Ladungs- und Stromdichte gs, iq der 1 Es ist niimlich der zum tmputsI die Anzahl der Zustinde

l"no:tg"

-!!--.. in Intervall d;', ' 4n3hd

st"o-,

^9-

--

ood

ll9 Nr. 6. V. Wprssropr:

elektronen dussert sich niimlich

auch dadurch, dass sie

bei Anwesenheit von Potentialen V, A einen Zasatz qoV und (io A) zar Energiedichte liefert, der ebenfalls abgetrennt werden muss. Diese Zusitze treten auf, da auch die Energie und der Impuls der von den Feldern noch unbeeinflussten bei Anwesenheit von Potentialen um 'T geandert u'ird. den Betrag eV bzw.

Vakuumelektronen

Die Annahmen I, und I, sind daher erst vollkommen erfrillt, wenn man die wegzulassendeBeitriige (30) folgendermassen modifiziert:

: -#"\,;l,y 1,*-7^1' u'o * *c+,y] lpl
,

(31)

Qo :

e f ,4;F;p)ap lpl
-r;: #*\6 lPl
"(i+ii)

/Q*?^)'

* mzcz

wobei auch wieder nur der von freien Vakuumelektronen mit Impulsen lp | ( P herriihrenden Teil angeschrieben ist. Nun ist noch die Bedingung I, zu erff,llen. Hierzu beachten wir, dass eine konstante feldunabhringige Polarisierbarkeit zu Gliedern in Energiedichte U(r) fiihrt, die proportional zu den Quadrat Ez (c) und B'(*) der Feldstiirken an der Stelle r sind. Ebenso frihrt sie zu einer Strom- und Ladungsdichte, die zu den ersten Ableitungen der Felder proportional ist, auf Grund der Beziehungen

i :

rotM*qO,

9:

divP

r20 Uber die Elektrodynamikdes Vakuums.

31

worin M und P die magnetischen und elektrischen Polarisationen sind, die im Falle einer konstanten Polarisierbarkeit zu den Feldern proportional sind. Um die Annahme I, zu erfiillen, mtissen daher in der Energiedichte U der Vakuumelektronen die zu .Ez und zu Bz proportionalen Glieder verschwinden und in der Strom-Ladungsdichte die zu den ersten Ableitungen proportionalen Glieder weggelassen'rn'erden. Es ist praktischer, die Form dieser Glieder nicht explizit anzugeben, sondern dieselben im Laufe der Rechnung an ihren Eigenschaften ztr erkennen. Als Erliiuterung berechnen wir die Ladungs- und Stromdichte des Vakuums unter dem Einfluss eines elektrischen Potentials i(s r)

(32)

V:VoeT*Vo*e

i (g.) h

und eines magnetischen Potentials

(33) mit

-> -- {i,t A:Aoeh

--> _dn h,

+Ao*e

-->_> (Ao,g):0

Hilfe der St6rungstheorie. Diese Berechnungen sind

bereits von HnrssNBERGr und noch viel allgemeiner von SeRneneund Paur,r u. Rosns durchgeftihrt worden und sollen hier nur als Illustration

zu unserer phvsikalischen Inter-

pretation der Substraktionsterrne dienen. Die Ladungsdichte g ist bis zur ersten Ordnung gegeben durch q:

g(0)+Q(1),

e,":"xxry#*conj irr

t Hnrsnrnnnc,Z. f. Phys.90, 209, 1934. ' R. Snnnrn, Fhys. Rev.48, 49, 1935. 3 W, Peur-r u. M. Rosr, Phys. Rev. 49, 462, 1936.

12r 32

Nr.6. v. wrrssxopr':

wobei die i flber die besetzten. die ft iiber die unbesetzten Zustinde zu summieren sind, und II,r. das Matrixelement der St6rungsenergie ist. Setzen wir das Potential (32) als Stdrung ein, so erhalten u'ir (II/(p) : ( 1 ): - e z t , o f , : 1 $ ' ( p )

e

t/p"l,FF)

w ( p * g ) - c z( p , p * g ) - m zc + ,

-

s " r 6 a - '")pl r n p 19 ' +, /dasselbe\l - s)i '" t cons \,,ri t

Entwickelt man dies nach Potenzenvon g'so erhiilt man tni

(1) e2I'n (' ,' oor';c2(p)'o e"': ;# )

cn(ps)'s' _4 tlpdls: _"'(pg)' _ c' sn *2s ' ' . . . I r nnni 8 wu(p) f 12 w'(p) 4w2(p) t6 t{'n(p) Welcher Teil dieser Ladungsdichte hat nun physikalische Bedeutung? Wegen der Annahme I,

muss 96 wegfallen; in g(t) sind die Glieder mit 92 proportional zur zrveiten Ableitung von V und somit zur ersten Ableitung der Feldstirken

und mflssen wegen I, weggelassen werden. Man

bemerke, dass auch nur diese Glieder zu Divergenzen ftihren. Der Rest liefert endliche Integrale und ist nach HusTNBERG in der Form .

,r,

(34) e"' :

1

pz / h\2

i;#;

(;")^

A v * h o h e r eA b l e i t u n g e n v o n v

zu schreiben. Die exakte Ausrechnung der ersten Niherung q'urde von Snnnnn und P.tulr u. Rosr l. c. geliefert. Als weiteres Beispiel betrachten wir die Stromdichte i in der ersten Niherung des Feldes (BB)

iu,-"|ry#!#*conj

122 Uber die Elektrodynamik

33

des Vakuums,

Es ergibt sich

ti,:-*#!ri

{sD eh

w w
\mit - sl)

.->

wobei n der Einheitsvektor in der Richtung A ist. Dieses ergibt naph g entwickelt: .(1) l.:

-#\,,w
(35)

* {t',o,

.Xtffr#*

c'(np)'-

c'g',

3 cn(pg)'

5 c 6( n p ) z( p g ) ' ,

2 -4wr@)-i--wn1oy--

4.u.hoherer Glieder o.d,'o.si,.s).

Hier treten auch von g unabhflngige, also nicht eichinauf. Diese sind aber mit den wegzulassenden variante Glieder --> + Beitrdgren ii aus (31) identisch: Entwickelt man nnmlich i'o nach A. so erhdlt man:

-,,: #,*!,itffi *Ra_"Wr.) +

-? 1 , f * : [+ ,cA (w 'Q) - c2( np) z*.' .]. n " ,-tp )d t w " .o = "'

Die Glieder erster Ordnung in 7 stimmen mit dgn von g unabhiingigenGliedernin (35) iiberein.Die zu 92 proportioder Rest nalen Gliedervon (35) werdenebenfallsweggelassen, ergibt das zu (34) entsprechendekonvergierendeResultat: Vidensk. Selsk. Math.-fys. Medd.xlV, 6.

123 34

Nr. 6. V. Wrrssxopr:

.(r)

ez / h\2

__->

; t: J A AA * h
Die beiden Beispiele sollen zeigen, dass bei einer Storungsrechnung die wegzulassendentseitriige unmittelbar erkenntlich sind und dass die iibrigen durch die Annahmen (I) nicht berfihrten Beitrdge der Vakuumelektronen

bei der

Summierung zu keinen Divergenzen mehr ftihren. Die angefthrten Beispiele beweisen dies zrn'ar nur in erster Ndherung. Die 0berlegungen lassen sich aber ohne weiteres auf h6here Niherungen ausdehnen. Die Behandlung zeitabhiingiger Felder ist im wesentlichen nicht von dem obigen Verfahren verschiecien. Es ist notwendig, die zeitabhiingigen Felder von einem Zeitpunkt fo an wirken zu lassen, an welchem die Vakuumelektronen in feldfreien Zustinden waren, oder in solchen stationiren Zustdnden, die sich einwandfrei in besetzte und unbesetzte einteilen lassen. Die zeitliche Verdnderung dieser Zustinde von diesem Zeitpunkt fe, an ldsst sich dann mit Hilfe einer St6rungsrechnung in Potenzen der AusserenFelder darstellen. Die Ausdrflcke (31) und die aus der Bedingung (Ir) folgenden Glieder kiinnen dann abgetrennt werden, wobei der rlbrigbleibende Rest nicht mehr zu Divergenzen fiihrt. Die Berechnung der Ladungs- und Stromdichte des Vakuums bei beliebigen zeithdngigen Feldern in erster Nffherung findet sich bei SennBn l. c. und bei Plur,r u. Rose L c. Die abzuziehenden Teile werden dort aus der HersnNsonc'schen Arbeit forrnal entnommen. Sie sind aber mit jenen, die aus den Annahmen I folgen, vollkommen identisch. Wie iussert sich nun die Entstehung von Paaren durch zeitabhingige Felder? Die Paare kommen bei der Berechnung der Energie-, Strom- und Ladungsdichte nicht unmittelbar zum Ausdruck. Die Paarerzeugung zeigt sich nur in einer

124 Uber die ElektrodynamikdesVakuums' proportional

35

zur Zeit wachsenden Gesamtenergie, die der

Energie der entstehenden Elektronen entspricht. Die Ladungs- und Stromdichte ist nicht unmittelbar von der Paarentstehung beeinflusst, da stets positive und negative Elektronen zugleich erzeugt werden, die die Strom-Ladungsdichte erst dadurch beeirrflussen, dass die Susseren Felder auf die entstandenen Elektronen je nach der Ladung verschieden einwirken.l Es ist daher praktischer, die Paarentstehung durch dussere Felder direkt zu berechnen als Ubergang eines Vakuurnelektrons in einen Zustand positiver Energie. Die Entstehungswahrscheinlichkeit des Elektronenpaares ist dann identisch mit der Zunahme der Intensitiit der betreffenden Eigenfunktion positiverEnergie bzw. mit derAbnahme derlntensitiit der entsprechenden Eigenfunktion negativer Energie infolge des Einwirkens der zeitabhiingigen Felder auf die Zustf,nde, die bis ntr Zeit fo geherrscht haben. Die Berechnung wurde von BBrne u. HrIrr-nnz. Hur,un u. JanceRB u. s, w. ausgeftihrt. Die Paarvernichtung unter Lichtausstrahlung liisst sich wie jeder anderer spontane Ausstrahlungsplozess nur durch Quantelung der Wellenfelder behandeln, oder durch korrespondenzmdssigeUmkehrung des Lichtabsorptionsprozesses. In

der bisherigen Darstellung wurden

die abzutren-

nenden Teile der Vakuumelektronen nicht explizit angegeben, sondern nur ihre Form und ihre Abhiingigkeit von t Die von Snnsnn belechnete Strom- und Ladungsdichte bei paarerzeugenden Feldern ist daher dem Mitschwingen der Vakuumelektronen zuzuschreiben und ist nicht etwa die )s176ug1g Strom-Ladungsdichte<. Die auftretenden Resonanznenner rrihren daher, dass dieses Mitschwingen besonders stark ist, wenn die iussere Frequenz sich einer Absorptionsfrequenz des Vakuums nihert. ' H. BnrnB u. W. Hcr:rr.nn, Proc. Roy. Soc. 146, 84, 1934. u H. R. Hur-uB u. J. C. Jascnn, ,Proc. Roy. Soc.

125

36

Nr. 6. V. WsrssropF,:

den flusseren Feldern bestimmt. Um sie explizit darzustellen muss man ein etwas anderes Verfahren N,dhlen, da diese Teile ja .divergente Ausdriicke enthalten. Hierzu eignet sich die von Dln.q,ceingefiihrte Dichtematrix, die dann vbn Drn.lc und spezieller von HrlseNnnnc l. c. auf dieses problem angewendet wurde. Die Dichtematrix -R ist dur.ch folgenden Ausdruck gegeben:

(r' , k' I Rl r" k") :

4

,lt: (*' , k') !t,(r,,.k,,)

wobei c'

und r" zwei Raum-Zeitpunkte, /c' und k" zwei Spinindizes bedeuten. Die Summe soll iiber alle besetzten ZustAnde erstreckt werden. Aus dieser Matrix kann man dann leicht die Strom- und Ladungsdichtel g und der EnergieImptrls-Ten sor' (Jf, bilden auf Grund den Beziehungen

i : . lim e| e: ,,

rt

(")L,p,,(*' k' I lll r" k")

lirne) (*' k' lRlr" k")

_ttt

i

fi

,,

.l

at

" u; : .,!f, , elrv rY{r')l} 7x'1+ L*-r-A;,1{ton aa :

Einheitsmatrix

Die Dichte-Matrix hat den Vorteil, dass fiir r' * r" die Summation tiber die Vakuumelektronen nicht divergieren, sondern einen Ausdruck el'geben der fiir s' : s" singulflr wird. Man kann nun aus den Annahmen (I) eindeutig angeben, welche Teile der Dichtematrix der Vakuumelektronen fiir 1 Der vollstdndige Energie-Impuls-Tensor besteht aus der Summe von Uf, lu:nd,dem Maxwslr.'schen

Energie-Impuls-Tensor

des Feldes. Oie Ul.

komponerrte ist daher nicht die gesamte materielle Energiedichte, sondern nur die kinetische.

126 Uber die ElektrodynamikdesVakuums'

37

x;'t in die wegzulassenden Teile iibergehen und gewinnt auf diese weise eine explizite Darstellung dieser Glieder. Der physikalisch bedeutungslose Teil der Dichtematrix

s' :

rnuss tlann aus jenen Gliedern bestehen, die von den Feldstflrken unabhengig sind' aus denen, die zu Gliedern in der Stromdichte fiihren, die zu den Ableitungen der Felder proportional sind und aus denen die zu Gliedern in der Energiedichte fiihren, die proportional zum Quadrat der Feldstiirke sind. Ausserdem muss der abzuziehende Teil der Dichtematrix noch mit dem Faktor

- vat)l e,ar, rr': €xp - f,"t ' fii ) lnc J*)E werden, wobei das Integral im Exponenten etin gerader Linie vom Punkte r' zs dem Punkt r" zt) strecken ist. Dieser Faktor fiigt zum abzuziehenden Energiemultipliziert

herImpuls-Tensor gerade die Beitrage hinzu, die davon im riihren, dass die noch ungestorten Vakuumelektronen F e l d e i n e Z u s a t z e n e r g i ee V u n d e i n e n Z u s a t z i m p u " ? i erhalten, und die infolge der Annahme It mit abgezogenwerden sollen. Da der abzuziehentle Teil der Dichtematrix bis auf den Faktor a' hochstens zweiter Ordnung in den Feldstdrken der ist. ldsst er sich durch eine Storungsrechnung aus im Dichtematrix der freien Elektronen gewinnen' Diese Prinzip einfache, in der Ausfiihrung jedoch sehr komplivon zierte Rechnung liegt tler Bestimmung dieser Matrix matheHrtsBunnnc l. c. ztt Grunde. Das Ergebnis liisst sich jeder Gr6sse matisch einfacher formulieren' wenn man bei stets das Mittel bildet aus der Berechnung mit Hilfe der vorliegenden Theorie und aus der Berechnung mit Hilfe einer Theorie in der die Elektronenladung positiv ist, und das negative

127

38

Nr. 6. V. Wnrssxopn:

Elektron als >Loch< dargestellt wird. Das Resultat ist ja ,in beiden Fillen dasselbe. Die Dichtematrix R wird dann durch R' ersetzt:

(*' k' IR'Id' k") :

(r' i t; q,* k')vt,(r"k") (r'k') r/o(a:"k')|, F r:

wobei die erste Summe iiber die besetzten,die zweite Summe iiber die unbesetzten Zustdnde zu erstrecken ist. Der abzaziehende Teil (r'1c'lSlec"/c") hat dann die Form

( * ' k ' l s l r " k " ): &' r^o*'

; 1-Vnr

*" l 1- 'rby l r ' _C | ' .

Hierbei ist So die Matrix R' fiir verschwindene Potentiale, 7 und b sind Funktionen der Feldstflrken und ihrer Ablei.tungen, C ist eine Konstante. Diese Grossen sind bei HnrsENBERGl. c. und bei HBrsEr.rBERG u. Eur,rn l. c. explizit angegeben. Fiir die Ausfiihrungen spezieller Rechnungen ist es praktischer, nicht auf den expliziten Ausdruck HBrsnNsrnc's zuriickzugreifen, sondern die wegzulassenden Glieder an ihrer Struktur zu erkennen. Dies ist vor allem deshalb einfacher, da die rlbrig bleibenden Ausdrrfcke nicht mehr bei s' : s" singulir werden, sodass man fiir die Berechnung derselben gar nicht das formale

Hilfsmittel

der Dichte-

matrix bendtigt. Die Summierungen uber alle Vakuumelektronen fiihren hierbei nicht mehr zu divergenten Ausdrticken. Allerdings eignet sich die explizite Darstellung HrrsnNnnnc's gut dazu, die relativistische Invarianz und die Giiltigkeit der Erhaltungssitze in dem Verfahren zu zeigen. Es ist hieraus ersichtlich, dass die hier beschriebene Bestimmung der physikalischen Eigenschaften der Vakuum-

128 Uber die Elektrodynamik

des Vakuums.

39

elektronen im wesentlichen keine Willkiir enthiilt, da ausschliesslich nur jene Wirkungen derselben weggelassenwerden, die infolge der Grundannahme der Positronentheorie wegfallen miissen: die Energie und die Ladung der von den Feldern ungestorten Vakuumelektronen, und die physikalisch sinnlose feldunabhiingige konstante Polarisierbarkeit des Vakuums. Alle physikalisch sinnvollen Wirkungen der Vakuumelektronen werden mitberiicksichtigt und fiihren zu konvergenten Ausdriicken. Man darf daraus wohl den Schluss ziehen, dass die L6chtertheorie des Positrons keine wesentlichen Schwierigkeiten fiir

die Blektronentheorie mit sich

gefiihrt hat, solange man sich auf die Behandlung der ungequantelten Wellenfelder beschrdnkt. Ich mochte an dieser Stelle den Herren Prof. BoHn, HerseNsnnc und RossNrpro meinen herzlichsten Dank ftir viele Disskussionen aussprechen. Auch bin ich dem RaskOrsted-Fond Dank schuldig, der es mir ermoglicht hat, diese Arbeit am Institut for teoretisk Fysik in Kopenhagen auszufiihren.

lntroduced lnto the This paper deals with the rnodiffcatlons of the vacuum by Dirac's theory of the posltron. electrodynarnlcs unarnbi$ously by of the vacuurrr can be described The behaviour number of electrons occuthe existence of an lnflnlte assurning pylng the negative energy states, provided that certain well d€but oirly those to whlch hhed effects of these elecEons are omitted, can be ascrlbed. It is obvious that no physical meaning The results are identical with these of Heisenberg's and Dirac's in positron method of obtaininE finite expressions rnathematical theory. A simple rnethod is glven of calculating the polarisability fields. of the vacuurn for slowly varying

P o p e r9

r29

Note on the Radiation Field of the Electron F. Brocn nNo A. Nonoslecr* StanJord University, California (Received May 14, 1937) Previous methods of treating radiative corrections in nonstationary processes such as the scattering of an electron in an atomic field or the emission of a B-ray, by an expansion i.n powers of e2/hc, are defective in that they predict infinite low frequency corrections to the transition probabilities. This difficulty can be avoided by a method developed here which is based on the alternative assumption that e2afmc3, hafmc2 and ha/cAp (a:angular frequency of radiation, APFchange in momentum of electron) are small compared to unity. In contrast to the expansion in pow'ers of e2fhc, this permits the transition to the classical limit fi:0.

I. INrnolucrroN -fHE

quanlum theory of radiation has been r successfully applied 10 radiative emission and absorption processes. If the methods which lead to these results are used to obtain more general radiative corrections, a characteristic difficulty arises. This difficulty is clearly visible in the formulae given by Mott, Sommerfeldl and Bethe and Heitle12 for the probability of scattering of an electron in a Coulomb field accompanied by the emission of a single light quantum. If the emitted quantum lies in a frequency range r,l to ufda, this probability is for small frequencies proportional to d.af a independently of the angle of scattering. Taking these formulae literally and asking for the total probability of scattering with the emission of any light quantum, one therefore gets by integration over o a result which diverges logarithmically in the low frequencies. The same difficulty appears in the radiative correction to t h e p r o b a b i l i t y o f B - d e c a y . 3a n d t o t h a t o f o t h e r nonstationary processes. This "infrared catastrophe" is obviously unrelated to the fundamental "ultraviolet" difficulties of quantum electrodynamics, exemplified by the divergent result for the self-energy of the electron. While the latter is already inherent in the class-l-N.ti*tResearchFellow.

External perturbations on the electron are treated in the Born approximation. It is shown that for frequencies such that the above three parameters are negligible the quantum mchanical calculation yields just the directly reinterpreted results of the classical formulae, namely that the total probabitity of a given change in the motion of the electron is unaffected by the interaction with radiation, and that the mean number of emitted quanta is infinite in such a way that the mean radiated energy is equal to the energy radiated classically in the corresponding trajectory.

ical theory, the former has no counterpart there, There is. however. a feature in the classical theory rvhich indicates the cause of the difficulty: If for simplicity one considers only frequencies which are small compared to the reciprocal of the collision time, the mechanism of emission may be described as follows. The amplitude of each Fourier component of the proper field of the electron before the impact retains its value after the impact.a The difference between the new field and the freld proper to the electron in its new motion is the emitted radiation. The significant point is now that 1,, the radiated intensity per unit frequency interval, does not approach zero5 as @+0. Hence I,f ha, which may be taken as an estimate of t'he mean number of light quanta emitted per unit frequency range, tends to infinity as o+0. Since the same result has to be expected in a rigorous quantum-theoretical treatment, one has to anticipate that only the probability for the simultaneous emission of infinitely many quanta can be finite; the probability of emission of any finite number of quanta must vanish.

t Thi, i. i" strict analogyto the mechanicalmodelof an o s c i l l a t oirn i ri a l l y h e l da w a y f r o m i t s e q u i l i b r i u mp o s i t i o n b y a c o n s t a nlto r c e ,t h e v a l u eo f w h i c hs u d d e n l yc h a n g e s . \ \ e s h a l ls e el a t e rt h a t t h e s a m ea n a l o g yh o l d si n q u a n t u m mechanics,whereit finds its mathematicalexpressionin 1N. F. Mott., Proc. Camb. Phil. Soc. 27, 255 (1931); the exnansion of the wavefunctionof an oscillatorwith one (1931). position in terms of the wave functions correAnn. Physik 11, 257 equililirium Sommerfeld, d. A. , H. Bethe and W. Heitler, Proc. Roy Soc. A146, 83 sponding to a new equilibriumposition. '5 (1934\. For i chareedoaiticlemovins in a oureCoulombfield, 3J . K . K n i p pan d G . E . U h l e n b e c kP,h y s i c 3a .4 2 51 1 9 3r6; /' - los 1 or a's@+0, where ii the iollision t ime; in a " than 1/r2 ior larger, I. F . B l o c hP , h y s .R e v . 5 0 , 2 7 2( 1 9 3 6 )s; e ep p . 2 7 6 - 7i n t h e field which falls off more rapidly approachesa finite value. latter.

54

t30 RADIATION

FIELD

In these circumstances the ordinary treatment of the interaction between electron and field by a consistent expansion in powers of e, the charge of the electron, is clearly inadequate, for such a procedure is based on the physical assumption that the probability of multiple light quantum emission decreases with ihcreasing number of emitted quanta. In fact, however, if one considers only frequencies above a certain minimum frequency oo, the expansion parameter of such a treatment is e2/hc.(v/c)'zlog E/han (E, tt the kinetic energy resp. velocity of the electron),6 which in the limit o0+0 violates the original assumption that it is small. In order to avoid this expansion, a method is developed below in which the coupling between the electron and the electromagnetic waves of low frequency is not considered as a small perturbation. The method is arfalogous to the classical expansion in powers ol ezotfmC, the ratio of the electron radius to the wave-length considered, in which in first approximation the motion of the electron is treated as given and the reaction of the field on the electron is taken into account in higher orders. We shall show how this can be formulated in quantum mechanics as the solution in successive approximations of a system of two simultaneous differential equations; of these approximations only the one of lowest order is here After thus having needed and investigated. treated the system electron plus electromagnetic field, transitions of this system due to external forces on the electron can be treated by the ordinary method of small perturbations.T II.

Foruure,uoN

oF rsB

MBrsoo

The Hamilton operator of the total system electron plus electromagnetic field, after elimination of the longitudinal parts of the electric field, is8 se : c { (a, p - (e/ c) A) | Brncl

Il. + (r /s").f 11nt')'z +1g1)d 6 This is the order of magnitude of the ratio of the probabilities of single quantum emission and radiationless process, 7 A case in which the action of a sattering field on the electron is not considered as a small Derturbation. is treated in the following paper by A. Nordiieck. 8 W. Pauli, Hand.buehder Physik, Yol.24, p.266.

OF

ELECTRON

JJ

We expand the vector potential A in the form A : 2c(rh/ $\Do,-le,r(P,r 'r

cos (k,, r) *0'r sin (k" r))'

where Q is the volume within which the cyclical boundary conditionsare applied, the summation index s characterizes the direction and circular frequency o" of the various waves with propagation vector k", tr their state of polarization; and e,r is a unit vector in the direction of polarization. The dynamical variables P,r and Or are related to the quantized amplitudes a(I, k) and @+(I,k)e accordingto 2-r(P,x*iQ,,:,):o(r, k,), iQ i =o+(tr,k,), "xand obey the commutation laws 2-t(P

[P,^' 0"',,']: -16""'drr' ; [p,r, p,,r,]: [0"r, 0,,r,] :0. We then obtain tc:c{(e, p- Ea"r[P,r cos (k,, r) f 0"r sin (k", r)f)*7mcl 1!!(P"r'?*0,f)/to", where

a"t,:2e(rh/Qo")\e,'r,.

(1) Q)

The matrix vector a of Dirac is to be interpreted physically as the velocity of the electron divided by c. If following the procedure in classical theory, we should in first approximation neglect the reaction of the electromagnetic field on the electron, we should be led to replace c in (1) by the c number v:Y/c,

(3)

where v stands for the (constant) velocity of the electron in its undisturbed motion. The operator p has then at the same time to be replaced by (l-p')\.But with c numberssubstitutedfor a and p in (1) the dynamical problem can immediately be solved. Indeed, the presenceof the interaction energy in (1) then causesno mathee W. Pauli, Hanilb*ch der Physih Yol.24, p. 250 ff.; the notation used in the presnt paper is es*ruially the sme as that of G. WentzLl, Hondbrch der Physih Vol. 24, p.

740fr.

t3r F.

56

BLOCH

AND

matical difficulty, since being closely analogous to the potential of a given constant force on each field oscillator, it can be taken into account by merely adding functions of r to the P,r and 0,r' Of course, the above replacement is not rigorously justifred. In order to enable one to take into account the error thus committed by successive approximations, we proceed as follows. The solution ry'(r, 0"r) of the wave equation

A. NORDSIECK We accordingly solve the equation cos (k,, r)

{c(g, p-la,r[P,r

*0"r sin (k", r)))*mc'(l*1I(P,r't0

p'))'

Elu:0. "x')ho"-

(10)

This can best be done by first making the canonicaltransformation

(4) P,r: P',r*o"r cos (k", r) ;

\t'l':EV

Q,1:Q,,r*a"r sin(k,,r) ; r:t' ; p: pt-lhk"a"1[P',1 cos (k", r)

can be uniquely separated into two parts

,! :{+ -trl,-

(5)

rt

by the help of the operator

* Q ' " r s i n ( k " ,r ) { } o , v l '

,{:(c, s)fP(l-p')l

(6)

and the conditions It{+=*+;

L{-:

-,1,-.

(7)

The correspondingtransformation of the s'ave function is u(t,Q"):exp

hB:2(l-p2)t-Blt

X [ 0 ' " r * ] a " r s i n ( k " ,r ) l ] ' u ' ( r , Q ' " x ) . ( 1 2 )

(8)

Choosing

and adding and subtracting Eqs. (4) and trJa!=879

{ i D o " r c o s( k " , r ) r)r

Using further the relations tro:2U-c,1;

(11)

o , ^ : ( u , a , r ) / n ( f r " - ( u ,k , ) )

(13)

@a) one obtains for u' the simple equation

we get {c(s, p - ! a,rlPrr cos (k,, r) * 0"r sin (k"' r) ]) '' *mc2ll - y\r+iE(P,x2lQ "x2)hu"TEl,l,' (9) "r = -t(u+0, p -mc(l-p2)-lg - Ea,r[P"r cos(k", r) f Q,1sin (k,, r)])r/-' +

lc(s, p')lmc'z(1

;p (P" "xl Q" "x)hu" "z;-11 - ( c/ 2 h ) 2 , G , a " ) " / ( k ( v , k " ) ) - E l u ' * o ( r 4 ) "with the generalsolution u' : "y(v)g-\ exp li I h(mc(l - uz)-tsl E, r) |

xflr-"r(0'"r), (1s)

sl

where the upper sign goes with the first superscript and the lower with the second. The two functions !+ and{- in Eqs. (9) correspond in the limit as the interaction between the electron and the field tends to zero, to motions of the electron in states with momentum mc(l-pz1-t9 and with energies+rnc2(l-p2)-", respectively. The approximation mentioned before of first assuming the velocity of the electron to be a given c number gc is equivalent to an approximate solution of the equations (9) in which the right-hand sides are neglected and /- is assumed to be zero. The latter assumption is made to obtain a state in which the energy of the electron is positive.

E ( g , m , x ,g ) : m c z ( l - p 2 ; - l 1 c ( u , 8 ) *L@ 'r

( ro.) - (cI 2h)E G, a")' / (k - (u, k")). " "x* ilh,'t" 3t

Here 7(p) is a noimalizedfour-componentamplitude satisfying the relation A"y: ?.tog is an arbitrary vector introduced so that for a fixed value of g the functions (15) form a completeorthogonal set. We shall below use only such functions z' for which 8:0, so that for vanishing interaction between electron and field we have to deal with an electron of momentum mc(|-pz)-to. ro .y may be taken to be proportional to any linear combination of the columns of (1*A).

132 RADIATION

FIELD

h^(r) is the normalized solution of the oscillator equation h*,, _ tczh*l (Zm* l) h^ : 0. Going back to the original function a and the original variables 0"^, u'e have from (11), (12), and (15) u(v, *"t):r(u)o-, tilo"x

J/

transition of the system between two states characterized by the velocities v:s, and w:rc of the electron and the quantum numb.ers zr"1 and z8r, respectively. The matrix element of a perturbation tr/ operating'on the coordinate and spin variables of the electron is given by

:lf

sin (k,, r)l]

.fIft."r(0"r-o"r sin (k", r)).

ELECTRON

7(U, n"r; v, n"r)

exp {i/h(ntc(I-p'z)-lu, r)

cos (k", r)[0,r-]",i

OF

(1?)

It is to be noted that the approximatevalidity of this solution is not conditionedby the smallnessof the fine structure constant e2fhc,sincewe have neglectedonly the recoil of the electron and can therefore make the solution (17) the more accurate the larger a mass we assign to the electron. In order to obtain the number which must be supposedto be small in order to have (17) valid, one may calculate ry'-in first approximation and compareits norm rl,'iththe norm of p+, wh'ch is unity. An estimate of norm (f-) gives norm (g-) - szc':t/ mc\U G) (hq / m c2) lg(p)(e'or/mc3)), where o1 is the highest frequency taken into account, and/ and g are functions of p alone such that for small,p,f-l l3r, C-ef 9r2)p2and,forlarge (r - p,)-t, I - ( / 4r) (r - rz), ge (r / 4r") (r - p,), log (1/(1 -p'z)). That the expansionparameteris not e2fhc may also be seen by considering the classical limit, in which D is assumed to be zeroi in this limit the shifts d,r cos (k", r) and d"r sin (k", r) of P"r and Q"1expressedin (17) correctly describe the superposition of the transverse part ol the field produced by the uniformly moving electron, ofi the field of the external light waves.rr IIL ApruceuoN To SMALLPsnrunserroNs oN THE ELEcTRoN;TnaNsrrroN PRoBABrLrrrEs AND MEAN ReoTerBp ENBncv The solution (17) can be used to calculate transition probabilities due to small external perturbations acting on the electron. Consider a u In fact, the quantities o,l can be obtained exactly by Fourier analysis of the transverse part of the classical vector potential oI a uniformly moving electron,

*

e x pt 1 - i / h ( n c ( -l p ' ? ) - t sr,) |

. 7*(u) I/6(v) exp li / h(mc(L- r')-tn, r I 'II1(k"; 6a\ttket,;r"^, z"^), (18) sI

where r"1 is the same function of r as o,1 is of g (see(13)), 6(v)is a four-componentamplitude for the state v corresponding to r(U) for the state g, and I(k;o,mtr,n): J

I d Q h ^ ( Q - os i n ( k , r ) )

c o s( k , r ) Q { l i ( c ' - r ' z )

Xexp {-i(o-z)

Xsin (k, r) cos (k, r)lh"(Q- r sin (k, r))

(19)

:exp l-i(m-n)(k,t)l.K with K(o, m ; r, n) : svp I - iG - r)'l X (m ln !)tl - i2- t (o - r11t^- "t I

l- +(a- r)'zfr

E r- (l- r) tr !(lm- nl -t () t

(20)

wherel':m if mfn and l:n otherwise.lzPutting the value (19) of 1 into (18), we can write the matrix element in the form I (21) l'(u, -,r ; v, ,"r) : x,mcxir tf(q)Il^f G, ">,,n,x), where

n@1: arexp{ *i(q, r) }7*(s)76(v) (22) I and q: (me/lr)l(l -p'z)-4u- (1- /)-lvl rL(ma,-n")k,. 12The integral over Q is best evaluated by'using generating function for the Hermitian polynomials.

(23) the

r33 BLOCH

F.

58

AND

From (21) follows the probability per unit time of the transition v, rts\+r, z"r in the form T(v, m"^ i v, a"r):

NORDSIECK

zero. In order to show this, consider first the case in which no light quanta are emitted, i.e', where ever] zu1:Q. We have then from (13) and (2)

Qr/ha')

XDlmc'1(l - Pz)-t -'/nc2(1-

f l e x'p |' - i ( o " r sr

v2)-\

* I(rle"r * z,^)i1o"l I F(q) l'? X fI j K(a"1, m

"r

'?, (24) i r"r, z"r) |

where D is Dirac's 6-function.r3 To simplify the discussion we shall from now on neglect the last (electromagnetic) terms in (23) and in the argument of D in (24)' We thus consider the changes in energy and momentum of the electromagnetic field to be small compared to the energy and the change in momentum of the electron, an assumption which is clearly justified for the treatment of the lorv frequencies.la Furthermore, we consider the state a, lnux for which all m"x:Ol this means that initially no free quanta are present. The probability per unit time for a transition in which nsx quanta of the kind s, tr are emitted and the electron after the transition moves in a direction within the element of solid angle sin SilBdq is by (24) and (20) U(gi,t, ni)

A.

sin &dtd.e

: ( l r'zhaA) sin Sdtde(m2vc / (1- v'z)) 2 (25) x I l . ( d | , I I l K ( c " r , o ; 2 " 1 ,a " 1 ) I str : $ l4r2h4a) sin Bdfds(mzr'c/(l - v'z))I F(q) | '

XfI exp{-}(""^-."r)t}

[*(o"r,-

'-i)']""^

(26)

n*1

The result confirms what has been anticipated in the introduction, namely, that the probability of the emission of any finite number of quanta is r3 In the arqument of the d-function there should stand, risorouslv speaking, E(u, z'r, 0) -Etv, u "r, 0) where the t i l u e s o i E i r e g i v C n b y ' ( J 6 ) . W e h a v e n e g l e c t e dt h e t e r m - G / 2 h E { J / - , a : i l ' z l ( 1 . - r U . k " ) ) i n ( 1 6 ) , r e p r e s e n t i n gt h e kinetic electromaqnetic self-energy of the electron, comoared to the mass term mc2ll - p2)-1.One may, in order to Le consisrent, imagine the lrequency spectrum to be broken ofi at an uDper limit or such that ezarlmC1ll. Since we are here concerned with difficulties arising from the low freouencies, this procedure, although connected with the fundamental difficulties of electrodynamics, does not affect our conclusions. r{ For thL treatment of higher frequencies one may use an expansion in powers ol e2fhc,

r " r ) ' z 1: e x p 1 ,- Q r e ' / h a 1 I

.',,-{##;-##;l'} (27)

The exponent becomes, after introducing the number of modes of vibration O/(2zr)3'dk" in the interval dk" and carrying out the sum over tr, f"

-(e,/4r,hc) tim o e o - l r |o

da" -"' ,"Jo

r"

I

f2n

sin d*dr}"I

'[(J;-*)'-(*-+)]

d,e,

,lo

(28)

where f,, p" are the polar angles of k, and p", z, are the components of g, v along k" and where rve have cut off the frequency spectrum at an upper limit or. The integration over the angles gives a finite positive factor; the integration over os, however, gives a result which is logarithmically infinite in the limit o0+0. Hence (27) vanishes. If a finite number of the a"r in (26) are different from zero, (26) differs from (27) in only a finite number of factors, thus remaining zero' However, the total probability of a transition g+v of the electron independently of the number of quanta emitted, which is given by the sum of (26) over all values of every a"r, is just (r /4r'zhaa) sin td$de(mzvc/ (1 - v')) | F(q) | ', (29) rvhich is the result which one would have obtained by neglecting entirely the interaction with the electromagnetic field. This result had physically to be expected since the change in momentum of the electromagnetic field has been neglected compared to that of the electron. From the facts that the probability of the transition u+r accompanied by the emission of a finite number of light quanta is zero and that the total transition probability is finite, it follows that the mean total number of quanta emitted is infinite. The mean radiated energy in the frequency interval d.<,:"and in the angular range d'8", d'9"

134 RADIATION

FIELD

when the electron isdeflected into the element of solid angle sin 3d,8d,p is given by (A/ (2rc)s)a"'zdot"sin 8"dO"d.e"

Xt

t l

n"aho"Usintittd.e (30)

a!tr:o

: (l / r,haQ) sin 8d.3d.,p(m2vc/ (l - vr)) I F(q) | , f / r ' ( . e 2 i ' 4 t 2 c () ;l - - . L\ 1-9"

e \t l 1- v"/

p" ," \ tl _ -l / -_ , | ldo-sin 3"d.8"d.p". (31) \ 1 -p" l- v"/ J

This has beenobtainedby makinguseof (26), (13)and (2) and of the formula i ne-,x"/nl:r. 4:0

On account of the extra factor ou in (30), the mean total energy, unlike the mean total number of quanta, is finite. Formula (31) is just the total probability that the electron be scattered into the element of solid angle sin $iL$dq multiplied by the amount of energy which it would radiate

OF

ELECTRON

59

classically in such a defldction. The expansion in powers of e2/hc in the limit of small frequencies, where alone Formula (31) may be supposed to be valid, leads to the same result for the mean energy radiated, though its results are entirely misleading so far as transition probabilities are concerned.l5 The above considerations can be applied almost literally also to cases such as the theory of p-decay, in which the external perturbation does not act on the electron coordinates, but creates

an electron. The only difference,so far as the electromagneticfield is concerned, consists in rgplacinggc by the velocity of the nucleusand vc by the velocity with which the electron is created. Here again the total probability of B-decay is unaltered by the interaction of thd electron with the low frequency radiation and the mean radiated energy is in agreement with that calculated by expanding in powers of e2fhc. 15This remark does not affect the cross section for the emission of a high frequency quantum as calculated by Belhe and Heitle-r, and expeiimbntally verified. Howevei, the cross section so derived has to be interoreted. not ai t h e p r o b a b i l i t y t h a t r h e h i g h f r e q u e n c yq u a n t u m a l o n e b e e m i t t e d , b u l a s t h e p r o b a b i l i t y t h a t t h i s h a p p e n sn o m a t t e r how many other light quanla are emitted.

I35

P o p e rI O

On the Intrinsic Moment of the Electron* H, M. FoLEYANDP. KuscH ColumbdaUniaersitl, New York, New Yorh December26, 1947 f N a previous letterr u-e have reported the observation zP:rz and zPtr I that the ratio of the g.r values of the states of gallium has the value 2.00344; the value 2 for this ratio follows from Russell-Saunders coupling and the conventional spin and orbital gyromagnetic ratios. If each of these states is exactly described by Russell-Saunders coupling, this observation can only be explained by setting (6s-262,):0.00229+0.00008, where the electron spin g and the orbital momentum g value is value is gs:2*6s, gr:l*6t. Since each of these atomic states may be separately subject to configuration interaction perturbations, the interpretation of this result was not entirely clear. A determination has now been made of the ratio of the g.r values of Na in the 25; state and of Ga in the 2P+ state, The e*pe.i-ental procedure was similar to that previously described.l The known hyperfine interaction constants of gallium2 and sodiums were employed in the analysis of the data. We find for this ratio the value 3.00732+0.00018 instead of the value 3. This result can be explained by making (as - 2az,): 0.00244+0.00006. The agreement between the values of (6s-262,) obtained by the two experiments makes it unlikely that one can account for the effect by perturbation of the states. The effect of configuration interaction on the g". value of sodium is presumably negligible.a To explain our observed effect without modification of the conventional values of gs or gz introduces the rather unlikely requirement that both states of galliurn be perturbeC, anci b1'amounts just great enough to give the agreement noted above. F rom any experiment in n'hich the ratio of the g"r values of atomic states is determined, it is possible to de[ermine only the quantity (6s-262,). If, on the basis of the correspondence principle we set 6, equal to zero, we may state the result of our first experiment as gs:2.00229 J.0.00008 and that of our recent experiment as

ss:2'00244L0.00006' It is not possible, at the present discrepancy

apparent conceivable

some

small

give rise to a discrepancy

would

These gestion ment

that

between

results

are not

time,

to state whether

these values perturbation

of

of the indicated

in agreement

by Breit6 as to the magnitude

is real. the

the It

is

states

magnitude.

the recent

sug-

of the intrinsic

mo-

with

of the electron.

* Publication assisted by the Ernest Kempton Adams Fund for Phvsical Researchof Columbia University. i P . K u s c h a n d H . M . F o l e y , P h y s . R e v . 7 2 , 1 2 5 6( 1 9 4 7 ) . 2 G. E. Becker and P. Kurch, to be published. r S . M i l l m a n a n d P . K u s c h , P j r r r s .R e v . 5 8 , 4 3 8 ( 1 9 4 0 ) . I M . P h i l l i p s ,P h y s . R e r ' . 6 0 , 1 0 o ( 1 9 4 1 ) . a br. J. Sc'hi'inger has verv kindli informed us in advance of publication of his conclusion from theoretical studies that 6, is zero lvtrereas ds may not vanish. 6 G, Breit. Phys. Rev. 72,984 (1947).

136

Poper I I

Fine Structure of the l{ydrogen Atom by a lVficrowave Method* ** Wrr,lrs E. Laur, Jn. eNn Roeenr C. RersEnronD Columbia Radintion Laborotory, Department oJ Physics, Columbia University, Neu I'orh. lieu, Yorh (Received June 18, 1947)

tf-HE

spectrum of the simplest atom, hydrogen, has a fine structuretwhich according to the Dirac wave equation for an electron moving in a Coulomb field is due to the combined efiects of relativistic variation of mass with velocity and spin-orbit coupling. It has been considered one of the great triumphs of Dirac's theory that it gave the "right" Iine structure of the energy levels. However, the experimental attempts to obtain a really detailed confirmation through a study of the Balmer lines have been frustrated by the large Doppler effect of the lines in comparison to the small splitting of the lower or n:2 states. The various spectroscopic workers have alternated between finding confirmationz of the theory and discrepanciessof as much as eight percent. More accurate information would clearly provide a delicate test of the form of the correct relativistic wave equation, as well as information on the possibility of line shifts due to coupling of the atom with the radiation field and clues to the nature of any non-Coulombic interaction between the elementary particles: electron and proton. The calculated separation between the levels 22Psand 22Pztzis0.365 cm-r and corresponds to a wave-length of. 2.74 cm. The great wartime advances in microwave techniques in the vicinity I

of three centimeters wave-length make possible the use of new physical tools for a study of the n:2 frne structure states of the hydrogen atom. A little consideration shows that it would be exceedingly difficult to detect the direct absorption of radiofrequency radiation by excited Fi atoms in a gas discharge because of their small * Publietion assisted by the Ernest Kempton Adarns Fund for Ph1'sical Research of Colurnbia Llniversit)', New York. ** Work supported bv the Signal Corps under contract number W 36-0.19sc-3200.1. I For a convenient account, see H. E. \iihite, Inl:roduclion Lo Alomit Spectra (McGraw-Flill Book Companl', New Y o r k , 1 9 3 4 ) ,C h a p . 8 . 'zJ. W. Drinkwater, O. Richardson, and \V. E. Williams, Prm. Ror'. Soc. 174, l6l (1940). 3W. V. Houston, Phvs. Rev. 5r, 446 (1937); R. C. W i l l i a m s , P h y s . R e v . 5 4 , 5 5 8 ( 1 9 3 8 ) ; S . P a s t e r n a c k ,P h y s . Rev. 54, 1113 (1938) has anall'zed these results in terms of an uprvard shift of the S level bl.about 0.03 cm-r.

population and the high background absorption due to ^electrons. Instead, we have found a method depending on a novel property of the 2251 level. According to the Dirac theory, this state exactly coincides in energy with the 2'P1 state which is the lou'er of the two P states. The S state in the absence of external electric fields is metastable. The radiative transition to the ground state 1'?S1is forbidden by the selection rule AZ: *1. Calculations of Breit and Tcllcra have shown that the most probable decay mechanism is double quantum emission with a lifetime of 1/7 second. This is to be contrasted r.ith a lifetime of onl1, 1.6X10-e second for the nonmetastable 22P states. The metastability is very much reduced in the presence of external electric fields5 owing to Stark effect mixing of the S and P levels with resultant rapid decay of the combined state. If for any reason, the 2251 level does not exactly coincide with the 22P1level, the vulnerability of the state to external fields will be reduced. Such a removal of the accidental degeneracy may arise from any defect in the theory or ma1' be brought about bl' the Zeeman splitting of the levels in an external magnetic field. In brief, the experimental arrangement used is the following: Molecular hydrogen is thermally dissociated in a tungsten oven, and a jet of atoms emerges from a slit to be cross-bombarded by an electron stream. About one part in a hundred million of the atoms is thereby excited to the metastable 22Sr state. The metastable atoms (with a small recoil deflection) move on out of the bombardment region and are detected by the process of electron ejection from a metal target. The electron current is measured u'ith an FP-54 electrometer tube and a sensitive galvanometer. If the beam of metastable atoms is subjected to any perturbing fields which cause a transition to any of the 22P states, the atoms will decay while moving through a very small distance. As a result, the beam current will decrease, since the

241

a H. A. Bethe ir Hantlbuch der Physih, Vol. 2411, $4.1. 5 C. Breir and E. -feller, Astrophl's. J. sl, 215 (1910).

137 242

!VILLIS

E.

LA]VIB, JI{.

AND

ROBERT

C.

RETHERFORD

Frc. 1. A llrpiel plot of galvanometer deflection due to interruption of the microwave radiation as a function of magnetic field. The magnetic field was calibrated rvith a flip-coil and ma1, be subject to some error which can he largely eliminated in a more refined apparatus. The wirlth of lhe curves is probably due to thc fr,llosing causes: (1) the radiative line l'idth of about 100 Mc/sec. of the rP slates, (2) hy'perfine splitting of the 25 state which amounts to about 88 Mcr'sec., (3) the use of an excessive intensity of radiation which gives increased absorption in the wings of the lines, and (4) inhomogeneity of the magnetic 6eld. have been No transitions from the state 2251(m= -!) observed, but atoms in this state mav be quenchbd b-v stral' electric fields because of the more nearly exact degeneracy with the Zeeman pattern of the'P states. detector state.

does not respond

Such

a transition

to atoms in the ground may

be induced

M A G N E T I CF I E L D - ( G A U S S }

by the

Experimental values for resonance magnetic Ftc,2. 6.elds for various frequencies are shown by circles. The solid show thre of the theoretimll], curves between source and detector. somewhere Transishifting variations, and the broken curyes are obtained by"*O".,"0 tions may also be induced by radiofrequency these down bl' 1000 Mc/sec. This is done merely for the sake and it is not implied that this would of comparison, radiation for which hr corresponds to the energy represent a "best fit." The plot covers onll'a small range difference between one of the Zeeman components of'the frequency and magnitic field scale covered by oir of 2251and any component of either 22P1or 22Ps12. data, but a complete plot would not shos up clearly on a small scale, and the shift indicated by the remainder of Such measurements provide a precise method for the data is quite compatible rvith a shift of 1000 Mc. application

to the

beam

of a static

electric

field

the location of the 2'9S1state relative to the P states, as well as the distance between the latter states. We have observed an electrometer current of the order of 10-14ampere which must be ascribed hydrogen atoms. The strong to metastable quenching effect of static electric fields has been observed, and the voltage gradient necessary for this has a reasonable dependence on magnetic field strength. We have also observed the decrease in the beam of metastable atoms caused by microwaves in the wavelength range 2.4 to 18.5 cm in various magnetic fields. In the measurements, the frequency of the r-f is fixed, and the change in the galvanometer current due to interruption of the r-f is determined as a function of magnetic field

strength, A typical curve of quenching ?errus magnetic field is shown in Fig. 1. We have plotted in Fig. 2 the resonancemagnetic fields for various frequencies in the vicinity of 10,000 Mc/sec. The theoretically calculated curves for the Zeeman effect are drawn as solid curves, while for comparisonwith the observedpoints, the calculated curves have been shifted downward by 1000 Mc/sec. (broken curves). The results indicate clearly that, contrary to theory but in essential agreement with Pasternack's hypothesis,sthe 2251state is higher than the 22P1byabout 1000Mc/sec. (0.033cm-l or about 9 percentof the spin relativity doublet separation. The lower frequency transitions 'S{m:l)+ zP{m: *}) have also been observedand agree

138 FINE

STRUC'TURE

OF

2St level. With the well with such a shift of the present precision, we have not yet detected any cliscrepancy between the Dirac theory and the doublet separation of the P levels. (According to most of the imaginable theoretical explanations of the shift, the doublet separation would not be affected as much as the relative location of the S and P states.) With proposed refinements in and magnetic field homogeneity, sensitivity, calibration, it is hoped to locate the S level with respect to each P level to an accuracy of at least ten Mc/sec. By addition of these frequencies and assumption of the theoretical formula 6v:]6a2R for the doublet separation, it should be possible to measure the square of the fine structure constant

THE

HYDTTOGEN

ATOM

243

times the Rydberg frequency to an accuracy of 0.1 percent. By a slight extensionof the method, it is hoped to determine the hyperfine structure of the 2'?S1 state. Al1 of these measurementswill be repeated for deuterium and other hydrogen-likeatoms' A paper giving a fuller account of the experimental and theoretical details of the method is being prepared, and this will contain later and more accurate data. The experimentsdescribedhere were discussed at the Conferenceon the Foundations of Quantum Mechanics held at Shelter Island on June 1-3' 1947 which was sponsored by the National Academy of Sciences.

P o p e rI 2

t39

The ElectromagneticShift of Energy Levels H. A. Brrnn Cornell Unia*sity, Ithaca, Nm Yorh (Receivedlune 27, 1947)

very beautiful experimenrs, Lamb and explained by a nuclear interaction of reasonable f[Y U Retherfordr have shown that the fine struc- magnitude, and Uehlings has investigated the ture of the second quantum state of hydrogen effect of the "polarization of the vacuum" in the does not agree with the prediction .of the Dirac Dirac hole theory, and has found that this effect theory. The 2s level, which according to Dirac's also is much too small and has, in addition, the theory should coincide with the 2p1 level, is wrong sign. actually higher than the latter by an amount of Schwinger and Weisskopf, and Oppenheimer about 0.033 cm-t or 1000 megacycles. This have sugg.estedthat a'possibleexplanation might discrepancy had long been suspectedfrom spec- be the shift of energy levels by the interaction of troscopic measurements.2'3However, so far no the electron with the radiation field. This shift satisfactory theoretical explanation has been comesout infinite in all existing theories, and has given. Kemble and Present,and Pasternackahave therefore always been ignored. However, it is shown that the shift of the 2s level cannot be possible to identify the most strongly (lineily) divergent term in the level shift with an electroI Phys. Rev. 72,241 (1947). I p. magnetic rzasseffect which must exist for a bound Plys. Rev. Sl, 446 (1937). !.-!!guston, 'R. C. Williams,Phys. Rev. 54, 558 (1938). as well as for a free electron. This effect should I E, C. Kemble md R. D, Pre*nt, Phys. Rev. 44. l03l (1932);S. Pasternack,Phys. Rev. 54, lliJ (f938). 0E. A. Uehling, Phys.Rev.48,55,(1935).

140 BETHE

340 properly be regarded as already included in the observed mass of the electron, and we must therefore subtract from the theoretical expression, the corresponding expression for a free electron of the same average kinetic energy' The result then diverges only logarithmically (instead Actheorl': in non-relativistic of linearly) cordingly, it may be expected that in the hole theory, in which the aeain term (self-energy of the electron) diverges only logarithmically, the result will be conaergent after subtraction of the free electron expression'o This would set an effective upper limit of the order of mc2 to the frequencies of light which effectively contribute to the shift of the level of a bound electron. I have not carried out the relativistic calculations, but I shall assume that such an effective relativistic limit exists. The ordinary radiation theory gives the following result for the self-energy of an electron in a state rn' due to its interaction with guantum transverse electromagnetic waves : gt:

-(2e2/3rhcs) fK

x I

hdhLlv,-l'/(8"-E,,+k), (1)

mentum, due to the fact that electromagnetic mass is added to the mass of the electron' This electromagnetic mass is already contained in the experimental electron mass; the contribution (3) to the energy should therefore be disregarded' For a bound electron, v2 should be replacedby its expectation value, (v2)-.. But the matrix elements of v satisfY the sum rule I lv-. I Therefore becomes

the relevant

wherc k = hct is the energy of the quantum and v is the velocity of the electron which, in nonrelativistic theorY, is given bY v:P/m:

(h/im)v.

(2)

Relativistically, v should be replaced by cc where c is the Dirac operator' Retardation has been neglected and can actually be shown to make no substantial difference.The sum in'(1) goesover all atomic states n, the integral over all quantum energiesI up to somemaximum K to be discussed later. For a free electron, v has only diagonal elements and (1) is rePlacedbY

yro: -(2e2/3rVr"'1 f nanu,1n.

(3)

This expression represents the change of the kinetic energy of the electron for fixed mo6 lt was first suggested b1' Schwinger and Weisskopf rhat hole theory must be ured to obtain convergence ln thrs problem.

(4)

Part of the self-energy

2e2

W':W-Wo:

3rhca rK

lv^"|,(E"- E-)

x I dkL----:-----' ; J" E"-E*+h

,_ , (s)

This we shall consider as a true shift of the levels due to radiation interaction. It is convenient to integrate (5) first over ft' Assuming K to be large compared with all energy differencesE*-E^ in the atom, K

-2e2 W':

Ilv-"1'(E,,-E^) 3rhct

J^

z: (v2)-^.

ln*

-'

(6)

lL"-L^l

"

(lf. E.-E^ is negative, it is easily seen that the principal value of the integral must be taken,.as was done in (6),) Since we expect that relativity theory will provide a natural cut-ofi for the frequency &, we shall assumethat in (6) K*mc2.

(7)

(This does not imply the same limit in Eqs' (2) and'(3).) The argument in the logarithm in (6) is therefore very large; accordingly, it seems permissible to consider the logarithm as constant (independent of n) in first approximation' We therefore should calculate

o:4 o"^:41P,^l'(E"-E^).(8) This sum is well known; it is

A:DIP" l'(8"-E^): -p' I g^*vv'v*-', :+h, - J fvrw*z2r:2nfuzszz,{r-2(o),

(9)

t4l ELECTROMAGNETIC

for a nuclear charge Z. For any electron with angular momentum l*O, the wave function vanishes at the nucleus; therefore, the sum 24.:0. For example, for the 2p level the negative contribution .4rgzr balancesthe positive contributions from all other transitions..For a state with l:0, however,

(10)

V*'(0): (Z/na)8/r,

where z is the principal quantum number and a is the Bohr radius. Inserting (10) and (9) into (6) and using relations between atomic constants, we get for an S state 81e21t

2a

K

SHIFT

341

This is in excellent agreement with the observed value of 1000megacycles. A relativistic calculation to establish the limit K is in progress.Even without exact knowledge of K, however, the agreement is sufficiently good to give confidencein the basic theory. This shows (1) that the level shift due to interaction with radiation is.a real effect and is oT finite magnitude, (2) that the effect of the infinite electromagnetic mass of a point electron an be eliminated by proper identifietion of terms in the Dirac radiation theory, (3) that an accurate experimental and theoretical investigation of the level shift may establish relativistic effects (e.g., Dirac bole theorv). These effects will be of the order of unity in comparison with the logarithm in Eq. (11).

If the present theory *t;'o(u.-r-h,' (11)should increase

is correct, the level shift roughly as Za but not quite so rapidly, becauseof the variation of (,8,-,8-)1, in where Ry is the ionization energy of the ground the logarithm. For example, for He+, the shift of state of hydrogen. The shift for the 2p state is the 2s level should be about 13 times its value for negligible; the logarithm in (11) is replaced by a hydrogen, giving 0.43 cm-r, and that of the 3s value of about -0.04. The average excitation level about 0.13 cm-1. For the x-ray levels Ll dnd energy (8"- E^)^, for the 2s state of hydrogen has LIl, this effect should be superposed upon the been calculated numerically? and found to be effect of screening which it partly compensates. 17.8 Ry, an amazingly high value. Using this An accurate theoretical calculation of the screen6gure and K:mc2, the logarithm has the value ing is being undertaken to establish this point. 7.63, and we find This paper grew out of extensive discussionsat the Theoretical Physics Conference on Shelter : (8"W"J 136lnlK/ E*)) Island, June 2 to 4, t947. The author wishesto :1040 megacycles. (12) expresshis appreciation to the National Academy of Science which sponsored this stimulating 7 I am indebted to Dr. Stehn and Miss Steward for the numeriel calculations. conference.

**':

u\il

P o p e rI 3

142

On Quantum-Electrodynamics a:rd tle Magnetic Moment of tle Electron JuLTAN SGNTNGER E o/!a\d u trild s;tJ, Canbridcc' M assa'hilscltt Dsember 30' t94?

correctlons

to elec-

The

of-a-radiative

exanple'

simplest

correction

is that

u"'"ttva;- n;ii5y*fjfj:ru'lf r''"",r'"l"Lio'"-i""n A',on phenomena ll

to evaluate

TTEIVIPTS

radiative

-

,*il':iltJ1fi.n!!1 -*#HilHil'':::ff:li 'ihe l";;";;;;;;;'n ;:A*1.'iti*T:-*'JTi::i:'s;:d:::E-J"fii "

.ig6!-3i" ;::'",,';:fu:i*:lT"TJ:'J:#i:"mt':.:"*:;t'ffi'l#Tih ru";Y:T"::: ;;#;;;i;;;u.""n.i'*i"piJ'.'io" l1i$f;*ffihJ'"!lll,1ll;'i,;l'i,1,""'ltTX"J""i-[i:jl5 are subject to modification

ov

from aspectsthat involv.e "*,

-o." " -"i""* ""ti"f."to.y

th"o.y,

andaie "^etgie'

rhi,'{*j:lim t,u"t*o.thv tbusrerativery l;:"f* transforming the Hamiltonian

o"Iii"

i"p".n""

;;d*;il

splitting

;tut"'iu*''

of the ground states ot atomrc

lT"il:ttt"t

;:H1"tJ:?t*Jl;

oro* *"::f:lHilJ"',T:*"""T,"1,T;;:;;,i" *o-"n,

l;'i*:lI*#:X*f'.'1":';'""':Xl*T; l'".,.-::lHi"A;f'3i:"i,'":'::,!.,:f,\T*Tl*1",fi ii:,:il#":X;'i: uu"oin"Jo''"i u'a ;;"';;il "*t.";o'. self-energv.of

a free.erectr::-^t1':r:,;

["::lH,*Jlt"'i:"::",]:1T;:..,'*"]'"a"'o"."aa ttvdtgg!1 to account {or the measured

1T:::;:Til?,';'J:'ilH;" " ira"7':booi These ?:ifi*iH.*tr*,*:',""**:**:tT:.,T'i a"/u: o'oorgr *0'00025' respectively' ;;.Oil;; cmbinathis i"""r"e theoreticalpredicmeaningfulstatementsof .rr" *".r, u'" not in disagreementwith the t=tit"' a free of mass is provided bv measuretion of masses,which is an" i.o-"r1*"r,"r iir. U"t" O*"i"e colnformatiot 'irl ,n", ffi,'"ir**, electron. It might ,OO".r, ,ri#1i," ;;;il;,;.";rth.i""t,o.'#,:li':ff+;"S*i*.:1.# *l*_":t:;,*i?..,i;-il ff3+:ill::i:i

electromagnetic

-;;;;";;.;;;

r*:""fijrug;t1!lll';;,;;;;.r--u,io.o*r"*#*fl ffi :.{:n'm:n;lt'r:i1Fil:T

**l'iltil:H:,yi*!T:l*:u*l )::;n'ffi*1;:ru**ry;:'J":$.'"r lil,',.,.m th""l".t.o-ugr"ticmassbeasmallcorrection to the mechaniel mass zo'

of elecatom", and modify the scatt€ring

\-\e'/nc)m0)

i, o"il.l*-iir." .-i;.;; H^-'tonian issuprio,;::h::,,f,,:,,]"ixi",J i:**i:;*n:f,"J1...",",',;H:: fi":,i:'Jil"#:::: inr values

vielded ion;'"a helium'5The essentiallYthre waYs: it mechaniel mass; il:';;i;;it;:^ta conjmtured rron mass,rather rhan ,t" unJi""*"tt" ;iffer.onlv slightlv from those ;;';;;;; in only field ,^a]".;"n no-'-relativisticcalculation'and an electron now interacts *t,-f, ii." is, only an accelerated ;; ffi;;;;;;t..i. "1 the presenceof an ""t"rr"t n"ia, inut

orabs'rb ;nr111;1*t1g ii*i-'"""-r'f i#t"tni:l;;iil"ff ;*Ji;.;'""',1*qq*lq action energYof an electron subject to a f,nite radiative.;;.'i;

i."". lit

ct"rtmb

frcld provides a satisfactory termrna-

iiT:T"f poi.t,I:rr^".,-,i:,il".il.;::n".l:,T';."j.;* ,ha,ast i::;r'"?l'#il,I-J""",ij"f'.:i T*j"";31,.?:i3;;";i;;*:ru n::trlJTi:lH"Ttri naa3{E:ffli.I;:,ijj',iffi1*l,i*":,?",*T with ".nnection-

il:i:n:J:-"":li#:i1{x** "",i".,*""r;ii;*#;l-U'jqffiit,?ffiir:triir Ho*"t"r,

such a term is it has long been recognized that

*.ft:*{,{{*rffi,11'*'iiJ$?Ji#i :H:l",tL1::'itr,ff#I"J[Tffi1::J,]il-,"JJ,l "Ti1;1Ii]"$i,?{*i,*'iri*;ru*l \2:';,t1"'nn *'lij:n:::1#1'li.fl'J::5'irJ:lx'il:iT:iru il;i;

rh.

"*p"ii-"n.ul

charge.Thus the inte.raction ;'g"J:lt#illm,,i:Rlt"-",311ifi;".*"\.72.r2s6(1e4?),andrurther

lactors' contained in the renormalization

On Radiative Corrections to Electron Scattering I7 a/laill

JULIAN ScgwrNcER U nio.rsit!, Cambrid.gc, M assaahw.u, January 21, 1949

ADIATIVE p corrmtions to the electromagnetic prop_ l\ erties of the electron produce energy level displace_ ments and modify electron scattering cross sections. Although the high accuracy of radiofrequency spectroscopy facilitates the measurement of energy level displace_ ments, as in the Lamb-Retherford experimentl and t}re evidence on the anomalous magnetic moment pf the electron,t nevertheless the correction to the cross section for scattering of an electron in a Coulomb field is not without interest, since it permits a comparison between theory and experiment in the relativistic region, as compared to the non-relativistic domain to which the energy level measurements pertain. The radiative correction to the cross sction for essn_ tially elastic scattering of an electron by a Coulomb field has been computed with the form of quantum electrodynamics developed in several recent papers.! In addition to the emission and absorption of virtual quanta. we include the real emission of quanta with maimum energy -1E, which is small in comparircn with l/:E-mc2. the initial kinetic energy of the electron, In other words, we treat only those inelastic events in which a small fraction of the original energy is radiated. The contribution of the remainder of the inelastic proce$es can be derived from the well-known bremsstrahlung cross section, and is not of principal interest, The result, expressed as a fractional decrease in the difierential cross section for rcatterine ttrrough an angle rl, is r/ F \ 6:2d/rL\toc;El)(I(0+K,) +iI( o- Kr+'IKz- L (mc2/E\2 s' , ' 1 - P 2 o^l sin?d/z"o1'

,\ r" ,

where Ko: f\/(1{ \r)tl log[(1]Ir) 5,: [(1f \r)/\,](o_ 1, Kr- [(1*\r)/r']Kr_ t,

l11l , \:

(p/mc) sin(s/2), Q)

and p: pc/8, Note that 6 diverges logarithmically in the limit AE+0. It is well known that this diflrculty stems froru the neglect of prcesses involving more than one low frequency quantum.. Actually the eswntially elastic satier" ing cross section approaches zero as AE*O; that is, it never happens that a scattering event is unaccompanied l,t' the emission of quanta. This is described by replacing tlrtt radiative correction factor 1-6 with e-6, which has the proper limiting behavior. The further ternrs in the serics expansion of a-'express the efiects of higher order proceses involving the multiple emission of soft quanta. However, for practical purposs such a refinement is unnecessrv. The accuracy with which the energy of a particle can be measured is such that the limit AE*0 cannot be realized, and d will be small in comparison with unity under presently accessible circumstances. In the norr-relativistic limit, p<<1, K

: (2n * 1)lB' sln'(o / 2), " lt / L=l(4/3)(os2-t)-!lp, sin,(o/D.

("/

| : (8al3r)B? sin, (3 / 2) Uog(mc2/ 2aE) + ( 19130) l,

(6)

and

which increases linearly with the kinetic energy of the particle. For a slowly moving particle, it is an elementary matter to include the additional scattering produced by the real emission of quanta with energies in the interual from AE to W. One thereby obtains the following fractional decrease in the differential cross section for scattering through an angle rl, irrespective of the final energy: 6: (8al3r)p'z sinl (o /2)llog(mc2/8W) + (19130) tan(d/2)f [cos r]/cosr(,]/2))logcsc(o/2)). *('-,t)

(7)

We may remark, parentJretically, that in the sme nonrelativistic approximation, the radiative correction to the energy of a particle moving in an external field with po. tential energy Iz(r) ist 6B : (c/3r) flog(mc2 / 2AW +Gt / 12a)f\h / mc),|v, ff ! (a/2r)(h/2mc)<- ipa.v V) /c\ = (a / 3 r) (h / mc), l(los (mc2/ 2^W ) + ( t 9/30) ) (v, y)' "/

and

L: (^'+i)67EY I:tlrs++#- !rgig#]4

{ip.L(r/r)(dv/dr))1,

_*t-"?a),itsff:Ju )

(3)

y : ll - sin2 (s/ 2)(1- d) ll

(4)

Here

where L is the in units of i, and system.l Applied and, 2zP1 levels

orbital angular momentum operator in A!/ is an average excitation energy of the to rhe relative displacement of the 2rS1 of hydrogen, this formula yietds 1051

144 LETTERS

2 mc/*c',

TO

experimental value? of to be compared with the

ttlttl"TiJ,,ll';"htivistic or(1)is rimit

THE

EDITOR

;:?rlr':*r4i:Yf 3fl #;::Hid{,ii*i determination of theenergy' sal

""'i""'*'"r'.i,".t91"1:*'StbhvJi::':T:i:'T" ' - *"' f h!,1*'i#,'l n*," t7z)+o(")r' (s) t-'ff:";j;1tlt.',' u *,rn angle,at moderate.energies' ;' rl!;t an addition of 4.4 104 to 6' I

where

(o/2) - { sin 6(o) [4i9P;['"",u,,,

**?$;'']

(9) alone' with fie a'limptotic formula t" *rt* "itai"d thecondition(f/6s)'z siil(d/2)>>1'vhich ft".".Jr ""*rts

.lii:ij;l1ji:'# ll*1":::,'; xp=$677p' tro) ::::lH^[;*;:

i1i*$1 **:;:::""$".1; Hiltl:ii*j:i1i,

rheinteg^rar,.i"*.::.::.'-il"i*i.ilrJlryrrt?1,";r:; i::E:#-;,ll:ff ;;;;t;ii"il*

l':: ili;;;;

:'",.'"1;liiu;:Tit:iT+

hasihecorrec'l which

at small angles'is provided bv

W and AE, the value 6:7.2 !Oa.

t-cos(B /2)

6Q)-O*@nfr+*@ifr *l.*za=*qar*t-co:(o/2t4t]

(11)

of 6

"1r'-"*,in*,iont !n"*o"o',liji[?l;" ::$l."t::"f: valuable conformation for t ,"a-Jii* "...*tions the electron.

properties of to the electromagnetic

l*";t*k*d:i-?$:i":**id;Y,":lxip;[1+T**tt ttltil (e)

;$3i;',ic rormura m:mtt j:.j;ir"i",,".fft;irr-**$3;*$*$$,3gff *.a"i.ii'"n",d":'n.",.y,

;trg1l;**+m*,*rff ftgi;'ffi 17:3.1

Mev, which corresPondt

['ltl,,tft;-"".iq;g1iffirf+i,glm #;;;;;,,,i*,di,io,"ioJ',;u:u'u.-x".i.'v :*;" "-r saei N.'i, ;g'',::l;%T:'1i1": *ll;"1;':X #*i :%;ll

ETECTRON THEORY Report to the SolvayConference for physics at Brussels,Belgium September27 to October2, l94g by f. R.OPPEI{HE|MER In this report I shall try to give an account of the deveropments of the last year in electrodynamics. It will not be usefur to eirre . complete presentation of the formalism; rather I shall try to pick out the essential logical points of the development, and raise at least some of the questions which may be open, and which bear on an evaluation of the scope of.the recent developmenrs, and their place in physical theory. I shall dlvide the report into three sections: (l) a brief summary of related past work in electrodynamics; (2) an account of the rogical and procedural aspects of' the recent developments; and (3) a series of remarks and questions on applications of these developments to nuclear problems and on the question of the closure of electrodvnamics.

t.

History

The problems with which we are concerned go back to the very beginnings of the quantum electrodynamics of Dirac, of Heisenberg and pauli.(r) This theory, which strove to explore the consequences of complementarity {br the electromagnetic field and its interactions with matter, red to great successin the understanding ofemission, absorption and scattering processes, and led as well to a harmonious synthesis of the description of Jtatic fields and of light quantum phenomena. But it also red, as was armost at once recognized,(2) to paradoxical results, of which the infinite displacement of spectral terms and lines was an example. one ,ecognirei an analogy between these results and the infinite electromagnetic inertia of a point electron in classical theory, according to which erectrons moving with different mean velocity should have energies infinitely displaced. yet no attempt at a quantitative interpretation was made, nor was the question raised in a serious way of isolating from the infinite displacements new and typical finite parts clearly separable from the inertial effects. In fact such a program could hardly have been carried through before the discovery of pair production, and an understanding of the far_reachinq differences in the actual problem of the singularities of quantum electrodynamics from the classical analogue of a point electron interacting with its field. In the

146 J. R. OPPENHEIMER clearly former, the field and charge fluctuations of the vacuum-which on the whereas part; a decisive have no such classical counterpart-play limit seriously so which prgduction, pair of other hand the very phenomena compared small for distances the electron of the usefulness of a point model they to its compton wave length hlmc,in some' measure ameliorate, though and of inertia electromagnetic infinite the do not ,.rolrr., the problems of first points last These distribution. the instability of the electron's charge were were made clear by the self-energy calculations of Weisskopf,(3) and that to Sakata,(5) and by Pais,(+) finding, by the still further emphasized by the return) to repeatedly have shall we the order e2 (and to this limitation and its electron's self-energy could be made finite, and indeed small, essentially and magnitude small of forces stability insured, by introducing arbitrar:ifitsmall range, corresponding to a new field, and quanta of arbitrarily high rest p255.(6) b1 th. other hand the decisive, if classically unfamiliar, role of vacuum in a highly academic situation fluctuations lvas perhaps first shown-albeit gravitational energy of the (infinite) -by Rosenfeld's- calculation(7) of the with the discovery of the view into light quantum, and came prominently the current fluctuations of to due of the self-energy of the photon piobleof the (infinite) problems related the the electron-positron field, and of renormalizanotion the time first for the polarizability of that field. Here refers in fact vacuum of polarization The infinite iior, ,rru, introduced. be possible should of charge definition just to situations in which a classical the linear finite, were polarizarion slowly varying fields) ; if the l*.uk, classically in any measured nor constant term could not be measured directly, charge induced gf and "true" interpretable experiment; only the sum linear infinite the ignore to natural could be measured. Thus it seemed the finite to significance attach tp constant polarizability of vacuum, but fields'(a) in strong and varying rapidly deviations from this polarization in they are in Direct attempts to measure these deviations were not successful; Lamb-Retherford the describe do which any case intimately related to those bulk of level shift,(e) but are too small and of wrong sign to account for the here philosophy and procedure renormalization this observation.(10) But the appliedtochargewastoprove'initsobviousextensiontotheelectron,s mass, the starting point for new developments' have In their application to level shifts, these developments, which could the required years, fifteen last the during been carried out at any time in other Nevertheless, verify. and impetus of experiment to stimulate identical with closely related problems, results were obtained essentially the Schwinger and shift Lamb-Retherford the those required to understand ratio' gyromagnetic corrections to the electron's Thusthereistheproblem-firststudiedbyBloch,Nordsieck,(1r)Pauli slow electron and Fierz,(lz) ofthe radiative corrections to the scattering ofa of electromagnetic The contribution z. (of velocity u) by a static potential 2

147 E L E C T R O NT H E O R Y inertia is readily eliminated in non-relativistic calculations, and involves some subtlety in relativistic treatment only in the case of spin 1/2 (rather than spinzero) charges.(13) It was even pointed 6st(1a) that the new effects of radiation could be summarized by a small supplementary potential

I.

-(+)(fleY L V l n ( ;)

(where e, li, m, c have their customary meaning) . This of course eives the essential explanation of the Lamb shift. on the other hand the anomalous g-value of the electron was foreshadowed by the remark,(15) that in meson theory, and even for neutral mesons, the coupling of nucleon spin and meson fluctuations would give to the sum of neutron and proton moments a value different from (and in non-relativistic estimates less than) the nuclear magneton. Yet until the advent of reliable experiments on the electron's interaction, these points hardly attracted serious attention; and interest attached rather to exploring the possibilities of a consistent and reasonable modification of electrodynamics, which should preserve its agreement with experience, and yet, lor high fields or short wave lengths, introduce such alterations as to make self-energies finite and the electron stable. In this it has proved decisive that it is zat sufficient to develop a satisfactory classical anaiogue; rather one must cope directly with the specific quantum phenomena of fluctuation and pair production.(o) Within the framework of a continuum theory, with the point interactions of what Dirac(lc) calls a ,,localizable,, theory-no such satisfactory theory has been found; one may doubt whether, within this framework, such a theory can be formed that is cxpansible in powers of the electron's charge e. on the other hand, as mentioned earlier, many families of theories are possible which give satisfactory and consistent results to the order s2. A further general point which emerged from the study of electrodynamics is that-although the singularities occurring in solutions indicate that it is not a completed consistent theory, the structure of the theory itself gives no indication of a field strength, a maximum frequency of minimum length, beyond which it can no longer consistently be supposed to apply. This last remark holds in particular for the actual electron-for the theory of the Dirac electron-positron field coupled to the Maxwell field. For particles of lower and higher spin, some rough and necessarily ambiguous indications of liryiting frequencies and fields do occur. To these purely theoretical findings, there is a counterpart in experience. No credible evidence, despite much searching, indicates any departure, in the behaviour of electrons and gamma rays, from the expectations of theory. There are, it is true, the extremely weak couplings of p decay; there are the weak electromagnetic interactions of gamma rays, and electrons, with the mesons and nuclear matter. Yet none of these should give appreciable

3

148 J. R. OPPENHEIMER of application; correctionsto the present theory in its characteristicdomains distances,and (nuclear) small very fo, ihut suggest they serve merely to will no longer be ,r.ry t igt energies, .1..i1., theory and electrodynamics theory of the separable from other atomic phenomena' In the ,o an alrnost closed' "t.u.iy electron and the electromagnetic field, we have to do with precisely to the absence almost complete system,in which however we look ofcompleteclosuretobringusawayfromtheparadoxesthatstillinhere in it.

2. The problem

Procedures

recognize then is to see to what extent one can isolate'

undportpo.tetheconsiderationofthosequantities'liketheelectron'smass infinite results-results which' und .hu.g., for which the present theory gives iffinite,-couldhardlybecomparedwithexperienceinaworldinwhich What one can hope to arbftrary values of the ratio izfhc cannot occt:Lr' .o*p',.withexperienceisthetotalityofotherconsequencesofthecoupling need to ask: does theory of .hurge and field, consequences of which we and in agreement with unambiguous finite, give for them results which are experiment ? as jlrdg'.d by these criteria the earliest methods musf be characterized They rested, as have to date all treatrnents ..r""orriugi.rg but inadequate' notseverelylimitedthroughoutbytheneglectofrelativity'recoil'and of a, going characteristically to pair formaiion, on a.t expansiott in powers out the calculation of the problem in question; the order ez. One carrij for the Lamb shift, I,amb and (for radiative scattering corrections, Lewis{u); for the electro{r's g-value' Kroll,trel Weisskopf ulrd F,.""h,(1e) Bethe(zo); order the electron's electroLuttinger(21)) ; one also calculated to the same induced by external fields' and magnetic mass, its charge' and the charge for the effect of these changes in the light quantum massl finaily one asked and sought to delete the charge and mass on the problem in question' Such a procedure would corresponding terms fro* ih" direct calculation' all quantities involved cumbersome-were no doubt be satisfactory-if ln In fact, since mass and charge,correctrons are finite and unambiguous' outabove divergent integrals' the general represented by logarithmically not necessarily unique or correct' but finite, obtain to Iin.d pro".dure serves an electron in an external field; reactive corrections for the behaviour of andaspecialtactisnecessary,suchasthatimplicitinLuttinger'sderivation oftheelectron'su.tomulot'sgyromagneticratio'ifresultsaretobe'not sound' Since' in more complex merely plausible, but unambigt'ot's at'd straip;htforward in calculations catried to higher order in a' this froUt.Inr, and and the results more depenprocedure becomes more and more ambiguous' of gauge, more powerful methods dent on the choice of Lorentz frame and steps' the first Their development has occurred in two are ,.qrrir.d. Schwinger'(zz) largely, the second almost wholly, due to

+

149 ETECTRONTHEORY The first step is to introduce a change in representation, a contact transformation, which seeks, for a single electron not subject to external fields, and in the absence of light quanta, to describe the electron in terms of classically measurable charge a and mass m, and eliminate entirely all " virtual " interaction with the fluctuations of electromagnetic and pair fields. In the non-relativistic limit, as was discussed in connection with Kramer's report,(z3) and as is more fully described in Bethe's,(24)1li5 112n5formation can be carried out rigorously to all powers of e, without expansion; in fact, the unitary transformation is given by

II.

[/:exp

],f<,Vl

where ,( is the (transverse) Hertz vector of the electromagnetic field minus the quasi-static field of the electron. When this formalism is applied to the problem of an electron in an external field, it yields reactive corrections which do not converge lor frequencies u>mc2fh, thus indicating the need for a fuller consideration of tvpical relativistic effects. This generalization is in fact straightlorward; yet here it would appear essential that the power series expansion in a is no longer avoidable, not only because no such simple solution as II now exists, but because, owing to the possibilities of pair creation and annihilation, and of interactions of light quanta with each other, the very definition of states of single electrons or single photons depends essentially on the expansion in question.(25) However that may be, the work has so far been carried out only by treating e2fhc as small, and essentially only to include corrections of the first order in that quantity. In this form, the contact transformation

clearly yields:

(a) an infinite term in the electron's electromagnetic inertia; (b) an ambiguous light quantum self-energy; (c) no other effects for a single electron or photon; (d) interactions of ord.er e2 between electrons, positrons, and photons, which in this order, correspond to the familiar Moller interactions and Compton effect and pair production probabilities; (e) an infinite vacuum polarizability; (f) the familiar frequency-dependent finite polarizability for external electromagnetic fields ; (g) emission and absorption probabilities equivalent to those of the Dirac theory for an electron in an external e.m. field; (h) new reactive corrections of order e2 to the effective charge and current diStribution of an electron, which correspond to vanishing total supplementary charge, and to currents of the order e:l/fc distributed over

5

150 J. R. OPPENHEIMER dimensions of the otd'er hlmc, and which include the supplementary potential -I, and the supplementary magnetic moment I

o2 \l

ph\/+\

\na)\r^,)\" ) as special (non-relativistic) limiting cases' in e, they Were such calculations to be carried further, to higher order to the mass, and of charge would lead to still further renormalizations correcreactive to and interactions, successive elimination of all "virtual" to the probabilities of tions, in the form of an expansion in powers of e2ffrc, Nevertheless' before etc' transitions: pair production, collisions, scattering, interesting new physically the such a prog.u-^ could be undertakdn, or The required' is development a new t..-, th; uLo,r. b. taken as correct' independent general in not are reason for this is the following: the results (h) ofgaugeandLorentzframe.Historicallythiswasfirstdiscoveredby energy in a uniform .oripu.iro', of the supplementary magnetic interaction magnetostatic field 1/

\r;n)\*)\' ') /

o2 \/

el\/*

z\

withthesupplementary(imaginary)electricdipoleinteractionwhich appeared*ithu,,electroninahomogeneouselectricfieldEderivedfroma static scalar Potential /

"2

\l

ph\

/

.\

\e;n)\r-,*)iP"\i:E) a manifestly non-covariant result' Nowitistruethatthefundamentalequationsofquantum-electrodynamics But-they have in a strict sense no soluare gauge and Lorentz covariant' these solutions, tions expansible in powers of e. If one wishes to explore no longer theory, in a later bearing in mind that cprtain infinite terms will, beinfinite,oneneedsacovariantwayofidentifyingtheseterms;andfor the whole method of that, not merely the field equations themselves, but This covariance' appro"imation and solution must at all stages preserve Lorentz a fixed imply *.url, that the fanriliar Hamiltonian methods, which Lotentz frame nor gauge frarne l: constant, must be renounced; neither a' all terms have been in can be specified until after, in a giv-en order identified,'andthosebearingonthedefinitionofchargeandmassrecognized andrelegated;thenof.o,,"t,intheactualcalculationoftransitionprobabilitiesandthereactivecorrectionstothem,orinthedeterminationof static, and in the reactive stationary states in fields which can be treated as coordinate system and corrections thereto, the introduction of a definite well-defined terms can gauge for these no longer singular and completely Iead to no difficultY.

l5r ETECTRONTHEORY It is probable that, at least to order e2, r;rore than one covariant formalism perturbation can be developed. Thus Stueckelberg's four-dimensional theory(26) would seem to offer a suitable starting point, as also do the related algorithms of Feynman.(27) But a method originally suggested by Tomonaga,(28) and independently developed and applied by Schwinger,{:2) would seem, apart from its practicality, to have the advantage of very great generality and a complete conceptual consistency. It has been shown by Dyson(ze) how Feynman's algorithms can be derived from the Tomonaga equations. The easiest way to come to this is to start with the equations of motion of the coupled Dirac and Maxwell field. These are gauge and Lorentz covariants. The commutation laws, through which the typical guarltum features are introduced, can readily be rewritten in covariant lorm to show: (l) at points outside the light cone from each other, all field quantities commute; and (2) the integral over an arbitrar2 space-like hypersurface yields a simple finite value for the commutator of a field variable at avanable point on the hypersurface, and that ofanother field variable at a fixed point on the hypersurface. In this Heisenberg representation, the state vector is of course constant; commutators of field quantities separated by timeJike intervals, depending on the solution of the coupled equation of motion, can not be known a priori; and no direct progress at either a rigorous or an apProximate But a simple change to a mixed solution in powers of e has been made.* Tomonaga and called .by Schwinger the in-troduced by that representation, "interaction representation," makes it possible to carry out the covariant analogue of the power series contact transformation of the Hamiltonian theory. The change of representation involved is a contact transformation to a system in which the state vector is no longer constant, but in which it would be constant if there were no coupling between the fields, i.e., if the glementary The basis of this representation is the solution of the uncoupled charge a:0. field equations, which, together with their commutators at all relative This transformation leads directly to positions, are of course well known. of the state vector F: for the variation the Tomonaga equation

III.

; h b J : - ! ; t ' ' ' , t rL .4' , , ' Y

"'" 6o

c'

Here o is an arbitrary space-like surface through the point P. d I is the variation in Pwhen a small variation is made il:io,localized near the point P; do is the four-volume between varied and unvaried surfaces; AOQ) rsthe * Author's note, 1956. Approximate solutionsof the Heisenbergequationsof motion were obtainedby Yang and liildman, Ph1ts.Reu.,79,972,1950;and Kdll'6n,ArkiuFi)rF2sik, 2,371,1950. 7

152 J. R. OPPENHEIMER operator of the four-vector electromagnetic potential at p; jp\et is the (charge-symmetrized) operator- of electron-positron four-vector current density at the same point. It may be of interest, in judging the range of applicability of these methods, to note that in the theory of the charged particle of zero spin (the scalar and not Dirac pair field), the Tomonaga equation does not have the simple form III; the operator on Fon the right invoives explicitly an arbitrary time-like unit vector. (30) Schwinger's program is then to eliminate the terms of order e, ez, and. so, in so far as possible, lrom the right-hand side of III. As before, o'ly the transitions can be eliminated by contact transformation; the real "viitual" transitions of course remain, but with transition amplitudes eventually themselves modified by reactive corrections. Apart from the obvious resulting covariance of mass and charge corrections, a new point appears for the light quantum self-energy, which now appears in the form ol'a product of a factor which must be zero on invariance grounds, and an infinite factor. As long as this term is identifiable, it must of course be zero in any gauge and Lorentz invariant formulation; in these calculations for the first time it is possible to make it zero. yet even here, if one attemp;ts to evaluate directly the product of zero factor and infinite integral, indeterminate, infinite, or even finite(er) values may result. A somewhat similar

situation obtains in the problem, so much studied by Pais, of the direct evaluation of the stress in the electron's rest system, where a direct calculation yields the value (-e2f2nhc)mc2, instead of the value zero which Ibllows at once as the limit of the zero value holding uniformly, in this order e2, for the theory rendered convergent by the y'quantum hypothesis, even for arbitrarily highy'quantum mass. These examples, far from casting doubt on the usefulness of the formalism, may just serve to emphasize the importance of identifying and evaluating such terms without any specialization of cooidinate system, and utilizing throughout the covariance of the theory. To order e2, one again finds the terms (a) to (h) listed above ; the covariance of the new reactive terms is now apparent; and they exhibit themselves again but more clearly as supplementary currents, corresponding to charge distribution of order eslhc (but vanishing total charge) extended throughout the interior of the light cones about the electron's position, and of spatial dimensions - hlmc; inversely, they may also be interpreted as corrections of relative order e2fhc and static range hfmc to the external fields. The supplementary currents immediately make possible simple treatments of the electron in external fields (where neither the electron's velocity, nor the derivatives of the fields need be treated small), and so give corrections for ernission, absorption and scattering processes to the extent at least in which the fields may be classically described(32); the reactive corrections to the Moller interaction and to pair production can probably not be derived 8

153 ELECTRONTHEORY without

carrying the contact transformation to order e4, since for these typical exchange effects, not included in the classical description of fields, must be expected to appear. At the moment, to my best present knowledge, the reactive corrections agree with the,S level displacements of 11 to about lo/o, the present limit oi'experimental accuracy. For ionized helium, and for the correction to the electron's g-value, the agreement is again within experimental precision, which in this case, however, is not yet so high.

3.

Questions

Even this brief summary of developments will lead us to ask a number of questions: (l) Can the development be carried further, to higher powers of a, (a) with finite results, (b) with unique results, (c) with results in agreement with experiment ? (2) Can the procedure be freed of the expansion in a, and carried out rigorously ? (3) How general is the circumstance that the only quantities which are not, in this theory, finite, are those like the electromagnetic inertia of electrons, and the polarization effects of charge, which cannot directly be measured within the framework of the theory ? Will this hold for charged particles of other spin ? (4) Can these methods be applied to the Yukawa-meson fields of nucleons ? Does the resulting power series in the coupling constant converge at all?

Do the corrections improve agreement with experience ? Can one expect that when the coupling is large there is any valid content to the Maxwell-Yukawa analogy ? (5) In what sense, or to what extent, is electrodynamics-the theory of Dirac pairs and the e.m. field-"closed" ? There is very little experience to draw on for answering this battery of questions. So far there has not y€t been a complete treatment of the electron problem in order higher than e2, although preliminary study(33) indicates that here too the physically interesting corrections will be finite. The experience in the meson fields is still very limited. With the pseudoscalar theory, Case(3a)has indeed shown that the magnetic moment of the neutron is finite (this has nothing to do with the present technical developments), and that the sum of neutron and proton moments, minus the nuclear magneton (which is the analogue of the electron's anomalous g-value) is of the same order as the neutron moment, finite, and in disagreement with experience. The proton-neutron mass difference is infinite and of the wrong sign; the reactive corrections to nuclear forces, formally

9

154 J. R. OPPENHEIMER analogous to the corrections to the Moller interaction, have not been evaluated. Despite these discouragements, it would seem premature to evaluate the prospects without further evidence. Yet it is tempting to suppose that these new successeso{'electrodynamics, which extend its range very considerably beyond what had earlier been believed possible, can- themselves be traced to a rather simple general As we have noted, both from the formal and from the physical feature. side, electrodynamics is an almost closed subject; changes lirnited to very small distances, and having little effect even in the typical relativistic domain p'-7n62,c,o1tld.sufnce to make a consistent theory; in fact, only weak and remote interactions appear to carry us out of the domain of electrodynamics, into that of the mesons, the nuclei, and the other elementary particles. Similar successescould perhaps be expected for those mesons (which may well also be described by Dir2ic-fields), which also show only weak nonBi-rt for mesons and nucleons generally, we electromagnetic interactions. are in a quite new world, where the sp$pial features of almost complete closure that characterizes electrodynamics are quite absent. That electrodynamics is also not quite closed is indicated, not alone by the fact that for finite ezlhc the present theory is not after all self-consistent, but equally by the existence of those small interactions with other forms of matter to which we must in the end look for a clue, both for consistency, and for the actual value of the electron's charge. I hope that even these speculations may suffice as a stimulus and an introduction to further discussion.

155 ELECTRON THEORY

References 1. Heisenberg and Pauli, /eits.f. Ph1sik.,56, l, 1929. 2. J, R. Qppenheimer, Ph1ts.Reu.,35, 461, 1930. 3. V. Weisskopf, <eits.f. Ph1sik.,9O, Bl7, 1934.' 4. A. Pais, Verhandelingen Ro1. Ac., Amsterdam,19, l, 1946. 5. Sakata and,Harz, Progr. Theor.Ph1s.,2,30, 1947 6. For a recent summary of the state of theory, see A. Pais, Deuelopments in the Theory of the Electron,Princeton University Press, 1948. 7. L. Rosenfeld,
1

:-l juAydnxlV(os) TILJ

'

I

In order to define the "exp", we have at present no other resort than to approximate by a power series,where the ordering of the non-commuting factors for JrAr at different points of space-time can be simply prescribed (e.g., the later factor to thb left). Cf. especially F. J. Dyson, Ph1ts.Rea., in press. 26. Stueckelbery, Anfu.derPh1s.,2l, 367, 1934. 2 7 . R . F e y n m a n , P h 2 s .R e a . , 7 4 , 1 4 3 0 , 1 9 4 8 . 28. S. Tomonaga, Progr. Theor.Ph1ts.,1,27,109, 19+6. 29. F. Dyson, Phys. Reu..in press. 30. Kanesawa .id To-orrga, Progr. Theor.'Ph1s.,3, 1, 107, 1948. 31. G. Wentzel. Phys. Rea..74. 1070. 1948. 32. See for insianie resulis reported to this.conference by Pauli on corrections to the Compton effect for long wave lengths, 33. F. Dyson. Phys. Rea.,in press. *Author's note, 1956. Questions l(a) and 1(b) were indeed answered by Dyson, Pi2s. Reu.,75, 1736,1949. 34. K. Case, Phys. Rea.,74, 1884, 1948.

P o p e rI 6

156

PHYSICS OF THEORETICAL PROGRESS 1946 Vol.l, No.2, Aug.-Sept., On a RelativisticallyInvariant Formulationof the QuantumTheorYof Wave Fields* by S. TOMOIIAGA L

The formalism of the ordinary quantum theory of wave fields

Recently Yukawa(1) has made a comprehensive consideration about the basis of the quantum theory of wave fields. In his article he has pointed out the fact that the existing formalism of the quantum field theory is not perfectly relativistic. yet ' Let a(rytz) be the quantity specifying the field, and A(x2z) denote its Then the quantum theory requires commutation canonical conjugate. relations of the form:

( lu(x2zt),u(x'1' a't)) -- l).(x1zt), )'(x'1' z't)l : 0 \lu (xyzt),)'(x'7' z't)l : i h6(x x') 6(7 1' ) 6(z z')'

(r)T

but these have quite non-relativistic forms. The equations (l) give the commutation relations between the quantities at different points (x7z) and (x'1' z') at the same instant of time l' The concept "same instant of time at different points" has, however, a definite ,n.u.ri.rg only if one specifies some definite Lorentz frame of reference. Thus this is not a relativistically invariant concept' Further, the Schrodinger equation for the ?-vector representing the state of the system has the form:

(a*l],)v:0,

(2)

where r? is the operator representing'the total energy of the field which is given by the space integral of a function of u and )'' As we adopt here Ih. S.h.odi.rger picture, a and.)" are operators independent of time. The vectogep.esenting the state is in this picture a function of the time, and its dependenceon I is determinedby (2). * Translated from the paper, Bull. L P. C. R. (Riken-iho), 22 (1943), 545, appeared in Japanese. originally -"i We assume that the field obeys the Bose statistics. Our consideraii. bl:2b-Ae. tioiriubplv also to the case of Fermi statistics. I

157 P R O G R E SOSF T H E O R E T I C APLH Y S I C S Also the differential equation (2) is no less non-relativistic. In this equation the time variable f plays a role quite distinct from the space coordinates x, 2 and 1. This situation is closely connectedwith the {bct that the notion of probability amplitude does not fit with the relativity theory. As is well known, the vector y has, as the probability amplitude, the following'physical meaning: Suppose the representation which makes the field quantity u(x2z) diagonal. Let ylu'(x2d ] denote the representativeof rp in this representation.* Then the representative yfa'(xyz)l is called probability amplitude, and its absolute square

Wlu'(xyz)l : lplu'(x7a)llz

(3)

gives the relative probability of u(x1,2) having the specified functional form u'(ry4 at the instant of time t. In other words: Suppose a planet which is parallel to the xyz-plane and intercepts the time axis at t. Then the probability that the field has the specified functional forrn u,(x2z) on this plane is given by (3). As one sees, a plane parallel to the x2<-plane plays here a significant role. But such a plane is defined only by referring to a certain frame of reference. Thus the probability amplitude is not a relativistically invariant concept in the space-time world.

'

2.

Four-dimensional form of the commutation relations

As stated above, the laws of the quantum theory of wave fields are usually expressed as mathematical relations between quantities having their meanings only in some specified Lorentz frame of reference. But since it is proved that the whole contents of the theory are of course relativistically invariant, it must be certainly possible to build up the theory on the basis of concepts having relativistic space-time meanings. Thus, in his consideration, Yukawa has required with Dirac(z) a generalization. of the notion of probability

amplitude to fit with the relativity theory. we shall now show below that the generalization of the theory on these lines is in fact possible to the relativistically necessary and suficient extent. our results are, * . Y. use the sqrrarebrackets to indicate a functional. Thus-tpla'(x1tz)]means that g is a functionalof the variablefunction u'(x24. ( ), a's .whenweuse ordinary-parentheses tp,(r'Ql)), we considerg as an o_rdinary functionof rhe functionut(.U,i). For example: th.eenergydensityis writtenas.i[(a(xl<\^, l(xld.).and this is alsoa functionol x.1, anl q, whereasthe total energyH=J'H(u(x2a),l(x1a))fuis afunctional of a(x1ta)and,),(il2) andii writtenasHlo(xl<),7(x1z)1. _f We call.a three-dimensionalmanifold in the four-dimensionalspace-timeworld simply a "surface."

158 S. TOMONAGA Dirac and by Yukawa' but are however, not so general as expected by is required by the relativity theory. it aS, far already sufficiently general i,-' so are only two fields interacting there that Let us suppose for simplicity of fields can also be treated number greater with each other. The case of a specilying.the fields' quantities the denote Let u1 and af in the same way. respectively' Then )"2 and )'1 are The canonically conjugate quantities relations between these quantities the commutation

( lu,(x1at), u,(x'2'1't)) :O

r, s:r,2 lii,fitril, ).,(x'2'a't):o ( : x' 6 (x i hd t)l y' ) 1' ! )6(z- z')6" zt),),,(x'z' lfr, 1*7 must hold.

('4)

The E-vector satisfies the Schrodinger equation

(s)

(tr,*a,tfrn*ik)r:o

the energy of the first and In this equation Fi1 and, E2rnean respectively of u1 and Er i, gL"'1 by the space integral-of a function iAa. ,i.'r.l""i of u2 and' 12' Further' f/12 is the 1r, Firby the space integral ol'a function of a^funcof thJ field' and is given by the space integral inl.ru.tion of H1'' "rl..gy integrand the (i) that assume We tion of both u,, /.1 and a2, 12' quantity' and (ii) that the i.e. the interaction-e.t.,g!- dt'-t'lty, is a scalar (but at the same instant of time) energy densities at twoliff""ttt'points the In general, these two facts follow from commute with each other' in the Lagrangian does not contain single assumption: the interaction term u2' the time derivatives of ul and then we have If this energy density is denoted by H t'r,

frrz:[Hp

(6)

dxd2dz.

AsweadoptheretheSchrodingerpicture'thequantitieszrand)'irrHv of time' H2 and H1, arc all operators independent the well-known facts' Now' as summarized Thus far we have merely thefirststageofmakingthetheoryrelativistic'wesupposetheunitary operator

Q)

u:.*p{!-(Er+trr)r} '[n )

andintroducethelbllowingunitarytransformationsofuand)',andthe corresponding transformation of 9:

!V,:Uu,U,

\

r, A,:UL"U-|

Y:u,p.

r:1,2

(B)

Asstatedabove'uand,).in(5)arequantitiesindependentoi.time.But of (B) contain I tlrrough U' Thus V and Aobtained fiom them by means thev dePend on / bY

!itii/,:v,Ei,-Fi,v, \in't.:A,H,-F,A,. 3

r:t,2

(9)

159 PHYSICS PROGRESS OF THEORETICAT These equations must necessarily have covariant forms against Lorentz transformations,becausethey are just the field equations for the fields when they are left alone without interacting with each other. Now, the solutions of these "vacuum equations," the equations which the fields must satisfywhen they are left alone, together with the commutation relations (4), give rise to the relations of the following forms:

(lV,(x1tzt),V,(x'7'z' t')l : A,,(x- x', ) -J', z - z " t - t ' ) -)', z - 2 " t - t ' ) : \ lA,(xl
(10)

where r4,", .B", and C,, are functions which are combinations of the so-called four-dimensional d-functions and their derivatives.(3) One denotes usually They are defined these four-dimensional d-functions by D,(x1
by I ei(k^,+-tr D,(x1at):#JJJ \-

k ^ , r + zz+ck/) *

ei(k xx+krr+kzz-.*rt).),,

(11)

jdk_ dk, dkz

with

(l 2 )

k,: \/ k*2+ krz+ kzz+x,2,

X, being the constant characteristic to the field r. It can be easily proved that thesefunctions are relativistically invariant.* Since (10) gives, in contrast with (4), the commutation relations between the fields at two different world points (x1tzt)and (*'j'<'t'), it contains no more the notion of same instant of time. Therefore, (10) is sufficiently relativistic presupposing no special frame of reference. We call (10) four-dimensional form of the commutation relations. One property of D(x2zt) will be mentioned here: When the world point (xy
(13)

for x2*)2+ z2-c2t2>0.

It follows directly from (13) that, if the world point (x'y'z'l') lies outside the light cone whose vertex is at the world point (x2zt), the right-hand *Suppose that a surface in thek*krkrk-space is defined by means of the equation Then this surface has the invariant meaning in this space, srnce 1rz:1t,zak,2+k,21x2, is invariant against Lorentz transformations. The area of the surface kr2+i!2+kz2-,t2 element of this surface is given by

dS:IW Now, since d,S has the invariant

.,

dk, dk"dk,

dlex drt dKz:x,--T-.

meaning, we can thus conclude ,h^,

invariant, and this implies that the function defined by (ll)

+

is invariant.

dktdf:dk" k

i" ^n

160 S. TOMONAGA sides of (10) always vanish. In words: Suppose two world points P and P'. When these points lie outside each other's light cones, the field quantities aI P and field quantities at P' commute with each other.

3.

Generalizationof the Schrodingerequation

Next we observe the vector P obtained from g by means of the unitary transformation U. We see from (5), (7) and (8) that 'this F, considered as a function of ,, satisfies

{l

orUrUrtat), A1@7at), V2Q2zt), A2Q4zt)) dxd1,dz+! 8r}*:,

(14)

One seesthat I plays also here a role distinct from x, 1 and e: also here a plane parallel to the x1z-plane has a special significance. So we must in some way remove this unsatisfactory feature of the theory. This improvement can be attained in the way similar to that in which Dirac(a) has built up the so-called many-time formalism of the quantum mechanics. We will now recall this theory. The Schrodinger equation for the system containing "Ay' charged particles interacting with the electromagnetic field is given by

+!ar}v:0. !,,s(q.)) {r",* tru,{l,,,

(ls)

Here FI,, means the energy of the electromagnetic field, Hn the energy of .F1"contains, besides the kinetic energy ofthe zth particle, the nth particle. energy between this particle and the field through q(q,), the interaction qn being the coordinates of the particle and s the potential of the field. p, in (15) means as usual the momentum of the zth particle. We consider now the unitary operator

z:exp

(; - ''

\io",tI

(16)

and introduce the unitary transformation of o: 2I:uau-r

( r 7)

and the corresponding transformation of g: @:ug.

(18)

Then we see that @ satisfiesthe equation

t))*! {lr,f,,0,,w(q^, !r}o:0.

(le)

In contrast with o, which was independent of time (Schrcidinger picture), 2[ contains I through z. To emphasize this, we have written t explicitly as the

5

l6'l P R O G R E SOSF T H E O R E T I C APTH Y S I C S argument of !I. We can prove that 2[ satisfiesthe Maxwell equations in vacuo (accurately speaking,we need special considerationsfor the equation div G:0). The equation (19) is the starting point of the many-time theory. In this theory one introduces then the function @(qttt, Qztz,' ' '' qN, tN) containing as many time variables ty t2, . . . ty as the number of the particles in place of the function @(qt,qz,"',4N,1) containingonly one time variable,* and supposethat this (D(qrtt,gztz,' ' ',1ytx) satisfiessimultaneously the following .lf equations: ( h A).. ' ', qNt.il:0 1 H , ( q , ,P , , 2 { ( q n/ ,, ) ) + ;t ;ult @ rl ( q t t t , 8 z t z , ' I

n:lr2r.

. .,N.

(20)

This @(11,t2,. . ., fy), which is a fundamentalquantity in the many-time theory, is related to the ordinary probability amplitude A(t) by

:q11, t, . . ', t).
(21)

(20) can be solved when and only

qztz,"

',1ytr):0

(22)

are satisfied for all pairs of n and n' . If the world point (g,1,) lies outside the light cone whose vertex is at the point (q,'tn'), we can prove that As the result, the function satisfying (20) can exist H,H;-H,'Hn:o. in the reeion where

( q , - q , ' ) ' - t 2 ( t , - t , ' ) z> o

(23)

is satisfiedsimultaneouslyfor all values of n and n'. A c c o r d i n g t o B l o c h ( 5 )w e c a n g i v e @ ( q 1 1 1Q, z t z , " ' , q r t N ) a physical meaning when its arguments lie in the region given by (23). Namely W ( q r t r ,Q z t z , .. . , Q w t u ) : l ( D ( q r t t , 4 z t z , '' ,' q N t t i l 2

(24)

gives the relative probability that one finds the value q1 in the measurement of the position of the first particle at the instant of time 11, the value q2 in the measurement of the position of the second particle at the instant of time 12, . . . and the value q]u.in the measurement of the position of the ,Mth particle at the instant of time try. This is the outline of the many-time formalism of the quantum mechanics. We will now return to our main subject. If we compare our equation (14) with the equation (19) of the many-time theory,.we notice a marked In (19) stands the suffix z, which similarity between these two equations. designates the particle, while in (14) stand the variables x,) ar\d e, which * Here we suppose the representation which makes the coordinates by a function of these coordinates. b is represinted the t ".toi

Thus

8r, Qz, "',4p

diagonal'

r62 S. TOMONAGA designate the position in space. Further, @ is a function of the "lf independentvariables Qt, Qz,''', QN,g, giving the"position of the zth particle, while F is a functional of the infinitely many "independent variables" u{x1z) (x7<)' and, u2@1tz), ulQyz) and u2Q1tz) giving the fields at the position 'd< stands d1 Corresponding to the trr- )F1, in (19) the integral lHpdx in (14). In this way, to the suffix z in (19) which takes the values 1,2, 3, . ., "ly' correspond the variables x,7 and z which take continuously all values from - co to + oo. introduce infinitely many time variables each for one position (x1z) in the space, variables, particle times, 11, t2, ' ' ', tv, difference is that we use in our case we have used "lf time variables in whereas variables many time infinitely theorY. many-time ordinary the Corresponding to the transition from the use of the function with one

Such a similarity Suggeststhat we 1,r., which we may call local time,* just as we have'introduced .M time The only each for one particle.

time variable to the use of the function of "ly'time variables, we must now consider the transition from the use of Y(t) to the use of a functional Ylt*rr] of infinitely many time variables 1,r.. We now regard t,rz as a function of (x2<) and consider its variation e,r. which differs from zero only in a small domain I/e in the neighbourhood' We will define the partial differential coefficient of of ihe point (roto
6V dt,oro,o

,,* Vlt,rrr e,r"f YLt,r"] u r r r -

lfi,ur

fJ

dx d2 da

(25)

We then generalize (14), and regard

{tr"1"7'

''

tt+l{}v:0,

(26)

the infinitely many simultaneousequationscorrespondingto the.ly'equations (20), as the fundamental equationsof our theory. In (26) we have written, ')' f o r s i m p l i c i t y , H r z ( r , ) , 1 , t ) i n p l a c e o f H r r ( V 1 Q 1 z , t ) , V 2 Q 1 z , t ) ," write will In general, when we have a function F(V, A) of V and A, we simply F(r, ), a,t) foy F(V(x1tz,t,rr), I (x22, trr)), or still simpler F(P)' P denoting the world point with the coordinates(x12,t,rr). Thus F(P') means F(*',J' , q', t') or, more precisely,F(V(x'2'<', tr,r'r'),(,i'<', t*y))' * The notion of local time of this kind has been occasionallyintroduced by Stueckelberg.t0)

r63 P R O G R E SOSF T H E O R E T I C APLH Y S I C S We will now adopt the equation (26) as the basis of our theory' For Vr(P), Vz(P),A1(P) andA2Q) in Hrrthe commutation relations(10) hold, where D(x2zl), has the property (13). As the consequence,we have H n ( P ) H n ( P ' , )- H r z ( P ' ,H) r , ( P ) : 0

(27)

when the point P lies a finite distance apart from P' and outside the light cone whose vertex is at P. Further, from our assumption (ii) the relation (27) holds also when P and P' are two adjacent points approaching in a Thus our system of equations (26) is integrable when space-like direction. the equations t:t,1a, considering txrz as a function by defined surface the of x, y and 1, is spaceJike. In this way, a functional of the variable surface in the space-time world is determined by the functional partial differential equations (26). Corresponding to the relation (21) in case of many-time theory, Ylt-rrf reduces io the ordinary pr(t) when the surface reduces to a plane parallel to the x1t<-plane. can be of any (space-like) form The dependent variable surface t:t,r, any Lotentz frame of presuppose not need we world,and in the space-time is a relativistically this F[1,r"1 Therefore, surface. a such to define reference be space-like rnust surface the that restriction The invariant concept. or time-like is space-like a surface that property the since makes no trouble, It is not system. reference of the choice special on a does not depend also admit to theory, relativity of the slandpoint the from necessary, time-like surfaces for the variable surface, as was required by Dirac and by Thus we consider that Ylt*r"] introduced above is already the Yukawa. sufficient generalization of the ordinary ?-vector, and assume that the state* of the fields is represented by this functional quantum-theoretical vector. Tben F is Let C denote the surface defined by the equation t:t*tz' C we take a On v7[C]. as this write we c. surface the a functional of C'which a surface suppose and (xyz,t,rr), are point P, whose coordinates of the volume denote P' We about domain in a small orr..lup, C except write may we Then dap' C' by C and lying between the small world (25) also in the form:

,Yltt _ r.rnYlc'l-Ylcl. d(tJp dLlp

(28)

c' -c

Then (26) can be written in the form:

(2e)

{o,u,,+!{}vra:o * Trhe word state is here used in the refativisfrcspace.time (second edition), $ 6.

B

meani:r$.

cf. Dirac's book

164 S. TOMONAGA This equation (29) has now a perfect space-time form. In the first place, H12 is a scalar according to our assumption (i) ;in the secondplace, the commutation relations between V(P) and A(P) contarned in Hp,has the four-dimensional forms as (10), and finally the differentiation

,fr

is

defined by (28) quite independently of any frame of reference. A direct conclusion drawn from (29) is that 9[C'] is obtained from YlCl by the following infinitesimal transfbrmation:

vg'1 :

{r

-LnH,,(p ) a.,} vyc1.

(30)

When there exist in the space-time world two surfaces C, and C2 a finite distance apart, we need only to repeat the infinitesimal'transformations in order to obtain VIC) from Ylcrl. Thus

Ylc,l :V{, -

p) Ytc,l. ;",r(P) da)

(31)

The meaning of this equation is as follows: We divide the world region lying between C1 and C2 into small elements dap (it is necessary that each world element be surrounded by two spaceJike surfaces). We consider for each world element the infinitesimal transformation I -ftHp(P)

do:p.

Then

we take the product of these transformations, the order of the factor being taken from Ct to Cr. This product transforms then F[C1] into P[Cr]. The surf,aces C1 and C2 must here be both space-like, but otherwise they may have any form and any configuration. Thus C2 does not necessarily lie afterward against Ct; Ct and C2 may even cross with each other. The relation of the form (31) has been already introduced by Heisenberg.(z) It can be regarded as the integral form of our generalized Schrcidinger equation (29) .

+.

Generalizedprobability amplitude

We must now find the physical meaning of the functional P[C]. As regards this we can follow a method similar to that of Bloch in the case of ordinary many-time theory. Besides the fact that in our case there appear infinitely many time variables, one point differs from Bloch's case: in (16) the unitary operator z is commutable with the coordinates ql, Qz, . . ., QN, while our [/is not commutable with the field quantities u{xyz) and u2@22). Noting this difference and treating the continuum infinity as the limit of a denumerable infinity by some artifice, for instance, by the procedure of Heisenberg and Pauli,(8) Bloch's consideration can be applied also here almost without any alteration. We shall give here only the results. q

165 P R O G R E SOSF T H E O R E T I C APLH Y S I C S Let us supposethat the fields are in the state representedby a vector [C]. We supposethat we make measurementsof a functionf (rr' r", 1r 1r) at every point on a surfaceC1 in the space-timeworld. Let Pl denote the variable point on C1, then, itf(Pt) at any two "values" of P1 commute rqith each other, the measurementsofl at each of these two points do not interfere with each other. Our first conclusion says that in this case the expectation value ofl(P1) is given by

f ( P r ) : ( ( Y l c l , f ( P r )v t c r l ) )

(32)

where;f(Pt) means-f(Vt(Pr), "') according to our convention on page '8, and the symbol ((A, B)) with double parenthesesis the scalar product of two vectors A and B. It is impossiblein casesof continr:ouslymany degrees of freedom to represent this scalar product by an integral of the product of two functions. For this purpose we must replace the continuum infinity by an at least denumerable infinity. More generally, we suppose a functional FL"fVr)l of the independent as a function of P1. Then the variable function -f(Pr), regarding f(P) expectation value of this F is given by

F[-f(Pr)]: ((v[cr], Fl"f(Pr)lv[cr])).

(33)

A physically interestingF is the projective operator Mlul'(P), uz'(P); Vr(P), Vr(Pr)l belongingto the "eigen:value" at'(P), rz'(P) of V{P), Vr(Pt). Then its expectation value Mlul'(P), az'(P); V{P), Vr(Pt)J : ((VICrl, Mlul'(P), uz'(Pr); Vt(P), Vz@r)lytctl))

(3+)

gives the probability that the field I and the field 2 have respectively the functional form u1'(P1) and u2'(P) on the surface C1. As C1 is assumed to be space-like,the measurement of the functional M is possible (the measurementsof Zr(Pr) and V2(P) at all points on C1 mean just the measurementof M). Thus far we have made no mention of the representationof YIC). We use now the special representationin which Vt(Pr) at all points on C1 are simultaneously diagbnal. It is always possible to make all V1(P) and Vr(Pr) diagonal when the surface C1 is space-like' In this representation tr[C1] is representedby a functional Vlul'(Pt),uz'(Pt); C1] of the eigenvaluesur'(P) and ur'(Pr) of V1(P) and V2(P1). The projection operator M has in this representation such diagonal form that (34) is simplified as follows

wlu|(p,),,yrrrll,j

(35)

In this sensewe can call Vlu y' (P 1), uz' (P r); C1] the " generalized probability amplitude." IO

166

S. TOMONAGA Generalized transformation functional

5.

We have stated above that between 9[C1] and YlCr) the relation (31) holds, where C1 and C2 are two space-like surfaces in the space-time world. We see thus that the transformation operator :2-/

i

rlc2;cl:l"l('-;Heaat)

\

(36)

plays an important role. It is evident that this operator also has a spacetime meaning. Just as the special representative of the ?-vector, the probability amplitude, has a distinct physical meaning, so there is a special representation in which the representative of the transformation operator TlC2; C1] has a distinct physical meaning. We now introduce the mixed representative of TIC2;C1] whose rows refer to the representation in which Z1(P1) and V2(P1) at all points on C1 become diagonal and whose column refer to the representation in which We denote this Vt(Pr) and V2(P2) at all points on C2 become diagonal. representation by

l, ," (P r) , uz" (Pz)lTlC r; C lllu 1'(P1), az'(P )1,

(37)"

' )1. l r t " ( P r ), a z "( P z ) l o r(' P r ) , u z V

(38)*

or simpler:

If we note here the relation (35), we see that we can give the matrix elements of this representation the following meaning: One measures the field quantities Z1 and V2 at all points on C2 when the fields are prepared and ur'(P) at in such a way that they have certainly the valuesal'(P) all points on C1. Then W l u 1 " ( P , ) , , z " ( P r ) ; a r ' ( P t ) , uz ' ( P r ) f : l l r r " ( P z ) , , z " ( P z ) l o t ' ( P t ) ,u z ' Q ) l l ,

(39)

gives the probability that one obtains the result ur"(Pz) and u2"(P2) in this In this proposition we have assumed thatC2lies afterward measurement. against C1. From this physical interpretation we may regard the matrix element (37), or (38), considered as a functional of u1"(P2), uz"(Pz) and u1'(P\), az' (P t) , as the generalization of the ordinary transformation function (g,r"lQr,'). As a special case it may happen that C 2lies apart from C1 only in a portion 52 and a portion 51 of C2 and C1 respectively, the other parts of C1 andC2 overlapping with each other. In this case the matrix elements of TlC2; C1] depend only on the values of the fields on the portions 51 and 52 of the surfaces Cl and C2. In this * As the matrix elementsare functionals of a(P), we use here the squarebrackets. II

167 P R O G R E SOSF T H E O R E T I C APLH Y S I C S casewe need lor calculating TlC2;Cl to take the product in (36) only in the closeddomain surrounded by 51 and Sr, thus

z[s2;s1]:tt(r-tor" ^)

(40)

The matrix elements of the mixed representation of this Z is a functional of u1'(p), a2'(p) and ut"(pr), az"(Pz), wherepl denotes the moving point of the portion 51, and p2.the moving point on the portion 52. This matrix is independent on the field quantities on the other portions of the surfaces Cl and C2. The matrix element of Z[S2;51] regarded as a functional of ur'(f), 0z'(p) and u1"(p), uz"(Fz) has the properties of g.t.f. (generalized transformation functional) of Dirac. But in defining our g.t.f. we had to restrict the surfaces ,S1 and S, to be space-like, while Dirac has required his g.t.f. to be defined also referring to the time-like,surfaces. As mentioned above, however, such a generalization as required by Dirac is superfluous so far as concerns the relativity theory. It is to be noted that for the physical interpretation of lu1"(P2), rr"(Pz)l ot'(Pt), uz'(P)l it is not necessary to assume C, to lie alterward against Cr Also when the inverse is the case, we can as well give the physical meaning for W of (39): One measures the field quantities Z1 and Viat aIl points on C2 when the fields are prepared in such a way that they would have certainly the values ut'(Pt) and ur'(Pr) at all points on C1 if the fields were left alone until C1 without being measured before on C2. Then W gives the probability that one finds the results ut"(Pz) and u2"(P2) in this measurement on c2.

6.

Concluding remarks

We have thus shown that the quantum theory of wave fields can be really brought into a form which reveals directly the invariance of the theory against Lorentz transformations. The reason why the ordinary formalism of the quantum field theory is so unsatisfactory is that it has been built up in a way much too analogous to the ordinary non-relativistic mechanics. In this ordinary formalism of the quantum theory of fields the theory is divided into two distinct sections: the section giving the kinematical relations between various quantities at the same instant of time, and the section determining the causal relations between quantities at different instants of time. Thus the commutation relations (l) belong to the first section and the Schrodinger equation (2) to the second. As stated before, this way of separating the theory into two sections is very unrelativistic, since here the concept " same instant of time " plays a distinct role.

r2

t68 S. TOMONAGA Also in our formalism the theory is divided into two sections, but now the separation is introduced in another place. One section gives the laws of behaviour of the fields when they are left alone, and the other gives the Iaws determining the deviation from this behaviour due to interactions. This way of separating the theory can be carried out relativistically. Although in this way the theory can be brought into more satisfactory form, no new contents are added thereby. So, the well-known divergence difficulties of the theory are inherited also by our theory. Indeed, our fundamental equations (29) admit only catastrophic solutions, as can be seen directly in the fact that the unavoidable infinity due to non-vanishing zero-point amplitudes of the fields inheres in the operator Hp(P). Thus, a more profound modification of the theory is required in order to remove this fundamental difficulty. It is expected that such a modification of the theory could possibly be introduced by some revision of the concept of interaction, because we meet no such difficulty when we deal with the non-interacting fields. This revision would then have the result that in the separation of the theory into two sections, one for free fields and one for interactions, some uncertainty would be introduced. This seems to be implied by the very fact that, when we formulate the quantum field theory in a relativistically satisfactory manner, this way of separation has revealed itself as the fundamental element of the theory.

PhysicsDepartment, Tokvo Bunrika lJniversitv.

References 1. 2. 3. 4. 5. 6. 7. B.

H. Yukawa, Kagaku,12,251,282 and322, l9+2. P. A. M. Dirac, Ph1ts./. U,SSR.,3, 64, 1933. W. Pauli, Solaa2Berichte, 1939. P. A. M. Dirac, Proc. Roy. Soc.London, 136, 453, 1932. F. Bloch, Phjts. /. USSR., 5, 301, 1943. E. Stueckelberg,Hela. Ph1s.Acta,11,225, $ 5, 1938. W. Heisenbery, l. Ph2s., 110, 25i, 1938. W. Heisenberg and W. Pauli, ,/. Ph1s.,56, I, 1929.

r3

P o p e rI 7

Erectro dvnamic s.,lll Quantum

J3:,?:fff;ilfr%:|;,Lfln;"t "'

169

ofthe Erectron

JULIAN SCEWINGER E araard, Uniorsity, Canbrid gc, LI assachusetls (ReceivedMay 26,1949) The discussionof vacuum polarization in the previous paper of this serieswas confrnedto that produced by the field of a prescribed current distribution. We norv consider the induction of current in the vacuum by an electron, which is a dynamical system and an eniity indistinguishablefrom the particles associated n'ith vacuum fluctuations. The additional current thus attributed to an electlon implies an alteration in its electromagneticproperlies q'hich will be revealed by scattering in a Coulonb field and by energy level displacements.This paper is concernedwith the computation of the second-ordercorrectionsto the current operator and the application to electron scattering. Radiative corrections to energylevelswill be treated in the next paper of the series. Follorving a canonical tmnsformation which efiectively renormalizes the electron mass, the correction to the current operator produced by the coupling with the electromagneticfield is developed in a power series,of which first- and second-orderterms are retained. One thus obtains second-ordermodifications in the current operator which are oI the same general nature as the previously treated vacuum polarization cu[ent, save for a contribution that has the form of a dipole culrent. The latter implies a fractional increaseof af2n in the spin magnetic moment of the electron,The only flaw in ttre second-ordercurrent co[ection is a logarithmic divergenceattributable to an infru-red catastrophe. It is remarked that, in the presenceof an external field, the fiFt-order cuuent correetion rvill introduce a compensating divergence.Thus, the second-ordercorrections to particle electromagnetic properties cannot be completely stated without regard for the manner of exhibiting them by an erternal field. Accordingly, we consider in the secondsection the interaction of three systems,t1ie matter field, the electromagneticfield, and a given current distribution. It is shown that this situation can be described in terms of an external potential coupled to the current

operator, as modified by the interaction rvith the vacuum electromagnetic field, Application is made to the scattering of an electron by an external freld, in which the latter is regarded as a small perturbation. It is found convenient to calculate the total rate at rvhich collisions occur and then identify the cross sections for individual events. The correction to the cross section for radiationless scattering is determined by the second-order correction to the current operator, while scattering that is accompanied by single quantum emission is a consequence of the first-order current correction. The 6nal object of calculation is the difierential cross section for scatteing through a given angle with a prescribed maximum energy loss, which is completely free of divergences. Detailed evaluations are given in trvo situations, the esseniially elastic scattering of an electron, in which only a small fraction of the kinetic energy is radiated, and the scattering of a slowly moving electron with unrestricted energy loss. The Appendix is devoted to an alternative treatmetrt of the polarization of the vacuum by an external field. The conditions imposed on the induced current by the charge conservation and gauge invariance requirements are examined. It is found that the fulfillment of these formal properties requires the vanishing of an integral that is not absolutely convergent, but naturally vanishes for reasons ot symmetry. This null integral is then used to simplify the erpression for the induced current in such a manner that direct calculation yields a gauge invariant result. The induced curent contains a logarithmically divergent multiple of the external current, which implies that a non-vanishing total charge, proportional to the external charge, is induced in the vacuum. The apparent contradiction witi charge conservation is resolved by showing that a compensating charge escapes to infinity. Finally, the expression for the electromagnetic mass of t}le electron is treated with the methods developed in this paper.

COVAnTAUT form of quantum elecrrodynamics I / \ h a s b e e n d e v e l o p e d , a n d a p p l ti e od twoelementary vacuum fluctuation phenomena in the previous articles of this series.rThese applications were the polarization of the vacuum, expressing the modifications in the properties of.an electromagnetic field arising from its interaction with the matter field vacuum fluctuations, and the electromagnetic mass of the electron, embodying the corrections to the mechanical properties of the matter field, in its single particle aspect, that are produced by the vacuum fluctuations of the electromagnetic field. In these problems, the divergencesthat mar the theory are found to be concealed in unobservable charge and mass renormalization factors. The previous discussion of the polarization of the vacuum was conaerned with a given current distribution, one that is not afiected by the dy'namical reactions ol the electron-positron matter fi.eld.We shall now consider the more complicated situation in which the origi-

na1current is that ascribed to an electron or positrona dynamical system, and an entity indistinguishable from the particles associated with the matter field vacuum fluctuations. The changed electromagnetic properties of the particle will be exhibited in an external field, and may be compared with the experimental indications of deviations from the Dirac theory that were briefly discussed in I. To avoid a work of excessiveiength, this discussion will be given in two papers. In this paper we shall construct the current operator as modified, to the second order, by the coupling with the vacuum electromagnetic field. This will be applied to compute the radiative correction to the scattering of an eiectron by a Coulomb field.'?The secondpaper will deal with the efiects of radiative corrections on energy levels. 1. SECOND-ORDERCORRECTIONSTO TI{E CURRENT OPERATOR We shall evaluate the second-ordermodifications of

-,1olinn s"h*iog"r, "euantum Electrodynamics. I,,, phys. Rev. f,4, 1439 (1948); "Quantum Electrodynamics. II," Phys. Rev. 75, 651 (1949).

the current operator produced by the coupling between 2A short account of ttre results has already been published, Julian Schwinger,Phys. Rev. 75,898 (1949).

790

170 OUANTUM

791

ELECTRODYNAMICS.

the matter and. electromagnetic fields. The latter is describedbY

6vIol ihc__:_:.:ss1s197r1,

III

obeys the Dirac equation for a spinor l,Ir-1[c](r)tr/[o] particle of iiss m: mo* 6m, the experimental mass.of ihe electron. Accordingly, the expectation value of the current operator can be computed as

Do(r)

(j,(x)) : (vlcf ,j r(r)v[o]),

(1.6)

mravlof:x(*)vlo),

(1.7)

where

1 rc(x): --j,(r)A,(x).

(1.1)

,ith the understri::i*,

the experimentalelectron

electromagneticmass of the electron, as contalned.,ln massis Lobe employed. the self-energy operator Kr,o(*). In order to descrtDe If a solution of th" Iott"r equation is constructed in the electron in terms of the experimental mass' we the form (1'8) ir,[o]: Y|"1Eo, wrire (1.1) as 6V[ol ;7,, _:-: : (K t.o(.r)+ ff (r)) vl"l, 6o(*)

Ir.zl

K(x):K(x)-'ct'o@)'

/i 2\ \r'rl

where

the expectationvalue oI the current operator becomes ( j,(")) : (v 0,u-tl6l j F@)u[o]v) : ('ro,i"(r) vo)' (1.9) in which the latter version describes the efiect of the coupling between the fie1dsby changing the current into operator '

The canonical translormation v[a]+trV[o]v[o],

j,(r):U-tlo)

j,(x)ulal'

(1.10)

obeys the equation of

6Ivlof ihcJ__Kr,o@)Wlo7,

(1.4)

6o(*)

then replaces (1.2) with 6Il'Io-l ;7rrlj:i=gr-tS-o)K(x)Wlolelof, 6o(*)

(1.5)

The unitary operator U[o] motion 6ulo) " ":y(x)Ulof, i7,, 6o(:r)

(1.11)

which may be supplementedby the boundary condition (1'12) Ul-a):|,

in accordance with the supposition that coupling between the two fields is adiabatically established in the be_9om.es while the operator represenring the currentpast' tll-rto]ru(*)tr42[o]. Now, as rve have shown in II, the remote The operator ju(*) can now be evaluated by remarking that

: i,@+ a,' ir@ [" J*rr-'rrt,,(*)ulo')) =i rA> a,' (J-Llo'fliu@),x(r'))ulo'). ; ["

(1.13)

This processcan be continued according to

[

"a,'u,7a1g-@),x.(*'))t] lc'): fuT i,t*),*t*)f (1'14) + !" a,'!" a;' J *rr-r,"lli,@),x.(r')lu[o"]),

and yields j"(r) in the form of an infinite series, it'zr' i.\f" / 1 (1'1s) i,t*):v,(*)+(-r)[1;ti,
t7l SCHWINGER

JULIAN

basedon the followingobservation, r*6r'6 I d,',lo,o')-(U-tlq' 0a

r-6

\I

U,(x)Ul"' )) : I a,' r-a Oq

)

(U-tlo'li,G) ULo'f) \fi )

+f'a,'*\qLr-Llo'fi,(x)ulq'l):(i,(x)_.j-(r))t(L(r)- u-tlalj,@)ul@l), (1.16) J*

or

6o,(x,)

i * j"(r):10,(rr*s 'j,(as)+( \ f'0,,,7,,",1u-Llo,)lj,(r),K(x,))uro,l, \ 2tu,tJ--

(1.12)

where S:U[€],

U[-o]:1

(1.18)

is the collision operator which describesthe real transitions that permanently alter the state of the system. This processcan be continued and finally yields

i,@): i @,@)-fS-'ftu(r).1),

(1.1e)

in which /

A,(*):7"(r)* ( -= \

L\f'

| t da'elo.o'lljr(rS,K6')f

zncl J-

/

*

it2n"

\- il

J_

j,(r),x(x')f,x(r"))*''' . (1.20) d,'du",la,c' lelo',o"Jll

The further terms in the seriesare not required to compule the second-ordercorrection of the current operator. The collision operator S can be constructed in a similar manner. Thus,

s-r: f-a,-ltbl:-!f"a,*(irbl, J_6 6o(r) hcJ_-

(r.21)

and

u[o]-t(s*r):l f- o.'"7,,r1]=ab'l:-: I a,',lo,o'fx(,'tulo'1, 2r _2hcr_a 60'(r")

o.2z)

rvhence,

| a,r1*;(s+r)+r( - =\'

s-r: |/--) \

zhcl.l_-

\

[* a,,t,',7o,o'1x(x)r(r')u[o'].

(1.2i)

2hc/ J *

Continuing in this manner, we obtain

( +)

-:

.. ['_a,xo* (- *)' ["-a,a,,ra,o'],r(r)r(x')*

(r.24)

Only the indicated terms need be retained for the desired degreeof approximation. fn view of the absenceof real first-order effects, as expressedby

f--*av':o' order: the leadingtermsin (1.24)areof the second i r" f i r'" S-t I -; = -;(r-.t')[r(.r),rc(x')]do'-rcro(r)l. ^,. | a.lL 4 h r JI - - e J

(1.2s)

(1)6)

lltcJ--

J+l

is of no consequence), According to (II 3.14) and (II 3.71), (the vacuum term 3Co,o

--

t,f"

J_*rt*-

r )[3c(i'),3c(#r)]d@':Kr,o(x)*K,,0(r)*3c', '(c),

0'27)

rvhence,

;:

-

* rc''r(*)ld'' o(r) # I:Fcz'

(1.28)

172 793

III'

ELECTRODYNAMICS.

QUANTUM

quantum. Since we.shall which describesthe real effects involving either two particles or one particle and a light quanta, such real l. .o.r."r.r.a oniy with second-ordereffits referrirg io a single partiile and the absenceof light "come into play and S is efiectively.-unity. Consiquently, the current operator is modified only by p."."rr* a" ""t future. Thus, to the desired order of and is iompleteiy symmetrical beiween p"it p.o."rr.s, liit*t "ttd approximation,

x

d,u' elo,o' ll j,(x), c (*)f* !.2h(J(,I r',lo,o'll j, (c),rc',o(*')1

i,(*) :,,)(r) - ..

_^

Zhc

i\2r' / - + ( | |

\

d r' dr" r;o,o' lelo',o" lll; u(t),tc(r')f

2tul r_^

'tc(r" ))'

(t.2e)

The correction to the current operator may now be written

i,(x) -i,(r):

(1.30)

(*)' 6i,or(5)f l7ut'?r

where xf-

j 6juilr (x) : | do'e(x x' )l uk),j,(x' )lA,(*' ), zkc'.t_and --(dl"(,)(-r))1,6: '

(1.31)

1r' | da'da"elo,a'lelo',o")llj,(x),j"(r')lA,(x'),jx(x")A1(r")lr,o

4hz(a.t_6

+'

l' (1.32) I a,'r(r-x')lj,(r),K,.0(r)

2hcJ,-

that we are only In the latter, the subscriptsemphasize respectively. corrections, are the first- and second-order (1'32),note that To simplily quanta. no light particle and involving a single effects with second-order lon."ro"d r llj,(x),j"(a')fA,(x'),jx(r")Ax(x"))',0:ilA"(*'),Ax@"))llir@),j"(*')),jx(*")l * LI A, (r' ), Ax(r") | ollj,(*),j,(*')l,y^(*")l', whence ir* (07,(r,(x))r.o: _ | dr' dr" 17*- x')D(*' - r") ll j,(r),j "(r')),j,(x" ) | t +hdr-* -

(1.33)

t

(* o,'o,"ela,o'felo',o"lDa)@'-l')llj,(x),j,(x')1,j,(f'D,a!ll, (1'34) f* a,',{*_ x')l'i,@),lr',vo(*) 2hcJ* thdJ-^

in consequenceof

- \e(x' - *")lA,(r'),At(*")f-

and

ihc6"xD(r' - r")

(Yt- *' t)' I A,(x'),Ax@")l o: 766,^P
(1.3s) (1.36)

The double commutator in (13a) is easily evaluated, - dcl{'(r)t us(*- r')til'(r') -{(x')7"s(r'

llj,(r),j,(*')f,j,(r")l:

: idc (0(it,S

- x)1u'!@)fi(x")7;l'(r"))

(x- r')t,s (x' - x")t,{@") *{ (r")t,S (r" - x')t,S (u' - x)t,{@) -{,(x')7,5(r'_

r)1uS(x-x")7,,!t(r")-{,(x")1,5(*"-r)7uS(x-x')t,t@))'

(1'37)

canbeconstructedinthemanneremployedinll.Wehaveonly T h e o n e - p a r t i c l ep a r t o l l l j r ( r ) , j " ( * ' ) ) , i , ( r " ) l a non-vanishingvacuum expectationvalue' Thus, to notice that ljr("),i,(*'))has \l j,(x),j,(x')f,j,(r")l'-

2lj,(r),j,(*' ))oj,(*" )

r)t,t!(*))',(0(u")t,{G"))']r, : - dc{(tr'(r)t,S(*-r')t"'1,@')-{'(x')t,S(r'x)t,{@) : - dC(0@)t,S(**')t,19@);0@")l otil,@")*{(r")t,l,l'G"),'l@)llrr,S(x'-|(r')t"S(x'-x)t,l{(*),{(*")lov,{(*")1{(*")t,l'l'@')fi@)\o,S(r-r')til(x'))t $'38)

173 JULIAN

SCHWINGER

794

and ll j,(x),i,(r' )f ,j "(x")l,: 2l i,(r),i,(r' ))oj,(x") * e3C(p(x)1uS(nst)1,gtr) (*t- *,)t,g(r,,)-{,1*,1.y,5o(x, _ rt)r"S(rt_r)74/t\r1 fi(r')1"5(*'-r)1,$
(6J*'''(r))',o: _ -

id r' nJ _.a,

-

;. I

a;'@(n')t,S(x'- r)"yrSttt (a- *,,)ys!(x,,)*.{t(r,)1,go; - x)7 uS(r_x,,)7s1,(x,,))1D(r, (x6, _ r,,)

da'd@',@(r')z,S(r,-.r)r"S (x- x,,)y,,1, (1,))1D,t,(r,_ ).,,)

te f'

+ - | d,' (0@).yf1.r-r,)41x,)a 6G')S(x'- J,t,,!(x))t -

'-[* ar,r{*- r,)({,(tc)l,1,!t(r),lcto*,)fll{@),tc,,o(*,)fiu*(r))x(1.40) 2hJ--

where (seeII (3.78))

0@): -

e' r'

"

I

do'7,(D(r - x'75 ot(r - r')| Dttt(n- x,)S(*- r,))y @,) ",1,

:6mc2L@).

(1.41)

The third term of (1.40) is derived from ld

-

f'

J ^d.'ar"

1r7o,o')- elo,o"f)elo',c")(f,(r,)7"5(x, - r)1uS(*- *,)7;!(x,,))prD (*, - r,,)

(1.42)

with the aid of the identity Qlo,a'f-elo,o"))elo',o"1:6lo,o,lelo,o,,l-1,

andthenullvalueof

(1.43)

I a,' a," 1.'1r'17,S(x'-*)7"S(*- i, )7,p@'')),ptrt(*,- *,,) lr*

:-;l

d a , d u ' , ( { , ( r , ) 7 , 1 , ! , ( r , ) , { ( x ) l y , { , 1 , @ ) , 0 @ , , ) W ^ V ( * , , ) ) r { A , ( * , ) , A x (( r1,.,4) 140) .

The latter is an immddiate conse4t'enceof (1.25), expressingthe absenceof real first-order transitions. The insertionof the expressionfor JC1,q(*),(tI (J.i7)),

rct,o(r): * (0@)o @)* 6@)* (r)),,

11 4ql

enablesthe last two terms of (1.40) to be combined into

de - (0@)t,xG) * xk)t r{ Qc)),, 2h

where

.
(1.46)

(1.4i)

It will first be observed thar the integrand of (1.a7) vanishes, in virtue of the relation 6@): 6mc2*(r), since t

-e(x- r' 11,1,1x),9(r')W(x'): - S(x- x')6mc2,!,(.r,). 2

(1 . 4 8 )

r74 QUC,NTUM ELECTRODYNAMICS.

7g5

On the other hand, integrals of the form

III

f^ I do'S(r-r')9(*..')

(1.4e)

J_6

respectively, obey homogeneousand inhomogeneousequations associated are divergent, since,y'(r') and s(r-r'), convenieni io e"p."rs the latter integral as the limit of the finite quantity It'is difiereniial'dperaior. *iir, tr.. in."6 equation satisfied by 'y'(o), in rvhich the mass parameter r is replaced tf," aif"i""tlul o? fv "lteration "" equations "fl"i".i The difierential bv x*Dr uod th" 1i-it 6x+0 is taken.

- or,r -rs(r- o'): 51s-'v';, : 0, ('.fi * -* * ) 0@') *; r.

.(1rs0)

imply the relation

!lSwhence

o)r,,t(r')l{ 6rS(*-*'){ (r'): 6(x- r')t@),

(1.s1)

n'1 -/(.t). I ar'S(x- r'),!(.r'):Ljm 6{-0

(1.s2)

6(

J_-

for xk) iI the spinor, oI which (1-'+7)is a linear It may be inferred that a non-vanishing value will be obtained xf 6r, and 6r is allowed to approach z e r c . l n parameter mass the with tirr. equatiir ;;;; iu$ffiil;j* (1.52), (1.48) and dded, according to r*

xk) : LrTJ d,'S(x-x')(6@)- 6nc\!,(r')) (1.s3)

:a6-141r)-6mN@D, ax_0dx or

I y

e2 c*

1

(1.s4) iJ-!,'r,tu1*-x5')$trt(x-l')!D(r)(r-#')S(r-j''\)7"'!(*')-tncztt(x)l'

-(.):l;S;L-

,\ 6/

--l\\

| /,-t\

with an altered mass parameter, {.+*(/'), A suitable representation of the solution of the Dirac equation provided by is oi the actualipinot,t,(r'), d 6r

in terms

(1.ss)

slnce

(1'56)

: - 6o1,,@'),

the validity of (1.5s)to the first orderin dx.The latter is so constructedthat'!"+a'(n):{-(r), whichestablishes whence,

-,

_6

-x')lDr1)(tt-tc')S(r-*))r,(q*1r1:L I a''r,(D(o-r')^Scrr(a ZxJ_

d

ri);;l(x')'

correctiouto the currentoperatoris for the second-order . The resultingexpression idr* i r(6J'(,)(r))10:: | ar'a""({,(x')K,(r'-x,x-r")rl(*"))r, I d,r',(r-x')li,G),i"({fo6A,(x')-"* 2hJ*

(1 57)

(1'58)

tnc"r4

where

and Here

lr' D(r- x')j,(r'), aA,(x): J^au' Ku(r'- *,x- tc"):I-uo\(x'-

r,*- *tt)! Krtz)(n'- x,r- fr")'

(r)t(t+r)*s(Ov,rs(?)DG)(f+?)h, D(€*r)*s0)(t)z,f K,o)(€,"r):r,(s(t)r*so)(a)

(1.se) (i.60) (1.61)

r75 SCHWINGER

JULIAN

796

and fdla KuQ)(t,n):-yu6({)----as,(D(4)Jio)(?)+rG)("r)S(,r))y,*: - - t r r' , ( D ('f-)' S c ) ( { ) * D o , ( € ) S ( { ) h , 6 ( ? h , . 2x 6qa 2r0h-

G.62)

An equivalent form can be given in terms of the functions

s1(r): S(r)+1sor1*;, z i D+(*): D(*)+-DQ\(tc). 2

Thus,

(1.63)

I

Kn(')({,,?): r,(S+(€)yos+(l)D+(t*'i) and

-S-(€)roS_('i) D_(t*d)^y,

,

(1.64)

ld 1 - D -('r)S_ ("r) K,Q) (t d : - t ud(€) )y, ^ :--z^ .r,( D +(r) S +(q) zK on^ I

1A 1 - D (i)s -(t))r,6('r)"y".(1.6s) 7zK ;tx1,(D+(OS+({) Ott\ t The first term of (1.58) is the current induced by the electromagnetic field that accompaniesa given current distribution, as discussedin II. It is the second part of (1.58), expressingthe additional efiects involved ivhen the current is associatedwith the matter field, rather than an external sysiem, that merits our attention. In we shall substitute Fourier integral representationsfor the various func_ order to evaluate Ku(r'-qr-r"), tions involved, (II (A.10), (A.31)), -lrl S(*):-

S ( r ) ( r ): -

| A h ) e , k ' ( i - y k -K ) - , h'l*' \'211"1 lr \ z Tr "

|

( dk ) e t k(, h h * x ) d ( 4 , { x : ) ,

lr1 l @p)4o' -, (2n)al k2

D("):

p,,,1,;:-t \ZT TJ

| {oo)r,r,u{u\,

(1.66)

in which the principal partof 7/(k2!12) andT/k2 is understood. We have employed the simplified notation oD to denote arbr, the scalar product of two four-vectors. The functions (1.63) have the following Fourier integral representations,

lr/1 \ t*,*, :,r,ro..| Gk)t' k'rfi k - x) - - - +r i 6(k:-f *) \ ), D*(-):

r (WJ

7

/l \ @h)e'k,\0,+ria(h,) ).

(1.67)

Theseexpressions can be written morecompactlyby observingthat

s*:rr,(*-'#) 1 : P-+*;.A(t\

t

(1.68)

176 797 whence,

III

ELECTRODYNAMICS.

QUANTUM

lrl *):_ , S*("v)=_ | 1dk1e,n'{i7k' PzlxzTit e"101 l"l D+(x):,^ * |(dk)etk'-, (2rf 't bz+ie

(1.6e)

in which the Iimit e+*0 is understood. The form obtained for Kro) from (1'61)' is

lr -'\ (k+ ('ri (k+k') @' ei k" \ ""' K lt ) (r,t- )t,rc- x,") : | @D {an) @n") \zr)"r p 6(E,r*xr) x t,Qt k' - x)t,(4 n" rh,l *U *+

6(kr) l r1 -1 1r- * -1 J' _1 ur* 1a 1u,4 Urp,,z1_,?) ,

It is convenient to replace ku' and ku" by f u':kulku' ,

, r^, \ r'' wi

(1 71)

?r":kutku",

is to be multithe Fourier integral. Since KntD(*'-r,x-x") which enter dir"ciiy i' the coordinate dependenceof p/ and'p," occur for which oi values .'.r.ft ,o / ptied by /(a,), rL@,,) andt"r"g.ii.J*iri'r.-rp".l ""iV ""a7, 0'72) p'2+ K2:p"2+ K2:0. As a result of this transformation, t{,(t}(r,-x,x-x'):*

\zr )""

f uu)lol)tol"leip'|(!'-!teip"(F'")"'/,(i'v(p'-h)-x)t,Qt(p"-h)-x)t' 6(fr'?) 6(hz-2k p") 5(h,-2kp,) l-

1 * u*,)a*\]'(1i3)

o*

"l*ffi,+ it as The last factor in (1.73) can be simplified by writing 11

I

- - 2hp') - 6(p211 (k'?

,, a' - t" )t w'$ that andobserving

I

I - 2hp'')- 6(k'?)) GG'z I'

(1.74)

t-61nr-znp1-6(e)):r f'ouu,1r":rupu1, o

(1.7s)

- 2bp''u)f' - 2kp'u)- 6'(h2 dul6'(k2

(1.76)

2hp whence (1.74) becomes

-

-n [,'

#

This, in turn, can be representedmore compactly as lnLFl -| da I udu6,,(k2-k(p'!p"l(p'-

Therefore,' 6 not(tc,- x,x - x", :

I ur*

f'

2J-r

Jo

ft

f

J _!, J, "o " J

(r.77)

P")a)u).

. . . t p'(''- r\ p'' (r- r"', e; @h)(d p')(dp" ) e

(h2- h (?,+ p" + (p' - p" ) t) u). xy,Qt @'_ k) - d"y,Ql (p,, k) x)7,6,, factors in (1'70) and (1'73) are replaced by If the expression (1.64) is employed for K"{r), the bracketed 11111111 -Irn,i""h'o+c-i*

kt'2+*-ia

-:-lffiP-ie

;

kz-2k/'-k

,-- .h2-2kq"-'i4

-' H-i'

( 1.78)

(1.7e)

t77 JULIAN

SCHWINGER

However,

1

1 1_ 1 t lr( _'\ h2-2kp' ie k2-2h!"-ie k,- ie 2k(f'- y',)LZ'Bp,\n,-znp, -ie h,- iel

-' (;,,_r-fr)1, ,'',

and,on extracling theimaginary partdividedb-vr, weagainencounter (1.74). The secondpart oI Ku, (1.62),canalsobe readilyexpreisedin Fourierintegralform I

,,. K,e)(x'-t,x-*"r:

f

*rJ

..lr .a /6((k-p')2+K,) @k)(dp,)(dp,,)e,p'(r'-!reip,,(hr,'rl;l^,;7,fuv,_rr__rr,(:l'-__1__

- p"r+ct * :# *),,+-,,)r (/,- ;)-,t^,, t'fu .@ 1a(e . ;# *)1. t,r,, To evaluate the derivatives with respect to pr, and 1r,,, we observe that

a px- (w(! - k)* K)(i1,(p- k)- K)f ((p- k),+,\ : Q, oh

( 1 . 8 2)

where/(r) is d(*) or 1/o. on difierentiating and multiplying to the left by iry@-k)-x.

we obtain

a -zkp) JG'z ,, ?x* (ir Q- h)- *)J((?- h)'I n"): (i7Q - k)- x)t7pQ7@- p1- *l-::-. oh

(1.83)

k2-2hp

Consequently,

nfu,
d(r)

(1's4) (1.8s)

Furthermore, 6'(k,-2bp\. 6(ft) a /6(k,_2kp) 6(,r')r - fl h2 Jo'a'a"&'-znput' Qrpy: ,g41\-nt* ): according to (1.75).Therefore, (1.81)becomes K,Q)(x'- x,.u- x", :

lf'r frrl' J,

udil

J

(1'86)

@k)(d p')(dp',)eie'e - i eie"G-r' \

rl

xfa"{n,- zn.p' ul;,W
A'- k)- r)i7p"(i7e" - k)- *h"al " (k"- zia/ t 14] (1.s7 ) fi "
ku-kr* (?r'* ?r"+ (pu'- p,")r)! 2

(1.88)

6" (k2+\2u2),

(1.8e)

now bringsthe delta-functionof (1.78)into the form

178 QUANTUM

799 where

ELECTRODYNAMICS'

III

^,:r(t*ff,t-"r),

in virtue of the relations

(1.e0)

- p',) :o o,(!,p,,)(p, ef)' . (+)'* *: *

(1.e1)

Asaconsequenceoithistransformation,thefactorinvolvingtheDiracmatricesin(1.78)becomes

p'+p" / t * o,-rj!---o' u_f ^r" uo1_.\.,,(r( r,(;r(l'--, 2 2 / / \ \?-

(1.e2)

-"/ z^0",o\--)t"-nn,.

(1.89), in connection with the I inthe symmetry of.the delta-function In writing this result we have exploited pr,,irit" following property of the Dirac The uy L6x,k]. *pl".i"gi^t, tesration, and discarded,..-.i'i""*r-i" (1.e3) -itri."t has also been used, vanish on integration, and by omitting the terms linear in o, which will The factor (1.92) can be further simpiified rearranging the remaining terms to obtan 1-rl u 1 - + -7' (?,' - p," ) I 2(!' - P" )\,\t 4x27ue - u - ! u,) t rk21- 2x(u r/i)o," " )

t+A(x(rlutr* p,")((i1p'lx)- Qt!" _-x))-2(r-*llt;"' !i.(r- uzt'?)(p,'-

i Pr"a;\!
t

1-14

- (it p'I x)t,(itP"+ -)+ ( ^(1+ u)ru* i pu'I i:

(f '' + P*") l4 P"+ *) (1e4) 1' )

-lJlflrx')*xop"111i"So"'fi{x'-x^'r-tc")'whichannihilates of the Dirac Now,aright-handlactori'vll"*risequivalent.to n annihilaies'y'(*')' As a consequence f.fr- Uunaiiito, ;tii Si*if"r'iy, parts. by integration on ,(r//) " equation,therefore, K!r)(x'-xfi-t"):+

J (2r)"J-rf'oof'uou[{oul{oo)lot"leip'(r'-r)e;p't(Fr")6"(h2+\'?uz) Jo

l-d ..- / r . P")\'lt-'+-* x(u-u')o,,(pi Pi')+(f'Xl2x\,(7-u-iuz)-Ltnkzl

\l ) l'

(1'9s)

E'*bu* p''u' and introduced in the two terms of (1'87), namelv Transformations analogous to (1.g8) can be Dittt matrices iht involving trtt-i"tt"i' a"(p'+niu')l J;i; u it* delta rt".il"", k,,+k,lput'u.Both "'o^" simplilY accordingto 1

x)' - x)7,+_ 2x2(r-u-+u')++k'-(2d(r-u-l\u')*ik"1-1;'vp*"k) t t(\"t\( K \,(trtp-, b- *)i'7pQ1Q-

(1'e6)

2x

wherefis..'orP"forthetwotermsof(1'87)'InconsequenceoftheDiracequation'therefore' ' (' K p Q ) ( * t - * " t - n " ) = - @ ' J o " ' " J *u

(a'-d ep"(t't') 5''l(tu2|a2u2) f @n)(a ' - ' p)(ap")eip' xrzx2,r(r-u-!42)-!1rk).

(r.e?) first two

TocombineK,o)andKr(,)'itissufficienttoperformanintegrationbypartswithrespecttooforthe terms of (1.95), as indicated bY A fr --. rr du,-^ 6" (h2*\2u2)' : 26" (hz+ *'u2)| I dr6,'(h2+x2u2) Jr do J-r

(1.e8)

179 SCHWINGER

JULIAN The integated terms precisely cancel-trr(2). If (1.95), the fr integral t}rus encounteredis

8OO

lle r differentiation is explicitly performed for the second term of

(dk)h,!r,{u,*^,u,1 ",-. J'tai)wu"'{n"+"*,,\:;l . : -zI onlyo2+\z7rz1.

(1.ee)

Eence, ... K,(r'-x,r-*"):_

rr r 1 f' f t | t u l u d . u l { a \"{ a "f l' {, a' O " ) e i t ( , ' - z ) e i p " ( r " ' , 1 ( p , - ,p-1' ") \, ( t (2r)ttJ-' Jo J L"

llaz\ u!-'ozl 4 /

! x(u - uz)o,"(p,, - p,,,) - 2x),(t - * - I O,!]t,

(h,+ ),, ur1. ( 1.I 00)

.o

The integration with respect to k may now be effected. According to the integral representation,

a{n + x,4 : !

wei. t*,+x"u\, [ _'-d

f

(1.101)

7r'

:e;-*' "'f pn1 J {aila"{n'+x'^ ; I _:4,";.x" rir' :

--=

|

w

2J *

d@-eiu^'u'

l.l (1.102)

A'U'

flowever, it should be noticed that we are then required to evaluate integrals with respect to a of the form

["'

:1 u*'au {ou)6''(k2+^2u\ u*'ou, [ f o'

(1.103)

in which n may be 0, I ot 2. For i:0, ttle integral is Iogarithmically divergent. fn order to ascertain the significanceof this divergenCe,we shall interchinge the operations used in obtaining _ (1.103), thus producing a more easily interpreted divergent fr integral. Fo. zJ0, (1.10s) reads l7

r'3

7n

k) du- 6'(k2+\' u2): 1aag 6'* l') - a'1b';1' J @ J, Dr', J

(1.104)

One may expressthis invariant integral, in three-dimensional notation. as

in which the delta-functions de_scribethe.energy-momentum relations of a Iight quantum, and of a particle wit} mass hi,,/c.On performing the &6integration, (i.105) becomes

:.J ( ;/ r,r.r[*-,-,_,,] {i, ; *4* )., ],

(1.106)

in which form it is evident that the divergence is.associated with zero frequency light quanta-an,,infra-red catastrophe." As we-shall later demonstrate, this divergence is entirely spurious, andis removed on properly including the efrectsof 61{tt(*), the first-order correction to the current operator. T'he divergent integrai(t.io6j can

t80 QUANTUM

801

III.

ELECTRODYNAMICS.

as be expressedin terms of an invariant minimum light quantum wave number, Ami'' I

rl

\

(1.107)

t/

^;('"tru*+

With the fr and ru integrations thus performed, K u(r' x,*- r" ) becomes

- r"): - x.x ^ rdp')(dp")*ol,e!! g'-,"r] .-n[;o"- r, Q-!::)] K,(x, #, I,' I K

f(p,_p,,), t+urt

(p,-p,,)r..

-. /. tos(t+-(t1++

"l?=(roc2-+ +

(?'-p"),

f)" . \ \ - -ic-, ))

ltl

(1'108) *(Pi - P,"t Irn*-,,1y u,yt _ r,1' ;*o

in which we have evaluated the third term of (1.100) by writing ((P'- P")'z/4l'z)(r-u'z)

t

\

p.art of (1.109), while reversing the inand performing the o difierentiation for the term obtaiileb f.g\ jl:,fr.j, be observedthat the integrand tesration by parts for the term producedby t he secondpart of (\rt\9;. It will now is then useful to introduJe th'elnewvariables otit.tOA) involves only !x,-t^,,.It I I !\'+Px" (1.110) py:px,,_px,, ,^:;, way, we obtain since the P integration can be immediately performed, yielding 6(*' r"), In this 11f( K r(i - x,r- r") : - 4,a{r' grz

- r"). ['l log (Fo(*-'v')*lqr('r_2h^i, x2_ L

r')) -t

r' )* lG (x -*')l+

* \F o(r- x')I F {r-

i ;a(*'

d - * ") o u-F o(t - x'),

(1'111)

where ?r

\

eLrr

r'

p"t..r:Jo ooo'^ @il rr,yJ t+tpy+*lrr_',t>

:rcc[,affi^(*,,.) andr,tTtog(l+(pr/4Kr)(l_

c("):

Finally, then,

J,da(t+d)_ J

(r.rr2)

a,))

@p),,,"

1+(p,/4*,)(r_t\

:,e' f"'offiI,', \l#(i-*) -^(#' )]

-'][0,,0,,,rr(x,)K,(x,x,x-*'\vG'\,::t*;,i:,lrFoG_

(1.113)

r')+F{r-x')li,@')da'

+|)e,jUns@-*')!F{r-x')*lG(r-x')fi,(*')d','l1ocu!,fno@-x')m,,(*')d'',

(1'114)

t8l SCHWINGER

JULIAN

802

:',r'hich e

(x))1 m,"(r) : - (1,(*)o,,,1,

- ll0 {r) {*)- {,'(x)o,;1,'(r)f. ",rt

(r.l rJl

LK

Expressedin the samenotation, the first term of (1.58)is (seeII (2.44)), i 7 -|do'e(x,x'r[7"(rr,j,(r';]qdA,(t:)2hc2J

-

al r --I'|[1,,(x-r,)-]Fz(x_x')fj,(x,)da,, J 4r x2'

:::m which we have omilted the charge renormaliza-,:lf, term, with the understanding that the value of e : io be correspondingly altered. A rederivation of thrs :::ult, employing methods akin to those presented in --:s paper, is given in the Appendix. Evidently the -,:,r contributions to the one particle current operator, :-: given in (1.114),are of the same generalnature as ::,e previously consideredefiect, (1.116),with the ex::piion of the last term in (1.114).This is an addition :l the current vector of the form

oI mu, over the vicinity of that point. If all quantities are slowly varying, relative to hf mc and hf mc2as units of length and time, an expansion in ascending powers of !2 can be constructed, as in II (2.47). For this purpose, it is sufficient to expand the denominator in the 6 r s t l o r m o f ( 1 . 1 1 2 ) ,t h u s o b t a i n i n g 1

r,,(.$:-51*1 tn-r l

(1.117)

c@/0x)"6mu,Q),

'rhere

f 6nu.(x\:af )r I l:nQ-tt)nr,(x')da'.

(1.11S)

-\ current vector of this type can be interpreted as a ::lole current, derived frorn an antisymmetricaldipole '...::or 6lttp which combines electric and magnetic dir, -: moment densities.The tensor 2", is that char,:-:iistic of the Dirac theory, in which intrinsic dipole :--omentsare related to the antisymmelrical spin tensor :-,, the factor of proportionality being

( 1 . 11 6 )

I{ence,

11 fl'6("r)1 "' (2nll)Qn*3) 2x, af

I

0r",(.'): ^-l nu,(x)!-^f2m",(")*. ZrL 6xz

..

1

l, I

(1.120)

(t.t2t)

and, under conditions that permit the neglect of all but the first term in this series, an electron will act as though it possessedan additional spin magnetic moment. 6p: (a/2r)pn.

0.122)

po:e/2x:eh/2mc, (1.119) The comparison of this prediction with experiment will be discussedin the sequel to this paper. ::e Bohr magneton.According to (1.118),the correcThe final result for the second-order correction to ::on to the dipole tensor at a point involves an average the one particle current operator is aKl tj,t2'(x))r.s:-log+T

'J LR\nin

[l"o(..-r' t I F'(x- r')]7u(r')do'

;

K"

q1 *;

;!' 4nx'

f a []1'-' (.r- x') f i F,(x - x') | lG (x - *)l -j,@' ) d.a'-l c- 6mr,(x). J| 0x"

Lnder conditionsoI slow variation ((l/n)l]rjr,

(,)(r)),,o: (67,

(1.123)

mr,((.jrmu), this reducesto

-1(.r*4

)

lp i r{o* L"!-,, r"l*),

(1.124)

-:r virtue of (1.120), and the analogousexpansionof G(r),

1 G(r): --;15i*1*.

.)("

conservation requirements, and the formal property that the operator of total charge commutes with ail at the January,1948meetingof ( 1 . 1 2 s ) 3This resultwasannounced

I! will be noted that the total charge computed from i.6jl'r@))to is zero, in agreement with evident charge

the American PhysicalSociely.The lormula is misprintedin a publisbednote,I. Sehwinger,Phys.Rev. 73, 416 (1948).Tbe misprint has unfortunately been copied by L. Rosenfeldin his book, Nuclear Forces(Intericience Pdblishe;s, Inc., New York, 1949), p. 438.

182 ELECTRODYNAMICS.

QUANTUM

803

III.

manner of one-particle operators. The apparent contradictioa be- completely stated without regard for the tween these siatements and tle existenceof the charge exhibiting them by an external field. We therefore turn renormalization term is discussed in the Appendix, to a discussion of the behavior of a single particle in where it is shown that a compensating charge is created an external field, as modified by the vacuum fluctuaat infinity, tions of the electromagnetic field. Our result, (1.123),is marred only by the appearance z. RADTATIVEcoRREcTIoNS To of the logarithmic divergence associated with zero ELECTRON SCATTERING frequency quanta. It should be remarked, however, thai (6r(')(s))Le is not a.complete descriptionof the we shall now be concernedwith the interaction of radiative corrections under discussio",.t: j: three systems-the matter field, the electromagnetic -"^tl".ti t l s n e c e s s a r y l h e c u r r e n t , t o ; m e a s u r et h e c o r r e c t i o n

nelg ;;;;;" anexternar irrt.;+iF;;;;;;; ff'k:*.-ffi1;:T:i::*'3:li:T;J}r".'#:l#iJ sion of quanta, as describedby 07"t"(r), among the : current distwo situations in which the reactiou on the tribution may have a negligible effect' A description of this state of afiairs, in the interaction representation, is given by

aiiltqtt"ty effects of which is o .o-p",t.liit'/i"J vergence.It will be apparent that, as a consequenceof irr"- ini.u-..a catastiophe,', the second-ordei corrections to particle electromagnetic properties cannot be

:l-:( i*(x) A,t,lvr"1 * r,(r)) ]

*H -:t

l#

",

- r)(j,(r) * t (r))ao"]v [o]: o, @'

(2.1a)

(2.1b)

system, respectively. where jr(x) and Jr(r) are the current vectors associatedwith the matter field and the external by Ar(*)' An equally valid characterized field, as electromagnetic to the are coupled distributions notfr.ut.*t matters is in terms of an external electromagnetic field acting on the matter field current *"y-"irr"ti"g distribution:

*l!:

6o(r)

lT rvhere

- i,@) (r)) (A,(x) r A,("' |L 1 ].r,tot, c

(2.2a)

: o' I.T "'l'r);(*)do']vrol

(2.2b)

-:

I a2Atk)(tc):-*Jy@'

aA'k)(r)

:O.

(2.s)

^

(2.1) by a canonical The equivalenceof the two descriptions is establishedby showing that (2.2) is obtained from t r a n s f o r m ai lo n , n a m e l Y , (2.4) .v-lol+e-;t@v-7"1, with J[o] determinedby 1 6lfol hc : :: --J,(x)A,(r). c 6o(r)

(2.s)

The {unctional -/[o] is explicitly exhibited as

r,'':-:[ ltc.r

J,(x')A,(x')du' ,

(2.6)

_a

electromagnetic field in which the choice of lower lirnit corresponrlsto selecting the retarded potentials for the state vector is generated by the given current clistribution. The equation of motion satisfied by the new

;hrY4+ inrrur,"'t "t*1"1:

l_.!r,

irr'r]vtoJ ror* r,neil t'tA pQ()e-

(2.7)

183 JULI.AN

Xow

804

SCHWINGER

ei r to1A y($)r i r t" : A,(*) * iU lol, A,(r)f - iQ lol, t-/trl,.4 ts(r) ll + . . .

lr, : Ar@)-| D(x-r')J,(r')da' CJ _q

- A,(r)IA,kt(r),

(2.8)

inwhich the seriesends after two terms since the components ol Ju(*) are mutually commutative, in view of the prescribednature of this current distribution. It is easily seen that

1,t"t(*) : obeys (2.3). Indeed,

lr" rJ ^D(x-

(2.e)

r')J,(*')du'

1 6 r " | D k - r ' \ 'J,(r,)do, . ll2ANGr(r):__ | e Ox,J_. |xi :__

1n |Db-r') _1,1*'1 | ao"'______ cJ o

6*i

1 : __J,(tc), c and

(2.10)

AA"Gr(x.) | r" d : da,-(D(x-r,)J,(x,)) -d*, | cJ+ dtt

:0. Furthermore,

t,.irtdt _:hc_:iheeirtal 6o(c)

(2.11)

6Jfol i.hcr dJlclr -+_l - Jl-al, l*... - -' 6o(r) 2 L 6c(tc) J

: --J,(tc)Ao@) c2c

(2.12)

J o(r)A r(.,(r),

and the transformed equation of motion therefore reads

&:#=l-i,@)(A*(x)rA,k)1*11-f,1161.a,t,k1]vt"1,(2.r3) describing the self-action of the given current which is equivalent to (2.2a), since the term -(l/2c)J*(r)Ar<'t(r), distribution, has no dynamical consequencesand can be omitted. In a similar way, the supplementary condition (2.1b) is transformed into

L,:

wherein er r't! llowever.'

-

- (i,(r) r,u .', r,-! @, : o, [.o x) * r u@Dao"]v[o] 1f,)

M ) ;.tr",-a A'(*' ) r *'; I

6*,,

or,,

tnn|D(x'-r")

1n'0

tt,1r'")d^" : -

;J ^

ut

A'(*'

A'(*'

) ] -0 ) -;J---l p' - ol.a L 6*u,l: t*,

J**(D

aD (*' - x\,

ur, ln

(*'- r") !,(x"))dot': -

J

''{*"10'"'

- *)J,(tc)ita,, "D@,'

(2.14) (2.1s) (2.16)

which verifies(2.2b). One can bitig Q.2) into a form which enables the results of the previous section to be utilized. The mass renormalization transf ormation f 6wqf , (2.17) vlopWla)vlo), ihc:-:tcto(x)Wlof, 6ck)

184 805

ELECTRODYNAMICS.

QUANTUM

replaces(2.2a)with

III

6v[al - lK ('c)+ Jc(')(r) lv[d], ihc-

(2.18)

6o(*)

where (see(1.3))

(2.1e)

cc(r) : Jc(r) - Jcr.o(r), 1 . . K k ) ( x ): - - j u @ A ) , \ " )( r ) ,

(2.20)

ic

and the Dirac equation for ry'(*) now involves the experimental mass. The further transformation

v[o]: U[o]o[o],

(2.2r)

6Ufa1 ihc -: r(x)U["], Ul- af:1,

(2.22)

where 6o(x)

of an external6eld, is the analogoI (1.8),savethat tD[c] variesin the presence DoIol :U-llqflcG)(r)U[o]o[o] ihc 6o(r) 1 -- --jF@),1p@(n)+lqf,

(2.23)

in responseto the coupling with the current operator ju(r). The latter contains the modifications produced by the vacuum electromagnetic field. The supplementary condition (2.2b) appears as

: o, u _.t- -,;jlu'raT# bl f "nr*' 1r;a"*]o["1 in consequence of these transformationsr

(2.24)

However,

u7o "r) ,-,raff ud }# : [_^,,rlr(r,1o,,1aE) :

n "r, rr'tluu*,*,'' .n.t*" tli,r*"l ut""r:1,I' 0,"Lo6'!' * -o, :

x" JiuG"))

lr

- x\i,(x)do,, ,J,D(x'

(2.2s)

conditionassociated with (2.23)is simply so that the supplementary AA"G') ------o[ol:

(2.26)

o

dT,

As the first application of (2.23), we shall consider the scattering of an electron produced by its interaction with an external field, in which the latter is regarded as a small perturbation.a We shall restrict the external potential to be that of a time independent field, which

will ever.rtuallybe specializedto the Coulomb field of a stationary uucleus. A solution ol (2.23) can be constructedin the form

r Radiative corrections to scattering have been discussedby many authors. That a finite correction is obtaioed after a renormalization of charge and mass was indepeqdently observed hy Z. Koba anJ S. Tomonega,Prog.Theor. Phys. 3, 2o0 (1048); ; n d J . S c h w i n g e rP. h y s . H . W . L e w i s ,P h y s .R e v . 7 J ,1 7 3( 1 q 4 8 ) a Rev. 73,416 (1948).Seealso R. P. Feynman,Phys.Rev. 74, 1430 (1948).

6R[ol " -: H(r)Rf of, ihc 6o(x)

a[o]:R[o]or,

Q.27)

where

(2.28)

^-j R[o]+l,

ot-

@.

(2.2e)

r85 JULIAN

SCHW]NGER

806

The state vector (F1characterizes the initial state of the system, composedof one electron with definite energy and momentum, and no light quanta. Thetotal probability, per unit time, that a scattering p.o."r. o..ur., .in be obtained by evaluating the time rate of decreaseof the probability that the system remain-in the initiat state,

, : -, I ^hl (o,,o[d]) l,: -, 1 r,r^,t(o,,R[c]o, t,.

(2.30)

The integration is extended over the surface'l:const., with dl the three-dimensionalvolume element. Now 6 (o,,R["]o,) lr: (o1,R-,[o]o,) (o1,.F/(s)R[d]o,)_ (o1,R[r]o1) (or,R-,[c]I/(*)o). ihc-l 6o(*)

(2.31)

In view of the treatment ol H(r) as a small perturbation, it is sufficient to write

Rt,l:1-: f' ,t*,rz,,,p-';"1:111f' u6,7a,,. hc.t_* hcJ-_

(2.s2)

It will also be useful to introduce

H' @): n 67- 1or,n(e)or),

(2.s3)

which possesses a vanishing diagonal matrlr element for the initial state, and obtain

Rr'r:*p[-

; ["_r,wr.,ttrv rl(, ; [' :' G')d,)

(2.s4)

The phase lactor evidently has no effect in (2.31), and can be omitted. The latter is also unafiected if I/(r) replaced by H' (r), Hence to the accuracy of first-order perturbation theory, we have

and

is

u,r"r tot r: J (tln,@) +!*u,a,v,, I' n,r{ a,, fr ,,.,,"r, Ir),

(2.3s)

*: L I a,a,, . +f-"' (*,)a.,,u, (rl*o,I-",, o,ro*0, arlr) ",

(2.s6)

We may now remark that a diagonal matrix element for a state of definite energy must be invariant with respect to time displacements,whence

and

"' : I o,o,'(.tltat r,aw;lt), I a,a,'(rl[ n't'>an'n'
(2.37)

(2.s8)

This result is perfectly equivalent to the more conventional perturbation formula in which the rate of transition from the initial state is expressedas a sum of transition rates to all possiblefinal states of equal energy. The dnergy conservation law is here expressedby the time integration, and the summation over all states oiher than ttre original is provided for by the removal from I/(r) of the diagonal matrlx element. Our basic formula for calculatins the transition rate for scattering of a particle by a time independent potential is thus

* : )- [ a,a,,oe(")(rx,(")(il ( tli,a> aw . f _'_i, ",lr)

(2.3e)

We have not indicated that the diagonal matrix element is to be subtracted from ju(r), since it is suif,cient to remove, in the finai result, those transitions in which no change of state occurs. We have shown in the first section that, to the secondorder in e,

(r) + 6, ('?) ('), i,@): j,(r) I 6j,(1)

(2.4o)

t86 807

ELECTRODYNAMICS.

QUANTUM

III.

where

uj,' (.):

j,(r')\A,(r')da' r')lju@), h I :@" - r' t (x')l 0@)t (x'- tc)tu,r@)frA (r')da', :f " "s [ roor S @ ) ",!

(2.41)

and (6j,r2t(26)) ,o:;s6

l!{r')r,{*-

[*

(2.42)

. r'),!'(x')l1d'o'

qK\ul

-2"-^E'(4 Fo(.r) * *Fz(r) * l G(x)) - i-o,"- - F s(x) ( fu(x) : --1u log^E'(Fo(r) * F' x) ) * . 4t r dx' 4r x2 2h^i^ x2 41

(2.43)

It is only the indicated portion of 67*('?)(r),referring to one particle and no light quanta,-that need be retained to compute ihe second-ordei correction to the scattering cross section for an external field, since only this part of 6jr('?'(r)is coherenLwith lu(*). The total rate of transition from the initial state can now be written as

(2.44)

1!:W\tWr,

where

7e0:fi[o,ororu,
wpL

I r /l rlr 1l6r(r)(r)| 6j,()(*')dr0'11 I dodo'A,<"t1r1A"(")(r')( | .t__ \l

hrcJ

(2'46)

l/

accounts for scattering that is accompaniedby single quantum emission. To indicate the manner in which the perturbation formulas are to be used, we consider the evaluation of

Ii rav' al I r)dx n'' [ --.rt

(2.47)

This can be written as

-acl_-_(11i,(n)t,96)0k)t,,!,(x'|)11)dr:.'|,

(2.48)

induces no change in state. in which it is understood that one omits the processesin which {(dt@),orlr@')rl,(x') Now ry'(r') can either annul the original electron, in which caseQ@')'!,(x') causesan electron transition to some nrat stute, or ry'(r') generatesa positron, in which event ry'(x'){(r') inducesthe creation of a pair' However, the latter processis'incompaiible with the energy conservation that is enforced by the time integration, and can therefo_re Le o-itt"d. Henie, it only occurs thit,!(r') annihilates the original electron, whence ry'(r')O1is a multiple of the vacuum state vector. The samecomment appliesto {t(y)@1:y4!t(r)Or. Therefore,only the vacuum expectation value of the operator t!t(*){(*') is required i" (2.+S). Furthermore, since only one state of the matter field is ini tially excited, as described by the wave function ileipt,we arrive at the result

J*

g11,1*11,t* )lt)d-{:#f

x)^ttuei(rs)'G'-'|), (dq)a(qoPo)6(q'z!x2)E7,Q1q-

() LO\

on employing the relation 7

r oee;ut'-'"t 1r,yJ,,ro@rla(g'?*x:)(1"y9-r) Before further simplifying this expression,we shall consider the analogousevaluations oI (f"(r)16(r'1)o: -rS"pi')(.r-r'):

-

r* t)dro' i,(r)(6i | (11 "')(*'))',ol

J_6

(2.s0)

(2.5la)

187 808

SCHWINGER

JULIAN

,/ | (r I ('a:;r,r(r)),oj,Qc,);l)dxo,

(2.s1b)

which describe the radiationless corrections to the scattering process.Now.(2.51a) can be written r'f - e2c2 dxo', do" (l p I @)1al,@)fi(*")t,(*' | I J 4 J

- r")!(*")

(2.s2)

l l),

whicb, according to the arguments presented in connection with (2.47),becomes f'f

- dc - *"1ueip(."- x, 170r,7*' I d*s' I do"aY,(t @),{,(*i' J4U' ezcz rt

r

:fiJ-.0^'J

?

^"

(2'53) Joorotodu{o'*x'z)'ilrt'(i'tq-x)r'(tc'-*")uedrcrc'-')'

On introducing the Fourier transform of I,(r):

r lp-c):

I

,,(rc)xr,('c)(to,

(2.s4)

we obtain

(2.ss) I"rrrr,urrr,r')(*')),.01L)d.*o':ffifaau;,<*-po)6(dtr,),r.t,Q\q-x)r,(p-q)u The result of combining (2.49) and (2.55) with the analogousevaluation of (2.51b) is expressedby

*r:

l dr r r d'qo@il6(qo-?dt{d+*l e;tt-t tA,<"t({)ito' ,-u;.1 1.;ta-at',7,<"t(r)ihtJ \zTr n'cr

'::# onperforming theintegration withrespect tosoand,r,, r"

directions of the vector q, other thah the incident direction:

"#;:;"j"

t'")T;

t:fi

J:?

^:*#l*lvlf.-,tn-nt',s,u,61aofrunor.,'tr"kt(/)da,d(tytr,@-.il)Qtc-*)@"*r,(?-d)u. (2.s7) This must be interpreted as the rate of transition from the initial state, expressedas tle probability per unit time for'a deflection into an arbitrary elemgpt of solid angle. A further simplifiiation can be introduc"i Ly averagiog (2'57) with respect to the.two spin stabbs in which thi incident electrin may ocflrr. For this putpos"ir" t.qirirJ the average of.u,ilp for the two polarizition states associated with a given Lnergy and momentum. It can bl inferred from the anticommutator

_I 19"b) fi e@')l: -.5a@- r' ) : -

1r O* J

ot* "'t, {aOWt + x)e(p)(i7p- x).uB;

(2.s8)

which exhibits, with equal weight, the contributions of all states of a particle, tJut

(2.5e)

\u"ae\:a6"-'*'*'

for a state with wave number four-vector &. The constant u4is conveniently evaluated for our purpose in terms of the expectation value of the particle flux vector in the iiritial state,

5<;"t: (1ldc(r/(*) rl@)),lt) :icil.(u,

(2.@)

Thus, Sunc): i c.(pdvc.\rti cA T rftv --4cAg,

p - x)

(2.6r)

188 809

III,

ELECTRODYNAMICS,

QUANTUM

so that

<""ail:-i

1 l S ( o *')' l (it\-x)"e. lpl

(2.62)

This leads to the iollowing expressionfor the total rate of transition from the initial state, Ze 2 |f f '-dz'] ITrl(it?-x)(t+*r-n(q-f))(itq--^)(7n+fn(2-q))1, f ,Jol f e,rc-ot J 4rr I lJ

I e2 no:,-lq5ru',")l gT2hz(z

(2.63)

in which we have also specializedto the Coulomb potential of a stationary nucleus. We may now infer that the difierential cross sectiorfor radiationless scattering through the angle r) into a unit solid angle is

y9:tl2]'+r,l{,rt-*ter-trn(q-p'))(itq-K)(14+r4t-q))f.

{2.6+)

L(p- q)'J

dA

The Fourier transform of Ir is conveniently written in the form

r o(p-q): -

t I (x)+-1. ip- q)Fo(\) J, 4,?414x'?,4 af

(2.6s)

where

(2.66)

,r^l:r**-ra(F,(r)+!',(r))++(F0(r)+3P,(r)+G(r)). *

l p - q l -l p l . srn-, tr: -: 2xr2 F .' ( I ) :

da log((11 tr'?)]rr) rt I (1*xz1l1 Jo 1 Ftr?(l-a,) fl

Fl(r):J

F,(r):

(2.67)

fdo

/

1\

(2.68a) I

(2.68b)

, 1 + x , ( 1 _ d , )( :1 + - / r o ( ^ ) 1

1\ zldo r' / 1l-r,/r'(\)--. Jo1a-q1_,,;: (

(2.68c)

matThe more complicated transform, G(I) is not required in the following development. The trace of the Dirac ricescontainedin (2.64)is easily computed: Za 2o r \ 1"Trl(itp_ x\(tq-llqh_ x ' l ( \ ) , f - - x ' ? X ' ? F o ( I ) (' z . O q ) il'(i"yg-n)(rnrrq(p_ qD):2lpo'z-^'r')(1-whence

(270 ry:(#.-,;)'(,-u,.*,;)1,-':"^^:-;*_o"no,(^)]

is the speedof the particle relative to c. in which g:lpl/p, To evaluate ihe rate at which transitions occur accompaniedby radiation, we consider tl

n*

lr

e4r

/l-

(rlay,u,1"ll' 51,u'1*'1a,j):-: I da"d,"'(tl&(*)r"Sr*-*")t&(x")rli" )rrS(r"-x)7u'p(x)l' \l r-6 hzr \l l/ xl

t-

l \

(2'71) dxo'10&)t,51*'-x"'11.,1,1x"')*tr,@")t,Sfx"'-x')t,'!(x')lrA^(x")A"(x"')ll) l/

value is Since the state vector iDris characterizedby an absenceof quanta, only the following vacuum expectation required for the electromagnetic field, hc r '' (2.72) (dk)a&')"0'n'-"" (Aa(x")A'(x"'))o: ihcS^'DH(n't- r"t): -5^" | (2r)3 J xo>o

189 JULIAN

SCHWINGER

8IO

The matter field operators are treated as before, with the result

tl

.

eahc r

l\

r'

r*

( t l a 7 " ( r ) ( r ) | l j u t t t ( x ' ) d x o ' l lt : - - _ @dAG,+x2t(dk)6(k)l dxo,ei(nr*ttt-,t | J-6 \ | h2(2r1sJoolnrn " " J_^ | /

x u(I,5 (q+ k)h*rrS(p - e)7,)(it q- O e 3 (p- hh xl nS (q* k)t u. (2.73) ") itklh)-x r S(q+r):J r-'ta'rtr$(x)du: (2.74) * !- k't- x st!- Pt : -it( 2pk areFouriertransformsof 8(r). Tle integrationwith respectto ca'imposesthe energyconsrvationlaw

Q'1s)

(2.76) Po: qo:-,ho, *'hich is evidently that of a light quantum emissionprocess.The integration with respect to qeand the magnitude -be performed, leaving one with an expressionfor ror in the form of an integial extended iver all !i u 9n now directions of the scattered electron, and all light quanta, as restricted by energy conservatiJn. On averaging with respect to the polarization of the incident electron, and specializingto the Cou'ilmb field of a nucleus, orr-.o"btui.r,

'" an(anup,t l![. _ ,l'+r,luro-*) "r:115,'*,1 r[* o > o n2 L' " lpll(p-q-k)'J h@|_il-*

../

"

(t,

i1(qfa)-x ./ ir|-k)-K --1^\ . -)\t^-L t^-r^-?,) )r;w-

hQ-k)-*

rou-x-'n--

--

1-l

J.

e.77)

It may then be inferred that the differential cross section for radiative scattering through the angle r9, in which the energy loss does not exceedA.E, is dot(S,LE) d

d.

wnere

:

fko:K.

lqlf

Za

I'

f

@h)5(h't11l,r-**lirrlthp-rt ",Joo-o

(, ( +_1) *r,5^ r r^]u..,, (irn-.,(, ^(o^- !'\+",4,,+?.4,^) 8) ) l, e.7 \ \ q R p k / z q B 2 p h / " \ \qk ph/ 2qk 2rk / l K:

(2.7e)

AF/h.

We'shall first consider the simple situation in which the emitted radiation exerts a negligible reaction on the electron- That is to say, we shall treat-the essentially elastic scattering of an electron, in wtr[tr only a small fractionof the electron kinetic energy is radiated. Under thesecircumstancei, lvhich are expressedby aEi
(t-o'?st"';)rnJr,:, uaaol\*-*)'

da

(280)

\ow

and

(L-+\',:+J',."r(|^__L_ l), \pk qh/ (ph)(qk\

\(pp)fct) (ph),1qp1z/'

I l\ d 7 rt /l -"t (q-p)k\n-a):;l ok)Qk): "-,1(P+1+e=,\k1'

(281)

1

from which one deduces,on integration by parts, that

L\ 2

'- L:- .,u :(?k)(qk)- (ph) (qk), f' J_

Q'82)

2 / l

'

' "l(T*'+)'l'

(2.83)

190 811

QUANTUM

Therefore,

r

/b

III.

ELECTRODYNAMICS.

q\2

r,da(@-q)'_a)f(dr)__1fr'l__

| { a l x n\ pl R.,- q.t tl : P - t- t r

'\

aal J

z*t

'

'1

1t+q

b-q \

'lz

(2.84)

ll;*;')ul

The & integration in the latter equation can be written as

I a tp!1+p::,\ _ "l^ak\ 2 /p+s,p-

6(k) - J f@ H - \ - r '

\ , *-7)

\i*Jr')u

: However,

| /ry*a:,) ' ' l

t''"r ,rylu'u"t ", "l-'!t
-,[ @n) u'{n): 19 L^*,: t rr#,(Y)* [ ffN-t,

(2.86)

so that

(\y -+) a\!-il' =t_:, f oa,

[r*'.-'

r+l(p- q)'z/4K'l(-&)

I l(a*oa'\ lll

, _ _ / . * 1 l f l .( 2 . 8 7 r al l\z \LU,r +J ( dr ) _l6( e) l I f l, ,**_^.- l,

t l\;*-t)- JJI

over the domain 0(&o(K' *if..which we have discarded terms that obviously vanish on integration in becomes n r, bracketed l"t.grui i" (ZSZ),-whe' etp.es."d in threeldimensional notation, r@k)dko

fn

[Y::::Nt,-na:r"

K

dho

(2'88)

tr^,^T:2otos+

quanlum wave number to characterize a logain which we have again introduced an invariant minimum Iight expressionof the second bracketed inAii-ilut catastrophei' the with associated "infra-red divergence rithmic (2'87) tegral in Yields. n

fl

l

_ lI

r.l ^.-(ry.T)

1 tatlatn'-n')l?-

t

I

2o* (p' [(p d'z/ a)(r o'z))*

:'"1'-ia-*4,rt**

1o_ (p'- [(p- q),/4](r-lr))t

Therefore. (dk)6(h')(p/pk-c/qk)' J| t 0 - 0

:*

f'./(p-q)'

J" ^ \t

-

1

R

f , t- t , o * t *"Fr-t l pE l' {+.'lrr aftos-+ zB n) *t,">y d\

(2.90)

I9r SCHWINGER

JULIAN

812

T:ere

B \l 1-sin,-(l-72) f . \2/

/ {:1

(2.e1)

\\-e may now employ the identity

'-- I 2P loe(r l- tst/ | - At) / t

1 log(1* [(n - flz/ 4 x2l,1- Ffi

11lb-il'z/adl!-t'z)

2

log(2ps/x)- 1

r*l(p-q),/4x2f(1-a2)

lfl(p-il,/ax,l,1-1d)

l. r_Bt -, :r cast(2.90)into the form

:rhere

*, 1

2 |tog l+Bt N'zBEL

I

l+BE

'oE

,

1-B€

I

" (p-- !,1)': *t:r,.' ",1[('"* ^rT * r) to"*F,)+F,++G+H]1, I r:, ron Nr,, "" tBt

|.

r+BEl

los,

1--B

|

I ":('**hJ,aL ** -;l-*;-h;/

r,,,^,r^lroc,

, *rl.*,

(2.e2) (2.e3)

t+BI

t"cr

l ,_B I

(1 0/.\

The function Il approachesa constant in the limit oI small velocities, B<11. 11=_(loez_1)_

(2.es)

s.

\t high energies,11 is approximated by

?o/x))l:':-!t<'1,

(2.e6)

pf

'rith

s, r+t . 1_{l ttno_

//,e\:-

1

r, |

tno_

lr '_z_' -_o 2 i

s i n B f2 J c o t y z )

l-l

| IIE

at

(2.e7)

) (*-cosz(f1211+

This integral can be performed analytically for 8:r,

(2.e8)

JG):"'/12,

lut must be evaluated numerically for other angles.An approximation in excess,which has the correct asymptotic iorm at small angles, is provided by

t - (**n*,* *)'['"rz*,.,-*1@*' ] "

r? 00)

l(r/2):1.167.

(2.100)

This formula is reasonablyaccurate even for rl :r/2, :he following result of a numerical calculation:

where the value yielded by (2.99) exceedsby only 8.6 percent

The total differential cross section for scattering through the angle rl, in which the energy loss does not exceed

18, is

do@ '^E) -dco.(,) +o'J!:ou) : ( 1",.*oo-\' (1-B, ,6,1) 11-6(r,^E)), ' ' de d{t da \2lplF z/ \ 2/'

(2.101)

where d(r},AE) is the desired fractional decreasein the cross section produced by radiative efiects. For essentially elastic scattering, we obtain -l

2q E 1 1 1 .-tf / xz/bo, \. - I,) : -s?f o z.'"tL 6(o,^E<<w) (F0+ Fl)+-Fo-F,a-p,-H+ Q.102) (.*G 7 ; ;ffi ^n, l,

r92 813

QUANTUM

ELECTRPDYNAMICS.

III.

from this on combining (2.70) with (2.80). It will be noted that lby only a fraction of a percent. It is evident to scattering the infra-red catastrophe, as characterized by fr-i" has numerical result that radiative corrections ' cross sections can be quite appreciable. For the pardisappeared. However, it is possible, in principle, to 'ticular conditions chosen, LE can be materiaily in6 diverge make would which AZ+0, limit the co.riid"r creased(but still subject to LE{1W), without seriously logarithmically. It is well known that this difficulty kev, 6:6.3 10-'?, stims from the neglect of processesinvolving more than impairing d. Thus, with AE:40 As to the energy one low frequency quantum'5 Actually, the essentially while AE:80 kev yields 6:5'1 10-'?. given accuracy elastic scattering cross section must approach zero as dependenceof d, we remark that with a 6 varies AE+0; that is, it never'happensthat a scattering event in the determination of the energy, AE/E, Thus, with is unaccompanied by the emission of quanta. This is linearly with the logarithm ol the energy. 10-', an increase in the total described by replacing the radiative correction factor AE/E:0.O4/3.6:1.1 an addition of 1-6 with ,-0, which has the proper limiting behavior energy by a factor of four produces 2 Mev, and as AE+0. The further terms in the series expansioq 4.4 l}-'z to D, whence 6:11 10 lor W:74 of e-6 express the efiects of higher order processesin- 6:15 10-' for llz:57 Mev. The angular dependenceof 6 at relativistic energies volving the multiple emission of soft quanta. However, formula for practical purposes,such a refinement is unnecessary. is not fully described by the asymptotic (Po/*)sin'l/Z condition, (2.105), underlying the since particle can a of energy the with which The accuracy dimhishing c9' Inbe measured is such that the Iimit A-E+O cannot be (1, cannot be maintained with deed, 6 is proportional to sirfSf2 at angles such that realized, and 3 will be small in comparison with unity (pt/x)sin|/2<
The radiative correction thus increases linearly with the kinetic energy of the particle, In the extreme relativistic region, on the other hand,

q))']' (n(pq)'; I^::,'," "T[,{i|l# Ir(p-

(2.107)

accordingtothenon-relativisticlimit of (2.78).Hered.oisanelementof solidangleassociatedwiththedirection of the unit vector n: k/fr0, and (2'108) 1ql : (p'z-2rpo)a. t n-sb.h andA. Nordsieck, Phys.Rev. 52,54(1937).

193 JULIAN

SCHWINGER

814

I'r performing the.integration over all emissiondirections of the light quantum, and introducing the new variable ( 2 . 1 0 7 )b e c o m e s : ; i r t e g r a t i o n ,* : l S l / l p l , :a / Zq \2

x

f \t-^E'/w)t

;tmr/l

llf

-2r

2rdr cos87-lc2

: (rffi**i)" rL-[ *ffi |:"i"!lr" "'

31

(2.10e)

Thus the contribution to d produced by emission of quanta with energiesin the range f.rom LE' to W is 8a dt 4IV rJ cosrl t I - -F'? sin'?- (r - J) t an csc-1. llog*" Z- *rr S/ 2lo8

(2.r10)

)n adding this to 6(d,AZ'), as given by (2.103),we obtain the desiredresult:

p<< 1: 6(r,rr/):

I +(o-,r),un ;

?:,'^,!rl^ *+

It may be remarked, finally, that the analogous :eso-nuclear phenomenon, the radiative correction to :.ucleon-nucleon scattering associated with virtual :eson emission, will be a relatively more significant :iect in view of the stronger couplings involved. This :lay well be the explanation of the discrepancybetween -,:e observedneutron-proton scattering crosssection for ::gh energy neutrons and the larger theoretical values :omputed from various assumedinteraction potentials.6

(2.r11)

and Ar(*) is the potential of a prescribed current distribution. The physical situation can be described as follows. In the remote past, the matter 6eld is uncoupled from the external electromagnetic 6eld, and the srare vector is that of the vacuum,

v[-

(A.3)

€ ]: v!.

It is supposed that the coupling is adiabatically srvitched on, and that the extemal field does not induce real pair creation. The latter restdction implies that the final state of the matter field, alter the coupling is adiabatically switched ofi, is simply Vq, whence

€]:(.t-1)vo:0. v[€]-v[A solutionof (A.2),in tle form

APPENDIX In this section, rve shall 6rst give an alternative treatment of :e polarization of the vacuum by an external 6eld, employing ::e methods developed in the preceding pages. It is desired to ::mpute the expectation value of 7r(r),

<j p(t)) : (v 0r), j p(e)v La)), -.!ere Vf']

"**. ; ]

#,

v[c]:

(A.s)

Y;"1E0'

may be constructed,where

( A . 1' )

mffi:-li,a)Au@)urat, u[- 6 ]:1.

u[o ]:.s, (4.2)

(4.6)

ano

obeys

n'ffi : -l i,u)lu(r) r![o],

(A.4)

(4.7)

The current induced in the vacuum is then wdtten as (j p(x)): (U-1lal j p(r) U laf) 6

(A.8)

\orv

+!]a.',f","' ljrru-,fo'lj,@)

ulo'): u-rlofj,(r) u lof-!(7,(e)f s-!u(r)s),

/a o\

-rhence

u-'Ilaljp@)ulol:,r(:r(r)*.t-'yp(r).s) +*I:-d.',L",,')u-,lo'fljulr),j,(a')fulc'lA"(r') o the dght side of (A.10), one obtains the 6rst approxination Jr placing Ulo'f:1 , acuum as small. Hence.

in a treatment

(A.10)

that regards the disturbance

: #I--a.' e(t - ! 1li,(x).j "(t',1't,tr' t, : vierv of (A.4) and the absence of a current in the unperturhed

vacuum. We shall, for convenience, rvrite this formula as

Gu,@- t') : iTrlSo (r' - e)%S(r- r'h,+S(,'-r)%S(1,( ihe introduction

(A.11)

<j,(.),:T f G*@- t )A"(r')d.,,

rbere, accordingto II (2.10),

of the Fourier integral representations

iTrtGitk'*r)yu(iryk"l

(A.12) t - | )t,1.

for the functions ,5(1)and S, combined with the trace evaluation

x)t,*Gitk"t

of the

x)ujtk'*x)t,f:

(A.13) (see II (2.10))

kp'k,"*h,'ht" - 6,"(k'k"- x2),

6 A summary is given by L. Rosenfeld,Nuclear Forces (IntersciencePublishers,Inc., New York, 1949),pp. 450, 454.

(A.14)

194 III

ELECTRODYNAMICS.

QUANTUM

815

yields the following expressionlor Gp,(r)' G,"@ :

;,), I

Gk' 114p" 1'u' *'"''6( !"n'l9lk;

k'" + k'' kN"- 6('(k' k" - c)7

by It is instructive to examinethe conditionsimposedo-nG"lr) gauge inthe-Jated requirements of charge conservation and variance.The former evidently demands that

! c',t,t : *=

dty

6r,

@ilr,uf \an',ft,"a(p",+*,), (A.20)

\zr) ' r

where hu:kr'Jk,i',

f {ad\n""uln'"+*):o'

(A17)

'

or that n ' ) -: f A . : 0, (A'18) - - d i \ ( r'a-' J #"*tr "'In(r')do' f c*tr tlffi

(4.22)

the value ol Althoueh the latter integral is strictly divergent' limttrng process,In zero is unambiguously obtained from any non-slngurar which the delta-function is replaced by a surtable In this sense, the requiremenls o[ charge conservatlon Iuction. the same that noted be are satisfied lt may i"t^ri"tce ln the unper"J*.t*" intee-ralls encountered in evaluating lhe current turbed vacuum, II (1.73),

ttre in which the absenceof an integrated term is a consequence-of future' aiiabatic re-oval of the coupling in the remote past and Evidently (A.18) is satisfiedin virtue of (A'16), since

T r^yrS tD(o)

\ j r(x)),:'{

*'t, | 6*"t*;' o{a"'+

: !\

(A 19)

Gu,@):Q*1r1.

(4 21)

rvhich is indeed zero if

The requirement of gauge invariance is that the induced current be unaliected by the gaugetransformation Ar(x)-A,(x)-S'

Eh;'6(k"2+x2) f @n)@*"),""'"

:2f

(A.16)

!c""1,1=o.

u

\zt)

(A.1s)

(A.23)

which must also be zero

On computingdGs,(r)/6tyfrom (A.15),we obtain

of 6r,(r), andutilizetheffi; We returnro theevaluation

byt

hh,

624)

p,u,,-*":zu2--\h,+p)T-(la+AT

(k"'lP)6(k"2*x2)'while no contribution in view of the null value of to simplify (A.15). The thild term,in the latter explession,makes the form' of Gr,(r) of contribution produces a term the seiond

(A.2s)

at x2),
(f ffi .Y+fle+'!1"u. G*a): h [ @k 11a*,,

P') )-(] - 5u,(kk'

(A.26)

introduced in the text: The delta-function factor can be simplified in the manner

t#H.'#+!

o* r)' x))=- *[' a't ( Y "*''' 1u"' =plp
(A.27)

p", as deined by The introduction of the new variables l, and

h;:tk,+(!,-;k) (A 28)

hr":+k,-(e,-;k), thenbringsGp,(r)into the form G,,(,):-#[:do(r-t)f(dk)(dP)etb(kpk,-.6pk')6'(t+c+f;o-*l)'

(A'2e)

andfpp' hasbeenreplaced on lt alone,termslinearin 1uhavebeendiscarded' of thedelta-function where,in virtueof thedependence by lit 12,It is therebYshownthat (A.30)

G,&): ea,*,- 6,,n)ct,l,

with

c(,):8a,-:-,yI'd,(L-hI@il(dp)atut(o'+*+!o-xl)' The divergent

and convergent palts oi G(t)

= G@) #,{

can be sepanted

(r'1 + p)t(e)-ri},(' @ t)6'!

by a partial

integration

with

(A.31)

respect to ?'

- *t) - ao {an)'*" ! {ao)a"Q"+ o+!
(A.32)

t95 TULIAN

816

SCHWINGER

l:: :r.ariant, logarithmically divergent integral that occursin the frrst term of (A.32) can be expressedin three-dimensionalnotation as

- 2.Lt-(r.c{'+I- p"- p): - +f r>an],ft il { @+ -'): - [
il:::e

Ps:(I2lP)s'

(A.33) (4.34)

l:: onvergent second integral of (A.32) is then obtained by difierentiating (A.33)with respectto x' and replacingthe latter by , : _ . h ,/ 4 ) ( 1 _ 8 ) , ' :i

f anu'Q"+r +f;o *t): oaa a)(ta)

these evaluations,

(A.3s)

G(r) becomes

r-i-(losP-!tr1),'(.) rf; ll'r.,i*r io,t,ll,

cr.r: -f*

(A.36)

I ::ie

r

"{

orlteib

x't:

J,-

tf "dt

^

Fiaally, we may insert (A.30) into (A.12) and t:::'::":1

h\ J ld t + fpy +*,I 0 _tl. parts to obtain

(A.38)

(j,{ t)) : 16"a.J G(x - e' t J,k' )d@', ':ete

(A'37)

Jr(r) is the current vector that generatesthe external electromagneticfield. The expressionof G(r) containedin (4.36) thenyields

- r')- t Flr - r'))I r(s')da', {r"k)): - f rr-( b{}l! - 1)r.@- *io' f
(4.39)

: rhich the fi6t term representsthe logarithmically divergent renormalization of charge. It should be remarked that the existirce of a charge renormalization term would appear to contradict the conservationof charge, :::ce it implies tlat a non-vanishing total charge is iiduced in the vacuum. Indeed, i iormal evaluation of the total induced charge

zero' rurd vierd

!
7p1a,,,i"a'lfd,ala-' (A.4o)

:0, of (fr(r)) as with the cuEentvectorat an arbitrarypoint.The expression ;:1cethe operatorof the total chargecommutes

: fr,ff t e@r)F*,(r')do',

(4.41)

-here AA F",:7A,-74,, dxy dtp : formally

(4.+2)

consistent rvith the result since

l1 *,

:ti I *u,-* I G(x- r')Fp,(*') d^' Q,, d

ln view of the theorem

(A.43)

(A'44)

[(o",Stav*fi"<"):0.

in Horvever,it is evident that these formal manipulations are only justified if the integrand in (A.44)-decreasessuEciently rapidly Jirrctions, which is not ful6ll"d fo, tire field strengths'ginerated by a chargi distributio.n of nm-vanishihg total charge' rp"l"-fif." ' total which-the This difficulty can be avoided by treating the actual electrimag"neticfield as the limit of a spatially confinedfield, for to inducedchargeis zero. A conveniint way tio accomplishthis is to introduce a finite light quantum mass,which is eventually allowed vanish, We thus mite the potentials generatedby the given charge distribution as

t# At@):! f D@- r,)r u@)da,,

: r,

a
(4.4s) (A.46)

The induced curent can then be exhibited in the form

: -# da',
:#

*tc k'D(k)4 @- t)@'k) (4)' luyf " 6)

(A.47)

in which the secondversion involves the Fourier transforms of the functions, G(x) and D(r),

ptu'(p,+ * +f o -at) (t - *)
D(k):\:I#.

(4.48)

r96 8I7

III

ELECTRODYNAMICS

QUANTUM

The total induced charge can then be calculated in terms of the total external charge, 1o which expression is independent

(A.4e)

Jlr-{)do,

Q:) of f. Therefore,

( (d * lf r:9*o I G@k,D th)6 h) h)

:Y*oyEffi"at)r,-" If e is placed equal to zero before evaluating charge

the Fourier transforms at &ts:0, we obtain the previousiy

(A.s0) computed non-vanishing

induced

: 6e:X'Qrc @Dou-o f;o f rao a'rt +*t

(A.s1)

On the_oth_er har.rt,if the limiting processr-0 is reservedto the end of the calculation, rve evidently find A0:0. The implicatio',s of this limiting processmay be further indicated by noting that, in the first term ot (e.:S), fu(r) will be replaced by Ju@)-e'cA'(x)'

(A 52)

in virtue of the difrerential ecuation

[J,- ads*1: -!41,1

(A.s3)

obeyed by the potential (A.45). Now (A.52) reducesto Ju@) atany point as €+0. Yet the total chargecomputed from (A.52)is zero. This is illustrated by the chargedensity associated,accordingto (A.52), with a point chargeat the origin:

li''(ai4-'l1l' .,0

(.{.54)

fit/

\

We may-concludethat in the processof'vacuum polarization, a non-vanishing,and indeed divergent chargeis attached to the original charge distributiqn, and a compensatingchargeis created at infinity. We shall finally apply the computational methods of this paper to evaluate the invariant expressionfor the electromagneticmass,

(D : -F2f tulD({- r,)s{')(r-f,,)+ D
(A.ss)

Theinsertionof the Fourierintegralrepresentations yields

u*ur<,>:-qrfi ,.f (dk)(d.k)exn+*ta-."').y,atk'-*h,(!g:!9+f 9,)*aw,.

(A'.s6)

This becomes

-t on introducing

k-o.v (t- k)- r,r(ry *L,I rorro,eb Fe^

-

ffi)r aw

(4.s8)

Pp: kplkp',

wNch is efiectively subject to the restriction

(A.s7)

.pr+p:0,

(A.se)

2thu)du, *16(k,-2 lk)- 6(k)1: - Jo |' 6'(k2' l?k-

(.{.60)

in view of the wave equation satisfiedby ry'(c,).Now

and

uGt Q - k) - x)tt:

whence

6 mi41'@= ^.

e r.-...- rt. (dh)(dp)Jod,ueiee/)G^rh-K)6'(hr-2rku),1'@')da', (WJ

in which we have employed the fact that lTpfr trunsfolmation

is equivalent Lo Lr(d/Ot/)lr

(A.61) (A.62)

applied to g(o,) and is thus efiectively equal to zero. The

kt*kr+PFlt

ttren yields 6nc,t (D :

- 2ti1 (t - h) +2K),

(4'63)

@k)(d.fi f I aastoa-a 1i,7pu - x)6,(kz! pu') l, (r,) da, &1 f

: -

(n) * rk) (r! u)6'(k ! au,),!, **- J| [

(A.64)

Hence,

u*/*:-fif"'lt+u)duf @.k)6'(k +r,uz)--fif

loulo'ro**,1+fifo'e,,+u")dudf @k)6,,(k,+ca,),

(A.6s)

and

a*n:"ur^('sff-1. according to tie integrals (A.33) and (A.35).

(4.66)

197

P o p e rI 8 On Infinite Field Reactions in Quantum Field Theory l'httts

S,\-rrrRo ]-oMoN^c^ . lailit.t., l okro Baaliho Doieukr, 7 .kro, Jdloa Jure r, rq$

thc level-shilt of the hyrJrogen atom in f \ irilerpreting I terms of the radiative reaction, Bether proposed a method of dealing with this problem without touchitrg the inherent divergency of the curreot quantum field theory. By bis theory jt has lrccome possible to treat the problem involving frckl rcactions for the 6rst time in cloe connFtion with the rcliablc cxErimental data. On the othcr hand, Lewis2 and Epsteins analyzed the infiniries ccurring cross-sction in the radiative correction to the sattering of an electron in an external field of force and found that als the* infinities could be got over by means of a pro' cedure similar to Bethe s. In a recent issue of the Psvstar p6inted out that in these problemSchuiingel Revrew, n-as ueful, brthe mettrod oi caoonical transformation wbich tlle separation from the fields of the parts belonging was per' eletrons to free photons and non-radiating formed, a method shich can be regarded as a relativistic generalization of the preedure used by Blch and Nordsicki as well as by Pauli and Fierz6 in tle discussions about the *lf-6eld of an electron. Almost the same line of atbck was taken independently authors also by our Tokyo group aod, of these.{merican mme results are now in course of publication in ourEnglisbl journal, Progress oJ Theoreliol Pfiysrcs' Irnder the un-, favorable conditions after uartime, houeler, it qill be al in print. So l' long time before these papers *ill apper should like to give here a brief summary of the state and views of our investigations. \Ye lirst treated the eff@t of field reactions in the collithe C-meson sion problem on the/-field theory of Paist
caus of this structu.e this infinity is to be attributed to the vacuum polarization effcr [n or,ler to sec the role to be played by this effect ln collision phenomena we analyzc(l the innnities ccurring in the e2-correction to the KleinNishiqa formula.rs In this problen, we found, besides thc infinities of the'types meiltioned above, an inl-rnity which is closely related to the above mentioned vacuum effect. Infinity of this kind el be, in fact, driven away froor the cross section when we substract beforehand thc inlinite term of the vacuum type from the Hamiltonian. But for this subtraction we cannot find a reasoning so natuml and plausible as that used in the case of mass-tlpe and chargetype infrnities, where the subtraction was considered as an result amalgamation. This is bcause it would-naes*rily in-a drastic chaoge of the Maxwell equation for the radiation. A way out of this difficulty was suggebted:la it might be possible to introduce some lields whiqh would give ri* to the vacuun effect u'ith the opposite sign so that a conrpensation method similar to the /-6eld theory might be used here, In fact, one 6nds, applying the same method, 6eld has this propert1.16 At that a Pauli-Weisskop{ possibilitf is Lo consider, in the style of Dirac's alterutive theory of the classical electron, that the "original equation" for the radiaiion contained, in the same way as the "original mass" of the electron, in itself an innnity with the opposite sign so that, supplemented with the iqnnity appearing as the result of the interaction, the equation for the obsewable 6e1d baomes just of the Maxwellian forn. The calculation of the level-shift of a bound electrol was also undertaken,ld This work is not yet completed but it was confirnred that the r€'sult converges by virtue of ortr subtraction prescription. \\'e fourrcl ftlrthcr, in agreelnclrt rvith Schwinger, that a part of thc radiative correcLiotr'to the energy can lrc interpreted as caused b1'an anotnalotls mornent o[ the electrfir the tsislenoe of whit:lt harl been cxpected by Brait.u We hope that various postwar difficultics rvill soon be settled atrd that our results will appear in print id the near future. I H. A. Bethe,Phvs Rev.72.339(1947). t H- w. Lewis.Phvs.Rev.73, 173(194E). r S. T. EDsrein,Phvs.Rev, 73, 177(1948).

lr.Ti::,;'SlillK'";,1il;ll'fn',,'"i'fl,'].,,*r'0.''. No. J I (lr)txr' .w. Pauli and M. tsierz, Nuovo Cimiento, 15. t A. Pds. Phvs. Rev. 6E, 227 (1946)

l "; ",?5'-!: Iii"; Lli""i il#?;"?# $'.?i\, *,. phvs.2.2t6.2tt

tt).till?r

t"-onasa,

*u

t*rure at the symposiuh on th€ th-e-oly

*ijTnl$tr#j;f;"*f il $ij*ti,:,:q,,':,:. ".

in.pros. ll X: lli'.1'ti."il[1'f,'lli"R1'l1l;,nr, to bepubrished

umeswa E lne I h@!. Phys. It was first Dointed.ou.lbv Sakata and Narcva U;iversitv that the PaulLweiskopf 6eld giv6 rise.to a rFrne to tiva lilf-.n€rd of a photon in coDtrast to a nesattve oqe auevrcum .lrtron field and this would tsult in the comrnsaton ot tne .fi(t mddon€d above '. V- Nambu. to be Dubtisbedln Prog. Theor. Phvq It G. Breit. Phys.Rev.72, 984 (1947). .\'ulp oE thc Aboti Lcttt: lil rrailsmirliDg to the i'hysical l{eview the acconrpaning review by Tomonaga of the remarkable work carried out in Japan in reent y@rs, there is one trchnical note that may be helpful. Tomonaga rernarks in the 6fth paragraph from the end I that in addition to the infinite ternrs which may be recognized as contriloutions to mass and cbarge, there are other infinities which appear, particularly in the corrections to forrnula.'l'hese the Klein-Nishina have to do with the familiar problem of the light quantum sdlf-energy. As long experience and the recent discussions of Schwinger and others have shown, the very greatest care must be taken in evaluating snch self-energies lest, instead of the zerq value which they should have, they give non-gauge'covariant, non-covariant, in general infinite results. From manu*ripts

kindly rent by Tomomga, I would conclude that the difficulties referred to in this note reult from an insufficiently eutious treatmeit, add therefore inadequate identi6cation, of light quantum self-energies. J. [t. OPPENHETMER Inslitute Jor Advanced Sttd'y Ptincelon, Nru Jersel

r98

P o p e rI 9

On the Invariant Regulafization in Relativistic Quantum Theory W. Paur,r aNu F. Vrr,r;,{ns SwissFederal Institule oJ Technolog, Zurich, Swilzerland' (ReceivedMay 10, 1949) 'llheformalmethodof regularization of sumsof productsof differenttipes expressions of mathematical

of 6-functionsis first applied to the exampleof vacuum polarization, It is emphasizedthat only a regularization of the rvhole eipiession without fictoriation leads to gaugeinvariant results. It is further shown, that for the regularizaiion of the expressionfor the magaetic moment of the electron, a single auiliary mass is suftcient, provided that difieient functions of the same particle (e.g., the photon functions D and f{r)) are regularizedin the same ray and that the regularization of products of two electron functions is never factoiized. The result is then the same as that of using Schwinger'smethod of introducing suitable panmeters as new integration variables in the argument of 6-functions,rvithout using any auxiliary masses.

by the argument tlat a time-like component of the current commutes with 7, at ail points of a spaceJike of the new relativistically f N spite of many successes general invariant I invariant formalism of quantum electrodynamics,r surface.a The specialization of the form of the commutator to this case, however, gives a which is based on the idea of "renormalization" of mass result proportional to and charge, there are still some problems of uniqueness left, which need further clarification. The most impor5ttt(s - yt ) (0 Lrtt/ 06), tant one seemsto us to be the problem of the self-energy which is indeterminate due to the singularity of of the photon, which was raised by Wentzel's2 remark ALD/A*, on the light cone, which has the form that the formal application of Schwinger's original -x"/(r"x,). The whole expression may therefore be technique of integration to the resulting integral gives written as for this self-energy. from zero a 6nite result difierent 6(r)(r_x')(0L(L)/6*"), . This problem is formally contained in the more general problem of the gaugeinvariance for the resulting current in agreement with the straightforward computation due to vacuum-polarization by an arbitrary external (see below). $2 field (not necessarilyby a light wave). Schwingershas The occurrenceof products of functions with a 6-type shown that this current is given by singularity and with a pole is typical of the new formalism and seemsto be the main source of the remaining ir (i,(")) : - | dax' (li r@), j, (x') ])oe(* - .r'),1,*t (a'1 uniquenessproblems. )J In order to overcome these ambiguities we apply,in the following the method of regularization of A-func*n"., .(r) : + t for I ] 0 ;,4,*i(r) is the vector potential help of an introof the external field and (Ur("), j,(*')l)o is the vacuum tions (or products of them) with the duction of auxiliary masses.This method has already expectation vaiue of the commulator of 7u(r) with j,Qc'),'Ihe condition for the gauge invariance of this a long history. Much work has beeh done to compensate electron with the expressionfor (j"(r)) (which includes the vanishing of the infinities in the self-energy of the help of auxiliary fields corresponding to other neutral the photon self-energyas a special case) is: particles with finite rest-massesinteracting with the aI axI (li u@),i,(r')l)oe(c- r') ) : 6' electrons.t some authors assumed formally a negative Schwinger tried to prove the validity of this condi- energy of the free auxiliary particles, while others did not need these artificial assumptions and could obtain tion, after reducingit to the form the necessarycompensationsby using the difierent sign j,(")l) (Lj,@), ol3e(x x' ) / ar,l: 0, of the self-energy of the electron due.to its interaction -ii. Toiln"*", Prog.Theor.Phys.1, 27 (1946).J. Schwinger, with difierent kinds of fields (for instance scalar fields Pbys.Rev.75,651(1949);Phys. ?s, vector fields). We shail denote these theories, in Phys.Rev. 74,1439\1948); are quotedin the following Rev. 75. 1912(1949).TheseDaDers as SI, SiI, SIII. Ou; nolationilbllorvascioselyaspossiblethosi which the auxiliary particles with finite masses and of tbesepaoers.For t}e definitionsand the propertiesof the positive energy are assumedto be observable in princiwith I in SII),anilah)seeparticua, A: (A is idenrical functions entering the Hamilof SII. In t}rispapernaturalunitsh:c:l arc ple and are describedby observables larlytheappendix F. T. Dvson.Pbvs.Rev.75,486(1949),and usei tbrodehout. . SII. Eo. {2.29). quotedas D[, and 1736(1949).In the following Phys.Rev."75, 6Compaiefor olderliterature(includinghisowncontributions), DU. , G. Wentrel,Phys.Rev.74, 1070(1948). A. Pais, The Dael.oDment oJ lhe Theory of lhe Eluhon (Princeton , SII, Eq. (2.19). University Press, Piinceton, New Jersey, 1948). sl. INTRODUCTION

L7L

199 435

RELATIVISTIC

QUANTUM

roni4n explicitly as "realistic," in contrast to "formalistic" theories, in which the auxiliary massesare used merely as mathematical parameters which may final1y tend to infinity. Recently the "realistic" standpoint was extended to the problem of the cancellation of the singularities in the vacuum polarization, due to virtual electron-positron pairs generated by external fields, by introducing auxiliary pairs of particles with opposite electric chargesand massesdifierent from tiat of the electron.6 It was shown that the signs oI the polarization effect allow compensation of the singularities only if the auxiliary particles are assumed to obey Bose-Statistics. Until now it was not possible to carry through the "realistic" standpoint to include al1 possible efiects in higher order approximations in the fine-structure constant, nor is it proven that this problem is not overdetermined. Presumably a consistent "realistic" theory will only be possible if, from the very beginning, all observablesentering the theory have commutation rules and vacuum expectation values free from singularities, i.e., difierent from the A and A(r) functions which obey a wave equation corresponding to a given mass value. Until now, however, it has not been possible to carry through such a program. At the present stage of our knowledge it is therefore of interest to investigate further the "formalistic" use of auxiliary masses in relativistic quantum theory. This was done independently by FeynmanT and by Stueckelbergand Rivier.6 The latter authors use (more generally) an arbitrary number of auxiliary masses, rvhile the former introduces only a single large auxiliary mass, rvhich was sufficient for his particular problem, the regularization of the self-energy of the electron. From the well-known expansions of the A- and A(r)functions near the light cone it can easily be seen (see $2) that in the linear combinations: La(r):t,;

cA(r;M r)

and [pttr (:c): f ; c;AQ(* ;M ;), the strongest singularities cancel, if Ltct:o' and the remaining singularities (finite jumps and logarithmic singularities) also cancei if in addition the condition lac;Mnz:g holds. If the first condition alone is sufficient to guarantee the regularity of a certain result, it is obvious that a single auxiliary mass M1 (besidesthe original electron 6G. Rayski,Acta Phys.Polonica 9, 129(1948).(Onlylieht

waves as external electromagnetic6eld are consideredin this paper.) Umesawa,Yukawa, and Yamada, Prog.Theor. Phys, 3, No. 3. 317 (1948). 7R. P. Feynman, Pocono Conference 1948; Phys, Rev. 74, 1439(1948).Applicationsby V. F. Weisskopfand J. B. French, Phys. Rev. 75,'t24o \1949).3 E. C. G. Stueckelberg and D. Rivier, Phys. Rev. 74,218 and 986 (1948).D. Rivier, Helv. Phys.Acta XXII, 265 (1949).

THEORY

massMs:1n) with cr: -40: - 1 is all that is necessary. It should not be forgotten, however, that Feynman's successin using a singie auxiliary mass in the problem of the self-energyof the electron implies the assumption that in the corresponding expressionsresulting from the invariant form of perturbation theory the photon functions D and Do) have both to be regularized with the same atxiliary mass. (A formal alternative would be to leave the photon-functions unchanged, but to regularize the electron-functions A and A(r) with the same auxiliary mass, or to regularize the whole expression without factorization and with one auxiliary mass.) The application of the formal method of massregularization to the problem of vacuum-polarizationro ($4) shows that not only the use of a single auxiliary mass is here insufficient, but that any regularization of A- or A(r)-functions as separate factors leads to results that are not gauge invariant. As was shown by Rayskirr only the regularization of the whole expressionfor the resulting current (without factorization) gives satisfactory results in this case. The formal use of continuous mass distributions is here particularly suited to illustrate the connection between the difierent results of Wentzel and Schwinger for the photon self-energy. In $5 the example of the correction to the magnetic moment of the electron, which is one of the main results of Schwinger, is treated from the "formalistic" standpoint of mass regularization. We agree with Schwinger that the use of auxiliary m&sses is not necessaryin this caseif the computations in momentum space are made with sufficient care (see additional remark A). In any case (difierent from the situation in the problem of vacuum-polarization), the use of a single auxiliary mass is here sufficient to avoid any ambiguity, provided that the same mass is applied both to the D and the D(r)-functions of the photon, analogous to Feynman's method for the self-energy of the electron, or that the regularization is applied to the products of two A- and A(r)-functions without factorizationr2 (seereference20). 'SII, Eqr (3.77)ancl(3.82). roDyson(seeDI I) appliesto thisproblema methodof regularizationwittroutuseof auxiliarymasses, whichis moresimilarto the methodsusedin the earlierstaees of oositrontheorv. rr Rayskimadethis proposalin tle sunimerof 1948,iuring his

investigations on the photon self-energy of Bosms (see reference 6). With his friendly consent we later resumed his work and generalizedthe method for arbitrary external fields (not nrcessarily light waves). u The problem of the magnetic moment of nucleons due to a mesonicinteraction, which shows a closeanalogy to tie problem of the mametic moment of the electron due to elecbomaenetic interactionl is not treated in this DaDer.Stueckelbere-Rivieisive (see relerence 8) a formula for'ihe masnetic moirent of-the neutron which ibey cbaracterize as noileading to a defDite uumerical value. A justification of t"hismay, in principle, be seen in the fact that the most general form of regularization with auiliary masses must always lead to an arbitrary value for integnls of this tlpe. On the other hand tle mentioned general analogy between ilie two eses makes it plausible that tbi same mathdmatical methods which lead to an unambizuous definition of tbe magnetic moment of the electron witl;ls lead to an unique definition of the value of the theoretical results for the magneticmoments of tle nucleons(at least for rcalar and pseudo-

200 \\T. PAULI

AND

Both groups of authors (Stueckelberg-Rivier8 and Feynman-Dysonrs) seem to ascribe to a particular combination of A-functions, which describes outgoing waves for the future and incoming waves for the past, an important or even fundamental significance.As this question can be left open for the purpose of this paper, rvediscussDyson's expressionfor the magnetic moment of the electron, in which the function A" for the electron and D" for the photon alone occur,la only in a brief additional remark (B; $5)' We believethat in order to investigate the range of applicability of the particular function A", the discussion of more complicated examples rvill be necessarY. Summarizing, one must a.dmit that the additional rules rvhich the "formalistic" standpoint has to use (e.g., to apply the same mass values for A- and A(r)futiciion., ind not to factorize the regularization of prod.ucts of A- and A(1)-functions corresponding to pairs of charged particles) could be immediately underitood from the "realistic" standpoint and appears as if borrowed from the latter.15 It seems very likely that the "formalistic" standpoint used in.this paper and by other workers can only be a transitional stage of the theory, and that the auxiliary masseswill eventually either be entirely eliminated, or the "realistic" standpoint will be so much improved that the theory will not contain any further accidental compensations. $2. THE BASIC CONCEPTSOF REGULARIZATION In an invariant perturbation theory, such as the one introduced by Schwinger into quantum electrodynamics, the trvo invariant functions, A and A(t), play an essentialrole, Vacuum expectation values of properly symmetrizedproducts of field operatorsare expressed in terms o14{t), rvhile A appearsin connectionrvith the covariant formulation of commutation rules. The handling of espressiollsinvolving A- and Air)K lI Case, Phys. Rev. 76, 1 scalar mesons). \Ieanthile 'from invariant.perturba1 1 Q 4 Q, o h t r i n s a n u n a m h i q u o u s r e s u l t rion lheory) for the maenelic nucleon momenls lvhlcn--agrees t h o s e o f L u t r i n g e r r - H e l v . . P h y s .A c t a X X I , 4 8 J io*pt"r"t1:sith 1 1 o 4 8 1 ) .H e d o e s n o l g i v e t h e d e l a l l s o l h l s e v a l u a t l o n o l t n e inteeiils, for which no auxiliary masses are needed. 'to-out" DI and Dll The lunclion in question is denoted riLh D. bv Stueckelberg-Rivier, with Dr L,y Dyson and uith Ar (asi. ii'e use the n-otation A" and ,. lor lhe corresponding hv eiectron and photon functions respectivell'. la DI. Section X. formula for l. 16The interesting problem of the "self-stress" of the electron rsee A. Pais. relerince 5: in-an unpubjished letter of last.year Pais gave the result thal in the lheor)'ol holes tne value oT tnls s e l f - s i r e s si s f i n ! t e , n a m e l y a / 2 n a t ( a : f i n e - s t r u c L u r e c o n s t a n t l . but not zero, as special relativill reqrtires lor the.lotal stress ot a closed system) may lhrorv more light on lhe relallons betseen rhe t$o strndDoints. Detailed calculations by one ot us (F'v ) cave the result lhat a formal regularization nith auxiliary masses Eoes not change the 6nite value of Pais for the seli-stress: one to consider the localization of energy in has therefore iithe, ipace und rime as a non-physical concept in quantum lheory and to admit only the energl-momentum ve'lor {whIch.ls alreaoy inteerated ouer space timel, or one has to ascrihe to the compens a t i i g a u x i l i a r l m a s s e sa p h ' s i t a l r e a l i t y s u c l t t h a t t h e i r c o n t r i b u t i o n t o t h e s t r e s s i r l h c i n t e r n l e d i a t e s l a t e s c o m D e n s a t e sl h e oth€r part of the self-stress of the electron.

F.

436

VILLARS

functions exhibit some characteristic difficulties, which may be summarized as follows: (a) The occurrence of indeluminale eipressrons as a consewith qu"n." oI the coincidence of the d-t'?e singularity of Ak) a properly defined Only the pole of A(t)(t) on the light cone limiting process may give them a definite meaning' (b) ihe necessity of taking into account, in the course of the calculation, the "covariance" oI some diverging (however formally too covariant) expression in order to'split ofi a 6nite part'.This may be done'in a proper way only after these expressions have process.

been made finite by a regulariation

Sinceboth dificulties are connectedwith the singular features of the A- and A(r)-functions on the light cone, an invariant elimination oi these singularities may be helpful in an attempt to escape the above-mentioned complications. Looking for such a device, one is guided bv tle dependenceof A and A(r) on the rest mass of the .o.t".pottdittg field. This dependenceis exhibited in the integral representations:

Ao)(r) : - (m,/ zfl

!,

al (1/ 4o)1, (la) da sinl)wz.

a

A(x): \n2/4r2)|

(lb)

do cosl)tm''o+(l/4")J,

Jo

where

(2)

)t: -ttpxc*,*

which show that both A(D and A are of the {orm: m2'Ju(\n'2). From this it follows that 6-t]?e singularities (6(\)) and f,rst order poles (1/I) are independent of ar', whereas finite jumps and logarithmic singularities are proportional to zz'. Since these are the only types of iingularities occurring in A and 6
5*trt:l;cA6(Mr),

(3)

cA(M),

where c; satisfiesthe conditions: (I)

I; cr:0

(Ia)

Z; c;M;2:O' In order to exhibit more clearly the effrcacy of these conditions we give the development of A(D and A for small tr (omitting all terms vanishing for tr:0): 1 t-)

A,r'(r)::l-fn, 4kt \

lllz

r

log11,n t r' ) r - ^ - + " z 2

or,r:1{aor+ (I. )"or} ,-(r):{l} * l. thJfoilo*iog ta: ias= iJ.

for r?0.

l

t1

(4)

201 437

RELATIVISTIC

It is easilyseenthat Aa vanishesfor tr:0, such that L,(x): - 2e(x)h,(x)

without factorization):

no:

is regularizedtoo. Ap(D,however,takes the value 1 -l;crMi2logM; 4r

THEORY

QUANTUM

I

anpl*1n1A(t)(K/), ro)(();A(x,),D(()) 6' :

for tr:0.

(71216)\,

or we may also regularize F only with respect to the D-function referring to one tlpe of field, e.g.:

It is the meaning of the regularization prescriptions that the first term in the series (3) represents the non-regularizedfunction itself, i.e., that

Fp :

I d x p 1 r 1 F 1 A\ n( t) , D i l , ( K ) ; A ( r r ) , D ( ( ) ) . J

Whenever in this latter caseF is linear with respect to the D-function to be regularized, (9) reduces to (3) and that all M; (i)0) shouldfinally tend to o (accord- (i.e., the introduction of individually regularized Ding to the "formalistic" standpoint, adopted in the functions), but implies the important additional rule following). The coefficients rr need hereby not remain that the same regulator p(r) has to be applied to both finite. We shall, however,imposethe condition D and D(L). If, on the contrary, F is bilinear with respect to the field in question, all these bilinear terms have to ( l c ; l ( 6 ) Li /an')-O be regularized without factorization and with the same regulator. In this latter case the conditions (I'), (I'a) rvhich ensuresthat are then, in general, not suficient to remove all singuif only Li cF(Mr2)-O larities from (9). They eliminate however the strongest lM;'zF(Mt2)1(,4 for all l)0. ones,especiallythoseof the tlpe 6(I)/\. The rule (9), interpreted in the above-explainedsense, For the purposes of a general discussion it may sometimes be advantageous to replace the discrete will be adopted in the follorving throughout. It is this spectrum of auxiliary masses by a continuous one rule that assuresthe gauge invariance of the polarization current in the problem of vacuum polarization(including,or course,the discreteas a specialcase): in contrast to the resultsof (3). fto One may object that this prescription sufiers from a (7\ Ae(x,: I d(pf(rA(rini. erc. lack of uniqueness,but this apparent deficiency affects J__. only the massand chargerenormalization terms. Hereby we mean, more precisely, that after mass and charge where x has the signification of the square of a mass. terms have been removed, all additional corrective The conditions(I, Ia) now read: terms shall beindependentof the way they are regularized, and.shall, of course,be i,ndefendentof lhe Paramelersci (r') and. (or pt(x)+O f or any fnite r). M; i.n the lim.i.tM "+a This is not the caseif in the form F, (9), the individual summands are regularized independently and difier(r'a) enily: Ian*r1r1:0. co:1,

Mn: 7n

Io-ou':o

On writing the form

p(x) as 6(x-m2)+pl(K)

condition (6) takes

F:FrtFzj

fr F R : I d K p " ( K ) F : ( x ) f* d r p o ( x ) F z ( x ) ; JJ

n* ,

dK( plK)t

x)-0.

a)o

(81

J d

An alternative possibility of regularization is contained in the prescription Fp:

f J

a quite arbitrary result may then be obtained, as will be shown later on (see $5, additional remark C). The charge and mass terms themselves, however, depend on the way they are regqlarized, since they depend on

l;cilogMi d x p ( x ) F ( A ' r ' ( xA) (. r t )

(0,

ivhereF representssome (bilinear or higher order) form in A, A(r), dL/il\, d!a) /d1,. The use of the prescription (9) needs further explanation: The form F may contain both A-(electron)and D-(photon) functions. We may then either regularize the expressionF as a whole (both A- and D-functions

,,

!

droQ)log*.

(10)

In connection with the use of the Fourier-integral representationof A and A(t) (as in (1a, b)) for computational purposes, it is convenient to have conditions (I', I'a) expressedin terms of the Fourier transformed o f p ( x ):

n1a1: a*p1r1r-, f

(11)

202 W. PAULI

The conditions(I', I'a) then read

(r")

R(0):0 R'(0):0

F.

AND

(I"a)

Sometimes it may be convenient to separate the electromagnetic field (and its sources)from the system under consideration and treat it as a given (c-number) field :,4 u"*t(r), satisfying :

whilethe integral(10) is transformeditito

-f,f.-Lu'o''' (e(o):41

438

VILLARS

s121,.*tp6):_J,."r(r).

(21)

As before, "states" are represented by a time-inde-

(10') pendent state vector Vr and a transformation formally

for a?0)'

S3. THE INVARIANT PERTI'RBATION THEORY

analogousto (19), introducing V5, exhibits the change, induced by the presence ol Ar'*t, in the expectation value of an operator O. The transformation zr, (19), shall be written, more precisely, as (22) xa(t):rist(t)Nfi-is2tt)N2"''

Let ,tt(r), 0@):{*@)9 be the quantized operators of the electron field and ArQc) the four potential representing the radiation f,e\d,.{,0 and Au arc supposed where S1(l) is thought to remove the first-order coupto satisfy the equations of the uncoupled fields: ling, Sz(l) the remaining second-orderinteraction, etc. Between these steps may take place renormalization (12) lt,@/ar,)-tm)*:0, transformations 1f, defining new matter field operators (12a) @p/ax)1"-nQ:0, which obey equations with adjusted mass. According to this program, 51 is defined by: (13)

l)'A,:0,

Sr:_E

(14)

(0An/6r,)*(t):0,

leaving for V1 the equation of motion

and accordingly, obey the commutation relations:

(1s)

l A r @ ) , A , ( r ' ) f : i 6 , , D( r r ' )

i@vt/ail:0/2lsr,El*..')!rr. Sz:i/2lsr,

1

- x') ) | : -s "a@ W"(r),0p@' 1/ a : . 1 t , ^ - - m \I a ( r - r ' ) . j\

lA,a1: 13-3a,

dr,

/

I A ,B l : A B + B A .

J

a:o+r[s,,o]-;[S'[s',o]l+ds,,ol ^t

o0):o(r)+rI atlu@,s(ul J_

f | d3xjr(x)A,(x)

:-;rla'*OG)tg,G).A,kt.

fr

ft

-+ | dl I d{,lHa')wu" ),a(t)ll JJ (18)

J The solution of (17) may be achieved in successive or approximations by a set of unitary transformations: {/:e-ds1(t)ilrr:e-ial(,)e-tsr(t)V2: ' . .: U0)Vx,

(24)

Restricting ourselvesto this order of approximation, a transformed operator fl, according to (20), may be written as follows:

(17)

Herein fl representsthe interaction energy: -

E*u

i/21s,,Hl.

The auxiliary condition (14) involves the state vector !I, of the system, whose equation of motion is given by

H:

Hl-

leading thus to a state vector V2 which is constant in time up to and including terms in e2,provided a mass ( 1 6 ) renormalization has taken place, removing .8""11from

"o

i(av/at):Hv.

(23)

Then

(19)

where Vr is time-independent in the desired approxi mation and thus representssome "free-particle"-state. The change in properties of an operator O referring say to the field o, due to the interaction of a with the vacuum of the field D, is then expressedin the vacuum expectation value (with respect to 6) of the transformed operator fl: (20) (fi)"."(6): ({/-ror)"*(b).

r' ti I + i I dt'l-[s,(/'1,H (t')l- H,(/),{t(t)| JL2J fr

o(r):a(r)+i I dtlH (/ ),{t(t)l J

- | at | (t/'lH\t")lH(t'),a(t)ll JJ

-i

J

,JtlH'(t'),a(l)}.(2s)**

** f*Ut i. this latter form a simpliflcation due to Schwinger has been used, which employs Jacobi's identity.

203 ;,i9

RELATIVISTIC

QUANTUM

With the help of the commutators (15), (16) and the :::rived relation: : : t) y +l'(r), 0 @') t',1,@')) : 1/ i 10Qc)"Y PS (n - t' ) t

respectto the gauge transformationAu-Au*(0h/0rr) grves a

f l: k' )

-'0 @')t"S(r'"l, - rh''t'Qc)|, (zo)

::.e transformed operator Q is easily evaluated. To -,:\'e its expectation value for a definite state (for _:stance the vacuum with respect to one of the flelds ---question) we need still know the vacuum expectation .:lues of the properly symmetrizedproducts: (l A,(x), A

( a- { ) " ( x ') l ) , : 5 , , P u t

(27),

{{"(r),fre@)l)o:_J"r{r)(*_o/).

dt'\lH"-t (t'), jp('il)o : -4e J

'.rhere

I dir.'A,,r.r-.r'rrd.\,,1.r,,r,

which requires: 0K,"(t)

:o

(33)

or, In momentum space,where (29) reads:

(j ,(p)): - 48K,,(?)A ,(p),

t d a xK' , " ( x - x ' ) A" " * t ( x ' ) ( 2 9 )

K*(p).p":0.

(.33)

On writing: K,,(1,) : K tpQ,+ K 25,,pxpx. Equation (33') is equivalent to the condition Kt: - Kz

j,(r,)])oe (r- x,).

J y""t(1,): (6,,hh - ? u?) A,.*' (?), condition (36) assuresthat: <je(D):4dKJ

(30;+xx

61 e'

(b) The d-radiative correction to the expressionfor :ie current associatedwith the matter field, According :r (25) this correction is contained in Fl

An evaluation of K' and (28) gives:

nl'

f

'arsn..,,1t), j,(r)1. (31)

The one particle part (for the defnition of the concept f the one particle part of an operator, see SII, page 'i 1) ot A,j r@) includes a term of the form

a

iu(d (r) : con51.t (i' @)ow,t'@.)),

$4. VACUUM POLARIZATION AND PITOTON SELT'-ENERGY In this section the tensor Kr,@-r,) (30) shall be .:vestigated. Invariance of the induced current with *** The factore(c-r') is introducedby writing: Ft

?F

fb J -d( : +J- dl 4t_/\ +LJ _ d(.,

ie secondintegral vanishes,if no real transition is induced by .re external field.

ltu

0x, 6ru

-t,(;

#-

n'^u'^).(3sa)

From the expansions(4, 5) it follows that this expression is indeterminate on the light cone, due to terms of the type d(I)/I, and the relation (33) yields, on account of (I2-ryr)a(r):0,

(fr-zr)A:

- d(r) ;

0K,,(r) __UO,',r,rr,

(s2)

:escribing an anomalous g-factor of the electron.

aa da(r)

3r,

(37)

F*t(lt).

with the help of relations (26)

dA dac)

K,,(x):-.-+-.-

- | ar I at'6a9")lH(t').j,(x)fl)nn* -i,(x): 'JJ_ "*

-;

(3s) (36)

I

K,,(r- r') : -1D,(x),

(.i4)

the condition (33) is equivalent to

(both Kr and Ke are still functions of the invariant prlr). Since ,4u"*t and the current -/r""t generating- ihis external field are connected through

^t

J

J

(2s)

.{s exampleswhich wili be discussedlater on, we give: (a) The current induced in the (matter-) vacuum by :r external electromagnetic field lu"*t, or its source _: ert. The desired approximation is achieved with Sr :rd yields immediately: ',rx : i D

THEORY

0r,

6xu

which is indeterminate, too. fn momentum space,where K", is given by Kp(p\:

7 7 6(hxhxlmr) ---l 2k"k,- k"b, , d4h-:-,( . h x -p ^ ) r l m , (2n)rJ -k,p,-

6,"(- kxpxlkxkx*mr)l

(JSb)

this ambiguity is less manifest, since Lf

K,,(p).p,:-' dokku6(k^hx+nr) ( 2 r ) aJ

(39)

204 W.

AN D F'

PAULI

440

VILLARS

(hereby terms of the form 6(kxkx*n2)'(kxkx*m2) have been omitted), an expressionwhich may well be put equal to zero lor symmetry reasons' Since, however, (r,(p) is represented by a divergent Fourier-integral, this property may be lost in the course of a direct computalion of Ku,. Note that conditions (I,I a) are just those necessary to make the integral (39) convergent ! An evaluation of K", is most conveniently done with the help of the Fourier-integral representations:16

respect to a and B) with

5ot(p):216(kxkxlm2)

It is at this step that our regularization device comes by R(afp), acinto play, -to replicing expli(atfin*) by (9) and (11), and m'zexpli(a!ilm'f cording (l/i)Ri(q+p).v The regularizedEq' @2) reads then:

-

rt, dae:ytlio(hft1ln'?tl |

(40r

J_

1 ^(k): Phxkx*m2 :t f-

(P:principal

12

x*(n:

f

f

+fz,yJJ

e(a{B)

- .,,r),,,ru,. dadtstetu)

qA

f

ll

-2oF

x expli;P^P^+ i {" +F)nr' ll r* r,P,P, '

/ oA *,_ \ l (+z) +a,"(,"*rp^p^_ ;u) l.

xo,*,:

12

f

f

or2,yJ J

e(a*0)

dada(e,"\+'e(0 )("+Bf

,*r(oLt^o^){ff;r"*ur

l 1) l * ' o \ r @ , e x p [ r 0 ( [ ^ a ^ * n ' ) (- 4

t

7 (- p,p,-r6r,Pxe) - u,"l#il,erl rR(a* d)

value). After introducing a new variable

h,: k_ (F/"_til p

-iRl"+B)ll ri!@j9) qla

and wrur

Jl

f = (ir'?/a'\e(a) I d a f re x P l l a k ^ & n ) J

Expressedin terms of the variables z:a*g

y: tlz("- il, : - 6,,(12/ 2o3\e(a)' l,la k k,h " e-rp(iafr^A^) we are left

(after

symmetrizing

which give e(a)!e($:16(2) with

the integrand

16' lYl=r'

and 0 elsewhere;and dad'P:llz)dzd'Y

16S. T. Ma tPhys. Rev. 75. 1264 r lq49rt has evaluated.Ku,(p) bv means oi elemeniary momenlum-space lntegrauons, uslng tne ar" to Pauli and Rose (Phys'. Rev. 49, 462 (1936)) His due lo tie "i.if'.J results are neither gauge nor Lorentz-invariant. ot an additionaI constant lerm I- in his expression Ior oiiiin." kii ti: l, 2, 3). As a consequence, this term appears also )n the lrace of Kr,: -3PPPrKr+31' Kw:

Ka(!') reads: 12 f-t f+*dz f z I - e ( z )e x p l ; - ( l - y ' ? \ P x f x f K a,u",: | ay | r * 2 4(2n)al-, I l-,rP x l R(r)j(I2

whereas, according to (35) and (361 K-"u should be proportional 1o (papg). But it is easily shovn thrt the lntroductlon ol a regucalculations makes the addirional lerm r vaoish. i.t.i''ii'nir lndeed, we have

i

1-y2

r 1l -u,,I^a> ,xtx* n@-iR'(z)ll'(43) n

(xrrrn-{2r,,;,.,../'d,aar&rartttttff 4o**rr-r).

r? It may DerhaDs be helpful to show how a factorized,regulator a".trcyi gitige iniariance. Taking a discrele spectrum of auxiliary masses, we have:

It iollows for a time like vector (px: o,iPi :

: pe'(2r)-32, (K11)3 ",[r'ffi ffi, _zrz,t "2,c,!"ff

p,p,-l 6pPxPx)

aK n.- :2r',tlLi 14,,6'u 1 1,4 r'a,i*f i'' 'Ld+ -!'r,,*,o,o,"'r-araa-4/)

6x' [o,:(ft,+,4/r,7r].

It is this second term tlat destroys covariance and gauge invariance: but since it can be written as

- (r / (2r)3)D ; c;(I( - trM i')x -we see that it vanishes ou account of the two conditions

OXt

-,u,a,airi"'} : zc,c,t u ,;- .vtt . { MgA. ,dxp o,u, oxp ) ,--.-/Y

(I, Ia)'

L

which never vanishes identiellY.

e^!

)

205 I

l 1

RELATIVISTIC

OUANTUM

fhe contribution of the last term in the bracket may :e rvritten as rJ

o*1

_.a,"t

-':T)'

:inr,whi,chgives,

J-r

ttze(z)

.J-6

Ed{',:q/2T.m2/lkl

- 2in2

f',

ADDITIONAL RXMARKS

- ilf

r R(z) f

(A) In (38') we may be tempted to omit t}re term

o&uu*J-,0'\; )," tz,f'rol' (44)

-{ccording to our regularization condition (I,,a) this 'erm is equal to zero and we are therefore leit with an erpression that has the required form (compare (35) and (36)): K o t, o (?) : K r(p^!^) | p,! _ 6u,?xhl " rrhereK1 is given by: _ r, f+* dz

Et(P^lx\:

I / Ih I

which is exactly Wentzel's result,

":lY*'lfo-rt^^ll :

together with

J dsrlA,(*). A r(r)fnn-orn: for a photon of momentum I

n*-

,tyl

THEORY

-'htn p1

nr* J--

6G!'+*') re): f d,h ,t

(hy- p^)ry rnz(hxkxlnz)

which means putting A(r)(-ar!io)(O+mr\o)(r)) equal to zero, or its regularized counterpart: J

drp(*)tt*;K)(-!rA("(r.

()+d,r)(r; x)):0.

(47)

An evaluation of Ia(p) along the lines of the above calculations yields: 'l

f z y, f*tl_ fz I : f{t f+'dz -Y'tP^?^f' * J-,0';"*nl'-rr (4s)I *(p) - J-, o,J_. -
-l

a

1 J

.JQ

rr+* dz -e(ztR(z\ *iPxPxl dyyl

l-

!,G)R(z).

|

zTr_a

JIJ^Z

(46)

Z

x""vfio-rtr^o^f

T!i1 may again be expressed,according to (10) and 1 0 ' ) ,a s a

ni*

6r: : I dxp(x)toglxl. JTr _a

(46',)

The connection with Wentzel,s result for the photon self-energy is now most easily established. Since the ield of a light wave is not connected with any current, j,(r))t"o vanishes according to (37), unlessK' has rroi :he.required form (35), (36). From (43) and (44) it is nade clear that in this case the induced current is siven by (r,(r)),"o : - 4r". -!P' (2,)n

Q1A u,^d(*).

The photon self-energy,defined as (ph)

n

"",,:

-

f

i

J

dsxl(i,(x)t;^aA u , . d( r ) 1 ,n r -o n n

.lecomesthus: (phj

H

e2

r: - ( ",t

- i R'0) )

f

J

\

"L(:++e^p^)oaxo'otf

sives the charge renormalization (compare (37)):

6e:aexio':--

t*yz

l/_2

i l r l A , * d ( r ) . . 4 u " . d( r ) . l ,o r _o o n .

:

(48)

dtR\z) ffl f'f z - tl t^t^t l exvfltr J_,o,J_. dze(zt l: ]l

+f

f+^ dz

J--

f*r

^ 4 2 ) R 't( z ) dl y

Z'

,

dt

iz

| t\l -0 - f) p,p^ x;f r(/ t -..rLl) | :2R'(0)+0:0

on account of (II"a).

Thus it may be seen that the identity (47) holds only because of olr.r slronger regularization condition. It is if special interest to mention that it holds lor a zero vector (2r2r:0) only if the limit lrA+0 is carried through after rhe integration. Purting 2^p^:0 ;r.,71o;, (48), from the very beginninggives rise to an additional term

-

[_*-*10,yr14"1*1,1

Since.R'(0) is.zero, the regularized photon self_energy In consequence of this the non-regularized expression ','anishes,as it should; without regularization R,(-0) 1(0)- becomes then inf.nite, whereas 1a(0) is ziro only

20,6 W,

PAULI

iI we put:

Ilo,/,'),1,)^{"):o

442

F. VILLARS

AND

the If, in the second, divergent integral we introduce regu)ator sbecial (49) | /r \l

i 'l - - r l l R(z):1-exP

or

L

/a*p(*)*

t o g l ^ :l 0 .

(49')

\?@o

/ -l

(which satisfies(r"): R(O):o' but not R'(0):0)' obtain

-ma1d:,/-t'-t' -";J:'t''(*o(;'TlHYT,ffiffi:I;,ffilff11$'.*:i:l'# )-"" thecasep^pi:0 r. a limiting to consider morenaturat noo-r"ro vector' than to introducethe new and thus ;;.;"i;

we

\ ozd II ):E^tl;,\

'"?$'itil$?;itllt*'ltailbrieflvconsiderttrecase.or tm:3nal2r(lloell/"v'ol**)' ';;, (SIl resutt ^(1)(p), Schwinger's A(&), is ; which r" ,h:"ffi;;-J.rt".gy. 3fir,(j;l/J;" oo**_ reads: Aaa op"'"to. lt'" ;i;),li;(pi rn rhis L. by Feynman.T ,-t*:Hl:1ililT:'T*.t

5*:!_ J a^n6rn+*t

*' ) r+ 6(Drfrr) d((&r*gr)'*ru') a^:- ^') (##h*uttu'*;1i'^ ) (k^lq)'*m' "

p(x); aresuraror iffi)';ffi"fiff1"".1',.,1,,#j:'-duce

into This expressionis readily transformed

(tnt)n:

I

Ln (x), d.xp(x)

and

-)"-r["(?)'] . !4"1 *:# [_'_" [','ar3

-"" am(x)::J-,,"t-',', t rila r+t

f1,

"""o["(ir'+''.(?)') *illlxl'i"".J.1.1;"'lu3;"'1l"o,tJ;i:'::i,"x"li"i mass

qi'T:ii:,':;#1J1"''(il; (xisadimensionress i:t*'frlg:;*yil':""#*"',"'",i.J-i:iil:'i ,:i the here' in tr'"1*""'""'la"red x; it follows used;

'rft:'.nfk

R(z)pieviously

iT':fit"fril'ltti:*g

tha'i i,ffif thererore ;:-"*:'at.:;1fi::?'*":1 *"'"',-*&Tiil

#Jlp"na.n,of

-,,"*(i' +rr+(7)' ) o*(-,: + I.'o,1, l#ili#'*stu.n'"'x{;* "^::+i',,a-,t :# I-'*'-sx]-1)2.(1#;h !dr
-me :-.F(() 8r

- 11. or1.. * Y [ !
- @o t a,)fn a,a- rt r.r(7)' : s't''' /+'1

-liThl

of the evaluatton rltu"tion is exactlythe samein the.case

r(1 1-to*+r(x);*o(x)

F(-):16 x))1' "*--t+o(1"*-) With the helP oI one auxiliarY mass:

,ra:ua>-u(--Y) then which is sufficient to satisfy (I'), we obtain

4fJ."*ull"lf il"l#;:it;f ;:i:::x";-:?'i#"li:ii:"":: (which corresPondsto (49')) 2i ctM;tlogMi:0 is again consideredas the is made, unless this particular ese limiting care of a non-zero vector'

;"fi;;it;;

*:#("+.:)

which is FeYnman's result'

207 +43

RELATIVISTIC

The d radiative correction A, given by (31) may be written as

to the current, as

F(pxp) -. dnq,alp+q6,u1qt ( 2 r \ a JI 1 +G(?t/l)f

n

^, I a'q"1pa6rrra(q), "(2r)al

(50)

:ere u(q),a(q') are the Fourier-amplitudesof ry' and f respectively,and 1Ie"): + (1F7'-.y,^t p). Foi small values of p, i andG may'be developedin powers of 2r2r (note that I is the difierence in momentum ascribed to f and ry',respectively). Whereas F(0) describes again a charge renormalization, the term correspondingto G(0) exhibits an extra current, which, in r-space, has the more familiar form.(32), and de_ scribes the radiative correction to the ele^ctron,smagnetic moment in a homogenerius externalfield. From the well-known-decomposition of the current due to Gordonre izu(p! dlru(q):

(fDl is the bracket in (53).) Since [D] the two parameters g and q,, or on

depends on

and, Q:q,tr,, !:q,_,1 7", and V may be written as: " T : 6 o* F,P,I t+ Q ", ",1 "Q,I 2 y,:e"Jz.

ln

)j,(p):

THEORY

OUANTUM

!5. TITE MAGNETIC MOMENT OF TIIE ELECTRON

e/2m[(!u{2q,),i,(p* flu(q1 h.a(P*dttu^tu(q)l

All invariants thus introducedare still to be considered as functions of pr2r (the invariant prQr reduces to the with thehelpof (51) 3m"t*r913;-(4m',*?xpx))' a(q')I Qu(q):2ina(q')u(q), we obtain from (54) and (55): .iea

a(q)v'pu(q):0

;,t-zi*ltztr(p)-Jr(il)ha(p+(lnru^)a.(q)+-^/p \zrr ' thus yielding

.2me" C(^h,\:_t)r u\p^px.: - -\:tt(f)*Jz0))

and, accordingto (52): @: e/ar:1/137)

(st)

it is seenthat the radiative correctionto the masnetic moment may be expresseain terms ;i;; ;;;;";i;;" " g-factor

(ss) (55a)

as:-4m'/r(2lz(0)-Jz@))(o/")'

(56) . The computation_of the integrals involved in 1z and J: be carried through in difierent ways. A regular3"{

^g:- @m/e)G(o) (s2)ffi":;j:nT. , ,ll",ffi:"if,:ia,:1 i};;n"y"iiiTiT cluctionof a specialcoordinate system,characterizedb1,

sincetlre unperturbedelectronis character.izedlly g:2. The relevant terms of (31) (containingG(Apr)) can be rvritten as follows:

p:t), in which

e:fu,2im)

ff

Iz(0):r/ m'(TtrTu), Jz(0):(1/2m)vt' (s7) -(ie3/2) a'ga',1{'Q*t?"[D(6-?)S(f).rrSo1-r, J| J| The most convenientregularizationmethod is that *D(E-a15<')16;?,,S(-?) whichregularizes [D] asi*rr"r., (lDf)o:Lo colDfn +r(l)(f -?)S({):'-s(-dl"yV@+d

or,in momentum (writing space q, forp!q), ie3

-:-to'l (2")t

t1$:::ti:-",:L"f]#.lH..JflTjionoof

f

r:k/m,

I a,naS7.SSD61a,&-q)L;o(k-q,) J

*Dt1,(k)6.(k-q)L(h-q'tlu(q) (s.t, where 9,(k):2;p,1;n.p-m)-2i(qu,!qt)(it.k)l-tp.

10Compare Handbuch der Physik XXIV/I.

23S.

r ( 1 ' ? ( 0 ) ) R-:

f, J0

I?.-flrz 2.1_e.:rl,.r:r --" r'atI'''{;," }rss

(J,tI)JR:L *art,r*1. f" 2m2Jo o'2gl'"

Let us introduce

v": a,nn,So1. I

a;:(lfr:fp;?)l

we are led to:

+D\t\'t'(k-q)6,(h-q')

r,,: a'nn,n,1o1 !

pi:M;/nL, Q;:(f!p])i,

EQ.(57)in

(54) /.,^\

(ssa)

Equation (58a) converges without regularization and yields Jz(o): r/2tn' a v a l u e w h i c h r e m a i n su n a l t e r e du n d e r r e g u l a r i z a t i o n , 20A coritrol calculation, carried throuch rvith a resulator afiecting only the electron A-functions in [D] gave exacily the sme results.

208 F.

PAULI.{ND

444

VILL.\RS

since all terms i)0 vanish at least as cif Mi2 (compare Thus 1r(0) is immediatelyisolated: (6)). This is not the case, however, for (58), which ftf becomes / r ( U r : . 1 | d u u r l , , , r , t ' k 6 " t k \ k ^ +u i t u : l l l l ' !t

a,Q) ^: #,(zZ,c,- t p',,c,). (se)

The expression(58) proves thus to be convergentwithe e e do n l y p u t r o : 1 , c , ( i ) 0 ) : 0 , out regularization;wn to obtailr I"(O):r/6m2 rvhereasthe properly regularized expressionyields

(1,(0)) /8m'. ": " 2IzQ)- JzQ): *r/6nt'

Therefore rvhereas

J

Jo

r

-

n

/.,\

.+u( :- I \M,!/" , 'r

M .2 8n2

The use of a regulator appears to be superfluousin this case, but from a general point of view it is doubtiul whether a translormation of variables as needed to obtain (62) can be justified a priori. (B) [r] may, accordingto Dyson, alsobe rvrilten as -lRe(D

2(1,(0))R- l,(0) : - r / 4n 2.

1

/"1

: 1,. I duu,_ Ja nt'tr!| 1

"(h)L"(k-

q)t"(k- q'))

where

The regularizedvalue, introducedin (56)' gives

: L F 2 iL : 2 16+ ( h 'I| n ? ) ' ( a n d D : D t t r- 2 i D ) ' " "(k) integralion, (complex) ,t-space With the elementary however, one immediately falls back on the formula

L

ag:

"/" in agreementwith Schwinger'sresult'21 From this example it should be made clear horv cautious one should be in handling divergent "covariant" expressions.Indeed, any decomposition, as done in (55), is by no means "covariant," as long as the expressionsunder discussion has not been made convergent by some invariant regularization, Once this has been done, one may quietly enjoy of all the facilities presented by a properly chosen coordinate system, as rvasdone in (57).

(s8),(s8a). (C) Finalty rve illustrate the possibilitl'of obtaining a quite arbitrary result by regularizing.in-a difierent rvay difierent terms of a form F(A, A(r)) Let- us just taie, as an example,the bracket [D], rvhich lor q':q may be written as follorvs: 1 6(A'Ai) '"-6'tk^k^-)k^'7rr:f/t']|[/'t,1 IDf -' (2kxqx)'z kxhx

ADDITIONAL REMARKS (A) The various possibilities oI evaluating the integrals 1r and -f2 arise from the difierent possibilities of rewriting the [D]-bracket in (5a), (5aa). It is easily seenfrom the definitions(40), (41) thattor q:qi,lDl may be written as fr

lDl:

l duu6"(kxhx-2hxqxu).22

(60)

vn

The regularization of [D], as a whole, as was done in the above calculations, consists then in replacing (60) by ^l

(l Df)o:

I d u u L ; r ; 6 "t h r h t - 2 h g p a M ; 2 t . 1 6 1

I

We need now only nole that if Rr and -R2are tw{) r e q u l a l o r sl .h e o p e r a t i o nf t : ] r R ' { R 1 t i ' a r e g u l a t o r t o i , * h e r " u . F : i 1 R , - R , r i s a n o p e r a t o rc,o r r e s p o r d ing to a mass spectrum rvhich satisfies(I' Ia)' but co;tains only auiiliary massesMd; rve mav norv rvrite Rr[Dr]*Rr[D:]

: R[D]+E([r'1-

[l'l)

T h e a d d i t i o n a lr e r m sd e p e n d i n go n , Qa . " b y n o m e a l t s zero in general, but depend on the structure of 'R which shall be characterized by a spectrum Mt and coemcients7;. In evaluating 1r(0) (!1), (55), as an example,the additional terms due to ,R are: rf 4m2(ltttu,? logpo-?),

Equation (61) may now be introducedinto (54):

an expressionrvhich is completely indeterminate'

f\?

t r,6"\hfty-)kvqxu*M,2 (T,,to: I auu I aonn"n,2, J Jn f\f

: I auuI d,krk,k,+u,q,q,t J Jn -i

XIr

rnNuz*M,2). (62) cr6"(,4r,4r*

. e v . 7 3 , 4 1(b1 q 4 8 / . . h 1 sR I . S . l * t i n * " rP r ;t l [ , . " i h i s m e r h - oi sdd u er o S c h w i n g er S

ACKNOWLEDGMENTS We have to acknowledgeProfessor J. Schwinger, Dr. F. J. Dyson, ProfessorV. F. Weisskopf,Dr' S' T' NIa, and Dr. K. M. Case{or making their publications accessibleto us prior to publication; Dr. G' Rayski for valuablehelp, and ProfessorE. C. G. Stueckelbergand Dr. G. Rivier for interestingdiscussions.

On Gauge Invariance and Vacuum polarization Juuel Scnwrwonr E arwrd U niorsity, Canbridgc,M ass@huselts (ReceivedDecember22, 19.50) This paper is bdsed on the elementary remark that the ex_ traction of gauge invariant results from a formally gauge invarianl theory is ensured if one employs methods of i't ut inrot* "otutiin only gauge covariant quantities, We illustrate this statement in connectior with the problem of vacuum polarization by a pe scribed eleclromagnetic freld. The vacuum current of a chaiged 6eld..whichcan be expressed in terms of the Green,. iun.fion lirac o r r n a t n e t d , l m p l r e sa n a d d i l i o n l o t h e a c l i o n i n l e g r a l o I the elec_ tromagnetic field. Now these quantities can be related to tJle dynamical properries of a,.particle,, with space_lime coorainui"s that depend upon a proper_time parameter. The proper time e q u a l i o n s o [ m o t i o n . i n v o l v e o n l y e l e c t r o m a g n e t i c6 e l d s t r e n g r h s , ano provlce a Sultabte gauge invarianl basis for trearing problems. Rigorous solutions of the equations of motion."n t oii"ln"Jlo" a constanl field, and for a plane wave 6eld. A renormalization " of hetd strength and charge..applied to the modilied lagrange func tron tor constanl fields, yields a 6nite, gauge invariant iesult which implies nmlinear properties for the electromagnetic 6eld in tlre vacuum. The contribution of a zero spin charged field is also staled. Afrer the same 6eld. strenglh renormalization, the modi6ed pnyslcal quanilues descnblng a plane wave in the vacuum reduce to lust those oI the maruell 6eld; there are no nonlinear phenomena lor a single plane wave, of arbitrary slrength and spictral com_ position. The results obtained for constant rihat is, slowJy varying 6elds), are then applied to treat the two-photon disinresrutio';-o?

I. INTRODUCTION electrodynamicsis characterized by QUaNfUU -<' severatrormat lnvanance properties, notably rel_ ativistic and gauge invariance. yet specificcalcul;tions by conventional methods may yield iesults that violate these requirements, in consequenceof the diversences inherent in present field rheories.Such di_fficultieicon_ cerning relativistic invariance have been avoided bv employing formulations of the theory that are explicitly invariant under coordinate transformations, a;d b; maintaining this generality through the course of caiculations. The preservation of giuge invariance has apparently been considered to be a more formidable task. It should be evident, however, that the two pro-blems are quite analogous,and that gaugeinvariance difficulties naturally disappearwhen methods of solu_ tion are adopted that involve only gauge invariant quantities. We shall illustrate this assertion by applying such a gauge invariant method to treat several ispect=sof the problem of vacuum polarization by a prescribedelec_ tromagnetic field. The calcuiation of ttie current asso_ ciated with the vacuum of a charged particle field involves the construction of the Gre;n,s function for the particle field in the prescribed electromasnetic field. This vacuum current can be exhibited is the variation of an action integral with respect to the potential vector, which action effectivelyadds to that of the maxwell field in describingthe behavior of elec_

a spin zero neutral meson arising from the polarization of the proton vacuum. We obtain approximate, gauge invariant ex_ pressrons tor lhe eHeclive interaction between the meson and the electromagnetic field, in which tie nuclear coupling may be scalar, pseudoscalar, or pseudovector in nature. fhe aiiect verification of equivalence between the pseudoscalar and pseudore.lo. inier acuons onty requlres a proper slatement of the limiting processes involved. 'For arbitrarily varying 6elds, perlurbation m""riod. can De apptred to the equations of motion, as discussed-in Appendix A, ot one cal employ an expansion in powers of the poiential vector. The.latter a_utomatically yields gauge invariani results, provroed onty tiat the proper-time integralion is-reserved lo lhe last. This indicates that the significant aspecr of the proper_time method is its isolation of divergences in inlegrals with'respecr to the proper-time parameter, which is independent of the coordrnale system and of the gauge. The connection beiqeen lhe proper-time melhod and the technique of ,,invariant regularization" is.discussed. Incidentally, the probability of actial pair creation is obtained from the imaginary part oi the electromagnetic held action integral. Finally, as an application of the Gieen.s lunctlon tor I conslant field, we construct lhe mass operalor of an a. weak, bomogeneous exrernal 6eld, and derive rhe T "1::1::" aoqrronat sprn magnetrc moment of a/2n magnetons by means of a perturbation calculation in which proper-mass pbts the cus_ tomary tole of energy.

tromagnetic fields in the vacuum. We shall relate these p r o b l e m sl o t h e s o l u l i o no f p a r t i c l ee q u a t i o n so f m o t i o n wrtn a proper-tlmeparameter.The equationsof motion, w h i c h _i n v o l v e o n l y e l e c t r o m a g n e l i cf i e l d s t r e n g l h s , provide the desired gauge invariant basis for our discusslon. Explicit solutions can be obtained in the tvrlo situations of constant fields, and fields propagated with the speedof light in the form of a plane wave.r For constant (that is, slowly varying) fields, a renormalization of field strength and charge yields a modifred lagrange function differing from that of the maxwell neta Uy terms that imply a nonlinear behavior for the electromagnetic field. The result agrees precisely with one obtained some time ago by other methods and a somewhat difierent viewpoint.t The modified physical quantities characterizing the plane wave in the vaiuum revert to those of the maxwell field after the same field strength renormalization. For weak arbitrarily varying fields, perturbation methods can be applied to thi equations of motion. This will be disiussed in Appendix A. The consequencesthus obtained are useful in connection with a classof problems in which gauge invarir-fhal

the Dirac equation can be solved exactly, in tle 6eld of

*"r", was recognizedby D. M. Volkow, Z. phvsik 94, 25 irB.!-"ill Heisenberg and H. Euler, Z. Physik 98, 714 (1936). -.2_W._ V. Weisskopf,Kgt. DanskeVidenskab.Selsliabs.Mat.-Iys.Medd. 14,No. 6 (i936).

664

2ro 66s

GAUGE

INVARIANCE

AND

VACUUi\{ POLARIZATION

ance difficulties have been encountered3-the multiple and 1 , ro)'ro' p h o t o n d i s i n t e g r a l i o no f a n e u t r a l m e s o n . W i t h o u l /] e\ €,(-r-- ' r. - ,),: 1 [ \z'd) i u r t h e r e r t e n s i v ec a l c u l a t i o nr, v e s h a l l o b t a i n a p p r o x i L Ir ' ae( - #s'| ' mate gauge invariant expressionsfor the interaJtion of we have u ,"roipiir, neutral meson t"i;h ;*; ph;;""q *tt"t" ttt" *o)*o' o r t h e i n t e r m e d i a l en u c l e a ri n l e r a c t i o nm a v b e s c a l a r , /( rv." (tr-i \t .rir.t-lt ^ . ,r1€ ,\ ,",ro , l ) . /\(.r-- r - , '\:- 1 { ' " t x l ' I a ? ' ) ' e q u i v a l c n lp s e u d o s c a l a rn d p s e u d o v e c t o. o, r p l i n g s . ro<;"0,. \ -Ou6W"fd, The utility of the proper-time. technique to be ex- There{ore ploited in this paper, apart from its value in obtaining (2.10) rigorous solutions in a few special cases,lies in its Ll{"(r),0e@)):(,1"@)0e@'))4@-r')lt-", isolation of the divergent aspects of a calculation in integrals with respectlo. the proper-time, a parameter provided one takes the average of the forms- obtained thaimakes no referenceto the coirdinate sysiem or the by letting r'approach r from the future, and from the per- past. The quantity o{ actual interest here is the expecgauge. Incleed, rve shall show that the customary -the tation value of lu(r) in the vacuum of the Dirac field, iurb"ation procedure of expansion in powers of potential vector does yield gauge invariant results, (2'11) (j-(*)):ie trl'G(*' r')ft"' provided only that the proper-time integration i, , , wnere reserved to the last. The technique of "invariant (2'12) G(x' r'):i(('l'@){(:c'))+)e(tc-r')' regularization,,arepresentsa partial realizationof this proper-time method through the use of specially and tr indicates the diagonal sum with respect to the rveighted integrals over the conjugate quantity, the spinor indices. square of the proper mass. The function G(r, r') satisfies an inhomogeneous Finally, in Appendix B we shall employ the Green's difierential equation which is obtained by noting that function of an electron in a weak, homogeneous,external field to calculate the seconi-order"electromag- ltGia-eA(x))-lmfG(*r'). (2'13) :6rl{(t),{(x')l}6(ro-ro')' netic mass, thereby providing a simple derivation 6f the second-order correction to the electron magnetic where the right side expressesthe discontinuous change moment' in form of G(x, r') as lrois alteredfrom ro'-0 to ro'*0' According to F'q. (2'2)' therefore, we have II. .ENERAL THEoRy (2'14) l7(ia-eA(r))tmfG(r'r'):6(r-*'); The field equations,commutation relations,and current vector of the Dirac field are given by5 . / \\ '/ \ , ,/ \ ^ 7:l-'o^'^-fi{l.:.lY'-,':.f,Y:l:"' \tdA-eAp\r))v\xt'futmv/\r):u' {/(r, ro), {(x',*o)l:'yo6(x-x'),

j,(x):iel{,(x), u{G)1, where *bu, t"\: and

'to: -ila,

(2.1 (2.2)

(2.3) *il','JtJ.ffilffregard element G(r,r,) asrhemarrix

- 6.

(2'4)

'to2:1.

(2..5)

The structure of the current operator' ju@):-rQ)uil*"(i,0e@1,

(2.6)

which arises from an explicit charge symmetrization, can be relatedto a time symmetrizationby introducing chronologically ordered operators, Thus, with the notation IA(xo)B(tct'), ro]ro' (z (x6)B(;ri))+: { lB(ro)A(ri, xo
that is, G(r, r') is a Green's function for the Dirac f i e l d . W e s h a l l n o t d i s c u s sw h i c h p a r t i c u l a r G r e e n ' s function this is, as specifiedby the associatedboundary c o n d i t i o n s ,s i n c e n o a m b i g u i r y e n t e r s i f a c t u a l p a i r creation in the vacuum doesnot occur, which we sha11

of an operator G, in which states are labeled by spacetime coordinates as well as by the suppressedspinor indices: G @ ,x , ) : ( r , l c l * , ) .

(2.15)

The defining difierential equations for the Green's function is then consideredto be a matrix element of the operator equation (2'16)

QIItm)G:l' where TIP: Pr- eAu

Q'17)

is characterizedby the operator properties (2.7)

3 H. Iukuda and Y. Miyamoto, Prog. Theor, Phys. 4, 347 ( 1949). aW. Pauli and F. Villars, Revs. Modern Phys.2l,434 (1949). 5 We employ units in which i:c:1. Note lhat also '!,:,1'170, since a67',-p:6, 1, 2, 3 form hermitian matdces.

ltcorl"):i6,,,

lr,,tl,1:;tpu,,

(2'18)

o"Au

Q.19)

and F,,:6uA"-

is the antisymmetrical field strength tensor. With this symbolism, it is easy to show that the

2tl JULIAN

SCHWINGER

666

vacuum current vector,

In virtue of the vanishing trace of an odd numberof (2.20) 7-factors, we have s obtained from an action integral by variation of ieTry6AG -{"(:r). This is accomplishedby exhibiting f* (j,(r)):i.e tryu@lGlr),

: -T16(zII)riI

J,

f

6W(n-,

@t)6A,(.r)(j,(x)):ieTry\AG

J

(2.21)

i1*.- (-yII)')sl (2.31)

(2.22)

and Tr indicates the complete diagonal symmation, including spinor indices and the continuous space-time coordinates.Now

"rO7-

l. f. I :t+tJ dss-rexp[-i(2,-(rn)')']1,

as a total difierential, subject to 6lu(r) vanishing at infinity. In the secondversion of 6Ul(1),d,4! denotes the operator with the matrlr elements

(*l aAu!x'S: 51*- *') 6Au@),

dt

which again involves the fundamental trace property (2.26).Thus,

sttt(x): liJ

f'

dss-rexp(*izr,s)tr(r.lU(s)l.r), (2.32)

I/(s): e*01-;*tr,

-e764:6(7nlm),

(2.23)

and

where I

G:

'y*l\+. nl

:i

JC: _ (?[)?: II!2_leo*F*,

f'

I dsexp{- i(7n*rr)sl, ro

Q.24) and ar,:iillu,

so that

y").

(2.33) (2.j4)

We now see that the construction of G(x, x') and 3{tr(r) devolves upon the evaluation of

ie Tr16AG f f:61 , I d s s - ,T r e x p { - i ( 7 n * z ) s l Lro

(r' I u (s)| r''): (r(s)'|{0)'')

l l, I

(2.3s)

( 2 . 2 s ) The latter notation emphasizes that U(s) may be regarded as the operator describing the development of a system governed by the "hamiltonian," JC, in the in virtue of the fundamental property of the trace, "time" s, the matrix element of U(s) being the transTrAB:TrBA. (2.26) formation function from a state in ri'hich 1:u(5:0) has the value ru" to a state in which ru(s) has the value *r'. Thus, to within an additive constant, Thus, we are led to an associateddynamical broblem i n w h i c h t h e s p a c e - t i m ec o o r d i n a t e so f a " p a r t i c l e " e' lV(Lt: i dss-te-iru Tr exp{ - iylls} depend upon a proper time parameter, in a manner I Jo (2.27) determined by the equations of motion dr,/ d.s: - ilt u, lCf : 21,, dll,/ds: - ilIlu, lt?f: e(F,"n"*n,F,,) * Leox,(aF x,/ dr) : 2eFu,rr, _ ie(dF,,/ ar,) | leox,(aFx,/0*,).

f

:J (dx)srr)(r), where the lagrange function Sttr(r) is given by f*

3(')(r): i I

drr-'.-t'" tr(rl expf- lTns|| *).

(2.28)

An alternative representation, and the one we shall actually employ for calculations, is obtained by writing G: (- tII-l n)lnz-

(:.,tt)zf-l :lm,(1Il121-tG tII*

: i t

(2.29)

id"(r(s)'j *(0)"): (r(s)'1rc (2.37) lr(0)"), (- i3,'- eAuQc')(r(s') | o(0)") : (r(s)'lnu(s)lr(0)"), (2.38) (i,0," - eAu(x")) (r(s)' Ir(O)" ) : (r(s)'ln"(0)| r(0)"), (2.3e) and,theboundarycondition

f'

C: (- tIl{. nt)i I uo

n)

(2.36)

The transformation function is characterized by the differential equations,6

ds exp[-i(zr,-

(rl])!)sl

d se r p [ - i ( 2 , - ( ? l t ) ! ) s ] ( - 7 r t + r n )

(r(s)'Ir(0)")l--6 : 6(r'- x"). (r.JU)

(2.40)

5 A proper time wave equation, in conjunction rvith the secondorder Dirac operator, has been discussed by V. Fock. Phr.sik. Z. Sou.jetynion 12, 4O4 tt937). See also y. Nambu, piog.-Theor. Phys. 5, 82 (1950).

2r2 INVARIANCE

GAUGE

667

We shall now illustrate, for the elementary situation Fe,:O, the procedure which will be employed in the following sections for constructing the transformation function. The equations oI motion read

POLARIZATION

and the Green's function is obtained as r' i I d,sexp(- im2s)

G(*', *"):

-

Jn

X ( * ( s )F 'l illm)l*(o)")'

(2.41)

drr/d,s:2l7,,

dllu/ds:0,

VACUUM

AND

: (4r)-zO1rr, *r'1

whence (2.42)

II,(s):tI,(0), and (r,(s)-*"(0))/s:2II,(0).

(2.43)

Therefore

f'

1 Jo

dss-2exp(-iza'?s)

/ (-"x ' - r " ) \ [ ( * ' - r ' ) ' l (2.s4) +m exvliL----]. ( ) " An equivalent,and more familiar procedure,is to employ the representationlabeledby the eigenvalues of IIr. Now (II(s)'I n(0)") : (n (0)'I U(s)| n (0)") :6(tI'-II") exp(-i["s), (2.55) "

Jc: il2 : +s-2(tt(s)- *(0))'?: ts-'?[c'z(s) - 2r(s) r(0) I r'z(0)l* is-'?[r(s), r(0)] : f, s-zlrz (s)- 2x(s)r(0) * r' (0)I - 2;'-r,

(2.44)

since - 2;' 5r,, 12.nt, [*,(s), r, (0)] : [*u(0) * 2sru(0), *"(0)l : Having ordered the coordinate opera,torsso that *(s) everywhere stands to the Ieft of *(0), we can immediately evaluate the matrix element of 3c in Eq. (2.37), thus obtaining id,(*(s)'I *(0)") : lf, s-2(r' - r'')z - 2i.s-tf(x(s)' | *(0)"),

while(r(s)'ltI(s)') is determined by (_ i6,'_ eAu@,))(r(s), il(s),) I I]n,(.r(s), I n(s),), (2.56) and the normalization condition

(2.46)

J

the solution of which is (r(s)' I r(0)//) : 6:(r', r") s-z explil@' - r")2 / sf . To determine the function C(r',r"),

Q.al)

we note that

(r(s)'ln,(s)|r(0)"): (r(s)'ln,(0)|,(0)")

: ((xrt_ r,t,) f 2s)(r(s),1*(0),,),

(nG)'l*G))(dr')(#(s)'lr(s)"):6(II'-n"), (2.s7)

to be (r(s)'lnG)'j fr''l :(2n)-2 explieI

(2.48)

dxA lexp(ix'Il'). (2.58)

LJo

J

which, in conjunction with Eqs. (2.3S) and (2.39), Therefore, implies that

(*(s)'l*(0)")

(-i.\u'-

eA,(r'))C(r' , x") - eAF(fi't))c(r', r"):9, : (i,aptt

:

12.491

or C(tc',x"):941*',

*"1,

fr''l Q(r', x"):eYP1i, I I

(2.s0)

.

Jn"

[

f

{a*) exp(i!x2/ s): | ;

X expli(r'-*")Il' -iII"i],

-

The line integral in Eq. (2.50) is independent of the integration path, since F",:0. Finally, the constant C is fixed by the boundary condition (2.40). It is evident that Eq. (2.47) does have the character of a deltafunction as s approacheszero, provided

c r-,

(r(s)'I uG)')(drl')(n(s)' Ir(o)") X(dr')(r(0)"lr(0)")

: (2r)-4A(*', *"1 6n, J

d*1,@) I' J

J

(2.sr)

and

(2.59)

G(x', r'' ) : i(2o)-a1 (r', *") f@f

X I ds | @n') expli(r'-r")fr') JoJ X(-

tn' * nr) expl- i(IJ'2!nt')sl,

(2.60)

which reduce to Eqs. (2.53) and (2.54) on performance of the rI'integration.

that is, C: -i.(4r)-2.

(2.52)

Therefore, (r(s)' I *(0) ") : - i(4r)-2y (x', x'') s-2 Xexplif,(r' - x")7s1,

III. CONSTANT flELDS The equations of motion (2.36) here simplify to

(2.53)

d.*,f ils:2IIu,

d'Ilrf ds:2eFu,Il,,

(3.1)

213 TULIAN -

-

*"

+.i-

n^i.

The solution of Eq. (3.15)has the form

f i^h

dr/ ds: 2n,

dfi/ d.s: 2eFll,

(3.2)

c(r,, x,,)

- -.: s1-mbolicsolution of theseequations

--:f ce

II(s): e?'r"[(0)' r(s) - r(0) : [(e?"F" - 1)/ eFIII(0),

fa"'l

:C(x") expfie I d.r(A(x)llF(x-x" t) l, (3.17) I L J",.

(3.3)

, . . i : e F ( e 2 e F s _1 ) - t ( r ( s ) _ * ( 0 ) )

:leFs-"r" sinh-r(eFs)(r(s)-r(o)), (3.a) :: I -. it : leF e"r" sinh-l(eFs)(r(s)- r(0)) : (r(s)-r(0))|eF€-eF! sinh-r(eFs). (3.5)

in which the integral is independent of the integration has a vanishing curl' path, since Ar(*)*iFu,Qc-r"), However, by restricting the integration path to be a straight line connecting x' and *", we may, invirtue of Eq. (3.6), simply write CQt" xtt):941'', *'r1,

F:-F,

(Fu,:-F,,).

(3.6)

',i

(*(s)-x(0))K(r(s)-r(0)), 1 1 : r n e z Fsz' n } - r ( ? F s ) .

and, with C a constant, attain the solution of (3.15) and (3.16).The constantC has the value /\ ? " " 7/ \

l: rearranging the order of these operators, the fol.:ring commutator is required: -t _:rs), r(0) I : [r (0)t QF) (sz"t " t ) u (o), r(o) ] :.i(eF)-t(e2eF3_1). (3.8) - hus

F: r(s)Kr (s)- 2r(s)Kr(0'#,:lfiij?,,, :-_i t eo

(3'10)

.rhich follows from Eq. (3.6). The resulting differential : q u a l r o n( l . J / , , j a , ( r ( s ) ' Ir ( 0 ) " ) : l - l e o F \ ( x ' _ r " ) K ( r ' - r " ) -li treF coth(eFs)l(r(s)' I t(0) "),

(3.11)

ras the solution .r(s)'lr(0)"): f (r' , r")q-tru't5-z Xexplf,i(r' - x") eF coth(eFs)(x' - r")l .exp(ileoFs), (3.12) Z(s):I

C: -i'(4r)-' since the limiting form of (r(s)'l*(0)") independent of the external field. Finally, then

(3.19) as s-0

(r(s)'| *(0)"): - i(4r)-2a(r' , r'/)e-L(s)s-2 Xexplil @'- r" ) eF coth(eFs) (r' - tc")f .explileoFsf,

(3.20)

the Green's function G(x', r") is obtained from (2.30) in the two equivalent forms, f*

G(r', r"):;

I

d se x p ( - i z ' ? s )

.lo

X[-r,(r(s)' ] n,(s)| *(0)") lmQc(s)'lx(0)")f r' : i I dsexP(- iz'?s) r0

x[- (f(s)'In"(o)l'(o)")r, Fm(r(s)' l*(0)")1, (3.21) which will be given explicitly on substituting Eqs. ( 3 . 1 3 ) ,( 3 . 1 4 ) ,a n d ( 3 . 2 0 ) . The lagrange function 3{tr(*) is now computed as

tr ln[(eFs) t sinh(eFs)].

To determine C(x',r"), we employ (r(s)' | rl(s) I r(0) rr) : lleF coth(eFs)f eF] f' X ( r ' - r " ) ( r ( s ) ' I r ( 0 ) " ) , ( 3 . 1 3 ) A0)(r):+i I drr-t exp(-int2s) and X tr(r (s)'Ir(O)")lt' t' -" coth(eFs)- eFl (rG)'l n(0) I r(0)'t):lleF X(r' - r")(r(s)' Ir(0)"), (3.la) n' : (l/32n') | drr-t exp(- iro'?s) ir conjunctionwith Eq. (3.12),to obtain the differential .ro equations xe-LG) tr exp(ileoFs). l- id,' eAu(r')- leF,,(r' *tt),]C(r' , r"):9, -\eFu,(x' - x")"]C(*' , r"): g. li6u" eAu(r")

is

(3.e) and

Tllere tr again denotes a diagonal summation, and we :-ave employed the fact that tr(F):6'

(3.13)

lr''l o(.r', .v"): expl ie I dxA(t) I' I L J,,,

-,re latter form involvesthe fact that

e now consider 'tlleoF:T12(s):

668

SCHWINGER

(3.15)

(3.22)

We may exhibit this more explicitly as a real quantity (3.16) by a deformation of the integration path, which is

214 669

GAUGE

INVARIANCE

AND

e f f e c t i v e l yt h e s u b s t i t u t i o ns + - i s : - -(l/32r') "C(i)(x)

f'

|

From the eigenvalue equation

and its equivalent accordingto Eq. (3.35), (3.38)

Fp,+'l',:-(l/F')g,l/r, (3.23)

l(s) : + tr In[(eFs)-r sin(eFs)]. Indeed, we could have initially employed the integral

we obtain by iteration: F,xFx,{,: (F')'{,,

(3.39)

F,s*Fx"*{": (l/ (F')')9"{,.

The identity (3.36) then yields the eigenvalue equation

t2 finn

n* l m z - 1 1 7 t 1 2 1 - r : I d se x p [ - ( r r ' - ( y l l ) ' ) r ] , uo

LIau,, ox,l: 6116,.-6r.d,r*leu,l.Ts,

rvhich has the solutions+F(r), +F(2), with pru:(i/t2)l(5-ttg)++(s-tg)rl, Fe': (i/\2)l$+ rg)r- (s- tg)rl.

sin(aFtrrs) e-I(s): (es)zF0)F{2)/sin(eFcr)s) 2(es)2F(t)F(2)

(3.25)

Fr,+:liep,x^Fxu

(3.27)

we have (!orF )2:

lF *21!7 5F,,Fp*,

(3.28)

thus introducing the fundamental scalar

5:lp*,,:l(H,-E ),

or e-tG): (es)29/ImcoshesX,

g:|FPFP*:E.H

(loF)'-2(sl76g),

f (r): __"

1r* I

dss-3exp(- rzrs)

6T. Jo

(3.2e)

Re coshrsX I r xl (eslzg_---t -Im '' L coshesX J

(3'44)

(3.30)

in which we have supplied the additive constant necessary to make 8(1) vanish in the absenceof a field. The first term in the expansionof 8(r) for weak fields is (3.31) e2f*

it follows that loF has the four eigen(+oF)': +(2(ff+ig))+.

(3.43)

where Im designatesthe imaginary part of the following expression. The final result for J(1) is

and pseudoscalar

constructed from the field strengths. Since

, (s.42)

- F('z))- coses(F(1)+F(2)) coses(F(r)

(3.26)

and e'r* is 1, or *1, if (prtrr) forms an even, or odd permutation of (1234), and is zero otherwise. In terms of the dual field strength tensor,

t\ 2 r l1 \ J rA

Expressed in terms of these eigenvalues,

where 'ts:i'Yr'Yz'ts'tt,'Ys2:-7,

(3'40)

(F')aj2s(P')z-gz:9, (3.24)

which exists in consequenceof the restriction on real pair creation. This, horvever, would have obscured the proper time interpretation. To evaluate the Dirac matrix trace, we employ the following spin matrix property:

and |uz: -1, values

(3.37)

FyQ,: F,{r, d s s - je r p ( - r ' s ) ye-i("r tr exp(!eoFs),

renrccpn

POLARIZATION

VACUUM

J(r,-__

| drr-' exp(_nrs)S. l2r2 Jo

(i.45)

(3.32) On separating this explicitly, and adding the lagrange function of the maxwell field,

Therefore, : 4 Re coshes(2(S+ig))l tr exp($eoFs) =4 Re coshesX, (3.33) where Re denotes the real part of the subsequent expression. Note, incidentally, that

x,: (H+iB),.

(3.35)

and &r*Pr,*-FurFr":26p,5.

(3.36)

(3.46)

we obtain the total lagrange function

(3.34) c :

The eigenvaluesof the matrix F:(Fr") are required for the construction of exp(-l(s)). They can be obtained with the aid of the easily verifiable relations, FlrFr,*: - D'9,

sro: _s:|(Ez_H), fe2r'I

-11+-

J

1r-

|

d s s' e x p ( - z ' ? s ) l $

dss-3exp(- zzs!

6T r'o

Re coshesX f 1 1- !(es)'?r Xf (es)'-l. (3.47) I L Im coshesX

215 JULIAN

l:,e logarithmically divergent factor that nultiplies the --::well lagrangefunction may be absorbedby a change :, :cale for a1l fields, and a corresponding scale change, :: renormalization, of charge. If we identify the quan: :ies thus far employed by a zero subscript, and intror:ie new units of field strength and chargeaccording to 5 I ig : (l

Ceo')($o*r9o),

ol the form (3.48),with and a renormalizalion C:Co+C\ lr' :_

+_

- 1- 3(es)'s] X [(es)'9(Re/Im) lf'

+_ | dss-3exp(_p2s) 7612Jo X [(es)'?g(1/Im)- 1*i(es)'?$]

Re coshesX r I '9-1 - $ ( e s ) ' ? rI x l ( e s )-Im I L coshesX

2az (h/mc)3 ' "

: ; ( E , - H 1' + -

2a2 (h/rnc)3

:+(Er-H,)+4J

"

In the latter expansion, the conventional rationalized units have been reinstated,and q:e2/4nhc. Incidentally, the addition to the lagrange function produced by a spin zero charged field is obtained Jrom Eq. (3.23)by omitting the Dirac trace,and multiplying by (-2). Thus'

*_

--

' _(E,_H)'+(D. I . .. (3.s4) H)'l+. l J

Tn: 6*t,- (6 e/ d&r)F,r : - (F,rF,r- 6u:4F x,') (d S / Au)

+d,,(J-s(dsla$)- gat'/ ag). (3.ss)

The maxwell tensor d s s - 3e x P ( - P z s )

I

(3'56)

Tula):P*P,^-Ur,rO^*'

Jo

is obtained from J: -5, the weak field approximation of Eq. (3.49). The next terms in the expansionof 3 yield

in rvhich p designatesthe mass of the spinlessparticle, and an additive constant has been supplied as in Eq. (3.4.+).The first term in the expansion for rveak fields is separatedexplicitly bY rvriting fou):

[(E2-H')'+7(E.H)']

90 K2L4

ko's -,.l, (3.s0) "fLIm coshesX I

ez

mc2-

The physical quantities characterizing the field are comprised in the energy momentum tensor

lr' IoT.

'

45

a2 (h/pc)3t7

mC"

x[(8,-H)'+7(E.ry'1*. . .. (3.4e)

€ " p i no ( r ' : -

dss*3exP(-az'?s)

|

6T. Jo

d s s - te x P ( - z r ' ? s )

I Jo

6T"

| dss'exp(_z,s), (3.S3)

1r-

f : -$--

dr r-t exp(-nrs),

'.'eobtain the finite, gauge invariant result

1r*

lr'

l2r2 Jo

yields

lr-

- : - 5-:"

dss-r exp(-g2s)

| 48rz Jo

(3.48)

e2: eo2/(1lCeoz), C: * | LZT'JO

670

SCHWINGER

f-

| 48n2 Jo

* -

dss texp(-prs)S

toT'

16 (h/mr)3 \ r ) 4i"' ,*, 2 (h/mc)a _ 6 u ,_ d z : _ _ , : ( 4 s , + 7 g r ) + . 4J mC'

exp(-p2s) f- dss-3

F.:

JuF(l),

f:n,*-

till.n.r-r+lr^)'s] (rs1)rvherezu is a null vector,

LIm coshesX

nu2:0,

If we take into account the existence of both spin 0 and spin ! charged fields, 8: C(o)+So(D+81(D,

. ..

(J.57)

TV. PLANE WAVE FIELDS A plane wave, traveling with the speed of light, is characterized by the field strength tensor

J o

"f.

/ 7,,:T,,M\1-

(3.s2)

(4.1)

(4 )\

and F({) is an arbitrary function. The constant tensor /",, and its dual /",*, are restricted by the conditions nrlo:0,

nrfr,':O'

/d ll

216 67I

GAUGE

INVARIANCE

AND

Jus*Js,*: -nrn,.

(4.4)

VACUUM

from which are derived Jrxfx'*:O,

POLARIZATION

On integrating Eq. (a.18) with respect to s, we find that

loJx,:

1 "tr"t dtl2c,eA()-nue2A2(l)

.r,(s)-x"(Q):-,

The latter statement also includes a convention con2(nll)2 J rrot cerning the scale of /r,. 1 nreF([)laJf{ 2D,s. (4.20) The proper time equations of motion in this external field, With the constant D, determined by Eq. (a.20), Eq. dr,f ds:2f1,, (4.18)statesthat (4.s) d.n,/ ds: 2eF(l) f ,,II,! n,eF' ([)lcpJ u, (r,(s)-*"(0)) ' . admit several first integrals. Thus IIu(s) : -=-d.(nrn,(s))/ ds:0,

(4.6)

d$,,*n,(s))/d,s:o.

(4.7)

(tG)-i(0))

2s

>(lzc,eA (t(s))- n,e2 A2(l(s))| n,eF(l(s))i oJl

and

S

dg*n"g))/ds:

-n,eF(l)dl/ds,

(4.s)

since df/ds:2nTt;

(4.9)

We canfinally evaluateCuas

and therefore d(Ju"rl"lnreA([))/ds:o,

C,: f,"fI,!n,eA (4.10)

where

dA(o/dt:F(0.

^E(!)

-.. agTzc_e,l1E1 : l (t(s)- t(0)), Jr(o) - n,e,A, ([) ] n,er(01" fl. (4.21)

In addition,

(4.11)

In arriving at Eq. (4.10), it is necessary to recognize that df/ds commutes with f, in virtue of

x,(0))

f*(x"(s)-

2s

nt

*-l

f lG)

{(s)-t(O) Jr(o)

dtea(d.(4.22)

The commutation properties of these operators are involved in the construction of the transformation (4.12) function. fg,n[l:ln,xnn,Il"]:inp2:0. As is already indicated in the commutativity Since zII is a constant of the motion, Eq. (4.9) can of f(s) and f(0), thesecommutation relations are greatly simplified by the special nature of the external field. be integrated to yield Thus to evaluate r"(s)], we employ F,q. @.21) (i(s)- f(0))/s:22n, (4.13) to expressru(0) in[r,(0), terms of rr(s), ilA(s), {(s), and {(0). Now from which we infer that

,-"' [f(s), i(0)]:2r[zrr, {(0)]:0. (4.11) ,l-lt}l",lJ;,t,?_;r'lT,J:[]5 3*; The constantvector encounteredon integration of Eq. ""., rr(0)]:[-2sllu(s), (4.24) r"(s)]-3;r. (4.10),

fu,Il,ln,eA(():C,,

(4.15)

has the following evident properties: n,C,:0,

Ju,+C,:0, Ju,C,: -n,nfl, Cr2:

(nll)2'

31: ls-:(.r;,(s) - 2r"(s).r,(0){i;r(0)) - 2ts-' ,\ a, ' ,r ^u ',l

The elimination of f,II" from the equation of motion, with the aid of Eq. (4.15),gives d.Ir,/ d.s: (d./dl)l2c,eA(t)-

n,e2A2(l) +nFeF(o+oJf,

(4.17)

flr: ldr,/ ds: (l / 2nn)l2C ,eA (l) - n,e,42 (l) D,, !n,eF([)iaJfl

(4.18)

whence

where Du is an integration constant. Note, incidentally, that JP*il':

Jp*D4

[rr(s), No other nonvanishing commutator intervenes in bringing 3Cto the form

(4.19)

which is independent of s, in agreement with Eq. (4.7).

1

,"tr'r

f(r) - {(0) Jr(o) 1 f flk) t2 -?rto:rrrll,LJ-,0, dteA(d (42s) l'

in which a constant added to l({) is without effect, as required by the correspondingambiguity of Eq. (4.11). The solution of the differential equation (2.37) is (x(s)'I r(Oj ") : C (!, f' 15-z expfl](x'- x"):/sl frll Xexpf-isz(i'-i") L

dfle,A,-eF[ofll J|t , , J

x*{;,(rza'- t"l ate)'),, g,26) ['',

217 JULIAN where t':ttrrr',

V. 1-DECAY OF NEUTRAL MESONS

(/. rJ\

t":nutcu"

In this section we shall apply the results of our proper-time method to compute the effective coupling b€tween a zero spin neutral meson and the electromagnetic field, as produced by the polarization of the proton vacuum. This interaction manilests itself in a spontaneous decay of the neutral meson into two photons. The lagrange function for a spinless neutral meson

The function C(r', r") is determined by the difierential equations (2.38) and (2.39), in conjunction with Eq. ( 4 . 2 6 )T . hus.

l-

- ro,1*'1 - f ,,(x'- e'),-:-(eA ou,' -

1 * -t <

G')

ft' \ I atuAQ) r"):0. | ' Jrlc(x', "f"

672

SCHWINGER

field, in scalarinteractionwith the proton-antiproton (4.28) field,is givenby s:-+l(d'6)'+p'6'l-cq{L0"tf'

(5 1)

The solution of this equation will be obtained in terms of a line integral which is independent of the integration path. Again choosingthe path to be a straight line, rve find simolv

To find an approximate expression for the resultant coupling betweenthe neutral mesonfield and the electromagnetic field, rve replace |[r1,9] by its vacuum expectation value, calculated in the presence of a homogeneous electromagnetic field. The use of the I C{r',r"):C dr",4,("r)l:Co(*', x"), (4.2s) latter to represent the photons emitted in the spon"*VlU [,,',, taneousneutral meson decay introducesa small error, which is measured by the square of the meson-proton in view of the antisymmetry of J*. It is evident that mass ratio, (p/M)'?:>1/40. On the other hand, by (4.30) ignoring the efiect of the meson field on the proton C: -i(lr)-'. vacuum, we obtain only the initial approximationof a Only the behavior of the transformation function for perturbation treatment. Norv xr't=rr" is of actual interest in applications to vacuum polarizationphenomena.Now for f'-f", (\t0@, {(,)f):i t r G ( x ,r ) -12 7 I rt' f ft' r' : -M t d s e r p ( - i M ? s )t r ( r l U ( s ) l r ) *, I dtA,(1-l ; I dEA(il| .-1

and

"t',

L l", -< J "t,, ^+(t' - E")'F',l+(t' + t')1, (4.31)

(E'- t")'F": (r'- r")unrF2n,(t'- x")" : _ (r, _ r,,) uF pxFy"(x,_ r,,)

Jo

- _ aao)(x)/aM,

(5 2)

according to Eqs.(2.30)and (2.31).Thus,the efiective

(4.32) lagrange function coupling term between the neutral ", meson and the electromagnetic field is given by (4.1) (4.4). accordingto Eqs. and Therefore,for rr'-ru" (r(s)' I r(o; ";-i(4r)-zo 1s',*r' 1'-z X exp[i]s-1(r'- r"),(6u,f ](es)'1P" ,fp)(x'- r"),) Xexp(|ieo,,Fu,s), (1.33) rvhich is identical with the transformationfunction for a. constant field, as simplilied by the special characteristics of the field now under consideration;namely,

s'(r): g6@)0ao(r)/aM, rvhich clearly also follorvs directly from the proton held equation of motion, Lt?ia-eA)IM*c6l{:0,

(s.4)

in the approrimation which treats {(r) as a rveak, slowly varying, prescribedfield. If rve retain only the G O-n (4.34) leading term in the expansionof 8(1) for u'eak fields, u - vn , J-v. Eq. (3,45),we have We can conclude,without further calculation, that r' the physical quantities characterizing the plane wave field, the components of the energy-momentum tensor O t ' t ' 0 M > ( e t / 6 n r l M I d s e x p ( - , l f r s ) J Jn 2",, wi{l be identical in form rvith those of a constant :(2a/3r)(I/M)s. (s.s) field that obeys Eq. (4.34).On referring to Eq. (3.57), rve see that Tu, Ior a plane wave is just that of the Therefore the effective coupling term is maxwell field, rvhich may be simplifred further to (5.6) s,':(a/3n)(g/LI)d(H'-D'), (4.3s) T u,: F ryF nrn,Pz1E1. "1: rvhich describesthe decay of a stationary meson,into Thus, there are no nonlinear vacuum phenomena for a two parallel polarizedphotons, at the rate single plane rvave, of arbitrary strength and spectral (5.7) hc)(p/M)'?(pc'/ h). 7/ r : (a2/ 744r3)(g'z/ comDosition.

218 673

GAUGE

INVARIANCE

AND

POLARIZATION

VACUUM

by A pseudoscalar interaction between the spinless meson and electromagnetic field is given neutri,l meson field and the proton field is described by e'(r)=(s/2M)a,6@)((1/2i.)l{'(r),*t,t@)l).. the term : (g/2M)a,SQt)tryn,$@,*)

c6@)El0@),%Ur))

in the lagrange function. For our purposes, this is replaced by

: s6@) s' (tc) Gli @),r{ @)l) :is6@) try6c(,c, *) :-s6@)M

nI dsexp(-iM'?s) ro Xtrvs(*luG)lr).

-is The transformation function (3.20)' with stituted for s, Yields g, : _ g6M (4r)-,

r' I

*)1, (5.15) + - (g/2M)g(x)6,ftr167,G(*,

(s'8)

where the last version represents the results of integrating by parts. We now remark that this derivative has the following meaning:

1 (0u" lieA r(n" ))l try 6vrG(r', tc"),

dss-z exp(- Mzs)e-t(a)

tr767,fi(x', *") en

Xtrzs exp(leoFs). (5.10)

1-i

f^ I ds exp(-M'zs)$ Jo

:(a/r)(g/M)OI"H.

"

1S

tr767uG(r', r") r/( t t\

:itr75|

ndsexP(-iMzs) .to

x (*(s)'l](n,(s)- II,(0))| r(0)")

(5' 13)

-tr1lopo

The pseudovector interaction term,

(r/ zi.)l{(x),t *,!(*)f, k/ 2M)a,g(x)

a' I Jo

ds exP(-iI'1'?s)

x(*(s)'l*(n,(s)*II,(0))|*(0)")' (s'18)

(s.14)

is formally equivalent to (5.8) for the problem under discussion, in the apploximation to which it is being treated. This is demonstrated by a partial integration, combined with the use of the Dirac equation (2.1). Yet it has been found difficults'? to verify the equivalence in the actual results of calculation. Such discrepancies between formal and explicit calculations may be produced by insufficient attention to the limiting processes imolicit in the formalism. We shall demonstrate that, wi[h appropriate care, the proper equivalence between the psiudoscalar and pseudovector couplings is indeed exhibited. The efiective Dseudovector interaction between the ? J. Steinberger,Phys. Rev. 76, ll80 (1949).

X(r(s)'lr,(0)lr(0)").(s.17)

The result of averaging thesetwo equivalent expressions

This effective coupling terrn implies the decay of a stationary neutral meson, into trvo perpendicularly polarized photons, at the rate 1/ r-(a2/64rs)(g'/ hc)(p/ M)'z(pc'z/h).

J^

X (r(s)'lr,(s) | r(0)//) r: -i trr,"vt'vp I ds exP(-iM2s) Jo

(5'11)

In view of Eq. (3.a3), we obtain, without further approximation, simPlY

ds exP(-iM'zs)

Lr757r1"I

"

Now, the eigenvaluesol laF, as related to those of 7s by Eq. (3.31)' give

g':g6Q2/4r2)M

(5.16)

in which the structure oI the right side is dictated by ( 5 . 9 ) the requirement that only gauge covariant quantities be employed. We shall verify that the straightforward sub- evaluition of Eq. (5.16)yields the pseudoscalarcoupling (5.12), without further dificultY. According to Eq. (3.21)

Jo

tr76 exp(feoFs) : - 4 Im coshesX'

lim l(0,' - ieAu@'))

6,f tr1 67uG(tc,*)]:

We shall be content to evaluate Eq. (5.18) in the approximation of weak fields. On referring to Eqs' (3'4)' in iS.S), ana (3.20), it is apparent that the leading term this approximation is lry6yrG(x', r") - r"),Fx,Q(t', n") : - (e/ 64r'z) try aaroox*(*' X I

dss-zexp(-iM2s) erp[il(.r'-*")'?/s]

: (e/ 8 12)F u,a(x' - x" ),Q(r', r" ) X I

dss-2exp(-iM2s) expli'a@'-x")2/

sl,

(5'19)

219 JULIAN

674

SCHW INGER

with the aid of Eqs. (3.25) and (3.27). Since we are concerned with the behavior of this quantity only for fi'-lc", we may evaluate the proper time integral by an appropriate simplification. For x'>:r",

The related operator

dss-'?erp(- iM'?s)erp[i](r'-*")fsl

I

iq dpterminerl

-

I/(s): I/o-'(s) t/(s),

(6.6)

U6(s): exp(- irc6s),

(6.7)

where

hv

(6.8)

ia"Z(s): Uo-r(s)ccrUo(s)Z(s)

f' | d.ss-,e*pUrn(x,_x,')r/sl Jo

and (6.e)

v(0): 1. t*

:

I

One can combineEqs. (6.8) and (6.9) in the integral equation

d(s-') erp[i] (.r,-.r.,,;zr-11

Jo

:4i/ (x'-tc")'.

i/< ?n\

f"

f(s): 1-i I

Jo

Therefore, try67uG(r', i') -(ie/2r2)Q(r' , r")Fu,*(r'- r")"(r'-

and construct the solution by iteration: r't)-2.

(s.21)

To obtain the quantity of actual interest, Eq. ( 5 . 1 6 ) , I / ( s ) : 1 - ; we observe that

x") l(0,' - i,eA u@'))-l (0,"t ieAu@"))16(x', XF,,*(r'- r"),(r'- x")-2: i.e1(r', x")

F r"* (x' - x"),F ux(x'- tc")7(x'- s")-2,

x

(3.3s), Fu,*(r'-r"),Frx(x'-r")a:g(*'-l'12, - (e2/2r2)g Iim

,Q(x',

s't:s'112,"',

(6.12)

we oDlaln Lneexpansron U(s): exp(-i3cs)

(a/r)(g/M)6f..H,

i'q ?(l fl

.. : %(s)+ (- ;s) | du,Uo((l- u,)s)tcrIlo(zrs)*. ro

VI. PERTURBATIONTHEORY

ft

We shall now discuss the approximate evaluation of drr -t exp(-iz'?s)TrU(s),

(6.1)

fl

*(-ts)' I ur"-tdut"' l d," Jo Jo X Uo((1-ar)s)Jcps(u1(l-u)s). . . X U o ( u t .. . u " - t ( l - u " ) s )

by an expansion in powers of eAu and aFr,. For this purpose, we write K:3fo*Kt

(6.2)

where Ko: Pz

(6.3)

and Kl:

(6.11) dr"Llo-'{r")JcJJo(s")+....

s':s,tt,

(5.24)

in completeagreementwith Eq. (5.12).

f*

fo

x")

Thus, Eq. (5.15)yields

W(1):ia I J0

ds'U0-'(s')JCrU0(s')

( 5 . 2 3 ) O n i n t r o d u c i n g n e w v a r i a b l e s o f i n t e g r a t i olnf ,,r , N z t . . . t according to

: - (2a/r)9.

e':

f'

I

Jo

ds'(Js-L(s')K1IJ + e D' s(s') I,

(5.22)

accordingto Eqs. (3.15)and (3.16).But, in view of Eq.

}ultrysluG(x, r)f:

(6.10) ds'LIo-t7s'lxcrUr(s')tr/(s'),

- e(pA+ Ap)-leoFle2A2.

(6.13)

Instead of taking the trace of this expressiondirectly, which would involve further simplification, 'we remark that TrU(s)-TrUo(s)

(6.4)

To obtain the expansion of TrU(s) in powers of 3C1,we observe that U(s) obeys the differential equation

ta"U(s): (3co*3cr)U(s).

XK:Uo(ut.' It"s)+....

: -is

fr I dr TrFcr exp(-i(Jco*I3cr)s)l

(6.14)

Jo

(6.5) and insert the expansion(6.13) for exp[-i(3co*tr3cr)s].

220 675

GAUGE

INVARIANCE

AND

VACUUM

POLARIZATION

Thus,

Therefore

TrU(s): TrUo(s)f (- rs)tr[rcrUo(s)J

2ie2 r. I f -is (dk) W(t\:;:: | . d s s - t e x p ( - z ' r z , s ) l J| (Zn)aJo t

f1

* i ( - r'), I tta T rlr JJo((r- u) s)KJto(u,s)l*. . . Jo r_.is),+r /.r

+'

fr

I uf-tda"'l

Jo

n*l

Jo

f

exp(-iP's) xA,(-k)A,(k) @.P) J

du"

f L f f

*i(-,t), I la, | @k), (dp)zpFa/-h) J_L J .'

XTr[3iluo((l-zr)s)JCr.' . XI(IUo(a..'u"s))+....

x exp(- i (pt \ k),1(1- a)sl (6.15)

x2.?,A,(k) expl- i(p - ik)'zi(1*u)sl

We shall retain only the first nonvanishing field dependentterms in this expansion:

?L?f

*t(-&), | +a,| @k)| (dp)I tr+oF(-k) J J .)-r xexpl- i(p-fih)'*(1 - o)sl*rF(ft)

r' W(L):+ie2 | drr-t exp(-imzs) Jo

xexpl- i.(f -ik;zi(r+r,)sJ l. x | -i, r.[l'

t-

,

exp(-a'r's)l

We thus encounter the elementary integrals fr

*1(-"s)'

(6.1e)

I id.aTrl(fA-l Ap) exp(- ip'zi$- t)s)

f

J_1

J@'PexvGif's):-;ort-2,

x (pA+ A p) exp(- ip'?|(1f z)s)l

J

tl

++(-?s)' I J 1

(6-20)

| (k'/ 4))st iPhs) @Dexvl- i(P'z : -'ir2s-2expl- i(k'? / 4)(l- d)sf, (6.21)

lao T:il|,'or "xp(-ip'z|(1-z)s) and

x*aF exp(-ip1;(1+,)rll. (6.16) t

J

For convenience, the variable zr has been replaced by *(1*z). The evaluation of these traces is naturally performed in a momentum representation. The matrix elements of the coordinate dependent feld quantities depend only on momentum difierences,

: - exp(- i|ft'?s)(ts)-'@ / ak,)(a/ak") x

J

@D exp(-i|2s+i|kos)

: - iozs-z(- i| s-r6rl lrzkuk ")

f

(p++klA,l p- +k): (2r)4 @r)e-ik'Au@) J =(2r)-'zAr(k),

p)p,p expl- i (p'zI @'1 / 4))s* i phtsl @. "

XexPl- iLk'(I- rf) sf. (6'22) (6.17)

and

It is convenientto replacethe 6r, term of the last integral by an expression whichis equivalentto it in virtue of the integrationwith respectto u. Now

f

(plA,'l !) : (2r)-a @x) A,,(*) J : (2il-'

[

@k)AF/-k)A,(k). (6.1s)

I

idn exvl-irlz(l-o'?)sl fr

: 1- is|kz |

J_l

+a*' expl- i.ik'(l-o2)sl'

G.23)

221 JULIAN rn

J

SCHWINGER

pfraefirrohr

fhqf

The field strength and charge renormalization con_ tained in Eq. (3.48) then produces rhe finite gauge mvanant result-6

@il !,?, e*vl- i (|'z+|k )s* iphas l : - |rz s-36r"- irz s-2!o2(krk,- 5 r,pz| Xexpl-

676

w:J

ilkz (r - *) sf .

(6.24)

@HiF,,(-i.)FPG)

o h, f, t 2 ( 1 - l-t 2 ) ..f. I - | dr Xl 1-l. J L 4r m2 o ll(hr/4n )(l_72)l

On inserting the values of the various integrals, and roticing that

(o.,tor

The restriction which we have thus far imposed. that no actual pair creation oicurs, correspondsto the re obtain immediately the gauge invariant form (with requirement that 1lG2/4mr)(1-rr1 never vanishes. This will be true if -k214m2, for all i, contained in the fourier representalion o f t h e f i e l d . T n d e e d .i t i s evident from energy and momentum considerations :r'",:-@* roul+o,,t-olr,61fo' o,{r-o") that to produce a pair by the absorption of a single [ quantum the momentum vector of the latter must be tim.e-likeand must have a magnitude exceeding2ru. We fd s s- I e x p l- l m z J l h z ( l - z , r l s ] . ( 6 . 2 6 ) shall now simply remark that, to extend our iesults to XJ pair-producing fields, it is merely necessaryto add an i n f r n i t e s i m an l e g a t i v ei m a g i n a r y c o n s l a n t t o t h e d e _ lnis has been achieved without any special device, nominator of Eq. (6.30) and inlerpret the positive ::her than that of reserving the proper-time integration imaginary contribution to l4l lhus obtained wirh rhe :: the last. statement that -{_significant.separation of terms is produced by a (6.31) I eiw12: e-2r-v : ::tial integration with respect to a, according to represents the probability that no actual pair creation fr r' o c c u r sd u r i n g t h e h i s t o r yo f t h e f i e l d .T h e i n 6 n i l e s i m a l I aul-t',1 Jo| drr-, exp{-lmz+lk2(l-zr)lsJ imaginary constant, as employed in h,k,- 6p"k2) A N(- k)A'(k) : _ +F p,(_ k)F p(k),

=3

f'

| Jo

(6.25)

11 lim-_:p_1o;51r;,

rl

dss-rexp(- z2s)-lhz I aaqaz-!"tr1 Jo X

f_ ds expl-ln2l![r(1-2,)]s]. I

(6.27)

--,'iing the action integral of the maxwell field, which : :rpressed in momentum spaceby w(o):-

.++u t_Ie

tepresents a familiar device for dealing with real processes.We obtain from Eq. (6.30) thai

2r,,,w:+d[@brFp"(-k)Fp"(kt\a*, I'

f

J

@k)iFp"(_h)Fp,(k),

(6.2s)

"('-{),['+

,,: obtain the modified action integral,

:

f-e'f-l = -Lt* a"-'t*01-'"'-, ,u'Jo

x {an)+r,t-n)n,"{n) [ *L

k)F,.(k) h, f toa+r,,(fI

v2(1-Ia2\

x t db-----:--' -. (6.2s) Jo m,jf,kz(l-a2)

(6.32)

I

lo-,,t]

f

t- DF"(- k)FwG) " I -r,,!11t x( r!*t't \ (-r)/

4 m '\ r -\'z+ eh\)

(6'33)

For the weak fields that are being considered,Eq. (6.33) is just the probability that a paii is crearedby tLe 6eld. It should be noticed, incidenialy, that

-1F,,(- k)F,,(h):+llB(h),- H(p) ,l I I I

(6.34)

€The corresponding result for a spin zero charsed6eld is ob.

itrJ',lilif{i'f iilijiilili::',.??i,i.r'"",;*lfl "?i,'r",/'ii rr-ir,,

rD xq. (o.JUJ.

222 GAUGE

677

INVARIANCE

POLARIZATION

AN D VACUUM

regularization." is actually positive for a pair-generating field. This time method and that of "invariant the action integral follows, for example, from the vanishing of the mag- The vacuum polarization addition to netic field in the special coordinate system where &, has has the general structure only a temporal component. k)K!"(k,m')A,(k). (6.42) w : An alternative version of Eq. (6.33) is obtained by "' [ {an).t,(replacing the field with the current required to generate this field, according to the maxwell equations The proper-time technique yields the coefficient ik,Fu'(k): -J'(h)' / 6 ? q \ Kn(k, tn2)in the form hFF,^(k)+ k,F ^u(At| frrFr,(A): 0. n* Now K u ( h ,n ! ) l o : I d se x p ( - i n : s ) K " , ( [ , s ) , ( 6 . 4 3 ) ro h^2F,,(- h)F,,lk) : 2k,F,,(- h\k$xp(k) /6 16) :2J,(-k)J,G), where K",(4, s) is a finite, gauge invariant quantity; so thate infinities appear only in the final stage of integrating s to the origin. In efiect' this method substitutes a lower z rmw:(o/\m,) [ @ktJ,(-h)JpG) limit, se, in the proper time integral and reserves the J -.,,>e^" limit, so+0, to the end of the calculation. If, on the X ( 1 - z ) a r * ( 2 * z ) ,$ . 3 7 ) contiary, the proper-time technique is rrot explicitly introduced, K*(h, m2) will be representedby divergent where ( 6 . 3 8 ) integrals which lead, in general, to non-gaugeinvariant 1:4m' /(_k ). resJts. The regulator technique avoids the difficulty by It is now appropriate to notice that the integral introducing a suitable weighted integration with respect (3.49), representing the lagrange function for a uniform to the square of the proper mass, thus substituting for field, hasiingularities,unless9:0, $>0, corresponding K.(k, m2), the quantitY to a pure mignetic field in an appropriate coordinate f' s v s t e m .T h i s i s t h e a n a l y t i ce x p r e s s i oonf t h e f a c t t h a t (6'44) K , , ( h , m z ) l p : I d * p ( x ) K , , ( h ,x ) . pairs are created by a uniform electric field' ln parJ_6 -25: E')0, which invariantly iicular, for g,:0, characterizesa pure electric field, the lagrange function The "regulator" p(x) must reduce to 6(x-m2), in an proper time integral, approp.iite limit, and will produce gauge invariant in this problem if the following integral condiresulti fditions are satisfied: g:i|z( l / 8 r ' ) | d r s - 3e x p ( - n 2 s ) Jo rl"(6 15) I d * P ( r ) : 0 , J,- . d R K P ( K ) : o ' X [ e d s c o t ( e E s ) -1 * i ( a 6 s ) ' ? ] ' ( 6 . 3 9 ) J-has singularities at s:s,:nr/eE,

n:1,2,""

Expressed in terms of the fourier transformed quan( 6 . 1 0 ) tities,

If the integration path is considered to lie above the real axis, which is an alternative version of the device embodiedin Eq. (6.32),we obtain a positive imaginary contribution to S, l @

2ImA:-

! s"*'zexp(-ds") 4n ":r

n(s):

['*a*e-*,oe),

K,,(k,s): lt/zd

(6.46)

x), !*.au*"Ku@,

we have

r,"@,*'D o:

:X*t,*"*,(#) (641)

s), !-.dsR(s)K,,(F,

(6.47)

while the conditions on p(x) appear as

This is the probability, per unit time and per unit volume, that a pair is created by the constant electric field. We must now consider, in the framework of this special problem, the connection between the proper , e ri-"f" example,to whichthis formulamay b-eapplied,is j=0--0 transitionJ R oppentrt".l*il,i" oi u pu'i.;nu nuclear Pbvs.Rev.56,1066(1s3e)' i;i;;;;"J J. Schwinger,

R(0):0,

R'(0):0,

R(s)+exp(-iz'?s)'

(6'48)

Now observe that the proper time method yields K,,(h,m2) in the form (6.47)' with (6'49) t'O' K"(fr' s;:g' and R(s): saPl-;rzs), :0,

s) s6 s(si.

(6.s0)

223 JULIAN

S CH W I N G E R

This R(s), and all its derivatives,vanishesat the origin, thus satisfying the regulator conditions as s0+0. It appears, then, that regularization is a procedure for inserting, into a calculation that does not employ it, enough of the structure provided by the proper time representation to ensure gauge invariant results.

On averaging the two forms, we find that QuQ\:

-ef-

X tr(r(s)' | *(tlu(s) * I1,(0)) lr(0)") ]",,,,',

(4.1)

In the absenceof a field, the equations of motions are solved by IIu(s):IIu(0),

ru(s):ru(O) |2IIu(0)s.

(A.2)

-\s a fi$t approximation for rveak fields, we accordingly write 4t7uG) / d.s: le/ (2r)'7f

II,(0) ] @n)n,,ln) koot,(o)+2il(oh),

+ e/ (2d' f @k) ik ptaxvF\e(k)eik(' (ot+2nrc)"). (A.3) On integrating rvith respect to s, one obtains

r,(s)-ru(o): [e/(2il,ff Gh)Fu,G) x f,' d{ 1"'ra o*'n (Dr'),rr,(o)] + e/ (2d' f @ilibp+o^,F x,G) ' < f " d " ' ' * - o ' " n ' o * ' r ' ( A4 ) ylelcls lntegrahon i Second - r.t_" /n\ - n u ( o t + c / t 2 r ) , JG h ' F p t k ) t;' xf" *4-

as'1-s 1s11et&('(o)+'tr(0)!'), 11,(0) |

\zr )' u

(4.9)

exists in the absence of a (A.10)

and, therefore, only the transformation function in the absence of a 6eld is required for the first order evaluation of Eq. (A.9). Now

trau,(r(s)' I *(nls) -n,(0)) r(o)") : l2e/ (2dlf @k)(aFp"/ ax")(il s rt,+d! r(0)"), (4.11) X (r(s)'lexp[z'(&r(s)](1*r)*r*(o)](1-a)ll in which the variable s' has been replaced by r, according to

(4.1.2)

s,: s(rlr) /2. Theoperators Ar(s)andtr(0) do not commute: [A*(s),]r(0)l: zs[&r(0),]r(0)l: -2is]'

(A.13)

We may, however, employ the easily established theorem, (A.14)

eA+B:eAeBe_u^,R1, for operators

A

atd

B that

commute

rvith

their

commutator

ft, Bl. Thus, exp[i(fr *(s)1(1*a)]rr(0)i(1- z))l -r)] : exp[t,4r(s)+(1 +?)] exp[r,tr(0)4(1 XexPl-ik2\(1-t')sl,

(A.15)

and

tra,,(r(s)'| ] (n,(s)- n,(0))| r(0)")1,,,,,,-"

f dhtik,iat'6t

: l2e/(zi,lf

@k)eitu(aF u,/dt,)(k)s f,"ld.t -t)/(az)'s'?. (4.16) Xexpl- ik2iQ- z2)sl(

A similartreatmentappliesto

therefore

tr(c(s)'| $(n,(s)f n,(0))l,(0)')1,,,,..-, '. ( : ]\ f tan or,,u\s f ldw(x(s'f Iex|[i/&r(sr]tf ot

:(uu(s)f lru(0) = t ' ( t ) r " u t o ) + , , " , ' l ' 1 a t r r , ,p 1 u zr

\zr )- v

\zT)-

*rr(0)*(1 - a))l, (c,(s)-r,(0))/2sI I r'(0))1,..,,,,.. (A 17)

" d s ' C - + / l e ' r ' i I 0 ) + 2 r0I r s ' , , xJ n,(0rl

With the aid of the commutation relations, ur, f ? i f t r ( 0 ' , ,ur ' . . r , r s ) ] : - f r r f i_ z ) s e i * r ' o r l , t i', r+ur, fe;h(s)l',+,),f,fo)]= F,(l +?)seitr

+j= f @ptia,+"rot \zrt' ! x../"" ar({-

;),'-'"'6'a2s'or').

(A.0)

The induced curlent is equivalently expressedby j,(r)):

- n,(0)) r(0) ") l''.,,' -'. |

Iim (c(s)'lr*(s)-ru(0)lr(0)"):a, r'-,"-+0

xJ" tar(l-{)'"('(o)+'tr(oF'), (A s) :rd

d.s exp(-im's)

X trou,(r(s)' | *(r,(s)

It may be noted here that no current field, since

APPENDIX A

@k)eik,F p"(h)

ds exp(-im2s)

-iefo-

It is our purpose here to use the proper time equations of motion 2.36) for the computation of the current induced in the vacuum :.. a weak, arbitrarily varying field:

F,,(x) : lr / (2fl \f

678

lr@\

e trTu\rI (7II - z)Jo ds exp(- in s\U (s)| x ) : eI- ds e*p(-i.^,s) trTu-y,(r(s)'lrr,( s)l*(o)t)1.,,.,,,", (A'7)

:nd /ltl\ 1,f r))- e tr-rr\* J o ds exp( - i n2s)U Gr{'rn - ut r,) | I : eI- ds e"p(-;.-'s) tra,.yu(r(s)'| II,( o)l s(o)")b.,,,-". (4.8)

(A.lg)

this reducesto tr(r(s)' I 4(IIu(s)*II,(0)) | r(0)")1,,,.,.-, : - l2ie / (2n)z1f 1dk)eik'(aF,, / ax,) (k), f XexplWe have thus obtained

ik2!(1 - il)sl( - i)/(4r)'zs'?. (A. 19) "tarr"

<jp(()>: - (d /2i (2d-, f @k)ett'(dF,,/ a'"){il fo' d.r{r- o,1 XJ'

dss-1expl -[z'+ifr'(1-?'?)]s],

(A.20)

in which the substitution s+-is has again been introduced. This is precisely the current derived from the action integral I7(r) of Eq. (6.26), and further discussionproceedsas in Sec.VI.

224 GAUGE

679

INVARIANCE

AND

APPENDD( B

.

VACUU.M

PO.LARIZATION

which gives

An electronin interactionwith its properradiationfield,and M(*, r') : v0516- r') +le, / (4d\foan external6el{, is desuibedby the modifiedDirac equation,lo .y,(-i0u-eA,(r)){,1x14 1a{a@, r),r'(r')=0. f

(8.1)

To the secondorder in a, the massoperator, M(c, r'), is given by (8.2) r')|ie27 uG(x, r')1nD*(r-r'). M(r, r'):ao51!Here G(r, *') is tJreGreen'sfunctionof the Dirac equalionin tle is a photon Green's function, ererternal field, atd Da(r-r') pressedby D+(x-rt):

(4il-rf-

du z explillr-/)2/tj.

(8.3)

We shall supposeibe external field to be weak and uniform. Under these conditions, the transformation function (r(s) | r(0)'), involved in the construction of G(r, r'), may be approximatedby (r(s) I r(0)')- -t(ar)-za(r, il')s-' XexpUiio - r')2 / s7 exp(iiuF) ; (8.4) that is, terms linear in the field strengths enter only through the Dirac spin magnetic moment. The corresppndingsimplification of the Green's function, obtain by averaging the two equivalent {orms in Eq. (3.21),is G(r, r')=(4r)'za(x,

*')Jn

Xexp[dI(*-*')'/r]t

(B.s)

The mass operator is thus approximately represented by

xrii{-rfr+

au-

l

n, exp(i!eo fl }^v1, @.6)

I

lt F i a- eA)* n - p'io Flg : s, nol @/2in

{-

6rr-'

(8.14)

f"

6.r-,12--

y11 (8.15)

representsthe mass of a free electron, and p, : (. / 2,) emi L-

ds f:

r')

Q.w/ s) (w/ s) (r - zu/ s) lerp[-jz,(s-a)]

(8.16)

dercribesan additional spin magnetic moment. B-othintegrals-are conveniently evaluated by introducing aad roking (B.g)

(B.ro)

employedhere will be dircussedat lengtl in

(8.17)

u:l_w/s,

@.7)

md employed properties oI the Dirac matrices, notably 'ytoptx:o. (B.e) We shall also write (r - r') u6(r, r' ) expAIk - x')2/ w1 :2u(-iap-eA p(&)-leF p,(r-r'),)a(*, r') expAi@-r'),/w) =l2w(-idueAr(*)) -2w,eF r,(-ia"-eA,(r))l

Xo(r, r') eali.iQ-r')2/wl,

(B.13)

we obtain

Xe*pL- im2(t-w)l

as-' ea1- in,'i

10The concepts later publications.

:exp(in'w)*(rl,

since ry'(r) is an eigenfunction oI 3C, with the eigenvalue -22. Therefore, on discarding all terms corftaining the operator of the Dirac equation, which will not contribute to a.

n:

drr. 2 explt'nk - x')'/wf fo' xL- +n - s-t'y(r- l) I Lilt! - x'), ieoPll, in which we have replaced I by the variable u, p_r:s_r+r_r, x f,-

|r (o)')(dr' )o@): f @l u (w)|x')(dr' ),te' ) f @@)

where

or M (r, x') : 6061*- *') lliaz / (4r)afo(*,

'u (8.12) tleia-eA), |oFf:z;"rp1"or. We now introduce a perturbation procedurein which the mass operato! assumes the role customarily played by ttre energy. To evaluate ;f(drt)M(r,*')9@), we replace {(r') by t}re unperturbed wave function, a solution of the Dirac equation associated witl tfie massz (we need not distinguish, to this approximation, between the actual ross z and the mechanical mass uo), The r' integration can be efiectedimmediately,

_

(- i.n\) *o X exp [ul" O,(1+)] r -^.t ---'\

in virtue of the relation

f \ai {a4* {")a {r,/ ),t(i),

d.ss-' exp(-inzs) 'l ( _^,t-_-'\ r,exp(r]@F)]. \-ff+

M (x, *') : m66(r- r') *licz / @r)afo(r, { fo- au- f-

dss-' exb(-iro")

x f" aryz*qz - * 1g * ew / s)(t G ia - eA) * m) - 2ni(l -w / s)iletF - iw(l*w / s) X It(- i.a- eA)*m, EUF l)(r(u) | x(o)), (B.11)

the replacement s+-rr

which yields

6: 6oa@/2flnf- 6n-,fo'dulL+r) exp(-n2us) :n"+(s*/4,)mll- a""-'"rp(-.,")*tu], (8.18) and d.s fo' auuG ut exp(-n2u) - : (a/ 2r) (eh/ 2nc). (8.r9) : (, / 2,) (e/ u) f otd.u(r u) We tlus derive the spin nagnetic momentof d/2r magnetotrs producedby second-order massefiats. electromagnetic p' : (a/ z4 en t-

225

P o p e r2 l

The Theory of Positrons Departnent

R. P. FevNeN oJ Physics, Cornell, Uni,wrsity,

Ithaca, Nw

Vorh

(Received April 8, 1949) The problem of the behavior of positrons and electrons ih given external potentials, neglecting their mutual interaction, is analyzed of the soluby replacing the theory of holes by a reinterpretation iions of the Dirac equation. It is possible to u'rite down a complete problem in the terms of boundary conditions on the solution of save function, and this solution contains automatically all the 'rossibilities of virtual (and real) pair formation and annihilation iogether lrith the ordinary scattering processes, including the correct relative signs of the various terms. In this solution, the "negative energy states" appear in a form irhich may be pictured (as b)' Stiickelberg) in space-time as waves rraveling arvay from the external potential backilards in time. Experimentally, such a lvave corresponds to a positron approaching the potential and annihilating the electron. A particle moving iorward in time (eiectron) in a pote[tia] may be scattered forward in time (ordinary scattering) or backward (pair annihilation). \\'hen moving backrvard (positrou) it nray be scattered backrvard

in time (positron scattering) or forward (pair production). For such a particle the anplitude for transition ffom an initial to a finai state is analyzed to any order in the potential by considering it to undergo a sequence of such scatterings. The amplitude for a process involving many such particles is the product of the transition amplitudes for each particle. The exclusion principle requires that antisymmetric combinations of amplitudes be chosen for those complete processes which difier only by exchange of particles. It seems that a consistent interpretation is only possible if the exclusion principle is adopted. The exclusion principle need not be taken into account in intermediate states. Vacuum problems do not arise for charges rvhich do not interact with one another, but these are analyzed nevertheless in anticipation of application to quantum eiectrodynamics. The results are also expressed in momentum-energy variables. Equivalence to the second quantization theory of holes is provecl in an apoendix.

1. INTRODUCTION as a whole rather than breaking it up into its pieces. is the first of a set of papersdealing with the It is as though a bombardier flying low over a toad fUfS I s o l u t i o n o fp r o b l e m s i nq u a n i u m e l e c r r o ' d y n a m i c ss. u d d e n l ys e e st h r e e r o a d sa n d i t i s o n l y w h e n t w o o I The main principie is to deal directly with the solutions them cometogetherand disappearagainthat he realizes passedover a long switchbackin a ro the Hamiltonian differential equations rather than that he has simply rith theseequationsthemselves.Here rve treat simply slngleroao' This over-all space-timepoint of view leads to conihe motion oi electronsand positronsin given extern;l problems'one can take .otentials.rn a secondpaperwe considertfreinteractions siderablesimplificationin many into account at the same time processeswhich ordiof theseparticles,thai is, quantum eleclrodynamics. The pioblem of chargesin a fixed potenti;l is usually narily would have to be consideredseparately' For example,when consideringthe scatteringof an electron :reated by the method of secondquantization oI the by a potential one automatically takesinto accounl the electronield, using the ideas of the theory of holes. effectsof virtual pair productions'The same€quation, Instead we show that t y u *ituut. .rroi."'."a i"[t _ r. o. .D; - D i r a c ' s , w h i c h d e s c r i b e s l h e d e f l e c t i o n o l t h e w o r l d i i n " *nI.n; .e p l r e l a i l o n o r I n e s o l u u o n so "r ^u,r_r a. -c:s. "e_q.u. a. ;l r, o of an electronin a nerd,can also describethe denection ;;;;;; whenit islargeenough "q,;it;ii,|u.;;;';;;;;.';;iJj= :undamentally nomore*-pri.","a ir'"' iir'.oi'i*..;.

Illl ll-il',t l':'l^pl"^l Tiil::t

nerhod ordearing withon.",'nio,""puiiiJr...'ir," i"'r: ::"1"^:::'-",t1:11:::":,',:,:1,5 L o p a l r a n n l n l n t l o n ' X:ll,llf'::i:T::lt. vuantum,mecnanlcall)' rus creation and annihilarion oDerators in the conven-

::"":Ponl

fielrlvieware requiredbecause the :T".Til:':rur;;"llit"XTi"lli."rl :iunalelectron

rs repraceo Dv rne

rumber of particles is not conserved, i'e', pairs may be This vierv is qrite different from that of the Hamil:reated or destroyed' on the..other hand.charge is tonian method which considersthe future as developing :onserved which suggeststhat if lve.follow the charge, continuously from out of the past. Here we imagine the rot the particle, the results can be simplified. entire soace_timehistorv laid out. and that we iust I n t h e a p p r o x i m a l i o no l c l a s s i c arl e l a t i v i s t i ct h e o r y b e c o . e a * a r e o f i n c r e a s l n gp o r t i o n so f i t s u c c e s s i v e l y . :he creation of an electron pair (electron,4, positron B) fn a scattering problem th]s over-all view of the com_ night be represented by the start of two lines plete scatteririg processis similar to the S-matrix view_world :rom the point of creation, 1. The world lines of the point of Heisenberg.The temporal order of events dur:ositron will then continue until it annihilates another lng the scattering, *fri.t is analyzed in such detail by :lectron, C, at a world point 2. Between the times lr the Hamiltonian difierential equation, is irrelevant. The :nd l: there are then three world lines, before and after relation of theseviewpoints will be discussedmuch more rnly one. However, the world lines of C, B, and A fully in the introduction to the secondpaper, in which :ogether form one continuous line albeit the "positron the more complicated interactions are analyzed. rart" -B of this continuous line is directed backwards The development stemmed from the idea that in non,r time. Following the charge rather than the particles relativistic quantum mechanics the amplitude for a :orresponds to considering this continuous world line given processcan be consideredas the sum of an ampli749

226 750

R.

P.

FEYNMAN

tude for each space-timepath available.r In view of the fact that in classicalphysics positrons could be viewed as electrons proceeding along world lines toward the past (reference 7) the attempt was made to remove, in the relativistic case,the restriction that the paths must proceed always in one direction in time. It was discovered that the results could be even more easily understood from a more familiar physical viewpoint, that of scattered waves. This viewpoint is the one used in this paper. After the equations were worked out physically the proof of the equivalence to the second quantization theorY was found.2 First we discuss the relation of the Hamiltonian difierential equation to its solution, using for an example the Schrddinger equation. Next we deal in an analogous way with the Dirac equation and show how the solutions may be interpreted to apply to positrons. The interpretation seems not to be consistent unless the electronsobey the exclusionprinciple. (Chargesobeying the Klein-Gordon equations can be described in an analogous manner, but here consistency apparently requires Bose statistics.)3 A representation in momentum and energy variables which is useful for the calcuIation of matrix elementsis described.A proof of the equivalence of the method to the theory of holes in second.quantization is given in the Appendix. 2. GREEN'SFUNCTION TREATMENT OF SCIIRODINGER'SEQUATION We begin by a brief discussion of the relation of the non-relativistic wave equation to its solution. The ideas will then be extended to relativistic particles, satisfying Dirac's equation, and finally in the succeedingpaper to interacting relativistic particles, that is, quantum electrodynamics. The Schrddinger equation ia{//At:H{/,

(1)

describes the change in the wave function ,tt in an infinitesimal time Al as due to the operation of an operator exp(-iHA't). One can ask also, if ry'(xr'h) is tlre wave function at xr at time lr, what is the wave function at time lzll? It can always be written as V g 2 , t z ) : f X G z , t r ; x r l, r ) * ( x r , / r ) d x r , J

(2,

where K is a Green's function for the linear Eq. (1). (We have limited ourselves to a single particle of coordinate x, but the equations are obviously of greater generality.) If 11 is a constant operator having eigenvalues.E", eigenfunctions d, so that ry'(x,lr) can be exC"O"(x), then 'y'(x, tr): exp(- iE"(tz- tr)) panded as f " XC"O"(x). Sirce C":-f6^*(x)ry'(xr, lr)dxr, one finds -, R E F"un.un.Rev.Mod. Phvs.20,367(1948). .Tbe equivalence of the entire proce,Jure (including photon interactions, witb the work of Schwinger and Tomonaga has been demonstrated by F. J. Dyson, Phys. Rev. 75,486 (1949). a These are speciaiexamples of the general relation of spin and statistics deducid by W. P1uli, Phys. Rev. 58, 716 (1940).

(where we write 1 for X1,11&rld 2 lor x2, lz) in this case K (2, l) :t,

d" (x') d"* (x') exp(- i E "(tz- t )),

(3)

f.or tz) ty We shall find it convenient lor tzlh to define K(2,1)=0 (gq. (Z) is then nol valid for lz(/r). lt is then readily shown that in general K can be defined by that solution of

Qa/ab-H,)K(2,r):i6(2,r),

(4)

which is zerof.orlz<.h, where 6(2, 7):6(t2-l)6(*2-r) and the subscript 2 on I12 means X6(yz-y)6(zz-z) that the operator acts on the variables ol 2 ol K(2' l). When 11 is not constant, (2) and (4) are valid but K is less easy to evaluate than (3).4 We can call K(2,1) the total amplitude for arrival at x2, 12starting from x1, 11.(It results from adding an amplitude,expiS,for eachspacetime path between these points, where S is the action along the path.I) The transition amplitude for finding a particle in state x(xz,tz) at time lr, if at h it was in ry'(x1,l1), is f

I x*\ztrQ,l

tP(1)d3x1drx,.

(5)

A quantum .Ln"ni.ut sysl-emis describedequally well by specifying the function K, or by specifying the Hamiltonian ,E from which it results. For somepurposes the specifcation in terms of K is easier to use and visualize. We desire eventually to discuss quantum electrodynamics from this point of view. To gain a greater familiarity with the K function and the point of view it suggests, we consider a simple perturbation problem. Imagine we have a particle in a weak potential U(x,l'), a function of position and time. We wish to calculate K(2,1) if {/ differs from zero only for I between h and tz. We shall expand K in increasing powers of [/: (6) K ( 2 , 1 ) : K o ( 2 , 1 ) + K ( ' ) ( 2 ' l ) + K ( 2 (' 2 , 1 ) + ' " . , 0 ( 2 ,l ) . 0 T o z e r o o r d e ri n U , K i s t h a t f o r a f r e e p a r t i c l e K To study the first brder correction K(D(2, l)' first consider the case that {/ differs from zero only for the infinitesimal time interval Al: between some time /3 andls+Ah(tL
r K--o,( 3 ,l ) r y ' ( l ) d 3 x r '

,

(7)

since from tr to t, tJe particle is free. For the short interval Als we solve (1) as ,!(x, fi! Att) : exp(- iH Lta){6, ta) : (l - iE oah- iu at)tlt (x, I z),

aFor a non-relativistic free particle, where C":exp(ip'x), (3) gives, as is well Lnown B":p,/2n, f ,. t t: exp[- (ip. x1- ip. x ) - if ( z- r t t 2nf&p(2r\-3 .l "12, : (lrin-t (rr h))-t srp(|in(xr- xr)'(tr- t J t) lor tz)lt atd Ko:O lor lallr

X

227 THEORY

Ile being the Hamiltonian where we put H:HstU, of a free particle. Thus ry'(x,laf Ah) difiers from rvhat it would be if the potential were zero (namely (l-iHo\ts){/(x,ls)) by the extra piece A!:

-iU(.n, tu).,!(4, tu)atx,

(8)

ryhich we shall call the adplitude scattered by the potential. The wave function at 2 is given by t l x z , t z \:

751

OF POSITRONS --><-7r

a

*

SdITER€O /slrLl,Lu:q^ -. rAvL ^6rz.r,

ffii -*t<)<:{t\

:+ffi:-== -:SS<=

\-.........R \ ! \ -\ . . . + \ t -

e \

xo{4J)CffiAlls\

Yxo(?''l

?8llJi8lq:j)5 >'! YI |

{\tilctoENt wAvEs SPACE (o, FIRST ORoER,EQ p) (bl SECONO OROER, Ea.0O)

f

K o , x , t " , x a ,l r * A 1 r ) * ( x a ,l 3 + A / r ) d x r .

Ftc. 1. The Schriidinger (and Dirac) equation can be visualized as describing the fact that plane waves are scallered successively by a potentill. Figure I (a) illustrates the situation in first order. since after ,r"+or, ,tr" particle is again free. Therefore Ko(2,3) is the amplitude for a free particle starling at point 3 to arrive at 2. The shaded region indicates the presence of the rhe change in the wave function at 2 brought about by potential 4 which scatters at 3 with amplitude -i,4(3) per (7) (8) (8) (substitute into and into cm3sec. (Eq. (9)). In (b) is illustrated the second order process rhe potential is (Eq. (10)), the waves scattered at 3 are scattered again at 4. Howthe equationfor ry'(xr,lr)): ever, in Dirac one-electron theory ll(4,3) would represent electrons both of positive and of negative energies proceeding from f 3 4. This,is difierent to remedied by choosing a scattedng kertrel i Ko(23 , JU 3 ) K o ( 3 , 1) r y ' ( 1 ) d x x : d 3 x r A / r . LVtz): I Ka(4,3), Fig. 2.

I

In the case ,i", ,ft" potential exists for an extended rime, it may be looked upon as a sum of effects from eachinterval Alg so that the total efiect is obtained by rntegratingover li as well as xr. From the definition (2) rf ( then, we find

1): -i 1{i,)(2,

I

r)d.rt, (9) r,e, s)u(.3)Ko(3,

lhere the integral can now be extended over all space and time, dr3:f,a*"61".Automatically there will be no rontribution if la is outside the range lr to lz becauseof r u r d e f i n i t i o nK , aQ,1):0 for tz1h. We can understand the result (6), (9) this way. We :an imagine that a particle travels as a free particle :rom point to point, but is scatteredby the potential U. Thus the total amplitude for arrival at 2 from 1 can re consideredas the sum of the amplitudes for various :lternative routes. It may go directly from 1 to 2 amplitude Ko(2, l), giving the zero order term in (6)). Or (seeFig. 1(a)) it may go from 1 to 3 (amplitude .\o(3, 1)), get scatteredthere by the potential (scatter:ng amplitude -iU(3) per unit volume and time) and :hen go from 3 to 2 (amplitude Ks(2,3)). This may rccur for any point 3 so that summing over these :lternativesgives (9). Again, it may be scattered twice by the potential Fig. 1(b)). It goesfrom 1 to 3 (r(o(3,1)), getsscattered :here (-iU(3)) then proceedsto some other point, 4, :n space time (amplitude K0(4, 3)) is scattered again -iU(4)) and then proceedsto 2 (Ks(2,4)). Summing rver all possible places and times for 3, 4 find that the :econd order contribution to the total amplitude r-c)(2, 1) is -it"

| | K o ( 2 ,4 ) I ' ( 4 ) K o ( 4 ,J ) X U ( 3 ) K o ( 3 ,\ ) d r & r +

(10)

This can be readily verifieddirectly from (1) just as (9)

was. One can in this way obviously write down any of the terms of the expansion(6).5 3. TREATMENT OF THE DIRAC EQUATION We shall now extend the method of the last section to apply to the Dirac equation. A1l that would seem to be necessaryin the previous equations is to consider 11 as the Dirac Hamiltonian, ry'as a symbol with four indices (for each particle). Then Ko can still be defined by (3) or (4) and is now a 4-4 matrix which operating on the initial wave function, gives the final wave function. In (10), U(3) can be generalizedto,4r(3)-c'A(3) where,4a, A are the scalar and vector potential (times e, the electron charge) and a are Dirac matrices. To discuss this we shall define a convenient relativistic notation. We represent four-vectors like x, I by a symbol rr, where p: 1, 2, 3, 4 and u-- t is real. Thus the vector and scalar potential (times e) A, Aa is Ar. The four matrices Bc, B can be consideredas transforming as a four vector ?r (ouryu difiers from Pauli's by a factor i for p: l, 2, 3). We use the summationconvention a rb,: aaba- otbt - azbz- atbt: a' D' In particular if a" is any four vector (but not a matrix) we write a:o,"yu so that o is a matrix associatedwith a vector (a will often be used in place of o, as a symbol for the vector).The 7, saLisly^tF"t,+'y,"rp:26r, where6q: * 1, 611:f22:f33:-1, and the other 6", are zero. As a consequenceof our summation convention 6r,au:au and 6,":4. Note that ab*ba:2a'b and that a2:arar :a.a is a pure number. The symbol 0f 0r, wlll mean 0/0t lor p:4, and -a/Ax, -d/Ay, -0/62 lor p:1, 2,j. Callg:1,0/0r,: B 3 / a t + f u . v . W e s h a l li m a g i n e o We are simplvsolvingby successive a n integral approximations and (4) equation ldedircibledirictiy from (1) with H:HIFU with 11:llo). ,r(2): -if KoQ,3)u(3)'t(3)d,s+ KoQ, Dg$)d.xxr, I where the 6rst integral extends over all space and all times 4 gleater than the rr appearingin the secondterm, and &)L.

228 R.

lJz

P. FEYNMAN e x p a n s i o no f t h e i n t e g r a le q u a t i o n

I II vl

K+6)(2, 1): K+(2, 1)

I

-i

(2:!), NEC.E

r

@*,,",

caraNs oNLr Pos E.

/

!c!'

a

e'

,

(.) VIRTUALPAIR 14>t3

t4< l3

sEooND oRoER,E0. (t4) The Dirac equation permits another solution Ka(2,1) fto.2. if one considers that lvaves scat lered by the poLential can proceed backwards in time as in Fig. 2 (at. This is interpreted in the second order processes (b), (c), bv noting that there is now the possi' bilitv (c) ol virtual pair production al 4, lhe positron going to 3 lo be annihilated. Tbis can be pictured as similar Lo ordinary scattering (b) except t}lat the electron is scattered backwards in time from 3 to 4. The waves scattered from 3 to 2'in (a) represent the oossibility of a positron arriving at 3 from 2'and annihilating the ilectron from l. This view is proved equivalent to hole tleory: electrons traveling backwards in time are recognized as positrons.

hereafter, purely for relativistic convenience, that d"* in (3) is replaced by its adjoint ,6":6""A' Thus the Dirac equation for a particle, mass m, in an external field A:Auyuis (iv-m),!:

Al/,

(11)

and Eq. (4) determining the propagation of a free particle becomes (iY z- m)K4(2, l) : i6 (2, 1),

(r2)

the index 2 on V: indicating difierentiation with respect to the coordinates *ru which are represented as 2 in K + ( 2 , 1 ) a n d 6 ( 2 ,1 ) . The function K+(2,1) is defined in the absenceof a field. If a potential / is acting a similar function, say K+G) (2, 1) can be defined. It difiers from Ka(2, l) by a 6rst order correction given by the analogue of (9) namely

f

t

K r Q , 3 ) . 4 ( . r ) 1 l , t e ' 113; ,7 " ,

which it ulro sutirJes. We would now expect to choose,for the special solution of (12), K+:Ko where Ko(2, 1) vanisheslor tzllt and for lz)lr is given by (3) where d, and.E" are the eigenfunctions and energy values of a particle satisfying Dirac's equation, and d,* is replaced by d". The formulas arising from this choice,however, suffer from the drawback that they apply to the one electron theory of Dirac rather than to the hole theory of the positron. For example, consider as in Fig. 1(a) an electron after being scattered by a potential in a small region 3 of space time. The one electron theory says (as does (3) with K+:K0) that the scattered amplitude at another point 2 will proceed toward positive times with both positive and negative energies, that is with both positive and negative rates of changeof phase. No wave is scattered to times previous to the time of scattering.Theseare just the propertiesof Ke(2,3). On the other hand, according to the positron theory negative energy states are not available to the electron after the scattering. Therefore the choice K+:Ko is unsatisfactory. But there are ot-l-rersolutions of (12). We shall choose the solution defining K-.(2, 1) so that K+(2,1) Jor lz>.h is lhe sunr oJ Q) overposiliaeenergy only. Now this new solution must satisfy (12) for sl.oles all times in order that the representation be complete, It must therefore difier from the o1d solution Ki by a solution of the homogeneousDirac equation. It is clear is the from the definition that the difierence Kq-Kf sum of (3) over all negative energy states, as long as 1z)lr. But this differencemust be a solution of the homogeneous Dirac equation for all times and must therefore be representedby the samesum over negative energy states also for lz(lr. Since Ko:g in this case, it follows that our new kernel, Ka(2, 1), Jor tzlh is lhe negatheof the sum (3) oaernegaliw energystales.That is,

K+(2,t) :L posu o"e)6 4 - , r ' 1 2L,- - , 1 n , 1 2 , 3 r . 4 1 3 r K1* rr r3l ,r 3 .( 1 3 ) "e) Xexp(- iE"(t2- t')) for t2>h : -Lwne e"6"Q)6"Q) to go from 1 to 3 as a free

representing the amplitude Darticle,qet scatteredthere by the potential (now the matrix CiS) insteadoI U(3)) and continue to 2 as free. The secondorder correction, analogous to (10) is

(16)

Xexp(-iE"(tz-t))

for

(17)

lz1tr.

With this choice of K+ our equations such as (13) and (14) will now give results equivaient to those of the ff positron hole theory, K +. @ ( 2 , 1 ) : | l l ( , ( 2 . 4 ) A ( 4 ) That (14), for example, is the correct second order JJ expression for finding at 2 an electron originally at 1 (14) X K+(4,3)A(3)Kn(3, r)d.rad4, according to the positron theory may be seenas follows (Fig. 2). Assume as a special example that l:)lr and and so on. In general K+(1) satisfies that the potential vanishes except in interval l:-lr so l n ) K + G ) ( 1 5 ) ( 2 , l):i6(2, l) ' (ivr- A(2) that lr and lr both Iie between h and 12. First supposelr>ls (FiC. 2(b)). Then (since h)/r) and the successiveterms (13), (14) are the power series

229 .IHEORY

OF

::e electron assumed originally in a positive energy ::3te propagatesin that state (by Kn(3, 1)) to position : whereit gets scattered(,4(3)). It then proceedsto 4, -,:ich it must do as a positive energy electron. This is :rrrectly describedby (14) for K+(4,3) contains only : rsitive energy componentsin its expansion,as taltz. .\iter being scattered at 4 it then proceedson to 2, ,:ain necessarilyin a positive energystate, as lzlle. In positron theory there is an additional contribution rje to the possibility of virtual pair production (Fig. - c)). A pair could be created by the potential.C(4) ::4, the electronoI which is that found later at 2. The : rsitron (or rather, the hole) proceedsto 3 where it .:nihilates the electronwhich has arrived there from 1. This alternative is already included in (14) as con::ibutions for which la(la, and its study will lead us to .r interpretation of Ka(4,3) for lr(lr. The factor -:,-(2,4) describesthe electron (after the pair produc::on at 4) proceedingfrom 4 to 2. Likewise K+(3,1) r.presentsthe electronproceedingfrom 1 to 3. K1(4, 3) lust therefore representthe propagation of the positron : hole from 4 1o 3.'lhat it does so is clear. The fact ::rat in hole theory the hole proceedsin lhe manner of :nd electron of negative energy is reflected in the fact ',nL Ka(4,3) for 1a(1r is (minus) the sum of only ::egative energy components. In hole theory the real :rergy of these intermediate states is, of course, 'Ihis is true here too, since in the phases ::,ositive. .xp(-iE"(ta-tu)) definingK+(4,3) in (17), E" is nega:ive but so is lr-lr. That is, the contributionsvary with :; as exp(-il.E"l(lr-lr) as they would if the energy rf the intermediatestate were i.E"l. The fact that the :ntire sum is taken as negative in computing K+(4,3) :s reflectedin the fact that in hole theory the amplitude :.as its sign reversed in accordancewith the Pauli :rinciple and the fact that the electron arriving at 2 :as been exchangedwith one in the sea.6To this, and ''o higher orders, all processesinvolving virtual pairs :rrecorrectly describedin this way. The expressions such as (14) can still be describedas : passageof the electronfrom 1 to 3 (K+(3, 1)), scatter:ng at 3 by .d(3), proceedingto 4 (,(+(.1,.3)),scattering :gain, ,4(4), arriving finally at 2. The scatteringsmay, lorvever, be toward both future and past times, an :lectron propagating backwards in time being recogrizpd

qc a nncifrnn

This therefore suggests that negative energy comronents created by scattering in a potential be con;idered as waves propagating from the scattering point :oward the past, and that such waves represent the propagationof a positron annihilaLingthe electron in the potential.T 6 It has often been moted that the one electron theory apparently :ives the same matrix elements for this nrocess as does hole theorv. Thr problem is one oI interprelalion,eipecially in a rvay thar will give correct results ior other processes,e.g., self-energy. "1so 7 The idea that positfons can be represented as electrons with roper rimc reversed relative ro true lime las heen discussed by he author 3nd others, parricularll Ly Sttickelherg. E. C. C.

POSITI{ONS

/JJ

With this interpretation real pair production is also described correctly (see Fig. 3). For example in (13) if the equation gives the amplitude that if at hltt{tz time lr one electronis presentat 1, then at time 12just one electron will be present (having been scattered.at 3) and it will be at 2. On the other hand if lr is less than lr, for example, il tr:1t11t, the same expressiongives the amplitude that a pair, electron at 1, positron a.t 2 w'tll annihilate at 3, and subsequently no particles will be present. Likewise if fu and 11exceed13we have (minus) the amplitude for finding a single pair, electron at 2, positron at 1 created by ,{(3) from a vacuum. If (13) describesthe scattering of a positron, Itlltltz, A1I these amplitudes are relative to the amplitude that a vacuum will remain a vacuum) which is taken as unity. (This will be discussedmore fully later.) The analogueof (2) can be easily worked out.8 It is,

9P):Ix-p,1)N(1),/(1)rrt'1,

(18)

where d3Ilr is the volume element of the closed 3dimensional surfaceof a region o{ spacetime containing

,'w IW 'w"W (b)

(c)

(d)

Ftc.3. Several difierent processes can be described by the same formula depending on the lime reialions of lhe variables /r, lr. Thus P,lK-r1'(2, l)1, is rhe probability that: (a) An electronar I will be scrttered at 2 (and no other pairs form in vacuum). (b) Electron at 1 and positron at 2 annihilatc leaving nothing, (c) A single pair at 1 and 2 is creared from racuum. (d) A positron at 2 is scattered to 1. (K+
230 P.

/.f +

FEYNMAN

point 2, and N(1) is .rtr,(l)"y,where Nu(1) is the inward drawn unit normal to the surface at the point 1. That is, the wave function r/(2) (in this case for a free particle) is determined at any point inside a four-dimensional region if its values on the surface of that region are specified. To interpret this, consider the casethat the 3-surface consists essentially of all space at some time say l:0 previous to lz, and of al1 spaceat the time Tllz.The Lylinder connecting these to complete the closure of the surface may be very distant from xz so that it gives no decreasesexpoappreciable contribution (as K{2,1) 'va: B, since nentially in spaceJike directions). Hence, if - B, be and' .lf will normals B drawn the inward ,1,\2):

I

K*(2, 1)0*(1)d3xr

tivistic calculations, can be removed as follows. Instead of defining a state by the wbve function /(x), which it has at a given time lt:O, we define the state by the function /(1) of four variablesxr, 11which is a solution of the free particle equation for all lr and is /(x1) for lr:0. The final state is likewise defined by a function g(2) over-all space-time.Then our surface integrals can 6e performed since -fK1(3, 1)p/(xr)d3xr:/(3) and I g6) B,l}x,K+(2,3) : AQ). There results

-;.[o1)A(3)f(s)d,",

(22)

the integral now being over-all space-time. The transition amplitude to second order (from (14)) is l"f

- | | stz:nQtx,t2,1\A(ltJJ-)drfi12, (23) JJ

-

f

K , Q . l ' ) p v 1 ' \ d " x 1, ' ( 1 9 ) for the particle arriving at 1 with amplitude /(1) is scattered (,4(1)), progressesto 2, (K+(2' 1)), and is w h e r e / 1 : 0 , 1 r , : I . O n l y p o s i t i v e e n e r g y ( e l e c t r o n ) scatteredagain (C(2)), and we ihen ask for the amplicomponents in 'y'(1) contribute to the first integral and tude that it is in state g(2). If g(Z) is a negativeenergy o n l y n e g a t i v ee n e r g y( p o s i t r o n )c o m P o n e n tosf r y ' ( l ' )t o state we are solving a problem of annihilation of elect h e s e c o n dT. h a t i s , t h e a m p l i t u d ei o r 6 n d i n ga c h a r g e tron in /(1), positron in g(2), etc. We have been emphasizingscatteringproblems,bur at 2 is determined both by the amplitude for finding an electron previous to the measurement and by the obviously the motion in a fixed potential I/, say in a arnplitude for finding a positron after the measutement' hydrogen atom, call also be dealt with. If it is llrst This might be interpreted as meaning that even in, a ti.*"d us a scattering problem we can ask for the problem involving but one charge the amplitude for amplitude, dr(l), that an electron with original free hnding the charge at 2 is not determined when the only wave function was scatteredfr times in the potential I/ thine known in the amplitude for finding an electron either forward or backward in time to arrive at 1. Then (or ipositron) at an earlier time. There may have been t h e a m p l i t u d ea f t e r o n e m o r e s c a l L e r i n igs no electron present initially but a pair was created in the measurement (or also by other external fields). The 6r+,(2):-iI x*<2,1)/(1)dr(1)drr. Q4) amplitude for this contingency is specified by the future. in the positron a u-plitrd" for finding An equation for the total amplitude fre can also obtain expressionsfor transition amplitudes, like (5). For exampleif at l:0 we have an elecdi(1) /(1):I tron present in a state with (positive energy) wave tr{ function /(x), what is the amplitude for finding it at for arriving at 1 either directly or after any number of with the (positive energy) wave function g(x)? t:I scatteringsis obtainedby summing (24) over all A Irom after anywhere electron the finding for The amplitude 0to@; the by ry'(1) replaced /(x), t:0 is given by (19) with second integral vanishing. Hence, the transition ele, t ( 2 ) :o \ Q ) - i K * Q ,1 ) V ( r ) V 0 ) ( t r 1(.2 s ) I ment to find it in state g(x) is, in analogy to (5), just (t2:T, h:o) Viewed as a steady state problem we may wish, for example,to find that initial condition do (or better just I u r * u na . Q , 1) 0 f ( x r ) d l x r d 3 x . , (20) the 0) which leads to a periodic motion of ,y'. This is most practically done, of course,by solving the Dirac since g*: *7B. eouation, Ii Jpotential acts somewherein the interval between (26) (,v - n\/!): V(I)t (r) ' first order Thus the K+(A). by 0 and ?, K1 is replaced z, iVrsides by (25) on both (13), by operating from is, deducedfrom efiect on the transition amplitude thereby eliminating the do, and using (12) This illus- ; I a ( x- , ) t s K , Q , 3 ) / ( 3 l K f ( 3I. r 6 l t x r t d x r d " x , . ( 2 1 ) trates the relation betweenthe points of view. |" For many problemsthe total potential AlV may be fixed one, Z, and another, i4' Expressions such as this can be simplified and the split convenientlyinto a If ,(+(1.)is definedas in a 3-suriace integrals, which are inconvenient for rela- consideredas Derturbation. |

23r 1'HEOltY

OF

(16) with V f.or A, expressionssuch as (23) are valid and useful with K1 replacedby K1(r) and the functions J0), C,(2)replaced by solutions for all space and time of the Dirac Eq. (26) in the potential I/ (rather than iree particle wave functions). 4. PROBLEMS INVOLVING SEVERALCHARGES We wish next to consider the case that there are two (or more) distinct charges(in addition to pairs they may produce in virtual states). In a succeedingpaper we discussthe interaction between such charges. Here we assume that they clo not interact. In this case each particle behaves independently of the other. We can expect that i{ we have two particles o and D, the amplitude that particle o goesfrom xl at lr, to xa at la while D g o e sf r o m x z a t / z t o x r a L / r i s t h e p r o d u c t K G , a ; 1 , 2 ) : K + " ( 3 , l ) K + b @ ,2 ) . The symbols a, D simply indicate that the matrices appearing in the K.' apply to the Dirac four component spinorscorrespondingto particle a or 6 respectively(the wave function now having 16 indices). In a potential Ka. and Kai, become K*,td) and K.'r(d) where K*o(a) is defined and calculated as lor a single particle. They commute.Hereafter the a, 6 can be omitted; the space time variable appearing in the kernels suffice to define on what they operate. The particles are identical however and satisfy the exclusionprinciple.The principJerequiresonly that one to get the net c a l c u l a t eK $ , a ; 1 , 2 ) - K ( 4 , 3 ; 1 , 2 ) amplitude for arrival of chargesat 3, 4. (It is normalized assumingthat when an integral is performed over points 3 and 4, for example,sincethe electronsrepresentedare identical, one divides by 2.) This expressionis correct for positronsalso (Fig. 4). For examplethe amplitude that an electronand a positron found initially at x1 and xr (say lr:lr) are later found at xr and xz (with . : : 1 r ) / r ) i s g i v e nb 1 ' t h es a m ee x p r e s s i o n

T'OSITRONS

/ JJ

m;m 3tt2t43\tZA

5H ORFh' (b) \/-/

Y-/

3\

A_]\ tlAl,l

v_11 l/

3\

12

\4

oR (c)

t2

/xC\ ( a )

\o/ l/

\4

Ftc. 4. Some problems involving two distinct charges (in addition to virtual pairs they may produce) : P" I K+(trJ(3, 1)K+6)(+,2) *1{+(d)(4, 1)K+\A)(3,2)1, is the probability that: (a) Electrons at 1 an,l 2 are scaltered to 3.4 iand no pairs arc fo.med). abt S t r r t i n g w i t h a n e l e c l r o n a l I a s i n g l ep a i r i s f o r m e d , p o s i l r o n a t 2 , electrons at 3, 4. (c) A pair at 1, 4 is found Lt 3,2, etc. The exclusion principle requircs that the amplitudes for processes involving exchange of two electrons be subtracted.

term (14). We shall see,however,that consideringthe exclusion principle also requires another change which reinstates the quantity. For we are computing amplitudes relative to the amplitude that a vacuum at lr will still be a vacuum at 12.We are interested in the alteration in this amplitude due to the presenceof an electronat 1. Now one process that can be visuaiized as occurring in the vacuum is the creation of a pair at 4 followed by a re-annihilation of the samepair at 3 (a processwhich rve shall call a closed loop path). But if a real electronis presentin a certain K + ( 1 ) ( 3 ,1 ) K + ( d ) ( 42, ) - K + ( ^ ) ( 4 , 1 ) K + ( 1 ) ( 3 ,2 ) . ( 2 7 ) state 1, those pairs for which the electronwas created The first term representsthe amplitudethat the electron in state 1 in the vacuum must now be excluded. We proceedsfrom 1 to 3 and the positron from 4 to 2 (Fig. must therefore subtract from our relative amplitude the +(c)), while the secondterm representsthe interfering term correspondingto this process.But this just reinamplitude that the pair at 1, 4 annihilate and what is states the quantity which it was argued should not found at 3, 2 is a pair newly createdin the potential. have been included in (14), the necessaryminus sign The generalization to several particles is clear. There is coming automatically from the definition of Kn. It is an additional factor K+(1) for each particle, and anti- obviously simpler to disregard the exclusion principle symmetriccombinationsare always taken. completelyin the intermediatestates. All the amplitudesare relative and their squaresgive No account need be taken of the exclusionprinciple in intermediate states. As an example consider again the relative probabilities o{ the various phenomena. expression(14) for tzlh and supposelr(lr so that the Absolute probabilities result if one multiplies each of situation represented(Fig. 2(c)) is that a pair is made the probabilities by P,, the true probability that if one at 4 with the electronproceedingto 2, and the positron has no particles present initially there will be none to 3 where it annihilates the electron arriving from 1. finally. This quantity P, can be calculated by normalIt may be objected that if it happensthat the electron izing the relative probabililies such that the sum of the createdat 4 is in the samestate as the one coming from probabilities of all mutually exclusive alternatives is 1, then the processcannotoccurbecauseof the exclusion unity. (For exampleif one starts with a vacuum one can principle and rve should not have included it in our calculate the relative probability that there remains a

232 756

R.

P.

FEYNMAN

vacuum (unity), or one pair is created, or two pairs, etc. The sum is P,-r.) Put in this form the theory is complete and there are no divergenceproblems. Real processesare completely independent of what goes on in the vacuum. When we come, in the succeedingpaper, to deal with interactions between charges, however, the situation is not so simple. There is the possibility that virtual electrons in the vacuum may interact electromagnetically with the real electrons. For that reasonprocessesoccuring in the vacuum are analyzed in the next section, in which an independent method of obtaining P, is discussed.

In addition to these single loops we have the possibility that two independent pairs may be created and eachpair may annihilate itself again. That is, there may be formed in the vacuum two closed loops, and the contribution in amplitude from this alternative is just the product of the contribution from each of the loops consideredsingly. The total contribution from all such pairs of loops (it is still consistent to disregard the exclusion principle for these virtual states) is Lzf2 Ior in Z2 we count every pair oI loops twice. The total vacuum-vacuum amplitude is then C":l-

L+L,/2- 1a/6*... :exp(-Z),

(30)

the successiveterms representing the amplitude from zero, one, two, etc., Ioops. The fact that the contribuAn alternative way of obtaining absolute amplitudes tion to C, of single loops is -Z is a consequenceof the is to multiply all amplitudes by C,, the vacuum to Pauli principle. For example, consider a situation in vacuum amplitude, that is, the absolute amplitude that which two pairs of particles are created. Then these there be no particles both initially and finally. We can pairs later destroy themselves so that we have two assume C,:1 if no potential is present during the loops. The electrons could, at a given time, be interinterval, and otherwise we compute it as follows. It changed forming a kind of figure eight which is a single difiers from unity because,for example, a pair could be Ioop. The fact that the interchange must change the created which eventually annihilates itsell again. Such sign of the contribution requires that the terms in C,, a path would appear as a closed loop on a space-time appear with alternate signs. (The exclusion principle is diagram. The sum of the amplitudes resulting from a1l also responsible in a similar way for the fact that the suchsingleclosedloops we call L. To a first approxima- amplitudefor a pair creationis -Ka rather than f K1.) tion Z is Symmetrical statistics would lead to 5. VACUUM PROBLEMS

l rr LtLr:-_ | lS/[K 2J J

C,:l+L+

( 2 ,1 1 . 4 r 1 ' X KaQ, 2)A(2)fd.rj.r r.

(28)

For a pair could be created say at 1, the electron and positron could both go on to 2 and there annihilate. The spur, Sp, is taken since one has to sum over all possible spins for the pair. The factor I arisesfrom the fact that the same loop could be consideredas starting at either potential, and the minus sign results since the interactors are each - lr4. The next order term would bee

L ' ' ) : * t i / 3 )l l

1 .' , 4 ( l ' f spt,<,(2 r t, x 1{+(1,3).4(3)1{* (3, 2)A (2)fd r Q.r zd.

etc. The sum of all such terms gives 2.10 gThta actually vanishes as can be seen as follows. In any t*a spur the sign oi all 'y matrices may be reversed. Reversing the o f ' y i n K 1 ( 2 . l ) c h a n g s 5 i t l o l h e t r a n s p o s eo f K , ( 1 , 2 ) s o sign that the order of all factors and variables is reversed. Since the integral is taken over all 11, 12, and zs this has no effect and we are left with (- 1)3 Jrom changing the sign of,4. Thus the spur equals its negative. Loops rvith an odd number of potential inteructors give zero. Physically this is because for each loop the electron can go around one way or in the opposite direction and we must add tlese amplitudes. 3ut reversing the motion of an electron makes it behave like a positive charge thus changing the sign of each Dotential interaction. so that the sum is zero if the number of inleracllons rs odd. lhls theorem is due to W. H. Furrv. Phvs. Rev. 51, 125 (1937t. 10A closed expression for Z in terms of K+La) is hard to obtain because of the factor (!/n). in the zth term. However, the perturbation in Z, At due to a small change in potential A,4, is easy to expless. The (1/n) is canceled by the fact that A.A can appear

L'/2:exp(*Z).

The quanLity Z has an infinire imaginary part (from Zo), higher orders are finite). We will discuss this in connection with vacuum polarization in the succeeding paper. This has no effect on the normalization constant for the probabilily lhat a vacuum remainvacuum is given by P,:

lC,)'1:exP(-2'real Part of Z),

from (30). This value agrees with the one calculated directly by renormalizing probabilities. The real part of Z appearsto be positive as a consequenceof the Dirac equation and properties of K.. so that P, is less than and conseone. Bose statistics gives C":exp(fZ) quently a value of P, greater than unity which appears meaninglessif the quantities are interpreted as we have done here. Our choice of 1(1 apparently requires the exclusion principle. Charges obeying the Klein-Gordon equation can be equally well treated by the methods which are discussedhere {or the Dirac electrons. How this is done is discussedin more detail in the succeedingpaper. The real part of Z comes out negative for this equation so that in this caseBose statisticsappear to be required for consistency.3 aftersunmingoverz by in.ny of th" z potentials. The resuLt (13),(14)andusing(16)is 1)-11+(1,1))ad(1)ldrr. (zs) ^L: -.if Sp((K+(r)(lJ to zero. The term1(*(1,1) actuallyintegrates

233 THEORY

OF POSITRONS

6. ENERGY-MOMENTUM REPRESENTATION The practical evaluation of the matrix elements in some problems is often simplified by working with momentum and energy variables rather than spaceand time. This is because the function K+(2,1) is fairly complicated but we shall find tbat its Fourier transform that is is very simple, namely (i/4t9)(p-nt)-1 f

KJz,l):Q/ar\

|

(P-n)-'exp(-if

-ru)dal,

Bl)

whete p. r2r:'p' fi 2- p' 11: p **2r- P&tw i : f rI p, and dap means (2r)-'zd,Pd?zd,!d.fu, the integral over all 2. That this is true can be seen immediately from (12), for tlre representation of the operator i,Y-m in energy (1n) and momentum (2t, 2,r) space is ! - nr and the transform of 6(2, 1) is a constant. The reciprocal matrix (l-n)-' can be interpreted as (Iltn)(F-tn2)-r for 1sa pure number not involving f-m,:(p-m)(!lm) 7 matrices. Hence if one wishes one can write K aQ, \ : i(;Y,1, m)I a,(2'1),

t2<*, Er@ is the Hankel function and 6(.f) is the Dirac delta function of s} It behaves asymptotically as exp(-im.s), decaying exponentially in spaceJike directions.12 By means of such transforms the matrix elements Iike (22), (23) are easily worked out. A {ree particle wave function for an electron of momentum y'r is where a1 is a constant spinor satisfying uexp(-i?t*) the Dirac equation pih:nt1h so that !12:77*, lfis matrix element (22) for going from a state fb ur to a state of momentum p2, spinor u2, is -4*i(tZ2a(q)u1) where -we have imagined 14 expanded in a Fourier rntegral f

,4(1): I a(q) exp(-iq.x,)daq, J

and we select the component of momentum e:?z-lr The second order term (23) is the matrix element between u1 and u2 of - a#i

where

757

f

|

(a(fz- I,-

q))(1,* q - m)-'a(q)daq, (35)

f

p, I +(2, l) : (2r)-, | (f - *,1-' exp(- i p. tcz)da

(32)

since the of momentum y'r may pick up q from "i".r.on the potential c(q), propagate with momentum y'1f g is not a matrix operator but a function satisfying (factor (p1f q-ar)-r) until it is scattered again by the potential, a(fz-lril, picking up the remaining mo( 3 3 ) ZrtI+(2,1)-m'zI+(2,1):6(2,1), mentum, !z-!rQ, to bring the total to y'2. Since all -[l: : (V,)z (6/ 0rz,) (6/ dxzr). where values of g are possible, one integrates over g. The integrals (31) and (32) are not yet completely These same matrices apply directly to positron probdefined for there are poles in the integrend when lems, for if the time component ol, say, pl is negative We can define how these poles are to be the state representsa positron of four-momentum -pr, l,-rr*:0. evaluated by the rule that ,n is considered.Io hooe an and we are describing pair production if p2 is an elecinf.nilesimal negoliueimaginary part. That is m, is re- tron, i.e., has positive time component, etc. placedby ttt - i6 and the limit taken as 6+0 from above. The probability of an event whose matrix element is This can be seenby imagining that we calculate K1 by (a2Mu) is proportional to the absolute square. This integrating on 1r first. If we call E:+(rn'+pf may also be writ"ten (ilLMu2)(E2MilL),where M is M then the integrals involve pr essentially as with the operators written in opposite order and explicit *?z'*P"1, 1f exp(-ipe(tz-tr))dpn(po'-E)-r which has poles at appearanceof I changedto -i(,41 is B times the complex pe:tE and pa: - E. The replacementol mby. m-i6 cbrrrjugatetransposeof BM). For many problems we are means that -E has a small negative imaginary part; the not concernedabout the spin of the final state. Then we first pole is below, the secondabove therlal axis. Now can sum the probability over the two z2 corresponding il t2-11>O the contour can be completed around the to the two spin directions. This is not a complete set besemicirclebelow the real axis thus giving a residuefrom cause,2 has another eigenvalue, -n. To pennit sumpole, or -Qn1-'exp(-iE(t2-t)). the pl:{E It ming over all stateswe can insert the projection operator lz-h10 the upper semicircle must be used, and (2m)- L(! zl m) and so obtain (2nx)-| (11tM (I 2+ rn)M u ) E at the pole, so that the function varies in each for the probability of transition from 101,u1, to p2 with !a: caseas required by the other definition (17). arbitrary spin. If the incident state is unpolarized we Other solutions of (12) result from other prescrip- can sum on its spins too, and obtain tions. For example if 2r in the Iactor (f -m2)-L is con(2nL)-,5pl@ t+ n )M (1,+ rn)M) (36) sidered to have a positive imaginary part K.. becomes replaced by Ks, the Dirac one-electron kernel, zero for for (twice) the probability that an electron of arbitrary lz(lr. Explicitly the function isrr (x, l:r21,) spin with momentum pl will make transition to pe. The /a(x, t) : - (4r)-r6(s)* (m/8rs) H {2t(ms), (34) expressions are all valid for positrons when p's with J

u If ttre -16 is kept with z here too the function Ia approaches where s:*(l-xz)l fs1 p;'*u and s:-i(rr-tr)l for lor in6nite positive and negative timc. This may-6e useful 111s(r,l)is (2i)-t(D/x,t)-iD(r,t)) whereD1 andD are the rero in general-analysesin avoidin-gcomplicationsfrom infinitell functions defiredby W. Pauli,Rev.Mod. Phys.13,203(1941). remote surfaces.

234 R,

/.! 6

P.

FEYNMAN

nesative energiesare inserted'and the situation interorJted in accoidancewith the timing relationsdisc"ssed (''l):1 ibove. 1We have usedfunctions normalizedto instead of the conventional (uBu):(u+u):1' On our scale (&Bu):enetgyf m so the probabilities must be corrected by the appropriate factors') The author has many people to thank for fruitiul conversations about this subject, particularly H' A' Bethe and F. J. DYson. APPENDIX

the Dirac Xexp(-i'-fotHdt'). As is well known v(x, l) satisfies i,iiff"t".tiate v(x, ,) with respectto ' and usecommuta"nrniil.. tion relaiions oI E and 9) (42) iavl,t)/0t:(a'(-iV*A)+/a+uB)v(x,')' {differequation Dirac ihe satisfy also must t) Consequenllvd(x, partsr' e n r i a t e( 4 1 )w i t h r e s p e ctto / . u s e( 4 2 )a n d i n t e g r a t eb y at trme That is, if 4(x, t) is thaLsolutionof the Dirac equatron and o*:/v*(x)4(x)d3x r *-rti.r,l! O&) at l:0, and if we define Oi*:Jfv*(x)+(x, T)d3x then O/*:SO*S-r, or (43) so*:o'*s,

a. Deduction from Second Quantization of this theory with In this section we sball show the equivalence lhe lheory ol secono the hole theory of the positron.2 Acording to if the electron field in a given potential,!3 the.state ""r"ii-ti". n"ra at any time is represented by a wave function 1 i-ifrir sacisfying

be The orinciple on which the proof will be based c&n now just one electron illustrated bya simple example.Supposewe have initially and finallY and ask for (44) 1: (1o*GSF*xo). S using (43)' operalor the through putting ,F* try misht We w a v el u n c t r o n sF*=F;*S, *hei"/';n p'*=7'!*1xy'(x)d3xis the at I arising from /(r) at 0. Then (xo*F'*GSxo), (45) ;^fGF'*Sxi : fc*(x)/'(x)d3x' C,use of the defiwhere the secondexpressionhas been obtained by niti.n (:S) of G and the generalcommutation relation |:

i6Y/01: E Y' - A) +,4 4* n 9) v (x) d|x v{.*l .uld l" where fI : Jrv' (x) (a. ( rV an electron at posirion x while V*(x) is un op"rutoionniftifating a siluation We contemplate operator' creation ih" io,r".ponding we have present some electrons in states reprein which at l:0 '. assumed sented by ordinary spinor functions /rtx),-/r(x), as noles ln orthosonal, and some positrons These are descrlbed fill the normally would which rons eiect *.rgy sea. the ii"-n-"?iit. wati functions pl(xr. p2(xr' " " we ask at time r ;;i;-f,;;i"* states 3r(x)' whar is thi amplitude ihat we 6nd electrons in . Iitheinitialand frnalstate rruf, ll. onahoiesatq'txt.gr(xt, we respectively' and are 1, li i,".ior, ,aor"tan,ing this situation wish to calculate the matrix element

GF*+F*G= f s*$)Ie)d\,

(the others are which is a consequenceof the properties of v(x) xo'F'* in the Iast term in F-G*:-G.Fri.Iiow ;;=-"CFil (45) is rhe complexconjugateof F'xo' Thus if /'-contarned,only iotponenti, F'xo would vanishand we wouldhave io.itiu" c,.'sut F', as worked.outhere,.d,oe; i.j*;;;i "n"tev t*t6iii.* potentlaLa contain negativeenergycomponentscreatedln tne and the melhod must be slightly modi6ed' -"e"f*"'priir"g thtll F* throu"ghihe operator.y." idd,t: l-1 { 3 7 ) anorheroDeratoiF"* arisinglrom a iunction/"(r) contalnrngo'l' n:(',-*n(-;;['aa,)',):,',"s'' that the resultins /' chosen so il;;;;;;;;;; ";-pon""t! we want ones.ThaL is"na zero only for times hasonlv bosiliae We assume that the potential '4 difiers from (46) times' at these be defrned can a vacuum that ? so between 0 and S(Fo**f F*""*) : Fp*'*S, negative energy lf 1g represents the vacuum state {thal is' all of the sign the of 'rpos" reminders as serve and having for where the "neg" .*r"t nfi.a, all positive energies emptyl. the amplitude we can now ao-po.".,. contained in the operators This a vacuum at time f, if we had one at l:0, is ".*gy the Jorm in use (38) (47) c,: (xo*sxo), SFo."*:Fo."'*S-SF".e"*' Our problem is to evaluale R and rel)lacesr by two suLstitulion writing 5 lor expl-ijforEdl) this problem eleclron one In our factor lnvolves show tbat it is a simple factor times C,, and that the terms previous sections' - (1s*GSF"","*xo). the K+(A) functionJ in the way discussed in the r: (xo*GFp"",*Sxo) To io this we first express xr in terms of xo' The operator

o*:J"v*(x)d(x)d,x,

(39)

creates an electron with wave function 4(x) Likewise'!:-fd*(x) annihilates one with wave lunction d(x)' Hence state iv(*)a'" js Gr*Gu* ' PtP, xo while the 6nal state rt l. rt:f'-frdefined like 6' in lOrOr: a" where Fi. Gi. Pi. Qi are operators pi, qj replacing d: for the initial.stale would i.ls),"bu, *i,tt r. 3i. tt Jt' J2' result from the vacuum i[ we created the electrons ' . Hence we must find noJ annittilated those in pr, pr, ('10) R:(xo*.. Qr*Qr*"'GzG$Ft*pt*"'P1Pr"'10)' beTo simplify this we shall have 1o use commutation relations u o* op"tu,ot and S. To this end consider exp( r"frHdl')o* *"* ,'ar*r11il'*(x)' and expand this quantily in terms of ;1;Aal1 (which defines d(x' f))' Now multrply siving /v*(x)O(x,l)drx, and find exp(-i./i'lldl') bv exp(|i-fstEdr't iti. ""o,iution

The 6rst of these reducesto

r: f c*G)ln*'$)d'r'c"

rs now zero,while the secondis zerosincethe as above.for FDo6'x0 p"","- giueszero when acting on the vacuum ;;;;';;."i"; the central idea of ri.i"-." tif negative e;ergies are full This is demonstntion. the *i;;;;;;il;';sented a function t""(x) Given bv (46) is this: r, ii."'rj. J n",i the u-outi, /.""", oI negativeenergy.component ol ulrac's equawhich must be ad,led in order that the solulion components' lion at time I will have only positiveenergy /::s.' Kernel^+ ' 15 tne This is a boundary value problemfor wluch the positiveenergycomponentsinitially' /o"* i.ti-""d w" t*; positiveoneshnally are nnJitt" t.gntiu" onesfinally (zero) The therefore (using (19)) (48) Io".'(*")=f K+@(2, 1)9/o*(xt)d3xr' (41) v*G, t)o1, t)d.x, ./'**1*y4i*)d3x:f are initially ones negative where lr: I, lr:0. Similarly, the where we have defined v(x, l) by V(x, l)=e1p11;76lEdl')v(x) (49) die 1..""(xz1: f X*cx(2, 1)p/p""(x1)d'x,-/p-(xt, Quontenexample, C. Wentzel, EinJuhtung in " S*J1943), ChapLeipzig, Deuticke, tFranz wit*iadiq The 'f"*(x:) is tnr*1i-'ai' where ,2 approacheszero from above, and L:0' ter V,

235 THEORY

OF

subtracted to keep in /."""(x) only those waves which return from the potential and not those arriving directly at l, from the K*(2,1) part ol K+6)(2, 1), as l*0. We could also have written J*!'(xr):

f

LK*tA)(2, 1) K +(2,1)l€l,*(xr)d3xr.

Therefore the one-electron problem, r :/g*(x)fo""' gives by (48) r : C,

(50) (x)d3x. C,,

POSITRONS

759

The value of C,(ro-Aro) arises from the Hamiltonian l1l6-at6 which difiers from 11,0 just by having an extra potential during the short interval At6. Hence, to 6rst order in Aro, we have -//.r\\ =\yo* C "(/o-410)

exp\-

i

R : ( x o * ' ' ' Q r * Q r *" ' G r G 1 F 1 p . " ' * S a r * ' ' P J z ' ' ' x o ) - (xo*''' Q " * Q t * ' ' ' G : G r S . F r ' " e " * F r * ' 'P f

z''' xd'

In the first term the order of F1u."'* a1d G1 is then interchanged, producing an additional term ,fg1*(x)1o."'(x)d3x times an expression with one less electron in initial and final state. Next it is exchanged with Gr producing an addition --fg,*(x)h"",'(x)dix times a similar term, etc. Finally on reaching tie Q1* with which it anticommutes it can be simply moved over to juxtaposition rvith xo* where it gives zero. The second term is similarly handled by moving Fr""""* through anti commuting ar*, etc., until it reaches Pr. Then it is exchanged with P1 to produce an additional simpler term rvith a tactot T;fp1*(x)1,,"""(x)d3x or + f pr* (.xz)K1(A) (2, 1) 8J1(x)d.3x\dsxrfrom (49), with rr: tr :0 (the extra /r(xg) in (49) gives zero as it is orthogonai to pr(xr)). This describes in the expected manner the annihilation of the pair, electron l, positron ft. The P""""* is moved in this way successively through the P's until it gives zero when acting on xo. Thus R is reduced, with the expected factors (and with alternating signs as required by the exclusion principle), to simpler terms containing two less operators which may in turn be further reduced by using F:* in a similar manner, etc. A{ter all the F* are used the Ql's can be reduced in a similar manner. They are moved through the S in the opposite dilection in such a manner as to produce a purely negative energy operator at time 0, using relations analogous to (a6) to (a9). After all this is done we are left simply with the expected factor times C' (assuming the net charge is the sane in initial and final state.) In this way we have written the solution to the general problem oi the motion of electrons in given potentials. The factor C, is obtained by normalization. However for photon fields it is desirable to have an explicit form for Co in terms of the potentials. This is given by (30) and (29) and it is readily demonstrated that rhis also is correct according to second quanLizalion.

b. Analysis of the Vacuum Problem We shallcalculateC, from secondquantizationby induction considering a series of problems each containing a potential distribution more nearly like the one we wish. Suppose we know C, for a problem like the one we want and having the same potentials for time , betwen some lo and l, but having potential zero for times from 0 to ro. Call this C,(r0), the corresponding Hamiltonian Ftq and the sum of contributions fo; all single loops, Z(ls). Then for lo:2 *" have zero potential at all times, no pairs can be produced, Z(T):0 and C,Q):1. For lo:9 we have the complete problem, so that C,(0) is what is defined as C, in (38). Generally we have,

)xo)

:(,t

*o(-t[,'

to -udt

)xo)

X ( - a A ( x , , 0 ) + 1 , 1rxo, ; ) v ( x ) d ,xx)l ;

as expected in accordance with the reasoning of the previous sections (i.e., (20) with 1{+(n) replacing,tn). The proof is readily extended to the more general expression R, (40), which can be analyzed by induction. First one replaces F1* by a relation such as (47) obtaining two terms

Hdl

_ r,"H

=(*. *n( ;.fi a,ar)lt-itrofv*,*,

K'(A) (2, t) Af(\1)d3xi3x2, J c* 6,)

//..T\\ c"(rs)=(ro+ "*p(*,J"

J,

,,a),),

since ,Ero is identical to the constant vacuum Hamiltoniat Hr lor I
we therefore obtain for the derivative

- dc,,Qo) / dto- - i(r,r

; "*v(- f,

of C, the expression

n,a)

X J f " v + , x . p . 4/rox),v / x l a , x 1 ) , i 5 1r which will be reduced to a simple factor times C,(to) by methods analogous to those used in reducing R. The operator V can be imagined to be split into two pieces gp." and rl".g operating on positive and negative energy states respectively. The Voo" on xe gives zero so we are left with two terms in the cuuent density, vp."*B/v""s and V""s*0.4V".c. The latter v"us*8.4{,"os is just the expectation value of p/ taken over all negative energy states (minus.F"""B:4V"""* which gives zero acting on xo). This is the effect of the vacuum expectation current of the electrons in the sea which we should have subtracted from our original Hamil tonian in the customary way. The remaining term il/p."*PlV".s, or its equivalent Vp."*0,4V can be considered as V*(x)fo""(x) where loo"(x) is written for the positive energy component of the operator p.4V(x). Now this operator, V*(x)feos(x), or more precisely just the V*(x) part of it, can be pushed through the exp(-iJnoruil.) in a manner exactly analogous to (47) when /is a futrction. (Aa alternative derivation lesults from the consideration that tie operator V(r, r) which satishes the Dirac equation also satisfies the linear integral equations which are equivalent to it.) That is, (51) can be written

by (a8),(s0),

/

-^

-dc,\ro\ldro:-i\xo*.J

x *v(- t fi

J *'txzlX.'n'tZ,tl

u a) n g7v 1x,1 ax,a"xzy)

//\ +;(x'-""0(-;..f'na)ff v-tx"ttx-,A'(2,l -f+(2,

1)1,{(1)V(r)d'x1d'x,1s),

where in the first term tz:T, and in the second t*to:tL'fhe (-4) in 7(*tet refers to that part of the potential .d after ,0. The 6$t term vanishes for it involves (from the l(+(r)(2,1)) only positive energy components of V*, which give zero operating into xot- In the second term only negative components of V*(xz) appear. If, then V*(xr) is interchanged in order with V(xr) it will give zero operating on xo, and only the tern,

- dC,(t,)/dt": + i.J s2[(r+(!)(1, 1) *r+(1, 1)).,1(1)ld3xr. C,(to), (s2) will remain, from the usual commutation relation of.I,* and V. The factor of C,(ro) in (52) times -Als is, according to (29) (reference 10), just .L(to-Aro)--L(io) since this difierence arises during the short time interval from the extra potential A.4:r4 (d L(t dto) C,(ti so that intesration Alo. Hence - dC d.t o: o) * / "(6 / lrom lo= f to lo:0 establishes (30). Starting from the theory of the electromagnetic field in second quantization, a deduction of the equations for quantum electrodynamics which appear in the succeeding paper may be worked out using very similar principles. The Pauli-Weisskopf theory of the Klein-Gordon equation can apparently be analyzed in essentiallv the same wav as that used here for Dirac electrons,

Paper 22

236

Space-Time Approach to Quantum Electrodynamics R. P. Frvruu Defa/lnml. oJ Physics, Cornell Unitasi.ly, Ithaca, Nw York (ReceivedMay 9, 1949) In this paper trvo things are done. (1) It is shown that a concan be attained in writing down matrix siderable simplification elements for complex processes in electrodynamics. Further, a yierv physical point of is available rvhich permits them to be problem, rvritten dorvn directly for any specific Being simply a restatement of conventional electrodynamics, hol'ever, the matrix elements diverge for complex processes. (2) Eiectrodynamics is

and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons, The simplification in rvriting the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation point of view is of this to the more conventional Hamiltonian discussed. It would be very dificult to make the modiflcation which is proposed if one insisted on having the equations in Hamiitonian form. The methods apply as rvell to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modifieiion like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used. The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix.

modifled by altering the interaction of electrons at short distances. All matfix elements are now finite, rvith the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, rvhich gives finite results for these matrices also. The only eftects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directll' observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modifrcation goes to zero. The results then agree rrith

those of Schrvinger. A complete,

unambiguous,

paper shoutd be consideredas a direct confUtS I t i n u a t i o n o I a p r e c e d i n go n e r ( [ ) i n w h i c h t h e motion o{ electrons, neglecting interaction, was analyzed, by dealing directly with the solution of the Hamiltonian difierential equations. Here the same technique is applied to include interactions and in that way to expressin simple terms the solution of problems in quantum electrodynamics. For most practical calculations in quantum electrodynamics the solution is ordinarily expressedin terms of a matrix element. The matrix is worked out as an expansionin powers oI e2/hc, the successiveterms correspondingto the inclusion of an increasing number of virtual quanta. It appears that a considerablesimplification can be achieved in rvriting down these matrix elementsfor complex processes.Furthermore, each term in the expansion can be written down and understood directly from a physical point of view, similar to the space-timeview in I. It is the purpose of this paper to describehow this may be done. We shall also discuss methods of handling the divergent integrals which appear in these matrix elements. The simplification in the formulae resuits mainly from the fact that previous methods unnecessarilyseparated into individual terms processesthat were closely related physically. For example, in the exchangeof a quantum between two electrons there were two terms depending on which electron emitted and which absorbed t-he quantum. Yet, in the virtual states considered,timing relations are not significant. Olny the order of operators in the matrix must be maintained. We have seen (I), that in addition, processesin which virtual pairs are produced can be combined with others in which only 1 R. P. Feynman,

Phys. Rev. 76,749

(1949), hereafter called I.

positive energy electrons are involved. Further, the effects of longitudinal and transversewaves can be combined together. The separations previously made were on an unrelativistic basis (reflected in the circumstance that apparently momentum but not energy is conserved in intermediate states). When the terms are combined and simplified, the relativistic invariance of the result is self-evident. We begin by discussingthe solution in spaceand time of the Schr
"..

2 For a discussionof this modiEcationin classicalpbysicssee R. P. Feynman, Phys. Rev. 74 939 (1948), hereafter referred to as A. 3A brief summary of the methods and results will be found in R. P. Feynnan, Phys. Rev. 74, 1430 (1948), hereafter referred to as B.

769

237 770

R.

P.

FEYNMAN

element for all real processesas the limit of that com- was still not complete becausethe Lagrangian method puted here as the cut-ofi width goes to zero. A similar had been worked out in detail only for particles obeying technique suggested by Pauli and by Bethe can be the non-relativistic Schrtldinger equation. It was then applied to probletns of vacuum polarization (resulting modified in accordance with the requirements of the in a renormalization of charge) but again a strict Dirac equation and the phenomenon of pair creation. physical basis for the rules of convergenceis not knolvn. This was made easier by the reinterpretation of the After mass and charge renormalizalion, tIe limit of theory of holes(I). Finally for practicalcalculationsthe zero cut-off width can be taken for all real Drocesses. expressionswere developed in a power seriesin e,,/rr. It T h e r e s u l t sa r e l h e n e q u i v a l e n t o l h o s eo f S i h u ' i n g e r . was apparent that each term in the serieshad a simple rvho does not make explicit use of the convergencefac- physical interpretation. Since the result was easierto tors. The method of Schwinger is to identiiy the terms understandthan the derivation, it was thought best to corresponding to corrections in mass and charge and, publish the results first in this paper. Considerabletine previous to their evaluation, to remove them from the has been spent to make these first two papers as comexpressionsfor real processes.This has the advantage plete and as physically plausible as possible without of showing that the results can be strictly independent relying on the Lagrangian method, because it is not of particular cut-ofi methods. On the other hand, many generally familiar. It is realized that such a descripticin of the properties of the integrals are analyzed using cannot carry the conviction of truth which would acIormal properties of invariant propagation functions. company the derivation. On the other hand, in the But one of the properties is that the integrals are infinite interest of keepingsimple things simple the derivaiion and it is not clear to what extent this invalidates the will appear in a separatepaper. demonstrations. A practical advantage of the present The possible application of these methods to the method is that ambiguities can be more easily resolved; various meson theories is discussedbriefly. The formusimply by direct calculation of the otherwise divergent las corresponding to a charge particle of zeto spin integrals. Nevertheless, it is not at all clear that the moving in accordancewith the Klein Gordon equation convergence factors do not upset the physical con- are also given. In an Appendix a method is given for sistency of the theory. Although in the limit the two calculating the integrals appearing in the matrix elemethods agree,neither method appearsto be thoroughly ments for the simpler processes. satisfactory theoretically. Nevertheless, it does appear The point of view which is taken here of the interthat we now have available a complete and definite action of charges differs from the more usual point of method for the calculation of physical processesto any view of field theory. Furthermore, the familiar Hamilo r d e ri n q u a n t u me J e cr to d y n a m i c s . tonian form of quantum mechanics must be compared Since we can write down the solution to any physical to the over-all space-time view used here. The first problem, we have a complete theory which could stand section is. therefore. devoted to a discussion of the by itself. It will be theoreticallyincomplete,however, relations of these viewpoints. in two respects.First, although each term of increasing r. COMPARISONWITH THE HAMILTONIAN order in e2fhc can be written down it rvould be desirable METHOD to see some way of expressing things in finite form to all orders in e2fhc at once. Second, although it will be Electrodynamics can be looked upon in two equivaphysically evident that the results obtained are equiva- lent and complementary ways. One is as the description lent to those obtained by conventional electrodynamics of the behavior of a field (Maxwell's equations). The the mathematical proof of this is not included. Both of other is as a description of a direct interaction at a rheselimitations will be removed in a subsequentpaper distance (albeit delayed in time) between charges (the (seealso Dysona). solutions of Lienard and Wiechert). From the latter Briefly the genesisof this theory was this. The con- point of view ligbt is consideredas an interaction of the ventional electrodynamics was expressed in the La- chargesin the sourcewith those in the absorber. This is grangian form of quantum mechanics described in the an impractical point of view because many kinds of Reviews of Modern Physics.6The motion of the field sourcesproduce the samekind of efiects.The field point oscillatorscould be integratedout (as describedin Sec- of view separates these aspects into two simpler probtion 13 of that paper), the result being an expressionof lems, production of light, and absorption of light. On the delayed interaction of the particles. Next the modi- the other hand, the field point of view is less practical fication of the delta-function interaction could be made when dealing with close collisions of particles (or their directly from the analogy to the classical case.2This action on themselves).For here the sourceand absorber a Schwinger,?hys. Rev. 74, 1439 (1948), Phys. Rev. 75, 651 are not readily distinguishable, there is an intimate J. (1949L,A proof of this equivalenceis given by F. J. Dyson, Phys. exchangeof quanta. The fields are so closely determined by the motions of the particlesthat it is just as well not Rev. 75. 486 (1949). 6 R . P . F e y n m a nR, e v . M o d . P h y s . 2 0 , 3 6 7( 1 9 4 8 )T. h e a p p l i c a . to separate the question into ttyo problems but to contion to electrodynamics is descrihedin detail by H. J. Croenewold, sider the process as a direct interaction. Roughly, the Koninklijke Nederlandsche Akademia van Weresrhannen.Pro ceedings Vol. LII,3 (226) 1949. field point of view is most practical for problems involv-

238 Q Li.\ N TU X{

E L E CT R O D Y N A N,I CS

ing real quanla, while the interaction view is best for the discussionof the virtual quanta involved. We shall emphasizethe interaction viervpoint in this paper, first becauseit is less familiar and thereforerequiresmore discussion,and secondbecausethe important aspectin the problems rvith which we shall deal is the effect of virtual quanta. The Hamiltonian method is not well adapted to representthe direct action at a distancebetweencharges becausethat action is delayed.The Hamiltonian method representsthe future as developingout of the present. If the valuesof a compleleset of quantilies are known norv, their valucs can be computed at the next instant in time. If particles interact through a delayed interaction, horvever, one cannot predict the future by simply knorving thc present motion of the particles. One rvould also have to knorv what the motions of the particleswerein lhe past in view of the interaction this ma1' have on the future motions. This is done in the Hamiltonian electrodynamics,of course,by requiring that one specify besidesthe present motion of the oarticles. the values of a host of ncrv variables (the ioordinatesof the field oscillalors)to keep track of that aspcct of the past motions of the particles which determinestheir future behavior. The use of the Hamiltonian forces one to choose the lield viewpoint rather than the interactionvielvPoint. In many problems,for example,the close collisions of particles, we are not inlerested in the precisetemporal sequcnceof evenls.It is not of interest to be able to say horv the situation would look at each instant of time during a collisionand how it progressesfrom instant to instant. Such ideasare only useful for events taking a long time anclfor which we can readily obtain informationduring the interveningperiod.For collisions it is much easierto treat the processas a lvhole.6The Mlller interactionmatrix for the the collisionof trvo electrons is not essentiallymore complicaledthan the nonrelativistic Rutherford formula, yeL the mathematical machinery used to obtain the former from quantum electrodynamics is vastly more complicated than Schrcidinger's equation wilh the e2frp inleraction needeclto obtain the latter. The differenceis only that in the latter the action is instantaneousso that the Hamiltonian method rcquiresno extra variabies,while in the lormer relativistic case it is delal'ed and the . Hamiltonian method is very cumbersorne We shall be discussingthe solutions of cquations rather than the time differentialcquations{rom which they come.We shall discoverthat the solutions,because of the over-all space-timeview that they permit, are as easv to understand when interactions are delayed as rvhenthey are instantaneous. As a further point, relativistic invariancewill be selfevident. The Hamiltonian form of the equations develops the future from the instantaneousprcsent. Ilut 6 This is the viewpoint bers.

of the theory- oi the S matrix

of }fuisen-

77r

for different observers in relative motion the instantaneous present is different, and corresponds to a different 3-dimensionalcut of space-time.Thus the temporal analysesof different observersis different and their Harniltonian equations are developing the process in difierent ways. These differencesare irrelevant, howe v e r , f o r t h e s o l u t i o ni s t h e s a m c i t r a n y s p a c et i m e frame. By forsaking the Hamiltonian method, the wedding of relativity and quantum mechanics can be accomplishedmost naturallY. We illustrate these points in the next section by equationfor nonstuclyingthe solutionof Schrcidinger's relativistic particles interacting by an instantaneous Coulomb potential (Eq. 2). When the solution is modified to include the efiects of delay in the interaclion and the relativisticpropertiesof the electronswe obtain an expressionof the laws of quantum electroclynamics (Eq. 1). 2, THE INTERACTIONBETWEENCHARGES We study by the samemethodsas in I, the interaction of two particlesusing lhe samenotation as I' We start by consideringthe non-relativisticcasedescribedby the Schrridingerequation (I, Eq. 1). The wave lunction at a given time is a function /(t., xu, l) of the coordinates 'Ihus catlK(x", x6,t',x"t,xb',l') xo and xoof eachparticle. the amplitude that particle o at xo' at time I' will get to x" at t while particle b at xb'at l' gets to xb at ,' If the particlesare free and do not interact this is K(x", x6,I ; x o', x6', l') : K p"(x t ; x t' ) K ot(xt,I ; xt', t' ) "', ", where Ko" is the ](o function for particle a considered as free. In Lhis casewe can obviously define a quantity like K, but for which the time I need not be the same f o r p a r t i c l e sa a n d b r l i k e w i s ef o r t ' ) ; e . g . , ( ')l . K o ( 3 ,{ ; 1 , 2 ) - K a . ( 3 , 1 ) K o D 2

(l)

can be thought of as the amplitude that particle o goes from x1 at 11to x3 at 13and that particle D goesfrom x2 ut tz to xt at t+ When the particles do interact, one can only define t h e q u a n t i t y K ( 3 . 4 : l . l ) p r e c i s e l yi l t h e i n l e r a c t i o n vanishesbetrveentt and lz and also betweenlr and lr. In a real physical system such is not the case.There is such an enormous advantage, however, to the concept that we shall continueto use it, imagining that rve can neglect the effect oI interactions between h and 12and between h ar^d ta. For practical problems this means choosirrgsuch long time intervals h-lr and tr-lzthat the extra interactions near the end points have small relative efiects. As an example, in a scattering problem it may rvell be that the particles are so well separated i n i r i a l l y a n d f r n a l l yt h a t t h e i n t e r a c t i o na t t h e s el i m e s is negligible. Again energy values can be defined by the avera€lerate of change of phase over such long time intervals that errors initially and finally can be neglected. Inasmuch as any physical problem can be defined in terms of scattering processeswe do not lose much in

239 i72

R.

P. FEYNMAN

o' +{4,6)

This turns out to be not quite right,T for when this interaction is representedby photons they must be of only positive energy, while the Fourier transform of 6(156-rs) containsfrequenciesof both signs.It should insteadbe replacedby 61(ls6-156)where r'

6*( + (5 ,l ) ELECTRONS Fro. 1. The fundamental interaction Eq. (4). Exchange of one quantum between two electrons.

6-(.):

|

ro

€ id,Jd r-lim c-u

(ri)-r -:6(r)+(rirr-r.

(j)

J_i€

This is to be averaged with /b6-t6+(-lse-rro) which arises when /5(16 and correspondsto a emitting the quantum which D receives.Srnce (2r)-\(6.,(t- r) l 6a( t- r)) : 61(P- r,),

this means rro 160s0)is replaceclby da(s662) where a general theoretical senseby this approximation. If it Smz:lda2-rsa2 is the square of the relativistically inis not made it is not easy to study interacting particles variant interval between points 5 and 6. Since in relativistically, for there is nothing significant in choosclassicalelectrodynamicsthere is also an interaction ng h:h if xrlxa, as absolutesimultaneity of events through the vector potential, the completeinteraction at a distance cannot be delined invariantly. It is essen(seeA, Eq. (1)) should be (1-(v6.v6)d1(s562), or in the tially to avoid this approximation that the complicated relativistic case, structure of the older quantum electrodynamics has been built up. We wish to describe electrodynamics as (1- o,. ca)6+(ssd) : 0"0a2",?a,6+(suut). a delayed interaction between particles. If we can make Hence we for electrons have obeying the Dirac equation, the approximationof assuminga meaningto K(3, 4; 1, 2) the results of this interaction can be expressedvery ff simply. K(r'(34 . ; 1 , 2 ) : - ; r z I t K . , ( 3 , 5 t K - r @ ,6 \ - y " , - y t , JJ To see how this may be done, imagine first that the interaction is simply that given by a Coulomb potential l) K+b(6,2)d.r6dr6, (4) X 6+(soer)K+.(5, e2fr where r is the distance between the particles. Ii this be turned on only for a very short time Als at time 10, where loA and 76! are the Dirac matrices applying to the first order correctionto -K(3,4; 1,2) can be worked the spinor corresponding to particles a and D, respecout exactly as was Eq. (9) of I by an obvious general- tively (the lactor 0"At being absorbed in the definition, ization to two particles: I E q . ( 1 7 ) ,o f K * ) . This is our fundamentalequationfor electrodynamics. K ( t ) ( 3 , 4 :1 , 2 ) : - ; r z | | K . , r J , 5 ) A ' ' b ( +6,) / s 6r It describes the effect of exchange of one quantum (therefore first order in e2) between two electrons. It X f o"(5, l) K x6(6,2) d3x5d3x6Lt n, will serve as a prototype enabling us to write down the correspondingquantities involving the exchangeof two where l5:t6:r0. If now the potential were on at all or more quanta betweentrvo eiectronsor the interaction times (so that strictly K is not defined unlesslr:ta and of an electron with itself. It is a consequenceof conh:tz), the first-order effect is obtained by integrating ventional electrodynamics. Relativistic invariance is on 16,which we can write as an integral over both lu clear. Since one sums over p it contains the effectsof and lo if we include a delta-function6(ts-16) to insure both longitudinal and transverse waves in a relaticontribution only when ls:16. Hence, the first-order vistically symmetrical way. efiect of interaction is (calling 16-16:lb6)' We shall now interpret Eq. (4) in a manner which will permit us to write dorvn the higher order terms. It ff - ic2 l can be understood (seeFig. 1) as saying that the ampliK("(3, 4; l. 2\: | | 1{0.f3, 5)Kn6(4, 6)156 tude for "a" to go from 1 to 3 and "b" to go from 2 to 4 is altered to first order because they can exchange a X6(t56)K0.(5, 1)Kob(6, 2)d,rsdra, Q) quantum. Thus, "o" can go to 5 (amplitude I(+(5,1)) where d,r:d.1xdl. 7 It, and a like term for the effect of a on b, leads to a theory We know, however, in classicalelectrodynamics,that the Coulomb potential does not act instantaneously, which, in the classical limit, exhibits interaction through halfadvanced and half-retarded potentials. Classically, this is equibut is delayed by a time 156,taking the speed of light valent to Durely retarded effects within a closed liox from which as unity. This suggests simply replacing r6;t6fts6) in no lighr e'capei (e.g.. see A, or J. A. Wheeler and I{. P. Feynman, (2) by something like r5;r6(fu6-lb6) to representthe Rev. Mod. Phys. 17, 157 (1945)). Analogous rheorems exisr in q-uantum mechanics but it would lead us too far astray to discuss delay in the efiect ol b ot a. Inem now.

240 Q U A N T U N { E L E C T R O D Y N A X ,I C S emit a quantum (longitudinal, transverse,or scalar once (either in emission or in absorption), terms like (f, Bq. (t+)) occur only when there is more than one "y.u)and then proceedto 3 (Ka(3,5)). Meantime "0" goes to 6 (K+(6, 2)), absorbs the quantum (ryru)and quantum involved. The Bose statistics of the quanta proceedsto 4 (K1(4,6)). The quantum meanwhilepro- can, in all cases,be disregarded in intermediate states. ceedsfrom 5 to 6, rvhichit doeswith amplitude 6n(su6:). The only effect of the statistics is to change the weight We must surr over all the possiblequantum polariza- of initiai or linal states.If there are among quanta, in tions p and positions and times of emission5, and of the initial state, some z rvhich are identical then the absorption 6. Actually if li>fu it would be better to weight oi the sta.teis (1/n l) of what it would be if these emits but no attention quanta were considered as dilTerent (similarly for the say that "o" absorbs need be paid to thesematters, as all such alternatives final state). are automaticallycontainedin (4). PROBLEM 3. THE SELF-ENERGY The correct terms o{ higher order in d or involving largernumbcrsof electrons(interactingwith themselves or in pairs) can be written down by the same kincl of reasoning.They will be illustrated by examplesas rve proceed.In a succeedingpaper they rvill all be deduced f rom conventionalquantum electrodynamics. Calculation,lrom (,1),of the transition element between positive energy free electron states gives the M6ller scattering of trvo electrons,rvhen account is taken of the Pauli principle. The exclusion prinr:iple for interacting chargcs is handled in exactly the same*'ay as for non-interacting charges (I). For example,for two chargesit requires o n l y t l r a t o n e c a l c u l a t eK ( . 3 , 4 ; t , 2 ) - K ( 4 , 3 ; 1 , 2 ) t o get the net amplitude {or arrival of chargesat 3 and 4. It is disregardedin intermediate states. The interferenceeffectsfor scatteringof electronsby positrons discussedby Bhabha rvill be seen to result directly in The formulasare interpretedto apply this formr.rlation. to positronsin the manner discussedin I. As our primary conccrn rvill be for processesin which the quanta are virtual we shall not include here the detailcd analysisof processcsinvolving real quanta in initial or 6nal state,and shall content ourselvesby only stating the rules applying to them.8The result of the analysisis, as expected,that they can be included by the same line oI reasoningas is used in discussingthe providedthe quantitiesare normalized virtual processes, in the usual manner to representsingle quanta. For example, the amplitude that an electron in going from 1 to 2 absorbsa quantum whosevector Ilotential,suitably normalized,is cuexp(-i,t r):Cu(r) is just the expression (I, Eq..(13)) for scattering in a potential rvith ,4 (3) replacedby C (3). Each quantum interacts only

Having a term representingthe mutual interaction of a pair of charges, we must inclucle similar terms to representthe interaction of a charge with itself. For under some circumstanceswhat appears to be trvo distinct electronsmay, accordingto I, be viervedalso as a single electron (namely in case one electron was createdin a pair with a positron destinedto annihilate the other electron). Thus to the interaction betrveen such electronsmust correspondthe possibiLityof the action of an electronon itself,e This interaction is the heart of the self energy problem. Considerto first order in e'!theaction of an electron on itse!f in an otherwiseIorcefreeregion.The amplitude K(z,t) Ior a single particle to get from 1 to 2 difiers from Kn(2, 1) to first order in e2by a term ff K r ' ( 21. ) : ' i i I I K , ( 2 . A ) t u K( , +3. ) y ,

(srt')' (6) XKa(3, 1)dr3d'rn6. It arises becausethe electron instead of going from 1 directly to 2, may go (Iiig. 2) first to 3, (Ka(3, 1)), emit a quantum (.yu),proceed to 4, (Ka(4,3)), absorb it ("yu),and finally arrive at 2 (K1Q,.Q). The quantum must go from 3 to + (6+(sca')). This is related to the self-energyof a free electron in the followingmanncr.Supposeinitially, time lr, \1.ehave an electron in state /(1) rvhich we imagine to be a positive energy solution o{ Dirac's equation for a free particle. After a long time /:-lr the perturbation will alter

-' at,lorgh in the expressions stemming from (4) thc quanta are virtual, tlis is not actually a theoretical limitation. One rvay to deduce the correct rules for real quanta from (4) is to note that in a closed system ali quanta can be consiclerecl as virtual (i.e., they have a knorvn source ancl are eventually absorbed) so that in such a system the present description is complete and erluivalent to the conventional one. In particular. the relation of the E i n s t e i n , 4 a n d B c o e l l i c i e n t sr a n b e ' l e , l u c e J . A m o r e p r c c t i c r l direct deduction of the expressions for rcal quanta rvill be givcn in the subsequent paper. It might be noted that (4) can be rewritten as describing the action on a, K(I)(3, l):if K*G,5) 1)d"e of the potential,4 u(5) : e'?-lK+(4, 6)6+(sro'])ru X l(5)r+(5, -A'zAp:4rju arising from Maxwell's equations XKa(6,2)dr6 from a "current" jr(6\:e2K-(4.6r7u4-r6.2r p r o r l u c e r Jb y p a r ticle D in going from 2 lo 4. This is virrue oi rhe facr thal 6+ satisfies - trl6a(s'11'?):{'512' 1;' (5)

Frc. 2. Intcraclion of an electron ilith itself, Eq. (6).

e These consideratiorrs make it appear unlikely that the conRev. Mod. Plys. tention of J. A. Wheeler and R. P.-ieynman, 17, 157 (1945), that electrons do not act on themselves, will be a successful concept in quantum electrodynamics.

241 114

P.

FEYNMAN

the wave function, which can then be looked upon as a superposition of free particle solutions (actually it only contains /). The amplitude that g(2) is contained is calculatedas in (I, Eq. (21)). The diagonalelement (g:,f) i. therefore f

f -

I f / ( 2 ) B K ' r ) ( 2 .l 1 B l ( 1 t r t 3 x f i 3 x 2 .

volume. If normalized to volume I/, the result would simply be proportional to l. This is expected,for if the efiect were equivalent to a change in energy A.E, the amplitude for arrival in J at t2 is altered by a factor exp(-iL,E(t2-tr)), or to 6rst order by the difierence -i(AE)I:. Hence, we have

(7)

JJ

L E : e 2" f| , (_u 1 u K , ( 4 . 3 ) 1 u r ) e x p ( i 2 . . r q r ) 6( s, r J ) d r r , ( a )

The time interval T: tz- h (and the spatial volume I/ over u'hich one integrates) must be taken very large, for the expressionsare only approximate (analogousto the situation for two interacting charges).l0 This is because,for example, we are dealing incorrectly with quanta emitted just before l, which would normally be reabsorbedat times after 12. Il Ko(2,1) from (6) is actually substituted into (7) the surface integrals can be perlormed as was done in obtaining I, Eq. (22) resulting in

integrated over all space-time drr. This expressionwill be simplified presently. In interpreting (9) we have tacitly assumedthat the wave functions are normalized so that (u*u):(u7au):1. The equation may therefore be made independent of the normalization by writing the left side as (LE)(A1 u), or since (rzTrrz): (E/ nx)(Au) and mLm: ELE, as Am('au) where Lm is an equivalent change in rnassof the electron. In this form invariance is obvious. One can likewise obtain an expressionfor the energy f f _ - i", | | l ( a ) y , K - ( a , 3 ) y , / ( 3 ) 6 + ( s o , r r d ' , d ' , . ( 8 ) shift for an electron in a hydrogen atom. Simply replace K1 in (8), by K+(v), the exact kernel for an electron in Putting for /(1) the plane wave z exp(-i2.r1) where the potential, V- Be2/r, of the atom, and / by a wave p" is the energy (1r) and momentum of the electron function (of space and time) for an atomic state. In general the AE which results is not real. The imaginary (f:m'), and ur is a constant 4-index symbol, (8) part is negative and in exp(-iA-El) produces an exbecomes ponentially decreasingamplitude with time. This is becausewe are asking ior the amplitude that an atom -ir' | | (u1uK*(4,3)7,u) initially with no photon in the field, will still appear after time ? with no photon. If the atom is in a. state X exp(ip' (r a- 4 )) 61(sa3,)dr3dra,which can radiate, this amplitude must decay with time. The imaginary part ol AE when calculated does the integrals extending over the volume I/ and time indeed give the correct rate of radiation from atomic interval I. Since K".(4, 3) dependsonly on the difference states. It is zero for the ground state and for a free of the coordinates of 4 and 3, ra3u, the integral on 4 electron. gives a result (except near the surfaces of the region) In the non-relativistic region the expression for AE independent of 3, When integrated on 3, therefore, the can be worked out as has been done by Bethe.rr In the result is of order VT. The effect is proportional to Z, relativistic region (points 4 and 3 as close together as a for the wave functions have been normalized to unit Co{npton waveJength) the K*{rr which should appear in (8) can be replaced to 6rst order in I/ by K1 plus K+(D(2,1) given in I, Eq. (13). The problem is then very similar to the radiationless scattering problem discussedbelow.

M O M E N T U pM- k , F A C T O(R9 - k - m

K, MOMENTUM FACTORk-2

TNTERACTTON,T p MOMENTUM

4. EXPRESSIONIN MOMENTUM AND ENERGY SPACE The evaluation of (9), as well as all the other more complicated expressionsarising in these problems, is very much simplilied by working in the momentum and energy variables, rather than space and time. For this we shall need the Fourier Transform of 6+(szr2)which is - 6 - ( s 2 ' 2 ) : a I. f e x p ( - i A ' r z t ) h 2 d 4 h ' I

FrG. 3. Interaction of an electron with itself Momentum space, Eq. (11).

10This is discussedin reference5 in which it is pointed out that tie concept of a wave function losesaccuracy il there are delayed self-actioni;.

fl0)

which can be obtained from (3) and (5) or from I, Eq. (32) noting that Ia(2,1) Ior mz:O is 61(sr1'?) from rrH. A. Bethe,Phys.Rev.72,339(1947).

242 QUANTUM

I lJ

ELECTRODYNAMICS

//

,"fr n-il,'-" 1,,.\, t,-tt

lP' a. Eq.lz

,K1

t'-ov lt'

f',

b. Eql3

c. Eq.l4

Frc. 4 Radiative correction to scat tering, momentum

Frc. 5. Compton space.

r or more precisely I, Eq. (3a). The ft-'?means (ft'[) tile limit as 6+0 of (&'E+tD)-r. Further da& means (2r)-2dkflkzdhdftr. If we imagine that qxanta are pariic1"s of ,".o mass, then we can make the general rule that all poles are to be resolved by considering.the massesof the particles and quanta to have infinitesimal

scattering,

Eq

(15)'

trated in Fig. 4(a), f,nd the matrix: @/ ri)

I

s)-\ "t,(P,- h- m)-r s(P'- r-

k' uk-2da

(rz)

For in this case, firstr2 a quantum of momentum ft is emitted (rr), the electron then having momentum (rtt- h- n).-t ' fr- ft and hence propagating with factor negative imaginary Parts. it is scatteredby the potential (matrix a) receiving Next flsing these results we see that the self-energy (9) is additional momentum g, propagating on then (factor the mairix element between u and u of the matrix with the new momentum until the quan(lz-k-m1-tl tlum is reabsorbed (y"). The quantum propagates from zd'ak, ( 1 1 ) emission to absorption (lr') and we integrate over all @/,DI u@- a- m)-'1rk ouanta (da[), and sum on polarizationp. When this is intesrated on frr, the result can be shown to be exactly to the expressions(16) and (17) given-in B for where we have used the expression(I, Eq' (31)) for the "quil proceis, the various terms coming from resithe self-energy the same for form This K.,. of Fourier transform duesof thl polesof the integrand (12). is easier to rvork with than is (9). Or again-iI the quantum is both emitted and reThe equation can be understood by imagining (Fig' 3) before thJ scattering takes place one finds that the electron of momentum f emits (7r) a quantum absorbeld (r'ie. +(b)) monow with its way makes and ft, of momentum mentum p-k to the next event (Iactor (!-h-*)-t) f h- m)'t"vpk-2d4k, (13) which is to absorb the quantum (anothervu)' The a(p1- m)-\,(IrtC/ I (There "D is a amplitude of propagation of quanta is k-'?' lacior d/ri for each virtual quantum)' One integrates p or if uottt" e-ission and absorption occur after the over all quanta. The reasonan electron of momentum scattering, (FiS. a(c)) rs rethe operator this is that propagates as l/(l-m) iioio.ut oi the Dirac equation operator, and we are (14) simply solving this equation. Likewise light goes as e / rD I t,(b"- h- rn)-'t t(fz- m)-' alrzd4k' 1/H, lor this is the reciprocal D'Alembertian operator of the wave equation of light. The first ?N represents t"..rl, are discussedin detail below. Tl the current which generatesthe vector potential, while "." have now achieved our simplification of the lorm We potenthe secondis the velocity operator by which this of writing matrix elements arising from virtual proclial is multiplied in the Dirac equationwhen an external esses.PrJcessesin which a number oI real quanta is fleld acts on an electron. given initially and finally offer no problem (assuming may problems other reasoning, of the same line Using "up normalization). For example, consider the Iorrect example, For directly in momentum space. be set efiect (Fig. 5(a)) in which an electron in state Compton considerihe scatieringin a potential A:Ar1yrvarying a quantum oI momentum q1, polarrzauon absorbs fr ^t".tor in spaceand time as d exp(-iq'r). An electroninitially so that its interaction is eLp"f r:eL, and.emits rr" in itate of momentum ft:?rr"yu will be deflected to a secondquantum of momentum -92' polarization-e2 The zero-order answer ls state pz whete bz:Pt*q. to arrive in final state of momentum pz. The matrix foi simply the matrix element of c between states 1 and 2' 't F1..,- **,, etc., here reler not to the order in rrue time but to We next ask for the first order (in e2)radiative correc- ttt. *ii"iti"" ntong the trajectory of the-eleclron' That iri"tlnit tion due to virtual radiation of one quantum' There are is, more precisely, to the order of appearance ol the matrlces ln several ways this can happen. First for the case illus- the expressrons.

243 776

R.

this processis ez(btlqt-m)-re1. the Compton effect is, then, er(f tl qr-m)-,et*

er(ltl

P.

FEYNMAN

The total matrix for qz- m)-tez,

(15)

for propagation of quanta of momentum ft is - F ' + ( h 2 ) :r ' t ,

f6

(ft-,-(ft,-trr)-r)G(I)dI,

Jo

the secondterm arising becausethe emission of ez may alsoprecedethe absorptionof e1 (Fig. 5(b)). One takes rather than k-2. That is, writing Fa (H): - r-rp-261p21, matrix elementsof this between initial and final electron states (pr-l-qr:02-Qz), to obtain the Klein Nishina - / - ( s 1 2 2 ) : . - t e x p ( - i i . 1 1 r ) f t - 2 c ( h 2 ) d 4 h .( l o r for'mula. Pair annihilation with emissionof two quanta, I J' etc., are given by the samematrix, positron states being those with negative time component of p. Whether Every integral over an intermediate quantum which q u a n t aa r ea b s o r b e do r e m i l t e dd e p e n d so n w h e l h e rt h e previously involved a fa"ctordak/ h, is now supplied with time component of q is positive or negative. a convergencefactor C(ft,) where 5. THE CONVERGENCEOF PROCESSESWITI{ VIRTUAL QUANTA These expressionsare, as has been indicated, no more than a re-expression of conventionalquantum electrodynamics. As a consequence,many of them are meaningless. For example, tl.re self-energy expression (9) or (11) givesan infinite resultwhen evaluated.The infinity arises,apparently, from the coincidenceof the d-function singularitiesin K*(4,3) and 6*(sa3r). Only at this point is it necessaryto make a real departure from conventional electrodynamics, a departure other than simply rewriting expressionsin a simpler form. We desireto make a modification of quantum electrodynamics analogous to the modification of classical eLectrodynamics described in a previous article, A. There the d(s12')appearing in the action of interaction s'as replaced by /(srl) where /(r) is a function of small ividth and great height. The obvious correspondingmodification in the quantum theory is to replace the 6a(sr) appearing the quantum mechanical interaction by a nelv function ,/*(s'?). We can postulate that if the Fourier transform of the classical/(srl) is the integral over all ft of F(h2) exp(-ih.rrz)d.nh, then the Fourier transform of ,f1(s'?)is the same integral taken over only positive frequencies kq for tzlh and over only negative ones for :2(/1 in analogy to the relation of 6a(sr)to d(sr).The iunction f(s'?):[email protected]) can be written* as r'6?

.,*.r':(2')'f v k':o

u

I sinlAnlxrlr X c o s ( K .x ) d A a d 3 K g (kk).,

ri'hereg(fr'b) is Eal times the density of oscillatorsand may be expressedfor positive Aaas (A, Eq. (16)) f'

c(h'):

t

(6(ft'?)-6(ft,-I?))G(I)dI,

Jo

rtere f-G(tr)dX:l and G involves values of X large :ompared to az. This simply means that the amplitude *.This_relation is given incorrectlyin A, equationjust pre:eding 16.

c(h\:

I:-),'(ft'?-x)-r6i1;41'

o7)

The poles are defined by replacing ft, by ft?f i6 in the limit 6-0. That is X, may be assumedto have an infinitesimal negativeimaginary part. The function ./*(su,) may still have a discontinuity in value on the light cone.This is of no influencefor the Dirac electron. For a particle satisfying the Klein Gordon equation, horvever, the interaction involves gradientsof the potential tvhich reinstatesthe 6 function if / has discontinuities.The condition that / is to have no discontinuity in value on the light coneimplies ft'C(&'?) approaches zero as ft2 approaches iniinity. In t e r m s o f G ( I ) t h e c o n d i r i o ni s

.f"

x,c1x)ax:0.

(18)

This condition will also be used in discussingthe convergence of vacuum polarization integrals. The expressionfor the self-energymatrix is now (e\/rit | 7u(p-h-mt-11ph 2d4kclh?\, (lq) J rvhich, since C(ft') falls off at least as rapidly as l/hr, converg€s. For practical purposes we shall suppose hereafter that C(ft,) is simply -Xr/(ftr-X!) implying that someaverage(with weight G(I)dX) over valuesof X may be taken afterwards. Since in all processesthe quantum momentu.m will be contained in at least one I representing extra factor of the form (tr-k-*) propagation of an electron while that quantum is in the field, we can expect all such integrals u'ith their convergencefactors to converge and that the result of all such processeswill now be finite and definite (excepting the processeswith closedloops, discussedbelotv, in which the diverging integrals are over the momenta of the electrons rather than the quanta). I noting The integralof (19) withC(e:): -Xr(ftr-tr) that P2:1n2, X)za and dropping terms of order tnf )r, is (seeAppendix A) (e'z / 2r) l+m (tn(\,/ n) * il - ! (.tn(x/ n) -l s/ +)1. Q0)

244 QUAN T U M

777

E L E CTRO D Y N A M I CS

When applied to a state of an electron of momentum p satisfyingpu:nu, it gives for the changein mass (as in B, Eq. (9))

We must now study the remaining terms (13) and (1a). The integral on ft in (13) can be performed (after multiplication by C(ft')) since it involves nothing but the integrai (19) for the self-energy and the result is Am:m(e,/2r)(31n(l'/m)Il). Ql) allowed to operate on the initial state u1, (so that prur:mur). Hence the factor following a(p\-nt) | will 6. RADIATIVE CORRECTIONSTO SCATTERING be just Az. But, if one now tries to expand l/(ft-m) : (pt+m)/ (br'-m2) one obtains an infinite result, We can now complete the discussionof the radiative correctionsto scattering. In the integrals we include the since p12:m2. This is, however, just what is expected convergence tactor C(E), so that they converge for physically. For the quantum can be emitted and ablarge ft. Integral (12t is also not convergentbecauseof sorbed at any time previous to the scattering. Such a the well-known infra-red catastrophy. For this reason processhas the efiect of a changein mass of the electron we calculate (as discussedin B) the value of the integral in the state 1. It therefore changesthe energy by A.E assumingthe photonsto havea small masstr-i"(rz(X. and the amplitude to first order in AE by - iLE t where The integral (12) becomes I is the time it is acting, which is innnite. That is, the major effect of this term would be canceledby the efiect of change of mass An. ( c / r i ) | ^ yp ( p , - h - n ) ' a ( f , - h - m \ -| The situation can be analyzed in the following manner. We supposethat the electron approaching the X 7,(ft?- tr-r"'?)-tdahC(k'z \^;,2'), scattering potential c has not been free for an infinite timei but at some time far past suffereda scattering by which when integrated (see Appendix B) gives (e2/2n) a pdtential b, If we limit our discussionto the effects times oI Lrn and of lhp virtual radiation oI one quantum between two sgch iscatterings each of the effects will be 2d \ t / m \/ finite, though large; and their differenceis determinate. I z( tn--1 l( l--)*d rend /\ Ltn20'/ L \ X-i" The propagation from b to d is representedby a matrix +

4ro-l | atanadalc Lan20Jo I 120 *-(qa4m

aq)-tra, sin20

a(!'-m)

(22)

|b,

(2s)

in which one is to integrate possibly over p' (depending on detailsof the situation). (If the time is long between b and a, the energy is very nearly determined so that p'2 is very near)y m2.) We shall compare the effect on the matrix (25) of the virtual quanta and of the changeof massArn. The efiect of a virtual quantum is

where (qz)\:2* sin0 and u'e have assumedthe matrix to operatebetweenstatesof momentum fu and Pz: Frl Q and have neglected terrns of order X;in/2, m/l', and g'/\'!. Here the only dependence on the convergence - h- 1n)-, (e"/ ri) aQ' - m) ]'yp(.p'I factor is in the term zc, rvhere [ )e u(!, - m)-'btt rd4bc(h2), (26) (23) /:h(\/m)+9/4-2tn(m/\,";). As we shall see in a moment, the other terms (13), (14) give contributions which just cancel the ra term. The remaining terms give for small q,

while that of a chanse of mass can be written a(P'-n')-tLm(P'-*)'b'

(27)

and we are interested in the difference (26)-(27). A tr 3\\ l q- ' / /l simple and direct method of making this comparison is dl ln -1 e ' ? / 4 n 1- t1q a - d g r | | l , t ) 4 1 just to evaluate the integral on ft in (26) and subtract Jn'? \ tr-i" 8/ / \2n from the result the expression (27) where Azr is given which sholvs the change in magnetic moment and the in (21). The remaindercan be expressedas a multiple -r(p/2) ot the unperturbedamplitude (25); Lamb shift as interpretedin more detail in B.rl rsThat the result given in B in Eq. (19) was in error was repeatedly pointed out to the author, in private communication, by V. F. Weisskopl and J. B. French, as their calculation, completed simultaneously with the author's early in 1948, gave a aifferent result. Freuch has finally shown that although the exp r e s s i o nf u r t h c r a , l i c t i o r , l e s s c a l t e r i n g B , E q . ( 1 8 ) o r ( 2 4 ) a b o v e i s c o r r e c t , i t w a s i r r c o r r e c t l yj o i n c d o n t o B e t h e ' s n o n - r e l a l i v i s t i c 1 = ln),-i" used by the result. He shows that the relatior ln2i-.=This results io author should have been ln2A-,*-5/6:lnl-ro. adding a term - (1/6) to the logarithm in B, Eq. (19) so that the result now agrees rvith that of J. B. French and V. F. Weisskopf,

-r(p4)a(p'-ry)-LS.

(28)

This has the same result (to this order) as replacing and the potentials c and b in (25) by (l-ir(P''))a **. ZS, l24O(lg4g) anclN. H. Kroll and W. E. Lamb, Phys.Rev.75,388(1949).Theauthorfeelsunhappilyresponsible "frr.. [or tbe very considerable delay in the pul,lication of French's result occasioned Ly this error. This lootnote is appropriately numberecl.

245 i78

P.

FEYNMAN

(.1-ir(p''))b. In the limit, then, as 0'2--+tn2the net effect on the scattering is -|ra where r, the limit of (assuming the integrals have an infrar(f'2) as f',-m' red cut-ofi), turns out to be just equal to that given in (23). An equal term - |rc arisesfrom virtual transitions after the scattering (14) so that the entire zc.term in (22) is canceled. The reasonthat r is just the value oI (12) when q'?:0 can alsb be seenwithout a direct calculation as follows: Let us call p the vector of length m in the direction oi p' so that il p'z:1r111r12 we have p': (1f e)1 and we rake e as very small, being oI order ?-l where I is the time betrveen the scatterings b and a. Since (p'-m1-t - (f' * m) / (P'"- m'z)- (p{ m) / 2m'?€,the quantity (25) rs o{ order e-r or T. We shall compute corrections to it only to its own order (e-t) in the limit e+0. The term , 27) can be written approximatelyraas o.

"it

tyu(p-h-n\ I a(p'-nD

|

X t,(!' - m) t 6 fz ia P(l(h2),

using the expression (79) for Lm. The net of the two efiects is therefore approximatelyrs - ( e 2 / r i 1| a l ! ' - m t ' t , l f - h - n )

lef(!-h-

m\-'

Xt ,(P' - m)-rblr2d4kc(k2), a term now of order 1/e (since (0'-tn)-'=(P+m) \(2nf e)-t) and therefore the one desired in the limit. Comparison to (28) gives for r the expression

,:7 * n / hn)

f

1*(P,- h- *)-' (f p-\) (p; h - m)-l llrlrzd.akC(h2).

(29)

The integral can be immediately evaluated, since it is the same as the integral (12), but rvith g:9, 1ot o rep)acedby Pt/*. The result is therefore r'(!r/m) q hich when acting on the statear1is jusl r, asPrur:mur in (29) is For the same reason the term (!t*m)/2m efiectively 1 and we are left with -r of {23).t6 In more complex problems starting with a free elec-

tt Th*pr"rrior of Aft is not exact because the substitution r v t h e i n t e g r a l i n r l a t i s . v a l i d o n l ) i f , o p e r a t e so n a . s t a l e b u c h 'ha1 t can be rcplacerl by u. lhe crror, no$ever, ls ol oruer rvhich is a',t I erp-m^P*m) a p')n1-tgt-ni11t'-zl-ib 'zm-a.But sincef :12?, ne havePQD-z) X(Glt)t'*m)!(2e+e) *n(b-m): = (f- z)1 so the ne t result is approximately arcl is not of order 1/e but smalier, so that its effect a(!-m)b/4nP irops out in the limit. td We have used, to lirst orcler, the general expansion (valid for any operators ,4, 4] A-t_ A_|BA t+A 181 tB.|-t_... ulB)_t: .,ri1a tr:p-k-m to expand the difierence of ancl B:b'-l:e! t and (b-h-n)-1. 'b ' - k - n \ 'u The r.normaliz-ation terms appearing B, Eqs. ( 14 r, (15r wbcn ;ranslated directly into the present notation do not Sive twice 29) but give this expression witi the central rrrt-r factor replaced 'y n1a/Et where E1-pru lor p:4. 1yL"n integraled it thereiore n)/2nor ra-ra(ntt/Et''(11erves rclQfr I n)/2n\(n1a/fr\ u h i c h g i v e s j u s t r s , s i n c ei t u t : m i l t ' S i n c ef 1 1 a f 1 4 f 1 : 2 E r )

tron the same t)?e oI term arises from the effects of a virtual emission and absorption both previous to the other processes.They, therefore, simply lead to the same factor / so that the expression (23) may be used directly and these renormalization integrals need not be computed afresh for each problem. In this problem of the radiative corrections to scattering the net result is insensitive to the cut-off. This means, of course, that by a simple rearrangement of terms previous to the integration we could have avoided the use of the convergencefactors completely (see for example LewistT). The problem was solved in the manner here in order to illustrate how the use of such convergence factors, even when they are actually unnecessary,may facilitate analysis somervhatby removing the effort and ambiguities that may be involved in trying to rearrange the otherwise divergent terms. The replacementof 61 by /* given in (16), (17) is not determined by the analogy with the classicalproblem. In the classicallimit only the real part of 61 (i.e., just 6) is easy to interpret. But by what should the imaginary part, l/(ris2), of D; be replaced?The choice we have made here (in defining, as we have, the location of the poles of (17)) is arbitrary and almost certainly incorrect. If the radiation resistance is calculated for an atom, as the imaginary part of (8), the result dependsslightly on the function [. On the other hand the light radiated at very large distances from a source is independent of /a. The total energy absorbedby distant absorbers will not check rvith the energy Ioss of the source.We are in a situation analogousto that in the classical theory if the entire / function is made to contain only retarded contributions (seeA, Appendix). One desires instead the analogue of (F),"t of A. This problem is being studied. One can say therefore, that this attempt to find a consistent modification of quantum electrodynamics is incomplete (seealso the question of closedloops, below). For it could turn out that any correct {orm of /1 which will guarantee energy conservation may at the same time not be able to make the self-energyintegral finite. The desire to make the methods of simplifying the calculation of quantum electrodynamic processesmore rvidely available has prompted this publication before an analysis of the correct {orm for /* is complete. One might try to take the position that, since the energy discrepanciesdiscussedvanish in the limit tr+@, the correctphysicsmight be consideredto be that obtained by ietting X+o after massrenormalization.I have no proof of the mathematicalconsistencyof this procedure, but the presumptionis very strong that it is satisfactory. (It is also strong that a satisfactory form for fi can be found.) 7. THE PROBLEM OF VACUUM POLARIZATION In the analysis of the radiative corrections to scattering one type of term was not considered.The potential u H. w. Lewis,Phys.Rev.73,173(1948).

246 QUANTU

N,I E LE CTRO D YNA

rvhich we can assumeto vary as au exp(-iq'r) creates -1r. This a pair ol electrons(seeFig. 6), momenta3., , i tt i n g x q u x n t u mQ : f o - ! " ' p a i r t h e nr e a n n i h i l a t eesm i v h i c hq u a n t u ms c a t t e r st h e o r i g i n r l e l e c l r o nf r o m s l a l c 1 to state 2. The matrir element for this process(and the others which can be obtained by rearrangingthe order in time of the various events)is - ( P / n i ) Q z t u u ' )l S P [ ( r . l q Xt,(P

"-

m)'

a,' m)-'7 u)daP.Y2C (q'?';

(30)

This is becausethe potential produces the pair with amplitude proportional to o,"y,, the electronsol moproceedfrom there to annihimerta p" and -(f"*q) Iate,producinga quantum (factoryu) which propagates (factor qaC(q'z)) over to the other electron, by which it is absorbed (matrix element of 7, between states 1 and 2 of the original eleclron (u27ru1)). All momenta p. and spin states of the virtual electron are admitted, which means the spur and the integral on dap" are calculated. One can imagine that the closed loop path of the positron-electronproduces a current

(31)

4ri r: J ,,a,,

which is the source cf the quanta which act on the . he quantity s e c o n de l e c t r o nT J , , : - t P , r i I I SP l (P 1 9 - m \ - t Xt,(F-

m)-'t,fdo ?,

(32)

is then characteristic for this problem of polarization of the vacuum. One seesat oncethat J", divergesbadly. The modification of 6 to / alters the amplitude with which the current j, will afiect the scatteredelectron,but it can do nothing to prevent the divergenceof the integral (32) and of its effects. One way to avoid such diflicultiesis appareut.Iirom one point of view we are consideringall routesby which a given electroncan get from one region of space-time to another, i.e., Irom the source of electronsto the apparatus which measuresthem. lirom this point of view the closedloop path leading to (32) is unnatural. It might be assumedthat the only paths of meaningare thosewhich start from the sourceand work their rvay in a continuous path (possibly conlaining many time reversals)to the detector. Closed loops would be excluded.We have already found that this may be done for electronsmoving in a fixed potential. Such a suggestionmust meet severalquestions,holvof the usual ever. The closedloops are a consequence hole theory in electrodynamics.Among other things, they are required to keep probability conserved.The probability that no pair is produced by a potential is

779

iVI CS

Frc. 6. Vacuurn polarization effcct on scattering, Eq. (30).

po+ !

not unity and its deviation from unity arisesfrom the imaginary part of J",. Again, with closed loops excludid, a piir of electronsonce createdcannot annihilate one another again, the scatteringoi light by light lvould be zero, etc. Although rve are not experimentally sure of these phenomena,this does seem to indicate that the closei1loops ate necess&ry.To be sure' it is ahvays possible that these matters of probability conservation. etc., will work themselvesout as simply in the caseof interacting particles as for those in a fixed the presumpl o t e n t i a l . L a c k l n gs u c ha d e m o n s t r a t i o n iion is that the difficulties of vacuum polarization are not so easily circumvented.18 An alternative procedure discussedin B is to assume tlrat the functiotr K+(2,1) used above is incorrect and is to be replacedby a modifiedfunction Ka'having no singularity on the light cone. The efiect of this is to p.otid" u convergencelactor C(p2-m2) Iot eaeryinte' sral over electron momenta.le This will multiply the integrandof (32) by C(f -m')C((fl d'-m'), sincethe and both integral was originally a6"-Irlq)d'f"day'a p" and po get convergencefactors. The integral now converqesbut the result is unsatisfactory'20 One ixpects the current (31) to be conserved,that is A l s o o n e e x p e c l sn o c u r r e n ti [ c ' qtir:o ot s,J ",:0. is a gradient, or o,:q, times a constant' This leadsto the condition Jr"Q,-O rvhich is equivalent to qrJr,:o since -/", is symmetrical. But when the expression (32) is integrated with such convergencefactors it does not satisfy this condition.By altering the kernel from K to another, K', which doesnot satisfy the Dirac equation we have lost the gauge invariance, its consequent current conservation and the general consistency of the theory. One can see this best by calculating J,q, direclly from (32). The expression within the spur becomes rvhich can be written as the @fS-m)-tq(t-m)-''t, (pIQ-m)-''vr. differenceof two terms: (f -"t)-tt,Each of theseterms would give the same result if the factor, for r","ga"tt"" dap werewithout a convergence rs It rvould be very interesting to calculate the -Lamb shiit accurateiy enough to be sure that the 20 megacycles expected p r e s e n t from vlcuum polarizationare actually re This technirrue also makes self energy and radlallonless scattering integralsfinitc cven without the modi6cation oi 6..to-/ for (and the consequent convergence {actor C(ft') for the riCiation the auanta). See B. m Added. to the terms given belorv (33) there is a term which is not gaugq for Crfrlr:-tr?tfr'?-tr'ir, I.^3,2Fr+:q?)iu, - / o added inveriant. t ln arl, lirion the charge rcnormallzal ron has to the losa;ithm.)

247 780

P,

FEYNMAN

the first can be converted into the secondby a shift o{ This doesnot result the origin of p, namely p':p+5. in cancelation in (32) however, for the convergence factor is altered by the substitution. A method of making (32) convergentrvithout spoiling the gauge invariance has been found by Bethe and by Pauli. The convergencefactor for light can be looked upon as the result of superposition of the efiects of quanta of various masses (some contributing negatively). Likewise if we take the factor C(p'z-m2) : -\2(12-m2-\2)-t so that (Pt-*") tc(p2-tn2) r we are taking the differ:(pr-nf)-r-(fr-mr-it') ence of the result for electrons of mass na and mass (X'+nOr. But we have taken this difference for each propagation between interactions with photons. They suggestinstead that once created with a certain mass the electron should continue to propagate with this mass through all the potential interactions until it closesits loop. That is iI the quantity (32), integrated over some finite range of p, is called J u,(m'?)and the corresponding quantity over the same range of p, but with az replaced by (nt+\'z)' is Ju,(il*X?) rve should calculate

It is zero for a free light quantum (9'?:0). For small q? it behavesas (2/15)q'(adling -f to the logarithm in the Lamb effect). For q2)(2m)2 it is complex, the imaginary part representing the loss in amplitude required by the fact that the probability that no quanta are produced by a potential able to proCuce pairs ((qt)'>z*) decreaseswith time. (To make the necessary analytic continuation, imagine m to have a small negativeimaginary part, so that (7-q'/4rn'z)t becomes -i,(q'/4nf -t)+ as 92 goes from below to above 4m2. Then 0: r / 2* iu tvhere sinhu: * (q'z/ !,mt- 1);, and - | / tanl = i tanhu - * i (q"- 4n2)r (q2) +.) Closedloops containing a number of qulnta or potential interactions Iarger than tlvo produce no trouble. Any loop with an odd number of intcractionsgiveszero (I, reference9). Four or more potential interaclionsgive integrals which are convergent even without a con'Ihe situation is vergence factor as is well known. analogous to that for self-energy, Once the simple problem of a single closeJ loop is solved there are n o f u r l h e r d i v e r g e n c ed i l f i c u l r i c s{ o r m o r e c o m p l e x processes.22

(32')

In the usual form of quantum electrodynlnics the longitudinal and transversewaves are grven seprr&te

J,"': '

I J0

lJ,,1rn'))-Ju,(m?+\')]G(X)dI,

8. LONGITUDINAL

WAVES

G(tr)satisrying.__J*6! therunctio. )11:,t"1tl [:?,f;"?1;*J':'1"j"'#3ffiJi,'j.}'.(j#,*]i":i l-G(r)r'dx:0. Then in 'h" are *ccesconsiderations

rangeof p integration*4"

:]p1:i:'."-",]31,11,i^ llll

i.e.ent fo.riro suchspecial

of the equation lury for *e aredealingwith thesorurions jntegral now converges.The"ll::9"1"t?ilnltY-::.*: result oI the integratlon "-l)tAu:4ni' rvith a current i' which is couservec ur of uvs(d(\) s\/\/ 'Br4r u'ul\ on dl over using this method is the rntegral

(see-Appendix c)

iJl"3:;i.,,1;iliJjl::::t.i];::'#;l!:!tt.':u

""0

To show that this is the caselve consider the amplif,,P:tude for emission(real or virtual) of a photon and show 'lhe that the divergence oI this amplitude vanishes. t h ep i n p u h r i z e J p h o t o n s t 1 l o r l o r e m i s s i o n amplirude l 4 n 2 + 2 q 2/ \ lf -l -rji) (1-lll d i r e c t i o ni n v o l v e sm a t r i x e l e m e n r so f 7 u . T n c r e f o r e rang'/ 9J) L 3qt \ w h a t w e h a v e t o s h o wi s r h a t t h e c o r r e s p o n d i t tmga t r i x elementsof qr"yu:Q vanish. For example' for a first wit]n q2:4mz sinz9. order effect n'e would require the matrix element oI q The gaugeinvarianceis clear,sincequ(quq,-g?6u):0. Operating (as it always will) on a potential of zero b e t w e e nt w o s t a t e s p , a n d p : = P r * q . l J u t s i n c e : divergence the (qpq"-6p,q2)o,is simply -qzau, the Q: Pz- f t and (uzpttt'r) - tn(uzu1) (a, ! 2ut) the matrix D'Alembertian of the potential, that is, the current pro- elementvanishes,rvhich proves the contention in this (essenducing the potential. The term -!(ln(),'z/m'z))(q,q, case.It alsovanishesin more complexsituations -16r) therefore gives a current proportional to the tially becauseof relation (3'1)'below) (for example,try current producing the potential. This would have the putttng e2:q2 in the matrix (15) for the Cornpton sameeffect as a changein charge,so that we would have Effect). To prove this in general,supposeat' i:1 to -V are a a difference A(e'?) between e2 and the experimentally observed charge, e2!L(e2), analogous !o the dif- set of plane wave disturbing potentials carryulg moferencebetween m and the observed mass' This charge menta gi (e.g.,somemay be emissionsor absorptionsof the same or different quanta) and consider a mrtrix for depends Iogarithmically on the cut-off, L(e2)fe2: - (2e'z/3r) ln(>,/m). After this renormalization of charge t h e t r a n s i t i o nl r c m a s t e t eo f m o m e n t u t nf o t o 1 r s u c h is made, no effects will be sensitive to the cut-off. 2 T h e r e c r t l o o p s c o n r t , l c L . l yw i l h o u t c x t e r n e l i n t e ' r c t i o n s F o r After this is done the final term remaining in (33), e x a m n l e ,a p a i r i s ' c r c r t t i v i r t u r l l y r l o n g s i t h a p h o t o n N c r t t h e y a n n i h i l a i e , i l r s o r l ' i n g t h i s p h o t o n S u c l r l o o p s a r e ' l i s r e g a r ' { e do n contains the usual effects2lof polarization of the vacuum' and lrc c2llx'z ( q u g , - 6 u , 9 ' )- l- l n T\3m2

21E. A. Uehling, Phys. Rdv. 48, 55 (1935),R. Serber,Phys. t{ev. 48, 49 (1935).

the crounds Ihet thcv do noL illterrct $lLh anylllllri t h g l e " b yq q m D l e t e l y u r i o L i c r v r L l e . A n y i n ' l i r c c t r ' f i c c t s t h e y m a y have vie rh. exrluiion principle have alrcady Leen inclu'led'

248 QUANTU

M

E L E C T R O D Y N A 1 4I C S

as dr ff ,-rN \ (b r- m)-t a ; whercP i: p i r* gr (and in the product, terms rvith larger i are written to the left). The most general matrix element is simply a linear combination of these. Next consider the matrix between statesp, and pr*Q in a situation in which not only are the c; acting but also another potential d exp(- iq.r) wherea: q. This may act previousto all c;, in n'hichcaseit givesarll(y'i* q-m) ta;(pn!q-m)-Lq r4rich is equivalent to +dNfl(p,+q-m)-ta; since -t(folq-nr)-tq is equivalent to (po*q-*)-t pp as is equivalent to m. acting on the X(Fo*q*m) initial slate. Likervise if it acts after all the potentials 'axll(f it givesq(pa-m') rm)-lc; rvhichis equivalent to - atll(bt-rn)-ra; sincepryf q - m giveszero on the linal state. Or again it may act betrveenthe potential dr and ci+r for each A. This gives J1

I

NI

(p;{q-nr)-tai(lrlq-m)-r

o" fI

x q(Pt,- n7)-t ak ii @,* *')-, o,.

78L

Incidentally, the virtual quanta interact through terms like yu. . .yrh-2dah.. Real processes correspondto poles in the formulae for virtual Drocesses. The oole o c c u r sw h e n f t ? : 0 , b u t i t l o o k sa t l i r s t a s t h o u g hi n r h e sum on all four values ol 1r,ol 1 r. . . 7u we would have four kinds of polarization instead of two. Norv it is clear that only trvo perpendicular to ft are effective. The usual elimination of lonsitudinal and scalarvirt u a l p h o t o n st l e a d i n gr o a n i n s l a n t a n e o uCs o u l o m b potential) can of coursebe performedhere too (although i t i s n o t p a r t i c u h r l yu s e i u l ) .A t y p i c a l l e r m i n a v i r t u i l t r a n s i t i o n i s 1 r . . . 1 r h 2 d a hw h e r e t h e . . . r e p r e s e n t someinterveningmatrices.Let us choosefor the values of p, the time l, the direction of vector part K, of &, and two perpendiculardirections 1, 2. We shall not change the expressionfor these ttvo l, 2 for these are representedby transversequanta. But we must find (lr"'7,)-(7r"'7x). Norv ft -41"y,-K16, where K: (K.K)1, and we have shownabove that ft replacing the 7u giveszero.23 HenceK7s is equivalentto AeTrand (t' "' 1'1- 1",*. . . ^tK) : ((K' - k t')/ K') (t,. . . t ),

Flolvever,

so that on multiplying by h-2d1h:(Jth(h;2-Kt)-t the net e f f e c ti s - ( 1 t . . . l ) d ' h / K 2 . T h e 7 1 m e a n sj u s t s c a l a r rvaves,that is, potentialsproducedby chargedensity. t- (!*tq-m)-r, (34) The fact that I/K2 does not contain lr means that Aa so that the sum breaksinto the differenceof tlvo sums, can be integrated first, resulting in an instantaneous the {irst of rvhich may be convertedto the other by the interaction, and the d3K/K, is just the momentum replacementof A by A-1. There remain only the terms representationof the Coulomb potential, 1/r. Ironr the endsof the rlnge oI summation, 9. KLXIN GORDON EQUATION ' Qr*q-m)-'q(Fo-ni) : (!*-m)

+ar

II ;:t

(p;-ttt)-ta;-ay

II

Q+q-il:)-tat.

i:t

These cancelthe t\yo terms originally discussedso that the entire elTectis zero. Hence any rvave emitted will satisfy d,1u,/dtu:0. l,ikerviselongitudinal ri,aves(that is, rvirvesfor rvhich Au-06/3r, or a:e) cannot be absorbedand lr,ill have no effcct, for the malrix elements for emission and absorption are similar. (We havc said little more than that a potential Au:0,pf 0x, has no ef{ecton a Dirac electronsincea transformation rl":exp(*ig),! removes it. It is also easy to see in coordinaterepresentationusing integrationsby parts.) 'I'his has a usefui practical consequencein that in computing probabilities for transition for unpolarized light one can sum the squared matrix over all four clirectionsrather than just the two specialpolarization vectors. Thus supposethe matrix element for some processfor light polarizedin direction euis euMu. Il lhe Iight has wave vector q, l'e knolv from the argument above that QrMu:0. Iior unpolarized light progressing in the z direction ive lvould ordinarily calculate M ,2| M rz.Butu,e can as well sum M ;,+ lf o2+],[ M t2 "2Ior qrMrimpltes Mt:tr1. since qr:g" for free quanta. This shows that unpolarizedlight is a relativistically invariant concept,and permits sornesimplification in computing crosssections{or such light.

The methodsmay be readily extendedto particlesof spin zero satisfyingthe Klein Gordon equation,2a ll\! - m2P: i0 (A,t/')/ 0r,l

iA u|t! / Ax,- A,A,,!.

(35)

'?3A little more care is rcquired rvhen both ar's act on the same particle. Dc6ne x:k*r] Ktr, and consider (4...x)*(x...ft). Exactly this tern would arise if a system, acted on by potential t ca.rying moncntum - ft, is clisturbed by an added potential t of nomentum ffr (the reversed sign of the momenta in the intermecliate factors in the second tein r...ft has no effect since we lriil latcr intecrate over all h). IIence as shorvn above the result is zero, but since (ft...r)+(r...k):kr,6r..t,)-r('(r<...rr) we can still conclude (16. . . yr) : AIK-z(rr. . ."yJ. 2aThe equations discusseclin this section l.erc deduccd fron the {ormulation of the Klein Gordon equation given in reference 5, Section 14.'fhe function / in this section has only one component and is not a spinor. An alternative formal method of making the equitions valicl for spin zero and also for spin 1 is (presumably) by use of the Kemmer-Duihn matrices B!, satisfying thc commutation relation p"p,l3 B pB,8'+ *: 6u,l3"I 6",4p. If rve interpret o to mean ap?p, ralher than op7p, for any au, all of the equations in momenlum space n'i1l remain formally iclentical to those for the spin 1/2; rvith thc exception of those in rvhich a | denominator (_0-rd-, has becn rationaiized to (!*m)t!'z-n'!) since p' is no longer equai to a number, f. p. BLrt P3 does equal (P.P)p so that (j-tn)-t may now be interpreted as (np{nf p)(.p p-m'?) tz 1. This implies tiat equatioDs in col!'-f orclinatc sJrace rvill be valid of the function 1i*(2, 1) is givcn as K*(2, 1) : [(iV,] n, - rr-r(vf+ n l)li1+(2, 1) lrith v:: BAir,la.rr!. This is all in virtue of the fact that thc neny compoDcnL \ra!e function ry' (5 components for spin 0, 10 for spin 1) srtislies (iv-n)t: A.l, which is lormally identical to the Dirac Iirluatiotr. See W. Pauli, ltev. NIod. Phys. 13, 203 (1940).

249 782

R.

P.

FEYNMAN

The important kernel is now 11(2, 1) definedin (I, Eq. (32)). For a freeparticle,the wave function ry'(2)satislies +O'z*-1n2{/:O. At a point, 2, insidea spacetime region it is given by VQ:

I lV\)aI)(2, 1)/d.t1u - (0,1,lxr) t I',, I 12, 1)ltil,,(1)(i3

sents the possibility of the simultaneousemissionand absorptionof the samevirtual quantum. This integral without the C(&'?)divergesquadraticallyand would not convergeil C(h'): -X':/(fr:-I?). Since the interaction occursthrough the gradientsof the potential, we must use a stronger convergencefactor, for example C(ft'?) :I4(4'?-X) 2, or in general(17) rvith,6-X'!G(X)dX:0. In this case the self-energyconvergesbut depends quadratically on the cut-ofi tr and is not necessarily small compared to m. The radiative corrections to scatteringafter massrenormalizationare insensitiveto the cut-off just as {or the Dirac equation. particlesone can obtain Bose When there are severaL statisticsby the rule that if two processeslead to the same state but with two electrons exchanged,their amplitudesare to be added (rather than subtractedas for Fermi statistics). In this case equivalenceto the secondquantization treatment of Pauli and Weisskopf in a rvay very much like that should be demor.istrable given in 1 (appendix) for Dirac eLectrons.The Bose statisticsmean that thc sign of contribution of a closed loop to the vacuum polarizationis the oppositeof what it is for the Fermi case(seeI). It is (Pu-.futC)

(as is readily shown by the usual method of demonstrating Green's theorem) the integral being over an entire .3-surfaceboundary of the region (with normal vector 1y'r). Only the positive frequency components of / contribute from the surfaceprececlingthe time correspondingto 2, and only negative frequenciesfrom the surfacefuture to 2. Thesecan be interpretedas electrons and positronsin direct analogyto the Dirac case. The right-hand side of (35) can be consideredas a sourceof nelv wavesand a seriesof terms written down o{ increasing to representmatrix elementsfor processes order. There is only one new point here, the term in Autlrby which two quanta can act at the same time. As an example, supposethree quanta or potentials, o u e x p ( - i q " . r ) , D ue x p ( - i q o ' * ) , a n d c , e x p ( - i q " ; r ) a r e e2f to act in that order on a particle of original momentum - : I p ' , , p , , "p . , 2 m and lrF"lqa; the final mo- J , " : ^ po1so that !.:/olq" | L , p " " rf " , ' r I J Itlm mentum being p":p6!q". The matrix element is the sum of three terms (p2: pr'p; (illustrated in Fig. 7) X (trr' - m') t - 6,,(f .t - nt't) t (! cl pt. c)(1t' - m'z)| (Pb'b+ P b) tld'P. -6,,(lu'-nf) "' ". x (Pa'- m'z)-t(Pa' a! ? o' a) (.16) t(b' - (p.' c+ a) grvlng, Pu'c)(P* m') - ( c 'b )( p - m " ) ' ( ? a l ! o 'a ) . "' "' ' t o'l,.'"'l' T h e f i r s t c o m e sw h e n e a c h p o t e n t i a la c t s t h r o u g h t h e J , " " - " . ' . v , r , , 6 " , q " JIl' n I - , ).|, t t O . r uT. h e s eg r a d i e n t 3gt \ trrun,/l p e r t u r b a t i o ni 0 ( A , , 1 ' \ A x , l i l " O t r Lo n2 operators in momentum space mean respectively the momentum after and before the potential ,4u operates. the notation asin (33).The imaginarypart for (,q')t>2m The second term comes from D, and a, acting at the is again positive representingthe lossin the probability same instant and arises {rom the,4u,4u term in (a). of frndingthe final state to be a vacuum,associatedrvith Togetlrer bu and a, carry monentum qouf q", so that the possibilitiesof pair production. Fermi statistics or lu would give a gain in probability (ancl also a charge alter b.a operatesthe momentum is lolq"*qt 'l'he final term comesJrom cu and bu operating together renormalizationof oppositesign to tirat expected). in a similar manner.The term rlrz1uthus permits a new type of processin which two quanta can be emitted (or absorbed,or one absorbed,one emitted) at the same time.'Ihere is no a'c term for the order a, D, c we have assumed.In an actual problem there rvould be other terms like (36) but with alterations in lhe order in which the quanta d, D, c act. In theseterms a'c rvould ei /r_o o.u-.)r o- r appear. I I As a further example the self-energyof a particie of \" \s" \e" momentum 1u is

A'U" i'

I',"r,

- *21..t (e"/ z,;m) ]z p - k),((P- P1z [

x ( 2P- k )p - 6 ,) d 4 h h ' z C( h ' ) ,

rvherethe duu:4 comesfrom the ArArterm and repre-

o.

,/'-"

'( \'

,/'"

o1/ /v"

hC.

Frc. 7. Klein-Gordon particle in three l)otcntials, Eq. (36). 'fhe coupling to the electromagnctic iieid is now, for example, arises, (b), of simultaneous interfa. o+ f.'a, and a \ew possibility 'l'he propagation factor is now action with two .iuanta a.6. (!.f -m") I Ior a particle of momentum 2a.

250 QUANTU

N,I ELE CTRO D YNAM

rO. APPLICATIONTO MESON THEORIES The theories rvhich have been developed to describe mesonsand the interaction of nucleonscan be easily expressedin the languageused here. Calculations,to lowestorder in the interactionscan be made very easily for the various theories,but agreementwith experimental results is not obtained. Most likely all of our presentformulationsare quantitatively unsatisfactory. We shall content ourselvesthereforervith a brief summary of the methodslvhich can be used. The nucleonsare usuaily assumedto satisfy Dirac's equationso that the factor for propagationof a nucleon of momentump ts (p-M) rwhere M is the massof the nucleon(which implies that nucleonscan be createclin pairs). The nuileon is then assumedto interact with mesons,the various theoriesdiffering in the form assumedfor this interaction. First, we consiclerthe case of neutral mesons.The is the theory of vector theory closestto electrodynamics mesonswith vector coupling.Here the factor for emission or absorptionof a mesonis 97, rvhenthis mesonis "polarizecl" in lhe p direction. The factor. g, the "mesonic charge," replacesthe electric charge e. The amplitude for propagationof a mesonof momentum q in intermediatestatesis (92-p') I (rather than,,q-2as it is for light) wherep is the massof the meson.Ihe necessary integrals are made finite by convergencefactors C(q'- p') as in electrorlynamics. For scalarmesonswith scalarcoupling the only changeis that one replacesthe 7u by 1 in emissionand absorption.There is no longer a directionof polarization,p, to sum upon. For pseudoscalar mesons, pseudoscalarcoupling replace 7u by l"or example,the self-energymatrix of "to:i"t,tfi""tr a nucleonof momentum p in this theory is k'/,i)

I

hQ

h- M) \5d.ak(h- u')'Clw- u,7.

Other types of meson theory result from the replacement of 7, by other expressions (for example by with a subsequentsum over all ,uand z i6n,-tt) for virtual mesons).Scalarmesonswith vector coupling result from the replacementof 7u by p-1qwhereg is the final momentum of the nucleon minus its initial momentum, that is, it is the momentum of the meson if absorbed,or the negativeof the momentum of a meson emitted. As is well known, this theory with neutral mesonsgives zero for a[ processes, as is proved by our discussionon longitudinal waves in electrodynamics. mesonswith pseudo-vectorcouplingcorrePseudoscalar sponds to 7, being replaced by ['luq while vector mesons with tensor coupling correspond to using (.2p)-t(trq-qt). These extra gradients involve the danger of producing higher divergenciesfor real procFor example,76qgivesa logarithmicallydivergent esses. interaction of neutron and electron.25 Althoush these divergenciescan be held by strong enoughconvergence ?5M. SlotnickandW. Heitler,Phys.Rev.75,1645(1949).

I CS

783

factors,the resultsthen are sensitiveto the methodused {or convergenceand the size of the cut-ofi values of \. For low order processesp-r?bq is equivalent to the pseudoscalarinteraction 2Mp-t-tu becauseif taken between free particle wave functions of the nucleon of momentap1 anclpz-prlg, rve have (u2y5qu)- (nq,(pz- p')ut1- - (urf ,tuut) (.uz'raf ru) : - 2M (ti zt su,) since "y5anticommuteswith pt and lt operatinq on the state 2 equivalent to M as is ,r on the state 1. This shorvsthat the 75 interaction is unusually weak in the non-relativisticlimit (for example the expectedvalue of 7s for a free nucleonis zero), but since au2:1 i5 oo1 s m a l l ,p s e u d o s c r l at hr c o r yg i v e se m o r ei m p o r t a n ti n l e r action in seconclorcler than it does in first. Thus the pseucloscalar coupling constant should be chosento fit nuclear forces incJudingthese important secondorder processes.:6 The equivalenceof pseudoscalar and pseudovector coupling which hoLdsfor low order processes thereforedoes not hold when the pseucloscalar theory is giving its most important effects.These theorieswill thereforegive quite different resultsin the majority of practical problems. In calculating the correctionsto scatteringof a nucleon by a neutral vector meson f,eld (.yu)due to the efiects oI virtual mesons,the situation is just as in electrodynamics,in that the result convergeswithout needfor a cut-off and dependsonly on gradientsof the meson potential. With scalar (1) or pseudoscalar("y) neutral mesonsthe result divergeslogarithmically and 'Ihe part sensitiveto the cut-ofi, so must be cut off. however, is directly proportional to the meson potential. It may thereby be removedby a renormalization of mesonicchargeg. After this renormalizationthe results depend only on gradientsof the mesonpotential and are essentiallyindependentof cut-off. This is in addition to the mesonicchargerenormalizationcoming from the productionof virtual nucleonpairsby a meson, analogous to the vacuum polarization in electrodynamics. But here there is a further difference from electrodynamicsfor scalar or pseudoscalarmesonsin that the polarizationalso gives a term in the induced currentproportionalto the mesonpotentialrepresenting therefore an additional renormalization ol the mass of lhe meson which usually depends quadratically on the cut-off. Next considerchargedmesonsin the absenceof an electromagneticfield. One can introduce isotopic spin operatorsin an obvious way. (Specificallyreplacethe neutral 7s, say, by r;76 and sum over i:1, 2 where rr--r++r-, r":i(r+-r-) and 11 changesneutron to proton (21 on proton:0) and r- changesproton to neutron.)It is just as easyfor practicalproblemssimply to keep track of whether the particle is a proton or a neutron on a diagram drawn to help write down the 36H. A. Bethe, Bull. Am. Phys. Soc. 24, 3, Z3 (Washington, 19,19).

251 t-84

R.

P.

FEYNMAN

matrix element. This excludescertain processes.For example in the scattering of a negative meson from q1 to {z by a neutron, the mesonq2 must be emitted first (in order of operators,not time) for the neutron cannot absorbthe negativemesonqr until it becomesa proton. That is,in comparisonto the Klein Nishinaformula (15), only the analogueof secondterm (seeFig. 5(b)) rvoutd appear in the scattedng of negative mesonsby neutrons, and only the first term (Fig. 5(a)) in the neutron scatteringof positive mesons. The source of mesons of a given charge is not conserved,for a neutron capableof emitting negativemesons may (on emitting one, say) becomea proton no longer able to do so. The proof that a perturbation q gives zero, discussed for longitudinal electromagnetic waves,fails. This has the consequence that vector mesons, if representedby the interaction .yu u'ould not satisfy the condition that the divergenceof the potential is zero. The interaction is to be taken2Tas ^/t- tr-24pe in emissionand as "y, in absorptionif the real emission of mesonswith a non-zero divergence of potential is to be avoided. (The correction lerm p-2t1rqgives zero in the neutral case.)The asymmetry in emissionand absorption is only apparent, as this is clearly the same thing as subtracting from the original "yu..."yp,a term 2q' . .q. That is, if the term lr rr2qu1is omitted the ;esulting theory describesa combination of mesons ol spin one and spin zero. The spin zero mesons,coupled by vector coupling q, are removed by subtracting the : . e r mp 2 q . . . q The two extra gradients9...q make the problem of diverging integrals still more serious (for example the interaction between two protons corresponding to the erchange of trvo charged vector mesonsdepends quadratically on the cut-off if calculated in a straightforward ray). One is tempted in this {ormulation to choose simply 7u..'.yu and accept the admixture of spin zero mesons.But it appears tbat this leads in the conventional formalism to negative energiesfor the spin zero component. This shorvs one of the advantages of the 27The vector meson fibl
9, satisfy 3 e"f l-tr) - p2eu: -1rs

u, '.rhere sr, the source for such mesons, is the matrix element of _ p r o r o n . o [ s l a t e s n e u t r o n a n , l Bv taliinc the,livereence !-,elween ?Jr, df,, so'rhat i J r r . o i b o t h s i , l e s ,t n n c l u r l e t h a t . 0 e , a r i : 4 r p 'ie orlglnal equallnn can I'P reqrrllen aS E' p p- p' p p: - 4n(s u! uao / or r(os,/ 6t,)\. T\e riqht henLl side givcs in 6o."n,um represcltation ?l - p a q u g , r , t h e l c f t y i e l J s r h c ( q " - u , r t t n d , f in a l l y { h e i n r c r a c ri o n g i v e s t h e L a g r a n g i a n i n thc 1p on ahsorl,rion. -ou Proceetling in tlris ray rintl gcnerally rhet prrticlcs of spin one r n b e r e p r e s e n r c r lL y a i u u r - v e c l o r u , l s h i c h , I o r a f r e e p a r r i c l e i momenlum g srlisfies g.l-0r. l'hc propagrLion of virlual q from state r to p is represented bv rarticles of momentum rq,o;) rullil,lication by the 4 4 matrix 1or tensor/ -fr,:(6!,-p ,..(q2- F2)-t. The 6rst-or,lcr inleracrion (frorn the Proca eouaiion) : ' i r h a n e l e c t r o m a g n e l i c l o t e n l i a l d c r o ( - i A . . r ) c o r r e s D o n d st o rultiplication hy the mairix .Eu:tqz.o1 qt. d)6p-qzya!-q1p, 'lhere qr and qt:qt+h are (he momenta bclore anrl after the :rteraction. Finally, two potentials o, b may acr simultaneously, aith matrix E'u,: -(a.b)6",*b"a,

method of secondquantization of meson fields over the presentformulation.There such errorsof sign are obvious while here rve seem to be able to write seemingly innocent expressionswhich can give absurd results. Pseudovectormesonswith pseudovectorcouplingcorrespond to using "y5(7u-p-zq,q) Ior absorption and.767u for emissionfor both chargedand neutral mesons. In the presenceof an electromagneticfield, whenever the nucleonis a proton it interactswith the field in the way describedfor electrons.The mesoninteractsin the scalar or pseudoscalarcase as a particle obeying the Klein-Gorclonequation.It is important here to use the method of calculation of Bethe and Pauli, that is, a virtual mesonis assumedto have the same"mass" during all its interactionswith the electromagneticfield. The result for mass p and for (p'*tr')! are subtracted and the differenceintegratedover the function G(X)dX. A separateconvergencefactor is not provideclfor each meson propagation betrveen electrornagneticinteractions, otherwisegaugeinvarianceis not insured.When the couplinginvolvesa gradient,such as 75q whereq is the final minus the initial momentum of the nucleon, the vector potential A must be subtracted from the momentum of the proton. That is, there is an adCitional coupling *75,4 (plus when going from proton to neutron, minus for the reverse)representingthe new possibitity of a simultaneousemission (or absorption) of mesonand photon. Emissionof positive or absorptionof negativevirtual nesons are representedin the same term, the sign of the charge being determined by temporal relations as for electronsand positrons. Calculationsare very easily carried out in this way to lorvest order in g2for the various theories for nucleon interaction, scattering oI mesonsby nucleons,meson production by nuclear collisionsand by gamma-rays, nuclearmagneticmoments,neutron electronscattering, etc., However, no good agreementwith experimentresults, when these are available,is obtained. Probably all of the formulations are incorrect. An uncertainty arises since the calculations are only to first order in 92, and are not valid if g2/hc is large. The author is particularly indebted to ProfessorH. A. Bethe for his explanation of a method of obtaining finite and gauge invariant results for the problem of vacuum polarization. He is also gratelul for Professor Bethe's criticisms of the manuscript, and for innumerable discussionsduring the development of this work. He wishes to thank Professor T. Ashkin for his carelul reading of the manuscript. APPENDTX In this appendixa methodwill be illustratedby rvhichthe simplerintegralsappearing in problemsin electrodynamics can be directly evaluated. The integrals arising in more complex processes lead to rather complicated functions, but the study of the relations of one integral to another and their expression in terms of simpler integrals may be facilitated by the methods eiven here.

252 QUA N TU M

E L E CTRO D YNA M I CS

As a tlpical problem consider the integral (12) appearing the 6rst order radiationiess scattering problem:

J'tu@"

ta(!1-h-n)-t^vuk

h-n)

2d1kc(h2),

in

(1a)

1 and -I'(h'-tr2) where rve shall take C(ft'z) to be typically We first rationalize the -factors d{} means (2r')-2dhdhzd.hika. : 1O- h* m) ((l - h)' - n')-1 obtainins, @ *-n1-' f

h* n) a(!1-

t y@,-

hl

x) v uh-'d.'hC (h2)

X ((Or-

h'12- ntt l-t 1.(!z- h)' - m'.)-r .

(2a)

The matrix expression may be simplifiecl. It appears to be best to B BA do so after the integrations are performed. Since AB:2A where A - B: A pB p is a number commuting with all matrices, find, A1rl2Ar, if iR is any expression, and .d a vector, sitce 1rA: - 4'uP"*'nO'

arARtr: Expressions between tlvo'y/'s tion. Particularly useful are

can be thereby

(3u) reduced by induc-

1 p " lp : 4 -24 tpAlt: auARTr:2(ABIBA)=aA'n tPABCT'= -2691

Qa)

(i.e., linear comu'here -4, B, C are any three vector-matrices binations of the four 7's). In order to calculate the integral in (2a) the integral may be written as the sun of three terms (since &:i"'y), 1 p(Qz* nt) a(i,*

n) t pJ r- f1 p1 1! m) v, "a(! ! 7 u(!z* m) at ot rfJ zl t fi

"at

n pJ z,

f

Q;h,;h,k.)h

(7a)

Lt:mt2-!]t etc., and we can consider dealing *here A:n2-!2, rvith cases of greater generality in that the different denominators need not have the same value of the mass z. In our specific problem (6a), p1':12r s6 that Ar:0, but we desire to rvork rvith greater generality. Now for the factor C(h2)/k2 rve shall use -I'(ft'-I')-tft-'!. This can be written as -\2/(h,

. )\2)k2: h 2C(k2t: -

O rrUr- ,rn. J'o^t

r8a)

and at the end inteThus we can.replace h-'C(h2) by (h2-L)n grate the result with respect to Z from zero to X2. We can for many practical purposes consider tr2 vety large relative to m2 or !2. the conWhen the original integral converges even without vergence factor, it will be obvious since the Z integration rvill then exists in the If an infra-red catastrophe be convergent to infinity. integral one can simply assume quanta have a small mass tr',in and extend the integral on Z from l2-in to \2, rather than lrom zero to tr2.

where by (1; k'; k'k,) we mean that in the place of this symbol either 1, or k", or kok, may stand in difierent cases. In more complicated problems there may be more factors (h'?-24'E-A;)-r or other powers of these factors (the (P- L)-2 may be considered Ar:Z) and Iurther as a special case of such a factor rvithy'r:0, factors like io,['lr. . . in the numerator. The poles in all the factors are made definite by the assumption that r, and the A's have in6niLesimal negative imaginary parts. We shall do the integrals of successive complexity by induction. We start with the simplest convergent one, and sholv

J'a'n&r-

tt 3=(8iz)-1.

(10a)

where the For this integral is f (2r)-2dkd3K(ta?-K'K-Z)-3 vector K, of magnitude K: (K. K)l is ky kz, kv The integral on lr shows third order poles at h: * (f'+ l,), and k4: - (K'+ L)'. Imagining, in accordance rvith our defrnitions, that Z has a small negative imaginary part only the first is belorv the real axis. The contour can be closed by an infinite semi-circle below this axis, without change of the value of the integral since the contribution from the semi-circle vanishes in the limit. Thus the contour can be shrunk about the pole .4a: { (K'?fZ)} and the resulting h integraiis 2zl times the rcsidueat thispole.Writing Ar:(l('*Z)l*e 3:e 3(el2(K'*L)l)-3 in powers of and expanding (h4'-K'-L) e, the resicluc, belng the coellcient of the term e-1, is seen to be 5 so our integral is 6(2(K,l Qr1

- $i / 32").1'-1" K'zd K (K' + L) 6t': (3/ 8i)(r/ 3L) from

in the

the symmetry

(11a)

$ t J ' ( ; k , ) d ' k ( h - z ) - 3 :( 1 ; o ) r - l

(6a)

That is for -Ir the (1; [d; lo]') is replaced by 1, for "/r by.t", and lor Jtby h,k". More complex processes of the first order involve more factors a corresponding increase in the number like ((?a-&)'-zf)-rand Higher of 4's which nay appear in the numerator, as k"k"k,"'. ortler processes involving trvo or more virtual quanta involve instead similar intcgrais but Eith factors possibly involving A*f of just &, and the integral extending on h 'zdlhc(h2)ht4d4k'C(k'2). 'l'hey can be simplified by methods analogous to those used on the first order integrals. The factors (p- p1z-roz may be written (P-k1z--z:Pz-20'U-^'

f o , a " ,n , a , ) a , a v - L ) 4 ( k , - 2 p 1 . k -^ t ) - l X ( k 2 - 2 P z . k -L r ) t , ( 9 a )

cstablishing (10a). L)-3:0 We also have fkddtk(HI space. We rvrite these results as

zd4kc(h,) /' (.Qz- h1t -,,r21-t1(fi 1- k)2 - m2) t.

We then have to do integrals of the Iorm

(5a)

where Ja;,;i:

/65

where in the Lrackets (1;,1o) and (1;0) corresponding entries are to be used. A shows that Substituting &: &'-p in (1 la), and calling Z-i2:

(sD 0 ; h h(h,- 2p k - ^)-3: (r t P,)(j'p+^)-l. (12a) J ")d^ both sidesof (12a) with respectto A, or rvith By cliflerentiating respect to p" there follorvs directly Q 4 i t f t | : h . ' , k " h , \ d ' \ k \ h 2 - 2 bh - N

.)

a

: -(1; !"; P,f,-ir",(r,+a))(l+^)

'. (13a)

successive integrals ingive directly differentiations Further cluding more i factors in the numeratol and higher polvers of (h'-2p.k-A) in the denominator. The integrals so far only contain one factor in the denominator' To obtain iesults for trvo factors we make use of the identity

o '6-,:

,,

(14a)

fo'd,r(ar*D(1-r))

t s u c q e s t c dL v s o m e \ v o r k o f S c h u i n g e r ' s i n v o l v i n g C a u s s i a n i n t e paraeruii. fni. represcnts the producr of two re.iprocals as a metric integral over one and will therefore permit integrals with powers oI two factors to be expressed in terms of one. For other a. D. we make use of all of the identities, such as

o-,6-,: J" 2r,]"lo*-I-D(1- r; ;-s declucible from (14a) by successive difierentiations toaorb. To perform an integral, such as ^.. t., . '8it.J rl:k"d1h(h'?-z\t

h-J,t

(1sa) respect

with

2\h2-2pth-Lr)'

t,

(l6a)

253 ;86

R.

P.

FEYNMAN

-'rite, using(15a),

f,st(l - r) dr ln(x(l -c)i) : - (1/4) find

t - 2p k - t 2"d.r1lc k,- 2Ft k - L r)n (H - 2! t. k - t,1-t: f ". "1 ",

$, f 0; k,)FC(E)d4k(h'-2p-k)-l

:(rnft+r,n(t#-))

'here and

!.:rprl(l-r)!z

so that the expression (16a) 3 which may now be

rote that A, is zol equal Lo n2-!,2) :: (.8i)-fo\2rdtI0;h,)d4k(W-2p,.k-A") :r'aluated by (12a) and is

(a{:

t'

(17a)

a,:ra'111-t;6r,

(1;p,.)2xd$(l),,+^,)t,

(18a)

; here/,, A, are given in (17a). The integral in (18a) is elenentary, -eing the integral of ratio of polynomials, the denominator of :econd degree in r. The general expression although readily ob:ained is a rather complicatecl combination of roots and logarithms. Other integrals can be obtained again by parametric differentiaof (16a), (18a) with respect to :ion. For example differentiation l2 or 12' gives 8D

f

-2pr.k0.i k,; k.k,)d|h(k,

: -

Lt)a(H-2p,.k-

L2)-2

f' $; t *; t *r ", i6',(t.'+^")) X2x(1* *)dx(!.'*,A")i,

(19a)

again leading to elementary integrals. As an example,consider the casethat the secondfactor is just tl.h2-L)a atd in the first put trt:|, A1:A. Then ,,:rrj l' :rA* (1-r)2. There results

so that substitutioninto (19) (after the (0-h-*) 1) gives replacedby (p-hln)(h'-2p.I)

(19): (d / 8n).yNl@I m) (2 h(\, / n ) I 2) - p(tn(\' /n') - il)t p : (d/82) [82(ln(I'/n')* 1) *!(2 h(]" /m')*5)f,

rf *p,p,-|a".(*f fo' 0t .i X2r(l

B. Corrections

l,r,)) (20a)

Integrals with three factors can be reduced to those involving two by using (14a) again. They, therefore, Iead to integrals with tlvo parameteE (e.9., see application to radiative correction to scattering below). The methods of calculation given in this paper &re deceptively simple when applied to the lower order processes. For processes of increasingly higher orders the complexity and difiiculty increases rapidly, and these methods soon become impractical in their prerent form.

A, Self-Energy

\h, -2pt h) lrhr-2pz h) ,: ro' dlrht-2p,'tl-', from (14a)where p!:rt+(-)iL

(e'/rlfau(j-k-m)-tarkad.akC(k2),

(22il: -

(1; *!u,i r2p,opp1 I' I: -tt',(dtr,\ 0 -r)L))2x(r-

- $i) k. h4 d^hc (h2)(h2- 2p I k)-t (h2- 2p2,k) | r

:z.fo' lo"f;'ar, {z+^)

since(p- k)2-mz: h2-2p. k, Lsf: m2.This is of the form (16a) pr:O, a':9, P2:P so that (18a) gives, since rvith Ar:r, p,:(1-r)P, L,:xL,

or performing

- {sil n,n,n-,a' (k2- 2pI k)-t(H- zp,-k1-' nc(h2) f : fo'!,"!o,!;'dt-4a.,J" avnlx'p,-)+i6"' (2sa) The integrals on I give,

- L) 4(h, . 2p. h)-1 -' 6i) . lf ( 't k")d|h(h,

0 ; {t * r) r

x)ds(fiu2* L(r- x))ady.

We now turn to the integrals on I as required in (8a). The 6rst term, (1), ir (\;k";k"k,) gives no trouble for large Z, but if Z is put equal to zero there results r 21r-2which leadsto a diverging integral on r as r*0. This infra-red catastrophe is analyzed by using tr-i.z for the lower limit of the,L integral. For the last term the upper limit of Z must be kept as X2.Assuming tr-i"'(lr'((tr2 the r integrals which remain are trivial, as in the *lf-energy case. One finds

- kl d t d4k(H'|- Da G' - 2 k)-t, P' f t u@

f

(22a1

which s'e will then integrate with respect to y from 0 to 1. Next we do the integrals(22a)immediately from (20a)with y':y',, d.:0:

(19)

so that it requires that u.e find (using the principle of (8a)) the integral on I from 0 to tr?of

:

Qra)

We therefore need the integrals

,_r^t",)-,a,nc,,_^^,,"7!,pr7;:r!, _(sr/(ft

The self-energrintegral (19) is

'

to Scatlering

The term (12) in the radiationlessscattering,after rationalizing the matrix denominators and using P12:p22:n2 requires the integrals (9a), as we have discussed.This is an integral with three denominators rvhich we do in two stages.First the factors (h'-2p( k) atd (k'-2p2. k) are combinedby a parametery;

$il J' ( t k"; k"k,)daplpt- 11t1ht-2pu.k)a,

x)dt(fl2-lA,)-'.

rro,

using (4a) to removethe ?p's. This agreeswith Eq. (20) of the text, and gives the self-energy(21) when f is replacedby z.

B l f 1 0 j h- '" i h- " h , t d 4 h t h , - I ) , \ h , 2 p - k - L ) z J

: -

I in (19) is

")zutx((l

- x)2m2 | r L) t,

the integral on Z, as in (8),

k)-, $i) f o; k.)d4kk4c(w)(k,-2p.

:..i['tt; rt -rr1,)z a, n-\",!!:,u!4 * \t

xrm'

Assuming now that tr2))rz2 we neglect (1-r),2, relative to ,X'in the argument of the logarithm, which then becomes r):1 (\'/m')(x/(l-r)'). Then since .fold.sln(x(|-x) and

F z\:4(n2 f' b,.-rdrln(b..rX-,^ -) . . . . -sin2 . . - t ) - r l t l n ( uI - i ; ' ) Jo ru l -Jo atan.da]' fo'fu,ru-"or:u{-2sin20)-r(!u*?2,), fo' !,o,P-!undt

(26a) (27a)

: 0(2m' sin2l)-t(p ul

?v) (f",1 f,,) (28a) 0 ctfi), !y'q,q"(1-

J'ot orrnl^"p,-,;:tn(|,z/n*)r2(t-0ctng).

(29a)

254 QUANTUM

These integrals on Jr'were performed as follows, Sincefi:lr*q rvhereq is the momentum carded by the potential, it follows irom L h a l z p t q : - q ' z s o l h a l s i n c el L : h L q ( l - y \ , ff:pf=m2 : b,2= m2 qzlt 1 yt. The substitution2),- I ta nal1an0 where d is definedby 4z' sin'd: q2is uselul for itmeans rtu2: m2sec2dfsec2q and rt, zd.y: (m2 sit20)-t4o *n"r" o goes f rom 0 to + 0. Theseresultsare substituted into the original scatteringformula (2a), giving (22). It has been simpiified by frequent use of the fact that p1 openting on the initial state is u, and likewise !z when it appearsat the left is replacableby z. (Thus, to simplify: - 2fPP, bY (4a',' i ttrzalfi t: : - 20 z- d a\Prl il = - 2(n - q\ a(n * c\' A term like qaq: -q2al2(a'q)g is equivalentto just -q'a since has zero matlix element ) The renormalization q:rr-Pt:m-n term requires the corresponding integrals for the special case q:o.

C. Vacuum Polarization The exirressions(32) and (32') lor Ju, in the vacuun polarization problem require the calculation of the integral J,,(.\

: -:i

I

s flt u(l - iq+ m) r(! l lq! m)ldap

(32) x ((l- tq)z- ilP)-t(Q* iq)'- n')-\, rvhere ne have replacedI by fi'trq to simplify the calculation somewhat.We shall indicate the method of calculation by studying the integral, - n2)-t. I (,n ) : I p !(@- rq)'- m')-t(l+rq)z "p,d' atd p2!p'q The factors in the denominator,F-?'q-n'++q2 - mr*tqt are combined as usual by (8a) but for symmetry we (1-r):{(1-a) and integrate r; from substitute ei(*n)' -1 to+1: *,' (30a) p, p,a' ! (12-,t f ' c - m' I f,q')adn/ 2. t 6") : f But the integral on y' will not be lound in our list for it is badly diversent. Ilowever, as discussedin Section 7, Eq. (32') we do not We can wish I(2,) but rather JiilI(m')-I(m'*X')lG(I)dI. by first calculating the calculate the difierence l(n2)-I(n2j\2) derivative f'(m'*L) of I with respect to m2 at m2lL and later integrating L lrom zero to tr'. By difierentiating (30a), with respect to 42 find,

where we assume tr2))z? and have put some terms into the arbitrary constant C'which is independent of \' (but in principle could depend on 4') and which drops out in the integral on G(I)dI. We have sel q2:41n2 9in20. In a very similar way the integral with u' in the numerator can be worked out. It is, of course, necessary to dif{erentiate this z'? also when calculating I' and 1". There results

- m2)-l - $l n a^ pgp- !q)2- nz)-t ((I+ Eq)2 f : 4n2 (1 - 0 ctnl) - q2/ 3 I 2 (\2 * n2)ln(1" ma + D - C' |)\'),

1 I " (rr" + L) : 3I : r' p ? dn! (.,!'- nI q - n2 - L + L q2)- d11 " (31a) +' - tr6 D-r)d.rt : - (8ilt (Lrr2q"q,D-z f ", convergesand hasbeen (whereD: |(a'- 1)q2lnzlL),whichnow evaluatedby (13a)with/:4nq a\d L:m2+L- 1q'' Now to get 1'we may integrate 1" with respectto t as atr indeinite integral ttdwe may cltooseany convenientarbi'trart constanl.Thisisbecatse a constant C in 1' will mean a term -C\2 i\ I(n'z)- I(m2+)\2) which vanishes since we will integrate the results times G(tr)dr This meansthat the logarithm appearingon atrd /i-r,G(\)d^:0. integrating , in (31a) presentsno problem. We may take r, (tu + L) : (8il1 f:' l\n q.q,D-'*tt,, rrDfdn* c6.,, a subsequent integral on Z and finally on 4 presents no new problems, There results - @D f dj prc) - *d' - n')' ((t+ id2 - mz)-L f "t 4 t n ' z - q ' ( .' d \ , , , \ ' l - t q 4 q t - r^d t q"' ft tl ' ) itn,ril 3o, \r ,^n011 n-2 + 1)- C']\'),

p')d4p((p- rq)' - n')-L((I + iq)2- m2)-l : (1,0)(4(1-d ctnd)*2 ln(I'?z-,)). (34a)

- @i) (\ f 'I'he

value of the integral (34a) times z' differs from (33a), of course,becausethe results on the right are not actually the integrals on the left, but rather equal their actual value minus their value lor m2:m2l)r2. Combining these quantities, as required by (32), dropping the constantsC', C" and evaluating the spur gives (33). The spurs are evaluated in the usual way, noting that the spur of any odd number of'y matrices vanishes and S!('1,n1:5p16tr1 1o, urbrtrary A, B. The S2(1):4 and we alsohave (35a) LSpl(b,1-m)(Iz- m2)): ?r P2- mtmz, \S pl(f'*

m') (l r w n)) n,) (!z- n,) (f "* m rnz)(Pt e- ntm) = (f r. i2I - lP t Px- nn) (! z'! t- mztn) * (f t. lr n'mo)(!r !t- mzm), (36a)

vherc Pi, ni are arbitrary four-vectors and constants, It is interesting that the terms of orclerX2ln\2 go out, so that the charge renormalization depends only logarithmically on X'. 1'his is not true for some of the meson theories. Electrodynamics is suspiciouslyunique in the mildnessof its divergence. D. More

Complex

Problems

Matdx elements for complex problems can be set up in a manner analogous to that used for the simpler cases' We give three illustrations; higher order correctionsto the Mlller scatter-

X

p - = p .- q

7v p^ +k

X /

\

\

/

\

/

,Ht rhl, /1t d.e.t

\

/1 /

q.

(32a)

(33a)

with another unimportant constant C". The completeproblem requires the further integral,

I' (m'+ D : .1:: p doI(l' - n!' q- m'z- Ll iq\ 3d,t. "! This still diverges,but we can difierentiate again to get

* d",[(I,*zl)ln(I'

787

ELECTRODYNAMICS

/

\ \/

/

/-< \ n.

\ /

/ l.

\

Frc. 8. The interaction between two electrons to order (d/hc)'?' of every figurc involving two virtual One adds the contribution quanta, Appendix D,

255 ;88

R.

P.

FEYNN,IAN

means that one adds with cqual weight the integrals correspoDding :-g, to the Compton scattering, and the interaction of a neutron ,.ith an electromagnetic 6eld. to each topologicaily distinct ligure, scattering, consider two electrons, one in state To this same order thcrc arc also the possibilities of Fig. 8d Ior the Mlller ,: ol momentum ,r and the other in state a2 of momentum .202. which give -1ter they are found in states a3, p3 and ua, pa. This may happen :rst orcier in e2/hc) because they exchange a quantum of momen1- h- m)-t11rt) @/., f @r,(fu- tu- n)-\-yuNp in the manner of Eq. (4) and lrig. 1.'lhe :;n q=pr-pz:pa-pz x(n41!1tr) h-,q-,d!k. :r3trix element for this process is proportional to (tratrslating (4) space)

: r momentum

(iqpu2)(hapur)q-'.

(37a)

.', e shall discuss corrections to (37a) to the next order in e,/bc. There is also thc possibiiity that it is the electron at 2 which ::ally arrives at 3, the electron at 1 going to 4 through the exThe amplitude for this ::ange of quantum of nomentum.iD3-rr. - :.cess, (i4y pu) (u* puz) (,!rbe subtracted from trz)-2, nust iia) in accordancc rviLh thc exclusiou principle. A similar situa:n exists to each order so that we need consider in detail only ::e corrections to (37a), reserving to the last the subLraction of r. samc terms rvith 3, 4 exchangec,, One reason that (37a) is modified is that two quanta may be ::,:changed, in the manner of Fig.8a. The total matrjx element : :: all exchanges of this type is ..,i)

f

k-

ei"1,,(p,-

m)-t^1,u,) (uaa,(pz! k-

n)-t\

. h ,(q-

4r,)

h)-,d4k,

(38a)

r.i is clear from the figure and lhe general rule that electrons of between inter-r -.mentum , contribute in amplitudc (j-z)-1 r::ions .yA, and that quanta of momentum ft contribute &-'z, In ::egrating on d{& and summing over p and /. we add all alterna.es oI th€ tlpe of Fig.8a. If the time of absorption, ?!, o{ the .irntum A by electron 2 is later ttran the absorption, a,, of q- k, being a positron (so ::: corresponcls to the virtual stale fl*k :t (38a) contains over thirty terms of the conventional method : analysis). ir integrating over all thcse alternatives s'e have considered all ::siblc distortions of Fig. 8a which preserve thc ordcr of cvcnts , ::rg the trajectories. We have not included the possibilities : ::responding to Fig. 8b, however. Their contribution is ,. ,il

f

h-m)

@"t,@,-

X (no r(lz*

11uur) q-

m) tyur)h

k-

2(q- h)'zd,ak, (39a)

:: is readily verified by labeling the diagram. The contributions ol :. possiblc $.ays that an event can occur are to be added. This

,.(

j,

,/

"/ /)^

L l L t ' *)) -/:./

o.

(

* 7 vll

,/.d . c

-t

b.

c.

/-& ^t

*J

4

,/

",/"',al

r*

tl

,/

g.

*, ,/

fr h.

Frc. 9. Ra.liative correction to the Compton scattering terD (a) oI Fig.5. Appendix D.

This integral on ft will be seen to be precisely the integral (12) for thc radiative corrections to scattering, which we have worked out. The term may be combined with the renormalization terms resulting from the difierence of the efiects of mass change and the Lerms, Figs. 8f and 8g, Figures 8e, 8h, and 8i are simiiarly analyzcd. -. Finally the term l"ig.8c is clearly relate,l to our vr(uum polarization probicm, and u.hen integraterl gives a term proportional to (?ia'rp?,!)(rsa&t)Jp,q a. If the charge is rcnormalizcd thc term ln(x/rn) in -/1, in (33) is omittcd so thcre is no remaining clependence on the cut-off. The only nelv integrals rve recluire are the convergeDt integrals (38a) and (39a). They can be simplificcl by rationalizing thc c1c' norninators and combining them Ly (14a). For cxarrple (38a) involves the f actors (h' - 2 p r k) | (h2 + 2 f 2' h) | h-'z(.q' I h2 - 2q h) 2. The first two may be combinecl by (14a) with a paraDrcter r, and the second pair by an expression obtainerl by differentiation (1.5a) rvith respect to 6 ancl calling the parametcr l'.'I'hcre results a h) a so Lhat thc intcgfais on Iactor (h2-2p.'k)a(h2ly'q'z-2lq d1,4now involve two faclors and can bc performed by lhe rncthocls given earlier in the alrpendix. The subsequent intcgrals on the paramcters * and 1t are complicatcd ancl have not been rvorked out in detail. Working rvith charged mesons there is often a considerable reduction of the nurnber of tcrms. 1'or example, for thc interacti(tr1 between protons resulting from thc cxchange oi tu'o mcsons onlv the tenn corresponding to Fig. 8b rcmains. Ternr 84, for cxamplc, is impossibJe, for i{ the 6rst proton emils a positive meson thc second cannot absorb it directly for only neutrons can absorb positive mesons. As a second exanple, consider the radiative correction to the Compton scattering, As seen from Eq. (15) and Fig. 5 this scattering is represented by two terms, so that we can consider the corrections to each one separately. Figure 9 shows the types of tcrms arising from corrections to the term of Fig.5a. Calling A tbc momentum of the virtual quantum, Fig. 9a gives an intcgral f

t rQr-

h- m) ter(Jpt+ qt-

h- m)-ts rQpr- k-

m)-17 uh-2dah,

convcrgent without cut-off and reducible by the mcthods outlined in this appendix. The other terms are relatively easy to evaluate. Terns 6 and c are closely rclatcd to radiative corrections (although of Fig.9 somewhat more difficult to evaluate, for one of the states is not Terms e, f are tetotnalizathat of a free electrot, (!r*Q)21n2). tion terms. From term d must be subtracted explicitly the efiect of mass Au, as analyzed in Eqs. (26) and (27) leading to (28) Terms g, I give zero since the b:er s:e2, rvith p':pr1q, vacuum polarization has zero efiect on free light quanta, qr'?:0, gz?:0. The total is inscnsitive to the cut-olT X The result shows ar infra-red catastrol)he, the largest part of the cffect. When cut-ofl at tr-i,,, thc effect l)rol)orlional ln(ru/X",i,) gocs as

@ / r ) lr(u h ^; ") (l 20 cLnzo),

to

(40a)

times the uncorrected amplitude, where (f z- I )2 : 4m' sit20.'fhjs is the same as for the radiative correction to scattering for a 'fhis is physically clear since the long wave deflection ft-br. quanta are not effected by short-lived intermediate states. The infra-red effecls arise'3 lrom a 6nal adjustment of the field from thc asymptotic coulomb fieltl characteristic of the elcctron o[ '!3F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937).

256 789

O U A N T U IVI E L E CT R O D Y N A M I CS momentum 11 before the collision to that chalacteristic of an electron moving in a new direction !z after the collision. The complete expression for the correction is a very complicated expression involving transcendental integrals. As a final example we consider the interaction oI a neutron with an electromagnetic 6eld in virtue of the fact that the neutron may emit a virtual negative meson. We choose the example of pseudoscalar mesons rvith pseudovector coupling. The change in amplit) determines rude due to an electromagnetic 6eld A:aexp(-iq rhe scattering of a neutron by such a 6eid. In the limit of small g qa-aq interaction of a patwhich represents the as ir rrill vary ticle possessing a magnetic moment. The first-order interactiotr betNeen an electron and a neutron is given b)' the same calculation b1'consiclering the exchange of a quantum between the electron and the nucleon. In this case o, is g-' times the malrix element of 7a between the initial and 6nal states of the electron, the states differing in momentum by g. The intcraction may occur because the neutron of momentum ,r emits a negative meson becoming a proton which proton interacts with the field and then reabsorbs the meson (Fig. 10a). Thc matrix for this process is @r:fr*{),

p- u)-ta0F h- l|)-t(tsk)(kz- uzltgap.1nrr, f 0,h)(trit may be the meson which interacts with the 6eld. Alternatively We assume that it does this in the manner of a scalar potential satisfying the Klein Gordon Eq. (35), (Fis. 10b) -

f

{",n"){I,-

n,-

M)-1(v5h1)(h22X(kz'a*kt

p2)-\ a)(h?-

p2)-\dahy

(42a)

where we have put tr: ftr*9. The change in sign arises because the virtual meson is negative. Finally there are two terms arising from the'yoc part of the pseudovector coupling (Figs. 10c, 10d) J', G^h)Q"-

A- V)-ttv

!a)(h',- F\-'\dth'

(43a)

and f

G"o)(t,-n-tt)'(7.h)(h2-p2)-1d1h.

('14a)

Using convergence factors in the manner discussed in the section theories each integral can be evrluated and the results on .iron combined. Expanded in powers of 4 the frrst term gives the magnetic moment of the neutron and is insensitive to the cut-off, the next gives the scattering amplitude of slow electrons on neutrons, on the cut-ofi. and depends logarithmically and combined somewhat The expressions may be simplifed before integration. This makes the integrals a little easier and also shows the relation to the case of pseudoscalar coupling. For M) example in (41a) the 6nal 'ybft can be written as r(kltl when operating on the initial neutron state' This is since rr:tf

ilEUTROi

grri

/

K!'l

"\o:-T4 x9.

J

\-\\/

i

,/ 2" 7sh( . IN,

XESON k

D - t -- rkl .I

I

to

| '/-

val

75\t f

\l

fr, b. /P"

%-!.fir %e*Y 'tp

c.

7.9-\ ,.-k!

,/,_

' ' - v )! hd.

Ftc. 10. According to the meson theorl a neutron inleracts witlr rn electromagnelic fotential c by 6rst emirring a virtual charged meson. The figure illustrates the case for a pseudoscalar meson with pseudovector coupling. Appendix D. since ?r anticommutes rvithrr and ft. The Qt-h-M)to*2M7s h- M) 1 and gives a term which just 6rst term cancels the (Itcancels (43a). In a like manner the leading factor ?sfr in (41a) is -2tr[ls-ts(lt-h-M), the second term leading to a written as factor and combining simpler term containing ao (Pz-k-M)-t with a similar one irom (44a), One simplifies the lsftr and 'ys&r in (42a) in an analogous way. There frnally results terms like (4la), (42a) but with pseudoscalar coupling 2Mzs instead oI 76fr, no terms like (43a) or (44a) and a remainder, representing the difierence in efiects of pseudovector and pseudoscalar coupling. The pseudoscalar terms do not depend sensitively on the cut-off, The difierbut the difierence term depends on it logarithmically. interaction but not the ence term afiects the electron-neutron magnetic moment of the neutron. Interaction oi a proton with an electromagnetic potential can be similarly analyzed. There is an efiect of virtual mesons on the electromagnetic properties of the proton ev€n in the case that the mesons are neutral. It is analogous to the radiative corrections to the scattering of electrons due to virtual photons. The sum of the magnetic moments of neutron and proton for charged mesons is the same as the proton moment calculated for the corresponding neutral mesons. In fact it is readily seen by comparing diagrams, q, the scattering matrix to frsl ordu in the that for arbitrart' elecbomagnelic Potenti.al, for a proton according to neutral meson theory is equal, if the mesons were charged, to the sum of the matrix for a neutron and the matrix for a proton. This is true, for any t]'pe or mixtures of meson coupling, to all orders in the coupling (ncglecting the mass difference ol neulron and proton)

P o p e r2 3

Mathematical

Formulation

of the Quantum Theory of Electromagnetic

257

fnteraction

Dep,,nee,of physul:!;,X,"u"I#i#,ity, rthac@, Nw york (Received June 8, 1950) The.validityoftherulesgivcnin-previouspapersforthesolutionofproblemsinquantumelectrodynamics , is established. Starting with Fermi's formulation of the field as a sei of harmonii oscillators, the effect of lhe oscillators is inlcgraled out in lhe,Lagrangirn- form of rluantum mcchanics. Therc resulrs rn exlressron for lhe effect of all virtual photons va.lid to all ordcrs inc/ic-lt is shown rhat evaluation ol this expressron as a por-er series h d/hc gives just the terms expected by the aforementionecl rules. In addition, a relation is established between the amplitude for a given process in an arbitrary unquantized potential and in a quantum electrodynamical-field. This relation"permits a simple g"r".uiri;t"."nt of the laws of quantum electrodynamics. A description, in Lagrangia.n quantum-mechenical form, of particles satisfying the Klein-Gordon equatron is given in an Appendix. It involvcs use of an extra parameter analogous to proper time to clescribe -lhe lhe trajectory oI the narticle in lour dimensions. A second Appendix discusses, in the special case of photons, the problem of finding rvhat real processcs are implied by the formula for virtual processes. Problems of the divergences of electrodynamics are not discussed.

r. INTRODUCTION

net effect of the field is a delayed interaction of the particles.It is possibleto do this easilyonly if it is not necessaryat the same time to analyze completely the motion of the particles. The only advantage in our problemsof the form of quantum mechanicsin C is to permit one to separatethese aspectsof the problem. T l r e r ea r c a r r u m b e ro f d i s a d v a r r t i g e sh,o w e v e r s, u c ha s a lack of familiarity, the apparent (but not real) n e c e s s i t yf o r d e a l i n g w i t h m a t t e r i n n o n - r e i a t i v i s t i c approximation, and at times a cumbersomemathematical notation and method, as well as the fact that a great deal of useful information that is known about operators cannot be directly applied. It is also possible to separate the field and particle aspeclsof a problem in a manner which usesoperators a n d H a m i l t o n i a n si n a w a y t h a t i s m u c h m o r e i a m i l i a r . One abandons the notation that the order of action of operatorsdependson their written position on the paper and substitutessome other convention (such thit ihe o r d e r o f o p e r a t o r si s t h a t o f t h e t i m e t o w h i c h t h e y r e f e r ) . T h e i n c r e a s ei n m a n i p u l a t i v e f a c i l i l y w h i c i r a c c o m p a n i etsh i s c h a n g ei n n o t a t i o nm a k e si t e a s i e rt o representand to analyzethe formal problems in electrod y n a m i c s .T h e m c t h o d r e q u i r e ss o m ed i s c u s s i o nh, o w ever, and will be described in a succeedingpaper. In this paper we shall give the derivations of the formulas of II by means of the form of quantum mechanics given in C. 'fhe problem of interaction of matter and 6eld will be analyzed by first solving for the behavior of the field in . N*t the California Institute of Technology, Pasadena, terms of the coordinates of the matter, and finallv California. r R. P. Feynman,Phys. Rev. 76, 749 (19+q),hereafrcrcalledI. discussingthe behavior of the matter (by matter is rnd Phys. Rev. 76, 769 (1949),herealrercallcd II. , Seein this connectionalso the Datlcrso[ S. Tomonaea.phvs. actually meant the electrons and positrons). 'fhat is to R e v . 7 4 , 2 2 4 { 1 q 4 8 ) i S . K a n e s a w aa n d S . T o m o n a s : a , ' P r o r . say, we shall 6rst eliminate the fietd variables from the T h e o r e t .P h y s .3 , t O l ( 1 q 4 8 ) ;J . S c h w i n g e rP, h y s . n e v . 7 6 , 7 V l r l q 4 9 ) ; F . D y s o n , P h y s . R e v . 7 5 , 1 7 J 6 ( 1 9 4 9 ) ;W . P a u l i a n d equations of motion of the electrons and then discuss F. Villars, Rev. Mod.. Phys. 21, 434 (1949). The papers cited the behavior of the electrons.In this way all of the -sive referencesto previous work. r u l e sg i v e n i n t h e p a p e rI I w i l l b e d e r i v e d . 3 R. Feynman, (1948), Rev. Mod. Phys. 20, 367 hereafter _P. called C. Actually, the straightforward elimination of the Iield 440 f N two previous papersr rules were given lor the r c a l c u l a l i o no f t h e m a l r i x e l e m e n tf o r a n y p r o c e s si n electrodynamics, to each order in e2/hc. No comDlete proof of the equivalence o f t h e s er u l c s t o t h e c o n v e n tional electrodynamics was given in these papers. Secondly, no closed expression was given valid to all orders in e2/hc. In this paper these formal omissions s-ill be remedied.2 In paper II it was pointed out that for many problems in electrodynamics the Hamiltonian method is not advantageous, and might be replaced by the over-all space-timepoint of view of a direct particle interaction. It was also mentioned that the Lagrangian form of quantum mechanics3was useful in this connection. The rules given in paper II were, in fact, first deduced in this form of quantum mechanics. We shall give this derivation here. The advantage of a Lagrangian form of quantum mechanicsis that in a system with interacting parts it permits a separation of the problem such that the motion of any part can be analyzed or soived first, and the results of this solution may then be used in the solution of the motion of the other parts. This separation is especially useful in quantum eiectrodynamics which represents the interaction of matter with the electromagnetic field. The electromagnetic field is an especially simple system and its behavior can be analyzed completely. What we shall show is that the

258 1'HI]ORY

OF

ELECTROMAGNETIC

variables will lead at first to an expression for the behavior of an arbitrary number of Dirac electrons. Since the number of electronsmight be infinite, this can be useddirectly to find the behaviorof the electrons accordingto hole theory by imagining that nearly all the negative energy states are occupied by electrons. But, at least in the caseof motion in a fixed potential, it ]ras been shown that this hole theory picture is equivalent to one in which a positron is representedas an electron whose space-timetrajectory has had its time direction reversed.To show that this samepicture may be used in quantum electrodynamicsrvhen the potentials are not fixed, a special argument is made basedon a study of the relationshipof quantum electrodynamics to motion in a fixed potential. Irinally, il is pointed out that this relationship is quite generaland might be used for a general statement of the laws of quantum electrodynamics. Chargesobeying the Klein-Gordon equation can be analyzed by a special formalism given in Appendix A. A fifth parameteris usedto specifythe four-dimensional trajectory so that the Lagrangian form of quantum mechanicscan be used. Appendix B discussesin more detail the relation of real and virtual photon emission. An equation for the propagation of a self-interacting electronis given in Appendix C. In the demonstrationwhich follorvswe shall restrict ourselvestemporarily to casesin which the particle's motion is non-relativistic,but the transition of the final formulas to the relativistic caseis direct, and the proof c o u l dh a v e b e e l rk e p l r e l a t i v i s t i ct h r o u g h o u t . The transversepart of the electromagneticfreld will be representedas an assemblageof independentharmonic oscillatorseach interacling with the particles, as suggestedby Fermi.aWe use lhe notation of Heitler.5 IN 2, QUANTUM ELECTRODYNAMICS LAGRANGIAN FORM

INTERACTION

441

same.) The Hamiltonian of the transverse field representedas oscillatorsis II

lr : -L L ((PK('))'z+ k'z(qK('))') " 2r, ,:t

where 2s(d is the momentum conjugate to qK(4. The longitudinal part of the field has been replacedby the Coulomb interaction,6

H":+L"L^e"e^/r"As is well known,a when this where r,",2:(x"-x-)t. Hamiltonian is quantized one arrives at the usual theory of quantum electrodynamics.To expressthese laws of quantum electrodynamicsone can equally well use the Lagrangian form of quantum mechanics to describe this set of oscillators and particles. The classicalLagrangian equivalent to this Hamiltonian is Lr -

Lf p L If , LLtf

| !.

Lf r/Ir I

.r.r' lhr !^r.\a

Lo:!1" m"ri,

(2a)

Lr:D" e"x'".Lt'(x") L : LLs. I, ((4K(") )') )' - [' (qK(d " L.:-iL"l^e"e,"/r^".

(2b) (2c) (2d)

When this Lagrangian is used in the Lagrangian forms of quantum mechanicsof C, what it leads to is, of course,rnathematicallyequivalent to the result of using the Hamiltonian 11 in the ordinary way, and is therefore equivalent to the more usual forms of quantum electrodynamics (at least for non-relativistic particles). We may, therefore, proceed by using this Lagrangian form of quantum electrodynamics,with the assurance that the resultsobtainedmust agreervith thoseobtained from the more usual Hamiltonian form. The Lagrangian enters through the statement that the functional which carries the system from one state to another is exp(iS) where

The Hamiltonian for a set of non-relativisticparticles (3) interacting wilh ratliation is, classically,II:Hp*IIr .t: I zdl:se+.tr+.t"+sr,. J + il "+ II h, whcre II ol II r : L " trm"-' (p"- e"A "(x"))' is tire H:rmiltonian of the particlcs of massrz,, charge The time integrals must be written as Riemann sums an, coordirale x, and momentum p" and their inter- with somecare; for example, action rvith thc transversepart of the electromagnetic 'Ihis field can be expandedinto plane waves field. (4) S':I I e"x'"(t)'4"(x"(t))dt : A ' " ( x ) ( 8 r ) 1 f , 6 [ e 1 ( 9 6 ( 'c) o s ( K . x ) + q K ( t )s i n ( K . x ) ) + e r ( g K ( ' !c) o s ( K . x ) + q K ( as) i n ( K . x ) ) l ( 1 ) becomesaccordingto C, Eq. (19) lvhereer and e: arc two orthogonalpolarizationvectors Sr: I" I; ie"(x", ;+r- x", )' (A"(x", +')*A'"(x", )) (5) at right anglesto the propagationvector K, magnitude so that the velocity x',,; which multiplies A"(x",) is A. The sum over K means,if normalizedto unit volume, gr(') be considered as the (6) cau each d3K/8r3, and x',, ;: ]e-l(x", ;1r-x" )*le-l(x", ;- x,, ;-r). lf coordinatcof a harmonicoscillator.(The factor I arises infinitebut must for the mode correspondingto K and to - K is the --n" t"* in thesum forn:m is obviously { E. Ierrni, Ilev. Nlod. I'hys. 1, 87 (19.12). 6 W. lleitler, The Quantunt Tlrcory oJ Rad'ialiox, sccond edition (Orford UniversityPress,London, 1944).

be included for relativistic invariance. Our problem here is to re-express the usual (and divergent) form of electrodynamics in the form siven in II. Modifications for dealing with the divergenccs a.idi..r.."d in II and we shall not discuss them further here,

259 t la

R.

P.

FEYNMAN

In the Lagrangian form it is possibleto eliminate the :ransverscoscillatorsas is discussedin C, Section 13. i)ne must specify,however,the initial and final state of :ll oscillators.We shall lirst choosethe special,simple :asethat all oscillatorsare in their ground statesinitially :nd finally, so that all photons are virtual. Later we lo the more general case in which real quanta are -rresentinitially or iinally. We ask, then, for the amplitude for finding no quanta presentand the particlesin .rlte x(, at time t", if at time l' tlie particles were in ;ttte r! y and no quanta were present. The method of eliminating field oscillators is de-.cribedin Section13 of C. We shall simply carry out the tlimination here using the notation and equationsof C. Io do this, for simplicity, we first considerin the next sectionthe case of a particle or a system of particles interacting with a single oscillator, rather than the of the electromagneticfield. entire assemblage 3. FORCED I{ARMONIC OSCILLATOR We consider a harmonic oscillator, coordinate q, interacting with a particle Lagrangian L:i(d-r'q') or systemof particles,action So, through a term in the Lagrangian g(l)"y(t) where 'y(l) is a function of the 'Ihe coordinates (symbolized as r) of the particle. rrecise form of 7(l) for each oscillator of the electronagnetic field is given in the next section.We ask for rhe amplitude that at sometime l" the parlicles are in statexy and the oscillator is in, say, an eigenstaterz (units are chosensuch that h:c:l) of energy,(mIil wher it is given that at a previoustime l/ the particles \rere in state /r, and the oscillatorin rl. The amplitude for this is the transition amplitude [see C' Eq. (61)] sr \ x v , p ^ l I It L p " ) s p + s + o :

ff

) ^ + ( q( , ) e x p i ( S o * S o { . S r ) | | x , , , * G , , ,e ' e"@/){ ( (r t)dx {'dr Ldq('dq I Dr(t) Dq(t)

(7)

rvhere* representsthe variablesdescribingthe particle, So is the action calculatedclassicallyfor the particles for a given path going from coordinatefit at tt to r(' at t", Ss is the action f i(4'z*o'zq')dtfor any path of the oscillator going from qt, at tt to qy at l'", while sr: J

I q(t)'y(td,,

(s)

the action of interaction, is a functional of both q(r) and *(l), the paths of oscilLatorand particles. The s y m b o l sD x ( 1 ) a n d D g ( 1 )r e p r e s e n at s u m m a t i o no v e r all possiblepaths of particles and oscillator which go between the given end points in the sensedefrned in C, Eq. (9). (That is, assumingtime to proceedin infinitesimal steps, e, an integral over all values of the coordinates r and, q corresponding to each instant in time, suitably normalized.)

The problem may be broken in trvo. The result can be rvritten as an integral over all paths of the particles only, of (expi.9,).G-": q " ) s " + s 0 + s r : ( x , , ,I G ^ " W r ) s o

Qu'e^lllt,

(9)

where G-" is a functional of the path of the particles alone (sinceit dependson 7(l)) given by

c^":(,^l*vi ilt)ilt)atl,c-) [ ,, :

*sr) p"(q/)dqfiq /,Dq(t) e^. (,t,,)expl(so

I :

v ^. (t,) e*pi:;l+,-'

I

'p

(,]n+' - qn)'- L,' q,'r qt ;l

ta-td(12'''a-rdqi "(rlo)dqoa-td.q

(10)

where we have written the Dg(r) out explicitly (and have set a:(2rie)t, t,,-t,:je, q/:q0, qL,:qj). The Iast form can be rvritten as G-

k t" qo,r'),p^(qo\d qodq ^* ; ": JI' 'e" ' k ) 1q,, ;

(11)

where ft(g;,t"; qa,t') is the kernel [as in I, Eq. (2)] for a forced harmonic oscillator giving the amplitude for arrival at q, at time l" if at time l' it was known to be at qo.According to C it is given by h(q1,l" ; qo,t'): (2rio-t sino(1"- l'))-i XexpiQ(q;, t"; qo,t')

(12)

where p(g;, l" ; qo,t') is tire action calculatedalong the c f a s s i c apl a t h b e t w e e nt h e e r r dp o i n t s q ; .l " i q 0 ,l ' , a n d is given explicitly in C.7It is tThat (12) is correct, at least insofar as it depcnds on q0, can be seerr clirectly as folioivs. LeL q(l) be the classical path rvhich satisfies the boundary condition q(l'):g0, 4(1"):qj. Thcn in the integral dcfining I replacc each of the variablcs qi by q;:Tr*!i, that is, usc the (iisl)lacement 1i from thc classical Qt:4(tt)), path q; a5 the coordinatc rathcr than thc absolute posilion. in the aclion With the substitution gi:Q;*]ri

, . -^ - ^ I. q)dt So* Sr:.,1r .(iQ' 1.,q, t :

t r f rLA" l-'q' | 1 Qtd I f \lr'? ib?Y'ir

the terms iinear in 1 drop out by intcgrations by parts using thc equation of motion 4:-o'Q+r(l) for the classical path, and the boundary condilions y(t'):yQ"):0. That this should occur should occasion no surprisc, for the action functional is an exq(t):q(r) tremum at so rhai it will only depencl to seconcl ordcr in the displacements 1 from this extrcmal ortrit 4(t). Furthcr, since the action fuoclional is quadratic to bcgin with, it cannot depend on jr' more than quadratically. Hence

.',|

So lSr:0

|

l'ttU'1-+@')P)dt

so that sincc dq;:dy;,

riOl,f *t,(i.f k1q;, ! " ; qs,t' t : r x1t

t ). rt u,-.'1,d,)D),(/

The factor following thc expiQ is thc amplitude for a frce oscillator I and does not thereto proc€ed lfom y:0 at t,:tI Lo !:O at l:l

260 oF

THEoRY

o:

;;r,,,

- nf

INTERACTION

ELECTROMAGNETIC

443

(seeC, Sec. 12). The functional 1, which is given by

(t"'- r')- 2q1q o te,'+n,) cosa

ff

I : 1 i @ ' t | | e x p ( - i o l r - s l ) r ( s ) r ( / ) d s d (r 1 5 ) JJ

*2:: *4 -

!,,'" [,,'"

,al sina(t-t,)dt - t)rtt ta) sina(t"

) [,,"' [,,',

<,lr(s)sino(r"-r) Xsin,(s-l')dsdr].

is complex,howeverI we shall speakof it as the complex action. It describesthe fact that the system at one time can affect itself at a different time by means of a temporary storage of energy in the oscillator. When there are severalindependentoscillatorswith difierent interactions,the effect,if they are all in the loweststate at l'and /", is the product of their separateGoocontributions. Thus the complex action is additive, being the sum of contributions like (15) for each of the ( 1 3 ) several oscillators.

The solution of the motion of the oscillatorcan now b e c o m p l e t e db y s u b s t i t u t i n g( 1 2 ) a n d ( 1 3 ) i n t o ( 1 1 ) 'I'he simplest case is for and performing the integrals. m, n:O lor which casej : @/ qoo(qo)

exp(- |rgoe) exp(- liat') ")I on so that the intcgrals qo,qr are just Gaussianintegrals. There results

c* : *p ( - 1,-,

"

uor- i a(t- s))1Q)^t(s)dds) 1,,' 1,,'

4. VIRTUAL TRANSITIONS IN THE FIELD ELECTROMAGNETIC We can now apply these results to eliminate the transverse fielcl oscillators of the Lagrangian (2). At first we can limit ourselvesto the caseof purely virtual transitionsin the electromagneticfield, so that there is no photon in the field at t' ar'd t". That is, all of the field oscillators are making transitions from ground state to ground state. The "yK(")correspondingto each oscillator gs{') is found from the interaction term Zr [Eq. (2b)]' substituting the value of A'"(x) given in (1). There results, for example,

a result oI fundamental importance in the succeeding r r ( " : ( 8 n ) , I . e , ( e , . x ' " )c o s ( K . x " ) developments.By replacingl-J by its absolutevalue \/ r1vA' \ jI" e.(er'x sin(K'x") ') zr(3': (8n) ll-sl we may integrate both variablesover the entire range and divide by 2. We will henceforth make the the correspondingresultsfor ?r(2),?r(n) replacee1by e2' results more general by extending the limits on the The complex action resulting from oscillator of @. o wishes to Thus if one to from * intesrals coordinate qr(t) is therefore with an of interaction particle study the effect on a oscillatorfor just the period l' to l" one may use

iklr-s])(e,.x'"(r)) exp(^, :Y++ I I e,e^

*,:.*(-_1I:I Xexp(-iolr-sl h(hft)dds)

X ( e r 'x ' - ( s ) ) ' c o s ( K 'x " ( t ) ) c o s ( K 'x - ( s ) ) r l s d r .

(14) The term h(3) exchangesthe cosinesfor sines,so in

imagining in this casethat the interaction 1(l) is zero outside these limits. We defer to a later section the discussionof other valuesof ar, z. SinceGoois simply an exponential,we can write it as exp(i1), consider that the complete "action" for the svstem of particles is S:Sot1 and that ore computes tiansition ilements rvith this "action" instead of So fore depend on 00, or, or a(/1, lrcing a function only of t"-t'| can bc dcmonstrated l-Thar it is acturllv i2rid I sinutt"-r')) Either lrv dircct iniesration oi lhe y varial'les or by using some normalizirrg property"of lhe kcrnels &. iur exampLethf,t 6@ for rhe casc 7":i] niusr'equal unity.] The.exprcssion.forQ,given,in C on page J86 is in crror, the quanllltes g0 and gi snoulc De interchansed. 8 It is irost convenient to dcfine the state q, rvith the phase and the 6nal state with the factor factor expf-io(z*i)l'] qp[-i.(r;+i)rt'l.so,t-hat the rcsults rvill not depend on lhe nartlcular tlmes r'. r' cnosen.

the sum 1K(t)+1K(3)the product of the two cosines, cos.4.cosB is replaced by (cos,4cos3{sin,4 sinB) or 'Ihe terms 1K(2)+1K(4)give the same cos(,4--B). r e s u l t w i t h e 2 r e p l a c i n ge r . T h e s u m ( e 1 ' V ) ( e 1 ' V ' ) ' ? ( K ' V ) ( K ' V ' ) s i n c ei t i s t(e,'V)(e''V') is (V'V')-fr the sum of the products of vector componentsin two orthogonaldirections,so that if we add the product in the third direction (that of K) we construct the complete scalar product. Summing over all K then, since ilK/873 we find {or the total complex action |K:+f of all of the transverseoscillators,

t',:iLL i

ft'

I

nJ.,

f('

dt I

J,'

f

d sI e " e ^ e x p ( - i i l f - s l ) ,l

x -(,))l x'"1r))(K' x t*'"tl' *-trl - P-r16' . c o s ( K '( x " ( , ) - x - ( s ) ) ) d , K / 8 r ' z h . ( 1 7 )

26r 114

R..P.

FEYNMAN

Tliis is to be added to So*S" to obtain the complete action of the system with the oscillators removed. The term in (K'x'"(l))(K'x'-(s)) can be simplif,ed by integratioa by parts with respect to I and with respect to s [note that exp(-lfrlt-sl) has a discontiruous slope at t:s, or break the integration up into trvo regions]. One finds 1s,:R-1"f

..r.here

16u,"1"o1

(18)

considered as varying with time). In this case, in the limit, 1s,u,"i".1is zero.e Hereafter we shall drop the transient term and considerthe range of integration of I to be from -o to *@, imagining, if oni needsa clefinition,that the chargesvary-with-time and vanish in the direction of eitheilimit. To simplify R we need the integral _ J:

f exp(_illli) I

cos(K.R)d3Kl8r,A

fl"rt"r

R:-iLLl a

ill

mJ(

f*

dsle,e^

J!

: I 'lO

J

exp(-ifltl) sin(kr)dk/2rr(22)

where r is the length of the vector R. Now Xexp(-iA Ir-s | )(1-x'"(l). x -(s)) ' cosK.(x"(t)-x^(s))d.3K/Sr,h (19) I

exp(-iA*)dA:

lim (-t(.u-ie)-r)

ro

and

: -ir-t ft"

/":-|I n

n J (

/'

I atle",^ (20)

comesfromthediscontinuity in slopeof exp(- ihlt-sl) at l:s. Since ff'

-, I co.(K. R)drKf4#k2 J o J

& r ) - , s i n ( k r ) d k / r- ( 2 r ) - l

this term 1" just cancelsthe Coulomb interaction term The term S":IL.dt. ltrr.sient: _lle"e* J

Xl I

r6+(x)

where the equation serues to define dn(:r) fas in II, nq. (:)].Hence, expandingsin(Ar) in exponentialsfind

J

X cosK . (x"(r)- x^(t))d3K/4tf k,

l16(r):

r d3K | ^ +7"R"

J: - (arr)-t((ltl -/)-1-

(lll +/)-,)

r l(_l r ) _ a ( | r l * r ) ) f (a;z)-'(6

: - (2r)-t(t, * 7z)-t+ (2i)-t6(t, - rr) : -|i6*(t'-r2)

(23)

where we have used the fact that 6 ( t 2 - r ' z ) : ( 2 r ) - t ( 6 ( l t l- r ) f

a(lil f r))

and that 6(ltl*r):0 s i n c eb o t h l t l a n d r a r e n e c e s sarily positive. Substitution oI theseresultsinto (19) gives linally, I R: -:tt

lexp(- ih(r."-r1)cosK.(x,(1")-x^(//)

IJr

r+- r+r,e-(l-x x f I "(t) ,(s)) Z n m J - 6 J _ 6 X d+((r- s),- (x

texp(-ih(t- t')) cosK.(x,(l)- x^(/)))dt (x ) - x^(t")) * (2&)-U[cosK. "(/' lcosK' (x"(l')- x-(l'))

"(t)

* x-(s)),) dtds. (24)

The total complex action of the system is thenro So*R. Or, what amounts to the same thing; to obtain

- 2 exp(-ih(t"-r')) cosK.(x,(l')-"-0',))l l. I

e1)

is one which comes {rom the limits of integration at l/ and 1", and involves the coordinatesof the particle at either one of these times or the other. If t' and t" ate consideredto be exceedinglyfar in the past and future, there is no correlation to be expected between these temporally distant coordinates and the present ones, so the effects of 11."n"iu,.,will cancel out quantum mechanically by interference. This transient was produced by the sudden turning on of the interaction of field and particles at l'and its sudden removal at 1". Alternatively we can imagine the charges to be turned on after tr'adiabatically and turned off slowly before l" (in this case,in the term 2", the chargesshould also be

'O,* obtain the 6nal result, that the total interaction is just R, in a formal manner starting from the Hamiltonian from which the longitudinal oscillators have not yet been eliminated. There are for each K and cos or sin, four oscillators qlK corresponding to the three components of the vector potential (p:1, 2,3) and the scalar potential (p:.1). It must then be assumecl that the rvave functions of the initial and final state of the K oscillators is the function (&/z) exp[- ]&(q1sr*c:rr]-qrdqn:rr)f. The rvave function suggested here has only formal signi6cance, of course, because the dependence on qng is not square intcgrable, and cannot be normalized, If cach oscillator were assumed actually in the ground state, the sign of the q{K term would be chuged to positive, and the sign of the frequency in the contribution of these oscillators l,ould be rcversed (they l'ould have negauve energyJ. r0 The classical action for this problem is just,t2+R'where R' is the real part of the expression (24). In view of the generalization of the Lagrangian formulation of quantum mechanics suggested in Section 12 of C, one might havc anticipated that n would have been simply 1l', This corresponcls, horvcver, to boundary condilions othcr rhrn no quanla l)rFscnt in past an'l Iuture. It is hardcr to intcrpret physiealll. For a systcrn cnclosedin a light tight box, hot'ever, it appears likely that both R and R'lead to the same lesults.

262 THEORY

OF

ELECTROMAGNETIC

INTERACTION

445

transition amplitudes including the efiectsof the field we must calculatethe transition elementof exp(iR):

can also be evaluated directly in terms of the propagation'kernel K(2, 1) [see I, nq. (Z)] for an electron m o v i n g i n t h e g i v e np o t e n t i a l . (2s) so r,lexPi'Rlt1' ',) \x The term x',.x'1 in the non-relativisticcaseproduces under tlie action .5, of the particles, excluding inter- an interesting complication which does not have an action. Expression(24) for R must be considereci to be analog for the relativistic casewith the Dirac equation. written in the usual manner as a Riemann sum and the We discussit below, but for a moment consider in expression(25) interpreted as definedin C [Eq. (39)]. further detail expression (26) but with the factor (1-x'".x') replacedsimply by unity. Expression(6) must be used for x'" at time l. The kernel f(2, 1) is defined and discussedin I. Expression (25), with (24), then contains all the effectsof virtual quanta on a (at least non-relativistic) From its definition as the amplitude that the electron systemaccorclingto quantum electrodynamics.It con- be found at x: at time t2,iI at \ it was at xl, we have tains the effectsto all ordersin e2/hcin a singleexpresK(xz, tz; x1,t) : (6(x-xr),, | 1 | 6(x-x,),,) I e (27) sion. If expanded in a power seriesin e2fhc, the various terms give the expressionsto the correspondingorder that is, more simply K(2, 1) is the sum of exp(rSr) over obtained by the diagrarns and methods of trI. We all paths which go from space time point 1 to 2. illustrate this by an examplein the next section. In the integrations over all paths implied by the symbol in (26) we can first integrate over all the x; 5. EXAMPLE OF APPLICATION OF EXPRESSION (25) variables corresponding to times l; from l' to s, not We shall not be much concerned with the non- inclusive, the result being a factor K(x", s; x,,, l') acrelativistic casehere,as the relativistic casegiven below cording to (27). Next we integrateon the variablesbeis as simple and more interesting.It is, however,very tweens and I not inclusive,giving a factor tri(x1,l; x", s) similar and at this stageit is worth giving ar.rexample and finally on those between , and i" giving to show how expressionsresulting from (2.5)are 1o be K(xy', t" ; x1, l). Hence the left-hand term in (26) interprcted according to the rules of C. For example, excludingthe x'1'x'" factor is consider the case of a single electron, coordinate x, either free or in an external given potential (contained -e a, a' t")K(xy,, t, ] x,,t)6+((ts)z for simplicity in So, not in't R). Its interaction rvith I I I V*(xy,, the field producesa reaction back on itself given by R - ( x , - x " ) ' ? ) . K ( x r , l ; x " , s ) K ( x " ,s x y , t ' ) as in (24) but in which we keep only a single term ; correspondingto nl:n. Assume the eflect of R to be Xt(xr, t')d3n,'d\xfi3xd3xy (28) small and expand exp(ift) as 1*iR. Let us find thc amplitudc at time l" of finding the electron in a state ry'rvith no quanta emitted, if at time l' it rvas in thc which in improvcd notation and in the relativistic case is essentiallythe result given in II. samestate. It is We have made use of a special case of a principle @ t , l I + i R l , l ' ( ) s o : \ { r ' l l l V r ) s e + i ( , ,t , , l R l V) s e which may be stated more generallyas where (,y',,,| 1 1ry',,)s,:gvp[- iE(t:'-t)l if l? is the ( x u , l F 1 * t , \ 1 x 2 , t 2 ; ' ' ' x 7 ,t 6 ) 1 , ! y ) s , energyof the state, and ft"

f/'

, , 1 , ,R ',l,l',)s":-[,'1I dtt Jr' J(

J s ( / , .( 1 - x , . x

:

"r

X d+((l- s)'?- (xr- x")?)] 'y'i).sn. (26)

f

x.6r,)x(*r,,

t " 1 x 1 , 1 1K) ( x 1 , t . ; x 2 ,t 2 ) .. .

Xd(xr,r, t.p 1; xy, te)K(x4 te; xy, t')

. F (4, 11;xz,tzj . . . xa, te),|,(xy) Here x":x(s), etc. In (26) rve shall limit the range of integrationsby assumings(1, and double the result. d.3xed1xy(29) Xd3x('d.3xd,3x2.,, The expressionwithin the brackcls ( )so on the (26) can be evaluatedby the methods where P is any function of the coordinatex1 at time lr, right-hand sideof describedin C [Eq. (29)]. An expressionsuch as (26) x2 at 12up to x6 tp, and, it is important to notice, we have assumedt") tt> lr> . ., tr>.t'. t1 Onc can shorv from (25) horv the correlated effect of many Expressionsof higher order arising for examplefrom atoms aL a.listzrnce pro.luces on a given system the effects of an (24) fielcls thc result that this cxternal potenliirl. Fornula R! are more complicatedas thereare quantitiesreferring potcntial is that obtained from Li6nard and Wiechert by retarded to severaldifferent times mixed up, but they all can be waves arising lrom the charges and currents resulting from the interpretedreadily. One simply breaksup the rangesof distant aloms making transitions. Assunre the $ave functions 1 ancl ry'can be split into proclucts of wave functions for system integrationsof the time variablesinto parts such that a n J ' l i s l r n l r l o m s a n d e \ l , a n , l c x t ) ( i n r a s s u r n i n qt h e c f i c c t o f a n v in each the order of time of each variable is definite. in lividual rlislcnl rtom is small. Couloml, porentials arise evcn from nearby particles if they are moving slorvly. One then interpretseachpart by formula (29).

263 116

R.

P.

FEYNMAN

As a simple example we may refer to the problem of lhe transition element

element of

II[](xn+'-xo) f ](x;-x;-)1. A(x;,r)

(*"1[,or,,tlat ! v61s1,oo,ir,,) arising, say, in the crossterm in U and 7 in an ordinary secondorder perturbation problem (disregarding radiation) with perturbation potential U(x,t)*V(x,t). In the integration on s and I which should include the entire range of time for each, we can split the range of .r into two parts, s
t lr r | \ ( x,,'lI u(x,,t)dtI v(x",s)dsll,, ) \

r

lr

nltt

: I

I

f

^t

at I as I yx(xy,)K(xy,, t,,;x1,t)

J!'

Jr

J

XU(4, t)K(a, r; x",s)Z(x",s)

ii

X[4(x;+'-x)*](x;-x;-1)l.B(x;,

+ | J(

f{'

dt I JI

(31)

In C it is shown that when converted to operator notation the quantity (x;-1- r;)f e is equivalent (nearly, seebelow) to an operator, (x ; a1- x ;) / e+ i (H.v-

(32)

xH)

operating in order indicated by the time index i (that is after oi's for /(i and beforeall *1,sfor /)l). In nonrelativistic mechanics i(Hr-rH) is the momentum operator 2, divided by the mass rz. Thus in (31) the e x p r e s s i o [n+ ( x r * r- x , ) + + ( x ; - x ; - 1 ) ] . A ( x ; , / ) b e c o m e s e(p.Al[.p)/2m. Here again we must split the sum into two regionsj(i and jli so the quantities in the usual notation will operatein the right order such that eventually (31) becomesidentical with the rieht-hand s i d e o f E q . ( 3 0 ) b u t w i t h U ( x 1 ,l ) r e p l a c e db y r h e operator

. K(x", s x y, t' ; )tlt (x y) d.3xy, d|x sdax dsxy ft"

r).

l/1

0

td\

^' 2 m( \.i ^A. \A ( * ' , l ) + A ( x , , t ) . . 1

f

i |xt/

ds I x*(x,,,)K(xr,, t',; x,, s) J

standing in the same place, and with the operator X 7(x,, s)K(x", s ; x6 t)U (x, t) 7/ld

so that the single expressionon the left is represented by two terms analogous to the two terms required in analyzing the Compton effect. It is in this way that the several terms and their corresponding diagrams corresponding to each process arise when an attempt is made to represent the transition elements of single expressions involving time integrals in terms of the propagation kernels r(. It remains to study in more detail the term in (26) arising from x'(l) . x'(s) in the interaction. The interpretation of such expressionsis considered in detail in C, and we must refer to Eqs. (39) through (50) of that paper for a more thorough analysis. A similar type of term also arises in the Lagrangian formulation in simpler problems, for example the transition element

/ lr r | \ ( x , , , 1| x ' ( t ) ' A 1 x ( t ) , |t )*d' (t ' ) . 8 ( * G');,a ' l / , , ) \

lJ

r

|

/

arising say, in the cross term in A and B in a secondorder perturbation problem for a particle in a perturbing vector potential A(x, l)*B(x, t). The time integrals must first be written as Riemannian sums, the velocity (see (6)) being replacedby x':|e-I(x;..1-x;) so that we ask for the transition *ie-r(x;-x;-r)

1

d\ r).- I - | . -.8(x,, s)+-B(x,,

. K (x 4 I ; x x, t' ) rlt(x y) d3xy,d.sxdxx1d3x y (30)

2m\i

6x"

i

0x"/

standing in the place of I/(x", s). The sums and factors e have now become lfdtifds, This is nearly but not quite correct, however, as there is an additional term coming from the terms in the sum correspondingto the specialvalues,j=1, j:ill and j:i-l.We have tacitly assumedfrom the appearance of the expression (31) that, for a given i, the contribution from just three such special terms is of order e2. But this is not true. Although the expectedcontribution of a term like (r;a1-r1)(4a-r) lor jli is indeed of order e2, the expected contribution of (r;ar-r;)2 is liem-L lC, Eq. (50)], that is, of order e. In nonrelativistic mechanicsthe velocitiesare unlimited and in very short times e the amplitude difiuses a distance proportional to the square root of the time. Making use of this equation then we see that the additional contribution from these terms is essentially hn-t el A,(x;,t ;) . B (x ;, t 7): ;p-r f I f iJ

"trl,

t) . B (x(t), t) dt

when summed on all i. This has the same effect as a frrst-order perturbation due to a potenlial A,.B/m. Added to the term involving the momentum operators

264 THEORY

OF

ELECTROMAGNETIC

we therefore have an additional termr2 -

i r'" f | d / | x * ( x , , . ) K ( x , , ' ,! " i x 6 / ) A ( x 1 /, ) ' B ( x r , r ) .K(t t, t; xr, t'){'(xy)d3x1,,d.3x1d3xy. (33)

fn the usual Hamiltonian theory this term arises, of course, from the term L2/2m h the expansion of the Hamiltonian H : (2n)-t (p- L)' : (2m)-t (p'- p. A- A. p{ Az) while the other term arisesfrom the second-orderaction of p.A*A.p. We shall not be interested in nonrelativistic quantum electrodynamics in detail. The situation is simpler for Dirac electrons. For particles satisfying the Klein-Gordon equation (discussed in Appendix A) the situation is very similar to a fourdimensional analog of the non-relativistic case given here. 6. EXTENSION TO DIRAC PARTICLES Expressions (24) and (25) and their proof can be teadily generalized to the relativistic case according to the one electron theory of Dirac. We shall discuss the hole theory later. In the non-relativistic case we began with the proposition that the amplitude for a particle to proceed from one point to another is the sum over paths of exp(iSo), that is, we have for example for a transition element ff

J

. r y ' ( x ) d 3 x s d 3 x 1 . . . d 3 x(p3 4 ) where for exp(iSr) we have written Oo, that is more precisely, (Do: fI;,4-r expiS(x;a1,x). As discussed in C this form is related to the usual form of quantum mechanics through the observation that n-t "xp[4"!(x;a1,

x)]

447

(or 4trX4il if we deal with ly' electrons) but the expression (34) with (36) is still correct (as it is in fact for any quantum-mechanical system with a suftciently general definition of the coordinate x). The product (36) now involves operators, the order in which the factors are to be taken is the order in which the terms appear in time, For a Dirac particle in a vector and scalar potential (times the electron charge e) A(x, ,), Aa(x, t), the quantity (xr+r I x').(1) is related to that of a free particle to the first order in e as (x;11j x).cir : (yr+r I x;).tor expl- i(eA ag., t ;) _ (x;a1_x;). A(x;, t;))f. (37) This can be verified directly by substitution into the Dirac equation.r3It neglects the variation of A and ,4a with time and space during the short interval e. This produces errors only of order e?in the Dirac case for the expected square velocity (x;a1-x;)2/e2 during the interval e is finite (equaling the square of the velocity of light) rather than being of order l/e as in the nonrelativistic case, fThis makes the relativistic case somewhat simpler in that it is not necessaryto define the velocity as carefully as in (6); (x;11-x)/e is sufficiently exact, and no term analogousto (33) arises.] Thus iDo{erdiffers from that for a free particle, iDrto), b y a f a c t o r I I r e x p - i ( e , 4 r ( x ; ,l ) - ( x ; 1 1 - x ; ) . A ( x u , l ) ) which in the limit can be written as (fl

expl- r I lA ^(x(t),4-x'0).A(x(r), Dldr| (38) l.tl

( x l t l 9 ) : l i m | " ' I x * ( x " ) o o ( x " , x r - r",' x o ) ,aJ

(x;11 | x).:

INTERACTION

(3s)

where (x;a1lx). is the transformation matrix from a representation in which x is diagonal at time t; to one in which x is diagonalat time rr+r:rr+e (so that it is identical to Ks(x;a1,t,a1;x;, tt) for the small time interval e). Hence the amplitude for a given path can also.bewritten Oo: Il;(x;a1lx;). (36) for which form, of course, (34) is exact irrespective of whether (x;1tlx)" can be expressedin the simple form

(3s).

exactly as in the non-relativistic case. The case of a Dirac particle interacting with the quantum-mechanical oscillators representing the field may now be studied. Since the dependenceof iDr(d)on A, -4a is through the same factor as in the non-relativistic case, when A, la are expressedin terms of the oscillator coordinates g, the dependence of O on the oscillator coordinatesq is unchanged.Hence the entire analysis of the preceding sections which concern the rcsults of the integration over oscillator coordinates can be carried through unchanged and the results will be expression (25) with formula (24) for R. Expression (25) is now interpreted as f

Qr,lexpiRlg"): xfl(o", -;;fiiJ*tiu"ly,.ote

l i m | 1 * ( x a ' ( r )x, v ' o ' " ) .aJ

. . . d|x,'@) d3xt"- c@) "(0)d3x,,,(n) 'exP(zR)ry'(xt'rtr, x"(2)' ' ')

(39)

that Eq. (37)is exactfor arbitrarilylarge

e if the potential l! is constant. For if the potential in the Dirac the equrtion is the qradient of a scalar function,4u-dx/d& potential may be removed by replacing the wave function by (gauge This alters the kernel by fl-" transformation). t* corresponding to this for theself-energv expression /,:sap1-i1)'y'' " (26) would give an integral over 6+((/-r)r-(x1-x1)r) rvhich is i f a c r b r e x p [ - i ( x ( 2 ) - x ( t ) l ] owing to Lhechange in the initial potential gradlent is ln the evidently in6nite and leads to the quadraticallydivergent self- and final wave functions. A constant energy.There is no such term for the Dirac electron,but there ol y=14ur" and can Le completely removed by this gauge trans' is for Kkin-Gordon particles.We shall not discussthe infinities formation, so that the kernel differs from that of a free particle (37). in this paper as they have already been discusscdin II. by the factor expf- i(A$pr-Arrp)lasi\

-

For a Dirac electron the (*n*tl*). is a 4X4 matrix

265 4-18

R.

P.

FEYNMAN

;here iDo,,(0),the amplitude for a particular path for rarticle z is simply the expression(36) where (xt*rlx). : : t h e k e r n e l K o , , ( x i + r ( " )l,i 1 1 ; x ; ( " ) , l t ) I o r a f r e e e l e c -,ronaccordingto the one electron Dirac theory, with ,he matrices which appear operating on the spinor ::dices correspondingto particle (z) and the order of :ll operationsbeing determinedby the time indices. For calculationalpurposeswe can, as before,expand -R as a power series and evaluate the various terms in :fe same manner as for the non-relativistic case. In :rch an expansionthe quantity x'(l) is replaced,as we :ave seenin (32), by the operator i(l1x-xI1), that is, :r this case by a operating at the corresponding time. fhere is no further complicatedterm analogousto (33) :rising in this case,for the expectedvalue of (rr*t-rn)' :s now of order e2rather than e. For example, for self-energyone seesthat expression 28) will be (with other. terms coming from those with s (t) replaced by c and with the usual B in back of :ach Ko because of the definition of Ko in relativity :reory) " f .! r'' R I tl'v)so: - e2| {rx\x (') K 0(x,", t" ; x 1,t) Ba, J

There ar:1,

TolB):(xu,l""piPl*r)

(40)

(41)

where P: -L ^J

/)-x.t"'().8(xt,,(t),t))dt | [ao1xr,11;,

by (38). We can write this as P: -I

I Blr,(il(t))iFtil(t)dt "J

where ra(!):, and i4:1, the other values of p corresponding to space variables. The corresponding amplitude for the same processin the same potential, but i n c l u d i n ga i l t h e v i r t u a l p h o l o n sw e m a y c a l l , T"zln):|r",1"o(iR)

.6..((/-s)r- (x1-x,)r)Kr(x,, t; x", s)Ba, . Ks(x,, s ; x y, t' ) p,tt(xy) d,3 x y,d,3 x fi,3x x yd.td. s, "d,3

the number of electronsis infinite, but that they all have the same charge,e. Let the states ry',,,Xr,,, represent the vacuum plus perhapsa numberof real electrons in positive energy states and perhaps also some empty negative energy states. Let us call the amplitude for t h e t r a n s i t i o ni n x n c x t e r n i r p l otentialBr,bul excluding drlual photons, TnlB), a functional ol BuQ). We have seen(38)

exp(iP)lt!v).

(42)

Now let us consider the effect on T"zlBl of changing the coupling e2ol the virtual photons. Differentiating (42) with respect to e2which appearsonly rain R we find

dr,2,a:ar.u,z and a sum on the repeated

::ii:hli*tl

ll,*i:?I',:*;;';i::?!;dr"ztBt/d(e'?1:(.''l-;++ I.fd,,,,u'"'1,;*u'''

In this manner all of the rules referrins to virtual o h o t o n s d i s c u s s e di n I I a r e d e d u c e d :b u t w i t h t h e thal r(o is used instead of K,. and we have di-fference rhe Dirac one electron theory with negative energy ;tates (although we may have any number of such electrons). 7. EXTENSION TO POSITRON THEORY Since in (39) we have an arbitrary number of elecrrons, we can deal with the hole theory in the usual manner by imagining that we have an infinite number of electrons in negative energy states. On the other hand, in paper I on the theory of positrons, it was shown that the results of the hole theory in a system with a given external potential ,4p were equivalent to those of the Dirac one electron rheory if one replaced the propagation kernel, Ke, by a different one, K4, and multiplied the resultant amplitude by factor C, involving ,4r. We must now see how this relation, derived in the caseof external potentials, can also be carried over in electrodynamics to be useful in simplifying expressionsinvolving the infinite sea of electrons. To do this we study in greater detail the relation between a problem involving virtual photons and one involving purely external potentials. In using (25) we shall assume in accordance with the hole theory that

. 6*((x (t)- r (5)),)expl(R* rll*,) "r,t "L-r

(43)

We can also study the first-order efiect of a change of B,:

6r "zlB)/ 6B,(r): - r(",,14

[

a,tpQ) 6a @"h)(t)- " ",,)

e.r;tn+rtl*,,)(44) where r". r is the field point at which the derivative with respect to Bu is takenrs and the term (current density) -L" I dti F(")(t)Ba(x"t"t (t) - x". 1) is just 6P/ 68 FG). The function 6a(x.t"t-*",t7 means 6(rr{"r-rr,t, la In changing thc charge e2 u'e meao to vary only the degree to which virtual photons arc important. We do not contemplate changes in the influcnce of the external potentials. If one wishes, as e is raised the strength of the potential is dccreased proportimes the charge e, is held tionally so that -Bs, the potential consran[. 16The functional derivative is defined such that if ?[B] is a number depending on the functions Bu(1), the 6rst order variation in ? produced by a change from Bp Lo Bp+ABF is given by

68p(1))LBy(r)dt1 rlB + ^Bf- rlB): J GrlBl/ r*r. overall four-space tie integralextending

266 THEORY

OF ELECTROMAGNETIC

X 6(rs(")- *3,r)6(rz("' - xz,t) 6(x{") - rr, r) that is, 6(2, 1) with ,".2:r"(")(l). A secondvariation of I gives, by differentiation of (44) with respect to B,(2), 6'T 68uQ)68,(2) "zlB)/

/lrr : -( xr,lLL I d/ | dsi!(n(t)i,(-)(s) \

I"-J

J

. an6.a:t(t)- x",r)64(n:r(") (s)-#p, t

xe*p;(n+P) lP,,). tf Comparison of this with (43) shows that ff

: i; 68 68 dr "zlBl/d(e'z) J J G'zr"zlaf/ Fg) FQ)) X6+(sn2)d.rdrz

(45)

where srz2: (ru,r - r u,2)(r r, 1- x r, z). We now proceed to use this equation to prove the validity of the rules given in II for electrodynamics. This we do by the following argument. The equation can be looked upon as a differential equation for I,z[-B]. It determines ?,2[B] uniquely if falBf is known. We have shown it is valid for the hole theory of positrons. But in I we have given formulas for calculating tq[B] whose correctnessrelative to the hole theory we have there demonstrated. Hence we have shown that the ?,2[B] obtained by solving (45) with the initial condition ?r[B] as given by the rules in I will be equal to that given for the same problem by the second quantization theory of the Dirac matter field coupled with the quantized electromagnetic field. But it is evident (the argument is given in the next paragraph) that the rulesl6given in II constitute a solution in power series in e2of the Eq. (45) fwhich for e2:0 reduce to the ?6[8] given in I]. Hence the rules in II must give, to each order in e2, the matrix element for any process that would be calculated by the usual theory of second quantization of the matter and electromagnetic fields. This is what we aimed to prove. That the rules of II represent, in a power segies expansion,a solution of (45) is clear. For the rules there given may be stated as follows: Suppose that we have a process to order k in e'z(i.e., having A virtual photons) and order n in the external potential Bu. 'Ihen, the matrir eletnentJor the processuith one more il.rtual pholon and lwo lesspolenlials i.slhat oblained. Jront 16That is, of course, those rules of II which apDly to the unmodified electrodynamics of Dirac electrons. (The limitation excluding real photons in the initial and 6nal stales is removed in Sec.8.) The same arguments clearly apply to nucleons interacting via neutral vector mesons, vector coupling. Other couplings require a minor extension of the argument. The modification to the (xi+r I xi)., as in (37), produced by some couolings cannot very easily be written without using operators in the exponents. These operators can be treated as numbers ii their order of operation is maintained to be always their order in time, This idea will be discussed and applied more generally in a succeeding paper.

INTERACTION

449

the preaious molri.st.by choosingJron lhe n lotentials a poir, soy B uQ) acting at 1 and.B,(2) acti.ngat 2, replacing them by i.e26r"6,,(spz), ad.d.ingthe resull,sfor eachway of choosing the pair, and. diri.ding by kl1, the presenl number of photons.The matrix with no virtual photons (E:0) being given to any n by the rules of I, this permits terms to all orders in e2 to be derived by recursion. It is evident that the rule in italics is that of II, and equally evident that it is a word expressionof Eq. (45). [The factor ] in (45) arisessince in integrating over all dr1 and dr2 we count each pair twice. The division by &fl is required by the rules of II for, there, each diagram is to be taken only once, while in the rule given above we say what to do to add one extra virtual photon to I others, But which one of the ft*1 is to be identified at the last photon added is irrelevant. It agreeswith (45) of coursefor it is canceled on difierentiating with respect to e2 the factot (ez1*+t for the (fti1) photons.l 8. GENERAIIZED FORMULATION OF QUANTUM ELECTRODYNAMICS The relation implied by (4.5) between the formal solution for the amplitude for a processin an arbitrary unquantized external potential to that in a quantized field appears to be of much q'ider generality. We shall discuss the relation from a more general point of view here (still limiting ourselves to the case of no photons in initial or frnal state). In earlier sections we pointed out that as a consequence of the Lagrangian form of quantum mechanics the aspects of the particles' motions and the behavior of the field could be analyzed separately. What we did was to integrate over the field oscillator coordinates first. We could, in principle, have integrated over the particle variables first. That is, we first solve the problem with the action of the particles and their interaction with the fleld and then multiply by the exponential of the action of the 6eld and integrate over all the field oscillator coordinates. (For simplicity of discussion let us put aside from detailed special consideration the questions involving the separation of the longitudinal and transverse parts of the field.e) Now the integral over the particle coordinates for a given processis precisely the integral required for the analysis of the motion of the particles in an unquantized potential. With this observation we may suggest a generalization to all types of systems. Let us supposethe formal solution for the amplitude for some given process with matter in an external potential -Bu(l) is some numerical quantity T6. We mean matter in a more general sense now, for the motion of the matter may be described by the Dirac equation, or by the Klein-Gordon equation, or may involve chargedor neutral particles other than electrons and positrons in any manner whatsoever.The quantity Io dependsof courseon the potential function Br(1); that is, it is a functional TolB) ol this potential. We

267 150

R.

P.

FEYNMAN

assumewe have some expressionfor it in terms of B, 'exact, or to some desired degree of approximation in rhe strength of the potential). Then the answer l"z[B] to the corresponding problem in quantum electrodynamicsis In[.4r(1)*Ar(1)] Xexp(lS) summed over all possible distributions of wherein 56 is the action for the field ield lu(l), So: - (Srd;-'f, F I (@A,/ dt), - (v A t)2)dsxdt the sum x p carrying the usual minus sign for space comporents. Il FlAl is any functional of A uQ) we shall represent nf F [ l ] e x p ( i . 5 ; 6 ) o v e r : y o l F [ l ] l o t h i s s u p e r p o s i t i oo ristributions of A, for the case in which there are no :hotons in initial or final state. That is, we have T,zlBl:xlTolA].Bflo.

(46)

The evaluation of 6lFft]1, directly from the dedni:ion of the operation ol lo is not necessary. We can dve the result in another way. We first note that the lperation is linear,

tions independent of Au. Then by (47)

,lpldll,:J,+ f ,e)nlA,(\lndr1 I +

where we set 611 | ,: 1 (frorn (48) with /!:0). We can work out expressionsfor the successivepowers of ,4u by differentiating both sides of (48) successively with r e s p e c tt o j u a n d s e t t i n g j u : 0 i n e a c hd e r i v a t i v e .F o r example, the first variation (derivative) of (48) with respecttolr(3) gives

I r r \l l - i A N @e x p\ ( - ir t j " 0 ) A " ( r ) d/ l,o, l l

ol

: - ir'

olPtlA)+F,lAll0: 0lFft ll 0+,1p,lAfl o (+7) .o that if F is representedas a sum of terms each term :an be analyzed separately. We have studied essentlally :re case in which F[,4] is an exponential function. In :act, what we have done in Section 4 may be repeated rith slight modification to show that

ff

(s0) J J f,,0,2)olAFQ)A"(2)lsdrd'r2r-"'

J

6a(qa2)j,(l)dr a

(s1) x""v(- +;", i.rtli,rrlu.au,)d,d,,). [ [ Settingj,:Q

giv".

oI -4,(3)I o: 0. Differentiating (51) again with respectto j,(4) and settingT,:0 shows

. "*v(-;I irrt)Are)d,,)|,

ie25p"6+(su2) ol A J3)A "(4)l o:

(s2)

="-n(-+;e (48)3iod;ilJ,Tl"l;ti:m5f;;"*:::;i:T ;,ffi1j: J J.iue)j,(2)6agn)d,Ld,,) /

f

f

\

^-r^^

lhere jr(1) is an arbitrary function of position and :;me for each value of p. Although this gi'res the evaluation of 6l lo for only : particular functional of A, the appearance of the :rbitrary functionju(1) makes it sufficiently general to :ermit the evaluation for any other functional. For it : to be expectedthat any functional can be represented :s a superposition of exponentials with difierent func::ons j,(1) (by analogy with the principle of Fourier -:tegrals for ordinary functions). Then, by (47), the :esult of the operation is the corresponding superposi:ion of expressionsequal to the right-hand side of (48) .;ith the variousT's substituted forjr. In many applications FlAl can be given as a power =eriesin ,4u: - -_ a l : I o f

/f\

16[8]:exp( -i t jp(r)Bp(l)drrl

(s3)

\J/

we have from (46), (48) that

r.ztrl: expf- +ie z] i,0)i -(2)6ag,z\d,td, [ [

*ol-,I i "otu"(t) d,,f.(s4)

., r.,",,,.,, J

in (46) is expanded in a power seriesand the successive terms are computed in this way, we obtain the results given in II. It is evident that'(46), (47), (48) imply that ?"2[B] satisfies the differential equation (45) and conversely (45) with the definition (46) implies (47) and (48). For if fo[B] is an exponential

Jt\r)AF\L)arr

Direct substitution of this into Eq. (45) shows it to be a

+ | | f,"(1,2)A/r)A"(2)drtd,,*... JJ

(4e) solution satisfying the boundary condition (53). Sincethe

','hereJ s, J *(l), J,,Q, 2)' .. are known numerical func-

differential equation (45) is linear, if Z0[B] is a superposition of exponentials, the corresponding superposition of solutions (54) is also a solution.

268 THEORY

OF ELECTROMAGNETIC

Many of the formal representations of the matter svstem (such as that of second quantization of Dirac eiectrons) represent the interaction with a fixed potential in a iormal exponential form such as the lefthand side of (a8), except that iu(1) is an operator instead of a numerical function' Equation (48) may still be used iJ care is exercisedin defining the order of the operators on the right-hand side. The succeeding papei will discussthis in more detail. bquation (45) or its solution (46), (47)' (48) constitutes a very general and convenient formulation of the laws of quantum electrodynamics for virtual processes' Its relativistic invariance is evident if it is assumedthat the unquantized theory giving Io[B] is invariant..It has been proved to be equivalent to the usual formulation {or Dirac electronsand positrons (for Klein-Gordon particles see Appendix A)' It is suggestedthat it is of wide generality. It is expressed in a form which has meani;g even if it is impossible to express the matter svstem in Hamiltonian form; in fact, it only requires t'he existenceof an amplitude for fixed potentials which obeys the principle of superposition of amplitudes. If Io[B] is known in power series in B, calcuLationsof f .rlEl i" a power seriesof d can be made directly ur1& In. italicized rule of Sec' 7. The limitation to virtrial cuanta is removedin the next section. On thi other hand, the formulation is unsatisfactory becausefor situations of importance it gives divergent results, even if f6[B] is finite. The modification proin (45), (48) by/+(su'z) posedin II of replacing6+(sr:'?) is not satisfactoryowing to the loss of the theoremsof conservation of energy or probability discussedin II at the end of Sec. 6. There is the additional difficulty in Dositron theory that even 16[8] is infrnite to begin *ith lrru.uo- polarization). Computational ways of avoiding these troubles are given in II and in the referencesof footnote 2.

{+J I

INTERACTION

in state zz, n and k is given in (12). The G-" can be evaluated most easily by calculating the generating function

(ss)

s(X,Y):L-I,"G^"X^Y"(m!nt)-t

for arbitrary X, Y. fi expression (11) is substituted in the left-hand side of (55), the expressioncan be simplified by use of the generating function relation for the eigenfunctions8of the harmonic oscillator L e"QiY"(nl)-'

: (o/ r)t exP(- !iot') (1 expl- iot')-

xexpl[oqo'-

(2a)tqo)'f

Using a similar expansion for the 9-* one is left with the Jxponential of a quadratic function of qs and g;' The iniegration on {o and g; is then easily performed to give (56) g(X,Y):GooexP(XY{iB*XliBY) from which expansion in powers of X and I comparison to (11) gives the final result Q-,:Qoo(ar.nt)-lt-

and

n! tnl -' (n-r)''rl (n-r)lrr'

xrtUB*)*-.(iil"-' (s7) where Goois given in (14) and ?'

rQ)exv?i,t)dt'

B:Q,1-t I "

" f'

0*:(2a)-,r tIt

(58)

u'

t(t) exp(+irt1a4

and the sum on r is to go from 0 to m ot to a whichever is the smaller. (The sum can be expressedas a Laguerre but there is no advantage in this') nolvnomial ' 9. CASE OF REAL PIIOTONS Formula (57) is readily understandable' Consider first a simple caseof absorption of one photon' Initially The casein which there are real photons in the initial we have one photon and finally none. The amplitude the beginning out from be worked can state final or the this is the transition element of Got:iB6oo ot for a case of the first consider We in the same manner.lT This is the same as would result if we Qy,li'BGool,l,r). From this svstem interacting with a single oscillator. element for a problem in transition to. the result the generalizationwill be evident. This time we "it "a but there was present a virtual are photons all which shall calcuiate the transition element between an initial oerturbins potential '(2u)-t7Q) exp(-iol) and we state in which the.particle is in state ly'/ and the reouired ihe first-order efiect of this potential' Hence oscillator is in its zth eigenstate(i.e., there are z photons photon absorption is like the first order action of a oscillator particle in with state final x/" in the field) to a ootential varving in time as 7(I) exp(-i@') that is with the when in zlth level. As we have already discussed, positive frequenry (i'e., the sign of the coefficientof,l coordinates of the oscillator are eliminated the result a the exponential corresponds to posltlve energyi' in is the transition element (xr,, I G-" | 9r) where The amplitude for emission of one photon ilvolv.es f which is the same result except-that the Grr:il#oo, G^": I e_*k)k(q;, t,,; qo,t),p"(qo)dqdci (11) ooi.niiut has negative frequency. Thus we begin by J interprering iB* is the amplitude lor emissionof one i0 is the amplitude for absorptionof one' where g-, g^arc tl:rewave functionss for the oscillator ohoton ' ior the geneial case oI z photons initially and Next r? For an alternative method starting directly from the formula m finally *. -uy understand (57) as follows' We 6rst (24) for virtual photons, see Appendix B.

269 452

R.

P.

FEYNMAN

neglect Bose statistics and imagine the photons as individual distinct particles. If we start with z and end with az this processmay occur in several difierent ways. The particle may absorb in total z-r of the photons and the final zr photons will represent r of the photons which were present originally plls m-r n"* photon, emitted by the particle. In this casethe z-r which are to_beabsorbed may be chosenfrom among the original n in n!/(n-r)lr ! different ways, and each contributes a far:tor iB, the amplitude for absorption of a photon. Which of the m-r photons from among thi m are emitted can be chosen in ml/(m-r)lr! difierent wavs and each photon contributes a factor i0* in amplitude. The initial r photons which do not interact *ith th" particle can be re-arranged among the final / in r ! ways. We must sum over the altematives correspondingto different values of r. Thus the form of G-" can be undersiood. The remaining tactor (m!)-\(n!)-l may be interpreted as saying that in computing probabiiities (which therefore involves the square of G^^) the photons may be consideredas independent but that if m arc actually equal the statistical weight of each of the states which can be made by rearranging the ar equal photons is only l/m!. This is the content of Bose statistics; that m eqtal particles in a given state represents just one state, i.e., has statistical weieht unity, rather than the ar! statistical weieht wh'ich would result if it is imagined that the parlicles and slates can be identified and rearrangedin ar! difierent ways. This holds for both the initial and final states of course. From this rule about the statistical weishts of states the derivation of the blackbody distribution law follows. The actual electromagnetic field is represented as a host oI oscillatorseachof which behavesindependently and producesits own factor such as G-,. Tnitial or final states may also be linear combinations of states in which one or another oscillator is excited. The results for this case are of course the corresponding linear combination of transition elements. For photons of a given direction of polarization and for sin or cos waves the explicit expressionfor B can be obtained directly from (58) by substituting the formulas (16) for the .y's for the corresponding oscillator. It is more convenient to use the linear cornbination correspondingto running waves.Thus we 6nd the amplitude for abso-rptionoI a photon of momentum K, frequency ft: (K.K)l polarized in direction e is given by including a factor i times px,e: (4r)t(2k)-tle" n

f'

I

exp(-iftl)

J,,

Xexp(lK.x"(t))e . x.

(Ss) "(t)dt in the transition element (25). The density of states in momentum space is now (2T)-3d3K. The amplitude for emission is just I times the complex conjugate of

this expression, or what amounts to the same thing, t"hesarde expressionwith the sign of the four vector i" reversed.Since the factor (59) is exactly the first-order efiect of a vector potential APE: (2n/h)tg exp(-i(ir-

K.x))

of the corresponding classicai wave, we have derived the rules for handling real photons discussedin II. _ Y9!.n expressthis directly in terms of the quantity T""lBl, the amplitude for a given transition withoui emission of a photon. What we have said is that the amplitude for absorption of just one photon whose classical wave form is ,4rPn(1) (time variation exp(-iht) corresponding to positive energy i) is proportional to the first order (in e) change produced in I",[.8] on changing B to B+ zAPE. That is, more exactly, f

J

(ar"zln)/aB,())AnPH(r)dr1

(60)

is_the amplitude for absorption by the particle system of one photon, A"". (A superposition argument ihows the expression to be valid not only for plane waves, but for spherical waves, etc., as given by the form of .4Pd.) The amplitude for emissionis the sameexpression but with the sign of the frequency reversedin ApH, The amplitude that the system absorbs two photons with waves Ar'H, and A,ea, i" obtained from the next derivative,

J J

( a ,r a t n ) / 6B , ( t ) 6B , ( 2 \ \A p pHt ( 1 )A , pH ,( 2 )d r L dr z ,

the same expressionholding for the absorption of one and emission of the other, or emissionof both depending on the sign of the time dependenceof APHL and APE2, Larger photon numbers correspond to higher derivatives, absorption of 11 emission of l, requiring the (/r*lz) derivaties. When two or more of the photons are exactfy the same (e.g., ArHr:trPt,) lde same expressionholds for the amplitude that lr are absorbed by the system while lr are emitted. However, the statement that initially z of a kind are present and za of this kind are present finally, does not imply I1:n and lr:1n. It is possible that only n-r:h werc absorbedby the system aid,m-r:lzemitted, and that r remained from initial to final state without interaction. This term is weighed by the combinatorial coefiicient , , ,._,/*\(n\ . u n a s u m m e do v e r t h e p o s s i b i l i t i e s lm:n:)-'\, )\"/rt for r as explained in connection with (57). Thus once the amplitude for virtual processesis known, that for real photon processescan be obtained by differentiation. It is possible, of course, to deal with situations in which the electromagneticfield is not in a definite state after the interaction. For example, we might ask for the total probability of a given process, such as a scattering, without regard for the number of photons emitted. This is done of course by squaring the ampli-

270 THEORY

tude for the emission of aa photons of a given kind and summing on all m. Actually the sums and integrations over thi oscillator momenta can usually easily be pedormed analytically. For example, the amplitude, starting from vacuurn and ending with zr photons of a given kind, is bY (56) just

453

INTERACTION

OF ELECTROMAGNETIC

then have a new type of wave function 9(r, z) a function oI five variables, r standing for the fourur. It gives the amplitude -for arrival ai point r, with a certain value of the parameter a We shall supposetbat this wave function satisies the equation (24) i0,p/du--\(id/0rr-Ap)2c

which is seen to be analogousto the time-dePendentSchrodinger equation, a replacing the time and the lour coordinats of spaceof space' (61) time r, replacing the usual three coordinates Q*o-- @!)-tGxnQB*)^. Sinie t-hepotmtials lp(r) are functions only of coordinatesru we and are indipendent of r, the equation is separable in r.r The square of the amplitude summed on ar requires can uite a specialsolution in the form e:exp(*dn2u){(r) .an{ where (the in the 0 (1A) expressions such 7(t) two and the the oroduct of o(r). a function of the coordinatesru only, satisfies of one and in the other wilt have to be kept separately) eigenvalue |z'conjugate to a is related to the mass n of the oa'rticle. Equation (2A) is therefore equivalent to the Klein'Gordon simmed on rr: Eq. (1A) provided we ask in the end only for the solution :L^ (rnl)-r?-$'*).-, Goo*Goo' of (1A) corresponding to the eigenvalue |zt for the quantity E- G^o*G^o' :GofGoo' exp(BP'*). conjugate to a. frJ-uy to* p.o."ed to representEq. (2A) in Lagrangian form over all oscillators in eenerai and without regard to this eigenvaluecondition' Only sum the In the resulting expression in'the final solutionsneed we apply the eigenvaluecondition' is easily done.-Such expressionscan be of use in the That is, if we have some special solution q(r,u) of (2A) we can analvsi; in a direct manner of problems of line width, *lect that part correspondingto the eigenvaluelz'1 by calculating

of the Bloch-Nordsieckinfra-red problem, and of staexp(- tdmzu)c(t, u) du * @) = Itistical mechanical problems' but no such applications will be made here. ud t"herebyoblain a solution * of Eq. (IA)' .. The author appreciates his opportunities to discuss Since (24) is so closely analogousto the Schriidinger equation' in C, simply described form Professer it is easily witten in thi Lagrangian these matters wilh ProfessorH. A. Bethe and at one Mr. M. Baranger with the by working by analogy. For example if 9(*, a) is.known of help the and Ashkin, J. value of a its value at a slightly larger vatue e+c rs Srven Dy manuscrrpf,, :

G' - 1'' )' -!('

-x'' \ (A,(r) -l A'(x'))]

expi,ls (x, a t e) f APPENDD( A. THE I(IEIN-GORDON EQUATION (3A) ' c(x', u)dart(2ti'e)-l(- 2tt€)J In this Appendixwe describca formulationof tlte equations : d t dt i dx,' dxz' rules (r* / the (r" x X' obtain firsl used to rr') \,. was (rrr"')' means r' where foia particleof spinzerowhich "' ue '' siven'inII for suchparticles.The completephysicalsignificance and the sign of the normalizing factor- is changed tor ln lrs hasnot beenanalyzedthoroughlyso that it may component since the component bas the reversed slgn 3i theequations our be orefeiableto deriveIhe rulesdirectlyfrom the secondquantr- ou"irutia coe{hcientin the exponential, in accordancewith otl* for-ulatioo of PautiandWeiskopf'This canbe donein a Jummation convention atbt= atbt- stbr- azbr-c#r' nquation -nnn". nnttogou*to tle derivationof the rulesfor the Dirac iiA), * *n b" u"tified reaiily as describedin C, Sec' 6, is equivaformulatiod i.nt'io At.t ota.t in €, to Eq. (2A). Hence, by repeateduse of this eiu"n-inI or fromtheSchwinger-Tomonaga rB "ooniion equation the wave function at 40:l€ can be representedln terms in'a mannerdescribed,for example,by Rohrlich The formulation qiven here is tberelore not necessaryior a descrlpuon or sprn of that at z:0 bY: iero oarticlesbut is givenonly for its own interestasan alternatrve r ^-- t.3 f/"":--""r)' e\x".",u\):Jexp-Zlrl\-.,/ to the formulation of secondquantization' We start with the Klein-Gordon equation (1.A') (i'd/aq-Ap)2'!'-m2, + e-r(r h i- r h i -) (Ar(a) +'ar(''-') )] for the wave function ,y'of a particle of massz in a given external ootential,4u.We shall (ry to renresenlthis in a manneranalogous io the fotniulation of quanlum mechanicsin C' That is, we try point.to io ,"nr.r"nt the amplitudefor a particle to get from-on€ ^.otb". as a sum over all trajectories of an amplitude exp(tJ) *i"i" S it the classicalaction lor a given lrajectory' To maintain the relativisticinvariancein evidencethe idea suggestsrtselt.ot d.t..ibing a trajectory in space-timeby giving the-four variables ,,ir) ut fin.tiont of somefrfth parameterr (rather than expressing which tl)"r, ,r i" lerms of J{). As we expect to rePresentpaths iltemselvesin time (to representpair production' -lu-t"t"o" u. i" I) this is certainly a more convenientrepresentation' "i.. ru(u) may be consideredas functions of a iot-'"ii lout'tu".ti""s .ter u (somewhat analogousto proper time) which increase iira i. *" go utong th" trajectory, whether the trajectory is prueeding t*i"ia ta"^ia">U or bactward (drq/du<0) in time'Ie We shall rs F. Rohrlich (to be Published).

.,p(r,.", o)"i @+r/ 41Peil. gL) That is, roughly, tlle amplitude for getting from -one point.to another with a given value of zo is the sum over all tr&Jectorles of exp(iS) where (sA) s=-I"'l+(drF/du)'*(dru/du)Au@)Uu, when sufrcient care is taken to define the quantities, as in C' i-Ui.-"o-pf"t"" the formulation for particles-in a fixed potential but a few words of description may be in order' In the first place in the slrccial case of a free particle we can uo define a kernai t(o)(r, uo;' t',0) for arrival from **', 0 to e.pal ovei all trajectories between these points of ;.- lh; ;texp-i .ft"1;(dxu/du)'da' Then lor this casewe have (64) s1r, uol: f k
i ,^':*:flF$i'i,tml:"J;:!"it"'*:$i:,:Jl:i:'1,":18!'.'iTf L'; :'iT: Hii:tif'f f::il'ii;:'l^z:rt:: ;?:l'ilt,i,:yB("|

and it is easily verified that &ois given by ptot(s,uni r' ,0) - (4duo'r)-r exp- i(rp- xp')'/2uo

(1e37).

(7A)

for ao)0 and by 0, by definition, lor un
271 154

R.

P.

FEYNMAN

kernel of importance when we select the eigenvalue|2, iso 2iI *(x, {

: -

nn (*, up; *', 0) exp(- \im2uo)duo

f f-,

.J o

,,

" u o, " i)-1 . . _ , exp - t i ( n2 u o* u o- | (r d u s(4 n2 r-

x r' 12) (8 A)

since le is zero for negative a6) ,the last extends only from r0:0 rhich is identical to the 1+ defined in II.,t This may be seen :adily by studying the Fourier transform, for the transform of ie integrand on the right-hand side is -f

@ rP u o'i)

-'

exp Qp. x) exp - li (m2u sl r p2f u a)dar. :exp-+iuo(m2-

pr2)

gives for the transform of f1 just :o that the t0 integration '-'(lr,-m") with the pole defined exactly as in II. Thus we are representing the positrons as trajectories with the :utonatically :ne sense reversed. is the amplitude for a If o(0)[r(r)]-exp-if{"t@rJdu)'d'u .-ven trajectory r,(u) lor a free particle, then the amplitude in : potential is c,(r) ['(e)

- i ] : orc) [r (u)) e\p

I"'

@.x! / du) A rQ) du.

(9 A)

i: desired this may be studied by perturbation methods by :soanding the exponential in powers of .4u. the integral in (9A) must be written as a For interpretation, !j.emann sum, and if a perturbation expansion is made, care must quadratic in the velocity, for the effect with the terms taken :: is not of order e'but is -l6u,c. The :l (16;4-rp,)(r,,t+r-cr;) .;e\ocity" drp/du becomes the momentw operator pr: +iA f Ob! .-erating halI beforeand half after,4u. just as in the non-relatiyric Schrddinger equation discussed in Sec.5. Furthermore, in ::actly ttre same manner as in that case, but here in four dimen: tts) a term quadratic in lu arises in the second-order perturba:rn terms from t}re coincidence of two velocitics for the same . :lue of a, -\s an example, the kernal p{e)@, uo; x',0) for proceeding from :- .0 Lo xr, uo in a pot€ntial l* differs from A(0) to first order in i. by a term - i

f"'duk
1r, uo; y, u)l(puA r(y) ! A r()

p)ktot (y, u; r', 0)ho

The kernel of importance :z f, here meanitg +i,A/A!* on r::cting the eigenvalue |z' is obtained by multiplying this by and integrating us from O to o. The kernel :-(.-|im2u) " : ..t, us; y, u) depends only on u' : uo- u aod in the integrals on . Ad us; -f.-d.u6fdrd.u exp(- !im\h) . .., can be written, on interthe order of integration and changing variables to u :::nging . . '. Now the integral on : ) u' , .fo'd.u,l-f,du' exp(- iim2(uIu')) " converts k\o)(x, u6; y, u) to 2iI *(1, ) by (8A), while that on I -_t (!, u i r', O) to 2i.I 1(y, r' ), so the result becomes er ts kQ) :.

2ir +(x,iepA p+A rpr)I a(y, x')d.ar, .f The sameprincipleworksto any orderso that the erpected. :

-- rs for a single Klein-Gordon particle in external potentials : .:n in II, Section 9, are deduced. The transition to quantum electrodynamics. is simpte for in :.{r we aheady have a transition amplitude represented as a ,- (over trajectories, and cventually a6) of terms, in each of , r:;h the potential appears in exponcntial form. We may make -:: of the general relation (54). Hence, for example, one finds :: The factor 2i jn front of 11 is simnly to make the definition : :- here agree with tbat in I and IL In II it operates with p as a perturbation. But the pcrturbation coming fron : \;A i" in a natural way by expansion, of the cxltonential is

- = ;. p . A * A . n ) .

I Expression (8A) is closely rclated to Schwingcr's parametric ':lral rel)resentation of thcsc functions. For example, (8A) :: aes formula (45) of F. Dyson, Phys. Rev. 75, 486 (1949) for -, = t$-2i\=2iJ + iI (2d)-t is substituted for 4o-

for the case of no photons in the initial and final states, in the prcsence of an external potential Br, the amplitudc that a particle proceeds from (t',0) to (tr,us) is dre sum over all trajectories of the quantity

"*

- nl;[,^ (#)' a,+ dfia,6671 au Jo"' '+! dlf a-{1ru1D*,(u' *; [,' "fi ))'1 )dildu']. ooA\

This result must be multiplied by exp(-lim2u) and integrated on 4o from zero to infinity to express the action of a Klein,Gordon parLicle acting on itself through virtual photons. The integrals are interpreted as Riemann sums, and if perturbation expansions are made, the necessary care is taken with the terms quadratic in velocity. When there are several particlcs (other than the virtual pairs already included) onc use a separate a for each, and writes the amplitude for each set of trajectories as the cxponental of -i times

i?L""'"'(L#)'o*+>!^'"'d"rl"'nu@,<*t(u))du a - .-.'t n t - . 6 ' d x y t ^ ( u \ d r y t . \ ( i l t ) latl l z ^nvr du du' (u) - srt^: (z/)),)dudu', X 6l(rr("t where rr(")(z) are the coordinates particle.z The solution should

(11 A)

of the trajectory of the zth depend on the a6{") a5

exp(- |irn22 , ao{")) Actually, knorvledge of the motion of a single chargc implies a great deal about the behavior of several charges. lor a pair which eventually may turn out to be a virtual pair may appear in the short run as two "other particles." As a virtual pair, that is, as the leve6e scction of a very long and complicated single track we know its behavior by (10A). We can assume that such a section can be looked at equally well, for a limitcd duration at least, as being due to other unconnected particles. This thcn implies a dehnite law of interaction of particles if the self-action (10A) of a single particle is known. (This is similar to the relation of real and virtual photon processesdiscusseclin detail in Appendix B.) It is possible that a detailed analysis of this could show that (10A) implicd that (11A) was correct for many particles. Thcre is even reason to believe that the lalv of Bose-Einstein statistics and the expression for contributions from closerl loops could be deduced by iollorving this argumcnt. This has not yet been analyzed completely, however, so we must leave this formulation in an incomplete form. The expression for closed loops should come out to be C,:exp*Z where Z, the contribution from a 2 T h e f o r m ( 1 0 A ) s u g g c s t sa n o l h e r i n r c r e s t i n q p o s s i L i l i t y f o r avoirling the divcrgences oI qucntum clcctro,lynrmies in 'his case. The divergences arise from the 6+ function wlrco u:u', We might rcslrict the intcgration in the doul,le inlegral such that qhere 6 is some finite quantily, very small comprrcd lu-u'l>6 with m 2. More generally, rve could keep the rcgion u:u' lrom contributing by including in the integrand a factor F(u-u') (e.g., where I(r)+1 for r large compared to some 6, and I(0):0 (Another way might F(r) acts qualitatively like 1-exp(-e?6-,). be to replace a by a discontinuous variable, that is, we do not use the limit in (4,4) as e+0 but set c:6.) The idea is that two interactions would contribute verv little in amnlitude if thev followed onc anolhcr too rapidly in r. It is eesily veri6ed rhit this makes the otherwise divergent integrals finite. But whethcr the resulting formulas make good physicai sense is hard to scc. The action of a potential rvould now rlepend on the value of a so that Eq. (2A), or its equivalcnt, woulcl nol be separab'e in a so that ]rz'woulcl no ionger lrc a strict eigcnvalue for all disturbanccs. High energy potentials coulcl excite states corresponding to other eigenvalues, possibly thercby corresponding to othcr masses. This note is mcant only as a sDcculation, lor not enough work has been done in this dircclion to makc sure that a reasonable physical theory can be developed along these lincs. (What little rvork has been c.lonervas not promising.) Analogous modifications can also be made for Dirac electrons,

272 THEORY

OF ELECTROMAGNETIC

singleloop,is L: 2 J

I (ul exp(- Ainl us)dun/ u s

"where l(ao) is the sum over all trajectories which close on themselves(r,(zo):*r(0)) of exp(zJ) with S given in (5A), and a final integntion dt.67 ot rp(O) is made. This is cquivalent to puttrng (r, uo; r, 0) - k@(x, uo; r, o))dr' t{uo): f &
B. THE RELATION OF REAL AND VIRTUAL PROCESSES

If one has a generalformula for all virtual processeshe should be able to find the formulas ancl states involveC in real processes. That is to say, we should be ablc to deducethe formulas of Section 9 directly from the formulation (24), (25) (or its generalized equivalent such as (46), (48)) without having to go all the way back to the morc usual formulation. We discussthis problem here. That this possibility exists can be seen from the consideration that what looks like a real proccss from one point of view may as a virtual processoccurring over a more extended time. appear ' For examplc, if rve rvish to study a given real process,such as the scattering oI light, we can, if we wish, include in principle the source, scatterer, and eventual absorber of the sciittered light in our analysis.We may imagine that no photon is present initially, and that the sourcethen emits light (the energy corning say from kinetic energy in the source). The light is then scattered and eventually absorbed (becoming kinetic energy in the absorber). I.rom .this point of view the processis virtual I that is, we start with no photons and end with none. Thus we can analyze the processby meansof our formula for virtuai processes,and obtain ihe formulas for real processesby attempting to break the analysis into parts correspondingto emission,scattering, and absorption.23 To put the problem in a more general way, considerthe ampli' tude for some transition from a state empty of photons far in the (r:1"). Suppose past -the (time l') to a similar one far in the future time interval to be split into three regions o, D, c in some convenientmanncr,so that region6 is an intervallr>r>rl around tlre present time that we wish to study. Region a, (hlt)t'), D, a\d c, (t">r>tr), follows D. We want to seehow it Drecedes iornes about that the phenomenaduring b can be analyzed by a of transitions study 8;i(D) between some initial state i at time rl (which no longer need be photon-free), to some other final state i at time ,r. The states i atd j are membersof a large classwhich we will have to find out how to specify. (The single index i is used to represent a large number of quantum numbers, so that different values of i will correspondto having various numbers of various kinds of photons in the field, etc.) Our problem is to represent the over-all transition amplitude, g(o, D,c), as a sum over various values of l, j oi a product of three amplitudes, (18) g(o,b,c):Z;21 goi(c)gir(D)go(a) ; 6rst the amplitude that during the interval o the vacuum state makes transition to some state i, then the amplitude that during t the transition to i is made, and finally in c the amplitude that the transition from i to some photon-frec state 0 is completed. 23The formulas for real processes deduced in this way are strictly limited to the case in which the light comes from sources which'are oriqinally rlark, and thal evenluilly cll light emitted is ahsorbcd rgain. We can only exlend it to the casc for which these restriction; do not hold by hypothesis, namely, that the details o[ the scattering Drocess are independcnt of lhcse charrcteristics of the liqht sourec rn,l of the eventual disposition ol the scrltcred lisht. Thc argumenl o[ the text givcs a mcthod for discovering foirmulas for ieal orocesses when no more thrn thc formula lor virtual Drocesses is at hand. But with this method bclief in the general validity o[ the resulting formulas must rest on the physical ieasonablcness ol the abovc-menLioned hypothesis.

INTERACTION

455

The mathematical problem of splitting g(4,6, c) is nade definite by the further condition that gi;(6) for given l, j must not involve the coordioatesof the particles for times correspondingto regions a or c, gio(a)must involve those only in region a, and 3o;(c)only in c. To becomeacquaintedwith what is involved, supposefirst that we do not have a problem involving virtual photons, but just the transition of a one-dimensionalSchrddinger particle going in a long time interval from, say, the origin t to the origin o, and ask what states i we shall need for intermediary tine intervals. We must solve the problem (1B) whereg(a,b,c) is the sum over all trajectories going lrom o at r' to o at l" of explS w6"ts $: if UL correThe integral may be split into three parts.S:S.*.ll*S" sponding to the three ranges of time Then exp(iS):exp(iSJ .ixp(rsi) exp(tsJ and the separation (18) is accomplishedby taking for gio(o) the sum over all trajectorieslying in a from o to someind point 11,of exp(iSJ, for g;;(b) the sum over trajectories in D of exp(iSo) betweenend points rir and r,r, and for gqi(c)the sum of exp(lS) over the section of the trajectory lying in c and going from ,,, to o, Then the sum on i atd j can be taken to be the integrals on itp tt, respectively. Hence the various states i can be taken to correspond to particles being at various coordinates r. (Of courseany other representationof the states in the senseof Dirac's transfolmation theory could be used equally well. Which, one, whether coordinate, momentum' or energy level representatiotr,is of course just a matter of convenienceand we cannot determinethat simply from (18).) We can consider next the problem including virtual photons' That is, g(a, b, c) now contains an additional factor exp(iR) over all time. Those where lQ involves a double integral .fJ parts of the index I which conespond to the particle states can Le taken in the same way as though R were absent, We study now the extra complexities in the states produced by splitting tJre ft. Let us 6rst (solely for simplicity of the argument) take the case that there are only two regions @,, separated by time ,o and try to expand g(a, c) =2t goi?)g;o(a). The factor exp(dlR)involves R as a double integral which en be for the -first.of ,-f split into three parts -f .-f "+L.f" "+-f which both l, s are in o, for the secoudboth are in c, for the third one is in a the other in c. Writing exp(dR)as exp(rtR-)'exp(d.R-) .exp(iR-) shows that the factors R", and R"o produce no new problems for they can be taken bodily into goi(c) and r1(o] iespectively, However, we must dfuentanglethe variables which are mixed up in exp(iR.). The expressionfor Ro" is just twice (24) but with the integral otr s extetrdingover the range o and that for I extending over c. Thus exp(lR-) contains the variables for times in o and in c in a quite;omplicated mixture. Our problem is to wite exp(if"") as a su- ovit possibly a vast class of states i of the product of two parts, like hik)ht(a), each of which involves the coordinates in one interval alone. This separation may be made in many different ways, conesponding 1o various possible repr€sentationsof the state of ihe electromagneticfield. We choosea particular one. First we can expand the exponential, exp(iR."), in a power series, as > i"(nt)-L(R"")". The states i can therefore be subdivided into subclassescorrespondingto an integer r which we can interpret as the number of quanta in the field at time ro. The amplitude for the casez:0 clearly just involves exp(lR.J and exp(z'R*) in the way that it should if we interpret theseas the amplitudes fo! regions o and c, respectively, of making a transition between a state of zero photons and another state of zero photons. Next consider the caseu: 1. This implies an additional factor in the transitional element; the factor Ro". The variablesare still mixed up, But an easy way to perform the separation suggests in R- as itself. Namely, expand the d+((r-s)'-(x"(l)-x*(s))') a Fourier integral as i f exp(- iklt-

sl) exp(-tK' (x"(r)-x -(s))d3K/4dh.

273 +56

R. P. FEYNMAN

For the exponential can be witten immediately as a product of exp*l(K.x.(s)), a function only of coordinatesfor times s in a (supposes(l), and exp-lK.r.(l) (a function only of coordinates dudng interual c). The integral on d3K can be symbolized as a over states I characterized by the value of K. Thus the sum state with z:1 must be further characterized by specifying a vector K, interpreted as the momentum of the photon. Finally in R"" is simply the sum of four parts .he factor (t-r'"(l).x'-(s)) ach of which is already split (namely 1, and ach of the three componentsin the vector scalar product). Hence each photon of nomentum K must still be characterizedby specifying it as one of four varieties; t}rat is, there are {our polarizations.2aThus in -dying to representthe efiect of the past e on the future r we are i€d to invent photons of four polarizationsand characterizedby a propagation vector K. The term for a given polarization and value of K (for z:1) is clearly just -p"Bo* where tle p" is definedin (59) but with ttre ime integral extending just over region o, wlile B" is the same erpressionwith the integration over region c, Hence the amplitude ror transition during interyal a from a state with no quanta to a :tate with one i! a given state of polarization and momentum is calculated by inclusion of an extra f.actor'ip"* in the transition element.Absorption in region c correspondsto a iactor iB.. We next turn to tie casez:2. This requires analysis of Ro"r. Ihe t+ can be expandedagain as a Fourier integral, but for each of the two 6a in jR"oz we have a value of K which may be difierent. Thus we say, we have two photons, one of momentum K and one aomentum K' and we sum over all values of K and K'. (Similarly ach photon is characterizedby its own independentpolarization rdex.) The factor ] can be taken into account neatly by asserting :hat we count each possiblepair of photons as constituting just rne state at time lo. Then the I arisesfor the sum over all K, K' and polarizations)counts eachpair twice. On the other hand, for -ie terms representing two identical photons (K:K') of like roiarization, the ; cannot be so interpreted. fnstead we invent ae rule that a state of two like photons has statistical weight I s great as that calculated as though the photons were difierent, This, generalized to z identical photons, is tle rule of Bose .tatistics. The higher valuesof z ofier no problem. The l/u ! is interpreted :ombinatorially for difierent photons, and as a statistical factor rhen some are identical. For example, for all z identical one rbtains a factor (nl)-t(-P"P"*)" so thar (n!)-t(i.5"*)" can be iterpreted as t}re amplitude for emission(from no initial photons) cf z identical photons, in complete agreemetrtwith (61) for Gao. To obtain the amplitude for tmnsitions in which neither the ritial nor the final state is empty of photons we must considet -ie more generalcaseof the division into three time regions (1B). Ihis time we see that the factor which involves t-hecoordinates .It is to be r an entangled manner is expl(R"6{R6}X."). .sp&nded in the lorm 2i2; lq"k)ki(b)hi@). Again the expan:ion in power series and development in Fourier series with a :olarization sum will solve tle problem. Thus the exponential is >, Zh >t, (i.R""),(iR"b)tt(iRb")1,(hl)-L(lr!)-1(r!)-t. tqqry the R are -rritten as Fourier series,one of the terms containing llflzlr ;ariables K. Since lr*z involve o, lz*r involve c and h*lz rvolve 6, this term will give the amplitude that lr+r photons :re emitted dudng the interval o, of those 11are absorbedduring i but the remaining r, along with 12new ones emitted during b go rn to be absorbed during the interval c. We have therefore photonsin the state at time rr when l begins,and u:lr*r r:Ltr :. ,, when 6 is over. They each are characterizedby momentum rectors and polarizations. When tlese are difierent the factors are absorbed combinatodally. When some are ;!)-t(lr!)-r(/l)-l :1ual we must invoke ttre rule of ttre statistical weights. For i Usuallvonlv two polarizationstransverseto the DroDasation rector K ire uied. This can be accomplishedby a'fuitb-er re:fangement of terms correspondingto tie reverse of the steps -3ding from (17) to (19). We omit the details here as it is wellsown that either forEulation gives ttre saEe !6ults. See II, :trtion 8.

example, suppose alI h+l2+/ photons ale identical. Then Ra:i9a?.*, Rt":ig"lt*, R*:i9"9,* so tlat our sum is -r 2 4 2t2 2, (l 1'!l','!r l) QP)h+r (i0 b)| | (i9 b*)h(ipa*)I:+'. Putdrg m:lzIr, n:h*r, this is the sum on m atd m of (i il- (n t)- tL2, (m tn !)| ((n - r) !(n - r) lr !)-t -, x (i p b*)^ r (dp b)"- / l(n t) (i,F ". "*) The last lactor we have seen is the amplitude for emissionof r photons dudng interval ir, while the first factor is the amplitude for absorption of z during c, The sum is ttrereforet-hefactor for transition from z to z identical photons, in accordancewith (57). We seethe significanceof the simple generatingfunction (56). We have tierefore found rules for real photons in terms of those for virtual. The real photons are a way of representingand keeping track of ttrose mpects of tle past behavior which may influence the future. If one starts from a theory involving an arbitrary modification of the direct interaction 6a (or in more general situations) it is possiblein this way to discoverwhat kinds of states and pbysical entities will be involved if one tries to representin the present all the information needed to predict ttre future. With the Hamiltonian method, which begins by assuming such a representation, it is difficult to suggestmodifications of a general kind, for one cannot formulate the problem without having a complete repr€sentation of the characteristics of the intermediate states, the particles involved in interaction, etc. It is quite possible (in the author's opinion, it is very likely) that we may discover that in nature the relation of past and future is so intimate for short duntions that no simple representationof a present may exist. In such a casea theory could not 6rd expressionin Hamiltonian form. An exactly similar analysiscan be made just as easily starting with the generalforms (46), (48). Also a coordinaterepresentation of the photons could have been used instead of the familiar momentum one. One can deduce the rules (60), (61). Nothing essentially difierent is involved physically, however, so we shall not pursue the subject furtier here. Since they implyt3 all the rules for real photons, Eqs. (46), (47), (48) constitute a compact statement of all the laws of quantum electrodynamics.But ttrey give divergent results. Can tJ)e result aJter charge and mass renormalization also be expressedto all orders in d/ic in a simple way? APPENDIX

C. DIFFERENTIAL EQSATION ELECTRON PROPAGATION

F'OR

An attempt has been made to find a difierential wave equation for the propagation of ar electroninteracting with itself, analogous to the Dirac equation, but containing terms representing ttre self-action.Neglecting all efiectsof closedloops,one such equation has beenfound, but not much has beendone witl it. It is reported here for whatever value it may have. An electron acting upon itself is, from one point of view, a complex system of a particle and a field of an indefaite number of photons. To frd a difierential law of propagation of such a system we must ask first what quantities known at one instant will permit tie calculation of these same quantities an instanl later. Clearly, a knowledge of the position of the particle is not enough. We should need to specify: (1) the amplitude that the electron is at , and there are no photons in ttre field, (2) tle amplitude the electron is at r and ttrere is one photon of such and such a kind in the field, (3) the amplitude tiere are two photons, etc. That is, a series of functions of ever increasing numbers of variables. Following this view, we shall be led to tle wave equation of the theory of secondquantization. We may also take a difierent view. Supposewe hnow a quantity a.2fB, r), a spinor function of rr, md functional of Bu(l), defined as the amplitude tlat an electron arrives at r, witl no photon in the field when it moves in an arbitrary external unquantized potential Br(1). We allow the electron also to interact with itself,

274 457 but,p.2 is the amplitude at a given instant that there happens to be tro photons prcscnt. As we have sccn' a complete knowledge of this functional rvill also tell us the amplitude that the electron arrives at r and therc is just one photon, of form lrPH(1) present' r) / 6B u$)) A pPIt (l) dn. It is, f rom (60), I $,b "218, Higher numbers of photons corrcspond to higher functional derivatives of Q.z. Thcreforc, 6,21]),r) contains all the inform&tion requisite for dcscribing the state of the electron-photon sl stem, and we may cxPect to frnd a dillcrential equation for it. Aciually ii. satisfies (V:7udl<1su, B:yuB), ( iv - n) Q,zlB, x):

B (r) o

"zlB,

xl

+i.87uJ 6*G,ilGs "zlB, r)/ 68 u1\d r r (1c) as may be scen from a physical argument.'5 The operator (iV-z) operating on the r coordinate of i"r should equal, from Dirac's equation, the changes in iF.2 as wc go from one position t to- a ."ighlroring position due to thc action of vector potentials' The term B(r),F"2 is the effect of the external potential. But'0"2 may ,, l-a*"^r"t vali,litv can also lre,lcmonstra(cdmrthematically irom r45t. fhc amplitudc for arriving at r with no photons in the fi.],1 \iith virrurl l,hoton coupling e2 is a transilion amplitude. r] for any r. lr rnust.lhcrciore,srtisiy (45r rvirlrf.z[B]=o.z[8, Hence show Lhat the quantitY C,zlB, *l:

(iv-nr-B(t))o"zlB

' t) - i*1,J'a ' rs,r'rla<',zlB, rf/

68t!\drr

in for f.z[B] also selisfics Eq. (45) by sul,stitu-ring-C.zfB..rl sutisfics ('15).IIcnce if r45) and usinc the lxct thxl 6"zfB,xl

b;i'o,;l -i , f"; -.i[r, ' 1: o ro'-"tt.'i But c.z[g, v 1- 6 q"ant i r ' t i b . j t f . * t s e r i s f i irsl t t . T h c r c f o r tch, s t s o l u i i o on l [ 8 , r ]

- /, - B( r r r+o[8. *] = 0 (t he prol )f, of (45) ;hich ;lso sal isfi.s (iV serion of r iree elcclron rithnut virtttri lrlr,'tonslis a solution of ilc) or *" wislre,l to shou. Equati,,n tlC) rnay be morc corrvetlicnt lhan (451 for sorne purposes for it does not involve , l i f f e r c n t i a t i o n w i t h r e s P e c tt o l h e c o u p l i n A c o n s t a n l , a n d i s m o r e analogous to a wave equatlon.

also change for at the first position r we may have had a photor that it was emitted &t another point 1 is present (amplitude 6O.2/68r(1)t which was absorbed at r (amplitude photon released at 1 gets to , is 6+(i,r2) where s,r2 is the squated invariant distance Efiects from I to r) acting as a vector Potential there (factor "ts). of vacuum polarizrtion are lcft out. Expansion of the solution of (1C) in a pouer serics in B and d starting from a free particle solution for a single electron, produces a series of terms rvhich agree with the rules of II for action of potentials and virtual photons to various ordcrs, It is another matter to use such an equation for the practical solution of- a problem to all ordcrs in d. It might be possible to represent the ielf-energy problem as the variational problem for u, stemming from (1C). The 6a will first have to be modifred to obtain a convergent result. We ire not in need of the general solution of (1C). (In fact, r] we have it in (46), (a8) in termi of the solution folts]:oL[B, of the ordinary Dirac equation (iv-m)aofB,rf:BaolB',). knowledge for complete The general solution is too complicated, of the motion of a self-acting electron in an arbitrary potential is essentially all of electrodynamics (because of the kiod of relation of real and virtual processes discussed for photons in Appendix B, extended also to real and virtual pairs). Furthermore, it is easy to see that other quantities also satisfy (1C). Consider a system of many electrons, and single out some one for consideration, supposing all the others go from some definite initial state i to some definite frnal state /. Lel Q"zlB, tl be the amplitude that the special electron arrives at t, thcre are no photons present' and the other electrons go from i to / whcn there is an external potential Bu present (which B! also acts on the other electrons)Then o"2 aiso satisfies (1C). Likewise the amplitude with closed loops (all other electrons go vacuum to vacuum) also stislies effects. The various (1C) inclucling all vacuum polarization problems correspond to different assumptions as to the d-ependence of *"zlB, xf on B, in the limit of zero d. The Eq. (1C) without Iurthei boundary conditions is probably too general to be useful'

Poper 24

275

The Radiation Theories of Tomonaga, Schwinger, and Feynman F. J. DysoN Institute for Ad.vanced Study, Princeton, New Jersey (Received October 6, 1948) A uni6ed development of the subject of quantum electrodynamics is outlined, crnbodf ing the main features both of the Tornonaga-Schwinger and of the Feynman radiation theory. The theory is carried to a point further than that reached b1' thesc arrthors, in the discussion of higher order radiative reactions and vacuum polarization phenomena, Horvever, thc theory of these higher order processesis a program rather than a definitive theorl', since no general proof of the convergence of these effects is attempted. The chief results obtained are (a) a demonstration of the equivalence of the Feynman and Schwinger theories, and (b) a considerable simplihcation of the procedure involved in applying the Schwinger theory to particular problems, the simplification being the greater the more complicated the problem.

I. INTRODUCTION disI S a resultof the recentand independent I I coveries of Tomonaga,l Schwinger,2and Feynman,s the subject of quantum electrodynamics has made two very notable advances.On the one hand, both the foundations and the applicationsof the theory have been simplified by being presentedin a completely relativistic way; on the other, the divergence difficulties have been at least partially overcome. In the reports so far published,emphasishas naturally been placed on the second of these advances; the magnitude of the first has been somewhat obscuredby the fact that the new methods have been applied to problems which were beyond the range of the older theories, so that the simplicity of the methods was hidden by the complexity of the problems. Furthermore, the theory of Feynman differs so profoundly in its formulation from that of Tomonaga and Schwinger, and so little of it has been published, that its particular advantages have not hitherto been available to users of the other formulations. The advantagesof the Feynman theory are simplicity I Sin-itiro Tomonaga, Prog. Theoret. Phys. 1, 27 (1946); Koba, Tati, and Tomonaga, Prog. Theoret. Phys. 2, 1di 198 (1947); S. Kaneswa aid S. T-omonaga, Prog. Theoret. Phys- 3, 1, 101 (1948); S. Tomonaga, Phys. Rev.74,224

(1e48). 'Julian Schwinger, Phys.Rev.73,416 (1948);Pbvs.

Rev..74, 1439.(1948). Several papers, giving.a complete exposition of the theory, are in cour* of publication. fR. P. Feynman, Rev. Mod. Phys. 20, 367 (L948); Phys. Rev. 74,939, 1430 (19a8); J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945). These articles describe erlv stases in the develooment of Fevnman's rheory, little irf which is yet published.

'I and easeof application, while those of omonagaSchwinger are generality :rnd theoretical completeness. The present paper aims to show how the Schwinger theory can be applied to specific problems in such a way as to incorporate the ideas of Feynman. To make the paper reasonably self-contained it is necessary to outline the foundations of the theory, following the method of Tomonaga; but this paper is not intended as a substitute for the complete account of the theory shortly to be published by Schwinger. Here thc emphasis will be on the application of the theory, and the major theoretical problems of gaugeinvariance and of the divergencies will not be considered in detail. The main results of the paper will be general formulas from which the radiative reactions on the motions of electrons can be calculated, treating the radiation interaction as a small perturbation, to any desired order of approximation. These formulas will be expressed in Schwinger's notation, but are in substance identical with results given previously by Feynman. The contribution of the present paper is thus intended to be twofold: first, to simplify the Schwinger theory for the benefit of those using it for calculations, and second, to demonstrate the equivalence of the various theories within their common domain of applicability.* * After this paper was written, the author was shown 2 letter, published in Prosress of Theoretiel Physics 3, 295 (1948)'bv Z. Koba anii G. Takeda. The letter is dated 'ivlav 22.i948, and brieflv describesa method of treatment of ridiative problems, similar to the method of this paper.

486

276 487

RADIATION

II. OUTLINEOF THEORETICALFOUNDATIONS Relativistic quantum mechanics is a special case of non-relativistic quantum mechanics, and it is convenient to use the usual non-relativistic terminology in order to make clear the relation between the mathematical theory and the results of physical measurements. In quantum electrodynamics the dynamical variables are the electromagnetic potentials 1u(r) and the spinor electron-positronfield *"(r); each component of each field at each point r of space is a separate variable. Each dynamical variable is, in the Schriidinger representation of quantum mechanics, a time-independent operator operating on the state vector aDof the system. The nature of O (wave function or abstract vector) need not be specified; its essential property is that, given the O of a system at a particular time, the results of all measurementsmade on the system at that time are statistically determined. The variation of (F with time is given by the Schrddinger equation lt' I (1) ihla/6tla:l IHG)d,la, IJ

I

where fI(r) is the operator representing the total energy-density of the system at the point r. The generalsolution of (1) is Ifl o(r):expll-it/hf IJI

I H(r)drlan,

(2)

with Ooany constant state vector. Now in a relativistic system, the most general kind of measurement is not the simultaneous measurement of field quantities at different points of space. It is also possible to measure independently field quantities at different points of space at different times, provided that the points of space-time at which the measurements are made lie outside each other's light cones, so that the measurements do not interfere with each other. Thus the most comprehensivegeneral type of measurement is a measurement of field quantities at each point r of spaceat a time l(r), Results of the application of the method to a calculation radiative correction to the Kleinof the rcond-oider Nishina formula are stated. All the papers of Professor which have yet been pubhis associates and Tomonaqa lished wJre completed before the end of 1946' The isolation of these Iapanese workers has undoubtedly constituted a erious loss-to theoretical physics.

THEORIES

the locus of the points (r, l(r)) in space-time forming a 3-dimensionalsurface c which is spacelike (i.e., every pair of points on it is separated by a spaceJike interval). Such a measurement will be called "an observation of the system on a." It is easy to seewhat the result of the measurement will be. At each point r' the field quantities will be measured for a state of the system with state vector iD(l(r')) given by (2). But all observable quantities at { are operators which commute with the energy-density operator 11(r) -every point r different from r', and it is a at general principle of quantum mechanics that if B is a unitary operator commuting with ,4, then for any state (b the results of measurementsof ,4 are the same in the state 6 as in the state BO. Therefore, the results of measurement of the field quantities at r'in the state 6(l(r')) are the same as if the state of the system were (fl o(a) : exp,l-b/hf lJl

I t(r)H(r)dr loo,

(3)

which differs from A(l(r')) only by a unitary factor commuting with these field quantities. The important fact is that the state vector O(c) depends only on o and not on r'. The conclusion reached is that observations of a system on d give results which are completely determined by attributing to the system the state vector a(o) given by (3). The Tomonaga-Schwingerform of the Schrddinger equation 'is a differential form of (3). Suppose the surface o to be deformed slightly near the point r into the surface o', the volume of space-time separating the two surfaces being tr/. Then the quotient la(a't

-o(a)f/v

tends to a limit as 7--+0, which we denote by Aa/aa(r) and call the functional derivative of Q with respect to o at the point r. From (3) it follows that (4) i.hclao/ao(t)l:fr(r)o, and (3) is, in fact, the general solution of (4). The whole meaning of an equation such as (4) depends on the physical meaning which is attached to the statement "a system has a constant state vector iDo." In the present context, this statement means "results of measurements of

277 F , J , D YSON field qqantities at. any given point of space are independent of time." This statement is plainly non-relativistic, and so (4) is, in spite of appearances,a non-relativistic equation. The simplest way to introduce a new state vector V which shall be a relativistic invariant is to require that the statement "a system has a constant state vector V" shall mean "a system consists of photons, electrons, and positrons, traveling freely through space without interaction or external disturbance." For this purpose,let

f/(r):rI (r)f/fi(r),

(s)

where IIs is the energy-density of the free electromagnetic and electron fields, and Ii is that of their interaction with each other and with any external disturbing forces that may be present. A system with constant V is, then, one whose f1r is identically zero;by (3) such a system coriesponds to a iD of the form o(c):71n;qo,

488

where ffr(*o) is the time-dependent form of the energy-density of interaction of the two fields with each other and with external forces. The left side of (9) representsthe degreeof departure of the system from a system of freely traveling particles and is a relativistic invariant; fl(ro) is also an invariant, and thus is avoided one of the most unsatisfactory features cjf the old theories, in which the invariant f/r was added to the non-invariant Ho. Equation (9) is the starting point of the Tomonaga-Schwingertheory. THEORY III. INTRODUCTIONOF PERTURBATION Equation (9) can be solvedexpiicitly. For this purpose it is convenient to introduce a oneparameter family of space-like surfaces filling the whole of space-time, so that one and only one member 6:(r) of the family passes through any given point r. Let oo,ot, d2, "' be a sequence of surfaces of the family, starting with 'steps steadily into do and proceeding in small the past. By Fa0

I n,@)a"

rG):*p{ -U/il[ li*,(,)d,i. (6) It is therefore consistent to write generally o(o):11n;E1o),

(7)

thus defining the new state vector V of any system in terms of the old A. The differential equation satisfied by V is obtained from (4), (5), (6), and (7) in the form i.hclav/ao(r)f:

(r(o))-rfl(r)I(c)v.

(8)

Now if q(r) is any time-independent field operator, the operator q(xi : (T(o))-'q(r) I(o)

vI

ol

is denoted the integrBl of I1t(rc) over the 4' dimensional volume between the surfaces ar and do;similarly, by

[^ r,{*)o*,!- ,,{*)o* are denoted integrals over the whole volume to the past of aoand to the future of o0,respectively. Consider the operator Lt: u(od :l

/r'o\ 1 -li/hcl \

l

./"r

H,(x)dx I /

is just the corresponding time-dependent operax (t -r;tna n,@)ax).. ., (to) [", tor as usually defined in quantum electrodynamics.r It is a functlon of the point rs of spacethe product continuing to infinity and the surtime whose coordinates are (r, ct(r)), but is the facesao,cr, ' . . being taken in the limit infinitely same for all surfaces o passing through this closetogether. Usatisfies the differential equation point, by virtue of the commutation of I/r(r) (11) ihclaU/aa(x)l:Ht@iu, with Ilo(r') for t'*r. Thus (8) may be written (9) and the general solution of (9) is ihcl6v/A6(r;l: H1(rs)9, 'See, for example, Gregor Wentzel. Einf'frhrungin die Quantenllnorie der WetrlcnJeldu (Franz Deuticke, Wien, l94J), pp. l8-26.

v(o): Y1"1Eo' rvith Vo any constant vector'

(r2)

278 489

RADIATION

Expanding the product powers of H1 gives a series fo0

U : 1-l (- i/hc) | H'(x')dx,f J_6

(10) in

ascending

(- i / hc)'z

THEORIES small perturbation as was done in the last section. Instead, I1i alone is treated as a perturbation, the aim being to eliminate -FIi but to leave H" in its original place in the equation of motion of the system. Operators S(o) and S(o) are defined by replacing Ht by Ht in the definitions of [/(o) and I/(o). Thus S(o) satisfies the equation

.... (13) ,,(*,)r,(*,)o*,t " I__*,f'*"

ihclaS/ao(xs)f:.Hr(tro)S.

Further, U is by (10) obviously unitary, and

Suppose now a new type of state vector O(c) to be introduced by the substitution

(i/hc), u-1: O: r+61n4 u$xt)d.rt1. J^ " I_":., I*"

(17)

v(o) :51";s1")'

. .. (r4) Ht(xz)Ht@)dxz*.

It is not difficult to verify that U is a function of ao alone and is independent of the family of surfaces of which o0 is one member. The use of a finite number of terms of the series (13) and (14), neglecting the higher terms, is the equivalent in the new theory of the use of perturbation theory in the older electrodynamics. obtained from (10) bv The operator U(-), taking cq in the irrfinite future, is a transformation operator transforming a state of the system in the infinite past (representing, say, converging streams of particles) into the same state in the infinite future (after the particles have interacted or been scattered into their final outgoing distribution). This operator has matrix elements corresponding only to real transitions of the system, i.e., transitions which conserve energy and momentum. It is identical with the Heisenberg S matrix.6 IV. ELIMINATION OF THE RADIATION INTERACTION In most of the problem of electrodynamics, the energy-density -F11(*6)divides into two parts-

H{xs):71t1*i*H"(xi,

(15)

Hi(xs):-lt/cfj,(xs)A,(xo),

(16)

the hrst part being the energy of interaction of the two fields with each other, and the Second part the energy produced by external forces. It is usually not permissible to treat H" as a 6Werner Heisenberg, Zeits. f. Physik 120, 513 (1943), 12O,673 (1943),and Zeits. f. Naturforschung f,608 (1946).

(18)

Bv (9), (15), (17), and (18) the equation of motion for O(o) is ihcl\o/ ac(x)l:

(S(o))-'Ir"(ro)S(o)o.

(19)

The elimination of the radiation interaction is hereby achieved; only the question, "How is the new state vector O(o) to be interpreted?," remains. It is clear from (19) that a system with a constant g is a system of electrons, positrons, and photons, moving under the influence of their mutual interactions, but in the absence of external fields. In a system where two or more particles are actually present, their interactions alone will, in general, cause real transitions and scattering processes to occur. For such a system, it is rather "unphysical" to represent a state of motion including the effects of the inter' actions by a constant state vector; hence, for such a system the new representation has no simple interpretation. However, the most important systems are those in which only one particle is actually present, and its interaction with the vacuum fields gives rise only to virtual processes. In this case the particle, including the effects of all its interactions with the vacuum, appears to move as a free particle in the absence of external fields, and it is eminently reasonable to represent such a state of motion by a constant state vector. Therefore, it may be said that the operator, Hy(rn):

(S(o))-'H'(ro)S(a)'

(20)

on the right of (19) represents the interaction of a physical particle with an external field, including radiative corrections. Equation (19) describes the extent to which the motion of a

279 F.

J.

single physical particle deviates, in the external 6eld, from the motion represented by a constant state-vector, i.e., from the motion of an observed "free" particle. If the system whose state vector is cor.rstantly g undergoes no real transitions with the passage g of time, then the state vector is callecl "steady'" if, and only if, it o is steady N{ore precisely, satisfies the equatiol-r

DYSON

490

mass, and should have been used instead of Ilo(r) in the definition (6) of I(a)' Consequently' the second bracket should have been used instead of I1r(r) in Eq' (8). The definition of S(o) has therefore to be altered by replacing I/t(r6) bY6

: rrr(xo): rri(xi t 1's(so) !"u*)lrr* r r* . (22) ", ",

The value of 6iz catl be adjusted so as to cancel 1 2 1 ) out the self-energy effects in S( o ) (this is only a formal adjustment since the value is actually is a general rule, one-particle states are steady and then Eq. (21) will be valid for infinite), and many-particle states unsteady' There are, no one-electron states. For the photon self-energy however, two important qualificatiol-rs to this such adjustment is needed since, as proved by rule. Schrvinger, the photon self-energy turns out to be First, the interaction (20) itself t'ill almost identically zero. alrvays cause transitions from steady to unsteady The foregoing discussion of the self-energy consists state initial if the states. For example, is intentionally only a sketch, but it problem of one electron in the field of a proton, llr rvill to be sufficient for practical applicafouncl be will electhe of have matrix elements for transitions theory. A fuller discussion of thc the of tions tron to a new state with emission of a photon, underlying this treatassumptions theoretical practice' in and such transitions are important be given by Schrvinger rvill problem of the ment Therefore, although the interpretation of the papers. Moreover, it must be forthcoming his in posnot it is states, theory is simpler for steady be realized that the theory as a whole cannot sible to exclude unsteady states from conput into a finally satisfactory form so long as sideration. occur in it, however skilfully these ii.r"rg"lt.i". Second, if a one-particle state as hitherto deare circumvented; therefore' the clivergencies rnust S(o) of fined is to be steady, the definition should be regarded as justified y,."."nt tr*rtment be moclified. This is because S(m) includes thc in applications rather than by its Ly it..uc.".. effects of the electromagnetic self-energy of the derivation. theoretical electron, and this self-energy gives an expectaThe important results of the present paper up tion value to S(o) which is different from unity point are Eq. (19) and the interpretation (and indeed inlinite) in a one-electron state, so to this vector O. The state vector V ol a state the of mistake The that Eq. (21) cannot be satisfied' interpreted as a wave function be can system that has been made occurred in trying to repreamplitude of finding any probability the g-iving its electromagsent the observed electron rvith numbers for the occupation of set netic self-energy by a rvave lield with the same iarticular free electrons, positrous' of states possible the "bare" characteristic rest-mass as that of g of a system with "a.iou. and phoions''fhe state vector electron. To correct the mistake, let 6zz denote . girren V on a given surface o is, crudely speakthe electromagnetic mass of the electron, i'e', had in in!, tn" V which the system would have the difference in rest-mass between an observed arrived at the given V on had past if it infinite the (5), divithe and a "bare" electron' Instead of a under the influence of the interaction f/r(*) sion of the energy-density 11(r) should have alone. taken the form The definition of Q being unsymmetrical beof state r/(r): (Ilo(r)* 6nxcv. @PIGI,] tween past and future, a new type - amc"g* (r)B{G)). vector Ot can be defined by reversing the direcOf tion of time in the definition of Q' Thus the The first bracket on the right here representsthe V is the given o a on V given a rvith of a system energy-density of the free electromagnetic and -l u*d' is tt"." S"t utinger'snotation !:'p*p electron fields with the observed electron restS(*)a:O.

280 491

RA D I ATIO N THEORIES

which the system would reach in the infinite future if it continued to move under the influence of HI(xo) alone. More simply, O' can be defined by the equation O/(o):S(o)A(s).

(23)

Since S( rc ) is a unitary operator independent of a, the state vectors O and g/ are really only the same vector in two different representations or coordinate systems. Moreover, for any steady state the two are identieal by (21). V. FUNDAMENTAL FORMULAS OF TIIE SCHWINGER AND FEYNMAN THEORIES The Schwinger theory works directly from Eqs. (19) and (20), the aim being to calculate the matrix elements of the "effective external potential energy" .F11'between states specified by their state vectors O. The states considered in practice always have O of some very simple kind, for example, g representing systems in which one or two free-particle states have occupation number one and the remaining free-particle states have occupation number zero. By analogy with (13), S(oo) is given by

s(as): 11 1-; Pr1

" I_:.,[__"i

['

n, 6,1d.x1t(-i/ hc),

p,1x,1H,q*)d.xz*. . ., (24)

and (S(cr))-l by a corresponding expression analogousto (14). Substitution of these series into (20) gives at once

satisfactory agreement with experimental results. In this paper the development of the Schwinger theory will be carried no further; in principle the radiative corrections to the equations of motion of electrons could be calculated to any desired order of approximation from formula (25). In the Feynman theory the basic principle is to preserve symmetry between past and future. Therefore, the matrix elements of the operator H7 are evaluated in a "mixed representation;" the matrix elements are calculated between.an initial state specified by its state vector O1 and a final state specified by its state vector O2'. The matrix element of 1{r between two such states in the Schwinger representation is

(26)

O:*f12'O1: Q2'*S( o )I/rOr,

and therefore the operator which replaces117 in the mixed representation is Hp(xs): S(a)H7@s) : S( o )(S(c))-tf1"(r6)S(c). (27) Going back to the original product definition of S(a) analogous to (10), it is clear that S(o) X(S(o;;-t is simply the operator obtained from S(c) by interchanging past and future. Thus, R(o) : S( o) (S(o))-L:r*(-i/hc) x,

f'

Hl(r)dx*(-i/hc),

Ja

f"

I

drr

,,d

x I

Hr(xz)Hl(xld.x,f.' .. (28)

"o(r)

The physical meaning of a mixed representation of this type is not at all recondite. In fact, @ a mixed representation is normally used to defo('d f'(rr) Hr@o):D Q/nQ"I dx, I dx,. . . scribe such a process as bremsstrahlung of an u-0 .t _d .J _6 electron in the field of a nucleus when the Born ro(xn-r) approximation is not valid; the process of dx*XIHI(x^), 7'.., IHI(xz), X I bremsstrahlung is a radiative transition of the (2s) electron from a state described by a Coulomb IHI(ry\, }1"(ro)ll...ll. wave function, with a plane ingoing and a spheriThe repeated commutators in this formula are cal outgoing wave, to a state described by a characteristic of the Schwinger theory, and their Coulomb wave function with a spherical ingoing evaluation gives rise to long and rather difficult and a plane outgoing wave. The initial and final analysis. Uslng the first three terms of the series, states here belong to different orthogonal systems Schwinger was able to calculate the second-order of wave functions, and so the transition matrix radiative corrections to the equations of motion elements are calculated in a mixed representaof an electron in an external field, and obtained tion. In the Feynman theory the situation is

281 F,

J.

analogous; only the roles of the radiation interaction and the external (or Coulomb) field are interchanged; the radiation interaction is used instead of the Coulomb lield to modify the state vectors (wave functions) of the initial and final states, and the external field instead of the radiation interaction causes transitions between these state vectors. In the Feynman theory there is an additional simplification. For if matrix elements are being calculated between two states, either of which is steady (and this includes all cases so far considered), the mixed representation reduces to an ordinary representation. This occurs, for example, in treating a one-particle problem such as the radiative correction to the equations of motion of an electron in an external field; the operator Hp(xo), although in general it is not even Hermitian, can in this case be considered as an effective external potential energy acting on the particle, in the ordinary sense of the words. This section rvill be concluded rvith the derivation of the fundamental formula (31) of the Feynman theory, which is the analog of formula (25) of the Schwinger theory. If F1(rcr),

F"(x")

are any operators defined, respectively, at the points 11, ' ' ' , x* of space-time, then

(2e) will denote the product of these operators, taken in the order, reading from right to left, in which t h e s u r f a c e sd ( x l ) , ' ' . d ( r . ) o c c u r i n t i m e . I n most applications of this notation F;(r;) rvill commute with F;(r) so long as .."rrand Jci are outside each other's light cones; u.'hen this is the case, it is easy to see that (29) is a function of the points 11, . ' ., r" only and is independent of the surfaces o(rr). Consider nou. the integral

t^:I_'-^,..J-"a*-rs"6s, IfI(x),

H'(x")).

Since the integrancl is a symmetrical function of the points rr, '' , r,, the value of the integral is just z! times the integral obtained by restricting the integration to sets of points rr, ' ' ', r" for which o(rr) occurs after o(r;*1) for each i.

492

DYSOn'

The restricted integral can then be further divided into (a*1) parts, the j'th part being the integral over those sets of points with the property that o(:;6) lies between o(rr-1) and o(r;) (with obvious modifications for j:1 and j:4 *1). Therefore, i+I

rd(ro)

I^:ntll i-r

.t _-

?o(

_6

d x , - t . . I.

x I

ttx,

a*,.. 1 J

dx,xlttlx,)"'

ud\x,)

"d(xo)

HI (x;)II"(xo)H'

(x). . . H' (x").

(.30)

Now if the series (24) and (28) are substituted into (27), sums of integrals appear which are precisely of the form (30). Hence finally II e (x o) : T. G i / hc)"lr / n tfl "

:L,en/r,)"lr/nt [_'.dx,I:." XP(H"(xs),

H'(x")).

HI(x),

(31)

By this formula the notation I1p(ro) is justified, for this operator now appears as a function of tlie point r0 alone and not of the surface o. The further development of the Feynman theory is mainly concerned rvith the calculation of matrix elements of (31) betu,'eenvarious initial and final states. As a special case of (31) obtained by replacing 11'by the unit matrix h (27),

' , , 1 1 " a . .r''f " a - ^ S(-):! 1-i lzcl,tr s

o

J_-

a/-6

X P ( H ( ( x 1 1 ," ' , A I @ " ) ) . ( 3 2 ) VI. CALCULATION OF MATRIX ELEMENTS In this section the application of the foregoing theory to a general class of problems will be explained. The ultimate aim is to obtain a set of rules by which the matrix element of the operator (31) betrveen two given states may be written dorvn in a form suitable for numerical evaluation, immediately and automatically' The fact that such a set of rules exists is the basis of the Feynman radiation theory; the derivation in this section of the same rules from what is

282 493

RADIATION

fundamentally the Tomonaga-Schwinger theory constitutes the proof of equivalence of the two theories. To avoid excessivecomplication, the type of matrix element considered will be restricted in two ways. First, it will be assumed that the external potential energY is H"(xo):-ll/cyu@o)A,"(xi,

(33)

that is to say, the interaction energy of the electron-positron 6eld with electromagnetic potentials,4.,t(rs) which are given numerical functions of space and time. Second, matrix elements will be consideredonly for transitions from a state .4, in which just one electron and no positron or photon is present, to another state B of the same character. These restrictions are not essen. tial to the theory, and are introduced only for convenience, in order to illustrate clearly the principles involved. The electron-positron field operator may be written

THEORIES integral of (31); let it be denoted by P". From (16), (22), (33), and (37) it is seen that P" is a sum of products of (n-11) operators {", (n+l) operators {* and, not more than n operators .A' multiplied by various numerical factors. By 8i may be denoted a typical product of factors ry'", {", and Au, not summed over the indices such as a and p, so that P" is a sum of terms such as Q", Then 0" will be of the form (indices omitted)

(xt)''''1, @;)'l' @; Q" : 0 @d V@to){'(x;),1, ") XA(xi'..A(xi^), (39) where do, it, . ' ', do is some permutation of the integers 0, 1, . . ., n, and ju ' ' ' , i^ are some, but n in not necessarily all, of the integers l, "', some order. Since none of the operators ,y' and )ry'commute with each other, it is especially impbrtq4t to preserve the order of these factors. 'fector of Q* is a sum of creation and Each operators by virtue of (34), (35)' annihilatioh

Gq, and so Q" itself is a sum of products of ""{ cldation and annihilation operators.' (34) Now consider under what conditions a product {"(x):Z 6""@)a", of creation and annihilation operators can give a where the 6""@) are spinor wave functions of non-zero matrix element for the transition ,4+-B. free electrons and positrons, and the a* are Clearly, one of the annihilation operators must annihilation operators of electrons and creation annihilate the electron in state -4, one of the operators of positrons. Similarly, the adjoint creation operators must create the electron in state B, and the remaining operators must be operator (3s) into pairs, the members of each pair divisible {"(x):L 6""@)arespectively creating and annihilating the same where d, are annihilation operators of positrons particle. Creation and annihilation operators reand creation operators of electrons' The electro- ferring to difierent particles always commute or (the former if at least one is a anticommute magnetic field operator is photon operator, the latter if both are electron--4 (36) (A",(x)b,-tA,"*(')6,), t -(i positron operators). Therefore, if the two single operators and the various pairs of operators in and where b, and 6, are photon annihilation the product all refer to different particles, the The chargecreation operators, respectively. order of factors in the product can be altered so current 4-vector of the electron field is as to bring together the two single operators and (37) j,(x):iec0@)t,'l'@); the two members of each pair, without changing the value of the product except for a change of be ought to strictly speaking,this expression sign if the permutation made in the order of the antisymmetrized to the formT positron operators is odd. In the (38) electron and (il j, (x) : |iec | {t "e, "(,c)!e@) ! e@){ "(x) | case when some of the single operators and pairs but it will be seenlater that this is not necessary of operators refer to the same particle, it is not hard to verify that the same change in order of in the present theorY.

Consider the product P occurring in the a'th ?See Wolfgang Pauli, Rev. Mod. Phvs. f3' 203 (1941)' Eq. (96), p.224'

factors can be made, provided it is remembered that the division of the operators into pairs is no longer unique, and the change of order is to

283 F.

J.

be made for each possible division into pairs and the results added together. It follows from the above considerations that the matrix element of Q"for the transition ,4+B is a sum of contributions, each contribution arising from a specific way of dividing the factors of 0" into two single factors and pairs. A typical contribution of this kind will be denoted by M. The two factors of a pair must involve a creation and an annihilation operator for the same particle, and so must be either one 'y' and one ry' or two A; the two single factors must be one ,y'and one ry'.The term M is thus specified by fixing an integer i, and a permutation r0, rb . . ., r" of the integers 0, 1, . . ., n, and a division (sr,tr), (sz,tr), . . ., (sr,lr,) of the integers jr, . . . j- into pairs; , has to be an even number; the clearly m:2h term M is obtained by choosing for single factors '{,(xr) and, and for associated pairs of t@,r), f a c t o r s ( r y ' ( r )s ! ( x , ; ) ) f o r i : 0 , 1, . . ., k-t, h + 1 , . . . , n a n d ( A ( x " ; ) , A ( x t ; ) )f o r i . : 7 , . . . , h . In evaluating the term M, the order of factors in Q" is first to be permuted so as to bring together the two single factors and the two members of each pair, but without altering the.order of factors within each pair; the result of this process is easily seen to be q

"'

: eP ($ (x o),*(x, o)) . . . p (:0 @ ),,1,@, ")) X P (A (x" ),4 (x t)) . . . P (A (x" 1"),A(xt1")), (40)

a factor e being inserted which takes the value :E1 according to whether the permutation of 0 and ry' factors between (39) and (40) is even or odd. Then in (40) each product of two associated factors (but not the two single factors) is to be independently replaced by the sum of its matrix elements for processes involving the successive creation and annihilation of the same particle. Given a bilinear operator such as Ar(x)A,(y), the sum of its matrix elements for processes involving the successive creation and annihilation of the same particle is just what is usually called the "vacuum expectation value" of the operator, and has been calculated by Schwinger. This quantity is, in fact (note that Heaviside units are being used) (A,(x) A,())

DYSON

not be given here, because it turns out that the vacuum expectation value of P(A,(x),A,(y)) takes an even simpler form. Nameh', \ P ( 4 , ( x ) , A , ( y ) ) ) s : | l r c 6* " D p ( : r- y ) ,

(41)

where Dp is the type of D function introduced by Feynman. Do(l) is an even function of r, with the integral expansion

D,(x) : -li/2,\

I"

expliax2lrla, (42)

where rc2 denotes the square of the invariant length of the 4-vector r. In a similar way it follows from Schwinger's results that \P ({ "(x),'1,e0)))o: }a(r,y)Sr a"@ - y7, rvhere Srp"(r) : - Q,@/ox,)*"g)p"Ao(r), ro is the electron, is earlier function

(43)

(44)

reciprocal Compton u'ave-length of the ,t@,y) is -1 or f 1 according as o(r) or later than a(1,) in time, and Ar is a rvith the integral expansion

ao(r) : -l.i/

2rl

expliax2- i rs2 /4alda. (45)

I,

Sqbstituting from (41) and (44) into (40), the matrix element M takes the form (still omitting the indicesof the factors 0, rlr,and A of Q") M :, II (tn @;,x,) Sr (r r - x, ;t t XII(.ihcD r(x" i - xt )) P ({,(xr),*(x,,)).

(46)

i

The single factors f (rr) and ,!t(x,1,) are conveniently left in the form of operators, since the matrix elements of these operators for effecting the transition ,4+B depend on the wave functions of the electron in the states A and B, Moreover, the order of the factors 'l@r,) and ry'(r'p) is immaterial since they anticommute with each other; hence it is permissible to write P ('{,(x*),*(x,r)) : n(xo,x,r){(xn)!(x,i. Therefore (46) may be rervritten M : e' II (*Sr(rr -r'))fJ ,lk

t ;P (* - y), n: j h c 6u "I D < t a |

where D(1) and D are Schwinger's invariant D functions. The definitions of these functions will

494

with

( l l t c D p ( x "I - x t) ) t

X{(xn),1'(x,r), (47) e' : ell4(x;,x,;). i

(48)

284 495

RADIATION

THEORIES

Now the product in (48) is ( - 1)r, where p is the number oT occasions in the expression (40) on which the ,l oI a P bracket occurs to the left of the ry'. Referring back to the definition of e after Eq. (40), it follows that e'takes the value f 1 or - 1 according to whether the permutation of tf

ing section it will be shown how this solution-inprinciple can be reduced to a much simpler and more practical procedure.

and ry'factors between (39) and the expression

Let an integer n and a product P" occurring in (31) be temporarily fixed. The points ro, rcr, ..., *o may be represented by (zfl) points drawn on a piece of paper. A type of matrix element M as described in the last section will then be represented graphically as follows' For with each associated pair of factors (f(ri),r/(r")) draw a line with a direction marked in it ilk, from the point rc; to the point rr;. For the single factors ry'(rr), r!(x,p), draw directed lines leading out from r* to the edge of the diagram, and in from the edge of the diagram to r'r. For each draw an unpair of factors (,4(r,;),A(xt)), directed line joining the points rs; ?nd rri. The complete set of points and lines will be called the "graph" of M; clearly there is a one-to-one correspondence between types of matrix element and graphs, and the exclusion of matrix lor i+k corresponds to the elements with rr:; exclusion of graphs with lines joining a point to itself. The directed lines in a graph will be called "electron lines," the undirected lines "photon

'0@it@,0)...9@){(tc,")

(4e)

is even or odd. But (39) can be derived by an even permutation from the expression

'L@o){(xi"'0@){(x"),

(s0)

of factors between (49) and the permutation and (50) is even or odd according to whether the permutation ro, " ' , rn of the integers 0, " ', z is even or odd. Hence, finally, e'in (47) is *1 or - 1 according to whether the permutation 16, '.., r* is even or odd. It is important that e' depends'only on the type of matrix element M xn; considered, and not on the points xo, "', therefore, it can be taken outside the integrals l n (J r , . One result of the foregoing analysis is to justify the use of (37), instead of the more correct (38), for the charge-current operator occurring in 11' and 11i. For it has been shown that in each matrix element such as M the factors { arrd {, in (38) can be freely permuted, so that (38) can be replaced bV (37), except in the case when the two factors form an associated pair. In the exceptional case, M contains as a factor the vacuum expectation value of the operator jr(r;) at some point 11; this expectation valrre is zero according to the correct formula (38), though it would be infinite according to (37); thus the matrix elements in the exceptional case are always zero. The conclusion is that only those matrix elements are to be calculated for which the integer and in these ri differs from i for every i*k, elements the use of formula (37) is correct. To write down the matrix elements of (31) for ft is only necessary to take the transition A+8, all the products Q,, replace each by the sum of the corresponding matrix elements M given by (47), reassemble the terms into the form of the P, from which they were derived, and finally substitute back into the series (31). The problem of calculating the matrix elements of (31) is thus in principle solved. However, in the follow-

VII. GRAPHICAL RSPRESENTATION OF MATRX ELEMENTS

lines." Through each point of a graph pass two electron lines, and therefore the electron lines together form one open polygon containing the vertices rck and'tcth, and possibly a number of closed polygons as well. The closed polygons will be called "closed loops," and their number tn denoted by L Now the permutation tn, "', of the integers 0, ' ' ', a is clearly composed of (l*1) separate cyclic permutations. A cyclic permutation is even or odd according to whether the number of elements in it is odd or even. Hence the parity of the permutation ro, ''', rn is the parity of the number of even-number cycles contained in it. But the parity of the number of odd-number cycles in it is obviously the same as the parity of the total number (z* 1) of elements. The total number of cycles being (l+ 1), the parity of the number of even-number Since it was seen earlier that cycles is (l-a). the e'of Eq. (47) is determined just by the parity of the permutation

rs, "',

rn, the above argu-

285 F. J, D YSO N ment yields the simple formula

496

to the identity operator and gives, accordingly, a zero matrix element for the transition ,4+8. (s1) Consequently, the disconnected G for which rr This formula is one result of the presenttheory and x,p lie in Gz give zero contribution to the which can be much more easily obtained by matrix element of (31), and can be omitted from intuitive considerations of the sort used by further consideration. When rp and *,1" lie in Gr, Feynman. again the C(G) may be summed over all G conIn Feynman's theory the graph corresponding sisting of the given Gr and all possible Gz; but to a particular matrix element is regarded, not this time the connected graph Gr itself is to be merely as an aid to calculation, but as a picture included in the sum. The sum of all the C(G) in of the physical processwhich gives rise to that this case turns out to be just C(G) multiplied by matrix element. For example, an electron line the expectation value in the vacuum of the joining rr to 12 represents the possible creation operator S(oo). But the vacuum state, being a cf an electron at rcr and its annihilation at x2, steady state, satisfies (21), and so the expecta:ogether with the possible creation of a positron tion value in question is equal to unity. Therezt xz and its annihilation at rt. This interpreta- fore the sum of the C(G) reduces to the single :ion of a graph is obviously consistent with the term C(G), and again the disconnected graphs nethods, and in Feynman's hands has been may be omitted from consideration. rsed as the basis for the derivation of most of The elimination of disconnected graphs is, --heresults, of the present paper. For reasonsof from a physical point of vierv, somewhat trivial, =pace,these ideas of Feynman will not be dis- since these graphs arise merely from the fact :ussedin further detail here. that meaningful physical processes proceed siTo the product Po correspond a finite number multaneously with totally irrelevant fluctuations rf graphs, one of which may be denoted by G; of fields in the vacuum. However, similar argu:ll possible G can be enumerated without dif- ments will now be used to eliminate a much iculty for moderate values of n. 'I'o each G more important class of graphs, namely, those :orresponds a contribution C(G) to the matrix involving self-energy effects. A "self-energy part" =lementof (31) which is being evaluated. of a graph G is defined as follows; it is a set of It may happen that the graph G is discon- one or more vertices not including 16, together :.ected,so that it can be divided into subgraphs, with the lines joining them, which is connected .ach of which is connected,with no line joining with the remainder of G (or with the edge of the : point of one subgraph to a point of another. diagram) only by two electron lines or by one or -n such a case it is clear from (47) that C(G) is two photon lines. For definiteness it may be :he product of factors derived from each sub- supposed that G has a self-energy part F, which :laph separately. The subgraph G1 containing is connected with its surroundings only by one :ne point 16 is called the "essential part" of G, electron line entering F at xt and another ire remainder Gz the "inessential part." There leaving F at xz; the case of photon lines can be :re now two casesto be considered,according to treated in an entirely analogous way. The points ;'hether the points xp and :trp lie in Gz or in Gr rc1dnd. rc2 may or may not be identical. From G lhey must clearly both lie in the same sub- a "reduced graph" G0 can be obtained by omit;raph), In the first case, the factor C(Gz) ol ting F completely and joining the incoming line i(G) can be seen by a comparison of (31) and at lc1with the outgoing line at *e to form a single 32) to be a contribution to the matrix element electron line in Go, the newly formed line being :i the operator S(o) for the transition .4+8. denoted by ),. Given Go and tr, there is conversely l{ow letting G vary over all possiblegraphs with a well determined set I of graphs G which are -re same Gr and different Sr, the sum of the associated with Go and tr in this way; Go itself.is ,ontributions of all such G is a constant C(G) considered also to belong to f. It will now be rultiplied by the total matrix element of S(co) shown that the sum C(l) of the contributions ior the transition A+8. But for one-particle C(G) to the matrix element of (31) from all the icatesthe operator S(o) is by (21) equivalent graphs G of I reduces to a single term C,(Go).

e,: (- 11t-"

286 RA DIAT I ON THEORIES

497

Suppose,for example, that the line tr in Go with Rr an absolute constant. Therefore the sum leadsfrom a point rg to the edgeof the diagram. C(f) is in this case just C'(Go), where C'(G6) is Then C(Go)is an integral containing in the inte- obtained from C(Gs)by the replacement srand the matrix element of (s6) {'(xi+R&(xs).

0"@,)

/c?\

for creation of an electron into the state B. Let the momentum-energy 4-vector of the created electron be 2; the matrix element of (52) is of the form (s3) ["(xs):a"expl-i(P'x)/hl

In the casewhen the line tr leads into the graph Go from the edge of the diagram to the point 13, it is clear that C(r) will be similarly obtained from C(Go) by the replacement (57)

g(x")+Rf,!(xs).

There remains the case in which tr leads from with a" independent of rcr. Now consider the one vertex ,s to another xa of. Go. In this case sum C(f). It follows from an analysis of (31) C(Go) contains in its integrand the function that C(f) is obtained from C(G) by replacing (58) $q(4,xe)Srp"(ra-xe), the operator (52) bV of the value expectation which is the vacuum oDerator

/n\ [* av,' t-:t" i"r -a/nd^t, '

xP('i,"@a),HI(y), . . ., H'(y")).

P(0"@z),,l,p?c)) (s4)

(This is, of course, a consequenceof the special character of the graphs of I.) It is required to calculate the matrix element of (54) for a transition from the vacuum state O to the state B, i.e., for the emission of an electron into state E. This matrix element will be denoted bv Z"; C(l) involves 2, in the same way that C(Go) involves (53), Now Z" can be evaluated as a sum of terms of the same general character as (47); it will be of the form

(se)

according to (43). Now in analogy with (54), C(r) is obtained from C(Go)by replacing (59) bv

i"er/nd^lr/n\f'_ay,I-:t" XP({,

Hr (t),''', "(x),,1,e@n),

HI (y")), (60)

and the vacuum expectation value of this operator will be denoted by ln @z,xt) S' r o"(x a x t) -

(61)

By the methods of Section VI, (61) can be expanded as a series of terms of the same charf* acter as (47) ;'tnis expansion will not be disKi"8(1ti-xt)YB\)d'y;, Z I " : L i J-n cussedin detail here, but it is easy to seethat it leads to an expression of the form (61)' with function Ki is a fact is that where the important So'(x) a certain universal function of the 4only of the coordinate differences betrveen y,' r. It will not be possibleto reduce (61) to vector (53), that this implids and *a. By a numerical multiple of (58), as Z" was in the (ss) previous case reduced to a multiple of f"' InZ":R"e@)Yp(4), stead, there may be €xpected to be a series exFrom considerations with R independent of ra. pansion of the form of relativistic invariance, R must be of the form S"o"(r) : (R:* or(!'z - Koz) + a2(a2- Ko2)z (P,t 6a"Rt(P\ * ) e"Rt(f2) , ' ' ) S r p " ( * ) * ( b t * b z(]'z- x0')+''') *' where p2 is the square of the invariant length of X(7,1d / ax,l- rs)BrSr'1"(r), (62) the 4-vector p. But since the matrix element (53) is a solution of the Dirac equation, p, : -hrro2,

(?n )p"Yp:,ih*oY ",

and so (55) reduces to

Z":RtY"Qci,

where !2 is the Dalembertian operator and the a, b are numerical coefficients. In this case C(f) will be equal to the C'(Go) obtained from C(Gr) by the replacement (63) Sr'(rr-rr)tS r'(xt-xo).

287 F. J. DYSON

498

Applying the same methods to a graph G with a self-energy part connected to its surroundings by two photon.lines, the sum C(I) will be obtained as a single contribution C'(Go) from the reduced graph Go, C'(Ge) being formed from C(Go) by the replacement

responding to that term will contain the point r; joined to the rest of the graph only by two electron lines, and this point by itself constitutes a self-energy part of the graph. Therefore, all terms involving f/s are to be omitted from (31) in the calculation of matrix elements. The inD r(x r - x n)-D p' (xz- x e). (64) tuitive argument for omitting these terms is that they were only introduced in order to cancel The function Drl is defined by the condition that out higher order self-energy terms arising from (6s) fli, which are also to be omitted; the analysis lltc6r"D pt(xs-xa) of the foregoing paragraphs is a more precise is the vacuum expectation value of the operator form of this argument. In physical language,the argument can be stated still more simply; since 6m is an unobservablequantity, it cannot appear in the final description of observablephenomena.

L,{- o/r,)"1, t,nI _'jy,. [_'-or, X P ( A N Q c a ) ,A , ( , c ) , H I ( t l ,

. . ., t{(y")),

(66)

and may be expanded in a series

D s' (x) : (Roa ctar+cz(ar)r*.

. .)D r(x).

(67)

Finally, it is not difficult to seethat for graphs G with self-energy parts connected to their surroundings by a single photon line, the sum C(I) will be identically zero, and so such graphs may be omitted from consideration entirely. As a result of the foregoing arguments, the contributions C(G) oI graphs with self-energy parts can always be replaced by modified contributions C'(Go) from a reduced graph Go. A given G may be reducible in more than one way to give various Go,but if the processof reduction is repeated a finite number of times a Go will be obtained which is "totally reduced," contains no self-energy part, and is uniquely determined by G. The contribution C'(Go) of a totally reduced graph to the matrix element of (31) is now to be calculated as a sum of integrals of expressions like (47), but.with a replacement(56), (57), (63), or (64) made correspondingto every line in Go. This having been done, the matrix element of (31) is correctly calculated by taking into consideration each totally reduced graph once and once only. The elimination of graphs with self-energy parts is a most important simplification of the theory. For according to (22), flr contains the subtracted part fls, which will give rise to many additional terms in the expansionof (31). But if any such term is taken, say, containing the factor fls(r) in the integrand, every graph cor-

VIII. VACUUMPOLARIZATIONAND CHARGE RENORMALIZATION The question now arises: What is the physical meaning of the new functions Dp' and Sr', and of the constant Rr ? In general terms, the answer is clear. The physical processesrepresented by the self-energyparts of graphs have been pushed out of the calculations, but these processesdo not consist entirely of unobservableinteractions of single particles with their self-fields, and so cannot entirely be written off as "self-energy processes." In addition, these processesinclude the phenomenon of vacuum polarization, i.e., the modification of the field surrounding a charged particle by the charges which the particle induces in the vacuum. Therefore, the appearanceol Dp', Spt, and R1 in the calculations may be regarded as an explicit representation of the vacuum polarization phenomenawhich were implicitly contained in the processesnow ignored. In the present theory there are two kinds of vacuum polarization, one induced by the external field and the other by the quantized electron and photon fields themselves; these will be called "external" and "internal," respectively. It is only the internal polarization which is represented yet in explicit fashion by the substitutions (56), (57), (63), (64); the external rvill be included later, To form a concrete picture of the function Dr', it may be observed that the function Dn(y-z) represents in classical electrodynamics the retarded potential of a point charge at y acting upon a point charge at z, together with the re-

288 RADIATION

499

tarded potential of the charge at z acting on the charge at y. Therefore, Dr may be spoken of loosely as "the elebtromagnetic interaction between two point charges." In this semiclassical picture, Dr' is then the electromagnetic interaction between two point charges, including the which each effects of the charge-distribution charge induces in the vacuum. The complete phenomenon of vacuum polarization, as hitherto understood, is included in the above picture of the function Dr". There is nothing left for S"' to represent. Thus, one of the important conclusions of the present theory is that there is a second phenomenon occurring in nature, included in the term vacuum polarization as used in this paper, but additional to vacuum polarization in the usual sense of the word. The nature of the second phenomenon can best be explained by an example. The scattering of one electron by another may be represented as caused by a potential energy (the Mlller interaction) acting between them. If one electron is at y and the other at z, then, as explained above, the effect of vacuum polarization of the usual kind is to replace a Iactor Dp in this potential energy by Dp'. Now consider an analogous, but unorthodox, representation of the Compton effect, or the scattering of an electron by a photon. If the electron is at y and the photon at a, the scattering may be again represented by a potential energy, containing now the operator Sr(y-z) as a factor; the potential is an exchange potential, because after the interaction the electron must be considered to be at a and the photon at y, but this does not detract from its usefulness. By analogy with the -vector charge-current density ju which interacts with the potential ,F, a spinor Comptoneffect density ua may be defined by the equation : A ,(x) (t ) , "e{ eQ) "(x) and an adjoint spinor by u

a"(x) :0p@)Q,)o"A u@). These spinors are not directly observable quantities, but the Compton effect can be adequately described as an exchange potential, of magnitude acting between the proportional to S.(y-z), Compton-effect density at any point y and the' adjoint density at z. The second vacuum polariza-

THEORIES tion phenomenon is described by a change in the form of this potential from Sr to Sr'. Therein fore, the phenomenon may be pictured physical terms as the inducing, by a given element of Compton-effect density at a given point, of additional Compton-effect density in the vacuum around it. In both sorts of internal vacuum polarization; the functions De and. Sr, in addition to being altered in shape, become multiplied by numerical (and actually divergent) factors Ra and Rr; also the matrix elements of (31) become multiplied by numerical factors such as RrRr*. However, it is believed (this has been verified only for secondorder terms) that all z'th-order matrix elements of (31) will involve these factors only in the form of a multiplier (eR2RJ)"; this statement includes the contributions from the higher terms of the series' (62) and (67). Here a is defined as the constant occurring in the fundamental interaction (16) bV virtue of (37). Now the only possible experimental determination of e is by means of measurements of the effects described by various matrix elements of (31), and so the directly measured quantity is not e but aRrRrl. Therefore, in practice the letter a is used to denote this measured quantity, and the multipliers R no longer appear explicitly in the matrix elements of (31) ; the change in the meaning of the letter e is called "charge renormalization," and is essential if e is to be identified with the observed electronic charge. As a result of the renormalization, the divergent coefficients Rt, Rr, and Ra in (56), (57), (62)' and (67) are to be replaced by unity, and the higher coefficients o, b, and c by expressions involving only the renormalized charge e. The external vacuum polarization induced by the potential /u" is, physically speaking, only a special case of the first sort of internal polarization; it can be treated in a precisely similar manner. Graphs describing external polarization effects are those with an "externai polarization part," namely, a part including the point rcq and connected with the rest of the graph by only a single photon line. Such a.graph is to be "reduced" by omitting the polarization part entirely

and renaming

with

the label ro the

289 F,

J.

point at the further end of the singlephoton line. A discussionsimilar to thoseof SectionVII leads to the conclusion that only reduced graphs need be considered in the calculation of the matrix element of (31), and that the effect of external polarization is explicitly represented if in the contributions from thesegraphs a replacement A u"(x)+A u"'(x)

500

DYSON

(68)

is made. After a renormalization of the unit of potential, similar to the renormalization oI charge,the modified potential,4.r" takes the form Au"'(x): (1+rrIr+rr(!z1z-t, . . .)Auc(x), (69) where the coefficientsare the sameas in (67). It is necessary, in order to determine the functions Dp', Sr', and Au'', to go back to formulas (60) and (66). The determination of the vacuum expectation values of the operators (60) and (66) is a problem of the same kind as the original problem of the calculation of matrix elementsof (31), and the various terms in the operators (60) and (66) must again be split up' represented by graphs, and analyzed in detail. However, since Dr' and Sr' are universal functions, this further analysis has-only to be carried out once to be applicable to all problems' It is one of the major triumphs of the Schwinger theory that it enablesan unambiguous interpretation to be given to the phenomenon of vacuum polarization (at least of the first kind)' and to the vacuum expectation value of an operator such as (66). In making this interpretation, profound theoretical problems arise, particularly concerned with the gauge invariance of the theory, about which nothing will be said here. For Schwinger's solution of these problems, the reader must refer to his forthcoming papers. Schwinger'sargument can be transferred without essential change into the framework of the presentPaPer. Having overcome the difficulties of principle, Schwinger proceeded to evaluate the function Dr' explicitly as {ar as terms of otder a:(e2/ Arhc) (heavislde units). In particular, he found for the coefficient q in (67) and (69) the value (- a/l5rro2) to this order.sIt is hoped to publish -s.h*irrg".'" results agree with those of the e-arlier, theoretieliv unsatisfactor-y trea tment of vacuu m polarizaiion. th" b."t uccount of the earlier work is V. F. Weisskopf, iigl. Oan"t<e Sels. Math.-Fys. Medd. 14, No. 6 (1936)'

in a sequel to the present paper a similar evaluation of the function Srf; the analysis involved is too complicated to be summarized here. IX. SUMMARY OF RESULTS In this section the results of the preceding pages will be summarized, so far as they relate to the performance of practical calculations. In effect, this summary will consist of a set of rules for the application of the Feynman radiation theory to a certain class of problems. Suppose an electron to be moving in an external field with interaction energy given by (33). Then the interaction energy to be used in calculating the motion of the electron, including radiative corrections of all orders' is

H n@i : L G i / h c ) " 1/1n t l J" n:0

:L,t-o/ra,u/nt[_'.a*, I_:." XP(H"(xi,

Hc(x), "',

Ht(x")),

(70)

with Ilt given by (16), and the P notation as defined in (29). To find the effective z'th-order radiative correction to the potential acting on the electron, it is necessary to calculate the matrix elements ol J, for transitions from one one-electron state elements can be to another. These matrix written down most conveniently in the form of an operator K" bilinear in ry'and ry',whose matrix are the elements for one-electron transitions same as those to be determined. In fact, the operator K" itself is already the matrix' element to be determined if the 0 and tlt contained in it are regarded as one-electron wave functions. To write down K", the integrand P" in J" is first expressed in terms of its factors {r' 9, and A' all suffixbs being indicated explicitly, and the expression (37) used for ju' All possible graphs G with (z*1) vertices are now drawn as described in Section VII, omitting disconnected graphs, graphs with self-energy parts, and graphs with external vacuum polarization parts as defiued in Section VIII. It will be found that in each graph there are at each vertex two electron lines and one photon line, with the exception of ro at which there are two electron lines only; further,

290 RADIATION

501

THEORIES

such graphs can exist only for even n. K" is the sum of a contribution K(G) from each G. Given G, K(G) is obtained from J" by the following transformations. First, for each photon line joining r and y in G, replace two factors A ,(x)A,(y) in P" (regardless of their positions) by

(7r)

llrc6u,Dr' (x !) ,

u'ith Dr'given by (67) with Rr:1, the function De being defined bv (a2). Second, for each electron line joining r to y in G, replace two factors in P" (regardless of positions) b1' {"@){e0) (72)

tS'ra"@-Y)

with Sr' given by (62) with Rz:1, the function Sr being defined bv @D and (45)' Third, replace in P" the remaining two factors P({,,(z),t{w)) by {rr(z),t'{w) in this order. Fourth, replace A u"((o) by A u"'(xo) given bY A u " ' ( x ): 4

-la/l5rxn')A'A u"7*)

u"@)

(73)

or, more generally, bv (69). Fifth, multiply the whole by (-1)t, where I is the number of closed loops in G as defined in Section VII. The above rules enable K" to be written down very rapidly for small values of z' It should be observed thatil K" is being calculated, and if it is not desired to include effects of higher order than tlre z'th, then Dr', Sr', and Au"' nr (71)' (72), and (73) reduce to the simple functions Dr, Sr, and Au'. Also, the integrand in /" is a symmetrical function of rr, ' ' ', r,; therefore, graphs which differ only by a relabeling of the vertices xt, '" ' x^ give identical contributions to K, and need not be considered separately. The extension of these rules to cover the calculation of matrix elements of (70) of a more general character than the one-electron transitions hitherto considered presents no essential difficulty. All that is necessary is to consider graphs with more than two "loose ends," representing processes in which more than one particle is involved. This extension is not treated in the present paper, chiefly because it would lead to unpleasantly cumbersome formulas. X. EXAMPLE-SECOND-ORDER CORRECTIONS

RADIATIVE

As an illustration of the rules of procedure of the previous section, these rules will be used for down the terms giving second-order writing

radiative corrections to the rnotion of an clectrolt in an cxternal field. Lct the energy of tlie erternal field be

-lt/cfj,(xo)A,"(xi.

g4)

Then there will be one second-order correction term 11: la / rS r x o2)11/ cfj u@) if' A," (t o) arising frorn the subst-itution (73) in thc zcruorder term (7a). This is the well-known vacuum polarization or Uehling term.' The ren-raining second-order term arises fronr the second-order part Jr of (70). Written in expanded form, J2 is

r* r" r'(r0). Jr: ietI dxt I dxrPr.0"(xo)(v^)"p/r(ro)l J-a .t-6 &,(x')(t ) 'tL{x)A p(x), Q"@,)0 ) <{ r (x,)A " (x,)). Next, all admissable graphs u'ith the three fi2 are to be drarvn. It is easy to vertices :tn,'"$1, see that there are only two such graphs, that G shown in Fig. 1, and the identical graph with lu1 and rg interchanged. The full lines are electron lines,the dotted line a photon line. The contribution K(G) is obtained from Jz by substituting accordingto the rules of Section IX; in this case l:0, and the primes can be omitted from (71)' (72), (73) since only second-orderterms are required. The integrand in K(G) can be reassembledinto the form of a matrix product, suppressingthe suffixesot, "', (. Then, multiplying by a factor 2 to allow for the secondgraph, the complete second-ordercorrection to (74) arising from Jz becomes

r(x,-xz)A,"(xo) L: - iles/vhcl !1.' I-!.,D @)' x{(r,)r,Sr(ro - x )t,S r (xz * o1r,9 A' Uehling, (1935); E' 49 a8, Rev. Phys. RJJS"rb"r, Phys.Rev.48,55(1935). -l

29r F.

J.

This is the term which gives rise to the main part of the Lamb-Retherford line shift,1o the anomalous magnetic moment of the electron,ll and the anomalous hyperfine splitting of the ground state of hydrogen.l' The above expression Z is formally simpler than the corresponding expression obtained by Schwinger, but the two are easily seen to be equivalent. In particular, the above expression does not lead to any great reduction in the labor involved in a numerical calculation of the Lamb shift. Its advantage lies rather in the ease with which it can be written down. In conclusion, the author would like to express his thanks to the Commonwealth Fund of New York for financial support, and to Professors Schwinger and Feynman {or the stimulating lectures in which they presented their respective theories. Notesad.dedin prooJ (To Section II). The argument of Section II is an over-simplification of the method of Tomonaga,land is unsound.There is an error in the derivation of (3); derivativesoccurring in 11(r) give rise to noncommutativity between I1(z) and field quantities at /' when r is a point on o infinitesimally distant from r'. The 10W. E. lamb

24r (1947).

and R. C. Retherford,

Phys. Rev. 72'

rr P. Kusch and H. M. Foley, Phys. Rev. 74, 250 (1948). uI. E. Nafe and E. B. Nelson, Phys. Rev. 73, 718 (194i|); Aage Bohr, Phys. Rev.73, i109i1948).

DYSON

502

argument should be amended as follows. ,D is defined only for flat surfaces t(r):t, ^n4 for such surfaces (3) and (6) are correct. V is defined for general surfaces by (12) and (10), and is verified to satisfy (9). For a flat surface, iD and rP are then shown to be related by (7). Finally, since I{r does not involve the derivatives in 11, the argument leading to (3) can be correctly applied to prove that for general c the state-vector v(c) will completely describe results of observations of the system on o. (To Section III ). A covariant perturbation theory similar to that of Section III has previously been developed by E. C G. Stueckelberg, Ann. d. Phys. 21, 367 (1934); Nature, 153, 143 (19+4). 'eflective potential" is not (To Section V.;. Schwinger's Ilr given by (25), but is I1r': QHTQ-|. Here Qis a "squareroot" of S(o) obtained by expanding (.9(€))l by the binomial theorem. The physical meaning of this is that Schwinger specifies states neither by O nor by 0', but by whose delian intermediate state-vector Att:Qtt:Q-rOt, nition is symmetrical between past and future. f/r'is also symmetrical between past and future. For one-particle states, 1{r and Hr' are identical. Equation (32) can most simply be obtained directly from the product expansion of S( < ). (To Section VII). Equation (62) is incorrect. The function Sr'is well-behaved, but its fourier transform has a logarithmic dependence on frequency, which rokes an expansion precisely of the form (62 ) impossible. (To Section X). The term I still contains two divergent parts. One is an "infra-red catastrophe" removable by standard methods. The other is an "ultraviolet" divergence, and has to be interpreted as an additional chargerenormalization, or, better, cancelled by part of the chargerenormalization calculated in Section VI I I

292

P o p e r2 5

The S Matrix in Quantum Electrodynamics F. J. DysoN Institrute Jar Aduanced. Stud!, princeton, New Jersey (Received February 21, 19+9) The covariant quantum electrodynamics of romonaga, Schwinger, and Feynman is used as the basis for a general treatment of scattering problems involving electrons, po.it.o.s, and photons. Scattering processes, including the creation and annihilation of particles, aie complctell, described by the S matrix of Heisenbcrg. It is shown that the elements of this matrix can be calcrilated. bv a consist"nt use of perturbation theory, {u anl de:irpLl order in rhe 6ne-srrucfure con.ranr. D.Lailed rules are given for carrying out such calculations, and it is shorvn that divergeoces arising from higher order radiative corrections can be removed from the S matrix by a consistent use of the ideas of mass and charge renormalization. Not considered ia this paper are the problems of extending the treatment to include bound-state phenomena, and of proving the convergrncc of the theory as the ortler of pe.turbatioo itself tends to infinity.

I. INTRODUCTION

JN a previous paperl (to be referrecl to in what I f o l l o , u =a s I J i h e r a d i a r i o nt h e o r y o f T o m o n a g a 2 and Schwingeri was appliecl in detaii to the problEm of the radiative cor."Ctio.,, to the motion oia sinsle electron in a given external field. It rvas shou'n th-at the rules of calcr:lation for corrections of this kind were identical with those rvhich had been derived by Feynmana from his own racliation theory. For the one-electron problem the radiative corrections lvere fully described by an operator H, (Eq. (20) of I) which appeared is the "effective- potential'; acting upon tire electron, after the intcractions of the electron with its own self_fielclhad been eliminated by a contact transformation. The differenc" between the Schwinger and Feynman theories lay only in the choice oia particulir representation i; which the matrix elements of IIr were calculated (Section V of I). The present paper deals with the relation betrvecn the Scirwingei and Feynman theories rvhen the restriction to one-electron problems is removed. In these more general circumitances the two theories appear as co;plementary rather than identical. The Feynman metirod is essentially a set of rules for the calculation of the elements of the Heisenberg S matrix corresponcling to any physical process] and can be appliecl with di.ectreis io all kinas oi

ates radiative corrections by exhibiting them as extra terms appearing in the Schrodinger equation o f a s ) s t e m o I p r r t i c l e s a n t l i s > u i r e , le s p c c i . r l l yt o bolnd-state problems. In spite of tl.re difference of principle, the two methods in practice involve the calculation of closely related expressions;moreover, the theory underlying them is in all casesthe sane. The systematic technique of Feynman, the erposition of which occupied the second half of I and occupies the major part of the present paper, is therefore norv available for the evaluation not only of the ,S rratrix but also of most of the operators occurring in the Schwingcr theory. The promincnt part which the S matrix plays in this paper is due to its practical usefulnessas the connecting link between the Feynman technique of calculation and the Hamiltonian formulation of quantum electrodynamics. This practical usefulness remains, rvhether or not one follows Heisenberg in believing that thc S matrix may eventually replace the Hamiltoniar.raltogether. It is still an unanswered question, whether the finiteness of the S matrix automatically implies the finitenessof all observable quantities, such as bound-state energy levels, optical transition probabilities, etc , occurring in clectrodynamics. An aflirmative answer to the question is in no way essential to the argr:ments of this paper. Even if a finite S matrix does not of

probrems'5 scattering t* t:^1:::^:'method evaru-[::i:,lliJi,-lilitr",tT.it"u"tL".1,lt'"Tii"T."liiT; lF Dvson, Phvs Rev T5'486 (1949)'

J' 2 S i n - l r i r o T o m o n a g a , P r o g . T h e o r . P h v . . l , 2 7 ( 1 9 1 6 ) ; cnite; to verify this, it will be necessary ro repear li the analysis of the present paper, keeping all the Koba, Tati, and lo-monaga] Prog. fhe6r. pil.. z, loi (1947) qnd 2,-198 (1947); S. Kanesawa^and.S.-Tomon_aga, time closer to the original Schrvinger theory than P r o g . T h e o r . P h y s . 3 , 1 ( 1 9 4 8 )a n d 3 , i 0 1 ( 1 9 4 8 ) ;S i n - I t i r o Tomonaga, Phvs,Rev. 74,224 Qa-48);Ito, Koba, and ro(1946); Nature, I53, 143 (1944); phys. soc. cambridge coomonaga, Prog. Them, Phys. 3, 276 (1918); Z. Koba and ference Report, 199 (19+i); E. C. C. Stueckeiberg-and D. S.-Tomonaga,Prog. Theor. Phys.3, 290 (1948). Rivier, Phys. Rev. 74, 218 (1948). Stueckelbergan'ticipatecl . I J-qlia3. Sg!,Ilngql, Phys. Rev. 73' 416 (194$; 7a, 1439 seyeral feaiures of the Feynman tireory, in partiiular the use (194-8);75,_651(1949). of the function Dr (in Sru;ckelberg'snotutioi pc) to represent a Richard P."Feynman, Phys. Rev. 7.4,1430 (1948). retarded (i.e., causally transnriried) electronagnetii inter6The idea of using standard elecLrodynamicsas a- starting actions. For a revierv-of the earliei part of this work, see point for an explicit calculation of the S matrix has been Gresor Wentzel, Rev. N,Iod.Phvs. l9'. 1 (1947). The use of previously_developed by Q Q. G.S!reckelberg, Helv. Phys. masi renormalizationin scaLteringp.oble-" is due to H. W A c t a , 1 4 , 5 1 ( 1 9 a 1 ) ; 1 7 , 3 ( 1 9 4 1 ) ; 1 8 , 1 9 5 ( 1 9 4 5 ) ; 1 9 , 2 4 2 L e w i s ,p h y s . R e v 7 3 , 1 7 3 ( 1 9 4 8 ) l

l 73 6

293 r737

S

NIATRIX

IN

QUANTUM

has here bccn possible. There is no reason for attributing a more fundamer.rtalsignificance to the S matrix than to other obscrvable quantities, nor \\'as it Heisenbcrg's intention to do so. In the last section of this paper, tentative suggestions are made for a synthesis of the Hamiltonian and Heisenberg philosophies.

ELECTRODYNAMICS

Here the P notation is as defined in Section V of I, and IL(x):

HI(x)*II"(x)

(s)

is the sum of the interaction energiesof the electron field with the photon field and rvith the external potentials. The Feynman radiation theory provides a set of rules for the calculation of matrix elements MATRIX THEORY AS AN S FEYNMAN THE II. of (4), between states composed of any number of THEORY ingoing and outgoing free particles. Also, quantities The S matrix was originally defined by Heisen- contained in (4) are the only ones with which the berg in terms of the stationary solutions of a scat- Feynman rules can deal directly. The Fcynman tering problern. A typical stationary solution is theory is thus correctly characterizedas an S matrix reprcsentecl by a time-inclcpendent s.ave function theory. V' rvhich has a lrart representing ingoing r,vaves One particular way to anal-vzeU( o ) is to use (5) rvhich are as-vmptotically of the form V1', ancl a to expand (4) in a seriesof terms of ascending order part reprcsenting outgoing rvaves rvhich are asymp- in 11". Substitution froln (5) into (4) gives totically of the forrn Vr'. The S matrix is the transformation operator S with the property that V:' : SiLr'

(1)

for cver)' stationary state V'. In Scction I I I of I an operator U( o ) rvas defined and stated to be identical with the S matrix. Since U( o ) was defineclin terms of time-clependentwave functions, a little care is needed in making the identification. In fact, the equation Vr: /(o)V1

Q) held, where Vr and Vz were the asymptotic forms of the ingoing and outgoing parts.of a wave function V in the V-representation of I (the "interaction representation" of Schwinger3). Now the timeindependent wave function V' corresponds to a t i m e - d e p e n d e n tw a v e f u n c t i o n expl(-i/h)Etlv' in the Schrodinger representation, where -E is the tbtal energy of the state; and this correspondsto a wave function in the interaction representation v:expl(li/h)l(110-E)lv',

(3)

rvhere -F1ois the total free particle Hamiltonian. However, the asymptotic parts of the wave function V', both ingoing and outgoing, represent freely traveling particles of total energy -8, and are therefore eigenfunctions of .I1owith eigenvalue E. This implies, in virtue of (3), that the asymptotic parts 9r and Vz of V are actually time-independent and e q u a l , r e s p e c t i v e l y ,t o V 1 ' a n d V r ' . T h u s ( 1 ) a n d (2) are identical, and I/(o) is indeed the S matrix. Incidentally, U( o ) is also the "invariant collision operator" defined by Schwinger.s There is a series expansion ol U(a) analogous to (32) of I, namely,

u@:Ea *l_'.*,[_'^d"" xP(I1r(rr), . . ', n'@")). (4)

" ur4: ^t:, .t,(;)* o I_-*, f-

x I I

d x , , - ^ P ( f i ' ( x , )" ,. , H " ( r ^ ) , xHI (x^a), ..', HI(x*"")).

(6)

In this double series, the term of zcro order in H" is S( o ), given by (32) of I. The term of first order is f-

Ur:Gi/hc) |

Hr!)dx,

A)

J.*

where -Flr is given by (31) of L Clearly, S( o ) is the S matrix representing scattering of electrons and photons by each other in the absenceof an external potential; Ur is the S matrix representing the additional scattering produced by an external potential, rvhen the external potential is treated in the first Born approximation; higher tcrms of the series (6) would correspond to treating the external potential in the second or higher Born approximation. The operator IIp played a prominent part in I, where it was in no way connected with a Born approximation; however, it rvas there introduced in a somewhat unnatural manner, ancl its physical meaning is made clearer by its appearance in (7). In fact, Hp maj be defined by the statement that (- i/h) (6t)(aa)H r(r) is the contribution to the S matrix that would be produced by an external potential of strength 11", icting for a small duration 6l and over a small volume 6o in the neighborhood of the space-time point r. The remainder of this section rvill be occupied with a statement of the Feynman rules for evaluating t/(o). Proofs will not be given, becausethe rules are only trivial generalizations of the rules

294 F.

J.

which were given in I for the evaluation of matrix elements of 11r corresponding to one-electron transitions. In evaluating U(o) we shall not make any distinction between the external and radiative parts of the electromagnetic field; this is physically reasonable since it is to some extent a matter of convention how much of the field in a given situation is to be regarded as "external." The interaction energy occurring in (4) is then H'(r)

: - ieA r@){ (x)t,,l, Qc)- 6mc,F@)'l'@),

(8)

where,4., is the total electrornagnetic field, and the term in 6m is included in order to allow for the fact that the interaction representation is defined in terms of the total mass of an electron including its "electromagnetic mass" 6m (see Section IV of I). The first step in the evaluation of U(o) is to substitute from (8) into (4), writing out in full the suffixes of the operators {r,, ,rp which are concealed in the matrix product notation of (8). After such a substitution, (4) becomes

U(*):

*loJ^

1738

DYSON

(9)

where -I, is an z-fold integral with an integrand which is a polynomial in {", te and .Auoperators. The most general matrix element of -I" is obtained by allowing some of the {'", {p and ,4.uoperators to annihilate particles in the initial state, some to create particles in the final state, while others are associatedin pairs to perform a successivecreation and annihilation of intermediate particles. The operators which are not associated in pairs, and which are available for the real creation and annihilation of particles, are called "free"; a particular type of matrix element of -I" is specified by enumerating which of the operators in the integrand are to be free and which are to be associatedin pairs. As described more fully in Section VII of I, each type of matrix element of J, is uniquely represented by a "graph" G consisting of z points (bearing the labels rr, . , ', r") and various lines terminating at these points. The relation betlveen a type of matrix element of -I" and its graph G is as follows. For every assothere is a ciated pair of operators (0@,{(y)), directed line (electron line) joining r to y in G. For every associated pair of operators (A(r)' A(y))' there is an undirected line (photon line) joining * and y in G. For every free operator 'y'(r), there is a directed line in G leading from rc to the edge of the diagram. For every free operator ry'(*), there is a directed line in G leading to r from the edge of the diagram. For every free operator -4(r), there is an undirected line in G leading from s to the edge of the diagram. Finally, for a particular type of

matrix element oI J, it is specified that at each point r; either the part of E{x;) containing -Au(r) or the part containing dzz is operating; correspondingly, at each vertex ,ci of G there are either two electron lines (one ingoing and one outgoing) and one photon line, or else two electron lines only. Lines joining one point to itself are always forbidden. In every graph G, the electron lines form a finite number m of open polygonal arcs with ends at the edge of the diagram, and perhaps in addition a number I of closed polygonal loops. The corresponding type of matrix element of J,has m free operators {, and m free operators ry'; the two end segments of any one open arc correspond to two free operators, one ty' and one ry',which will be called a "{ree pair." The matrix elements of -r" are now to be calculated by means of an operator -r(O, which is defined for each graph G of z vertices, and which is obtained from -I, by making the following five alterations. First, at each point r;, 111(:r;)is to be replaced by either the first or the second term on the right of (8), as indicated by the presence or absence of a photon line at the vertex r; of G. Second, for every electron line joining a vertex r to a vertex y in G, two operators {r"(x) and {p(y) in -I", regardless of their positions, are to be replaced by the function

tsra"@-Y),

(10)

as clelined by (44) and (a.5) of I. Third, for every photon line joining two vertices x and y of G, two operators.4r(r) and A,(y) in -I,, regardlessof their positions, are to be replaced by the function lhc6,,Dr(x-y),

(11)

defined by (42) of I. Fourth, all free operators in -I" are to be left unaltered, but the ordering by the P notation is to be dropped, and the order of the free rf and ry'operators is to be arranged so that the two members of each free pair stand consecutively and in the order {tt; the order of the free pairs among themselves, and of all free -du operators, is left arbitrary. Fifth, the whole expression -I" is to be multiplied by (12) (_t1"-t-^. The Feynman rules for the evaluation of U(*) are essentially contained in the above definition of the operators -f(O. To each value of z correspond only i finite number of graphs G, and all possible mairix elements of {/( o ) are obtained by substituting into (9) for each -/" the sum of all the cort"rpo-tlding J (G). It is necessaryonly to specify how the matrii element of a given -I(G) corresponding to a given scattering process may be written down' The matrix element of J(G) for a given process mav be obtained, broadly speaking, by replacing

295 1739

S

MATRIX

IN

QUANTUM

each free operator in -r(G) by the wave function of the particle which it is supposed to create or annihilaie. More specifrcally, each free rf operator may either create an electron in the final state or annihilate a positron in the initial state, and the reverse processes are performed by a free ry' operator. Therefore, for a transition from a state involving .4 electrons and B positrons to a state involving C electrons and D positrons, only operators J(G) (Bf C) ft"" pairs contribute containing (A+D): matrix elements. For each such -r(G), the (.4*D) free ry' operators are to be replaced in all possible combinations by the ,4 initial electron wave functions and the D 6nal positron wave functions, and the (B*C) free ry'operators are to be similarly replaced by the initial positron and final electron wave functions, and the results of all such replacemtlnts added together, taking account of the antisymmetry of the total wave functions of the system in the individual particle wave functions. In the case of the free ,4, operators, the situation is rather different, since each such operator may either create a photon in the final state, or annihilate a photon in the initial state, or represent merely the external potential. Therefore, for a transition from a state with,4 photons to a state with B photons, Iree Au any J(G) with not less than (AIB) operators may give a matrix element. If the number these of free A, operators in J(G) is (,4.*B*C), operators are to be replaced in all possible combinations by the (.4 f B) suitably normalized potentials corresponding to the initial and final photon states, and by the external potential taken C times, and the results of all such replacements added together, taking account now of the symmetry of the total wave functions in the individual photon states. In practice cases are seldom likely to arise of scattering problems in which more than two similar particles are involved. The replacement of the free operators in -r(G) by wave functions can usually be carried out by inspection, and the enumeration of matrix-elements of I/( o ) is practically complete as soon as the operators J(G) have been written down. The above rules for the calculation of U(o) describe the state of affairs before any attempt has been made to identify and remove the various divergent parts of the expressions. In particular, contributions are included from all graphs G, even those which yield nothing but self-energy effects. For this reason, the rules here formulated are superficially different from those given for the oneelectron problem in Section IX of I, which described the state of affairs after many divergencies had been removed. Needless to say, the rules are not complete until instructions have been supplied for the removal of all infinite quantities from the theory; in Sections V-VII of this paper it will be

ELECTRODYNAMICS

shown how the formal-structure of the S matrix makes such a complete removal of infinities appear attainable. Another essential limitation is introduced into the S matrix theory by the use of the expansion (4)' All quantities discussed in this paper are expansions of this kind, in which it is assumed that not only the radiation interaction but also the external potential is small enough to be treated as a perturbation. It is well known that an expansion in powers of the external potential does not give a either in problems satisfactory approximation, involving bound states or in scattering problems at low energies. In'particular, whenever a scattering problem allows the possibility of one of the incident particles being captured into a bound state, the c a p t u r e p r o c e s sw i l l n o t b e r e p r e s e n t e di n U ( o ; , since the initial and final states for processes described by U(o) are always free-particle states' It is the expansion in powers of the external potential which breaks down when such a capture process is possible. Therefore it must be emphasized that the perturbation theory of this paper is applicable only to a restricted class of problems, and that in other situations the Schwinger theory will have to be used in its original form. III. TIIE S MATRI'( IN MOMENTUM SPACE Both for practical applications to specific problems, and for general theoretical discussion, it is convenient to express the S matrix [/( o ) in terms of momentum variables. For this purpose, it is enough to consider an expression which will be denoted by M, and which is a typical example of the units out of which all matrix elements of I/( o ) are built up. A particular integer n and a particular graph G of a vertices being supposed fixed, the operator J(G) is constructed as in the previous seetion, and M is defined as the number obtained by substituting for each of the free operators in J(G) one particular free-particle wave function. More specifically, for each free operator ry'(a)in J(G) there is substituted (13) {(k)eikr"t, where fr, is some constant 4 vector representing the momentum and energy of an electron, or minus the momentum and energy of a positron, and where t(k) is a constant spinor. For each free operator l(*) there is substituted {,(k'12-;tr''u,

(14)

where ry'(fr')is again a constant spinor. For each free operator Ar(x) there is substituted Au(k't)etkr""r' where Au(k")

(15)

is a constant 4 vector which may

296 F.

J.

7740

DYSON

appear in M a 4 vcctor variable of integration, which nay be denoted by put,l: 1, ' . . , F. However, after this substitution is made, the space-time variables Jrl, ..., Jraoccur in M only in the exponential factors, and the integration over these variables can be performed. The result of the integration over rci is to give (2r)a6(q), F r G .1 . represent the polarization vector of a quantum whose momentum-energy 4-vector is either plus or minus ,br"; alternatively, Ar(k") ntay represent the Fourier component of the external potential with a particular wave number and frequency specified by the 4 vector h". There is no loss of generality in splitting up the external potential into Fourier components of the form (15). When the substitutions (13), (14), (15) are made in -r(Q, the expression M which is obtained is still an z-fold integral over the whole of space-time, and in addition depends parametrically upon E constant 4 vectors in momentum-space, where E is the number of free operators in "r(G). The graph C will contain E external lines, i.e.. lines with one end at a vertex and the other end at the edge of the diagram. To each of these external lines correspondsone constant 4 vector, rvhich may be denoted by kr!, ,i:1, ' . ., E, and one constant spinor or polarization vector appearing in M, either ,l,Gt)or 0&\ or Ar&t). Suppose that G contains F internal lines, i.e., lines with both ends at vertices. To each of these lines corresponds a Dr or an Sr function 1n M, as specified by (11) or (10). These functions have been expressed by Feynman as 4-dimensional Fourier integrals of very simple form, namely

(20)

where the 6 represents a simple 4-dimensional Dirac 6-function, anrl qi is a 4 vector formed by taking an algebraic sum of thc ki and pi 4 vectors corresponding to those lincs of G which meet at rc7. The factor (20) in the integrand of M expressesthe conservation of energy and momentum in the interaction occurring at the point nr.. The transformation of Minto terms of momentum variables is now complete. To summarize the results, ,44 now appears as an F-fold integral over the variable 4 vectors prd in momentum space, In the integrand there appear, besides numerical factors; (i) a constant spinor or polarization-vector,,!(kt) or,!(hi) or Au(ki), corresponding to each external line of G; (ii) a factor (2r) D,(pu):6*((Pt)') corresponding to each internal photon line of G; (iii) a factor S r(pt) : l*ip,utu-

*ol6*((pu')rl *or')

(22)

corresponding to each internal electron line of G; (iv) a factor 6(qi) corresponding

(2i)

to each vertex of G;

(t) u zu operator, surviving from Eq. (8), corresponding to each vertex of G at whic.h there is a ,, @): (16) photon line. e-- u'.a*(pz)d.p, * f The important feature of the above analysis is that all the constituents ol M are now localized and t? associatedwith individual lines and vertices in the S"(r) :rof | e-'',,,llipuy,graph G. It therefore becomes possible in an 4rr J or X6*(p2! xn2)dp, (17) unambiguous manner to speak of "adding" "subtracting" certain groups of factors in M, when where 16 is the electron reciprocal Compton wave- G is modified bv the addition or subtraction of certain lines and vertices. As an example of this length, p2:pppt:ptr+pz2+pt2_p02, ( 1 9 ) method of analysis, we shall briefly discuss the treatment in the S matrix formalism of the "Lamb and the 6a function is defined by shift" and associatedphenomena. Suppose that a graph G, ol any degree of com11r^ plication, has a vertex rr at which two electron k ' d z . an(a) .,ra(a){^-:( 1 9 ) | e lines and a photon line meet. These three lines may ZTIA ZT J n be either internal or external, and the momentum S u b s t i t u t i n g f r o m ( 1 6 ) a n d ( 1 7 ) i n t o M w i l l 4 vectors associated with them in M may be either introduce an F-fold integral over momentum space. pi or kil these 4 vectors are denoted by tt, t2, t3 as Corresponding to each internal line of G, there will indicated in Fig. 1. The factors in the integrand of

297 S

I t4l

MATRIX

iN

QUANTUM

above analysis applies equally to an expression M

M arising from the vertex rr are - iey - 1z- 7t', , r(2n)t5(1t

the matrix ele(24) which may occur anywhere among

rhe two spinor indices of the 7u being available for matrix multiplication on both sides with the factors in .41arising from the two electron lines at :c1. Now suppose that G' is a graph identical rvith G, except that in the neighborhood of *r it is modified by the addition of two new vertices and three new lines, as indicated in Fig. 2. With the three new lines, which are all internal, are associated three 4 vector variables P', !', P3, which occur as variables of integration in the expression M' formed from G'as M is from G. It can be proved, in view of Eqs. (21), (22), (23), that M' may be obtained from J4 simply by replacing the factor (24) in M by the expression io3

-Lpo1'l I lap,ap,ap' NC

JJJ

6(11- p' + p3)6(p' - pt - t3)6(p1- p3- t ) o2to xo)t,(I

n(*'i4

i4,tt, - *o)n

a*((2r)r* ro,)61((2')r+ ror)6+((23)r). (25) (The factorial coemcients appearing in (4) are just compensated by the fact that the two new vertices ways, of G' may be labelled ra, xi in (n!7)(nl2) where n is the number of vertices in G.) In (25), two of the 4-dimensional D-functions can be eliminated at once by integration over pr and p'?,and the third then reduces to the d-function occurring in Q\. Therefore M'canbe obtained from Mby replacing the operator 1u in (24) by an operator L,: Lr(t', t2) :2a

I dpltx(*i(po*tot)to-

ELECTRODYNAMICS

xit,

x(+i(p"+t,'h, - ro)yrl x d+((P+t)'+*o') Ko')6+(2).Q6) X6+((P+t'z)'z+ Here o is the fine-structure constant, (e2/4rhc) in Heaviside units. The operator Zu can without great difficulty be calculated explicitly as a function of the 4 vectors t\ and t2, by methods developed by Feynman. In the special case when Fig. 1 represents the graph G in its entirety, M is a matrix element for the scattering of a single electron by an external potential. Figure 2 then represents G'in its entirety, and M' is a second-order radiative correction to the scattering of the electron. In this case then the operator Lu gives rise to what may be called "Lamb shift and associated phenomena." However, the

ments of U(o), and may represent any physical processwhatever involving electrons, positrons and photons. There will always appear in U(o), together rvith M, terms ,41' representing secondorder radiative corrections to the same process; one term M' arises from each vertex of G at which a photon line ends; and M'is always to be obtained frorn M by substituting for an operator 7u the same operator Zn. Furthermore, some higher radiative corrections to M will be obtained by substituting Lu for y, independently at two or more of the vertices of G. By a "vertex part" of any graph will be meant a connected part of the graph, consisting of vertices and internal lines only, which touches precisely two electron lines and one photon line belonging to the remainder of the graph. The central triangle of Fig. 2 is an example of such a part. In other words, a vertex part of a graph is a part which can be substituted for the single vertex of Fig. 1 and give a physically meaningful result. Now the argument, by which the replacement of Fig. 1 by Fig. 2 was shown to be equivalent to the replacement of 7u by Lu, can be used also when a more complicated vertex part is substituted for the vertex in Fig. 1. If G is any graph with a vertex nl as shown in Fig. 1' and G' is obtained from G by substituting for 11 any vertex part V, and if. M and M' are elements of I/(o) formed analogously from G and G', then M'can be obtained from Mby replacing an operator "y, by an operator L,:L.(V,

tr, t2),

(27)

dependent only on 7 and the 4 vectors tl, t2 and independent of G. To summarize the results of this section, it has been shown that the S matrix formalism allows a wide variety of higher order radiative processesto be calculated in the form of operators in momentum space, Such operators appear as radiative corrections to the fundamental interaction between the photon and electron-positron fields, and need only to be calculated once to be appligable to the various special problems of electrodynamics.

298 1742

F. J. DYSON rv. FURTIIER REDucTIoN oF TIIE s MATRD( It was shown in Section VII of I that, for the one-electron processesthere considered, only connected graphs needed to U""tuL"" i"tt account' In no constructing the S matrix in general, .this is loneer the case; clisconnectedgraphs give matrix

M by replacing a factoilf (11)by (30)

0(t))>(W,1t)Sr(11).

a a special case' trZ ma-v consist of a single point; As tnen ai this point it is the term in 6m of the interaction (8) *hith it operating' and ) reduces to a

*j-."f."."ntini t*o oi morecol- constant' #;";;^;ie among >(w, t'): -2ri(6ruc/h): -2ri6xo' ii"i;;'^;;.;"r*. o...,i.ittg simultaneouslv

(31)

."purui" groups of particles, and such.proc€sses in a general permiss..ble to The operator lu'(ll). in (28).describes have physical reality. lt.;';i; -f".Utg o m i t a d i s c o n n e c t e c l g r a p h - h " n o n " o f i t s c o n - w a y t h e s e c o n d - o rto d ethe r c ophenomenon n t r i b r r t i o n tcalled o t h e e"vacuum lectron in.external ,elf-"nergy and nected compon"nt. i. in Section VIII of kind" second of the potuti'oiio" "r,ti"iy will lines; such a component wrthout exteinal lines contribution is supposed to be i. The self-ene-r-9y lq;!; give rise only to..o,rr,u'i"ilriilii..ir"" cancelled by (31); the constant 6xo being a power is ancl u(-) matrlx every factor in ","-*i.i series in o' the linear term only is required to therefore devoid of physical significance' cancel the self-energy part of (28)' and the higher VII in Section t.".t?"nt On the other hand, the ,,."ri"n".gv terms are available'for the cancellation of selfparts,, applies of I of graphs with a l m o s t w i t h o u t c h a n g e t o t h e g e n e r a l S m a t r i x e n e r g y e l T.|tr" e c t sSf r omatrix m . o p e]ormalism rators>(W , t ' ) o f hclear i g h ethe r makes ,,self-energy part,, of a graph is a o.d"i. formalism. A >(W'tt) operators the since that, fact i-po.t""t connecteclpart, consisting Jf vertices an'd internal of the graph G' tttl .,-,iaateof are universal operators independent lines only, which can O" i""".t"i^ini. sellenergy effects will be formally itt" meaningful gi"" a .. r.1. graph G a single line of a "l"tttonby a constant independent of the lancelled '6xo graph G'.In Fig. 3 i. =tto*nl't"t-umple of tuttiu-tt effects occur' the which in situation physical t-"t,lr insertion made in one or t;; li;;;ot rig. t. Wt is inserted into a part self-energy . ivh".t of the and M, beexpressionsa"rir"a i., the mJnner for example the line giiph G' of a line photon of which ; previous section from $" c;il 1"i:C' i.s"i"a 13in Fig. 1, then-the modification produced for Suppose 3. 1 and rrgs. 1n shown Darts are bv a function l(W"tz) i" ,tlr rnov be ogiin d-escrib^ed defrniteness that the 5r" ru'u"il"J ri'i, oi'i"t"rn"t if the 13 line in G i;i;;"";;1-G' contribute (22) \twill .specificallv' ,11 by replacing a line of G; then according to from is obtained .lf iniernal, is gv." to that sim,ilar *gr;ent a factor Sr(rr) in az. now factor Da(13) by f".al"g t" iZO;, it.t.t be shivn that M'may (32) bv so(t') replacing M by frlom DFUz)Tr(w',szSDr(t3). t" ouiuin.j a If the ,3 line is external, the replacement is of factor ,4*(13)by (33) Ap(t3)Tr(Wt, tz)DF(13).

5.(1r)I/(t1)Sr(11) f _ : S r ( / r ) 2 q J dp l t x ( * i t , l P , * / o ' )

- *o)r^l

6+(1')'sr(r1). (28) x a+((2+,1)'?+ Ko?) G by In the same way, if G' were obtained from then part tr/' self-energy any line li the in irr".titg *ouid b" obtained from M by replacing Sr(") by l.|,1,

(2e)

S"(l'))(W, 11)Sr(/'),

W and tl where ) is an operator dependent only on rvere an line 11 if the Moreover, G. on not and line of G, then ,41'would be obtained from ":tt".not i !+,

i' t'. ----rr- Pt ":t]->--tr --"-

4---^---.-a'

Frc. 3.

--'\

In addition to terms of the forn appear tcrns such as

(33), there will

A,(t3)t,stplItt(W' , t3)DF(t3);

(34)

g-aug.e these hor.vever are zero in consequence of the will l'31u3 in terms Similar ,4,. by satisfied .o"ai,lo" case rlro opp"". with the expression (32); in-this the extia terms can be shown to vanish in conof the equation of conservation of charge ;;;t;;." satislieclby the electron-positron lield The functions selfn ( W ; , r " 1 d e s c r i b e t h e p h e n o m e n o no f p h o t o n a n . l t h s " v l c u u m p o l r r i z a t i o n o - f- t h e f i r t t "l n . i"nrJr .:r; o f S t c t i o n Y l l l o f l . F o l l o r v i n g5 c h w t n g e r ' tr." .1o". not explicitly subtract awal' the div-ergent pt oton ..tt-"n"ig1 eff"crs from the Illl/" tr)' but o n e t s s c r l s l h r t t h e s e c f f c tl s a r e z c r o a s a c o n s e oi the gauge invariance of electrodynamics' ou""." '-it S".tiot Vff of t, it rvas shown how self-energy be systematicatly eliminated from all p".i. -tfa

299 I l+J

S

MATRIX

IN

QUANTUM

graphs, and their effects described by suitabll' modifying the functions Dp and Sr. The analysis was carried out in configuration space, ancl rvas confined to the one-electron problem. We are now in a position to extcnd this method to the whole S m a t r i x f o r m a l i s m ,w o r k i n g i n r r r o m c n t u m. p a , e . and furthermore to eliminate not only self-energy parts but also the "vertex parts" defined in the last section. Every graph G has a uniquely defined "skeleton" Go, which is the graph obtained by omitting all self-energy parts and vertex parts from G. A graph rvhich is its own skeleton is called "irreducible;" all of its vertices will be of the kind depicted in Fig. 1. From every irreducible Go, the G of which it is the skeleton can be built by inserting pieces in all possibleways at all vertices and lincs of Go; these G form a well-defined class l. The terrn "proper vertex part" must here be introduced, denoting a vertex part which is not divisible into two pieces joined only by a single line; thus a vertex part rvhich is not proper is essentially redundant, being a proper vertex part plus one or more self-energy parts. The graphs of I are then accurately enumerated by insertir.rgat some or all of the vertices of G0 a proper vertex part, and in some or all of the lines a self-energy part, these insertions being madc independently in all possible combinations. Suppose that M is a constituent of a matrix element of U(a), obtained froln Go as described in Section III. Then every graph G in f will -vield additional contributions to the same matrix elenent of U(a); the sum of all such contributions, including M, is denoted by M u- As a result of the analysis leading to (27), (29), and (32), and in r.iew of the statistical independenceof the insertions made at the different vcrtices and lines of Go, the sum Ms will be obtained frorn ,41by the following substitutions. For cvcry internal electron line of Go, :r firctor ,So(pn)of i/ is replaced by (35) s r , ( p t \: 5 , 1 p t , + s a ( p ' ) ) ( 2 ' ) s n ( p ' ) ,

ELECTRODYNAN,{ICS

w h e r e . A , u ( llr:,) i s t h e s u n t o f t h e ^ u ( V , t r , l 2 ) o v e r all proper vertex parts I,'. The matrix elements of t / ( o ) t v i l l b e c o r r c c t l y c a l c u l a t e d ,i f o r . r ei n c l u d e s contributions only frorn irreducible graphs, after n . r a k i n gi n e a c h c o n t r i b u t i o n t h e r e p l a c e m e n t s( 3 5 ) , (36),(37),(38). To calculate the operators -{u, 2 and II, it is necessarl,to ri'rite dorvn explicitll' the integrals in m o m e n t u n s p a c e ,e x a m p l e s b c i n g ( 2 6 ) a n d ( 2 8 ) , corresponding to every self-energy part W or proper vertex part I/. \\'hen considering effects of higher order than the seconcl,the parts l,tr/and I/ rvill therlselves often be reducible, containing in their intcrior self-energy and vertex parts. It will again be convenient to oilrit such reducible I/ ancl W, and to include their effects by n.raking the substitutions (35) (38) in the ir.rtegralscorresponcling to irrcducible V and W.ln this u'ay one obtains in general not explicit formulas, but integral equations, for l\u, ) and II. For example, A': a1'('\' )' II)

(39)

where 1, is an integral in which l\u, ) and fI occur e.rplicitly. Fortunatcly, the appearance of c on the right side of (39) makes it easl' 1e solve such equat i o n s b y a p r o c e s so f s u c c e s s i v es u b s l i t u t i o n , t h e forrns of Au, ), ancl II being obtained correct to order a" when values correct to order a"-r are substituted into the integrals. The functions Dr' and Sr'of (35) and (36) are the Fourier transfornts of the corresponding functions ir.r I. The interpretation of these functions in Section VIII of I can be extencled in an obvious wa1' to inclucle the operator fu. Since Q7,,1,is the charge-current 4 vector of an clectron rvithout r a d i e r i v cc o r r e c l i o n s r, y ' f u , m 2 , ' . 1 ' ei n t e r p r r t e da s a n " c f f e c l i v e . t t r r e n t " c a r r i e d b 1 , a n e l c c t r o t t .i n cluding the elTectsof exchangeinteractions bctrveen t h e e l e c t r o na n d t h e e l e c t r o n - p o s i t r o n f i e l d a r o u n c li t . An additional reduction in thc number of graphs cffectively contributing to U(o) is obtained from a theorenr of Furrl'.6'fhe thcorem lvas sholvn by u'here )(pt) is the sum of the >(tr/, p') over all Feynman to be an elcgant consequence of his electron self-energy parts I4l. For every intcrnal theory. 1n any graph G, a "closed loop" is a closcd photon line, a factor D o(pu) is replaced by clectron poll'gon, at the vertices of u'hich a number D F ' , ( p t :) D F ( p t )+ D r ( p r n ( p i ) D F ( p t ), ( 3 6 ) p of photon lines originate; the loop is called odd or $'here [(2d) is the sum of the II(i{z', pi) over all even according to the parity of p. If G contains a photon self-energy parts W'. For every external c l o . e dI o o p .t l r e n t h e r eu ' i l l b o a n o r h e lg r a p h O ' a l s o line, a factor ,1,(ht)or 0(k') or Au(kr) is replaced by contributing to I/( o ), obtained from G bv reversing the scnseof the electron lines in the loop. Norv if ,i1,1 v , @ t ) : s F ( k t ) > ( k t ) v ( h t )+ v k t ) , r n t l M r r e c o r r t r i L r r t i o n sf l o m C r n d G , M i s ( 3 7 ) cleriveclfrom M by interchanging the roles of elec0'(ht):{6'1>1kt)SF(hr+0(ht), A , ' , ( h t ): A , ( k t ) r 7 ( . h t ) Dt s ( h t+) A , ( k t ) , tron and positron states in each of the interactions respectively. For every vertex of Gs, rvhere the at the vertices of thc loop; such an interchange is incident lines carry momentunl variables as shown c a l l e d " c h a r g e c o n j u g a t i o n . " I t w a s s h o w ' n b y Schs'ingcr that his theory is invariant under charge in Fig. 1, an operator 7u is replaceciby I p(tt , t2) : 7 u!

Lu(.tr, t2) ,

(38)

6 \\IenclcllH. Furry, Phys. Rev. 5t, 1,25(1937)

300 1744

F. J. DYSON conjugation, provided that the sign of a is at the same time reversed (this is the well-known charge symmetry of the Dirac hole theory). It is clear from (8) that the constant, appears once in M for each of the p loop vertices at which there is a photon line; at the remaining vertices only the constant 6zzis involved, and 6ru is an even function of e. Therefore the principle of charge-symmetry implies (40) M:(-r)eM.

example (26) and (28)) ; in thesecasesitis legitimate to replace each 6a function by a reciprocal, making a separate detour in the po integration for each pole in the integrand, provided that no two poles coincide. Thus every constituent part M of U( o ) can be written as an integral of a rational algebraic function of momentum variables, by using instead of (21) and (22) 1 DF(Pi):-, 2ri(pt)2

(44)

Taking p odd in (40) gives Furry's theorem; all ( i P u ' lu - ^ o ) contributions to U(o) from graphs with one or (4s) Jf (r') --more odd closed loops vanish identically. 2 r i((p')2I ro2) By an "odd part" of a graph is meant any part, consisting only of vertices and internal lines, which This representation of D r and Sr as rational functouches no electron lines, and only an odd number tions in momentum-space has been developed and graph. of photon lines, belonging to the rest of the extensivell, used by Feynman (unpublished). The sirnplest type of odd part which can occur is a There may appear in ,4.{infinities of three distinct single odd closed loop. Conversely, it is easy to see kinds. These are (i) singularities caused by the that every odd part must include within itself at coincidence of two or more poles of the integrand, least one odd closed loop. Therefore, Furry's (ii) divergences at small momenta caused by a parts be graphs to all with odd allows theorem factor (44) in the integrand, (iii) divergences at Lr(o). omitted from consideration in calculating large momenta due to insu{iciently rapid decrease of the whole integrand at infinitY. THE DTIIERGENCES IN OF' INVESTIGATION V. In this paper no attempt ivill be made to explore S MATRIX the singularities of type (i). Such singularities occur, The 61 function defined by (19) has the propertl' for e-rample. when a many-particle scattering that, if b is real and /(o) is any function anall'tic process may for special values of the particle in the neighborhood of b, then momenta be divided into independent processes involving separategroups of particles' It is probable f f O ) a - ( o _D a a : ( / z n 'i J)[ -J r a ) ( t r t-ab ) ) d a , G t )that all singularities of type (i) have a similarly J"" clear physical meaning; thesesingularities have long energy dewhere the first integral is along a stretch of the real been known in the form of vanishing perturbation theory, and axis including b, and the second integral is along nominators in ordinary the same stretch of the real axis but with a small have never caused any serious trouble. A divergence of tf'pe (ii) is the so-called "infradetour into the con.rplexplane passing underneath to be caused D. In the matrix elements of I/(o) there appear red catastrophe," and is well knor,vn by the failure of an expansion in powers of e to integrals of the forn describe correctly the radiation of low moment"un.t quanta. It would presumably be possible to elimidPF(P)6*(P," + c'), (42) nate this divergence fron.r the theory by a suitable +P,'+Pt"- Po" I adaptation of the standard Bloch-NordsieckTtreatintegrated over all real values of pr Pz, Ps, Po. B,r ment; we shall not do this here. From a practical (41), one may write (42) in the form point of view, one may avoid the difhculty by arbitrarily writing instead of (44)

*,1,

F(p)

- Po'+c') ( pt'+ pr'+ P32

(43)

in which it is understood that the integration is alorrg the real axis for the variables pr, Pz,Pz, and for 2e is along the real axis with two small detours, one p a s s i n ga b o v e t h e p o i n t + t p 1 r + P " 2 + p , 2 - 1 2 r l ,a t t d o n e p a s s i n gb e l o r v t h e p o i r ] t - ( P t " * p r ' + p t ' + r ' ) t . To equate (42) rvith (43) is certainly correct, when F(p) is analytic at the critical values of po. In practice one has to deal with integrals (42) in which F(2) itself involves d1 functions (see for

Dr(D):

2ri((pt)'z*)()

where X is some non-zero momentum, smaller than an-v of the quantum momenta which are significant in the particular process under discussion.E -t p . 6 1 o " ra. n d . { .N o r d s i e cpkh, y s .R e v . 5 2 , 5(41 9 3 7 r .

3 The device of introducing I in order fo avoid lnlrd-red divergences must be used with circumspection. Schwinger lunpu"bli'hed' has shown that a iong -sranding discre-pancy be$vecn trvo alternative crlculations ol rhe Lamb shllt w3q due to ca.eiess use of tr in one of them

30r

!

I

i

t74s

S MATRIX

IN

QUANTUM

It is the divergences of type (iii) which have always been the main obstacle to the construction of a consistent quantum electrodynamics, and which it is the purpose of the present theory to eliminate. In the following pages, attention will be confined to type (iii) divergences; when the word "convergent" is used, the proviso "except for possible singularities of types (i) and (ii)" should always be understood. A divergent M is called "primiti.ve" if, whenever one of the momentum 4 vectors in its inteerand is held fixed, the integration over the remaininq variables is convergent. Correspondingly, a primitive divergent graph is a connected graph G giving rise to divergent M,bfi such that, if any internal line is removed and replaced by two external lines. the modified G gives convergent M. To analyze the divergences of the theory, it is sufficient to enumerate the primitive divergent M and G and to examine their properties. Let G be a primitive divergent graph, with z vertices, .E external and F internal lines. A corresponding ,41will be an integral over F variable pi of a product of F factors (44) and (45) and n factors (23). Since G is connected, the d-functions (23) in the integrand enable (z-1) of the variables pi to be expressed in terms of the remaining (F-n-tI) pi and the constants &t, leaving one o'-function involving the frt only and expressing conservation of momentum and energy for the rvhole system. An example of such integration over the 6-functions was the derivation of (26) from 25). After this, the integrations in M may be arranged as folto-ws; the-fourth components of the F-nll) independent pi are written

ELtrCTRODYNAMICS defined in (47). In view of (43), we take the integration variables in (48) to be real variables, with the exception of a which is to be integrated along a contour C deviating from the real axis at each of the 2F poles of R. As a general rule, C will detour above the real axis for c)0, and below it for a(0; the reverse will only occur at certain of the poles corresponding ro denominators (49) for which (p't)'z+(prt)'+(pto)'*p,{.(cu)t.

(51)

Such poles will be called "displaced." The integration over a alone will always be absolutely convergent. Therefore the contour C may be rotated in a counter-clockwise direction until it lies alone the imaginary axis, and the value of 11 will be unchanged except for residues at the displaced poles. Regarded as a function of the parameters &i describing the incoming and outgoing particles, i4 will have a complicated behavior; the behavior will change abruptly whenever one of the ct has a critical value for which (51) begins to be soluble, for pti, Pzi, Pti, and a new displaced pole comes into existence. This behavior is explained by observing that displaced poles appear whenever there is sufficient energy available for one of the virtual Darticles involved in M to be actually emittecl is a real particle. It is to be expected that the behavior of rl1 should change when the processdescribed by M begins to be in competition rvith other real processes. It is a feature of standard perturbation theory, that when a process .4 involves an intermediate state ,I which is variable over a continuous range, and in this range occurs a state 11 which is the final state of a competing process, then the matrix element for .4. involves an integral over .1 which has a singularity at the position 11. In p qi : ipsi :,iar ni, (47) standard perturbation theory, this improper inte:nd the integration over a is performed first; sub- gral is always to be evaluated as a Cauchv prin:3quently, integration is carried out over the c i p a l v a l u e , a n d d o e s n o t i n t r o d u c e u n y r " r i d i . , " r i n d e p e n d e n t p r i , p r i , p " i , a n d o v e r t h e gence into the matrix element. In the theory of the i(F-nll) present paper, the displaced poles give rise to ratios of the r6t. ,LI then has the form l-z) similar improper integrals; these come under the heading of singularities of type (i) and will not be ff* M : I d p t ; d p z t d p t i d r o,i R a F - " d q , ( 4 9 ) discussedfurther. u J-If pri, Pz;, p:t satisfying (51) are held fixed, then :'here R is a rational function of a, the denominator the value of p4i at the corresponding displaced pole is fixed by (50). The contribution to M from the ': rvhich is a product of F factors displaced pole is just the expression obtained by (Prt)'l (P"t)'I (Pti)2* p2- (oro;lci)2. (49) holding the 4-vector 2t fixed in the original integral Il[, apart from bounded factors ; since ,41is primitive :lcre the constants roi, ci are defined by the con- divergent, this expression is convergent. The total ::tior that contribution to M from the I'th disolaced oole is pj:.ipni:i(arsilci), j : 1 , 2 , . . . , F . ( 5 0 ) the integral of this expressionover th; finite sphere (51) and is therefore finite. Strictly speaking, this lius the ct corresponding to the (F-af 1) inde- argument requires not only the convergence of the :cndent pd are zero by (47), and the remainder are expression, but uniform convergence in a finite :ear combinations of the ki; also (n-l) of the 16, region; however, it will be seen that the convergent :-e linear combinations of the independent ret integrals in this theory are convergent for large

302 t746

F. J. DYSON momenta by virtue of a sulllcient preponderanceof large denominators, and convergence produced in thii way will ahvays be uniform in a finite region' ,41is thus, apart from finite parts, equal to the ,z;a in (48) and integral ,11'obtaincd b1' replacing a by (49). Alternatively, M'is obtained from the original by substituting for each pot integral ,i1,1 iPtil (1-i)ci,

(52)

i n c l e p e n d c n t1 u ' , anclthen treating the 4(F-z*1) p : 1 , 2 , 3 , 4 , a s o r d i n a r y r e a l v a r i a b l e s 'I n ' t I ' t h e denominators of the integrand take the form (.pl)' + (p,i)' + (! {)" t p' t (Pni- (l I i15;'12, (53) and eire uniformll' large for large values of p"d' The convergenceof ,il/'can now be cstimated simpll' by counting powers of prt in numerator and denominator of the integrand. Since M' is knol'n to converge whenever one of the 2iis hcld fixed and integration is carried out over thc others, the con,,".g"r." of the whole expression is assured provided that K:2F-F"-4lF-nl1l)1.

(s4)

Here 2F is the degree of the dcnominator, and F" that of the nul.nerator, rvhich is by (44) and (45) equal to the number of intern:rl electron lines in G. Let E, and Eo be the numbers of cxternal electron and photon lincs in G, and lct z, be the number of verti;es rvithout photon lines incident. It follorvs from the structure of G that

light by light" or the mutual scattering of two photons. Further, (55) shows that the divergence ivill. never be more than logarithmic in the third and fourth cases,more than linear in the 6rst, or more than quadratic in the second.Thus it appears that, horvever far quantum electrodynamics is clevelopedin the discussionof many-particle interactions and higher order phenomena, no essentially ner- kinds of divergence u'ill be encountered. This gives strong support to the view that "subtraction ph-vsics,"of the kind used by Schwinger and Feynnran, will be enough to make quantum electrod1'namics into a consisterlt theor)" IN THE !'I, SEPARATIONOF DNTERGENCES S MATRTX First it ivill be shown that the "scattering of light by light" does not in fact introduce any clivergence into the theorl'. The possible primitive divergent , l l 1i n t h e c a s eE " : 0 , E o : 4 r v i l l b e o f t h e f o r r n 6(k1+ k, + k3+ k1)A t.(h1)A Jk') A, (k3)A o(ha)I xp o, (56) rvhere 1;r,, is an integral of the t-vPe

I

n^,,,{u',k', h3,h4,Pt)d'P",

(s7)

at most logarithmically divergent, and R is a certain rational function of thc constant kt and thc variable p t . I n a n y p h y s i c a ls i t u a t i o n l ' h e r e , f o r e x a m p l e ,t h e A(k) are the potentials corresponding to particular incident and outgoing photons, there l'ill appear i n U ( o ) a m a t r i x e l e m e n tr v h i c h i s t h e s u r n o f ( 5 6 ) 2F:3n- n"- E"- Ep, and the 23 similar expressions obtained by per2 uP' muting the suffrxesof /1u,oin all possible rval's lt and so the convergencecondition (52) is may therefore be supposed that at the start R1u,, ( 5 5 ) has been symmetrizedby summation over.all perK:+E"+Ee+n"-4)r. mutations of suflixes; (56) is then a sum of conThis gives the vital inforrnation that tl.re only tributions lron 24 or ferver (according to the possible primitive divergent graphs arc those with clegreeof s)'mmetry existing) graphs G. E":2, Ep--j,1, and rvith E":0, Ee:l' 2,3' 4. i f , u n c l e rt h e s i g n o f i n t e g r a t i o ni n ( 5 7 ) , t h e v a l r t e i s s u b t r a c t e df r o r n F u r t h e r , t h e c a s e sE " : 0 , E p : 1 ' 3 , d o n o t a r i s e , R ( 0 ) o f R l o r h t : k 2 : h 3 : b a - 0 -t since these givc graphs rvith odcl parts l'hich x'ere R, the intcgrancl acquires one extr:r porver of I prt l shog,n to be harmiess in Section l\''. It should be for large I f"rl , and the integral becomesabsolutel-v observed that the course of the argument has been convergent at infinitlr' Therefore "if E" and -Eoclo not have certain small values' then (58) Ixr"o: Ixu,r(A)l Jx,o, the integral ,4f is convergent at infinit-v;" there is integrations no objection to changing the order of rvhere /(0) is a possibly divcrgent integral indein ,lf as rvas done in (48), since thc argument pendent of the Ei, and .I is a convergent integral r e q u i r e st h a t t h i s b e d o n e o n l J ' i n c a s e sl ' h e n M i s ' vanishing rvhen all fri's are zcro. To interpret this in fact, absolutelY conr.'ergent. result phl'sicalll', it is cotrvenient to \\'rite (56) The possible prinitive ciivergent graphs that a g a i n i n t e r m s o f s p a c e - t i m ev a r i a b l e s ; t h i s g i v e s have been found arc all of a kind {aniliar to p h y s i c i s t s .T h e c a s e E " : 2 , E p : 0 c l e s c r i b e s - s e l f , (5q) uf t ^ ' , " r o s e ^ 1 . r , t , 4 * ( x ) , 4 , r r ) r' 4r ) c 1 r r I y ' tnergy effects of a single electron; E"--0, Ee:2 J s e l f - e n e r g ye f f e c t so f a s i n g l e p h o t o n ; E " : 2 , E e : l involving dethe scattering of a single electron in an electromag- rvhere N is a convergent expression respect to space and $'ith ,4(t) of the rivatives o f " s c a t t e r i n g E p : 4 t h e E , : 0 , a n d l i e l d ; netic

303 1747

S MATRIX

IN

QUANTUM

time. Now the first term in (59) is physically inadmissable; it is not gauge-invariant, and implies for example a scattering of light by an electric field depending on the absolute magnitude of the scalar potential, whicl.rhas no physical meaning. Therefore 1(0) must vanish identically, and the whole expression (56) is convergent The fact that the scattering of light by light is finite in the lowest order in which it occurs has long been known.e It has also been verified by Feynman by direct calculation, using his own theory as described in this paper. The graphs which give rise to the lowest order scattering are shorvn in Fig. 4. It is found that the divergent parts of the corresponding ,44 exactly cancel when the three contributions are added, or, what comes to the same thing, when the function Rrp,, is symmetrized. It is probable that the absence of divergence in the scattering of light by light is in all cases due to a similar cancellation, and it should not be diffrcult to prove this by calculation and thus avoid making an appeal to gauge-invariance. The three remaining types of primitive divergent M are, in fact, divergent. Horvever, these are just the expressionsrvhich have been studied in Sections III and IV and shorvn to be completely described by the operators Ar, ), and II. Nllore specifically, when E":2, Ep:g, M will be of the form

'p(h')>(w,hr)v@\,

(60)

where trZ is some electron self-energy part of a g r e P h .W h e r rE " : 0 . E p : 2 , M w i l l b e A,(kt)r(w"

hr)AP&t)'

(61)

with W' some photon self-energypart. When E":2, Ep:1, tuI will be {(kt)A,p(V, bl, k'z),|,(k'?)A/ht-k'z),

(62)

with tr/ solne vertex part. Therefore, if some means can be found for isolating and removing the divergent parts frorr Au, ), and fI, the "irreducible" graphs defined in Section lV will not introduce any fresh divergences into the theory, and the rules of Section IV will lead to a divergence-free S matrix. ), and fI in Section IV it was In considering -A.u, found convenient to divide vertex and self-energy parts thernselves into the categories reducible and irreducible. An irreducible self-energy part W is required not only to have no vertex and self-energy parts inside itself; it is also required to be "proper," that is to sa1', it is not to be divisible into two eH. Euler and B. Kockel, Naturwiss. 23, 246 (1935); H. Euler, Ann. d. Phys. 26, 398 (1936). In these early calculations of the scattering of light by light, the theory used is the Heisenberg electrodynamics, in which certain singularities are eliminated at the start by a procedure involving non-diagonal elements of the Dirac density matrix. In Feynman's calculation. on the other hand. a fioite result is obtained without subtractions of any kind,

ELECTRODYNAN{ICS pieces joined by a single line. In Section IV it was shown that to avoid redundancy the operator .{, should be defined as a sum over proper vertex parts I/ only. By the same argument, in order to make (35), (36), (37) correct, it is essential to define 2 and fI as sums over both proper and improper self-energy parts. However, it is possible to define Sr'and D7l in terms of proper self-energy parts only, at the cost of replacing the explicit definitions (35), (36) by implicit definitions. Let >*(pt) be defined as the sum of the 2(W, pi) over proper electron self-energy parts W, and let II*(pi) be defined similarly. Every W is either proper, or else it is a proper W joined by a single electron line to another self-energl' part which may be proper or improper. Therefore, using (35), Sr' rnay be expressed in the two equivalent forms S F '( p t ) : S F ( p t )+ S F ( p t ) z +( p t )S F '( p t ) : sr(pu) * s.,(2 t12+(.pt)sr(pt).

(63)

Similarly, D e'(pt): D e(pt)+D F(pt)il*(pr)DF'(pu) : D F ( p t )+ D F ' ( p t ) n * ( p t )D F ( p t ) . ( 6 4 ) It is sometimes convenient to work with the ) and II in the starred form, and sometimes in the unstarred form. Consider the contribution >(W, tt) to the operator )*, arising from an electron self-energy part W. It is supposed that W is irreducible, and the effects of possible insertions of self-energy and vertex parts inside I,/ are for the time being neglected. Also it is supposed that W is not a single point, of which the contribution is given by (31). Then I4l has an even number 21.of vertices, at each of which a photon line is incident; and >(W,fr) will be of the form

p;1ap,, e,'f n1t,,

(6s)

where R is a certain rational function of the lr and pt, and the integral is at most linearly divergent. The integrand in (65) is now written in the form R(t', pu): R(o, pu)

/aR \ +r-,(--(0,D l*n"1t,p,7, (66) t dlr,

t\r'

ri

''TtJ'

'i

k:i \{

\

r*i:" l*1

tY /L<-{

i''*

,

;ki

k"\

ib

ui\, ir Frc. 4.

L4\\

304 1748

F. J. DYSON and for large values of the | 2"il the remainder term R" will tend to zero more rapidly by two powers of lputl than R. Therefore, in complete analogy with (58),

2(w, t'1:

",t7a+B&Fl+2"(W'

tl)f'

(67)

where .1 and.B, ate constant divergent operators, and Z"(W,lr) is defined by a covariant and absolutely convergent integral. >"(W,t') must, on grounds of covariance, be of the form R1((tt)2)lRz((tt)'z)tr1rr

(68)

with Rr and Rz particular functions of (tl)'?;for the same reason, .8u must be of the form 87, with B a certain divergent integral. Now if 11happens to be the momentum-energy 4 vector of a free electron, (tt)':

- ro',

tutyr:ixo.

(69)

It is convenient to write >"(W, t 1: a'18'(t,17,-ixn) + (tpti F- iio) S(W, tt),

(70)

where .S(l/, tr) is zero for lr satisfying (69), and to include the first two terms in the constants,4 and B of. (67).; since all terms in (70) are finite, the separation of S(W,11) is without ambiguity. Thus an equation of the form (67) is obtained, with (71) tt) : (t,\y,- ixo)S(W, tt). "(W, Summing (67) over all irreducible llz and including (31), gives for the operator 2*, >

>* (t') : A - 2ri 6xnl B (t*1''o- i *o) (7 2) a Q,ty,-4ro)S "(t1). Hence by (63) and (a5)

and derive instead of (67) tr) l. (74) t1) : e2tlA + B i p1+ C,,tuLt,L+ n "(w " " The A, Bu, C' are absolute constanc numbers (not Dirac operators) and therefore covariance requires that Br:9, Cu,:C5,,. n.(Wt'tr) is defined by an absolutely convergent integral, and will be an invariant function of (lr)2 of a form n (w

TI"(w' ' tt) - (tt)'D(W"

t1),

(7s)

where D(W' , tL) is zero for 11satisfying

(76)

(r')':0

instead of (69). Summing (74) over all irreducible I4l"s will give 11*11t): Atl

C(tt),+ (tt)rD

"(tt),

(77)

and hence by (64) and (44) 1 D r' (t1 : a' p rTtL)DF' (tr) +-.CD ZTx

F' (r) 1

D r',(tt). + D FQt)+-D 2ni "(tt)

(78)

In (77) and (78), D is zero for lr satisfying (76), and " ,-'' is divergence free. The constant A' in (77) is the quadratically divergent photon self-energy. It *'ill give rise to matrix elements in t/( o ) of the form f

M:A' J

I A,(x)Au(x)ilx,

(79)

which are non-gauge invariant and inadmissable' Such matrix elements must be eliminated from the S F '( t 1 ): ( A - 2 r i 6 r ) S r ( l t ) S r ' Q r ) theory, as the first term of (59) was eliminated' by the statement that A'is zero. The verification of 11 +-Bsr'(rl) +so11)a-s"(/r)Sr'(t1). (73) this statement, by direct calculation of the lowest 2r 21 to A', has been given by order contribution Schwinger.s'ro ln (72) and (73), ,4 and B are infinite constants, The separation of the divergent part of.Au again and S".a divergence-free operator which is zero follows the lines laid down for )x. Since the integral when (69) holds ; ,4, B, an&S" are power series in e analogous to (65) is now only logarithmically starting with a term in e2. In (72) and (73), howdivergent, no derivative term is required in (66)' ever, effects of higher order corrections to the and the analog of (67) is > ( W , t ' ) t h e m s e l v e sa r e n o t y e t i n c l u d e d . parts may be (80) A similar separation of divergent Lu(V, tL, t2): e2rlLu* L,"(V, f , t')), made for the lI(W',11), when W' is an irreducible rlp" and photon self-energy part. The integtal (65) may now where Zu is a constant divergent operator, be quadratically divergent, and so it is necessary to is convergent and zero for tr:tz:O. In (80)' Zu can only be of the form L7u. Also,if tt:t2 and lr satisfies u s e i n s t e a do f ( 6 6 ) (69), (Lr.will reduce to a finite multiple of 1u which / a-_(0, R \ can be included in the term L1u. Therefore it may p) pn): R(0, RU, P) *'u'( | be supposed that 4," in (80) is zero not for lr : 12:0 \ al,r / Uut fbi l':p satisfying (69). The meaning of this

pr, ++,1,;(#,(0,p,l) +n"(r',

10Greeor Wentzel. Phvs. Rev. 74, 1070 (1948)' presentsthe ese agaTnstSchwinger'i treatment of the photon self-energy'

305 1749

S MATRIX

]N

QUANTUM

physically is that.{r" now gives zero contribution to the energy of a single electron in a constant electromagnetic potential, so that the whole measured static charge on an electron is included in the term Z7r. Summing (80) over all irreducible vertex parts tr/, and using (38), ^p(tt , t2): Lt ul L,"(tt , t2), ly(t1, t2): (1lL)7,1Lu"Qt,

(81) t )

(82)

In (81) and (82), effects of higher order corrections to the tru(I/, t\,t2) are again not yet included. Formally, (82) differs from (73) and (78) in not containing the unknown operator I" on both sides of the equation. VII. REMOVAL OF DIVERGENCESFROM THE S MATRD( The task remaining is to complete the formulas (73), (78), and (82), which show how the infinite parts can be separated from the operators f", Su', and Dp', and to include the corrections introduced into these operators by the radiative reactions which they themselves describe. In other words, we have to include radiative corrections to radiative corrections, and renormalizations of renormalizations, and so on ad inf,n'itum, This task is not so formidable as it appears. First, we observe that -4.r,)x, and II* are defined by integral cquations of the form (39), which rvill be referred to in the following pagesas "the integral equations." \{ore specilically, consider the contribution Lp(V,tt, r'?) to ,,l.! represented bV (80), arising from a vertex part Ir with (21f 1) vertices, I photon lines, and 2l electron lines. This contribution is defined by an integral analogous to (65), with an integrand which is a product of (211-1) operators 7u, I functions Dr, and 2l operators Sp. The exact A,p(V,tL,t'?)is to be obtained by replacing these factors, respectively, by lr, Dr', Sr', as described in Section IV. Now suppose that Sr/ in the integrand is represented, to order e2^ say, by the sum of Sr and of a finite number of finite products of Sr with absolutely convergent operators S(W, tt) such as appear in (71); similarly, let Dp' be represented by Dp plus a 6nite sum of finite p r o d u c t so f D r w i t h f u n c t i o n sD t W ' , t t ) a p p e a r i n g in (75); and let fu be represented by the sum of 7, and of a finite set of nr"(I, t|,t2) from (80). Then the integral Lp(V,tt,l'!) will be determined to order s i n c et h e o p e r t l o r sS t f i , t ' , . D l l T ' , 1 t 1 , e2n-?/:and ^u,(V,tt,1:) always have a sulficiency of dcnominators for convergence,the tl.reoryof Section V can be applied to prove that this Lp(V,tl,tz) wlll not be more than logarithmically divergent. Therefore the new ,,\.u(l/,tt,t2') can be again separated into the forn (80). The sum of these l\/V,tt,t) will then be a ll.u(11, l':) of the form (81), with con-

ELECTRODYNAMICS qL

a! s__--->-, -\-4-l-

L'

Frc. 5.

stant I and convergent operator,4.r, determined to order e2"*2.Thus (82) provides a new expressionfo l, determined to order e2"+2. The above procedure describes the general method for separating out the finite part from the contribution to Iu arising from a reducible vertex part Vp. First, 116 is broken down into an irred_uciblevertex part T/pl us various i nserted pa r ts'W, W', V; the contribution to lu from tr/ais an integral M(Vn) which is not only divergent as a rvhole, but also diverges when integrated over the variables belonging to one of the insertions W, W', t, the remaining variables being held fixed. The divergencesare to be removed from M(Vn) in succession, beginning with those arising from the inserted parts, and ending with those arising from Iz itself. This successiveremoval of divergences is a welld e f i n e dp r o c e d u r e ,b e c a u s ea n y t l r o o f t h e i n s e r r i o n s made in Z are either completely non-overlapping or else arranged so that one is completely contained in the other. In calculating the contribution to )* or II* from reducible self-energy parts, additional complications arise. There is in fact only one irreducible photon self-energy part, the one denoted by W'in Fig. 5; and there is, besides the self-energy part consisting of a single point, just one irreducible electron self-energy part, denoted by trZ in Fig. 5. All other self-energy parts may be obtained by making various insertions in trZ or W'. However, reducible self-energy parts are to be enumerated by inserting vertex parts at only one, and not both, 'W of the vertices of or IZ'; otherwise the same self-energy part would appear more than once in the enumeration. And the contribution M(Wn) to )* arising from a reducible part Wn will be, in general, an integral which involves simultaneously divergences corresponding to each of the ways in ivhich I4ln might have been built up by insertions of vertex parts at either or both vertices of trV.This complication arises because, in the special case when two vertex parts are both contained in a self-energy part and each contains one end-vertex of the self-energy part (and in no other case), it is possible for the two vertex parts to overlap without either being completely contained in the other. The finite part of It(Wn) is to be separated out as follows. In a un.ique way, Wn is obtained from l 4 l b y i n s e r t i n ga v e r t e x p a r t 7 . a r a , a n d s e J f - e n e r g y parts 17" and W"' in the two lines of trU. From M(Wn) there are subtracted all divergences arising l r o n t / " , W " , W i : l e t -t h e r e m a i n d e r a f t e r t h i s s u b -

306 F.

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1750

DYSON

Next, l7n is considered as traction be M'(Wil. built up f.rom W by inserting some vertex part Vt at b, and self-energy parts W6 and Wt' in the two lines of W. The integral M'(Wa) will still contain arising from 7a (but none from ffa and divergences 'Wt'), and these divergences are to be subtracted, leaving a remainder M"(Wa). The finite part of M" (Wn) can finally be separated by applying to the whole integral the method of Section VI, which gives for M"(Wa) an expression of the form (67), with )" given by (71). Therefore the finite part of M(Wd is a well-determined quantity, and is an operator of the form (71). The behavior of the higher order contributions to )* and II* having now been qualitatively explained, we may describe the precise rules for the calculation of 2* and II* by the same kind of inductive scheme as was given for.{u in the second paragraph of this Section. Apart from the constant term (-2r'i6xo), 2* is just the contribution >(W, tr) from the trZ of Fig. 5; and >(W,t') is represented by an integral of the form (65) with l:1. The integrand in (65) was a product of two operators ?ts,one operator Dr, and one operator Sr. The exact 2(W,11) is to be obtained by replacing Dr by Dr', Sr by Sr', and one only of the factors 7, by lu, say the 7u corresponding to the vertex a of W. Suppose that Srl in the integrand is represented, to order ez",by the sum qf Sr and of a finite number of finite products of Sr with operators S(W,tt) such as appear in (71);and suppose that De'and I, are similarly represented. Then >(IZ, 11) will be determined to order e2"*2. The new Z(W,|L) will be a sum of integrals like the M'(Wn) of the previous paragraph, still containing divergences arising from vertex parts at the vertex b of. W, in addition to divergences arising from the graph IUp as a whole. When all these divergences are dropped, we have a >"(W, tt) which is finite; substituting this >"(W, tt) for )* iri (63) gives an Sr'which is also finite and determined to the order 42"+2. The above procedures start from given Sp', Dp' and Iu represented to order e2" by, respectively, Sr plus Sp multiplied by a finite sum of products of S(fr,tt), Dr plus Dr multiplied by a finite sum of products of D(W',/'), and 7, plus a finite sum of Lr"(V,tr,l'z). From these there are obtained new expressions for St' , D r' , fu. In the new expressions there appear new convergent operators S(W,tr), D(W', tr), l\r"(V, tr,l'z), determined to order e2"+2i in the divergent terms which are separated out and dropped from the new expressions, there appear divergent coefficients A, B, C, Z, such as occur in (73), (78), (82), also now determined to order e2"+2. After the dropping of the divergent terms, the new l" by (82) is a sum of 7" and a finite set of LN(V,tt,l'); the new Sr' bV (79) is Sr plus '5r multiplied by a finite sum of products of S(trf' tr);

and the new Dr'by (78) is Dp plus Dp multiplied by a finite sum of products of D(W',lr). That is to say, the nerv lu, Sr', Do'can be substituted back into the integrals of the form (65), and so a third set of operators ly, Sp', D p' is obtained, determined to orderuzn+a, and again with finite and divergent parts separated. In this way, always dropping the divergent terrns before substituting back into the integral equations, the finite parts of lu, Sr', Dr', may be calculated by a process of successiveapproximation, starting with the zero-order values 7", Sr, Dr. After n substitutions, the finite parts of l, S v' , D v' will be determined to order e2". It is necessary finally to justify the dropping of the divergent terms. This will be done by showing that the "true" Iu, Sr', Dr', which are obtained if the divergent terms are not dropped, are only numerical multiples of those obtained by dropping divergences, and that the numerical multiples can themselves be eliminated from the theory by a consistent use of the ideas of mass and charge renormalization. Let l"r(e), .Srr'(e), Dn'(e) be the operators obtained by the process of substitution dropping divergent terms; these operators are power series in e with finite operator coefficients (to avoid raising the question of the conver$ence of these power-series, all quantities are supposed defined only up to some finite order a2x)' Then we shall show that the true operators fu, Sr', Dr' are of the form lu:Zatlut(et)

(83)

Sr':ZrSrt'(et)

(84)

Dr':ZrD"r'(er),

(85)

where Zr Zz, Zsare constants to be determined, and a1is given by (86) er: ZtrZzZs\e' This er will turn out to be the "true" electronic charge. It has to be proved that the result of substituting (83), (84), (85) into the integral equations So', Drl, is to reproduce these exdefining I, pressions exactly, when Zu Zz, Za, and 6xo are suitably chosen. Concerning the I a@) , S rt' (e), D4'(a), it is known that, when these operators are substituted into the integral equations, they reproduce themselves with the addition of certain divergent terms. The additional divergent terms consist partly of the terms involving A, B, C, Z, which are displayed in (73), (7S), (82), and partly of terms arising (in the case of Sr' and D p' only) from the peculiar behavior of the vertices b, b' in Fig. 5. The terms arising from b and, b' have been discussed earlier; they may be called for brevity b-divergences. Originally, of course, there is no asymmetry between the divergences arising in )* from vertex parts inserted at

307 775r

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the two ends o and b of W: we have manufactured an asymmetry by including the divergencesarising at a in the coefficient Zr I of (83), rvhile at b the operator 7, has not been replaced by l, and so the D divergenceshave not been so absorbed. It is thus to be expected that the effect of the 6 divergences, like that of the n divergences, will be merely to multiply all contributions to )* by the constant Z1-1. Similarly, we expect that divergences at b' rvill multiply II* by the constant Z;r. It can be shown, by a detailed argument too long to be given here, that these cxpectations are justified. (The interested reader is recommended to see for hirnself, by considering contributions to 2* arising from various self-energy parts, how it is that the finite terms of a given order are always reappearing in higher order multiplied by the same divergent coef6cients.) Therefore, the complete expressions obtained by substituting fur(e), Srr'(e), Det'(e), into the integral equations defining ,{u, )*, [*, are

Therefore the substitution gives

-2n;6xrSr

Z rZ z*le12, and the remaining factor of >o(trU) i. explicitll' a function of e1and not of e. Therefore (90) is Z122-t2{W, e),

Sr)* +z,-,(.ne)s,+!; a* !; "r,>),(s8)

D rrIJ (e): r,' (L c fA* l;

(90)

Z.tZzZ3>o(W,

where )n(I4l) is the expression (65) obtained by s u b s t i t u t i n g f u 1 ( € r ) , . S r 1 ' ( e 1, )D t r ' ( e r ) , w i t h o u t t h e Z factors. Norv the Z factors in (90) combine with the e2of (65) to give

where )1(trU, e) is the expression obtained by substituting the operators fu1(e), Srr'(a), Dpt'(e) into >(W, t').Thus the )+(tl), obtained by substituting from (83)-(85) into (65), is identical with the result o f s u b s t i t u t i n g t h e o p e r a t o r sl u 1 ( e ) ,S 4 ' ( e ) , D r t ' ( e ) , and afterwards changing e to er and multiplying the whole expression (except for the constant term (87) in 616)by Z1Z2'r. More exactly, using (88), one can say that the )* obtained by substituting from (83)-(85) is given by

ttu1(e) : L u"@) { L (e) 7,, Sp)1x(a) :

ELECTRODYNAMICS

"A)

(8e)

Here A(e), B(e), C(e), L(e) are well-defined power series in e, with coefficients which diverge never more strongly than as a power of a logarithm. The f i n i t e o p e r a t o r s A u " ( e ) ,S " ( e ) , D " ( e ) , w i l l , w h e n a l l divergent terms are dropped, lead back to the f u 1 ( a ) ,S p 1 ' ( e ) ,D " t ' ( e ) , f r o m w h i c h t h e s u b s t i t u t i o n s t a r t e d ; t h u s , a c c o r d i n gt o ( 3 8 ) , ( 6 3 ) , ( 6 4 ) , I ur(e): t ul h*(e) ,

7 Sr''(e): Sr* ^ S"(e)Sp''(e).

(87',)

: - 2ri6roSr

/l1r + Z z - ' l A ( e ) S r * - B ( e , ) * - & ( ? r )l . ( 9 1 ) \212n/

Further, the Sr' obtained by substituting from (83)-(85) into the integral equations is given by (91) and (92)

Sr':Sr*Sr)*Se'.

It is now easy to verify, using (88'), that Sr'given bv (91) and (92) will be identical with (84), provided that 1 Zz:I|-B(e,),

(93)

ZT

(88)

ZT

1 6ro:

Z'-tO"''

'

(94)

2r'i

D rl (e): D oa-P"(e)D p{ (e).

(se')

In a similar way, the Dr' obtained by substituting from (83)-(85) into the integral equations E q u a t i o n s ( 8 7 ) - ( 8 9 ) , ( 8 7 ' ) - ( 8 9 ' ) , d e s c r i b ep r e c i s e l y can be related with the II1+(e) of (89). This Dr' the way in which the lpr(a), Sv1'(e), Dp1'(e), when will be identical with (85) provided that substituted into the integral equations, reproduce 1 themselves with the addition of divergent terms. (95) Zs:l|.C(e,). And from these results it is easy to deduce the 2m self-reproducing property of the operators (83)-(85), Finally, the Ip obtained by substituting fronr rvhen substituted into the same equations. (83)-(85) can be shown to be Consider for example the effect of substituting from (83)-(85) into the term 2(W,11), given by 7u:7u! Z;l.Lrt(e), (65) with l:1. The integrand of (65) is a product of one factor lu, one 7r, one Sr', and one Dr'. with -r1,r(e)given by (87). Using (87'), this l, will tTx

308 f

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terminacy is removed, and one must take

be identical with (83) provided that Zt:l-L(e).

(96)

Therefore, if Zt Zz, 23, 6xn are defined by (96)' (93), (95), (94), it is established that (83)-(85) give ihe'correcl forms of the operators I.u, Sp', Dp', including all the effects of the radiative corrections which these operators introduce into themselves and into each other. The exact Eqs. (83)-(85) give a much simpler separation of the infinite from the finite parti of these operators than the approximate equations(73), (78), (82). Consider now the result of using the exact operators (83)-(35) in calculating a constituent M oi U1*;, where M is constructed from a certain irreducible graph G6 according to the rules of Section IV. Go will have, say, F, internal and E" external electron lines, Fo internal and E" external ohoton lines, and 4: P"llE":2Fe!fle

t7 52

DYSON

(97)

vertices. In M there will be {E, fac\ors-{'(fri)' }8, factors {,'(ki) and Eo facLors .4,'(frt) $ver\b4 (37). In t'(kt), fti is the momenlum-energ\4 vbctor ot an electron, which satisfies (69), and thE S"(fr') in (73) are zero at evety stage of the inductive definiiion of Srr'(a). Therefore (84), (35)' (37) give in turn S Ft(kt) : Z 25F(kc), >(k\:2"(2"-t)(h,iy,-ixn), {sel {' (ko): V (k') + 2r(Z z 1)S r(ht) (k,ct u- ixit(ht). The expression (98) is indeterminate, since (hri1r-ixs) operating on 'lt(kt) gives zero, while operating on Sr(ftt) it gives the constant (1/2r)' Thus, aCcording to the order in which the factors are evaluated, (98) will give for ry''(frc)either the value /(fri) or the value Zr{(h'). Similarly, 0t(ki) is indeteiminate betweent(F;) and Z\LG;), and, excluding for the moment .du(ftt) which are Fourier components of the external potential,. Ar'(kr) is indeterminate between A,(ki) and Z3Ar(kc). In any case, considerations of covariance show that the '!r'(ho), 0' @n), A r' (hi) are numerical multiples of the ,r@\, {'(kd), At (ftt) ; thus the indeterminacv lies onlv irr a constant factor multiplying the whole expression M. There cannot be any indeterminacy in the maenitude of the matrix elements of I/( - ), so long as ihis operator is restricted to be unitary. The indeterminacy in fact lies only in the normalization of the electron and photon wave functions ry'(fti), '{/(kt), A,(ht), which may or may not be regarded as altered by the continual interactions of these particles with the vacuum-fields around them. It can be shown that, if the wave functions are everywhere normalized in the usual way, the apparent inde'

{/' @c): z r+{/(kt), {,,(ht) : z.i,(kt) , A''(kr):Zt+au1Pt1'

(ee)

It viill be seen that (99) gives just the geometric mean of the two alternative values of \l/t(ki) obtained from (98). Wherr Au(frt) is a Fourier component oJ the external poteniial, then in general (hi)z + o' ad A,i \!:) is not indeterminate but is given by (37) and (85) in the form A,', (ki) :2o;2tD F|' (et)(kt)2Au(kr.

(100)

However, the unit in which external potentials are measured is defined by the dynamical effects which the potentials produce on known charges; and these dynimical efficts are just the matrix elements of U1-; i" which (100) appears. Therefore the factor Zzin (lO0) has no physical significance,and will be changed when.C, is measured in practical units' The which appears when practical .or.".t used is 231; this is because the photon units are"on.tunt potentials Au in (99) were normalized in terms ot practical units; and (100) should reduce to (99) when (Et)'?+O,if the external ,4, and the photon 'dts are measured in the same units. Therefore the correct formula fot Ar', covering the cases both of photon and of external potentials, is i), A,' thil : 2ri2 D r r' (er) (ki)' A r(h A , ' 1 h t 1 : 2 r t n 7 "\ P;])', (fr')'-O.

(frr)' I 0' ( ] 101) J

In M there will appear F, factors S7', Fo factors Dr', and. z factors f*, in addition to the factors of the type (99), (101). Hence bv (97) the Z,factors will occur in M pnly as the constant multiplier Z;"22"23\n' By (36), this multiplier is exactly suffi-cient to convert the factor d", remaining in M from the original interaction (8), into a factor er". Thereby, both e and Z factors disappear from M, leaving only their combination er in the operators lur(er), Sr1(er), Der'(er), and in the factor ern' If now er is identified with the finite observed electronic charge, there no longer appear any divergent expressions in M, And since M is a completely general constituent of t/(@), the elimination of divergences from the S matrix is accomPlished. It hardty needs to be pointed out that the arguments of this section have involved extensive manipulations of infinite quantities. These manipulations have only a foimal validity, and must be justiEed a posterioriby the fact that the)'ultimately iead to a clear separation of finite from infinite expressions. Such in a posteriori justi6cation of dubious manipulations is an inevitable feature of

309 f/JJ

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any theory which aims to extract meaninqful r e s u l t s f r o m n o t c o m p l e t e l y c o n s i s t e n tp r e m i s e i . We conclude with two disconnected remarks. First, it is probable that Zt:2t identically, though this has been proved so far only up to the order e2. If this conjecture is correct, then !ll charge-renormalization effects arise according to (86) from the coefficient Zz alone, and the arguments of this paper can be somervhat simplified. Second, Eqs. (88'), (89'), which define the fundamental operators Srt', Drr', may be solved for these operators. Thus

equivalent to the following: each factor Sr in M is replaced by Sp1'(e),each factor Dpby Dp1'(e), each factor 7, by lr1(e), each factor -4uwhen it represents an external potential is replaced by A ur(hi) :2o;p ,y'(e) (hc)2Au(kt),

(102)

factors ry',ry',.1, representing particle wave-functions are left unchanged, and finally e wherever it occurs in M is replaced by er. The definition of M is completed by the specification of Srr'(a), Dot'(e), Iu1(e) ; it is in the calculation of these operators that the main difficulty of the theory lies. The method of obtaining these operators is the process of successive substitution and integration explained in the first part of Section VII; the operators so calculated are divergence-free, the divergent parts at every stage of the calculation being explicitly dropped after being separated from the finite parts by the In electrodynamics, the S" and D" are small radi- method of Section VI. ative corrections, and it will ahvays be legitimate The above rules determine each contribution M and convenient to expand (88") and (89") by the to {/( o) as a divergence-freeexpression,which is a binomial theorem. If, however, the methods of the function of the observed rnass nx and the observed present paper are to be applied to meson fields, charge e1 of the electron, both of which quantities with coupling constants which are not small, then are taken to have their empirical values. The diverit will be desirable not to expand these expressions; gent parts of the theory are irrelevant to the calin this way one may hope to escape partially from culation of U(o), being absorbed into the unobthe limitations which the use of weak-coupling servable constants 6m and e occurring in (8). A approximations imposes on the theory. place where some ambiguity might appear in M is in the calculation of the operators Sp1'(e),Dp1'(e), VI[. SUMMARY OF RXSULTS Iu1(e), when the method of Section VI is used to The results of the prcceding sections divicle separate out the finite parts S(W,tt), D(W',tr), (67), (74), (80). themselves into ts'o groups. On thc one hand, there ^ ! " ( V , t t , l ' ? ) ,f r o m t h e e x p r e s s i o n s Even in this place the rules of Section VI give unamis a set of rules by lvhich the element of the S matrix corresponding to any given scattering process may biguous directions for making the separation; only question whether some alternative direcbe calculated, rvithout mentioning the divergent there is a expressions occurring in the theory. On the other tions might be equally reasonable.For example, it is possible to separate out a finite part from >(W, tt) hand, there is the specification of the divergent (67), and not to make the further step expressions, and the interpretation of these ex- according to pressions as mass and charge renormalization of using (70) to separate out a finite part S(W, tt) which vanishes when (69) holds. Actually it is easy factors. The first group of results may be summarized as to verify that such an alternative procedure will not change the value of M,but will only make its follows. Given a particular scattering problem, with specified initial and final states, the corresponding evaluation more complicated; it will lead to an (infinite) part oi the matrix element of I/(o) is a sum of contributions expression for M in which one mass and charge renormalizations is absorbed into from various graphs G as described in Section II. A particular contribution M from a particular G is the constants 6m and e, while other finite mass and charge renormalizations are left explicitly in the to be rvritten down as an integral over nomentum formulas. It is just these finite renormalization variables according to the rules of Section III; the integrand is a product of factors V&t) , |Gt) , Ap(ht), effects which the second step in the separation of S(14/,tt) and hr"(Y,11,l'?) is designed to avoid. SF(pt), Di(pt), 6(q), t,, the factors corresponding Therefore it may be concluded that the rules of calin a prescribed way to the lines and vertices of G. According to Section IV, contributions M are only culation of U(o) are not only divergence-freebut to be admitted from irreducible G; the effects of unambiguous. As anyone acquainted with the history of the reducible graphs are included by replacing in M Larnb shiftll knows, the utmost care is required the factors V, 'tr', A, Sr, Dr, 7u, by the corres p o n d i n g e x p r e s s i o n s( 3 7 ) , ( 3 5 ) , ( 3 6 ) , ( 3 8 ) . T h e s e trH. A. Bethe, ElectrcmlgneticShilf o.f Energy Leaels, replacements are then shorvn in Section VII to be Report 1o Solvay Confercnce,"Brus'ets(1048t.

s",'r,t:Ir-5"k1]s",

(88,,)

o,,'at:lt_.!n"at] o,.

(8e,)

3IO F. J, DYSON

1754

the Ar and B; are logarithmically divergent before it can be said that any particular rule of where coefficients, independent ol m and ev numerical given in this rules The lalculation is unambiguous' each thal sense the unambiguous, in ouo", IX. DISCUSSIONOT' FURTIIER OUTLOOK . r " n t i t v"r" t o b e c a l c u l a t e d i s a n i n t e g r a l i n m o al The surprising feature of the S matrix theory, mentum-space which is absolutely convergen-t infinitv: such an integral has always a well-defined as outlined in tiis paper, is its success in avoiding v a l u e . - H o w e v e r , t h e r u l e s w o u l d n o t b e u n a m - difficulties. Starting from the methods of Tomonaga' Uigrou" if it were allowed to sptit the integrand.into Schwinger and Fej'nman, and using no new ideas ,"?"..1 putr. and to evaluate the integral by inte- or techiiques, one irrives at an S matrix from which g r a t i n g ; h e p a r t s s e p a r a t e l ya n d . t h e n a d d i n g t h e the well-known divergences seem to have conspired results; ambiguities would arlse lt ever Lne parrlat to eliminate themselves. This automatic disapDearanceof divergences is an empirical fact, which ini"sr^1. were-not absolutely convergent' A splitting parts m u s t b e g i v e n d u e w e i g h t i n c o n s i d e r i n gt h e f u L u r e of tf,e integrals into conditionally convergent in the context of the present prosDects of electrodynamics. Paradoxically op.ry """-l,tnnatural paper, but occurs in a natural way when.calcuJa- posed to the fini{enessof the S matrix is the second iact, that the whole theory is built upon a Haqll; iiol" are ba""d upon a perturbation theory in which (8) electron and poiitron states are considered sepa- tonian formalism with an interaction-function from each other' The absolute convergence rvhi.tt i. infinite and therefore physically meaningrately 'th" ittt"gtul. in'the present theory is essentially less. oi th" fict that the electron and The arguments of this paper have been essenconnected'*ith are field electron-positron tially mathematical in character,being concerneo oo.itron parts of the lever separated; this finds its algebraic expression *itn tit" consequencesof a particular mathematical in the statement that the quadratic denominator in formalism. ln attempting to assessthelr slgnlncance (4.5) is never to be separated into partial fractions' for the future, one must pass from the language ot the absenci of ambiguity in the rules of mathematics to the language of physics' O.ne.m.ust ift"r"fot" that the mathematlcal torassume provisionally caic,llatio" of U(o) is achieved by introducing '"o.t".ponds to something existing in -uli.into ttt" theory what is really a new physical tylottt".;., namely that the electron-positron field nature, and then enquire to what extent the paraui*uyt u.a. as a unit and not as a combination of Jo"i.ui .".utt. of the formalism can be reconciled l*o i"putut" fields. A similar hypothesis is made for *ittt t".tt an assumption. In accordance with this theel"ctrom.gnetic field, namely thal- this field also prografl, we interpret the contrast between the finite acts as a unit and not as a sum of one part repre- ii"E g"nt Hamilto;ian formalism and the qnother part repreS *u?.i" as a contrast between two pictures of the senting photon emission and world, seen by two observers having a different senting -Photon absorPtion' it must" bL said that the proof of the choice qf -ea.uri.tg equipment at their disposal' n"iili, f i n i t e n e s sa n d u n a m b i g u i t y o f U ( o ) g i v e n i n t h i s itt" nt"t picture ii of a- collection of quantized by oaoer makes no pretence of being complete and fields with^localizable interactions, and is seen general a fictitious observer whose apparatus has no atomlc iiglrru". It is most desirable that these should as soon as possible be supple- structure and whose measurements are llmlteo ln u.?u-"nt. only by the existence of the fundamental mlnted by bn explicit calculation of at least one ;;;t no .on"turrt, c ind' h. This observer is able to make fourth-order radiaiive effect, to make sure that freedom on a sub-microscopic s9]9 -ittt .o-pf"t" unforeseen difficulties arise in that order' ihe recond group of results of the theory is the iti" f.i"J oi observations which Bohr and Rosenfeldr2 (86)' i" a more restricted domain in their ciassic identification 6f. im and e bv (94) and ai.iu".ion of the measurability of field-quantities; Although these two equations are strictly meaning- "rnotou as the i""r, boih sides being infinite, yet it is a satisfactory and he will be referred to in what follows feature of the theory that it determines the unob- iid"al" obs"rver. The second picture is of. a.colpower lection of observable quantities (in the terminology servable constants 6m and a formally as real itt the observable er, and not vice versa' There of Heisenberg), and is the picture seen by a ano ""ii"" i, thn" tto objection in principle to identifying e1 observer, whose apparatus consists ol atoms are *ittt ttt" bbseived electronic charge and writing elementary particles and whose measurements not only by c and I but also by in accuracy limited ( 1 0 3 ) (e,2/4rhc):q:1/137' olher constants such as a and m' The real observer (94) by (8) then' are in appearing The constants ', N. B"ht and L. Rosenfeld, Kgl. Dansk'Vid'.Sel:,Math paperbv folr.an: and (86), - 12,No. 8 (1933).A secondPhvs.Medd. o, (104) ir'i""r"ii i. to'bepubiished 6m:m(AplAzq2*'''), t,:i, fo1' l&,f;:,tr u:",Xo;i' booklet by A. Pais,.DmbPwn

e: e r ( l* B P - l B z a 2 - l ' ' ' ) ,

(1os) (Princeton University Press,Princeton, 1948)'

"i

3ll

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IN

QUANTUM

makes spectroscopic observations, and performs experiments involving bombardments of atomic systems with various tl'pes of mutu3lly interacting subatomic projectiles, but to the best of our knowledge he cannot measure the strength of a single field undisturbed by the interaction of that field with others. The ideal observer, utilizing his apparatus in the manner described in the analysis of the Hamiltonian formalism by Bohr and Rosenfeld,l2 makes measurements of precisely this last kind, and it is in terms of such measurements that the commutation-relations of the fields are interpreted. The interaction-function (8) will presumably always remain unobservable to the real observer,whoisabletodeterminepositionsofparticles only with limited accuracy, and rvho must always obtain finite results from his measurements. The ideal observer, however, using non-atomic apparatus whose location in space and time is l<nown with infinite precision, is imagined to be able to disentangle a single field from its interactjons with others, and to measure the interaction (8). In conformity with the Heisenberg uncertainty principle, it can perhaps be considereda physical consequence of the infinitely precise knowledge of location allowed to the ideal observer, that the value obtained by him when he measures (8) is infinite. If the above analysis is correct, the divergencesof electrodynamics are directly attributable to the fact that the Hamiltonian formalism is based upon an idealized conception of measurability. The paradoxical feature of the present situation does not then lie in the mere coexistence of a finite S matrix with an infinite interaction-function. The empirically found correlation, between expressions which are unobservable to a real observer and expressionswhich are infinite, is a physicalll. intelligible and acceptable feature of the theory. The paradox is the fact that it is necessary in the

ELECTRODYNAMICS

present paper to start from the in6nite expressions in order to deduce the finite ones. Accordingly, what is to be looked for in a future theory is not so much a modification of the present theory rvhich will make all infinite quantities finite, but rather a turning-round of the theory so that the finite quantities shall become primary and the infinite quantities secondary. One may expect that in the future a consistent {ormulation of electrodynamics will be possible, itself free from infinities and involving only the ph1'sical constants m and ey, and such that a Hamiltonian formalism with interaction (8), with divergent coelficients 6m and e, may in suitably i d e a l i z e d c i r c u m s t a n c e sb e d e d u c e d f r o m i t . T h e Hamiltonian formalism should appear as a limiting form of a description of the world as seen by a certain type of observer, the limit being approached more and more closely as the precision of measttrement allowed to the observer tends to infinity. The nature of a future theory is not a profitable subject for theoretical speculation. The future theory rvill be built, first of all upon the results of future experiments, and secondll' upon an understanding of the interrelations between electrodynamics and mesonic and nucleonic phenomena. The purpose of the foregoing remarks is merely to point out that there is now no longer, as there has seemedto be in the past, a compelling necessity for a future theory to abandon some essential features of the present electrodynamics. The present electrodynamics is certainly incomplete, but is no longer certainly incorrect. In conclusion, the author rvould like to express his profound indebtedness to Professor Feynman for many of the ideas upon which this paper is built, to Professor Oppenheimer for valuable discussions,and to the Commonwealth Fund of New York for financial support.

3r2

P o p e r2 6

TIIE LAGRANGIAN IN QUANTUM MECHANICS. BA P. A. M. D'irac. (Receivecl November 19, 1932).

Quantum mechanics\ryasbuilt up 0n a foundationof analogy with the Hamiltonian theory of classical mechanics. This is becausethe classical notion of canonicalcoordinates and momentawas found to be one with a very simplequantum analogue,as a result of which the whole of the classical Hamiltonian theory, which is just a structure built up on this notion, could be taken over in all its details into quantum mechanics. Now there is an alternative formulation for classical dynamics, provided by the Lagrangian. This requires one to work in terms of coordinatesand velocities instead of coordinates and momenta. The two formulationsare, of coursc, closely related,but there are reasonsfor believing that the Lagrangian one is the more fundamental. In the first place the Lagrangian methoclallows one to collect together all the equationsof motionand expressthem as the stationary property of a certain action function. (This action funrrtion is just the time - integral of the Lagrangian). There is n0 correspon.lingaction principle in terms of trre eoordinates and momenta of the Hamiltonian theory. Secondly the Lagrangian method can easily be expressed lelativistically, 0n account of the action function being a relativistic invariant; while the Hamiltonian method is essentially non - relativistic in form, since it marks out a particular time variable as the canonical conjugate of the Hamiltonianfunction. For these rcasonsit would seemdesirableto take up the questionof what correspondsin the quantum theory to the Lagrangian methotl of the classical theory. A litile considerationshows,however,that one cannot expectto be able

3r3 P. A. M. Dirac, The Lagrangian in Quantum Mechanics.

65

to take over the classicalLagrangian equationsin any very direct way. These equations involve partial derivatives of the Lagrangian with respect to the coordinatesand velocities and.no meaning can be given to such derivatives in quantum mechanics. The only differentiation process that can be carried out with respect to the dynamical variables of quantum mechanicsis that of forming Poisson brackets and this processleads to the I{amiltoniantheory.t We must therefore seek our quantumLagrangian theory in an indirect way. We must try to take over the i d,e a s of the classicalLagrangiantheory,not the equations of the .classicalLagrangian theory. Contact Transf ormations. I,agrangiantheory is closeiy connectedwith the theory of contact transformations. We shall thereforebegin with a discussionof the analogy between classicaland quantum contact transformations. Let the two sets of variablesbe pr, .er and Pr, Qr, (r : 1, 2 ... n) and supposethe q's and. so that any function of the dynaQ's to be all ind,ependent, mical variables can be expressedin' terms of them. It is well known that in the classicaltheory the transformationequations for this case can be put in the form pr:

ds oq.r,

fD y :_ _

ds dQr,

(1)

where S is some function of the q's and Q's. 1 Prouessesfor partial differentiation with respect to matrices have been given by Born, Heisenberg a n d J o r d a n ( Z S . f . P h y s i k3 5 , 561, 1926)but these processes do not give us means of differentiation with respect to dynamical variables, since they are not independent of the representation chosen. As an example of the difficulties involved in differentiation with respect to quantum dynamical variables, consider the ,three components of an angular momentum, satisfying ffi*ffis-rn1rne-

i,hm",

lN'e have here m, expressed explicitly as a function of nt,* and mu, bat we can give no meaning to its partial derivative with respect to rnn or rna.

314 P. A. M. Dirac,

66

In the quantum theory we may take a representationin which the q's are diagonal, and a secondrepresentationin which the Q's are diagonal. There will be a transformation function (q'lQ') connectingthe two representations. We shall now show that this transformationfunction is the quantum analogueof etslh. If e is any function of the dynamical variables in the quantum theory, it will have a ,,mixed." representative @.'I o I Q'), which may be defined in terms of either of the (q' l"l q"), (Q'l" i 0") by usual representatives ( q ,l " l Q ' ) : I @ ' i o l q " ) d q , , ( q , , I Q ' ) :@ , I q , , ) d , Q , , l(oQl Q ,,,). I these definitions From the first of we obtain

(q'I q, I Q'): q',(q.'I Q')

(2)

(q'Ip,l Q'): - ih !4;{n,Ia,)

(3)

and from the second

@'I Q,iQ'): Q',(q' I Q')

@)

@ ' I P , I Q ) : i ' hu *Y r ( q ' l U ) .

(5)

Note the differencein sign in (a) and (r). Equations(2) and (a) may be generalisedas follows. Let f(q) be any function of the q's and g(q any functionof the Q's. Then

r]")dq"(q"i Q')d,Q" @'i f@)s \01 Q'): I I fr' 1f@)'t Q' l s (ql Q') : f (,t')g Q') @'I Q'). X ' u r t h e r ,i f f * ( q ) a n d g n ( Q ) , ( k : 1 , 2 . . . , n t ) d e n o t et y o s e t s of functions of ghe q's and Q's respectively, (q' lrn f,,(q)gu(q Q') : Er frc(q')u*(Q').@'I Q). Thus if a is any function of the dynamical variables and u'e suppose it to be expressed as a function a (qq of the q's and - orclered" wB,f, that is, s0 that it consists of Q's in a ,,wr:11 a sum of tenns of the form f(q)g(Q), we shall have

@'i o (q0 | Q'): o (q' Q')@'I Q).

(6)

315 The Lagrangian in Quantum Mechanics.

67

This is a rather remarkable equation, giving us a connection between which is a function 0f operators, and "(qQ), o ( q ' Q ' ) ,w h i c h i s a f u n c t i o n o f n u m e r i c a l variables. Let us apply this result for &: gr. Putting (7\ @'lQ')-eiuth' where U is a new function - of the q"s and Q"s we get, from (3)

@,lp,l Q): !!I^q:a) oe, @,,I Q). By comparingthis with (6) we obtain d u(qQ) Pr: dq., as an equation between operators 0r dynamical variables, which holds provided dUldq, ts well- ordered. Similarly, by applying the result (G) for ct- p, and using (b), we get

, g q. P,: - du dQ' '

provided duldQ, is well-ordered. These equations are of the sameform as (t) and show that the U defined by (T) is the analogueof the classicalfunction S, which is what we had to proYe. Incidentally, we have obtained another theorem at the same time, namely that equations(1) hold also in the quantum theory providedthe right-hancl sidesare suitably interpreted, the variables being treated classicallyfor the purpose of the differentiations and the derivatives being then well-ordered. This theoremhas been previouslyproved by 1 Jordan by a differentmethod. The Lagrangian

and the Action Principle. The equationsof motion of the classicaltheory causethe dynamicalvariablesto vary in sucha way that tireir valuesq;, pt at any time t are connectedwith their values er, pr.aL any other time T by a contact transformation,which may be put into the form (t) with e, p:et, Ft; Q, P:er, py aod. S equal to the time integral of the Lagrangianoverthe range r .l o r d a n, ZS. f. Phys, 38, 518, 1926.

316 P. A. M. I)irac,

68

? to /. In the quantum theory the Qt, pt will still be connected with lhe q7, pr bY a contact transformation and there will be a transformation function (q.rlqr) connecting the two representations in which the 8r and the q7 are diagonal respectively. The work of the preceding section now shows that

,i

I

to exp le Ldtitl ) ' (qr\qr) corresponds J

(8)

T

where -L is the Lagrangian. If we take T to differ only infinitely littie from l, we get the result (9) to expli'Ldtlhl. corresponds (q.t+arlqr) The transformationfunctions in (8) and (9) are very fundameutal things in the quantum theory and it is satisfactory to fincl that they have their classicalanalogues,expressible simply in terms of the Lagrangian. We have here the natural extensionof the well - known result that the phase of the wave function conespondsto Hamilton's principle function in classical theory. The analogy (9) suggeststhat we ought to considerthe classical Lagrangian,not as a function of the coordinatesand velocities,but rather as a function of the coordinatesat time / and the coordinatesat time t + dt. in this section tr'or simplicity in the further d"iscussion although freedom, we shall take the case of a singled.egreeof the argument applies also to the general case. We shall use the notation t

rfl

e x p[ i

J

Ldtlh]-t(tr),

T

so that AUf\ is the classical analogue of (q.tlqr). Suppose we divide up the time interval T --t into a large number of small sections T-tt, tr-tz, .. ., tm-r't,o, t*--'t hy the introd,uction of a sequence of intermed.iate times tt, tz, ... t*. Then

(10) a(tT) - a(tt*) a (t*t*-i . .. a (trt')a(tJ). Now in the quantumtheorYwe have (q.rlq.t)d,q,,(qt q.*-r)dq.*_'... (q.rlq.r):l@rlq*) dq^(q.*l lqr), (tt)

317 The Lagrangian in Quantum Mechanics.

69

where q7,denotesq at the intermed"iate time tu,(k- 1, 2... m). Equation (11) at first sight doesnot seem to correspondproperly to equation (10), since 0n the right-hand side of (rr) we must integrate after doing the multiplication while 0n the right - hand side of (10) there is no integration. Let us examinethis discrepancyby seeingwhat becomes of (t1) when we regard f as extremely small. From the results (8) and (9) we see that the integrand iq (rr) must be of the form eiFihwhere F is a function of q.r,er, ez. . . e*, et which remains finite as h tends to zero. Let us now picture one of the intermediate e's, say e*, as varying continuously while the others are fixed. Owing to the smallness of h, we shall then in general have Fth varying extremely rapidly. This means that ei,Fthwrll vary periodically with a very high frequency about the value zero, as a result of which its integral will be practically zero. The only important part in the domain of integration of q.ois thus that for which a comparativelylarge variation in Qrcproducesonly a very small variation in -F. This part is the neighbourhood of a point for which F is stationary with respect to small variations in q7r. We can apply this argument to each of the variables of integration in the right-hand side of (tr) and obtain.the result that the only important part in the domain of integration is that for which, r is stationary for small variations in all the intermediate q's. But, by applying (s) to each of the small time sections,we see that f' has for its classical analogue tt*t2ttt

I rat+ I Ldt+...+ [ rat + I Ld,t: f rat, tln

tm_t

t;

f

f

which is just the action function which classical mechanics requires to be stationary for small variations in all the intermediate q's. This shows the way in which equation (11) goes orrer into classical results when h becomes extremely small. We now return to the general case when D is not small. We see that, for comparison with the quantum theory, equa-

3t8 P. A. M. Dirac,

70

tion (10) must be interpreted.in the following way. Each of the quantities ,4. must be consideredas a function of the q's at the two times to which it refers. The right - hand,side is then a function, not only of qr and Qa but also of er, ez, . .. em, and in order to get from it a function of q., and q; only, which we can equate to the left - hand side, we must substitutefor et, ez .. . em their valuesgiven by the action principle. Thls processof substitution for the intermediate q's then correspondsto the processof integrationover all values of these q's in (11). Equatiou (11) containsthe quantumanalogueof the action principle, as may be seenmore explicitly from the following argument. From equation(11) we can extract the statement (a rather trivial one) that, if we take specifiedvaluesfor q7 and qt, then the importance of our consideringany set of values for the intermediateq's is determined by the importance of this set of values in the integration on the righthand side of (11). If we now make fu tend to zero,this statement goes over into the classicalstatementthat, if we take specified. values for er and qr, then the importance of our q's is zero considering any set of values for the intermed,iate unless these values make the action function stationary. This statement is one way of formulatingthe classicalaction principle. Application

to F ield' Dynamics.

'W-e may treat the problem of a vibrating medium in the classical theory by Lagrangian methodswhich form a natural generalisationof those for particles. We chooseas our coordinates suitable field quantities or potentials. Each coordinate' is then a function of the four space- time variables fr, !/, H, t, conesponding to the fact that in particle theory it is a function of just the one variable /. Thus the one independentvariable / of particle theory is to be generalised variablesfr, U, H, t.l to four intlepend,ent 1 It is customary in field dynamics to regard. the values of a field. quantity for two different values of (n, y, z) bat, the same value of f as two differeut coordinates, instearl of as two values of the same coordi-

3r9 The Lagrangian in Quantum Mechanic:i.

7l

We introduce at each point of space- time a Lagrangian density, which must be a function of the coordinates and their first derivatives with respect to fr, U, z and. t, correspond"ingto the Lagrangian in particle theory being a function of coordinates and velocities. The integral of the Lagrangian d,ensity over any (four - dimensional)region of spacetime must then be stationary for all small variations of the coordinates inside the region, provided the coordinateson the boundary remain invariant. It is now easy to see what the quantum analogue of all this must be. If S denotes the integral of the classical Lagrangian density over a particular region of space- time, we should expect there to be a quantum analogue sf st'stnssvresponding to the (qrl qr) of particle theory. This (qr I q.) i* a function of the values of the coordinates at the ends of the time interval to which it refers and so we should expect the quantum analogue gf si'stnto be a function (really a functional) of the values of the coordinates on the boundary of the space- time region. This quantum analogue will be a sort of ,,generalizedtransformation function". It cannot in general be interpreted, like (qrl q_r),as giving a transformation betwsen one set of dynamical variables and another, but it is a fourdimensional generalization of (qt t er) in the following sense. Corresponding to the composition law for (q1lq7) (12) q r )d q , ( q r l q . r ) , ( q , l q r ): [ @r[

the generalized transformation function (S.t.f.) rvill have the following compositionlaw. Take a given region of spacetime and divide it up into two parts. Then the g.t.f. for the whole region will equal the product of the g.t.f.'s for the two parts, integrated over all values for the coordinates on the commonboundary 0f the two parts. Repeatedapplication of (12) gives us (11) and repeated t g.t.f.'s will enable nate for two different points in the domain of inrlepenclent variables, antl in this way to keep to the idea of a single inclepenclent sariable /. This point of view is necessary for the Hamiltonian treatment, but for the Lagrangian treatment the point of view atlopted in the text seems preferable on account of its greater space - time symmetry.

320 72

P. A. M. Dirac. The Lagrangian in Quantum Mechanics.

us in a similar way to conneot the g.t.f. for any region with the g.t.f.'s for the very small sub - regions into which that region may be divided. This connection will contain the quantum analogue 0f the action principle applied to fielcls. The square of the mod.ulus of the transformation function (qtlqr) can be inter'preted as the probabllity of an observation of the coordinates at the later time I giving the result et for a state for which an observation of the coordinates at the earlier time ? is certain to give the result 4r. A corresponding meaning for the square of the modulus of the g.t.f. rvill exist only when the g.t.f. refers to a region of space- time bounded by two separate (three - rJimensional) surfaces, each extend.ing t0 infinity in the space directions and lying entirely outsid.e any light - cone having its vertex on the surface. The square 0f the mod.ulus 0f the g. t. f. then gives the probability of the coordinates having specified valrres at a1l points 0n the later surface for a state for which they are given to have definite values at all points on the earlier surface. The g.t.f. may in this case be considered as a transformation function connecting the values of the coordinates and momenta 0n one of the surfaces with their values ou the other. We can alternatively consider l(q.rlqr)l'as giving the' probability of any state yielding the rerelatj.ve a pliori observations 0f the q's are made at' when sults er and et time Z and at time I (account being taken of the fact that the earlier observation will alter the state and affect the later observation). Correspondingly we can consider the square of the modulus of the g.t.f. for any space- time region as. probability of specified results giving the relative a priori being obtained when observations are made of the coordinates at all points on the boundary. This interpretation is inore general than the preced.ing one, since it does not retluire a restriction on the shape of the space - time regionSt John's College, Cambridge.

Poper 27

32r

Space-Time Approach to Non-Relativistic Quantum Mechanics R. P. Fnnnten Cornell Unhersity, Ithaca, Naa York Non-relativistic quantum mechanics is formulated here in a difierent way. It is, however, equivalent to the familiar formulation. In quantum mechanics the probability mathematielly of an event which can happen in several difierent ways is the absolute square of a sum of complex contributions, one from each alternative way. The probability that a particle will be found to have a path r(l) lying somewhere within a region of space time is the square of a sum of contributions, one from each path in the region. The contribution from a single path is postulated to be an exponential whose (imaginary) phase is the classical action (in units of i) for the path in question. The total contribution from all paths reaching r, ! from the past is the wave function 'y'(*, r). This is shown to stisfy Schroedinger's equation. The relation to matrix are indicated, in particular to eliminate the and operator algebra is discussed. Applietions coordinates of the field oscillators from the equations of quantum electrodynamics.

classicalactionBto quantum mechanics.A probaamplitude is associated with an entire bility fT is a curious historical fact that modern particle as a function of time, rather I quantum mechanics began with two quite motion of a simply with a position of the particle at a than different mathematical formulations: the differparticular time. ential equation of Schroedinger, and the matrix The formulation is mathematically equivalent algebra of Heisenberg. The two, apparently disto the more usual formulations. There are, similar approaches, were proved to be mathetherefore, no fundamentally new results. Howmatically equivalent. These two points of view ever, there is a pleasurein recognizing o1dthings were destined to complement one another and from new point of view. Also, there are proba to be uitimately synthesized in Dirac's trans- lems for which the new point of view offers a formation theory. distinct advantage. For example, if two systems This paper will describe what is essentially a A and,B interact, the coordinates of one of the third formulation of non-relativistic quantum systems, say B, may be eliminated from the theory. This formulation was suggestedby some equations describing the motion of ,4 ' The interof Dirac'sl'2 remarks concerning the relation of r. INTRODI'CTION

I P, A. M, Dirac, Thz Principles of Quanlum Muhanics (The Clarendon Press, Oxford, 1935), second edition, Section 33; also, Physik. Zeits. Sowjetunion 3, 64 (1933). r P. A. M. Dirac, Rev. Mod. Phys. 17, 195 (1945).

I Throushout this paper the tem "action" will be used along a path. for the time integml bf the lagnngian When this path is the one actuallv taken by a particle' moving chisically, the integral shiluld more properly be called Hamilton's first principle function.

367

322 368

P.

Fi.:\'NN{.\N

action with E is represented by a change in the formula for the probability amplitude associated with a motion of .4 . It is analogousto the classical situation in which the effect of B can be represented by a change in the equations of motion of A (by the introduction of terms representing forcesacting on,4.). In this way the coordinates of the transverse, as well as of the longitudinal field oscillators, may be eliminated from the equations of quantum electrodynamics. In addition, there is ahvays the hope that the new point of view will inspire an idea for the modification of present theories, a modification necessaryto encompasspresent experiments. We first discuss the general concept of the superposition of probability amplitudes in quantum mechanics. We then show how this concept can be directly extended to define a probability amplitude for any motion or path (position us. time) in space-time. The ordinary quantum mechanics is shown to result from the postulate that this probability amplitude has a phaseproportional to the action, computed classically, for this path. This is true when the action is the time integral of a quadratic function of velocity. The relation to matrix and operator algebra is discussedin a way that stays as closeto the language of the new formulation as possible. There is no practical advantage to this, but the formulae are very sugggstirfe if a generalization to a wider class of action functionals is contemplated. Finally, we discuss applications of the formulation, As a particular illustration, we show how the coordinates of a harmonic oscillator may be eliminated from the equations of motion of a system with which it interacts. This can be extended'directly for application to quantum electrodynamics. A formal extension which includes the effects of.spin and relativity is described. 2. THE SUPERPOSITION OF PROBABILITY AMPLITUDES

changes in physical outlook required by the transition from classical to quantum physics. For this purpose,consider an imaginary experiment in which we can make three measurements successivein time: first of a quantity 1., then of B, and then of C. There is really no need for these to be of different quantities, and it will do just as well if the example of three successive position measurementsis kept in mind. Suppose that a is one of a number of possibleresultswhich could come from measurementA, D is a result that could arise from B, and c is a result possible from the third measurement C.aWe shall assume that the measurementsA, B, and C are the type of measurementsthat completely specify a state in the quantum-mechanical case. That is, for example, the state for which B has the value b is not degenerate. It is well known that quantum mechanicsdeals with probabilities, but naturally this is not the whole picture. ln order to exhibit, even more clearly, the relationship between classical and quantum theory, we could supposethat classically we are also dealing rvith probabilities but that all probabilities either are zero or one. A better alternative is to imagine in the classical case that the probabilities are in the sense of classical statistical mechanics (where, possibly, internal coordinatesare not completely specified). We define P"6 as the probability that if measurement,4 gave the result o, then measurementB will give the result b. Similarly, Pu" is the probability that if measurementB gives the result b, then measurement C gives c. Further, let Po, be the chance that it A gives o, then C gives c. Finally, denote by P"6" the probability of all three, i.e., if ,4. gives o, then ,Q gives b, and C gives c. If the events between a ar.d,b are independent of those between & and c, then Pou:PotPu

(1)

The formulation to be presented contains as This is true according to quantum mechanics when the statement that B is D is a comDlete its essential idea the concept of a probability amplitude associatedwith a completely specified specification of the state. motion as a function of time. It is, therefore, a For our discussion it is not important that certain worthwhile to review in detail the quantum- values of a, b, or c misht be excluded bv-For ouantum mesinplicity, mechanical concept of the superposition of proba- chanics but not by clissiel mrchanic-s. assume the values are the sme for both but that the bility amplitudes. We shall examine the essential probability of certain values may be zero.

323 NON_RELATIVISTIC

OUANTUM

In any event, we expect the relation p"":l

Pa*

Q)

b

This is because,if initially measurement.4gives a and the system is later found to give the result r to measurement C, the quantity B must have had some value at the time intermediate to .d. ar'd C. The probability that it-was D is P"0". We sum, or integrate, over all the mutually exclusive alternatives for 6 (symbolized by Ia). Now, the essential difference between classical and quantum physics lies in Eq. (2). In classical mechanics it is always true. In quantum mechanics it is often false. We shall denote the quantum-mechanical probability that a measurement of C results in c when it follows a measurement of ,4 giving a by P""t. Equation (2) is replaced in quantum mechanics by this remarkable law:6 There exist cornplexnumbers eab pbct 9o"such that Pa:lp,ul',

P6":leul',

and P".c:le*l'.

(s)

The classical law, obtained by combining (1) and (2), (4) P"":L PdPb" b

is replaced by go":Z

gotg,,",

(5)

b

lf (5) is correct,ordinarily (4) is incorrect.The logical error made in deducing (4) consisted, of course, in assuming that to get from a t6 c the system had to go through a condition such that B had to have some definite value, D. If an attempt is made to verify this, i.e., if B is measured betweeri the experiments A and C, then formula (4) is, in fact, correct. More precisely, if the apparatus to measure B is set up and used,but no attempt is made to utilize the results of the B measurement in the sense that only the A to C correlation is recorded and studied, then (4) is correct. This is becausethe B measuringmachine has done its job; if wi wish, we could read the meters at any time without 5 We have assumed 6 is a non-degenerate state, ancl that therefore (1) is true. Presumably, if in some generalization of quantum mechanics (1) were not true, even for pure states 6, (2) could be expected to be replaced by: There such that Po"= lpd,l'. The anaare complex numbers ,pou" log of (5) is then eo":Zu e*.

MECHANICS

369

disturbing the situation any further. The experi ments which gave o and c can, therefore, be separated into groups dapending on the value of b. Looking at probability from a frequency point of view (4) simply results from the statement that in each experiment giving a and, c, B had, some value. The only way (4) could be wrong is the statement, "B had some value," must sometimes be meaningless. Noting that (5) replaces (4) only under the circumstance that we make no attempt to measureB, we are led to say that the statement, "B had some value," may be meaningless whenever we make no attempt to measureB.o Hence, we have different results for the correlation of a and c, namely, Eq. (a) or Eq. (5)' depending upon whether we do or do not attempt to measure B. No matter how subtly one tries, the attempt to measure B must disturb the system, at least enough to change the results from those given by (5) to those of (4).7 That measurements do, in fact, cause the necessary disturbances, and that, essentially, (4) could be false was first clearly enunciated by Heisenberg in his uncertainty principle. The law (5) is a result of the work of Schroedinger,the statistical interpretation of Born and Jordan, and the transformation theory of Dirac.s Equation (5) is a typical representation of the wave nature of matter. Here, the chance of finding a particle going from o to c through several different routes (values of D) may, if no attempt is made to determine the route, be represented as the square of a sum of several complex quantities-one for each available route. 6 It does not help to point out that we could' have measured .8 had we wished. The fact is lhat we did not. t How (4) actually resrrlts from (5) when measurements disturb the'system has been studied particularly by J. vorr Neumann (Mathemtisc he Gr und.logen der Quantennechanik (Dover Priblietions, New York; 1943)). The effect of perturbation of the mmsuring equipment is effectively to chanse the phase of the interferinq components, by 4,, sy, so th"at (5)'becomes go":!u eieip"oc*. However, as von Neumann shows. the ohase shifts must remain unknown if B is measured so that the resulting probability P* is the square of 9o. averaged over all phases, da. This results in (4). 8 II A and B are the operators corresponding to measurements r4 and B, and if tL" and 9a are s6lutioni of A,9': a,1," and Ba6:!"u, lhet 9,b: Jixb*{.dx:(x0", 'y'"), Thus, ,p.ais an element (a lD) of the transfomation matrix for the in which A is transfomtion f.on a.eotesentation diagonal to one in which B i! diagonal.

324 370

R.

P.

FEYNMAN

Probability can show the typical phenomena of interference, usually associated with waves, whose intensity is given by the square of the sum of contributions from different sources.The electron acts as a wave, (5), so to speak, as long as no attempt is'made to verify that it is a particle; yet one can determine, if one wishes, by what route it travels just as though it were a particle; but when one does that, (4) applies and it does act like a particle. These things are, of course, well known. They have already been explained many times.e However, it seemsworth while to emphasizethe fact that they are all simply direct consequencesof Eq. (5), for it is essentiallyEq. (5) that is fundamental in my formulation of quantum mechanics. The generalization of Eqs. (4) and (5) to a Iarge number of measurements,say A, B, C, D, . , ., K, is, of course, that the probability of the a, b, c, d, ' ' ., & is sequence Paud...n:I v*"a..*12.

Assume that we have a particle which can take up various values of a coordinate r. Imagine that we make an enormous number of successive position measurements,let us say separated by a small time interval e. Then a succession of measurementssuch as A, B, C, . '' might be the successionof measurements of the coordinate * times 11,tz,tu ' ' ' , where t;+t: t;* e. at successive Let the value, which might result from measurement of the coordinate at time li, be ri. Thus, if ,4. is a measurement of.x at h then rr is what we previously denoted by a. From a classical point of view, the successivevalues, *r, fib xs, ' ' ' of the coordinate practically define a path r(r), Eventually, we expect to go the limit e-+0. The probability of such a path is a function o f .x y x 2 , . . . , l C i , . ' ' , s a y P ( " ' * n , x i + r ,' ' ' ) . The probability that the path lies in a particular region R of space-time is obtained classically by integrating P over that region. Thus, the probability that r; lies between a; and b;, and r;11 lies betweenoi+r ?nd D+r, etc., is

The probability of the result a, c, k, for example, it b, d, " . are measured,is the classicalformula:

,,"r:14'..r*"0...r,

(6) :

while the probability of the same sequenceo, c, & if no measurements are made between A and C and between C and K is

frrr...

r c i , t c i +" l ,. ) . ' . d . x " d . x * t . . . ,( s )

the symbol fi meaning that the integration is to be taken over those ranges of the variables ( 7 ) which lie within the region R. This is simply P*oo:l!t "'p"u"o...ol'. Eq. (6) with a, b, . . . replaced by tc1,tc2,' ' ' and integration replacing summation. probability pabcd...kw€ the can call The quantity In quantum mechanics this is the correct amplitude for the conditionA:0,, B:b, C:c, D:d, . ", K:k. (It is, of course,expressibleas formula for the casethat rr, xcz,''', sr' '' ' were actually all measured, and then only those paths a product ,p"ugo,p,a'"pio') lying within R were taken. We would expect the 3. THE PROBABILITYAMPLITUDEFOR A result to be different if no such detailed measureSPACE.TIMEPATH ments had been performed. Supposea measure'Ihe physical ideas of the last section may be ment is made which is capable only of deterreadily extended to define a probability ampli- mining that the path lies somewherewithin R' The measurement is to be what we might call tude for a particular completely specified spacetime path. To explain how this,rnay be done, we an "ideal measurement." We suppose that no shall limit ourselves to a one-dimensional prob- further details could be obtained from the same lem, as the generalization to several dimensions measurement without further disturbance to the is obvious. system. I have not been able to find a precise definition. We are trying to avoid the extra e See, for eremple, W. Heisenberg, Thz Physhal Prinuncertainties that must be averaged over if, for ciples oJ the Quantum Theory (University of Chicago Press, example, more information were measured but particularly IV. Chapter Chiego, 1930),

325 NON-RELATIVISTIC

not utilized. We wish to use Eq. (5) or (7) for all r; and hdve no residual part to sum over in the manner of Eq. (a). We expect that the probability that the parricle is found by our "ideal measurement" to be, rndeed,in the region R is the square of a complex rumber le@)lt.The number e(R), which we rray call the probability amplitude for region R is given by Eq. (7) with a, b, ... replacedby ::;, x41t . , . and summation replaced by in:egration:

QUANTUM

MECHANICS

371

shall be normalized to unity. It may not be best to do so, but we have left this weight factor in a proportionality constant in the secondpostulate. The limit e+0 must be taken at the end of a calculation, When the system has several degreesof freedom the coordinate space n has several dimensions so that the symbol r will represent a set of coordinates (x.(L),x@, . . . , rc(/.))for a system with E degrees of freedom. A path is a sequence of configurations for successive times and is described by giving the configuration rr or f ( x ; Q ) , y r ( z ) , . . . , x i ( k )i).,e . , t h e v a l u e o f e a c h o f :(R):Lim I .--'O a, R the D coordinates for each time tr. The symbol dri X ! D ( .. . r r , r t + r . . . ) . . . d x " d , x . i +. -t.. ( 9 ) will be understood to mean the volume element in fr dimensional configuration space (at time t). The complexnumber @(.'.rcu,ri+r. . .) is a func- The statement of the postulates is independent :ion of the variables r; defining the path. of the coordinate system which is used. 'Ihe -\ctually, we imagine that the time spacing e appostulate is limited to defining the results :roaches zero so that i[ essentially depends on of position measurements. It does not say what -Jreentire path lc(l) rather than only on just the must be done to define the result of a momentum '"-aluesof rcl at the particular times l;, rr:x(t). measurement, for example. This is not a real iVe might call (Fthe probability amplitude func- limitation, however, because in principle the :ional of paths r(l). measurement of momentum of one particle can .We may summarize these ideas in our first be performed in terms of position measurements :ostulate:. of other particles, e.g., meter indicators. Thus, I. If an id.eal measuremenl is performed, to an analysis of such an experiment will determine :elermine whethera particle has a path lying in a what it is about the first particle which deter.egion of space-time,then the probability that the mines its momentum. .esult will, be afi,rmatiae is lhe absolute square of a 4. TIIE CALCIJLATIONOF TIIE PROBABILITY :um oJ complex contributions, oneJrom eachpath AMPLITUDEFOR A PATH in the region. The statement of the postulate is incomplete. The first postulate prescribes the type of .ihe meaning of a sum of terms one for "each" mathematical framework required by quantum :ath is ambiguous. The precise meaning given mechanics for the calculation of probabilities, ,e Eq. (9) is this: A path is first definedonly by The second postulate gives a particular coRtent :he positions r; through which it goes at a to this framework by prescribing how to compute :equenceof equally spaced times,rok:t;-tle. the important quantity iD for each path: lhen all valuesof the coordinateswithin R have II. The paths contribute equal,l,yi,n magnitude, 'fhe ;,r equal weight. actual magnitude of the but the phase of thei,r contribution is the cl,asshal, ;eight depends upon e and can be so chosen aclion (in units oJ h); i.e,, the t;imeintegral,oJ the --:ratthe probability of an everit which is certain Lagrangian taken olong the palh. That is to say, the contribution O[*(l)] from a roThere are very interestinq rnathematical oroblems given path r(l) is proportionalto exp(i/h)Slx(t)1, :volved in the aftempt to a-void the subdiviiion and . niting proceses. Some sort of complex measure is beins where the action S[r(f)]: Jf L@Q),:r(t))dt is the :-sciated with the space of functions r(l). Finite resulti :=n be obtained under unexpected circumstances beeuse time integral of the classical Lagratgian L(i, x) ':e masure is not positive everywhere, but the contributaken along the path in question, The Lagrangian, -:ns from most of the paths largely cancel out. These which may be an explicit function of the time, :rrious mathemaliel problems are sidesteppedbv the sub':vision process. However, one feels as-Cavafieri must is a function of position and velocity. If we ':ve felt caiculating the volume of a pyramid before the :,Yennon ol Glculus. suppose it to be a quadratic function of the

326 372

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velocities, we can show the mathematical equivalence of the postulates here and the more usual formulation of quantum mechanics. To interpret the first postulate it was necessary to de6ne a path by giving only the successionof points r; through which the path passes a( successivetimes l;. To compute $:f,L(i,x)d,t we need to know the path at all points, not just at rr. We shall assume that the function r(l) in the interval between t; and,t6,r1is the path followed by a classicalparticle, with the Lagrangian -L, which starting from ru; at ,i reaches x;q1 dt l;+r. This assumption is required to interpret the second postulate for discontinuous paths. The quantity @(...tn, x+y ...) can be normalized (for various e) if desired, so that the probability of an event which is certain is normalized to unity as e+0. There is no difficulty in carrying out the action integral becauseof the sudden changesof velocity encountered at the times ,d as long as Z does not depend upon any higher time derivatives of the position than the first. Furthermore, unless Z is restricted in this way the end points are not sufficient to define the classical path. Since the classical path is the one which makes the action a minimum, we can write S:E

S(*+r,r),

(10)

where f';+r

S(xrar,rr) : Min. I .,

L(i(t), x(t))dt. (11)

ti

Written in this way, the only appeal to classical mechanics is to supply us with a Lagrangian function. Indeed, one could consider postulate two as simply saying, "iF is the exponential of i times the integral of a real furrction of r(l) and 'fhen the classical its first time derivative." motion might be derived later as equations of the limit for large dimensions. The function of r and o then could be shown to be the classical Lagrangian within a constant factor. Actually, the sum in (10), even for finite e, is infinite and hence meaningless (because of the infinite extent of time). This reflects a further incompletenessof the postulates. We shall have to restrict ourselves to a finite, but arbitrarilv long, time interval.

Combining the two postulates and using Eq. (10), we find ,p(R):Lirrr l' ._-,, Jn

li

1

dx;rtdxi

xexnf-E,.s1'"',r.)J.'. A i-

, (12)

where we have let the norrnalization factor be split into a factor 1/.4 (whose exact value we shall presently determine) for each instant of time. The integration is just over those values )ct,xct+r,. . . which lie in the region R. This equation, the definition (11) of S(r,4r, r;), and the physical interpretation of I e(R) l'z as the probability that the particle will be found in R, conrpleteour forrnulation of quantum mechanics. 5. DEFINITION OF THE WAVE FUNCTION We now proceed to show the equivalence of these postulates to the ordinary formulation of quantum mechanics.This we do in two steps. We show in this section how the wave function may be defined from the new point of view. ln the next section we shall show that this function satisfies Schroedinger's differential wave equation. We shall seethat it is the possibility, (10), of expressingS as a sum, and hence iDas a product, of contributions from successivesections of the path, which leads to the possibility of defining a quantity having the properties of a wave function. To make this clear, let us imagine that we choosea particular time I and divide the region R in Eq. (12) into pieces, future and past relative to l. We imagine that R can be split into: (a) a region R', restricted in any way in space, but lying entirely earlier in time than some l', suilt that tt ti (c) the region between t' and t" in which all the values of * coordinates are unrestricted, i,e., all of space-time between t' and,1", The region (c) is not absolutely necessary.It can be taken as narrow in time as desired.'However, it is convenientin letting us considervarying I a little without having to redefine R' and R". Then le(R',R")1'? is the probability that the

327 NON-RELA'IIVIS'TI

C QUANTUM

i\,{ECHANI CS

J/J

path occupiesR' and R". BecauseR' is entirely previous to R", considering the time I as the present, we can expressthis as the probability .a JR, that the path had been in region R' and will be in regionR". If we divide by a factor, the probar,-r fi ldxr i drr-z ': (l.s) X e x p f- I S ( . r ; , r , . r |' ; ) bility that the path is in R', to renormalizethe A Lhi-* J A probability we find : I e(R', R") l' it the (relative) probability that if the system were in region R' and r fi ' it will be found later in R". c x 1 , l- ! S ( . r , ' ' , l):Linr 'lhis x * ( x u , I is, of course,the important quantity in ",)] ' fr JR" Ll1 i k predicting the results of many experiments.We I d.xpal tlxpy2 prcparethe system in a certain way (e.g.,it was (16) in regiolt R') anrl then measurc some other A AA I)roperty (e.g.,will it be found in region R"?). 'fhe symbol R' is pla<:ed on the integral for ry' What does (12) say about computing this quantity, or rather the quantity s(R', R") of to indicate that the coordinates are integrated over the region R', and, for lr between t' and t, which it is the square? Let us suppose in Eq. (12) that the time I over all space. In like manner, the integral for 1t correspondsto one particular point & of the sub- is over R" antl over all space for those coordinates corresponding to tinres betweetr t and ,". The rlivisionof time into steps e, i.e., assumel:lp, the index &, of course, depending rtpotr the asterisk on 1* denotes contplex conjugate, as it 'fhen, the exportentialbeing the will be found ntore cottvettient to define (16) as subdivisione. cornplex conjugate of some quantity, l. exponentialof a suttr tnay be split into a product the'Ihe quantity f depends,or-rly upon the region of two factors

)-ll,:,,r

R' previous to l, and is completely defined if that region is known. lt does not depend, in any way, upon what will be done to the system 'Ihis I Ln i:k latter information is contained after time l. ry'and 1 we have separated the in Thus, with 1. *-r 1 li ."*oL; s(*r+r, x)J. (13) past history from the future experiences of the ,l system. This permits us to speak of the relation of past and future in the conventional manner' The first factor contains only coordinates with Thus, if a particle has been in a region of spaceindex ft or higher, while the second contains only time R' it may at time I be said to be in a certain coordinates with inclex ft or lower. This split is condition, or state, determined only by its past possible because of Eq. (10)' which results essen- and described by the so-called wave functiotr tially from the fact that the Lagrangian is a ry'(rc,l). This function contains all that is needed function only of positions and velocities. First, to predict'future probabilities. For, suppose, in the integration on all variables x; for 'i)k can another situation, the region R'were different, be performed on the first factor resulting in a say r', and possibly the Lagrangian for times function of rc* (times the second factor). Next, before I were also altered. Bttt, nevertheless, the integration on all variables xr f.or i Lk can suppose the quantity fron-r liq. (15) turned out 'fhctr, at:corrling to (14) the be performed on the second factor also, giving a to be the same. function of r7". Finally, the integration oo xk can probability of cn
ti* c*pl _ I

I S ( r ' + ' ,. t ) |

328 374

R.

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FEYNX{AN

Likewise, the function 7*(x, t) characterizes the experience, or, let us say, experiment to

which the system is to be subjected.If a different

r" anddifferent L;;;;;;;il;;;;;; region, to give the same 1s*(x,t) ttia Eq. (16), as does region R", then no matter what the preparation, r/, Eq. (1a) says that the chance of finding the system in R" is always the same as finding it in r". The two "experiments" R" ar^d r" are equivalent, as they yield the same results. We shall say loosely that these experimentsare 10 determine with wh:at probability the system is in state 1. Actually, this terminology is poor. The system is really in state ry'.The reason we can associate a state with an experiment is, of course, that for an ideal experiment there turns out to be a unique state (whose wave function is y@, t)) for which the experiment succeedswith certainty. Thus, we can say: the probability that a system in state ry'will be found by an experiment whose characteristic state is 1 (or, more loosely, the chance that a system in state ry'will appear to be in 1) is

t),t@, lf ur., Ddrl.

we were to compute I at the next instant of time:

t'

rx

9(x*+rtte):J., *oL;,I-

L

I

s("'*" "l dx* d.r*-r

*n

o'-..(15')

This is similar to (15) except for the integration over the additional variable ri and the extra term in the sum in the exponent. This term means that the integral of (15') is the same as the integral of (15) except for the factor (l/A) exp(i/h)S(r;,ar,r*). Since this does not contain any of the variables x; for i less than i, all of the integrations on dr; up to drr-r can be performed with this factor left out. However, the result of these integrations is by (15) simply t@6 t). Hence,we find from (15') the relation t(x*r'

llr) rfi-l

: | t)dxr1A. (18) :S(rr+,,x,,)l,l'@r, J L h"*pl I 'fhis

relation giving the development of f with time will. be shown, for simple examples,with suitable choice of ,4., to be equivalent to These results agree, of course, with the prin- Schroedinger'sequation.Actually, Eq. (18) is not 'l'hey ciples of ordinary quantum mechanics. are exact, but is only true in the limit e+0 and we a consequenceof the fact that the Lagrangian shall derive the Schroedingerequation by assumis a function of position,velocity, and time only. ing (18) is valid to first order in e. The Eq. (18) needonly be true for small e to the first order in e. 6. THE WAVEEQUATION For if we consider the factors in (15) which carry To complete the proof of the equivalence with us over a finite interval of time, I. the number the ordinary formulation we shall have to show of factors is T / e. lf an error of order ezis made in that the wave function defined in the previous sec- each, the resulting error will not accumulate tion by Eq. (15) actually satisfiesthe Schroedinger beyond the order e2(T/e) or Ie, which vanishes wave equation. Actually, we shall only succeed in the limit. We shall illustrate the relation of (18) to in doing this when the LagrangianZ in (11) is a quadratic, but perhaps inhomogeneous,form in Schroedinger's equation by applying it to the the velocitieso(r). This is not a limitation, how- simple caseof a particle moving in one dimension ever, as it includes all the cases for which the in a potential Z(rc). Before we do this, however, Schroedinger equation has been verified by ex- we would like to discusssome approximations to the value S(r;+r, r;) given in (11) which will be periment. The wave equation describesthe development sufficient for expression (18). The expressiondefinedin (11) for S(r;a1,r;) is of the wave function with time. We may expect to approach it by noting that, for finite e, Eq. (15) difficult to calculate exactly for arbitrary e from permits a simple t'ecursive relation to be de- classical mechanics,Actually, it is only necessary veloped. Consider the appearance of Eq. (15) if that an approximate expressionfor S(r;."1, *1) be

(r7)

329 NON_REI,A'I'IVISTIC

QUANTUM

375

MECHANICS

For this example, then, Eq. (18) becomes

used in (18), provided the er.rorof the approxi-

o':a)' iJL'lr:1".:','J# J,:"1xl1T;T:i"" fi,:l1"J,ff : (x,+,,, e * * ) - L :l Z t 2!/ \ *l | J " isaquadratic,butperhapsinhomogeneous,form e / in the velocities i(r). As we shall see later, the paths which are important are those for which xi+t- x; is of order ei. Under these circumstances, it is sufficient to calculate the integral in (11)

- I/(r:r1r) AoUo. frrl l]vf*r,

overthe classicalpath taken by aJrie particie'lr Let us call ***t:x'i and r1E1-rc4:f so that Then (23) becomes the path of a free x*:x-t. In Cartesiancoord.inatest2 particle is a straight line so the integral of (11) imt, f can be taken along a straight line. Under these t!(x,l-te): I expJ e'2lt circumstancesit is sufficientlvaccurateto replace rule by the trapezoidal the integral -ieV(x) Jt ."*pQ4) h-'PQ-t,t)i. e /x;a1-x; \ S(rrpr,xJ: Ll-,xivrl 2\ e / The integral on I will converge if t@, t) falls off sufficiently for large r (certainly if - '', *,) (re) f 9*@)'!(x)d.x:1). In the integration on {, since ri r.(')" e is very small, the exponential of.itnEz/2heoscil/ 2\ t lates extremely rapidly except in the region about f:0 (f of order (he/m)t). Since the funcor, if it proves more convenient, tion ty'(*-f, l) is a relatively smooth function of f (since e may be taken as small as desired), /X;+t-X;,rilr+Ji\ rr) : ,Ll S(xr+r, l. (20) the region where the exponential oscillatesrapidly \e2/ will contribute very llttle becauseof the almost complete cancelation of positive and negative These are not valid itr a general coordinate contributions. Since only small f are effective, system, e.g., spherical. An even simpler approximay be expanded as a Taylor series. {(x-t,l) mation may be used if, in addition, there is no Hence, vector potential or other terms linear in the 1-ieV(x)1 tl'(x,t-le):*o( velocity (see page 376): u / /x;+t-x; \ --:, x,*r S(x,ar, .r;) = cLl'-:l. t \ e

Thus, for the simple example of a particle of mass tn moving in one dimension under a potential V(x), we can set me f X,,t_

.s(r.1 ; a,x , ) = ; ( = - a

X;"

)

f x)exp

Ql)

,*%)lro,D-YYl? .:'J#-

.fotro.(2.5)

Now f-

-,v.,,,,).

(2thci'ln)t' (zz) J -"e"oQnt'/2he)d|:

1r It is assumed that the "forces" enter throllgh a scalar and vector potential and not in terms involving the square of the veloiity. More generally, what is meant by a free ymrticle is one for which the Lagrangian is altered by omission of the terms linear in, and those independent of, the velocities. t2 More generally, coordinales for which the terms quadratic in the velociry in L(r, r) appear with constant coeflicients.

r' * f , 2he){d| : o, [ -*.*"*r I

(26)

exp(inl'z/2h e)t'zdt : (hei/n)(2rh ei/ m)r,

while the integral containing {3 is zero, for like

330 376

R.

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FEYNMAN

the one with f it possessesan odd integrand, This example shows that most of the contribuand the ones with fa are of at least the order e tion to t@*+r,t*e) comesfrom values of 4 in smaller than the ones kept here.raI{ we expand *(xn, t) which are quite close to ru;..1(distant of the left-hand side to first order in e, (25) becomes order el) so that the integral equation (23) can, in the limit, be replaced by a differential equation. a{@, t) The "velocities," (rs4r-r*)/e which are im0@,t)*e-portant are very high, being of order (h/me), at which diverges as e+0. The paths involved are, - i' v (*) hei/ m)t therefore, continuous but possessno derivative. : o( l?r ' \ "" They are of a type familiar from study of h / A Brownian motion. h e i0 2 ' ! ( x , 4 I t It is these .large velocities which make it ( 2 7 ) xl{tx,D+so necessary to be careful in approximating ** I S(r;a1,q,) from Eq. (11).15To replace V(xr*r) In order that both sides may agree to zero order by V(x1")would, of course, change the exponent in e, we must set in (18) by ielV(x*) - V(xn+t)f/h which is of order A: (2trhei./m)t. (28) e(x*+r-xr), and thus lead to unimportant terms of higher order than e on the right-hand side Then expandingthe exponentialcontaining I/(*), of (29). It is for this reasonthat (20) and (2 1) are we get equally satisfactory approximations to ^S(rr+-r, *;) av/i,\ when is no potential. there vector A term, linear 9&,t)te-,:\r--v(x) ) in velocity, however, arising from a vector potential, as Aid,t must be handled more careltei O\Lt / (2e)fully. Here a term in S(r4r, rr) such as A(*o*t) xl *(*,r+-l. \ 2rn 0x2/ X(x*at-x*) differs from A(tck)(xk+L-tck)by a term of order (r41-4)'z, and, therefore, of Canceling {t(x, t) lrom both sides, and comorder e. Such a term would lead to a change in paring terms to first order in e and multiplying the resulting wave equation. For this reason the bv -lt/i one obtains approximation (21) is not a sufficiently accurate 11ha1z ha{ ___ :_l __ | p+v.t*),t,, (30) approximationto (11) and one like (20), (or (19) from which (20) differs by terms of order higher i at 2th\i ax/ than e) must be used. If A representsthe vector which is Schroedinger'sequation for the problem potential and p: (h/i)V, the momentum operin question. ator, then (20) gives,in the Hamiltonian operator, The equation for 1* can be developed in the a term (r/2m)(p- (e/c)A). (p- (e/c)A), while same way, but adding a factor ilecreases the time (21) gives (r / 2m)(p. p - (2e/ c)A. p { (e'z /c'z)A.A). by one step, i.e.,.x* satisfiesan equation like (30) These two expressions differ by (he/2irnc)V.A but with the sign of the time reversed. By taking complex conjugates we can conclude that 1 is, there would be no state into which a system may be put satisfies the same equation as ry',i.e., an experi- for which a pa.rticular experiment gives certainfo' foi a result. The class of functions 1 is not identical to the class ment can be defined by tlre particular state 1 to of available states rr, This would result if, for example, 1 mtisfied a different equation than 'y'. which it corresponds.la 15Equation (18) is when (11) is used for tr Rellv, these intesrals are oscillatorv and not defined. but they may be defiied by using a convergence factor. Such a factor is automatically provided by {'(x-t,t) in (24). lt a more fomal procedure is deired replace ft by for enmple, where 6 is a small positive number, h(l-i6), and then let &*0. rr Dr, Hartland Snyder has pointed out to me, in private conversation, the very interdsting possibility thai there may be a generalization of quantum mechanics in which the states mesured by experiment ennot be prepared ; that

actually enct S(*;11, *;) for arbitrary c for eses in which the potential does not involve * to hieher powers than the second (e.g., free particle, hamonic osiillator). It is necessary, however, to use a more accurate value of l. One can define ,4 in this way. Assume classiel particles with l degrees of freedom start from the point ri, ti with unifom density in momentum space. Writti the number of particle having a given component of momentum in range dp as p dp/?owith 1o conslant. a6sn tr:(2rhi/po)bt2p-r,where is the density in I dimensional coordinate space ri+r of these particles at time l;ar.

33r NON_RELATIVISTIC

QUANT'UM

which may not be zero. The question is still more important in the coemcient of terms which are quadratic in the velocities. In these terms (19) and (20) are not sufficiently accurate representations of (11) in general. It is when the coeflicients are constant that (19) or (20) can be substituted for (11). If an expressionsuch as (19) is used, say for spherical coordinates, when it is not a valid approximation to (11), one obtains a Schroedinger equation in which the Hamiltonian operator has someof the momentum operators and coordinates in the wrong order. Equation (11) then resolvesthe ambiguity in the usual rule to replace p and q by the non-commuting quantities (h/D(a/ad and q in the classical Hamiltonian H(P, q), It is clear that the statement (11) is independent of the coordinate system. Therefore, to find the differential wave equation it gives in any coordinate system, the easiest procedrtre is first to find the equations in Cartesian coordinates and then to transform the ioordinate system to the one desired. It suffices,therefore, to show the relation of the postulates and Schroedinger's equation in rectangular coordinates. The derivation given here for one dimension can be extended directly to the case of threedimensional Cartesian coordinates for any number, K, of particles interacting through potentials with one another, and in a magnetic field, described by a vector potential. The terms in the vector potential require completing the square in the exponent in the usual way for Gaussian integrals. The variable , must be replaced by the set,(r) to r(s.K)where rc(l),)cer,,c(a)are the coordinates of the first particle of mass rnt, xQ), r(6r, x<6)of the second of mass mz, etc. The ' ' 'dr<arl, and symbol dr is replaced by d.xrr)d,v
MECHANICS

377

describewhat can be describedby non-relativistic quantum mechanics, neglecting spin, 7. DISCUSSTON OF Tr{E WAVEEQUATION The Classical Limit This completes the demonstration of tlre equivalence of the new and old formulations. We should like to include in this section a few remarks about the important equation (18), This equation gives the development of the wave function during a small time interval. It is easily interpreted physically as the expressionof Huygens' principle for matter waves.'In geometrical optics the rays in an inhomogeneous medium satisfy Fermat's principle of least t'ime, We may state Huygenst principle in wave optics in this way: If the.amplitude of the wave is known on a given surface, the amplitude at a iear by point can be consideredas a sum of contributions from all points of the surface. Each contribution is delayed in phase by an amount proportional to the time it would take the light to get from the surface to the point along the ray of least lime of geometrical optics. We can consider (22) in an analogous manner starting with Hamilton's first principle of least action for classical or "geometrical" mechanics. If the amplitude of the wave ry'is known on a given "surface," in particular the "surface" consisting of.all x at time l; its value at a particular nearby point at time l{ e, is a sum of contributions from all points of the surface at L Each contribution is delayed in phase by an amount proportional to the action it would require to get from the surface to the point along the path of least action of classical mechanics.r6 Actually Huygens' principle is not correct in optics. It is replaced by Kirchoff's modification which requires that both the amplitude and its derivative must be known on the adjacent surface. This is a consequenceof the fact that the wave equation in optics is second order in the time. The wave equation of quantum mechanics is first order in the time; therefore, Huygens' principle fs correct for matter waves, action replacing time. ro See in this connection the very interesting Schroedinger, Ann. d. Physik 79,489 (1926).

remarks of

332 R.

378

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FEYNMAN

The equation can also be compared mathematically to quantities appearing in the usual formulaiions. In Schroedinger'smethod the development of the wave function with time is given by

-h

at

:w,

First we note that the wave function at x" at time t't can be obtained from that at r' at time t'by f ''' f *(x", t"): Lim | I

(31)

fi i-r

I

s)l Xextf-! Stx;+r,

i3t which has the solution (for any e if H is time independent) (32) ,!'(x,t* e): exp(-iell/h){(x' t)'

dxo d.h

dx i,r

x'l'@',t')TT' A,

(r5)

:t" -t' where we ptJt xn=vt arrdx j:-tctt where je no reassume we (between the times t' and t" striction is being put on the region of integration) ' This can be seen either by repeated applications of (18) or directly from Eq. (15). Now we ask, as y'z+0what values of the intermediate coordinates ri contribute most strongly to the integral? These will be the values most likely to be found by exoperator equation periment and therefore will determine, in the ii-it, th" classicalpath. If h is very small, the (33) (ieH/h). x' :exp(ieE/h)xexP exponent will be a very rapidly varying function the positive The transformation theory of Dirac allows us to of any of its variables:ct.As ri varies, exponent the e) of t* e ' contributions ' timetl negative at ,rlt(x' and considerthe wave function rcicontributes which at region representation The in a nearly cancel. state a representing as -in phase of the which r' is diagonal, while 'y'(r, l) represents the most strongly is that at which the is exponent varies least rapidly with *; (method of r which in representation a in state same in the exdiagonal. They are, therefore, related through the stationary phase). Call the sum relates (*'lr). which s; function Donent traisformation

Therefore', Eq. (18) expresses the operator by an approximate integral operexp(-i&l/h) ator for small e. From the point of view of Heisenberg one considers the position at time l, for example, as an operator x. The position x' at a latet time I * e can bL expressedin terms of that at time I by the

these rePresentations: f

l'(x',tIe): J

5:

a*' {*'l*),,1'{*,t)

Therefore, the content of Eq' (18) is to show that for small € we can set (34) exPliS(x',x)/h) (x'lx),:(L/A) with S(r', r) definedas in (11)' The close analogy between (r'lr)' and the pointed out on ouantity exp(lS(x', x) /h) has been we now see fact, In Dirac'r by occasions several quantithat to sufficientapproximationsthe two to-eachproportional be to taken be ties may oth".. Di.."'" remarks were the starting point of the present development' The points he makes concerning the passageto the classical limit Z+0 are very beautiful, and I may perhaps be excused for brieflY reviewing them here'

!

S(r4r, r).

(Jo,

i-0

Then the classical orbit passes,approximately, through those points rr at which the rate of change of ^Swith rr is small, or in the limit of small i, zeto, i.e., the classical orbit passes through the points at which ES/\x;:O for all ri' Taking the limit e+0, (36) becomes in view of (11)

s:

I ,,'

L(*(t),x(t))dt

(37)

We see then that the classical path is that for which the integral (37) suffers no first-order change on varying the path. This is Hamilton's principle and leads directly to the Lagrangian equationsof motion'

333 NON_RELATIVISTIC

8. OPERATORALGEBRA Matrir Elements

QUANTUM

MECHANICS

379

we discusshow it changeswith changesin'the preparation R'or the experiment R//. The state at time l'is defined completely by the preparation R'. It can be specified by a wave function t@',t') obtained as in (15), but containing only integrals up to the time t/. Likewise, the state characteristic of the experiment (region R") can be defined by a function y(x,,,t,,) obtained from (16) with integrals only beyond t,/. The wave function {(x",t") at time t,t can, of course,also be gotten by appropriate use of (15). It can also be gotten from,lr(r', l') by (35). According to (17) with ,// used instead of t, the probability of being found in x if prepared in ry'is the square of what we shall call the transition amplitude -f x*(x" , t")t(x,, , t,,)d.x,, . We wish to expressthis in terms of yat t" and,ry'at l/. This we can do with the aid of (35).Thus, the chancethat a system prepared in state tlty at time l' will be found after t" tobe in a state 1,,, is the squareof the transition amplitude

Given the wave function and Schroedinger's equation, of course all of the machinery of operator or matrix algebra can be developed. It is, however, rather interesting to express these concepts in a somewhat different language more closely related to that used in stating the postulates. Little will be gained by this in elucidating operator algebra. In fact, the results are simply a translation of simple operator equations into a somewhat more cumbersome notation. On the other hand, the new notation and point of view are very useful in certain applications described in the introduction. Furthermore, the form of the equations permits natural extension to a wider class of operators than is usually considered (e.g.,onesinvolving quantities referring to two or more different times). If any generalization to a wider class of action functionals is possible, the formulae to be developed will play an important role. ff We discuss these points in the next three \ x , ' 1 t 1 9 , , 7 " : l ' "f J x * G " , t " ) J sections. This section is concerned mainly with definitions. We shall define a quantity which we (3s) X c x p ( i S z f r ) r y ' (0xY' ,. . . d 2 6 * , call a transition element between two states. It AA is essentially a matrix element. But instead of being the matrix element between a state ry'and where we have used the abbreviation (36). another 1 corresponding to the same time, these In the language of ordinary quantum metwo states will refer to different times. In the chanics if the Hamiltonian, H, is constant, following section a fundamental relation between -tt)H/h}l,@,1') so that (38) {(x, t"):expl-i(lt transition elements will be developedfrom which is the matrix elementof expl-i(t" -t')H/hlbethe usual commutation rules between coordinate tween states xr,, and {{. and momentum may be deduced. The same If F is any function of the coordinates ni for relation also yields Newton's equation of motion t'1t;1t", we shall define the transition element in matrix form. Finally, in Section 10 we discuss of F between the states { at t' and y at t" Ior the the relation of the Hamiltonian to the operation action S as (x" =xi, r'=xo): of displacement in timel We begin by defining a transition element in .rf terms of the probability of transition from one Q,'1119,'1': ,rgJ J state to another. More precisely, supposewe have a situation similar to that described in deriving X x * ( x " , t " ) F ( x o ,x r , . ' ' x ) i17). The region l? consistsof a region R/ previous f i i-r 1 dx^ dcc'-, to t', all spacebetweent' and.t,,and the region R, .e*pf E s(rr+r,ri) ...-6*t. 1j91 : IVQ',t) . af.tert". We shall study the probability that a Lhi:o .l A A -rystemin region -R' is later found in region Rrr. This is given bV (17). We shall discussin this In thelimit e+0, Fis a functional of the path r(l). .ection how it changeswith changesin the form We shall seepresently why such quantities are rf the Lagrangian betweentt and,t',. In Section l0 important. It will be easier to understand if we

334 P.

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FEYNMAN

stop for a moment to find out what the quantities correspond to in conventional notation. Suppose F is simply ,e where & correspondsto, some time tr:1*. Then on the right-hand side of (39) the integrals from rcoto xk-l may be performed to Inlike produce!(xu t) or expl-i(t-t')H/hj!v. give manner the integrals on rci for j)i>k -t)H/h)xt',1*. Thus, the 7x(x*,t) or {exp[-i(1" transition elemtnt of rc*,

I x t,,* e-

(i I DH ( t " - t )x e- | i I h)H1t- t' )1lty d.x

:

r

t)x'!(x, t)dx I x*(t,

\ x u ,nl 1 9 , ' 1 " ' /

|

iei

I

\

|

h,-t

I

r'

: ( r , , , l F e x pL - U l x , t, ' ) l V/ rs> $ 2 ) which is obtained from (39). Incidentally, (41) leads directly to an importan.t perturbation formula. If the effect of [/ is small the exponential can be expanded to first order in U and we find

Q,'lFW,)u :

restricted class because the action must remain a quadratic function of velocities. From one observable functional others may be derived, for example, by

(40)

s,: (xu'I t1'1' 4" k y' | | 1,1,,,) i

tU(x;, t')l'1"'l' (43)

* .\xr'll is the matrix element o", * ., ,t-" l:l* between hi from t at time r develop would which { state the. Of particular importance is the case that xt" is a at t' and, the state which will develop from time I matrix element the state in which ry's,would not be found at all were therefore, lt'tsi t". at to Xt,, it not. for the disturbance,{/ (i'e., (xr'1t1,1,,'1u of x(l) between these states. :0) Then the Likewise, according to (39) with F:r*+r, transition element of rp;r is the matrix element of 1 x(r* e). The transition element of F: (x*+t x*) / e' -lQr,lL (44) ) r) sl, e t l ( s r ,t n W or oI h2i is the matrix element of (x(if e)-x(t))/e as is easily shown from (a0). We i(Hr-xE)/h, is the probabilityof transitionas inducedto first can call this the matrix element of velocity i(l). orderby the perturbation.In ordinarynotation, which problem second a we consider Suppose differs from the first because, for example, the (x ," I eU(xa, t) l'ttv)s T potential is augmented by a small amount U (' xt) . rt r l. Then in the new problem the quantity replacing : | { | xt,,*e-titDE(t"-t)ge-(itilII(t-t')tltydxld.t S is S':S*I ie[J(xt,l). Substitution into (38) ' J IJ leads directly to so that (44) reduces to the usual expressionlT for

\ x v ' l r l 9v )s '

time dependent perturbations.

(41) : (",1*r;t' u@n,r,rl,a,)".

9. NEWTON'S EQUATIONS The Commutation

Thus, transition elements such as (39) are important insofar as F may arise in some way from a change 6,5 in an action expression. We denote, by observable functionals, those functionals F in which can be defined, (possibly indirectly) terras of the changes which are produced by possible changes in the action S. The condition be observable is somewhat that a functional similar to the condition that an operator be Hermitian. The observable functionals are a

Relation

In this section we find that different functionals may give identical results when taken between any two states. This equivalence between functionals is the statement of operator equations in the new language. If F depends on the various coordinates, we 0F/0xa can, of course, define a new functiotal tt Pi. it4. Dirac, ThePrincipl,esoJ Quantum Mechanics (The Clarendon Press, Oxford,1935), second edttlon' Section47, Eq. (20).

335 NON_RELATIVISTIC

MECHANICS

QUANTUM

381

by differentiating it with respect to one of its potential, or force. Then (47) becomes variables, say cc*(0(& ( j). If we calculate h dF f /xt+t-xt kr'laF/axelgr,)s by (39) the integral on the Fl -ml 'i6x*s L right-hand side will contain |F/Ex*. The only \ € other plabe that the variable r/c appears is in S. Thus, the integration on rr can be performed by parts. The integrated part vanishes (assuming wave functions vanish at infinity) and we are left with the quantity -F(a/drx) exp(iS/h) If F doesnot depend on the variable 16,this gives in the integral. However, (d/dxt) exp(iS/h) Newton's equations of motion. For example, if F : (i,/h)(dS/ax*) exp(iSlZ), so the right side repre- is constant, say unity, (aS) just gives (dividing bv .) sents the transition elementof -(i/h)F(AS/Ax), i.e.. lrl / Xp11-.xy .rt -Jck .r \

-T)

-iQ''l'fflr'\, ("l*l'-),: (4s)

This very important relation shows that two different functionals may give the same result for the transition element between any two states. We say they are equivalent and symbolize the relation by hdF ds ___eF_, i |xr s 0x*

(46)

0.: --l s

e \

-,v,(*)] (4s)

--

t

e

/

I

- l-'(.rk).

Thus, the transition element of mass times acceleration f(rcaa1-x)f e-(xa-xr)/rJ/u between any two states is equal to the transition element of force -V'(xk) between the same states. This is the matrix expression of Nervton's law which holds in quantum mechanics. What happens if F does depend upon rua?For example, let F:xx. Then (48) gives, since 0F/6xr:1, -

the symbol

e

emphasizing

the fact that func-

tionals equivalent under one action may not be equivalent under another. The quantities in (46) need not be observable. The equivalence is, nevertheless, true. Making use of (36) one can write h AF

f d S ( x p p 1 ,1 6 )

'i 1xt s

L

arl

dS(x*, :c*_r)l

+-

Ext

I

l. ({7)

'Ihis equation is true to zero and first order in e and has as conrequences the commutation relations of momentum and coordinate, as well as the Newtonian equations of motion in matrix form. In the case of our simple one-dimensional problem, S(rc+r, rui) is given by the expression (15), so that dS(rr+r, x*) / 6x*:

- rn(xaa1- x4) e, /

and 0 S (x*, tcx-) / 6x* : * rn(xr- x t-r) / e- eVt (x r"); where we write

V'Qc) for the derivative

of the

xt, r* f*-r\ - ;lt? ' t -| m t /. -xr,+t.)-u ''(.j,]

I

or, neglecting terms of order e,

* (r:a,). ^- * (Yi:-:).,, r". (4s) In order to transfer an equation such as (49) into conventional notation, we shall have to discover what matrix corresponds to a quantity such as :rhtck+r.It is clear from a study of (39) that if F is set equal to, say, f(x)g(x*+), the corresponding operator in (40) is e - ( i t i l ( t , , t , z ) Hg ( x ) e r;t n t , uJ ( 1 )e ( i| i l ( t _ t , r l J , the matrix element being taken between the states 1,,, and rlty. The operators corresponding to functions of *1a1 will appear to the left of the operators corresponding to functions of 11, i.e., the order oJ terms in a matrir operator product corresponds to an order in time of the conespond.ing Jactors in a functional. Thus, if the functional can and is written in such a way that in each term factors corresponding to later times appear to the

336 R.

382

P,

FEYNMAN

left of factors corresponding to earlier terms, the corresponding operator can immediately be written down if the order of the operators is kept the same as in the functional.lS Obviously, the order of factors in a functional is of no consequence. The ordering just facilitates translation into conventional operator notation. To write Eq. (49) in the way desired for easy translation would require the factors in the second term on the left to be reversedin order. We see,therefore, that it corresPondsto

the derivative of which gives an expression like (51), But the changein zechangesthe normalizacorresponding to d'xx as tion constant lfA well as the action. The constant is changed 67 rot f,rom (2rhei/m)-t to (zrh"i/m(l|6))-t effect total The 6' in order first to *6(2trhei/m)-+ of the change in mass in Eq. (38) to the first order in 6 is

9r-xP:h/i where we have written p for the operator mir' The relation between functionals and the correspondingoperators is defined above in terms of the order of the factors in time. It should be remarked that this rule must be especially carefully adhered to when quantities involving velocities or higher derivatives are involved' The cor' rect functional to represent the operator (i)2 is actually (xt"+t-xt")fe' (x1,-x1"4)fe rather than as l(xx+t-x*)/e)2' The latter quantity diverges lf , as ,-0, This may be seen by replacing the secondterm in (49) by its valuex4tm(x*at-ecx) /e calculated an instant e later in time. This does not change the equation to zero order in e' We then obtain (dividing bY e)

(:y=)""_*;

(s0)

(x,,, I t ariml(x,+t- xn)/ ef'/ h* Eal{ r)' We expect the changeof order 6 lasting for a time e to be of order 6e.Hence, dividingby 6ei/h, we can define the kinetic energy functional as 1{..9:$rnl(x*+t-

x) / elz*h/2ei'

(52)

This is finite as e+0 in view of (50)' By making useofan equation which results froin substituting m(x*+t-xn) f e for F in (48) we can also show that the expression(52) is equal (to order e) to -

/xr.r-xr\/r*-rr-r\,...,,

t
,

)

(rr/

That is, the easiest way to produce observable functionals involving powers of the velocities is to replace thesepowers by a product of velocities' each factor of which is taken at a slightly different time. rO. TIIE IIAMILTONIAN Momentum

This gi-es the result expressedearlier that the 'tvelocity" (rx+t-x*)/e root mean square of the between two successivepositions of the path is of order e-i. It will not do then to write the functional for kinetic energY, saY, simPlY as *ml(xx+t-x) /c)'

(51)

for this quantity is infinite as e+0. In fact, it is not an observable functional. One can obtain the kinetic energy as an observable functional by considering the first-order change in transition amplitude occasionedby a change in the mass of the particle. Let m be changed to m(lJ-6) for a short time, say e,around le.The changein the action is Edeml(x*+t-xn)f c]2 rs Dirac has also studied operatorscontaining quantities referring to different times. Seereference2.

The Hamiltonian operator is of central importance in the usual formulation of quantum mechanics. We shall study in this section the functional corresponding to this operator' We could immediately define the Hamiltonian functional by adding the kinetic energy functional (52) or (.53)to the potential energy' This method is artificial and does not exhibit the important relationship of the Hamiltonian to time' We shall define the Hamiltonian functional by the changes made in a state when it is displaced in time' To do this we shall have to digressa moment to point out that the subdivision of time into egual' intervals is not necessary.Clearly, any subdivision into instants li will be satisfactory; the limits are to be taken as the largest spacing' t*t- li, approacheszero' The total action 'Smust

337 NON_RELATIVISTIC

QUANTUM

now be representedas a sum S: E S(*r+r,t41i:t;, t),

(s4)

i

where fl

S(ni+r,l;+r ; x;, t;) : J

L(i(t), x(t))dt,

I

(55)

[2\l'+r-li/

(xlt l/)u-(xl rl,l,a)sa: (s7) ,(xlH*l*)s, here the Hamiltonian by

functional

dS(r6ar, t41; x1",ta)

S(x4y tat; x;, t)

:l:l

is changedto 5(16.1,t*+tix*, rr-d). The constant l/A f.or the integration on iltchis also altered to (2rhi,(tr*t- tt * 6)/rn)+. The effect of these changeson the transition element is given to the first order in 6 by i6

tl

the integral being taken along the classical path between rt at t; and r;11 at l"11. For the simple one-dimensional example this becomes, with sufficient accuracy,

I m /x;a1-x;\'

383

MECHANICS

I7h:I

l -v(x+')l(r+r-l);

0h

(56)

J

the corresponding normalization cons,tant for integration on d.x; is / : (2rhi(t ar- t ;)/ m)-*. The relation of.H to the changein a state with displacement in time can now be studied. Consider a state ry'(r)defined by a space-time region -R'. Now imagine that we consider another state at time t, {r(t), defined by another region Rr,. Supposethe region R5' is exactly the same as R, except that it is earlier by a time 6, i.e., displaced bodily toward the past by a time d. All the apparatus to prepare the system for Ro, is identical to that for R' but is operated a tirne 6 sooner. If Z dependsexplicitly on time, it, too, is to be displaced, i.e., the state /6 is obtained from the Z used for state ry'except that the time I in Z6 is replaced by t+6. We ask how does the state ry'a differ from rlt?In any measurementthe chance of finding the system in a fixed region R// is different for R/ and Rr', Consider the change in the transition element (1l1lrl)sa produced by the shift 6. We can consider this shift as effected bv decreasingall valuesof t; by 6 for i ( i and leaving all l; fixed for i>k, where the time I lies in the interval between t*+t and ll.re This change will have no effect on S(r;..1, l;11 ; xt, t;) as defined by (55) as long as both l;*1 and rdare changedby the sameamount.On the otherhand,S(*1"a1,fua1;x1,, t1") te From the point of view of mathematical rieor. if 6 is h.nite, as F0 one gerl into difficulty in that, foi exbmole, the rnteryal ,i+r-rI is kept finite. This can be straishteiled out by 9.sslming 6 to vary with time and to be tuined on moothly belore ,:rr and turned off smoothly aftet t-h. Inen keeplng the ttme variation of 6 fixed, let e0. Then seek the trrst-order change as &*0. The result is essentiallv the same as that of the crude procedure ued above,

f/*

is defined

k

(58)

2i(th+t-t)

The last term is due to the change in I/A and servesto keep If4 finite as G-r0. For example, for the expression (56) this becomes

::(T:)' * (x "r # ^* v *,,), which is just the sum of the kinetic energy functional (52) and that of the potential energy V(x,,+r). The wave f unc tion { 5@,t) represents,of course, the same state as rl(x, t) will be after time d, i.e,, {t(r, t} 6). Hence, (57) is intimately related to the operator equation (31). One could also consider changesoccasionedby a time shift in the final state 1. Of course,nothing new results in this way for it is only the relative shift of y and { which counts. One obtains an alternative expression dS(*1a1,fua1irc;,ta) h I7*: -----+__. dln+t 2i(tpat-l*)

(59)

This differs from (58) only by terms of order e. The time rate of change of a functional can be computed by considering the effect of shifting both initial and final state together. This has the same effect as calculating the transition element of the functional referring to a later time. What results is the analog of the operator equation h. -f :Hf -fH. The momentqm functional pi can be defined in an analagous way by considering the changes

338 R.

384

P.

FEYNMAN

made by displacements of Position: 7A

(xIr I/)u- (xIr l,l, t)u: ". ;(xlp*lV>

The state ry'6is prepared from a region Ra'which is identical to region R/ except that it is moved a distance A in space. (The Lagrangian, if it depends explicitly on ,t, must be altered to for times previous to ,.) One Lt:L(n,*-A) finds2o dS(xl+r,rc*) dS(lrar, rr) (oui -b*:_ 6xt dx*at Sincega(r, l) is equal totlt(x-A,l), the closeconnection between 2& and the r-derivative of the wave function is established' Angular momentum operators are related in an analogousway to rotations. The derivative with respect to t+t of S(r;11,l;,'1;*;, li) appearsin the definitionof H;. The derivative with respect to r41 defines p;. But the derivative with respect to fr+r of S(*+r, l+r;r;, l;) is related to the derivative with respectto x+1, for the function S(r4r, l4r i rc;,l) defined bV (55) satisfies the HamiltonJacobi equation. Thus, the Hamilton-Jacobi equation is an equation expressingfll in terms of the fact that the pr. In other words, it expresses time displacementsof states are related to space displacementsof the same slates. This idea leads directly to a derivation of the Schroedinger equation which is far more elegant than the one exhibited in deriving Eq. (30).

needs, in addition, an appropriate measure for the space of the argument functions r(l) of the functionals.lo It is also incomplete from the physical standpoint. One of the most important characteristics of quantum mechanics is its invariance under unitary transformations. These correspondto the canonical transformations of classicalmechanics. Of course,the present formulation, being equivalent to ordinary formulations, can be mathematically demonstrated to be invariant under these transformations. However, it has not been formulated in such a way that 1t is physicall'y obvious that it is invariant. This incompleteness showsitself in a definite way' No direct procedure has been outlined to describe measurements of quantities other than position' Measurements of momentum, for example, of one particle, can be defined in terms of measurementsof positions of other particles. The result of the analysis of such a situation does show the connection of momentum measurementsto the Fourier transform of the wave function. But this is a rather roundabout method to obtain such an important physical result. It is to be expected that the postulates can be generalizedby the replacement of the idea of "paths in a region of space-timeR" to "paths of classR," or "paths having property R." But which properties correspond to which physical measurementshas not bedn formulated in a general way. 12.A POSSIBLEGENERALIZATION

The formulation suggestsan obvious generali zation. There are interesting classical problems which satisfy a principle of least action but for The formulation given here suffers from a seri- which the action cannot be written as an integral ous drawback. The mathematical conceptsneeded of a function of positions and velocities. The are new. At present, it requires an unnatural and action may involve accelerations, for example' cumbersome subdivision of the time interval to Or, again, if interactions are not instantaneous,it make the meaning of the equations clear. Con- may involve the product of coordinates at two The siderable improvement can be made through the different times, such as "fx(t)x(t*T)d't. use of the notation and concepts of the mathe- action, then, cannot be broken up into a sum of matics of functionals. However, it was thought small contributions as in (10). As a consequence' best to avoid this in a first presentation. One no wave function is available to describe a state' Nevertheless,a transition probability can be demWe did not immediately substitute A from (60).i-nto (47) because (47) woutd then no longer have been valid to 6ned for letting from a region R' into another 6oth zero order and the first order in c. We could derive R". Most bf the theory of the transition elements relations, but not the equations of ihe commutation (xu,lF?ulu can be carried over. One simply motion. The two expressions in (60) represent the momenta ai Lch end of the interual t; to t;+r. Th-ey differ by eV'(xta) invents a synrbol, such as (R"lFlR')s by an e' rT. INADEQUACIESOF THE FORMULATION

because of the force acting during the time

339 NON-nELA'frvrsTrc

QUANTUM

MECHANTCS

395

equation such as (39) but with the expressions (19) and (20) for ,! and, y substituted, and the more general action substituted for S. Hamiltonian and momentum functionalscan be definedas in section(10). Further detailsmay be found in a thesisby the author.2r

could also be eliminated. This presents an almost insurmountable problem in the conventional quantum mechanics. We expect that the motion of a particle o at one time depends upon the motion of D at a previous time, and uice aersa. A wave function {t(x",xt,;l), hr-rwever, can only describe the behavior of both particles at one 13.APPLICATIONTO ELIMINATE time. There is no way to keep track of what D did FIELD OSCILLATORS in the past in order to determine the behavior of One characteristic of the present formulation is o. The only way is to specify the state of the set that it can give one a sort of bird's-eye view of of oscillators at l, which serve to "remember" the space-time relationships in a given situation, what b (and a) had been doing. Before the integrations on the rr are performed in The present formulation permits the solution an expression such as (39) one has a sort of of the motion of all the oscillators and their comformat into which various F functionals may be plete elimination from the equations describing inserted. One can study how what goeson in the the particles. This is easily done. One must quantum-mechanical system at different times is simply solve for the motion of the oscillators beinterrelated. To make thesevague remarks some- fore one integrates over the various variables r; what more definite, we discussan example. for the particles. It is the integration over nj In classical electrodynamics the fields de- which tries to condense the past history into a scribing, for instance, the interaction of two single state function. This we wish to avoid. Of particles can be representedas a set of oscillators. course, the result depends upon the initial and The equationsof motion of theseoscillatorsmay final states of the oscillator. If they are specified, be solved and the oscillators essentially elimi- t h e r e s u l t i s a n e q u a t i o n i o r ( a y , l l l r l 1 , ) l i k e ( 3 8 ) , nated (Lienard and Wiechert potentials). The but containing as a factor, besides exp(iS/h) interactionswhich result involve relationships'of another functional G depending only on the the motion of one particle at one time, and of the coordinates describing the paths of the particles. other particle at another time. In quantum We illustrate briefly how this is done in a very electrodynamicsthe field is again representedas a simple case. Suppose a particle, coordinate r(l), set of oscillators. But the motion of the oscillators Lagrangian L(i, x) interacts with an oscillator, cannot be worked out and the oscillators elimi- coordinate q(t), Lagr angian | (Q,- o, q,), throu gh nated. It is true that the oscillators representing a term t@, t)q(t) in the Lagrangian for the longitudinal waves may be eliminated. The result system. Here 7(r, l) is any function of the is instantaneous electrostatic interaction. The coordinate r(l) of the particle and the time.22 electrostatic elimination is very instructive as it Suppose we desire the probability of a transition shows up the difficulty of self-interaction very from a state at time l', in which the particle's distinctly. In fact, it shows it up so clearly that wave function is /r,and the oscillator is in energy there is no ambiguity in deciding what term is level n, to a state at t" with the particle in 11,, incorrect and should be omitted. This entire and oscillator in level m. This is the square of

processis not relativistically invariant, nor is the omitted term. It would seemto be very desirable if the oscillators, representing transverse waves, 2! The theory of electromaqnetism described bv I. A. Wheeler and R. P, Feynmn, Rev. Mod. phvs. iZ. tSz (1945) can be expressed in a princiole of leasi action involving the coordinates of particl'es alone. It was an attempt to quantize rhis theoiy, without reference to the fields, which led the author t6 study the fomulation of quantum mechanics given here. The extension of the ldeas to cover the case of more qeneral action functions was developed in his Ph,D. thesisl',The princiole of leasr action in quantum mechanics" submittid to' princeton Universitv.1942.

Qr e-l I lt ue*) sr'"so*t, ff

: t "'I JJ

p^"(q)x,"*\*t)

:

X exp-(.S,1Su *Sr)*,, (ro)e^(Co) h

.'r" 9. . .d2. AoAa

de,-rd* re,. (61)

2 The generalization to the case that 7 depends on the velocity,.t, of the particle presents no problem.

340 386

R.

P.

FEYNMAN

Here p"(g) is the wave function for the oscillator in state z, So is the action ,--l

!

Sr(lc;11' 'r;)

be split into Riemann sums and the quantity 7@i, t i) substituted for tQ). Thus, Q dependson the coordinatesof the particle at all times through the'y(*i, lt) and on that of the oscillatorat times l' and t" only. Thus, the quarrtity (61) becomes

calculated for the particle as though the oscillat<-rr ( x * e ^ l 1 l t l , ' e * ) s r + s ur + ,: were abselt, [ eo:

i _ r ; - e, z q | a r _ 9 , \ 2

.Su:tl

lj:--|*-9","1 ;-oL2\ e

.,1

2

/

J

(where 7;:7(*r, r)) is the action of interaction between the particle and the oscillator. The normalizing constant, a, for the oscillator is (2rei/h)a' Now the exponential depends quadratically upon all the q;. Hence, the integrations over all the variablesg;Ior 01i1j can easily be performed. One is integrating a sequence of Gaussian integrals. The result of these integrations is, writing T : t " - t', (2ilh siru':T/ ot)-l expi (Sr' | 0 (at, qi) / h, where Q(qi, qo) turns out to be just the classical action for the forced harmonic oscillator (see reference 15). ExplicitlY it is

- 2qilo n* aT)(qi' * qo')

;*rf

*+[,',,

tQ) sina(t-t')d't

.?:

f ,:

-: ['

1Q)sina(t"-t)d't

I,:

* r ' * \r )G ^,,

: \ x r ' I G * " W, ) s o which now contains the coordinates of the particle only, the quantity G;," being given by

E rrqi i-0

Q(qi, qo):

[

,.*'(j,)*,,(*i+!:rd*,

that of the oscillator alone, and Sr:

"'

- t) r(r)r(s) sinct(t'" I X sino(s- l')dsdl l.

It has beenwritten as though 7(l) were a continuous function of time. The integrals really should

rf G^,,-- 12rih sinuT1 u1-t | | e** @) JJ

Xexp(i.Q(qi,qo)/h) e"@o)dsiso. Proceeding irr an analogous manner one finds that all of the oscillators of the electromagnetic field can be eliminated from a description of the motion of the charges. 14. STATISTICALMECHANICS Spin aud RelativitY Problems in the theory of measurement and statistical quantum mechanics are often. simplified when set up from the point of view described here. For example, the influence of a perturbing measuring instrument can be integrated out in principle as we did in detail for the oscillator' The statistical density matrix has a fairly obvious and useful generalization. It results from considering the square of (3S). It is an expressionsimilar to (3S) but containing integrationsover two setsof variables d'x; and. d'ri. The exponential is replaced by expi(S-S')/h, where S' is the same function of the *t'as S is of ri. It is required,for example, to describe the result of the elimination of the field oscillatorswhere, say, the final state of the oscillators is unspecifiedand one desiresonly the sum over all final states zr. Spin may be included in a formal way. The Pauli spin equation can be obtained in this way:

341 NON-RELATIVISTIC

One replaces the vector potential interaction term in S(r4r, rc),

QUANTUM

MECHANICS

J6/

sponding time. The Lagrangian used is 4

ee (xr+r - xr)' A(x)

2c

f' f

-(x;1t

-an)

. A(xrar)

2c

| (e/ c)(d.xu / dr) A f(dxr / d.r)'z

"f,

where r4, is the 4-vector potential and the terms in the sum for p= 1, 2,3 are taken with reversed sign. If one seeksa wave function which depends e upon r periodically, one can show this must -'-(c' (xir r -x;)) (a.Atx,)) 'lhe satisfy the Klein Gordon equation. Dirac 2c e equation results from a modification of the * , ( o . A ( r ; + r ) ) ( ' .( x n + , - x ) ) . Lagrangian used for the Klein Gordon equation, 2c which is analagous to the modification of the Here A is the vector potential, xr+r and xd the non-relativistic Lagrangian required for the vector positions of a particle at times t+t arrd tt Pauli equation. What results directly is the and c is Pauli's spin vector matrix. The quantity square of the usual Dirac operator. iF must now be expressedas II; expiS(rc;+t, x t)f h These results for spin and relativity are purely for this differs from the exponential of the sum of formal and add nothing to the understanding of S(r41, r;). Thus, iFis now a spin matrix. these equations. There are other ways of obThe Klein Gordon relativistic equation can taining the Dirac equation which offer some also be obtained formally by adding a fourth promise of giving a clearer physical interpretation coordinate to specify a path. One considers a to that important and beautiful equation. 'fhe "path" as being specified by four functions author sincerely appreciates the helpful 'fhe scGr(r)of a parameter r. parameter r now advice of Profes'sorand Mrs. H. C. Corben and of goes in steps e as the variable I went previously. Professor H. A. Bethe. He wishes to thank The quantities x(t\(t), )cQ)(t),s(3)(r)are the space ProfessorJ. A. Wheelerfor very many discussions coordinatesof a particle and *(a(l) is a corre- during the early stagesof the work. arising from expression(13) by the expressiorr

342

P o p e r2 8

The Theory of Quantized Fields. I* JurreN SqnwNcen H arwrd Unitasi.ly, Canbridge, M assuhusells (ReceivedMarch 2. 19.51) The conventional correspondencebasis for quantum dynamics is here replaced by a self-containedquantum dynamical principle from uhich the equationsof motion and the commutation relations can be deduced. The theory is developedin terms of the model supplied by localizablefields. A short review is frrst presentedof the general quantum-mechanicalschemeof operatorsand eigenvectors, in which emphasisis placed on the difierential characterization of representativesand transformation functions by m@ns of infinitesimal unitary transformations. The fundamental dynamical principle is stated'as a variational equation for the transforrialion function connecting eigenvectorsassociatedwith difierent spdaelikesurfaces,which describesthe temporal development of t}le system. The generator of the infinitesimal transformation is the variation of the action integral operator, the spacetime volume integral of the invariant lagrangefunction operator. The invariance of the lagrangefunction preservesthe lorm of the dynamical principle under coordinate transformations, with t}te exception of those transformations which include a reversal in tle positive senseof time, where a separatediscussionis necessary. It will be shown in Sec. III that tlte requirement oI invariance under time reflection imposes a restriction upon the operator properties of 6elds, which is simply the connection between the spin and statistics of particles. For a given dlmamical system, changesin the transformation function arise only lrom alteratiods of the eigenvectorsassociatedwith tie two surfaces,as generated by operators constructed from field variables attached to those surfaces.This yields the operator principle of stationary action, from which the equations of motion are obtained. Commutation relations are derived from the generating operator associated rvith a given surface. fn particular, canonical commutation relations are obtahed for thosefield componentsthat are not restricted

by equations of constraint. The surlace generating operator also leads to generalizedSchrtidingerequationsIor the representative of an arbitrary state. Action integral variations rvhich correspond to changingthe dynamical system are discussedbriefly. A method for constructing the transformation function is described, in a form appropriaie to an integral spin field, which involves solv-in-g Hamil[m-Jacobi equations for ordered opelators. In Sec. III' t1le excepiional nature of time reflection is indicated by the remark that the chargeand the energy-momentumvector behavc as a pseudoscalarand pseudovector,respectively,for time reflection iransformations. This shows, incidentally, that positive and negative charge must occur symmetrically in a completely covariant theory. The contrst between the pseudo energymomentum vector and the proper displacement vector then indicates that time reflection cannot be described within the unitary transformation framework. This appears most fundamentally in the basic dynamical principle. It is important to recoenizehere that the contributions to the lagrangefunction of hallintegral spin freids behave like pseudoscalarswilh respect to time reflection.The non-unitary transformation required to representtime reflstion is found to be the replacementof a state vector by ils dual, or complex conjugatevector, together witll the transpos-ition of all opirators The fundamental dynamical orinciole is tien invariant under time reflection if inverting the order of all operators in t-helagrangefunction leavesan- integral soin contribuiion unaltered,and reversesthe sign oI a half-integral siin contribution. This implies the essential commutativity, or anti-commutativity, oI integral and half-integral field components, respectively, which is the connection between spin and statistics.

I. INTRODUCTION

tuting a single quantum dynamical principle for the conv;tional array of assumptions based on classical hamiltonian dynamics and the correspondenceprinciple.l We shall find it useful, however, first to review briefly some aspects of the mathematical formalism that hnd repeated application in the construction of our theory. The simultaneous eigenvectorsof some complete set of commuting hermitian operators, V(a'), provide a description of the arbitrary state i! by means of the representative

extensivedevelopmentsin the concepts f-lE'SeffO L-t and techniques of the theory of quantized fields, quantitative successhas been achieved thus far only in the restricted domain of quantum electrodynamics. Furthermore, the existence of divergences, whether concealed or explicit, serves to emphasize that the present quantum theory of fields must, in somerespect, be incomplete. It is not our purpose to propose a solution of this basic problem, but rather to present a geneml theory of quantum field dynamics which unifies several independently developed proceduresand which may provide a framework capable of admitting fundamentally new physical ideas. Quantum mechanics involves two distinct sets of hypotheses-the general mathematical schemeof linear operators and state vectors with its associatedprobability interpretation and the commutation relations and equations of motion for specific dynamical systems. It is the latter aspect that we wish to develop, by substi-ith" tle hospitalityof the uitho. wishesto acknowledge Brookhaven National Laboratory, which is under the auspicesof the AEC. The general program of this serieswas initiated there durine lie earlv summer of 1949,and lhe presentpaper was largef written it ttris Laboratory during the iummer of 1950.

(a'l):(v(a'), v),

(1'1)

which has the interpretation of a probability amplitude' Two such representations' associated with difierent complete sets of commuting operators, are related by

@'D:[@'lB)dP'@'l), wherc -fd|'indicates

integration

and summation

(r.2) over

1 Although our attention will be focusedon -6elddynamics,.tle analogousdevelopment of particle quantum dynamlcs should De evident.

914

343 THEORY

915

Therefore,

the totality of eigenvaluesB', and (a' I0'): (v(a'), v(0'))

(1 . 3 )

is the transformation function, .{s a special example o{ E q . ( 1 . 2 ) ,w e h a v e

r.,,"

| \q'l1')dB'(p'i|7'),

@'ll'):

F IELDS

OF QUANTIZED

(1.4)

or

a(a'l) : (i/ h)(v (o'), ?'v) : (i/ h)(c'I r | )' f

(h/i)6(q'l): | (o'lFlo")da"(q"l),

( 1.13) (1.14)

which is a difierential equation for the representative

[1llt"]]"1tili'.]1:,iHrui'u'i#H.:]:'.1'jil1".'ll; rawortransrormation themurtipricari,".oipo.i,ion setsc and into a-6o and the two commuting

functions. The set of commuting hermitian operators a:UaU-t,

(1.5)

f"::15"4:"ri*,two

rrom a withtheaidorthearbirrary isobtained which

unitary operator U, has the property that its eigenvalues are identical with those of a, and that its e i g e n v e c t o rasr e g i v e n b y

v(a'):Us1r''),

(1.6)

where d'and q'are the sameset of eigenvalues.Conversely, two sets of operators that possessthe same eigenvalue spectrum are related by a unitary trangformation. Noti that the tiansformation function (d'l a") may also be viewed as the matrix of [/-r in the original eigenvector system, (6'Ia'.): (UV(a'), V(c")): (v(a'), U|V(a")) : (at lU-rlatt). The unitary

B-dB, B infinitesimal generating operators

a<"'ls'llflfffi,riJi]'ilfirtr'' u*(fl)

(r.rr)

or (h/i)6(d'l7'):

f ("'lr"l q")dd"(a"l9') |

J

t"'1fl')a|"@"lFBlB).(1.16)

II. QUANTUM DYNAMICS OF LOCALIZABLEFIELDS

A localizablefield is a dynamical system characterized by one or more operator functions of the space-time coordinates, {"(r). Contained in this statement are the assumptions that the operators tr, representing (r.7) position measurements,are commutative, lrr, r,]:0,

operator

(2.r)

( 1 . 8 ) and furthermore, that they commute with the field u-t:11Q/h)F, u:r-(i/h)F, operators, (2.2) in which P is an infinitesimal hermitian operator, l*,, Q"f:O, induces an infinitesimal transformation in the com- so that _ (2.3) muting set of operators, @l 6" I r, ) : 6(r r, ) 6" @).

(1.e)

The difficulties associated with current field theories may be attributable to the implicit hypothesis of ( 1 . 1 0 )localizability. However, our development of quantum Ff. i.h6a: qF-Fa:lq, field dynamics will be confined to such fields. It remains If the system is such that it is possible to obtain to be seen whether other systems can be included operators'Da that commute with the complete set d, within its scope. one can treat the da as arbitrary, infinitesimal numbers, The problem of constructing a complete set of comand !I'(d') provides an eigenvector of a with the muting operators, that is, of simultaneously measurable to This evidently corresponds a'*6a. set eigenvalue physical quantities, necessarily involves specific propthe special circumstance of a having a continuous erties of the fields. Nevertheless,as a general principle eigenvaluespectrum. associated with relativistic requirements, we must The concept of infinitesimal unitary transformation expect such mutually commuting operators to be formed can be used to provide a differential characterization from field quantities at physically independent spacefor the representative of a state, or for a transformation time points, that is, points which cannot be connected function. The change in the representative (c'l) when even by light signals. A continuous set of such points is altered by the of operators set the commuting form a spacelikesurface,which is a geometrical concept unitary transformation generated by the infinitesimal independent of the coordinate system. Therefore, a hermitian operator F, is given bY base vector system, V(f', c), will be specified by a 6 ( a ' l ) : ( ( a - d a ) ' l ) - ( a ' l ) = ( d ' r ( a ' ) , v ) , ( 1 . 1 1 ) spacelike surface a and by the eigenvalues f' of a complete set of commuting operators constructed from where field quantities attached to that surface. A change of 6*(a')= gs 1"'1- v (a')= - Q/h)Fv(a')' (1.t2) represeotation will correspond,in general, to the introa:udu':d-oq,

where

344 JULIAN

SCHWINGER

cluction of another set of commuting operators on a clifierent spacelike surface. Of particular importance is the.transformation12' or+rr, o1,in which fr and f2 are similarly constructed operator sets which possessthe same eigenvalue spectrum_andare _thereforerelated by a unitary transformation[Eqs. (1.5) and (1.6)], ( r: U u( zU rz-r, 9((1t, o1)-IJtrg(( 1, c"), yr': yr',

(2.4)

so that [Eq. (1.7)] ( l r ' , o t l ; r " , or ) : ( { r ' , o 2 lL lp \ I | 2 " , c 2 ) .

(2.5)

9t6

theinlinitesimalgeneratingopcratorssatisfyanadclitive law of compositiin. our basii assumptionis that dllr1, is obtained by variation of the quantities contained in a hcrmitian operator l/rz, whi& must have the general form

IV,,: 0/ d f

(dt)slxf, "',,

(2.r2)

accordingto the additive requirement(2.11).Individual systems are described by stating I as an invariant hermitian function of the fields and their coordinate derivatives,

A description of the temporal development of a system is evidently accomplished by stating the relac[r]: c(4"(r), 6,f QD, 6,"G):0,6"@). (2.r3) tionship between eigenvectorsassociatedwith difierent spacelikesurfaces,or, in other words, by exhibiting the In conformity with theL classical analogs, we shall transformation function (2.5). Accordingly, we may call trtrland I the action integral and lagrange function expect that the quantum dynamical laws will find their operators, respectively. The invariance of the lagrange proper expression in terms of the transformation function.and thereforeof the action integral,guarantees Iunction. A differential formulation of this tvpe will that our fundamental dynamical principle, now be constructed. The operator I/12 I describesthe developrnent of the 6 ( ( ' ' , o r l ( r " , o , ) system from d2 to or and involves, not only the detailed : ( . i / h ) ( f i , o 1 l 6 W e l y 2 ' , ,o 2 ) dynamical characteristics of the system in this spacetime region, but also the choiceof commuting operators, : (i/ hc)(t{,,,1 a lr and fr, on the surfaces oy a'rd o2. Any infinitesimal @*)sl r 2",oz), (2.r1) [.",,' change in the quantities on which the transformation function depends induces a corresponding alteration is unaltered in form by a change in the coordinate in {/p-r, system.An exceptionmust be made,however,for those d(11, | | 2", oz): ly r', orl 6u u-t I f 2", az). Q.6) coordinate transformations that include a reversal in "r the positive sense of time, which require a separate Now it is a consequenceof the unitary property that discussion. We shall see that the requirement ol iUrzEUn-L must be hermitian. Accordingly,we write invariance under time reflection imposes a general 6U,r-r: (i/ h)U t2-tAWn, (2.7) restriction upon the commutation properties of fields, which is simply the connection between the spin and where 6tr212is an infinitesimal hermitian operator, statistics of elementary particles. and obtain If the parameters of the system are not altered, the 6 ( (r ' , o r l ( 2 " , or ) : ( i / h ) ( l r ' , o L I6 W\ 2 1( 2 , , ,dr ) . ( 2 . g ) variation of the transformation function in Eq. (2.14) arisesonly from infinitesimal changesof f1, o1and f2, o2. The composition larv r.rftransformation functions fEq. Such transformations may be characterized by infini(1.4)1, tesimal generatingoperators,F(o) and F(o), which act on the eigenvectorsV(11, and V(fr", oz), ar.rd \!r, olll3 ' dJ,/ are therefore expressedin terms"r) of operators associated lvith the surfaces c1 and d2, respectively. On referring : ( i r ' , q t l ( , " , o i d f " ( ( r " , o r t! " " ' , o , ) , ( 2 . 9 ) | t o E q . ( 1 . 1 5 ) ,w e o b t a i nf o r s u c h v a r i a t i o n s , imposesa restriction on 6M, the generating operator of infinitesimal transformations. Thus,

6Wn: F1o'7- Oror.,.

(2.1s)

This is the operator principle of stationary action, for it states that the action integral operator is unaltered (ft', otl6I,Yttlf a"', qr) by infinitesimal variations of the field quantities in the interior of the region bounded by o1 and o2, being : | {f r', a,)dl"(t,", o"lir"', o,) dependent only on operators attached to the boundary "rl6wrrl(r", surfaces. The equations of motion for the field are o ) d ! " t ( 2 " . o 2 , 6 l l ' 2 i \ ' 3 " 'o, r ) , ( 2 . 1 u ) contained in this orinciole.2 * | (lt', I "tllr", t T n t h e f o l l o r v i n gd i s c u s s i o n so, n e s h o u l , l k e e p i n m i n d r h a t r h e

or 6 I V1 : 6 t Vy l 6 W6 ;

l a g r a n g e f u n c t i o n s o f t h e s i m p l e s y s l e m s u s u a l l y e o n s i d e r e da r e (2.rr) no more than quadratic in the comDonents o[ indir idual field".

345 917

THEORY

OF

The evaluation ol 6Wv involves adding the independent efiects of changing the field components at each point by dod'(c), and of altering the region of integration by a displacement 6r, of the points on the boundarv surfaces.Thus.

6Wt : (r/ c) I

wnere

FIELDS

QUANTlZED

tesimal coordinate transformation fiu'-

fru:

-

(2.2r)

5Yr,

where 6*r: er- eui$u, er": - e,u:016r,,

(2,22)

the field components sufier a linear transformation, as expressedby

6os "",,(d'c)

+( /,)( f .,-I do,u* * ") ",

(2.23)

Therefore, o"' @)- o"@.): a,Q"@)6*,* (i/ h)\e,"s u,"0$0(r), (2'24)

60J : (d s/dd")6 oO"+ @a / a6 f) a t606"

and

: l(as / a0) - aJa8,/ 06f)1606" (2.r7) t / 60p")600"f. + a Nl{d This expressionfor 6oJ is to be understoodsymbolically, since the order of the operators in I must not be altered in the course of efiecting the variation. Accordingly, the commutation properties bf 6e4" are involved in obtaining the consequencesof the stationary requirement on the action integral. For simplicity, we shall introduce here the explicit assumption that the commutation properties of dod" and the structure of the Iagrange function must be so related that identical contributions are produced by terms that differ fundamentally only in the position of 60d". We may now infer the eouations of motion

d,@e/a0,"):ae/a0".

6'@) - 6"@): (i/h)Le,,5,"468(r).

(2.16)

(2.18)

From the resulting lorm of 6Wu we obtain the infinitesimal generating operator F(o), which acts on eigenvectors associatedwith the surface r,

dd"(*) : 8od"(r)* OF"@)6r I p""Ffg(e). (2.25) + (i/ h)+a ,.6xc"S With the introducl"ion of the total variation, the infinitesimal generating operator F(o) assumesthe form F(o) :

I do uln,"6|"+ 0/ c)&6xt-TI f6""6x" - (i/ zh)nf S\,"PQB}xdr,f, (2.26)

where

(2.27)

cIl,":6 s166u".

To simplify the last term of.Eq. (2.26), we define 7,^, :

-

fx- : (i / zh)llJ p"S\,"P6E+ rI,"S1u"r40 +IIr".t,!"rdtl,

(2.28)

and obtain (i / 2h)tt f S\,"8 083 s6r,= J,y"0 x6r " ": dr(/,r,6*)a axf4,6x"'

(2.29)

since the last two terms of /"r, are symmetrical in tr and tt F ( o ) : ( 1/ c ) a o , l l a s 1a 6 , " ) 6 0 C " *s 6 * " 1 . ( 2 . 1 9 ) r, and therefore do not contribute to Eq. (2.29), in | view of Eq. (2.20). We now remark that, in virtue of Jr7,": -fxr". is composed additively The total variation, 6@"(*), (2.30) of the variation 6od"(c) at the point *, and of the f ,ao,a^U,^,6*,):0, change in {"(*) produced by moving from the point * on o to **dr on o*6a. In evaluating the latter, we provided/ur,6*, efiectively approacheszero, with suffishall take into account that the field components {"(o), cient rapidity, at infinitely remote points3 on o. Finally although stated in terms of some fixed coordinate then, in relation considered are most advantageously system, to the local coordinate system provided by o at the point e. Only such motions are contemplated that correspond to a local rigid displacement of the surface o. This restriction is expressedby 0 '6x": - 6 '5*"

(2'2O)

being the condition that an infinitesimal space vector on o be mapped into one of equal length on af6o. The displacement induced change in {'(r) may be obtained by an alteration in the coordinate system that reduces, in the neighborhood of *, to the equivalent local coordinate transformation. Thus, under the infin!

f

r(o):

I do,[n!"6d"+(1/c)7,,M,f,

(2.31)

where (t/ c)7,,: (l/ c)s6 p-IIy"e""-

lxfxp,

Q.32)

is the stress tensor operator. As we shall demonstrate, this tensor has the property of being symmetrical. T_T 3All

(2.s3)

such characlerizations o[ a spatially closed system, in terms of an operalor approachiug zerc aL in6nity, are to be understood as a restrictio;l lo slatea for which the matrix elements of the operator have this plopetty.

346 IULIAN

as an expression of the conservation of angular momentum. Conservation laws are associated with variations that leave the action integral unchanged, since 51ry,r:p(6r)- F(o2):0

918

SCHWINGER

(2.s1)

generating operator is 161(o):

-(ie/hc)

f | da,Tlu"e"g"6\ Jo

(r/c)QG)6\,

wnere implies the constancy of the corresponding generating oDerator. The mechanical conservation laws for an Q@): (r/c) do,j, ) isolated system are derived by considering a rigid displacement of the entire field, or equivalently, of the and jF: _(iec/h)IlN"c"O". coordinate system, which is described by a common infinitesimal translation and rotation of the surfacesor The implied conservation law, and o2, (2.35) 6*r: eu- eu"x", Epe:- GtpT QG)_QG)=0,

(.2.46) (2.4i) (2.48) ()

LO\

combined with the field variation 6d":0. The dis- is that of the total charge in the system. It is important to notice the ambiguity in the placement generating operator is then given by lagrange function that is associated with given equa(2.36) F u ( t ) : 2 , P , 7 o 1 { . l e , " J, , ( o ) , tions of motion. Thus, two lagrange functions that are where related by

e"6):1t/)

[,a,,r,",

(2.37)

provide action integral operators that difier by surface integrals:

and

J ,,(o): (r/c) f do,M^,,, I

(2.38) Mar,: xrTp-rcuTst. Accordingly, P"(a)- P,(oz):o' and Ju,G)-J,"(a,):0,

'E(6",0,"): s(0",6,")+cAf ,@",O,f) (2.s0)

w,":w,,+(1,,I,)*r"

(2.s1)

Therefore, the principle of stationary action for lTrz is ( 2 . 3 9 ) automatically satisfied by the equations of motion deduced from I{zrz,a1d (2.40) (2.s2)

fr,,!l-(,)-F("t,

l which are the conservation laws for the energy-momen- wnere tum vector, and the angular momentum tensor, respec(2.s3) 0"r,. F: Ftbu. ': tively. Since the surfaces rr and 12 are arbitrary, we [ conservation laws, difierential infer the corresponding (2.4r) Hence, augmenting a lagrange function by the diver0 u Tu , : 0 , gence of an arbitrary vector does not affect the equaand (2.42) tions of motion, but modifiesthe infinitesimal generating 0xMxr,:O, operator associated with a given surface o. However, ,vhich, in conjunction, imply the symmetry of the stress this ambiguity of the lagrange function corresponds precisely to the possibility of subjecting the commuting tensor: (2.43) set of operators bn o to an arbitrary unitary transforOxMt*:T*-7"':0 mation. The conservation law oI charge can be obtained from We verify this statement by specializing the general the required invariance of the hermitian lagrange func- transformation theory to unitary transformations on a phase multitransformations-the constant tion under given surface. Let us introduce i, a new set of complication of mutually hermitian conjugate pairs of field muting operaton on a, which are obtained from f by a components by exp(+ir). We consider infinitesimal unita'ry transformation, phase transformations and, for convenience,write (2.54) !t,(i,, o):9E11,, o;, (2.M) 7:Q/hc)6\,. where tl is characterized by an infinitesimal hermitian Thus, we postulate the invariance of t under the generating operator 62, according to infi nitesimal transformation (2.S5) 6,u_t: (i/ h),tt_tiw. 66": - (ie/hc)e"6^0", Q.45) As the analog of Eq. (2.8) we have, therefore, where e' is characteristic of the field component C", (2.56) 6(f', o| 1", i)= (i/h)(l', ol\wl9", and may assume the values 0, or :t1. The associated "1;

347 919

THEORY

OF

QUANTIZED

FIELDS

but

To obtain the proper interpretation of F56 or ,F6n, it is necessaryto recognizethat some of the II. can be identically equal to zero. Ihis expresses the possibilitl' rvhere ,1.anrl Ii are, respectively,the operatorsgenerthat derivativesin timelike directionsof someof the 6" ating infinitesinraltransformrtions of f and i. 1'his is just of the fonn (2.53); anrl conversely,b1. enrploying may not occur in the lagrangefunction. Accordingly, we shall divicle the quantities d" and Il" into two sets: a particular z, we obtain from Iiq. (2..56)a differential arld II", called the canonicalvariables,and 6a, II4, equation to determinethe transformationfunction that d" termed the constraint variables, in which the second definesthe new representation. set is characterizedbv The commutation relationsof our theory are implicit j in the significance of ,F as an infrnitesimal generating (2.66) II =0. operator. We shall consider first those transformations that do not alter the surface o. so that 6n,:0. It is The name ascribedto the @drefersto tlie fact that, for these quantities, the equations of motion (2.60) deconvenientto write generateinto equationsof constraint, p,7o, 6w: I,'-l',

(2.si)

do u:

(2.s8)

rvherenuis a unit timelikevector and do is the numerical measure of the surface element. To avoid irrelevant geometricalcomplicationsin the following discussion, we shall henceforthrestrict d to be a plane surface,so that zu is constant on o. Note, incidentally, that coordinate derivatives can be decomposedinto components normal and tangentialto o, 0r : n u 6 * J 0 4 , 3o:-nuOu, 64:(iu,fnun")0,,

(2.59)

n pI7p"

null,A- n,II,A : (i/ h)il"S p""A.

(2.60)

We have here introducedthe notation fI":

(2.6i)

(2.68)

On multiplication with rz,,we obtain from this equation that

and that the equationsof motion read

A"n":(1/c)@s/ae")-A,;rf,

(.1/c)(A a/dOa): A*tFA

that is, relationsamong the variableson o. The nature of these relations can be made more apparent by exploiting the requirement that Eq. (2 66) be independent of the coordinate system. We shall later show that the implied restriction on the structure of "C is expressedby

(2.61)

for a quantity which, more precisely,should be rvritten 11"(r, o). The generatingoperatorF now becomes

(2.6e) wlrich enablesthe constraintequationsto be written (r/c)(0 s/064):

(i/ h)dil"S p"An,.

(2.70\

We shall now assumethat it is possibleto solve the left side of Eq. (2.70) lor di, thus exhibiting explicitly the constraint variables as functions of the canonical varinor: (2.62) ables. This excludes systems for which the d/ are fundamentally ambiguous in consequenceof the exIa,n"r,o" istence of gauge transformations. The latter situation Another significant form, associatedwith a different will be discussedsubsequentlyin terms of the familiar basevector system,is obtained from Eq. (2..5.3) with example provided by the electromagnetic field. It is evident from theseconsiderations, and from the -rr,"0". (2.6.\) I,: structure of the generatingoperators, Indeed, we have ff

E|,ro;1,:-6|,ton"4" JJ

:-

[0,6"u0"+6n"c9, e.64)

so thal

ror: Ia,rcw", ron:- a,awo", f

(27r)

that only the canonical variables are dynamically ( 2 . 6 . 5 ) independenton o. Accordingly,1i56is to be interpreted as the generator of that infinitesimal transformation of [a'an"o" the commuting operator set f on o which is produced It should be emphasizedagain that these operator by changingd" into 4"-56". Similarly, F6n is regarded expressionsare symbolic in the sensethat the actual as generatingthe infinitesimal transformation of I in p o s i t i o n si n w h i c h D d " a n ( l 6 l l ' a p p e a r r l e p e n du p o n which II" is replacedby Ii'-6II". Thus, S" and IIo are t h e s t r u c t u r eo f t h e l a g r a n g ef u n c i i o n . special examples of a set of independent field coordi-

F':Far:-

348 JULIAN

920

SCHWINGER

nates; and the most generalpossibility is implicit in the become transformatior (2.53). Associated with any such set of operaton is the conjugate set appearing in F, as II" is : in6o.(.r), I a;Lo"trl,tt6(.r')]+6d6(.r.') conjugate to f', and -d" to II'. We shall now examine the change in the matrix of G, 'ao'gn"{r1, an arbitrary function of field variables on o, which is : o, tt 6(r')]+adb(x') f produced try the infinitesimal transformation generated by Faa, say. Thus, we have (2.78) 6((, olGll", o): (v(f/, o),GtV(|", a)) + (dv(f', o), Gv((" , o)) : - (i/ h)((, F,+711,,, o). (2.72) "llG, On the other hand, we have 6 ( l ' , a l G l l " , o ) : ( ( ' , o l 6 a G l ( " ,o ) ,

(2.73)

f ro'un'(*)[d'(r'), ,,'(x)]+: iiDn'(r').

J"

g'(.r) jr = o, fao'un o1.r'i;601r''), where

(2.7q) where 66Gmeansthe changein G produced on increasing IA, B)_: AB- BA 6" by 6Q". This simply expressesthe fact that replacing and (2.80) lA, B)+: AB+BA. O" by O"-60" in both G and f leaves the relation between them, and t}terefore the matrix, unaltered. Since the 6f' and 6II" are quite arbitrary, we have We thereby infer the commutation relation, derived the fundamental commutation relations, (2'74) lG, n'r1:;7'5t6' [4'(r), rlb(r')]a : ih6"$,(x- r'), (2.81) n'(r')]*:6. [0"(*), d'(*')]+:[n'(r), with its evident generalization in regard to the field coordinates, including, in particular, Here d,(r-a') denotes the three-dimensionaldeltafunction, which is defined by (2'75) lG' F5"1:;7t5"6' Of special importance are the results oblaine(l from Eqs. (2.74)and (2.75)with G:dd and IId: tfl I o ' ( r ) , I d 6 ' I r b ( r )' 6 o b ( n ' )l : i h 6 o " ( x ) , J. L I Q'76) I r i I n"{r), I do'n'(r)Dd'(r') l:0, J" I L and

{"a,

a,1r-r')t1r'):J@),

(2.82)

where/(r) is an arbitrary function. The commutation properties of the 61 can then be obtained from their explicit expressionin terms of the canonical variables. Thus, accordingto Eq. (2.70), (r / c)ld e / A6AQ), d"(r') l* : a,a * r' ) S, An,, (2.83) "' "(r and (2.81) l0s/06a(r),II'(r')l+:0.

f r I | | a " ' o n a 1 x ' 1 6 r ( r ' )r,I . ( . r rl : ; i a n " { . ' ; ,

In the requirementthat commutatorsbe employed, for components of an integral spin field, and anti( 2 ' 7 7 ) commutators for components of a half-integral spin field, we have the connection between the spin and r r 1 statistics of particles. We shall note here that the | | d 6 ' 6 t r b ( x ' ) 6 b ( . rd' ).,( . r ) l : 0 , tJI commutation properties of a Bose-Einstein system, that is, an integral spin field, can be represented by in which we have invoked the dynamical independence meansof difierentialoperators.Accordingto Eq. (1.13), of the @'and the II'. a suitable representative of an arbitrary state obeys To extract explieit commutation relations among the properties of 6$" d" and II" we must know the operator 6 ( r ', o l ) : G / h ) ( l ' , o l F o o l ) and 6II". The requirement that the formalism be / | f l\ invariant with respect to time reflection supplies the : ( i / h ) 1 l ' , o l t d o n " 5 6I"'l r . l . s s ) desired information. It will be shown in Sec. III that \ l l lr' 6dd(*-') and dIID(r') comrnute with all field quantities lJ-

l

o"(r) and I-I"(r), on q, except when both o and, b in which a is not altered. Now the characteristic designatecomBonentsof fields that possesshalf-integral property o{ an integral spin lield is that d@"commutes spin, in whic\ event they anti-commute. Accordingly, with all dynamicnl variables and can therefore be the commutation relations of Eqs. (2.76) and, (2.77) treated as a number, The representation involved in

349 921

THEORY

OF QUANTIZED

Eq. (2.85) is evidently that which is Iabeled by the continuous eiginvalues of the O"(r) at all points of a. In terms of the notation f

6(d" dl): I do66.'kc)(6/66"'(r))(d" ol), (2.s6) J.

The contribution of the secondterm to d-/,, is evaluated as follows,

50/c) @6,rfi-tlo,r"&) J :

we obtain (h/ i) (6/ 66"'Qc))(6', ol) : (6', o | il'(r) | ),

FIELDS

l ldo"xrAx$l 1"dS')- do,r"d1(Irr"6d")

)

(2.87)

:

and similarly, ih(6/ 6fid (x)) $I', a l) : (rr',

6"@) l). "1 We shall make further application of the general commutation relations (2.74) and.(2.75) by successively placing G:P,, Ju", and Q. According t9 Eq. (2.30), the Iast term of Eq. (2.32) makes no contribution to P,, so that

P":-

ff

, don"a"6*0/c) | d.o,s.

doxk*a,- r,dJ(I[r"dd")

J (2.88)

(2.89)

JJ

F

+ | @6il,"-do,npr6o" J

:

- *,a") (n.6d9 a's{*,u,

f

| (n uIr,"- n,rIu") 66r.

Therefore,

Q.g4)

I

In the evaluation of 6P, we'encounter f

(1/t) | do"6t: JJJ

6J-:

.f

f

J

tlo 6II"l(x,6,- t,6 p)O"+ Q/h)S*"F 6El

, do,aJnN"6d)= | dop,(nF 6O.) f

+,

: a,o"tt"to't, I

d6 l ( n , a , - ' t

(2.e0)

f

.J

(2.e1)

16"Qc),P"f: (h/ i)6 "6l1c), ln (r), P,l : (h/ i.)0,T1"(t),

('c,(h/la r"(h/ | a)6* "J Itt", u,l= @u(h/i)a,- r:,(h/la;)rI"-

[

a,n"l@&,- x,a,)o + e/ h)s p""p 6lf *0/d

f

:

S,,"lf , n]s/,b,

! (h/ i) (n,TI - n"r,"), "" 0 : - II",s/""i+ ( lt/ i) (n,It - n "A "',t7,A),

(t''o)

which exhibit -/u, in the role of the rotation generator

(2.92) and illustrate the formation of Ju, as the superposition

in virtue of the commutativity of P, with 6d" and 6II", which is a consequenceof tire fact that half-integrai spin field components must appear paired in the vector P,. Incidentally, a commutator [F(t), F(o], which has been evaluated by considering the efiect on F(2) of the transformation generated by FG), can equally well be viewed from the reversestandpoint. Thus, the relations (2.92) also exhibit P, in the role of the translation generator. \ The angular momentum tensor -Ir, is easily brought into a form analogousto (2.89),

t,,: -

We have thus derived the commutation relations lO", J *):

The reariangeme4ts of Eq. (2.90) have involved Eq. (2.17), as simplified by the equations of motion, and the assumption that the system is spa{ially closed. We thereby obtain

- Q /h ) T I b S B,b'

.! n urr,;- n,rru"f66". (2.9.51

whence

5P,:ldo@"il"do"-Dlr"d,d").

" a) n '

6',x,e-do,s,t). (2.e3)

of orbital and spin angular momenta. The third equation of Eq. (2.96), the statement that IId=0 is a property independent of the coordinate system, has already been employed in Eq. (2.68). According to Eqs. (2.47) and (2.48), the charge operator is given by

O: - U?/h\ I

rorTocata.

(,2.97)

Therefore, rve have f 6Q:-Qe/h)I a"(,1n..,0"+n"*ao"), Q.e8)

from'which we obtain the commutation relations

l+,Ql:r,"0., [n,, Q]: -ee"Il". (z.ss)

3s0 IULIAN

These indicate the significance of e as the elementary charge, and exhibit Q in the role of the phase transformation generator. Note, however, that the derivation of Eq. (2.99) from the latter viewpoint is not restricted to the canonical variables, although nothing new is obtained thereby. The general infinitesimal operator (2.31) describes the transformation from the commuting operator set f, o to f -Df, r{6o, as indicatedbY v((f - 3f )', o-l 6o): lr - (i / h) Ffv G', o)'

v((r- 6r)" o): 11- (i./h)Fa'lv(r" "), (2.ro2) F5":Q/e) d.a,r,"6r" f

0:6.G(6)-(i/h)lG(o),F6.f'

(2.10e)

lc(o),P,):(h/i)5,G(a),

(2.110)

whence

( 2 . 1 0 1 ) and

where F66 induces, via d{". a change in the commuting operator set defined in relation to the local coordinate system provided by a lixed o, while

independent ol o, since the relation between G(o) and the I on o is unchangedby an alteration of the surface. The components of Pr(o) referred to axes based on o are oI this naturel and, consequently, the matrix o{ P"(a) in Eq. (2.107)jr:volves the orientation of o relative to the coordinate system, but is otherwise independent of o. Commutation relations between Pu, J u, and G(a) follow from this property of the G(a) matrix. Thus, we have

(2'100)

The operator F is additively composedof two parts, F:FtolFa,,

922

SCHWINGER

(2.103)

generatesthe change in o described by 6r,, for a fixed set of commuting operators defined relative to o,

lc(a), t *l:

(h/i)6,,G(o).

(2.1rr)

As the first of several illustrations of these commutation relations,we chooseG(o):$"(*). According to Eq. (2.25), we have lo"@), P!l:(h/i)]u6"@),

(2.112)

and 16" @), J, "f : (r,(h / i) 0, u(h / il a,) 6" @) fS,""BSd(r),

(2.113)

which is in agreernentwith Eqs. (2.92) and, (2'96), but without the restriction of the latter to the components !r,(f,, o*do) : [t- (i/ h)F *fv((" o). (2.104) {'. A particularly simple example is provided by the total charge. Since this operator is Consistent with our restriction to plane surfaces,we G(t):Q, consider only rigid displacements of o, for which the independent of o, we have () 11J\ generating operator has already been given in Eq. IQ, P,]:IQ, J U'f:O' (2.36). Difierential equations that describe the change in which state, inversely, that P, and Ju, are unafiected the representative of an arbitrary state, as produced by phase transformations. The effect of a displacement of o on the quantity G(o):P^r^1"1, where el(o) is an by rigid displacements,are inferred from arbitrary vector that is rigidly attached to o, comes ( 2 . 1 0 5 ) ( i / h ) ( f ' , o ) , v ) : 6 , ( f ' ,o l ) : ( d " v ( ( ' , entirely from the rotation of the vector e1(o), "lF,"l). In terms of the notation

6"(Prer(a)): - e,,P,e'(o)'

5 , ( f ' , o l ) : 6 u 5 , 1 1o' ,l ) * i e , , 6 , , ( ( ' , o l ) ,

(2.106)

we obtain generalized Schrddinger equalionsa for translations, ) : (r" o IP,(o) l) "|

( h / i ) 6/ r "

:

, l), | ( g ' , o l P , ( o ) l ( " . o \ d ( " ( ( "o

and rotations, ( h / i . ) 6 , " ( l ' ,o | ) : ( f ' , o I J :

",(o)

Therefore, we have

(2.11s) (2.116)

lP,, P,):0, and lPx, J -f:ih(6;P,-

6,rP).

(2.117)

where Our last example, G(o):1^,r^
l)

| (f', olJ,"(o)11",o\,11"((",ol\. (2 108) I

An operator G(o), which is constructed from field q u & n t i t i e so n o , h a s a m a t r i x ( l ' , o l c ( o ) | t " , o ) t h a t i s I Note that these Schrtirlingcr equations have becn obtained from the Heisenberg picture, in wlrich the arbitrary state vector is fixed. P. A. M. Dirac, The Principles oJ Sadnlilm Mechunics (The Clarendon Press, Oxford, 19,17),thir(l edition, Sec.32.

J s^es(t)p,\2) :

f (1 / c) | d o,lxxe x(') T,^e,(2\

"

-r,e.c)fprdr(r)] (2.118)

involves space-time coordinates, in addition to field variables.The necessaryrevisionof Eq. (2.109)is 6,G(o): (i/ h)lc(o), F,,l-t 0.G(o),

(2.11e)

where d,G(o) denotes the displacement induced change in G(o), associated with the explicit appearance of space-time coordinates. In the example provided by

35r THEORY

923 Eq. (2.118), d$(o) not rotation) of a,

OF QUA NTIZED

arises from the translation (but

: (erP. - e,P1)s^{t)e.tz).(2,120) 0,(J peatt)6,
FIELDS

Now, the difierencebetween the results of the two ways in which these transformations can be successively applied may be regardedas the eftect of a third, related transformation, _ 6(r)60))!r(r"o) (6(t)6(r) : {/((f + (6o)6(r)- d(r)6(t))f )/, a} (6order 6t,rlor) 6) -v(l', c) (2.rzs) : - (i/h)Flt2tvt,' 6).

lJ t,, J u"l: ih(6r,Jr,* 6irJ.,* 6*J xul 6,,J,x). (2.122) Therefore, (2.126) lFor, F(2tl:ihFrtzl We may remark, as an example of a general procedure of Eq. integrability necessary to the for constructing representations of operator commuta- is a condition (2.124). A simple illustration of this viewpoint is tion properties, that the identitY provided by rigid displacements:

;;4",Jr.l,J -l-ll0", t -t,!ti",lr\,,

leads to analogous commutation r€lations for the representativesof orbital and spin angular momentum in Eq. (2.113). In a final comment concerning commutation relations. we observe that the commutators of generating operators are of significancein connection with integrability conditions for the infinitesimal transformations generated by these operators.sIf I(I) and F(2) are two such generators of infinitesimal transformations, we have 6 ( t ) v ( r / ,o ) : v ( ( i - a u , r r , ,

5(L'2\rr:er(t'2)-el"(t'z)&v,

r pl) (2.123)

6f 6ttro)-v(f" o) : -(i/DFn\,v((" o).

6a){/(f/, o) : q/((l- ao,rr', 6f f ero)- V(f/, o) : -(i/h)Fov(r" o). 6 ( { t ' ,o r l ( r " , 6,):

0.124)

. slnce

e.l\i\

Fo'(t2t:eatr'2\Pp+rer"02)J!,,

(f {t)f t:) -_ fit2)dtrr)*,-

- e ,,0) e,Q)I

euoQ) e,Q)

_ ( _ eu^O) 5^,tzl1 cul(2)er,0))*, :6"1r2)-6u,U21x6,

(2.128)

is another rigid displacement, The ensuing commutal ion relations are just Eqs. (2.116, (2.117), and' (2.122)' In our discussions of the variational principle (2.14), we have dealt with the properties of a given dynamical system. The principle is also applicable, however, to variations in which the system is altered, as characterized by a change in the structure of the lagrange function. For a variation of this type, we have

(i/ hd (o*)(r,', o1l6 slrfl Y2", I,",,'

or),

(2.r2e)

or

6 ( f i , o l l , " , o z \ : G / h c t J , , ( r t * U ( f ' ' , o l f " , o ) d y T " , o l d s f . r l l g 6 , 6 ) d ( b ( l h , o 1 f t " , o z \ ,( 2 ' 1 . ] 0 ) variationsof this natureare appliedsuccessivelr' where the surfaceo containsthe point r. If two independent we obtain

6 ( : ) 6 ( 1 )o( il,l,y, ' , , , o 7 ) : ( i f l t o f . ' , , ' u . r l l I r , , ( l , " o , l l . ' o ) , 1 1 , ( 1 . ' 6 ] 6 r r r ! [ r ] l f z " ' a z ) +

Cl,orl 6(I)a[ir]lfb,o)dtb6('?)(l', "ldi', "t]

[

: {r/ nd, {axlf "taotr,, o,I D(2)s[r']6(r) slt)llz", [ "',, / +

!

oz)

-cl*' fl l r", or)]' -8[t]6('?) )
(2'13I )

We shall introduce here a notation for chronologically ordered operators, A(x)B(t'),

Gk)B(r')),: Gllli""".nl",

xolro'

(2.ri2)

(.r), .rn') xn, B("r').4

H. \\ eyLThe Theoryol Grorlt and euanhrtn l[ echanics(E. P. Dutton end Company, Inc., Nerv York, 1931),p 1?7.

3s2 JIILIAN

SCHWINGER

924

which is an invariant conceptprovided tbat the operatorsinvolved commute when r-r'is a spacelikeinterval and that the positive senseof tine is preserved.Thus we may write Eq. (2.131)more compactlv as u l r r , ; {(:i r / , o 1I l : / / , d " ): 6 ( ! ) d (('1) , r ,o , I i r , r , o r )

: U/ rL|' (di ( d x ' ) ( ( i , o 1 l( 6 ( ' ) 3 [ r ] 6 r r r " C [ r ' ] ) *1l , " , o , ) . 1.",,'I,',,' These results will find frequent application in later lvork. We shall conclude this section by indicating, in connection with a Bose-Einsteinfield, a method for constructing the transformationfunction ((r', qtl(2", qr), which has as its classicalanalog the Hamilton-Jacobi theory of field mechanies.The actual motion of the svstemis implicit in the forn assumedby the variation of the action integral,

uv,,--l "n"ag.{ e,p ue) } 1,,,r, "{ df, (2.|34) [,a "'",,

(2.133)

quenceof the noncommutativity of the g" on or and on 02 in a manner which depends upon the location of these surfaces.Thus, if the operator ltrzrris first ordered and then varied, the result will differ from what is obtained by ordering 61112.We now turn to the differential characterizationof the transformation function labelled by eigenvaluesof 6" on oy zlrd o2, 6 ( s ' , o 1 1 6 " ,o 2 ) : ( i / h ) ( 6 ' , a , l a w 1 6 r , a r ; $ 2 , c z ) 1 6 , ,o, . ) , ( 2 . 1 3 6 ) and observe that, in virtue of the ordering in 61V, the operators d" on or and on q2act directly on their respective eigenvectorsand can be replaced by the associated eigenvalues:

in which we continue the restriction to plane spacelike surfaces.It is implied by Eq. (2.13a)that I/u can be exhibited as a function of o1, c2, and of the do on these 6 ( 6 ' , a 1 l g " , o 2 ) : (i/ h) 6W(Ot, o | ; 0", 62)(6', orl e", o (2.137) surfaces,and therefore that the II., P, and ./u, associ"). ated with each surface can also be so exhibited. With the aid of commutation relations between the @" on The transformationfunction is thereby obtaineclas? dr and on or, it will be possible to order the operators @ ', o ' l O " , o , ) : e x p [ ( i / , 4 ) 1 \( t6 ', o y ; g " , o ) 7 , ( 2 . 1 . ] S ) in Eq. (2.13.1)so that the Q" on o1 everyrvherestand to the left o{ the 0" on or. The differentialexpression,thus rvherethe constant of integration, which is additively ordered, shall be denoted by 6W(@r,or;dr, or), from containedin 11t,can be determinedfrom the condition which we obtain difierential equationsconnectingthe lim,(d', dr]0", or):6(6'-6"). (2.13s) various ordered operators, (6/6d.(*D)w:n,(rr), 6utuw:2r1o-;,

(6/66"(r))w:

-n'(rr),

III. TIME REFLECTION

6 p Q ) 7 , 9 : _P p ( q 2 ) , ( 2 . 1 3 5 )

The general physical requirement of invariance with respect to coordinate transformations applies not only 6p,(2r1V:- J u,Qz), |ulr)nfi: J r"(o), to translations and rotations of the coordinate system, uhere 11 and rs are arbitrary points on a1 arrd o2, but also to reflections o{ the coordinate axes. Among respectively. In conjunction with the commutation relations (2.81), theseHamilton-Jacobi operator equa- which should be compared with the hermitian action integral w D : +wj t : (.n / 2t) (r (t ) - x (t ))' tions serve to determine the ordered operator : (n /2t)fr2(t) - 2r(t ) s(t,) | f (.t))- |i.h. (Cr, 1V or ; 62, cz), to within an additive constant. Incidentally, the analog of Eq. (2.138) is It is important to recognizethat\g lW p, and indeed, (r', tylr", t,):expl(i/b'W ((', i:', Df that W is a non-hermitian operator.0This is a conse: (A t)-| exprGm / 2ht) (r' - t, )2), 6 The elementary example of a one-dimensional free particle sill sullice to illusrrate rhis. The Hamillon-lacobi ecuations lor the construction of 1Ptrtt,t, *1tzt,!), !:t1-i2, are

( a / a r ( t , ) ) W : - ( a / a x ( D ) n $: P , - G / a t ) W : f / 2 , ' . r\ccording

to the solution of the equations of motion,

r(t.t)- i(tr): (t/n.\p, rre have

Ir(t), r(t,)): -iht/m, whence - @/ at)\9 : (m/2t,)(r(,r)-r(,r)), : (n/2t2)l*(tJ -2x(t)r(t,)+

f(h)f- ilt /2t The solutionof the Hamilton-Jacobioperatorequationsis \p : (1n/ 2t)lx2(t1\_ 2r(t) r(t) + i (t')f+ iirl blu t),

rvhere the constant.4 is determined to be A:2rih/m {rom the analog of Eq. (2.139), lim(r' , h r" , h):

66'-

t").

T fhe exponential form of Eq. (2.138) is familiar as a basis for establishing a correspondence connection with classical HaniltonJacobi particle mechanics. Dirac employed this form in a discussion of unitary transformations and recognized, in part, th&t the Hamilton-Jacobi equations are rigorous as relirlions anrong ordered operators (see the end of the section quoted in reference 4). In Feynman's version of quantum mechanics lR. P. Feynman, Revs. Ilodern Phys. 20. 3oi (tS48rl, the exponenlial form is employed for inEniiesimaltime intervais,rith the rerl part of 1V defined as the classical action integral.

3s3 e23

THEORY

OF QUANTIZED

FIELDS

the latter transformations, time reflection has a singular position. Its special nature can be indicated by the transformation properties of some integrated physical quantities. Thus, the expectation value of the energymomentum vector,

with respect to time reflection.8If we w'ereto consider only such a half-integral spin field, the basic dynamical equation would preserve its structure under time reversal, but at the expense of violating the general transformation properties of all physical quantities; charge would remain unaltered, and energy would reverse sign under time reflection. The latter difficulty (3.1) simply :(l,/c) indicates that, on inclusion of the contributions f"d;,g,,), of integral spin fields, the various parts of I would is actually a pseudovector with respect to time reflec- transform difierently, thus emphasizing again the tion. With thd plane surface c chosen perpendicular to general failure of Eq. (2.la) to admit time reflection as the time axis, the components of (P,) are obtained as a unitary transformation. 'three-dimensional To aid in investigating the extended class of transforvolume integrals, mations that is required to include time reflection we f shall introduce some notational developments. The (Po):G/c) d'o(rsn). scalar product of two vectors, Vo and V6, can be written | (3'2) (3.4) (al6) = V"'Va: Voilr.', t,

(Pt):0/c) aolr*), k:r,2,3, |

thereby being regarded as ttre invariant combination of € vector Vr with the dual, complex conjugate vector induces iI,.* We allow operators to act both on the left and on ,"n.ltio., tco---fi0, rl-rr and the ti-" (P)+(P6), Qrl--(Pr), according to the transforma- the right of vectots, V and iP*. Thus, an operator tion properties of tensors. This difiers in sign from a associated with A, the transposed operator .4"; is proper vector transformation. In particular, the energy defined bys does not reverse sign. under time reflection. More (3.5) A*:tAr, V*A:Arg', generally, this property of (P,) is obtained from the pseudovector character of drr, which expressesthe or by (3.6) pseudoscalar nature of a four-dimensional volume (alAlb):v"'Avt:9uAr9"'. element with respect to time reflection. Similarly, the We also define the associated complex conjugate expectation value of the charge operator.4", ff

\Q):0/c)

| dc,(j,):(1/c)

JJ

| d.o(j]

(,4v)..:1-v'.

(3.3)

behavesas a pseudoscalarunder time reflection. Hence, this transformation interchanges positive and negative charge, and both signs must occur symmetrically in a covariant theory. Indeed, for some purposes the requirement of charge slmmetry can be substituted for the more incisive demand of invariance under time reflection. The significant implication of these properties is that time reflection cannot be included within the general framework of unitary transformations. Thus, on referring to the Schrddinger equation for translations (2.107), or the analogousoperator equation (2,110), we encounter a contradiction between the transformation properties of the proper vector translation operator du and of the pseudovector Pu, This difficulty appears most fundamentally in our basic variational principle (2.14). With 8 behaving as a scalar and (d.r) as a pseudoscalar,reflection of the time axis introduces a minus sign on the right side of this equation. However, it is important to notice that the scalar nature of S cannot be maintained for that part of the lagrange function which describes half-integral spin fields. Indeed, such contributions to 3 behave like pseudoscalars

(3.7)

The connection with the hermitian conjugate operator ,4i is obtained flom the definitioir of the latter,

(,4v)':E'7t, Iio-*-0"*.n,al

(3.8)

invarianrof a spin | fretdisw:{/.l!{/.

R*, cat The transformation that represents time re8ection, {': be obtained from its equivalence with a rotation through the Accordingly, angle r in the (45) plane; R: exp?rtro*):ias.

.9,,9,=9tn-tyon{:-W,

which indiates tle pseudoscalar character of the spia | field laqranqe function, witi respect to time reflection. The correspondins behavior oI 6eldswith other spin valuescan be obtained fiom tb6 observationthat a spinor of-rank z contains6elds of baiic invariant and lime reflection spin lz, -{z-1,. .". mnk tr are operator lor a splnor ol"Th..

W={11ft,0ot9, and

f

1"

L .-,

J *..

R:erpl i"| 2" d{"(t) l: fJ idr6*) Therefore,

{/'*' : {t R-tiI r&' R! : (- r)"6V, which shows the pseudoxalar nature of the lagrange function for all haltinteral spin ffelds, eNote how the familiar property of transposition, (,'18)r *B"r{?, foflowsfrom dripdefinition:',4B'l!:A(*BT\:\ltBrAr.

354 JULIAN

namely, t1-

t.T

(3.9)

Conventional quantum mechanics conte;plates transformations only within the V vector space, and contragradient transformations within the dual V* space. We shall now consider transformations that interchangethe two spaces,as in (3.10)

!["+V;:i{r"-' The efiect of Eq. (3.10) is indicated by

( a l a ) :v " - v r : v ; v ; ' : ( 6 l a ) ,

926

SCHWINGER

R will now be chosen to oroduce that linear transformation of the f", RQ"R-L:R'808,

which compensatesthe efiect of the gradient vector transformation. Thus. we have

E:(+)er(e"r, A,O.r),

( 6 1A r I e ) .

(J.2J)

where the (+) sign here refers to the fact that the structure of the lagrange function, for half-integral (3.11) spin ficlds, can be maintainedonly at.the expenseof a change in sign. We now see that if

and

( a l A l b ): v j A * a : v ; , 4 i h ' :

(3'22)

(3.r2)

t': e($"r, apo'r'),

(3'24)

the form of our fundamental dynamical equation rvill have been preserved under time reflection, since Eq. ( 3 . 1 3 ) (3.20) will then differ from Eq. (2.14) only in the V;-:Rt[", substitution of {"? for d" as the appropriate field where R is a unitary operator, we have variable, and in the interchange of or and o2, which ( 3 . 1 4 ) simply reflects the reversed temporal sense in rvhich ( a l b ) : ( 6 1 d . ) , ( e l A l b ) : ( 6 1A l a ) , the dynamical development of the system is to be in which traced. (3.1s) Invariance under time reflection thus requires that A:1ntp-'1, inverting the order of all factors in the lagrangefunction Now, we have leave a scalar term unchanged,and teverse the sign of (RAR-')': 8,4, (s.to) a pseudoscalarterm. This can be satisfied, of course, tn : lAtnn-,1r: (RBR-t)r by an explicit symmetrization or antisymmetrization ol and therelore the various terms in !. When the lagrange function, (3.17) thus arranged, is employed in the principle of stationary @llA, nflb): - (6llA, Blld). action, the variations Dod"will likewise be disposedin a We have here precisely the sign change that is required symmetrical or antisymmetrical manner. We must now to preservethe structure of equationslike Eq. (2.110) recall that the equationsof motion (2.18), which do under time reflection. not depend explicitly on the nature of the field commuWe now examine whether it is possible to satisfy the tation properlies, have been obtained by postulating invariance under time reflection of by requirement the equality of terms in 60"8that difier basically only means of transformations of the type (3.13). When we in the location of 60d", Since such terms appear with introduce the coordinate transformation the same sign in scalar components of 3, and with ( 3 . 1 8 ) opposite signs in pseudoscalarcomponents, we deduce io:-*0, ip:16 h:1,2,3, a correspondingcommutativity, or anticommutativity, in conjunction with the eigenvector transformation between 6od" and the other operators in the individual (3.1e) terms of 6o!. v-(i', o):Rll(l', "), The information concerning commutation properties the fundamental dynamical equation (2.14) becomes that has thus been obtained is restricted to qperatorsat common space-time points, since this is the nature oI 6(i2", orlil', a) the terms in 8. Commutation relations between field quantities located at distinct points of a space-like :(i/ttc)(ti',,,1afatlulfu', oJ, (i.2u) surface are implied by the general compatibility reI J quirement for physical quantities attached to points "L with a spacelike interval. Components of integral spin where fields, and bilinear combinations o{ the components of c : (nsR-'; r : 3r ((ROdR-t)r, + A!(Rd"R-) "). (3.21) half-integral spin fields, are the basic physicial quanIn the last statement, the * sign indicates the efiect of tities to which this compatibility condition applies. By the coordinate transformation (3.18) on the components consideringthe generalpossibilities of coupling between of the gradient vector, while the notation J"( ) the various fields, we may draw from these two expressymbolizes the reversal in the order of all factors sions of relativistic invariance the consequencethat the induced by the operation of transposition. The operator variations 64b(r'), and therefore tle eon=jugatevarialVlore generally, if

355 927

THEO,RY OF QUANTIZED

tions 6llt(*f), iommute or anticommute with d"(r), ll,(a) for all r and *' on a given o, where the relation of anticommutativity holds when both a and b refer to componentsof half-integral spin fields. The consistency of this statement with the general commutation relations that have already been deduced from it is easily verified. By subjecting the canonical variables in Eq. (2.81) to independent variations, wei'bbtain

FIELDS

and statistics of particles is implicit in the requirement of invariance under coordinate transformations.r0

r0The'discussion of ttre spin and statistia connection by W. Pauli fPhvs. Rev. 58, 716 (1940)l is mmewbat more negative in charac'ier,although bawd on closelyrelatedphysiel requirements. Thus, Pauli remarks that Bose-Einstein quantiation of a halfinteeial spin field implie au enerEythat posrcse no lower bound, and-that Fermi-Diric quantization of an integral spin field leads to an alcebraic contradiction with the commutativity of physical oumtitiis located at points wilh a spacelike interval. Anotier dostulate which bas Leen emploved, that of charge symmetry g, i-W. p.uti and F. L Belinhnti, Phvsie 7, 177 (1940)l' suffices -m lO"@), d@6(r')la: [II'(r), a4b1r';1*: determine the niture of tbr commutation relations for suffrci[C{r), 6ilb(r') ]+: [rl'(o), drl6(r')]*: 9, 13.25; entlv simple systems.As we have noticed, it is a consequenceof timi refleition invariance.The commentsof Feynman on vrcuum rvhich is valid for all r, d on r. In addition, Eq. (2.81) polarization and statistics [Phys. Rev.76, 749 (lq+9)] aPpearto suce a properly states that all physical quantities commute at be an illustration of the charge symmetry requirement, contradiction is establishedwhen ttre chargesymmetrical concept distinct points of o. of t-hevmum is applied to a Bo*-Einstein spin | field, or to a We conclude that the connection between the spin Fermi-Dinc spin 0 field,

356

P o p e r2 9

The Theory of Quantized Fields. II JurtaN Scurvttcrn II qrutrd Unioersil,t,, Canbrid.ge, fufassdchuseils (ReceivedFebruary 19, 1953) 'Ile arguments leading to the formulation of the action principle for a general field are presented. In associationsith the complete reductiou of all numerical matrices into symmetrical and antisymmetrical parts, the geaeralfield is decomposedinto two sets,which are identified with Bose-Einsteinand Fermi-Dirac fields. The spin restriction on the trvo kinds of fields is inferred from the time reflection invariance requirement. The consistencyof the theory is verified in terms of a criterion involving the various geireratorsol infinitesimal transformations.Following a discussionof chargedfields,the electromagneticfield is introduced to satisfy the postulate of general gauge invarimce. As an aspect of the latter, it is recognizedthat the electromagneticfield and charged fields are not kinematically independent.After a discussionof the field strength commutation relations, the independent dynamical variables of the electromagnetic freld are exhibited in terms of a special gauge.

tfHE

general program oI this seriesl is the conI struction of a theory oI quantizedfields in terms of a single fundamental dynamical principle. We shalt first present a revised account of the developments containedin the initial paper. THE DYNAMICAL PRINCIPLE The transformation functions connecting various representationshave the two fundamental properties

6W"fi:6W"8; the infinitesimal operators 6trV"0are Hermitian. The 6W"p possessanother additivity property referring to the composition of two. dynamically independent systems. Thus, if I and II designate such systems, (ot' au' IAt'Fn' ) : (ar' I f.r) (ar' I 9n'), and if 6ll"pr and 6IV are the operators characterizing "6rr ininitesimal changes of the separate transformation functions, that of the compositesystem is

f

("' lt't : I la' lP'\lB'@'11').

@'1ts')*:@'1"');

dll

where{d,B' symbolizesboth integration and summation over the eigenvaluespectrumi If 6(a'lB') is any infinitesimal alteration of the tr4nsformation function, rve may write 6(q,lB,):i(q,l6Iy"pll3,), (1) which serves as the definition of the infinitesimal operator 6W"p. The requirement that any infinitesimal alteration maintain the multiplicative composition law of transformation functions implies an additive composition law for the infinitesimal operators, ' 6W (2) "r:51Y "ua51, ur. If the a and B representations are identical, we infer that 6W"":0, which expressesthe fixed orthonormality requirements on the eigenvectors of a given representation. On identifying the o and 7 representations,we learn that 6ilrp": -51y"u. The second property of transformation functions implies that

-i(d'l6w

or

611'"rtr517"utr. "6: Infinitesimal alterations of eigenvectorsthat preserve the orthonormality properties have the form 6{,(a): -iG.V(q'), ffI,(a')I:iV(d')1G",

,

where the generator G" is an infinitesimal Hermitian operator which possesses an additivity propery'yfor the composition of dynamically independent systems.If the two iigenvectors of a transformation function are varied independently, the resulting change of the transformation function has the general structure (1), with 6W"e:6"-6u' The vector V (a')+6.P (d') : (1- iG

(a'), ")v can be characterized as an eigenvector of the operator set a: (l - iG a(1 * iG.) : q- 6a, ") wrtn ule ergenvalues d . Ilere Ea: -ile,

G"l.

This infinitesimal unitary transformation of the eigenvector V(a/) induces a transformation of any operator F such that (a'IFIa"): (a'|FIa").

-i(B' l6w "plp')*: "ptla') :i.(9'l6Wp,l q'),

IJ. Schwinger,Phys. Rev. 82, 914 (1951), Part I ,IJ

357 JULIAN

714

SCHWINGEIT must have the additive form

We write this in the form (a' I F I a") - (a' I F I a") : (a' | (F - F) | a"),

c : f a o c , o ,J{I"^a,' :, c , r , 1 , J,

or, in virtue of the infinitesimal nature of the transformation, 6 ( a ' l F l a ' t ) : ( a ' l 6 FI a " ) ,

where rJo is the numerical measure ol an element of (*) is to be regarded as the timespace-likearea and G10) rvhere the left side refers to the change in the eigen- Iike component of a vector in a local coordinate system vectors for a fixed F, while the right side provides an based on o in order to give the surface integral an invariant form. If one can interpret Gu@) oll tr1, and equivalent variation of the operator F, given by on 02, as the values of a vector defined at all points, -ilF,G"). 6F:F-F: the difference of surface integrals in (6) can be transIf the change consists in the alteration of some formed into the volume integral paramete-.rr, upon which the dynamical variables de?6r pend, and which may occur explicitly in F, we have (d.r)duG,(r), 6ll'r,: | J,, (6F)" F:F:F+6,F- a.F, @ u : a / a x , )' where 6"F is the total alteration in F, from which is A second type of transformation function alteration subtracted 0,F, the change in F associated with the is obtained on considering that the transformation conbe cannot explicit appearance of r, since the latter necting f1, o1,and f2, d2 can be constructed through the produced by an operator transformation. We thereby intermidiary of an inhnite successionof transformations obtain the "equation of motion" with respect to the relating operators on infinitesimally neighboring surparameter r, faces.According to the general additivity property (2), (3) 6,F: a,F+ilF,G,l. For dynamical systems obeying the postulate of local action, complete descriptions are provided by sets of physical quantities, f, associated with spaceJike surfaces,o. An infinitesimal alteration of the general transformation function (fr'orl(r"or) is ch4racterizedby (4) ""oz). Here the indices 1 and 2 refer both to the choice of complete set of commuting operators f, and to the spaceJike surface o. We can, in particular, consider transformations between the same set of operators on difierent surfaces,or between different sets of commuting operators on the same surface, as in 6(lt'orlf z"oz):i(f t'otl6Wplf

6(('oli'c):i.(l'ol6wl('o).

(5)

One type of change of the general transformation function consistsin the introduction, independently on rr and on oz, of infnitesimal unitary transformations of the operators, including displacements of these surfaces. The transformations will be generated by operators G1 and G2, constructed from dynamical variables o1rd1 &nd d2, respectively' and 6W12:5t-6''

(6)

When the transformation function connects two different sets of operators on the same surface, which are subiected to infinitesimal transformations generatedby C ana G, respectively, we have, referring to (5),

oI

6W.rr:2 6W"aa'.,, a modification of the where 6tr4/,16",, .hu.".r""iir.. transformation function connecting infrnitesimally differing complete sets of operators on the infinitesimally separated surfacesa a.nd of-d'a' If the choice of intermediate operators depends continuously upon the surface, we shall have 5W ".,:0, and, referring again to the dynamical independenceof phenomenaat points separatedby a space-likeinterval, with the consequent additivity property, we see that will have the general form 6W "+a",, Votdc

(dr)ac(rt.

6W,va".,: | Therefore 'al

6tvu: | ,

(dr)6JG).

(s)

The combination of these two types of modifications is describedby

'' 6w,r: 6,- 6r* f

1r*;u"ir,,

(7) 6w:G-G. which involves dynamical variables on the surlaces o1, o2, and in the interior of the volume bounded by these points on a spacephenomena at distinct Since physical the like surface are dynamically independent, a generator G surfaces.On the other hand, we can write this as

358 THEORY

OF

QUANTIZED

7t5

FIELDS

where 6llrrr:g(I,trzrz)and the objects of variation here are 61, 62, and the dynamical variablesof which J is a function. { , / r ' ) [ a u c , f] .+r 6 t ( . f r ] . 6ll'rr: | The latter statement is the operator principle of stationaryaction.ft assertsthat l,tr/r:must be stationary which indicates, conversely,that any part of 6A(r), with respect to variations of the dynamical variables possessingthe form of a divergence,contributes only in the interior of the region definedby o1 and o2, since to the generation of unitary transformations on o1 Gr and G: only contain dynamical variablesassociated with the boundariesof the region.This principle implies and o2. The fundamentaldynamicalprinciple is containedin equationsof motion for the dynamical variables,that the postulatethat there existszrclassof transformation is to say, field equations,and providesexpressions for function alterationsfor rvhich the characterizingoper- the generatorsGr ancl Gr. The class of variations to ators 6llrz are obtained by apyrropriatevariation of a rvhich our postulatereferscan nos, be clefinedthrouglr the requirementthat this informatior.rconcerninglield singleoperator IIl12, equationsand infinitesimalunitary transformationsbe 6lV t':5111t ' self-consistent. "1 'I'here existsmuch freedomrvithin this class,as may Of course,this principle must be implernentedb). the be inferred from the remark that trvo Lagrange funcexplicit specificationof that class. tions, difiering by the divergenceof a vector, describe The operator LIlg, the action integral operator, evi the same dynamical system. Ihus dently possessthe form v^l',hp

intaor'l

?61

c (r): s (r) - d'l" (r)'

'ar

It',,: I

J",

(r.r. id.r;"e

frn:If

The Hermitian requirementon 6lI/12is satisfiedif IIlr: is Hermitian, which impliesthe sameproperty for J(r), the Lagrangefunction operator.In order that relations betweenstateson or and o2be invariantly characterized, the Lagrangefunction must be a scalarwith respectto the transformations of the orthochronous: Lorentz group, rvhich preservethe temporal order of o1 and a2. A dynamical system is specified by exhibiting the Lagrange function in terms of a set of lundamental dynamical variables in the infinitesimal neighborhood of the point r. Contained in this Lagrange function n'ill be certain numerical parameters,rvhich may be {unctionsof :r. Any changeof theseparametersmodilies the structure of the Lagrangefunction and is thus an alteration of the dynamical system. Accordingly, inlinitesimal changesof the dynamiczrlsystem are clescribedby 'or

6ll r' -

|

1 d . r t 6 lr . r r ,

n'here 6A:6(J), ancl thc numerical parameters are thc object of variation. This lorm is in agreement rvith (8). For a fixed dynamical system, tr/12 can be altered by displacing the surfaces 6b q2 and by varying the dynamical variables contained in the Lagrange function. 'Ihe transf ormation function ({ r' o rl l r" o r) describes the relation bets'een two states of the given system so that a change in thc transformation functior.i can oniv arise from alterations of the states on or and or. Hencc, for a fixed dynamical system we must have --ff*

yields

6IY tt:1; t- 1;"'

*u. \ras suggestediry H. J. Bhabha, Revs. ]Iodern P h y s . 2 1 , 4 5 1( 1 9 4 9 ) .

n'hcrc

n.

n".h

,"- (tTrWz),

(9)

.".f""" ff

r:ldo,l,-ldoJo. t-

J 6

. \ c c o r t l i r r g l l ' .t h c . t r L i o n a r y a c t i o n p r i r r , i l ' l c f o r [ - 1 : is satisfied if it is obeyed b.v ltrl12,since

6W":Gr-G. Here

6 l l ' , : 6 : , , G , , 6 l lr : 5 . . t ; r ,

dehne dr and dr, rvhich are new generatorsof infinitesimal unitary transformationson 01 i1r]do2Jrespectively. The latter equationspossessthe form (7), and thus characterizetransformationfunctions connecting trvo different representationson a cornrnon surface. Indeed,rvith a suitably elaboralenotalion) t'e recognize in (9) the additivity property of action operators, r) : 1l'(iror, fi or)* IIl ((,o,, l ro "o ") llt;(ist, rvhere,{or example, II/ (f s', i

i:ot),

I t r ' r - 1 Y 1 7 t o r ,f r o r ) : l l " ( f r o r , i r o r ) , and lVz:IL'1.1,or,7rorr. To be consistentwith the postulalc of local action, the field equations must be differential equations of linite order.One can alwaysconvertsuchequationsinto systemsof iirst order equationsby suitableadjunction o{ variables.\Ve shall designatethe fundamental dynamical variablesthat obey first-order field equations by x,(r), which {orm the componentsof tire general lield operator x(.D. With no lossin generality,rve take

359 7t6

JULIAN

SCHWINGER

ment on I is satisfied if JC is a scalar,

x(r) to be a Hermitian operator,

K(Lil:K(i,

Y'(x)t: Y'(x) ' If the Lagrange function is to yield field equations of the desired structure, it must be linear in the first derivatives of the field operators with respect to the space-time coordinates. Furthermore, if these field equations are to emerge as explicit equations of motion for field components,that part of the Lagrange function containing first coordinate derivatives must be bilinear in the field components. With these prelimiuary remarks, we write the following general expressionfor the Lagrange function, s:l(721,0,y-0ux2lux)-K(x),

(10)

in whiclr a matrix notation is employed, a2Iuoua: y, (21,),,a,a ".

and if

/irs)lI

-$l

(1J)

Note that ?Iu" and $* also obey Eqs. (12) and (13), respectively, and that these equations can be combined into Z-'($ r?I,)Z:r,,(S t?I,), in view of the nonsingularcharactero{ $. For an ininitesimal Lorentz transformation, '

ep,: tr:'r_ 'rr'r* ep, The derivative terms have been symmetrized witl.r respect to the operationof integrationby parts, a process the matrix I can be written which adds a divergence to the Lagrange function, and L:l_iie*Sr,, is thus without efiect on the structure of the dynamical syst€m. In order that S be a Hermitian operator, the where generalfunction JCmust possessthis character, Sr,*: -Sr,1 u' v:0,

rc(il1:3c(il' and the numerical matrices ?L,; p-0, 1, 2, 3 (ra:ixs, ? t 4 : t U o Jm u s t b e s k e r v - H e r m i t i a n , 2 l p 1 : ? I r h x :- \ p t

p:(), 1,2,3.

(12)

Lr"WuL:ru,4,,.

lhe same \\'e shall supposell)at the source possesses transformation properties as the field. The condition for the source term of the Lagrange function to be a scalar is then siven bv

-

cvpt

(14) 3.

(15)

The infinitesimalversionof (13) is - S",h: SSr'$-t-

Su,t

or

(ss!,)1: ($s,,),

in rvhich the cornplex conjugate statements refer to the Although we are interested in complete dynamical componentsindicaied in (15). Similarly, systemsJit is advantageousmathematically to employ (16) ?I,s,x- ^t,It?Iu:i (6,r?I,-r,,9Ir) devices based upon the properties of external sources. Accordingly,rveadd to (10) a term designedto describe and 1 ? 1 ,S, " r ] : i ( 6 u i S ' ? I , - 6 , , . S - ' 9 I x ) . the generation of the field 1(r) by an exlernal source [E f(r), which is to be regardedas a field quantity of the If one viervs'a: (1-i\e ,,5) x as a field in the original samegeneralnature as 1(r), coordinate system and thus subject to the same de(i1) pendence upon that coordinate system as 1, it is s"""""":+(€Sx+x$t). inferred that if is operator a Hermitian matrix, This is a Hermitian $ L-rS,,L: r rxr,rSxr. mi-m For inlinitesimal transformations, this reads For the sourceconcept to be meaningful, all componi[^S,,,Sr"] : d,-S,x- 6,^S,x*d,rSr*-durS,-. ents of 1 must occur coupled with the source components in (11), which requires that E be a nonIn performing the variation of the action integral, singular numerical matrix. we shall treat the two types oI quantities, coordinates An orthochronousLorentz transformation and ireld variables, on somervhat the same footing, tx r although the former are numbers and the latter operr : p , r , + IF , ators. We introduce an arbitrary variation of the cort"r:I, ru>-0, ordinates, 6r!, throughout the interior of the region, but subjectto the conditionthat the boundariesremain induces a linear transformation on the lielcl complane surfaces, ponents, tr: ['a:aLv, (r7) d u d r ,{ d , 6 r ' r : 0 , rvhereZ must be a real matrix, f *- r '1. The scalar requireto maintain the Hermiticity of

on o1and or. The field components x,(r) are dependent both upon the coordinate system and the "intrinsic field." Ltnder a rotation of the coordinatesystem, the field components are altered in the manner described

360 THEORY

OF OUANTIZED

by (1a). Accordingly, we write the general variation of the field as the sum of an intrinsic field variation, and of the variation induced by the local rotation of the coordinate system, 6(x) :6x-

where the antisymmetry of S' ensures that only the rotation part of the coordinate displacementis efiective. For the source field, a prescribed function of the coordinates, we have

^f

r

fanc^r

: +(?1,a,+?{,du). ?{(ua,) 68 : 6x?lrapx- d"x?1"0x- arc+ I (dxSt+ f$6x) *6rui (xsauE+ a,t$il * a"[] (x?{u61- 61?[u1)]. Hence, on applying the principle of stationary action to coordinate and field variations, separately, we obtain

(18)

0,T u,: I (yE 6,1* A,fEx), and

We also remark that

6?,C:6xaplpx- A,x2lu6x++(6xst*f$6x),

6(d*) : (d.r)6 ultc,, 6(0): - (a,6r,)a,, whence 1 1q \

The Lorentz invariance of 3 produces a significant simplification, in computing the contribution to d(S) from the coordinate induced variation of 1. Thus, if 316*,were antisymmetrical and constant, its coefficient in the variation of the Lagrange function would vanish identically, save for the source term since the rotation induced change of f is not present in (18). Accordingly, for the general coordinate variation of (10), there remains only those terms in which d"6o, is differentiated, or occurs in the dilation iombination, 0r6x,!0,6x,. Both types are contained entirely in (19), which leadsto r,! 6,6r)| QA u04- 0,yWu7) 6(r) : 6"c- + (Ap6 - i.L(a ra,6r)tx(21,,S,^-pS,.t21";" - r+ (dts6r,)({SS,,x xSu,t$t).

G-

-DxU*r)*1,,6r,]l | do"[ (\U"6x J n

The operator 3Cis an arbitrary, invariant function of the field 1. If its variation is to possessthe form (21), with D1appearing on the le{t and on the right, the latter must possesselementary operator properties, characterizing the class of variations to which the action principle refers. Thus, we should be able to displace 01 entirely to the left, or to the right, in the structure of DJC, 6K : 5x(6r\c/ Ox) : @,tc/ dx) 6x, which defines the left and right derivatives of 5C with respect to 1. In view of the complete symmetry between left and right in the processof multiplication, we infer that the expressionswith dx on the left and on the right are, in fact, identical. The field equations, therefore, possessthe two equivalent forms 2AuOuy: (afic/ai-Et, -ou72A,u: G,K/ax)-tE,

In virtue of the symmetry of the secondderivative, (dud,3rr)x (?IuS,r*S,ri?lu)x : (d" (ddrrf drdr,))1 (21,S,1f S,1t?L)1 +-

where the last step expressesthe result of'an integration by parts, for which the integrated term vanishes, since the dilation tensoris zero on the boundaries(Eq. (t7)). Collecting the coeficients of dudr, into the tensor ?r,, we have fol

J

-

(dr)[6f+(d,6.r.,)r,,1

d,

fol

|

and G can be equivalently written

(d,611{ dldr,) ap[x (?l,Srr+^S,r1?I)x],

6(Wn\: |

Ql)

rvhile the surface terms yield, on oy and o2, the infinitesimal generator

and

6 ( d , x ): d u d ( x ) - ( 0* 3 r , ) 0 , y

and we have employed a notation for the s)'rnmetrical nr"f

The expressionfor df' is

iL(auor,)s",x,

6(f):6*ud,{.

717

FIELDS

(dr)[6"c-6.r,a,ru"lou(7,,6x"\f,

J n,

where T * : s,6* - t Q\I'1no,1v-o pa{',1v) -ii(g$S,a-15",t$f) *lldr[x(9lr,sx,r*sr1,i?I,;)1], Qo)

f

G:

I rlo,[19i,01f I",6i',] (2',2)

:

f a",l-aer,,1f

?u,6r,1.

In keeping with the restriction of the stationary action principle to fixed dynamical systems, the external sourcehas not been altered. If we now introduce an infinitesimal variation of f, and extend the argument of the previous paragraph to df, we obtain the two equivalent expressionsfor the change induced in tr/12,

6ilu:

fot

|

J o,

(d:)D€$r:

fol

|

{a';1559.

Jo'

The correspondingmodification in the relation between

36r 718

JULIAN

SCHWINGER

stateson or and on 02 can be ascribedto the individual states only if one introduces a convention, of the nature of a boundary condition. Thus, we may suppose that the state on 02 i: unaffected by varying the external source in the region between or and qz. In this "retarded" description, Dtltr/r: generates the infinitesimal transformation of the state on or. An alternative, -dgtr/12generat"advanced" descriptioncorrespondsto ing the changein the state on oz,with a fixed state on or. These are just the simplest of possible boundary conditions. The suitability of the designations,retarded and advanced, can be seen by considering the matrlt of an operator constructed from dynamical variables on some surface o, intermediate betrveena1 a.\d o2,

to plane spaceJike surfaces,limiting displacements to infi nitesimal translati4ns and rotations, 6ru: e'!e "ru' with the associatedoperators,the energy-momentum veclor f

p,:

I rtouT'r,,

and angular momentum tensor f

J u,: I doxMx,,. J

Mxuo: :rrTx'* r,Txu.

(.({ o 'l F (a) ll r" 02)

The operator G, evidently generatesthe infinitesimal produced by the - | , l i o l f ' o \ d l ' t f ' o F ( o \ l l " q \ ' ' l f " ( f " o 1 2 " o 2 \ . transformation of an eigenvector, displacement of the surface to which it refers. With the notation An ininitesimal change of the source I produces the aJ, (1,") : (e,6,* |epdu)V (i,o), following change in the matrix element, we have 6t(ir'orIF(o)|lz"oz) l5,v(1/c) : P,!Ir(f/o), -i6,V(f'o) i:{/ (l'o) rP,, : (11o, | (a f (o) { ibsWuF (6) + iF (6)6rW, 2)| f 2"oz) : ( r 1 " ' | ( a f ( o ) * i ( F ( a ) 6t w n ) + ) l l z "qz ) , and '") i6,N (f ' q) : J pN (f ' o), - du,v (f t : v (l' o) tI u,. in which we have allowed for the possibility that F(o) may be explicitly dependentupon the source,and introIf F(o) is an arbitrary function of dynamical variables duced a notation for temporally ordered products. The on o, and possibly of nondynamical parameters dematrix element depends upon the external source pendent on o, we use the notation through the operator F(o), and the eigenvectorson o1 gets for various expressions 6gF(o), D,F(o): (e,6,1 I e,,6,,)F (o), and or. One thereby depending upon the boundary conditions that are a,F (o) : (e,0,1 1,,,A,) F ("), adopted. Thus, if the state on a2 is prescribed, we find to distinguish between the total change on displace6 s f ( o) 1 , " ': 6 , p 1 o ) * i ( F ( o ) D t l Zrr) +- i 6 l / r z F ( o ) ( 2 J ) ment, and that occasionedby the explicit appearance of nondynamicalparameters.On referring to Eq. (3), : atF (o)* ilF (") , 6Jv ',f, we seethat which only involves changesin the source prior to, or 'Ihe 6,F(o): a,F(q)i ilF(c), P,), opposite convention yields the analogous on o. result 6u"F('o): 6,,P 1") -l ilF (o) , l ,'f 12 drF(o)1,a"- dtF(o) {i(F(o)Daltr/r,)1- iF (o)61tr' The proper interpretation of the generating operator : atF (o) _ i.lF ("), dy'{'r"]. G, can be obtained by noting its equivalence with an appropriately chosen inflnitesimal variation of the exNote that ternal source. Consider the following infinitesimal surdsF(o)1."i-6gF(")1.a": i[P("), 6rII/'r]. face distribution on the negative side of o, The operator G of Eq. (22) consistsof two parts, (24)

So€:21,ru"u,r,rr,

G:G,*G,, lvhere

c,:

f"ao,y\t,a": ["rto,6ylr,a,

which is not incompatible with the operator properties of these variations. We have assumed, for simplicity, that the equation of the sur{aceo is r:1oy:0.With this choice,

and

do,T,,a*,:e,P"lle,,J u, G ": f of the restriction The latter form of G, is a consequence

6{v,,:

f

Gx d.oa2Iqo16x: "

The change that is produced in 1 can be deduced from

362 THEORY

OF qUANTIZED

719

FIELDS

decomposition

the variation of the field equatons, 2Wro,62y- 6E(61K/ ai : - E5 t : -?Irordxd(rror).

?I,:?J*tr)f!1,{:r, ?lro)t- -?IP(t),

E:S(t)+S('?), $ori.:9u,, s(2)h: -s(2),

2lP(2)r-?l!(2)' Evidently there is a discontinuity in 6rx, on crossing the surfacedistribution 6f, which is given by The matrices of the first kind are real (p:6, ' '3;, and those of the second kind are imaginary. We shall -2116y61. 2?lrordrxl: not write the distinguishing index when no confusion is In the retarded description, say, 6g1is zero prior to the possible. According to this reducibility hypothesis, the field source bearing surface, so that the discontinuity in 6g1 is the change induced in 1 on (the positive side of) o. equations in the two equivalent forms Thus, the surface variation of the external source 2?1,0,a: (agt /Ax)-EE, simulates the transformation generatedby G", in which -2?It"a &: (A'tc/Ax) -E "t' on o is replacedby ?[
?Iror *?Irord i : 2Irorx * :2lrorx- 12[
separate into the two sets

(2s)

The matrix ?lrorhas been retained in this statement since it is a singular matrix, in general. The number of components of 1 that appear independently in (25) and this is the number equalsthe rank of the matrix ?1101, of independent component field equations that are equations of motion, in that they contain timelike derivatives. The expression of (25) in terms of the generator G* is [?Irorx,G"]:i]?lrordx.

(a &/ ad : @,K/ ad,

221u6 uq: (arclad)-Sf, and

Gfic/aV): - (a,K/a{').

221,6u{: (afic/a,!)-En, Furthermore, the generator G,:J

f-f d"vU,q6r:J do( U.",t'6rtv,

decomposesinto G6fG,1, where

Q6)

The factor of ! that appears in this result stems from the treatment of all componentsof ?Irolxon the same a n d footing; we have not divided them into two sets of which one is fixed and the other varied.* If I is an arbitrary function of ?I1o;1on a, we rvrite

f -. od G ^ : I d o 6 2 ( ,D JJ

f l do{?{'0,6d)d. Q7)

cu: ' J J la,g\ts,a{,:I dor_ltu6,l,t{.

, lF , C,l:;15n7,: i15P

These resulls reflect the form assumedby the Lagrange {unction,

in which the componentsof 2lro;x are the objects of variation. When the field equationsthat are equations of constraintprove sufficientto expressall components of 1 in terms of ?{rorx,we can extend (26) into

a,6l++lvah a,vl- K(4,,1,), J : +{d?I,, +i{rs,0)*4[4$,*].

lx' G,f:iLax. Of course,one must distinguish between these variations, in which only the ?lrorxare independent,and the independent variations of all components of 1 which produce the equations of constraint from the action principle. In order to facilitate the explicit construction of t-he field commutation relations, we shall introduce a reducibility hypothesis, which is associatedrvith the Lorentz invariant process of separating the matrices ?[u,S into symmetrical and antisymmetrical parts. We require that the field and the source decomposeinto two sets, of the first kind lttr:6, €(r):f, and of the secondkind, x@:V, tQ):n, as a concomitant of the * Note addedin hool:-Further discussionof this point rvill be found in a paper submitted to the PhilosophdcalIlagazine.

The equivalencebetween left and right derivatives of the arbitrary function 5C,with respect to field components of the first kind, and of the trvo expressionsfor G6, showsthat D{ commuteswith all fields at the same point. It is compatible with the field equations to extend this statementto fields at arbitrary points,

@),66@')f : 0, lo @),6oG)l : 1,1, provided the source components are included, [l(r'), 6d(r')]:

["r(r), ad(r')] : o.

It lollows from (27) that the relation between ry'and dry'is one of anticommutivity. The opposite signs of the left and right derivatives of 5Cwith respect to ry'is then accountedfor by [ O ( r ) , 6 9 ( r ' ) ] : { , 1 ' Q ) 6, V Q ' ) l : 0 , provided only that JC is an even function of the vari-

363 720

JULIAN

SCHWINGER

ables of the second kind. The inclusion of the source components

which requires that the real, symmetrical matrix illroy{rr-rO" positive definite.

rr('),De(r')t:{n@),6te)t:0, coI;:11i.":?1i:Y:::?ISL"J#:o"i*Jf insures compatibility with the field equations. We have now obtained the explicit characterization of the class of variations to which our fundamental postulate refers. Let us also notice that "'ra.rlt$"agr*. rat -Jd' 'u!'\\us'^' f

r,r 6twrr: I td'rq$ag:

'

r".

into6rwnt6,w12, lvhere crecomposes f"t

f"'

6 r l lr : : I

J o t

and

r d . r r 4 t l a EI :

rd.rtr$69'6.

(dr)r/864- |

rtxll-Sta,g.

J o ,

llror(2),must be even, 2n@. Let us imagine that, by a suitable real transformation, !I1o1o)is brought into diagonal form. If the number of components in r[ is odd, the product of all these componentsat a given point commuteswith t at that point. Thus, as far as a l g e b r ao I o p e r a l o r sa t a s i v e n p o i n t i s c o n c e r n e d ' :nt

*:H:i'::J'-:,il,1i'?li,.'J^^",:1';.:1'J,1 :l:1*T,i:fri':;:lL:i'.it"::lffii:Lthe

The relation betweeninvariance under time reflection, and the connectionbetweenspin and statistics,may be noted here. The time reflection transformation

' 6"1',,: f

assump-

txq:

-14,

tatr:ao,

induces a transformation of the field We can conclude that source variations have the same operator properties as field variations, as already exploited in Eq. (2q. The operatorpropertiesof ?Irotxon a given o can now be deducedfrom (26), with the results

'x: Ltx' such that

Lqr.l(tLr- -lI4

La'"!loLr:s5 r,

(2e)

and ZI"SZ4:E,

K(Lq):K(i,

[Urord("), d(r')?lior]: ri2l10)6"(r- *'),

However, this preservation of the form of the Lagrange (28) [ ? I t o r d ( r )9' ( r ' ) 2 1 , 0 , ] : 0 , function is only apparent,for fields of the secondkind. Since -l9l,rtzl is a non-negativematrix, one can only {?Itor/(r), /(r')?lror} - i}?IrorD,(rc-r'), satisfy the first equationof (29) rvith an imaginary Zai?) ir rvhich 6,(r-r') is the three-dimensionaldelta func- rvhich produces skerv-Ifermitian field components '1('!). tion appropriateto the surfaceo. The numericalforms But the invarianceof the Lagrangefunction is not the of these commutators and anticommutators insures c o r r e c tc r i l e r i o n f o r i n v a r i a n c eu n d e r t i m e r e f l e c t i o n . their consistencywith the operator properties of 6?11qS The reversal of the time senseinverts the order of o1 variables first and The dynamical o{ the 6?I1nyry'. and and o2, and thus introducesa minus sign in the action second kind thus describeBose-Einsteinand Fermi- integral, which can only be compensatedby changing Dirac fields, respectively, rvhich are unified in the the sign of i in (4). We shall describethis as a transgeneral field 1. formation from the algebra of the operators 1 to the Since the rank of the antisymmetricalmatrix 2116;(1)complex conjugate algebra of operators 1*. Since the is necessarilyeven, there are an even number of inde- linear transformation designedto maintain the form of pendenf field componentsof the first kind, say 2n(L). s(0, a,O;{, dury') has efiectively replaced a with One can always arrange the matrix ?l
kinematically invariant under time reflection.In order that it be dynamically invariant, 3Cmust be such that Jc(0, i,l,)*: ut(6*,,t/*). Since 3C is an even function of the componentsof ry', the ialter are to be paired with the aid of imaginary matrices, characteristicof the variables of the second kind. The sourceterm is invariant if sourceand field transform in the same rvay. The correlationbetweenspin and statisticsenterson

364 THEORY

observing that an imaginary Za is characteristic of halfintegral ipin fields. We can prove this by remarking thaiall the transformation properties of Z4 are satisfied by -riS'+)lr, L+: exp(- tniSu)1-r exp(irrSu) : exp( where Zq is the matrix describing the reflection of the irrst space axis. The latter form is a consequenceof

721

FIELDS

OF QUANTIZED we have

f"l

J *ro1t-J r,(o2\: |

(dxl'Ir51'.,r,-x,d,*i'su,){

Jot

+ (tcu0,- s,0 ut i,S,,)€Ex].

In the absenceof an external source,I*, is symmetrical and P,, Ju, are conserved'.For and divergenceless, simplicityfwe shall confine our verification to the situation of no source, in which the infinitesimal 6f is distributed in the region between o1 and o2. Hence

Zr-1S1aZ1:-'!t'' The essential point with regard to the reality oI Ir is that Su:iSro is a real matrlx, whence

6 a P , G 1 t -:

|

(dr')a,YSDt.

Jat

and exP(2riS u) L t' f" ( d . r tl r u d , - . t , 3 , * i . s " , ) \ 5 6 { . 6 2,J, r o r t : - | say' as.9r2' same eigenvalues Now S1amust possessthe Joz spin field, which implies that /-r is real for an integral and imaginary for a half-integral spin field. The re- The consistencyrequirement quiremenl of time reflection invariance thus restricts fol (d.rr(6\)$6t:6'G,' (6Gr),=| fields of the first (B.E') and second (F.D') kind to This corre. t . t spins, respectively. half-integral integral and lation is also satisfactory in that it identifies the doublethen demands that valued, half-integral spin fields rvith fields of the second - (6x),: ,,0"x|!er'@,0 x"0 u-liS,,)x, (30) kind, of which I is an even function. "We have introduced several kinds of generators of which is indeed true in virtue of the equivalence beinfinitesimal transformations. A criterion for consistency tween (d1(*)),, induced by the displacement 6f!, and of the evaluations alternative the from is obtained ' x @ ) - x @ ) , i n d u c e db y the coordinatetransformation generators, such commutator of two '*r: rri_6xu. - i (6Gi. (6G Alternative forms of P, and /n, are convenient {or i G t: bl: lG ", ") the consistencYof G. and Gr. The following testing namely (6G,)b+(6Gt.:0. relationsderived {rom (16), Ir* : exp(riSr) h:

ix (2lI.Sr,- S't2l x)3xx, y2I r7_ all,a "7: "0 - Su't?Il)x, id rx ( ?IxSu, 0,xll,x: 0 uv2l "7-

As a first example, we consider the two generators e , P , ( q ) * | eu , J , " ( . or ) ,

G.:

enableus to write fp, as

and fo'

G= J

|

T. : S6r-

(d'r11$56.

o,

where

in the retardeddescription.In preparationfor the test, rve remark that l',to.t ,,t", -

:

f"'

J.,

syu,: -suy,:if,1(2?I1"Sra*2Srr,i?Irr _ -?|S",-S,,i?[)x, and

t,1.r\3.1",

r*o,a,rra.ttx), /"' tr'o+

P,-

"",

1,,: (dr')[r"3^rr, - r,01TsulT,"- ' r

7,"- T,u: - i+(tES,,x- xS",i$t),

(1U"a,v- a"r?l,r)*p",1. | d ou l s : 6 , , - |

but doesenter in

(dx)0xLIx,, f

: | Since

p,": -iilS u,x@&/ax)+(a;tt/ax)S,,xl.

In virtue of the antisymmetry of sr* in the first two indices, drsr", is automatically divergencelessand does not contribute to the energy-momentum vector P,,

and that J r,(or)-J r,(or)-

t Q}l u0;x- 3 "x?I,x)I d1s1",{Pp,,

1

I

- r,o,l i'9,,) a ^l- ix}Ix(:c,a, x " !!

(r,0,- x

"0,*iSu)1?Lx*runr,+

r,Pxpl

ra',.*,- tlo,x,\!. [

365 722

SCHWINGER

JULIAN

The components of P, in a local coordinate

system are

t

3"(r-r') and therefore vanish when multiplied by x1p1*fr1*1t . Furthermore,

p,o,: -x?I,*,dt,x- I (t8xix$t)], I dafK J

(31) P 1a ,:

y?I1o'd,*'1{p,6'1p,].

while those oI J,

ti(a fic/ 3y)6,(x- r'),

from which we obtain llc, N e1Gll: 2ip(or6r : 0.

arc f | dor1a,[JC | (19{,,,d,r,t J

- o utxllutx)* i(fSx* xSt)l -|i

[rc(*), x (r')]?Iro, : li(a,.tc/ ay)6"(r - x'), ?lror[x(r'), K(r)l:

f I dolJ

J r o , r * r: * ( n , P m -

and

With this information, the proof is easily extended to all componentsof pu,. The consistencyof the generators G, and G* requires that

f I do1(2l,ouS1o,ar*Sror.rrtlror)1, (.i2)

i 6 ^ ( e , P , f l e u , J , , ): -

faofa*),?t,o,a*.

or I o tt,:

t?{to, (*,r' d1ly r11,di*1* t51r114)X I daf

6\P,:

I do6"y22lq,6x,

J

- r1i;p1o;11;]. f r1t;p1oy17y The quantity p", is closely related to the infinitesimal expression of the scalar character of 3C, K (y- ii eu"S,,y) -cc (x) : 0.

f

6\ J u , : ,

do(t,0,- :,0 u|rSu,)1Zlfro,dr,

J

which can now be verified from the expressions (31) a n d ( 3 2 ) ,w i t h p 1 o ; i 1 y : 0 .

We can, indeed, conclude that

CHARGEDF'IELDS

pp":0, if 3C is no more rhan quadralic in the components of various independent lields. We shall also prove this without the latter restriction, but, for simplicity, with the limitation that there are no equations of constraint. The commutation relations equivalent to (30),

la, P,l: lx, J -f:

-ia,y, -i(x,\'-xd,JiS,")v,

imply that [x, tr'u,]:S,,x, where

Our considerations thus far specifically exclude the electromagnetic field (and the gravitational field). We introduce the concept of charge by requiring that the Lagrange function be invariant under constant phase (special gauge) transformations, the infinitesimal version of which is ,r: 11_i.6t6)x. Here 6X is a constant, and 6 is an imaginary matrix which can be viewed as a rotation matrix referring to a space other than the four-dimensional world. The invariance requirement implies that

N r,: J u,-*uP,!x,Pu.

6i:s6s-1,

This enablesone to expressthe scalar requirement on JC or in the form and [3c' N!"]:o' The components and that d l o r l r r:

I d o '( r 1 r ' r 1 r , ' ) [ f f ( r ' ) I ( r ? [ r i , a , r r ' t

- oai x?Iatld- l(4Sx*x$t)

($a;t:55' [6, E-l?I,]:

[6, S,,]:0,

3(.(x-i6\Ex)-K(i:0. We now write the general variation as 6() : 6y - ii @,6x,) Su - i6r.Ey, "a

where 6tr, characterizing a local phase transformation, f - l i d o r ( ? I , o , S , o , , r , * S i o ; 1 r 7 t ? [ , q is ) 1 ,an arbitrary function of r, consistentwith constant I values on or and on oz. The additional contribution to 6(a) thereby producedis do not involve the unknown ptoltrt. According to our jf dts6l- t; (tS6x- xS6€)DX, simplifying assumption of no constraint equations, the commutators (anticommutators) of all field components rvhere at r and *' contain the three-dimensionaldelta function i ,: - ix?Ir8x

366 THEORY

OF OUANTIZED

is the charge-current vector. The stationary action principle requires that

(33)

a,jF-i+(tEEx-xE8$, and yields as the phase transformation generatror f

I dol*6\:Qdl,

f1:

J"

where p is the charge operator. The integral stntement derived from (33),

QG,) - QG,) :

[,','

(d.r)diQ8 6t - t8 8x),

becomes th'i conservation gf charge in the absenceof an external source. If an infinitesimal source is introduced in tne region bounded by o1and o2,we then have, in the retarded descriPtion, 6g0(",):-,

|

723

FIELDS

algebras are involved, the field contains particles with charges0, *a. To present 6 as a diagonal matrix, we must forego the choice of Hermitian field components. Thus, for the example of a charged F.D. fiild, where the field components decomposeinto ry'1i,9(2), corresponding to the structures (34) and (35), the mutually Hermitian conjugate operators rlts; : tltss- i9 et, t et : t ot * i{ pt, are associated with eigenvalues *e and -e' respectively. On introducing these field components, the derivative term in the Lagrange function, the electric current vector. and the commutation relations, respectively, read Il*<-tw,,du*r*rl*1&r+r?Iu, 0r!<-tf, -iei(tePlr{<+t-{+flIu*<-),

(36) (37)

and } {?Iot! e>@), {c+r (o',)?Iror :{2t,0),y'1_;(r),/1-,(*')U10,}:0, (38) {?I
(ditdi$61

J oz

:ilQ@),Gi' W'rence

lx,Ql:Ex. This commutation relation also follows directly {rom the significanceof G1, indicating the consistency of the latter with Gs. We shall supposethat the matrix $ is an element of the algebra generatedby S-121* and Sr,. It follows that $ commutes with 6, and therefore that the latter is explicitly Hermitian, Et: E.

There is evident symmetry with respect to the substitux+-e' tionr!6<+!6; Since/1..yand p1-yare Hermitian conjugate operators' we can arbitrarily select one as the primary nonHermitian field. We shall write ca-l9fa u --;^, 'lPt <

and

te>E:,1,18:0.

to:{,

yields the following forms for (36), (37), and (38): possesses This

Such an antisymmetrical, imaginary matrix real eigenvalues which are symmetricaliy distributed about zero; nonvanishing eigenvalues occur in oppositely signed pairs. Since 6 commutes with all members of the above-mentioned algebra, the charge-bearing character of a given field dependsupon the reducibility of this algebra. Thus, if the algebra for a certain kind of field is irreducible, the only matrix commuting with all members of the algebra is the symmetrical unit matrix. Hence 6:0, and the field is electrically neutral. If, however, the matrix algebra is reducible to two similar algebras, as in

,r:(? ;r),

, ^

. r

tL!^t ,, xdpV)-

i . . ^

i

' l

iLxd tY"t t, V ),

e+10'Yr'!l'

(3e)

and : I0 @)rtot'/(r"')r
{": (-s.-').F,

(41)

(34) and state this ryrnmetry as invariance under the substi-

the matrix 6 exists and has the form (with the same partitioning) r* """'- o' / 0" _ i ", l (35) s: r( ) \i 0 / This describes a charged field, composed oI particles with charges +e, the Jigenvaluesof 6. If threi similar

i.:.ttionrlte{", ee-e. T h e m a t r i c e s" l p i P : 0 , ' ' ' 3 ,

obeY

'Y'i:E7*t-t' and ?ut':

-S7rS-r,

(42)

since they are purely imaginary matrices' ore sbould also recall that E is an antisymmetrical, imaginary matrix. If we were to depart from these special struc-

367 724

JULIAN

SCHWINGER

tures by subjecting all matrices to an arbitrary unitary trairsformation, we should find that the only formal changes occur in (41) and (42), where the matrix !] appears modified by an orthogonal, rather than a unitary transformation. Hence, in a general representation these equations read ,lr":C{, "t,r' : - C-11,C , where C still exhibits the symmetry of $, appropriate to the example of a half-integral spin field,

THE ELECTROMAGNETICFIELD The postulate of general gauge invariance rnotivates the introduction of the electrornagnetic field. If all fields and sources are subjected to the general gauge fra nqfnrm

c tinn

'1: exp(- iI (r)d)x: x exp(iI(r)6), the Lagrange function we have been considering alters in the following manner, 'f : S-firdrX. The addition of the electromagnetic field Lagrange function,

The commutation relations (40) are in the canonical form which corresponds to the division of the independent field components into two sets, such that one has vanishing anticommutators (commutators, for an integral spin field) among members of the same set. The generator of changes in ry'and ry',Eq. (27) in the notation of the charged half-integral spin field example, is C hl',{,) : Li I do (0t, 0,6{- a,i't,",V\.

C " ^ r : i U u , A p l - I { F w , . 0N A , - 0 , A u l ++FN'2+JpAp,

(43)

provides a compensating quantity through the associated gauge transformation 'A':Au-3'\' The term involving the external current -ftsis effectively gauge invariant if o'J u:0'

which can be deduced directly from the Lagrange function derivative term (39). Associated with the freedom of altering the Lagrange function by the addition of a divergence, are various expressions for generating operators of changesin the field components. Thus, we have the following two simple possibilities for the derivative term and the associatedgenerating operator,

+lh* id,vl, Ghl'):i J

and

I d"{to,6*,

-+lidih,"1'f, _f

Ghlr\: - i I do6,l,tp,{.

since the modification is in the form of a divergence. In the same sense, there is no objection to employing a form of the Lag^rngelunction in which the secondterm of (43) is replacedby (44)

+ { a p F P "A, , l . We write the general variation of ,4" in the form

6 ( Au ) : A 1 , - 1 6u ' r " 1 O , :6A u- l(3 uin,- A,6r,)A |(0,6t,t 3,5r)A,, "which ascribesto ,4, the same transformation properties as the gradient of a scalar, thus preserving the possibility of gauge transfonnations under arbitrary coordinate deformations. In a similar way, 5(F ) : 5Pu,- (dudir;)Fr,- (d,irJF"x.

Evidently G(ry'),for example, in the generator of alterations in the components z(0)ry',with no change in f7,r. The associatedcommutation relations, lt a{, G (,1')f: it ot6,!, lhat,G(0):0, are satisfied in virtue of (40), and, conversely, in conjunction with the analogousstatements for G(l), imply these operator properties of the field components. The connection with the generator in the symmetrical treatment of all field components is given by

(,1,) (,1'), c ({',,r): +G + +G which indicates the origin of the factor (1/2) in the generalEq. (26).

With .regard to the derivation of the electromagnetic field equations from the action principle, it should be noted that general gauge invariance requires that the sources of charged fields depend implicitly upon the vector potential -4u. We expressthis dependenceby ?

6 r ! { . r ' ): ,

Gx \ 6 t G ' \ / 6 A , ( r ) ) d l

"(r).

Since the infinitesimal gauge transformation, 6,4r: -dr6tr, must induce the change6t:-?dtr6f, we learn that a,QtQ')/6Au@)): -i8t@)6(x-r',).

(4s)

One obtains the following field equations on varying

368 THEORY

OF QUA NTIZED

If 6"Iuhas the explicitly divergencelessform

F r" and A, in the complete Lagrange function, F,":3u'4'-0'A,, 3,Fr,: j,!kplJ

p,

FIELDS

(46) (47)

6J':0'6M"'

M":-M'u'

(s3)

where6Mu, vanisheson o1 atrd 02,we find that

where

^61

f

ku ( s ' -) ) , I t d r ' r I l a g t r ' t .6 1 , { r r 1 $ x 1 v ' 1 " lx(r')S(a{(r')/6,a,(r))1,

6tW,,:

| Joz

(d)'\LbMpr,,,

which makes it unnecessary to introduce an external is the contribution to the total current vector associated source that is directly coupled to the field strength tensor Fu,. with charged field sources.We derive from (45) that The special nature of the electromagnetic field3 is a,hr:it(tSEx-xEE0. apparent in the form of the operator (52) generatilig Ilut ttie total current vector is divergencelessin conse- changesin the local electric field components.Since one quenceof the electromagneticfield equations. Therefore of the field equationsis the equation oI constraint

a,ir: -i+(ttb'x-x$49,

d 1 l y F 1 o y 1j lro1r:* k < o t l J < o t ,

(54)

the three variations DFlo;1*;cannot lle arbitrarily aswhich is in agreementwith (33). A{ter removing the terms in d(8"-) that contribute signed; the electromagnetic field and charged fields are not kinematically independent. This is evidently an to the field equations, we are le{t with aspect of the gauge invariance that links the two types : (s"*, A 6PQ"P6A')| 6 u6'J'6x' +{6j,, A } of fields. Alternatively, s'e see from (51) that ,410)is - \ (6 rbx,! 0 (+ A,j - + lFur, F,r) ) not a dynamical variable subject to independent varia"6r,) U,, * ( a u 6 r , ) 1 , A , , ( 4 8 ) tions. But there is no field equation that expresses,4loy in terms of independent dynamical variables, in virtue in which of the arbitrariness associated lvith the existence o{ -i6yl/',E{a, A pl: - i{A $ y}2l,Eby. A , l : +16j,, gauge transformations. Furthermore, a variation of -,1gr in the form o{ a gradient, that is, a gauge transThis term alters the field equations of charged fields, formation, yields a generating operator which, in con2 2 1 , ( A , x -i , E + l A, , 7 \ ) : ( AL K /A ) - $ L sequenceof (54), no longer contains electromagnetic - (auxliL{x, A p}E)2WP: (a,K/a).i- t!3. field dynamical variables. Thus, in either {orm, (51) or (52), there are only two kinematically independent We have anticipated that not all componentsof '4, variations of the electromagnetic field quantities. is as The tensor ?r, now obtained commute with 1. We now apply these generators to deduce commuta( 4 9 ) tion properties for the gauge invariant fieid strength Tp,:...+ilFt^, F,r)- j{jr,, A,Jl-J,A,, comporlents. According to the efiect of a variation where .. . stands for (20), but with "8 the complete 61 1r;,upon the local componentsof F' we have Lagrange function. The action principle supplies the [ F 1 o ) 1 rG y ,e ] : 0 , difierential equation A ot), G tl: i (d c16A 6 - o s16 at, lF tot (50) A pa,J,. auTu,: i(x8a,t*a,tEx)l whence 'I'he divergence term in (48) yields the infinitesimal (55) generator [ i ' 1 q 1 r y ( r )i,' 1 0 ) 1 r ) ( r ' ) ] : 0 , and

cn: -

ao,n,"aa,:-

!

!

(s1) a'a
rvhile the Lagrange function with the derivative term (44) would give Qr:

ff do,oF,,A":

J

dabFo,,r,Aur.

(52)

J

The change in the action integral produced by a variation of the external current _/, is given by

5rw,,:

I', @.r)EJ,A,.

[l'orro (r), Fcorr-r(*')] - 63r<-rdr.r)6,(r - r'). : i (6rp<,,rdro

(56)

In using Gr, we must restrict the electricfield variation according to dllyDFlqlry:0, w h i c h i s i d e n t i c a l l ys a t i s f i e do n w r i t i n g 6 F< o t t o :3 < t t 6 Z < n p 1Z' 6 1 6 : - Z o t 1 o . 3 Papers dealing tith the situatior peculiar 1o the electromaenetic 6eld are" legion. Of the older literature, the closesl irl spiiit to our proceduie is {hat of W. Pauli, Handbuch der Physik (Edwards Brbt}ters, Ann Arbor, 1943), Vol. 24.

369 726

SCHWINGER

JULIAN

Th ic viAl/le f ha fnrm

A changein the external current, of the form (53), yields -

f

Qp: I do)2F6s8266.

: 3pOx6M,x-0"3x6Mux, (59)

fr'r 6rF,,(r): if r,"G), |

[ F 1 1 ; 1 aG, r ] : 0 , Gp]: idFlo)1ly, [F1q)11y,

:

f . .," d o l i \ n 1 nD , F G t1 1- 16 mo , 1 1 , . F 1 0J y. 1 0 , J

The alteration produced in the field components follorvs from the lield Eq. (47), and the {orm of (46) given by

(s8)

Thus, 0e) (6Fo a)- 6Mrorro)= - dordlr 116 | Os16M61pl, d l o y D F q r:; 1d71; 4 D F 1 s-y 1d;1y1 ; 6 F 1 e 1 1 r ) ,

the

on

c,rrfare

6 F 1 e ; 1 4 ]d- 1 r ; 6 r u 1 r r o , 6 F& 1e ) f : A( D 6 q o ; 1-r yd 1 r ; 6 m 1 0 y 1 4 . In the retarded description, these discontinuities are the actual changes in the field components on o. On referring to the general formula (23), we obtain 661tn 61s1: ilFrorrzr,G-], - dlrydm : ilF d14dzz1o;1ry G^].
( d x . , ) ! 6 M 1 , ( r , ) q y e _x , 1 i

|

XIF -,(r),1"^,(r')1, and r11is the discontinuousfunction

for which the associatedgenerator is

which yields the iollorving discontinuities in 6I'

I

fdt

(s7)

gM u,:6m,A(rp),

rrnscino

(dr')l6M^^(jr )!.'^^(r') |

Jot

where the latter is equivalent to (56). An alternative derivation employs an infinitesimal change in the external source,distributed on (the negative side of) d, r(o):0r

dpF,r+A,Fr!+dxF,,:0.

Jot

L

then provides the commutation properties

^ G^ :

d,/bjr J I

where, in the retarded description

The expressionof changesinducedby 61"1n;11y,

(r')]: g, [Frnrro(r), l'1,,;1"1 lFcorr.r(r), F utot@')l : i ( d 1 1 ;d11a- 1- 6 o rr . rd < u ) 6(,r - r ' ) .

o ^'o il r, pe- d u0M J,+

na(r- x'):1, :0,

5 0 2* o r fio
We have a similar expressionlor 6xj,(r). On comparing the coefficientsof 6Mp(x') in (59) (our two treatments employing external sources are thus distinguished by surfaceand volume distributionsof 6M ,,, respectively), 1vefind - 0 f 11 a@- r')ilF, Q), F y"@')f - 0,a1 - *' @ )ilj, (r), ry^ (r:')l * 0q1Q - r' )i,lj" (r), P1-(o')l : ( 6 , 1 d r d * -D , " d u d 1 - 6 * 1 d , d * *Du-a,dr)6(r:-*'). (60) The value oI ilF,"(x), F^.(r')], for equal times, is then obtained from the coemcient of the difierentiated delta function of the time coordinate, $'ith the anticipated result. In the approximation that neglects the dynamicai relation between currents and fields at points in timelike relation, the differential Eq. (60) has the solution q1Q - r' )ilF,,(x), Fr. (s')l : (6,r4!d.- d,.d,3r - 6rrd,d** 6u-A,a x)D."t(r:- r-'), tvhereD""1(r-*/) is the familiar retarded solution oI - di'D.",:61r-t';

(61)

Had we employed the advanceddescription,4+ would be replacedby -4 , where 4 (r-r'):0' :1,

1'n1'o' r:o(ro',

and the advancedsolution of (61) rvould appcar. Subtracting these trvo results, we find ilF,,(x), F^"(r')l : ( 6 , r d t s a _6-, ^ a r d r - d , 1 d , d " f6 g * 3 , d 1 ) D ( . r : - . u , ) , in which D(r.-x') provided by

is the homogeneoussolution of (61) D: D*t- D.a".

370 THEORY

OF

141

FIELDS

QUANTIZED

'I'he kinematical relation between the electromagnetic The resulting commutator lield and charged 6elds, on a given o, is most clearly ilA t?r 61 @), F10)1a (irl)] indicated in a special choice of gauge, the so-called :3ruut6"(x- r') - 0*16si D'(rradiation gauge, : (611y1n6"(r;r'))(r), (62) 3o>Ao>:0'

r')

With this choice, the constraint equation for the electric Iield reads d l r v F l o r t r l :- 6 t r t ' A < o t i:< o t I J < o > ,

is also consistentwith the transversenature of '41t1'The remaining commutation relationsare

so that the scalar potential is completely determined by the charge densitY,

We shall use the device of the external current to derive the commutation relations between the electromagnetic field tensor and the displacement generators P,,-J u". According to (49) and (50),

.,1ro (r) :

-' (r') ), f a"' o " (* *' I Uror(*')*J,0,

(") (r')] : 0' : lA 6 Q),,4 1a(r')] [F.,,0, (r), Florcl(")

Jt

rvhere D,(x-

r') : (l / 4r)l(r1*1- x
f6l

P,(qr)-P"(o)-

|

U*ll" 'lAfi"J^7,

Jcz

does not commute with the components Evidently, -410y fo' of chargid fields. In this gauge, then, the dependence J (dx)[ " *.4r(r,0"- x,6,)Jx u,G)- J,"(o,\: | Joz of the ilectric field upon the charged fields is made +A.J,-A,JF], explicit through the decomposition of the electric field inio transverse and longitudinal parts, in which we have indicated only the terms containing Prortrl: - drol'A t*1- 0 614P1 the external current. We consider an infinitesimal : p,o,1py("){F1oy1o(r). change in the latter possessingthe form (53)' In the description, lhe resulting changes of P, and The inference that the transverse fields are the inde- retarf,ed ate o1 on J u, pendent dynamical variables of the electromagnetic held in this gauge is confirmed on examining the generf"t (dx)tz6Mx"a"Fx,, ators Ge and Gr. Indeed, a"P,(a): - | Jo

ff doF oxtf)6A o, daF s0t16Aut: G^: J J and 5o:

f J

6,t. (" ) : -

^-1*ua"- r "0u)Fx, 1a*174tM

[:"'

d o 6 F < o t t t t A -o . | do6F'o"u's'A'r" J

in view of the transversenature of A61,FQ' (62)' We can now derive the commutation properties of these dynamical variables from lAsl,Gal:i6A6,

*6M x"Fr,-6MxrF,xf. When expressedin terms of the generator

"

fF1ol1*r(t),Ge]:o'

lArrt,Gr]:0, fFqoylo("),Grf:i6Fo>ots), restrictions the into d.ccount on taking : 0, dtll6.4rlr : dr6DF(o)(*)(")

*:

I.',,'

(dx)|6M x,Fx,

the following commutatorsare encounlered, ilFx' P,l:a,Fx,, i.LF*, J u"): (* u6,- x,0)F a^J 6,^F,t -dp-F,r*6grF* -6"xFv*

oroducedby the transversenature of thesequantities' The Lagrange multiplier device permits us to deduce Finally, we remark that the extension of (31) to electromagnetic field, in the radiation include'tie that gauge, is (rr)]: 56rtoD"(r-r')*dro'Ig>' i.lA 111{i:r),F1oy1;t"r The divergencelesscharacter of the transverse electric field supplied the information 6 6t3"(r- r'), 61t,'2)r1*1: whence

I
^

P\o\:

f

J

,,',, , \ , t t t ' (F-i8Ao)x

- J A (0)-+(€Sx+xSf)l' <*t*L(i p)+"I(0))'4

371 728

JULIAN

SCHWINGER

and

.'

of the independent fields yiel
P@:

J

dol+l\ 0.114(r), F1*rrrr)-xglrordr*rx].

ln arriving at the expressionfor p1ny,the noncommutivity of lrol with 1 must be taken into consideration, but producesno actual contribution, A variation of each

ar,:

f

a"7arru1111e)0sA1p1-6.111116,1,,1n111110) -6y2[610p1al'

which confirms the consistencyof the translation generator with the various field variation generators.

P o p e r3 0

372

The Connection

Between

Spin and Statisticsl

W. Peur-r Physikaksches Institul, Eidg. Technischen Hochschule, Zilrich, Switzeildnd and Institilte for Adtanced, Staly, Princeton, New Jersey (Received August 19, 1940) In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity ol Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a spaceJike distance are commutable, the necessity ol Einstein-Bose statistics for particles with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classeswhich have a commutable multiplication like *l, - 1' * e. - e with e2: 1.

(e.g. Sr:iSo, the digit 4 among the i, k, "' Sr*: lSo*). the requirements of the relativity QINCE Dirac's spinors u rwith p : 1,''', 4have always u theory and the quantum theory are fundaa Greek index running from 1 to 4, and uoa mental for every theory, it is natural to use as means the complex-conjugate of ao in the ordiunits the vacuum velocity of light c, and Planck's nary sense. constant divided by 2zr which we shall simply Wave functions, insofar as they are ordinary denote by Z. This convention means that all vectors or tensors, are denoted in general with quantities are brought to the dimension of the capital letters, [J;, (J;n.... The symmetry charpower of a length by multiplication with powers acter of these tensors must in general be added of. h and,c. The reciprocal length corresponding explicitly: As classical fields the electromagnetic to the rest mass rn is denoted by r:mc/lt'. and the gravipational fields, as well as fields with As time coordinate we use accordingly the rest mass zero, rtake a special place, and are length of the light path. In specific cases,how- therefore denoted with ,the usual letters pi, ever, we do not wish to give up the u.se of f ;p: -fm, arrd gik: gki, respectively. the imaginary time coordinate. Accordingly, a tensor I;r is so deThe energy-momentum tensor index denoted by small Latin letters r, fined, that the endrgy-density trZ and the morefers to the imaginary time coordinate and mentum density G7,are given in natural units by runs from 1 to 4. A dpecial convention for de- W : - T a r a n d ' G * : - i T * a w i t h k : 1 , 2 , 3 .

51. Uxrrs AND NorATroNs

noting the complex conjugate seems desirable. Whereas for quantities with the index 0 an asterisk signifies the complex-conjugate in the ordinary sense(e.g., for the current vector Si the quantity Sox is the complex conjugate of the charge density Sn), in general U*;*...signifies: the complex-conjugateof Ur*... multiplied with (-1)", where z is the number of occurrencesof

TENSoRS. DsrrNrrrox

$2. InnenucralE

oF SPINs

We shall use only a few general properties of those quantities which transform according to irreducible representations of the Lorettz group.2 The proper Lorentz group is that continuous linear group the transformations of which leave the form 1

I This paper is part of a report which was prepared by the

Lx*':x'-xo'

..tr'Jri5iiil'"'"s5il;;'i;;;;" -since "been 1e3eand in whichslight

imorovements have made. In view of lhe uniavorable times, the Congress did not take place, and publication of the reports has. been postponed for an the indehnite lengtl of time. The -retation befreen the pr.esent discussion of the connection between spin and statistics, and the somewhat less general one of Belinfante, based on the concept of charge invariance, has been cleared up by W. Pauli and F. J. Belinfante, Physica 7, L77 (1940).

k:r : rnvariant

and

in

condition

that

they

addition

to

have

the

that

satisfy

determinant

the *

1

lSee B. L. v. d. Waerden, Die gruppentheoreLische Method.einiler Qaantentheorie (Berlin, 1932).

716

373 SPIN

I Ll

AND

and do not reverse the time. A tensor or spinor which transforms irreducibly under this group can be characterized by two integral positive numbers (2, q). (The corresponding "angular momentum quantum numbers" (j, ft) are then given by P:2j+1, S:2k+1, with integral or half-integral j and, k.)* The quantity U(j, fr) characterized bv U, k) has p.q:(2j+l)(2k+l) independentcomponents.Hence to (0,0) corresponds the scalar, to (+, +) the vector, to (1,0) the self-dual skew-symmetricaltensor, to (1, 1) the symmetrical tensor with vanishing spur, etc. Dirac's spinor zo reduces to two irreducible quantities (],0) and (0, !) each of which consists of two components. lf U(j, &) transforms according to the representation

ui:

(2i+1\ (2k+1)

E

L,"u",

l:1

then U*(k, j) transforms according to the complex-conjugate representation rl*, Thus for k:j, .{*:d. This is true only if the components of UQ,k) and U(k, j) are suitably ordered.For an arbitrary choice of the components, a similarity transformation of ,t and A* would have to be added. In view of.$1 we represent generally with U* the quantity the transformation of which is equivalent to A* if the transformation of U is equivalent to A. The most important operation is the reduction of the product of two quantities U{jt

k').Uz(j,, kz)

which, according to the well-known rule of the composition of angular momenta, decomposeinto several I/(j, &) where, independently of each otherj, & run through the values ' ' ', lj'-j,l j:jrIjr, jtljz-r, k:krIkz, kt*kz-1, . . ., lkr-krl. By limiting the transformations to the subgroup of space rotations alone, the distinction between the two numbersj and I disappearsand U(j,k) behaves under this group just like the product of two irreducible quantities U(j)U(k) which in turn reduces into several irreducible *. In the spinor.calculus this is a spinor with 2j-undotted and 2& dotted indices.

STATISTICS

U(l') each having 2llt

components, with

t:j+k,j+k-r, ..., li-kl. Under the space rotations the t/(l) with integral I transform according to single-valued representation, whereas those with half-integral I transform according to double-valued representations.Thus the unreducedquantities T(j , k) with integral (half-integral) j*& are singlevalued (double-valued). If we now want to deterrnine the spin value of the particles which belong to a given field it seemsat first that these are given by l:j+n. Such a definition would, however. not correspond to the physical facts, for there then exists no relation of the spin value with the number of independent plane waves, whieh are possible in the absenceof inteiaction) for given valuesof the components &r in the phase factor exp i(kx). In order to define the spin .in an appropriate fashion.s we want to consider first the case in which the rest mass m of all the particles is different from zero. In this case we make a transformation to the rest system of the particle, where all the .spacecomponents of. k; are zero, and the wave function dependsonly on the time. In this system we reduce the field components, which according to the field equations do not necessarilyvanish, into parts irreducible against spacerotations.To eachsuchpart, with r: 2s* 1 componentsl belong r different eigenfunctions which under space rotations transform among themselves and which belong to a particle with spin s. If the field equations describe particles with only one spin value there then exists in the rest systerri only one such irreducible group of components. From the Lorentz invariance, it follows, for an arbitrary system of reference,that r or lr eigenfunctions always belong to a given arbitrary h. The number of quantities Uff, fr) which enter the theory is, however, in a general coordinate system more complicated, since these quantities together with the vector &i have to satisfy several conditions. In the case of zero rest mass there is a special degeneracybecause,as has been shown by Fierz, this casepermits a gauge transformation oi the sSee M. Fierz, Helv. Phys. Acta 12, 3 (1939); also L. de Broglie, Comptes rendus 208, 1697 (l%9):209, 265 (193e).

374 W.

PAULI

718

We divide the quantities U into two classes: (1) the "*1 class" with j integral, fr integral; (2) the "-1 class" with j half-integral, F halfintegral. The notation is justified because, according to the indicated rules about the reduction of a product into the irreducible constituents under the Lorentz group, the product of two quantities point of view is concerned because the total of the *1 class or two quantities of the -1 angular momentum of the field cannot be divided class contains only quantities of the *1 class, by up into orbital and spin angular momentum whereas the product of a quantity of the i1 measurements. But it is possible to use the class with a quantity of the - 1 class contains following property for a definition of the spin. -1 class. It is important If we consider, in the q number theory, states only quantities of the conjugate I/* for whichj and & complex that the not present, all then where only one particle is are interchanged belong to the same class as [/. of the square of the the eigenvalues j(j*l) angular momentum are possible. But j begins As can be seen easily from the multiplication rule, tensors with even (odd) number of indices reduce with a certain minimum value s and takes then the values s, s*1, " "a This is only the case only to quantities of the *1 class (-1 class). i s n o t p o s s i b l e The propagation vector ftr we consider as befor m:0. For photons, s:1;i:0 Ionging to the -1 class, since it behaves a{ter for one single photon.s For gravitational quanta s : 2 a n d t h e v a l u e s j : Q a n d j : 1 d o n o t o c c u r . multiplication with other quantities like a quantity of the - 1 class. In an arbitrary system of reference and for We consider now a homogeneous and linear arbitrary rest masses, the quantities U all of equation in the quantities [/ which, however, which transform according to double-valued does not necessarily have to be of the Iirst order. (single-valued) representations with half -integral Assuming a plane wave, we may put ft1 for (integral) j*ft describe only particles with half-i0/3xr Solely on account of the invariance integral (integral) spin. A special investigation is the proper Lorentz group it must be of against decide required only when it is necessary to

second kind.* If the field now describes only one kind of particle with the rest mass zero and a certain spin value, then there are for a given value of kr. only trvo states, which cannot be transformed into each other by a gauge transformation. The definition of spin may, in this case, not be determined so far as the physical

whether the theory describes particles with one single spin value or with several spin values' $3. Pnoor oF THE INoBrtNttB Cnan-lcrnn op firB Cnenco rN Clse oF INrEcnel aNo OF THE ENENCY IN CASE OF SpIN Heln-INrrcnel We consider first a theory which contains only I/ with integral j-lk, i.e., which describes particles with integral spins only. It is not assumed that only particles with one single spin value will be described, but all particles shall have

the typical form

(1) LVU-:LU+. Lku+:DU-, This typicalform shallmeanthat theremay be as many different terms of the same type present, as there are quantities [/+ and U-. Furthermore, among the (J+ may occur the [/+ as well as the ([/+)*, whereas other [/ may satisfy reality conditions [/: U*..Finally we have omitted an eaen number of & factors. These may be present in arbitrary number in the term of the sum on the left- or right-hand side of these equations' It is now evident that these equations remain in-

integral spin. variant under the substitution -Ei-*u*"-transformal - t r a n s f o r m a t i o nion o[ the frrstkind" rveunderU- { ct" U+* U*c-i" wirh an s t a n d -a [J++(J+, ht+-k;., [(t/+)*+(I/+)*]; arbitrarvspaceandtime functiona. By "gauge-transformaU-+-U-, l(U-)*--(U-)*1. tion of ihe secondkind" we understanda transformation

(2)

of the tYPe l.da ,pr-et--r6it a s f o r t h o s e o f t h e e l e c t r o m a g n e t i cp o t e n l i a l s a T h e g e n e r a lp r o o f f o r t h i s h a s b e e n g i v e n b y X [ . F i e r z . Helv. Phvs. Acta r3, 45 (1940). 6 See for instance W. Pauli in the article "Wellenmechanik" in the Handbuch der Phvsik, Yol, 24/2, p. 26O'

Let us consider now tensors I of even rank (scalars, skew-symmetrical or symmetrical tensors of the 2nd rank, etc.), which are composed quadratically or bilinearly of the U's. They are then composed solely of quantities with even j

375 719

SPIN

AND

STATISTICS

and even ft and thus are of the typical form

T-LU+U++LU-U-a2Lr+ku-,

(3)

where again a possible even number of & factors is omitted and no distinction between U and U* is made. Under the substitution (2) they remain unchanged, T+I. The situation is different for tensors of odd rank S (vectors, etc.) which consist of quantities with half-integral j and half-integral b. These are of the typical form

s_uu+ku++Lu_ku_+LU_ (4) and hence change the sign under the substitution (2), S+-S. Particularly is this the case for the current vector sd.To the transformation k;+-h; belongs for arbitrary wave packets the transformation xi--xi and it is remarkable that from the invariance of Eq. (1) against the proper Lorentz grolp alone there follows an invariance property for the change of sign of all the coordinates. In particular, the indefinite character of the current density and the total charge for even spin follows, since to every solution of the field equations belongs another solution for which the components of s6 change their sign. The definition of a definite particle density for even spin which transforms like the 4-corfponent of a vector is therefore impossible. We now proceed to a discussion of the somewhat less simple case of half-integral spins. Here we divide the quantities [/, which have half-integral j-tk, in the following fashion: (3) the "*e class" with j integral A half-integral, (4) the " - e class" withj half-integral A integral. The multiplication of the classes (1), . ' ., (4), follows from the rule e2:1 and the commutability of the multiplication. This law remains unchanged if e is replaced by -.. We can summarize the multiplication law between the different classes in the followins multiplication table:

We notice that these classeshave the multiplication law of Klein's "four-group." It is important that here the complex-conjugate quantities for whichj and & are interchanged do not belong to the same class, so that U+', (U-e)* belong to the f e class -e class. U-,, 1U+"1x We shall therefore cite the complex-conjugate quantities explicitly. (One could even choose the [/+' suitably so that oll quantities of the - e class are of the form (U+")*.) Instead of (1) we obtain now as typical form

.)*: I u-.+t( u*r* Lhu+,+Lh(u

Lku,+Lh(u1,)* :lu+,ll1t1-'7*,

(r.)

since a factor k or -i6/0x always changes the expression from one of the classes *e or -e into the other. As above, an even number of & factors have been omitted. Now we consider instead of (2) the substitution ht--k;; u1c+iu+e. (u+.)*+-i(ll+c)*.

(u-,)*+i(u,)*. .. u,--iLI

,t\ \u/

This is in accord with the algebraic requirement of the passing over to the complex conjugate, as well as with the requirement that quantities of the same class as U+,, (tl-e)* transform in the same way. Furthermore, it does not interfere with possible reality condftions of the type U+.:(U-,)* or U-,: ([/+.)*. Equations (5) remain unchanged under the substitution (6). \\'e consider again tensors of even rank (scalars, tensors of 2nd rank, etc.), which are composed bilinearly or quadratically of the I/ and their complex-conjugate. For reasons similar to the above they must be of the form

T -LU+, U+,+LU-, U-"+DU+,kU-,+LU+.( U-r*+ t U,(U+")x a2(U-,)*ku-,

+L(U+,)*ku+,*l(u-,sxp1u+r*+ u-,)*+>(u+r*(u*r*. a) I ({/-.)*(

Furthermore, the tensors of odd rank (vectors, etc.) must be of the form

x1 2 u-,(g-,sx s -, u+,ku+"+ L u-,ku-,+ L u+"u-,+ L u+,k(u-,)*+ L u-,k(u+,1 + >.u+,(u+,)tt + L (u-,)*k(u-,)** L tu*,1*,(u+,)*+ r, (u-, *( u+.)*. (s)

376 W.

PAUL I

720

The result oJ the substihr.tion (6) is now the op|osite oJ the result of the substitution (2): the tensors of even rank change thei,r sign., the tensors oJ od.drank remain unchanged:

is an indication that a satisfactory interpretation of the theory within the limits of the one-body problem is not possible.* In fact, all relativistically invariant theories lead to particles, which in external fields can be emitted and absorbed in T+-T:; (9) S+fS. pairs of opposite charge for electrical particles In case of half-integral spin, therefore, a and singly for neutral particles. The fields must, positive definite energy density, as well as a therefore, undergo a second quantization. For positive definite total energy, is impossible. The this we do not wish to apply here the canonical latter follows from the fact, that, under the above formalism, in which time is unnecessarilysharply substitution, the energy density in every space- distinguished from space, dnd which is only time point changes its sign as a result of which suitable if there are no supplementary conditions between the canonical variables.? Instead, we the total energy changesalso its sign. It may be emphasized that it was not only shall apply here a generalization of this method unnecessary to assume that the wave equation which was applied for the first time by Jordan is of the first order,* but also that the question is and Pauli to the electromagnetic field.8 This left open whether the theory is also invariant method is especially convenient in the absence with respectto spacereflections (x/ : - x, s6l : aq). of interaction, where all fields (/(") satisfy the This scheme covers therefore also Dirac's two wave equation of the second order component wave equations (with rest masszero). nui, - K2u(,r:0, These considerations do not Drove that for integral spins there always exists a definite where energy density and for half-integral spins a lAzA2 definite charge density. In fact, it has been shown _:a__ n : t by Fierz6 that this is not the case for spin )1 b:r atrk2 6xo2 for the densities. Th6re exists, however (in the c number theory), a definite total charge for half- and r is the rest -u., .lf th" particles in rrnitshfc. integral spins and a definite total energy for the An important tool for the secondquantization integral spins. The spin value ] is discriminated is the invariant D function, which satisfies the through the possibility of a definite charge wave equation (9) and is given in a periodicity density, and the spin values 0 and 1 are dis- volume I/ of the eigenfunctions by criminated through the possibility of defining a definite energy density. Nevertheless,the present 1 sin &orn D(x, xJ:--l exp [i(kx)] -----_-i (10) theory permits arbitrary values of the spin vho quantum nrrmbersof elementaryparticlesas well as arbitrary valuesof the rest mass, the electric or in the limit 7+charge, and the magnetic moments of the particles. I r sin A^r^ D(x, xo): _ * | d3&exp [i(br)]- : (11) (2r)BJ rnB oF Frer.os rN TnB Aeho $4. QuewrrzarroN sENcEoF INrBnecrroNs. CoNNpcrroN * The author therefore considers BBrwBsN SprN anp Starrsrrcs as not conclusive the The impossibility of defining in a physically satisfactory way the particle density in the case of integral spin and the energy density in the caseof half-integralspinsin the c-numbertheory * But we exclude operations like (b2+K2)r, which operate at finite distances in the coordinate space, 6 M. Fierz, Helv. Phys. Acta 12,3 (1939).

original argument of.Dirjc, according to which the fleld equatlon must be ol the lirst order. 7 On account of the existence of such conditions the canonical formalism is not applicable for spin )1 and tnerelore tne dtscusslon about the connection between spin and statistics by J. S. de Wet, Phvs. R.". Si. O+6 (19,1O),which is based on that formalism is not general enough. 8 The consistent development of this method leads to the "many-time formalism" of Dirac, which has been given.by..P. A..M..Dirac, Quantun Mechanics (Oxford, second edition. 1935)-

377

72r

SPIN

AND

STATISTICS

Ry &6 we understand the positive root hs:l(k2lx2)\.

In

(r2)

D(x,0):9;

it

follows

114 D{x, xn):- - -FJ.r, xi 4rr 0r

The D function is uniquely determined by the conditions: llD-rczD:0;

general

f o r x o > .r lNoli(t{or-rr)+) F J r, x o): 1 - iH o\I)li K(rz- x o2 )',f f o r r l x o > . - r for -rlxo. llroJ"l*oL r=;+;

(#)^=":'o,(13)

(18)

Here -ly'ostands for Neumann's function and For r:0 we have simply .I1o(Dfor the first Hankel cylinder function. The strongest singularity of D, on the surface of the D ( x , * o ): { 6 ( r - r ) - 6 ( r * x o ) l / 4 r r . ( 1 4 ) light cone is in general determined by (17). We shall, however, expressively postulate in This expressionalso determines the singularity the following that al,l physical, quantities ot f,nite of.D(x, 16) on the light cone for x{0. But in d,istances exterior to the light cone (for l*o'-xo" l the latter caseD is no longer different from zero l lx' -x" l) are commutaDle.* It follows from this in the inner part of the cone. One finds for this that the bracket expressions of all quantities regione which satisfy the force-free wave equation (9) can be expressed by the function D and (a finite 1A -F(r, xo) D(x, xo) number) of derivatives of it rvithout using the 4nr 3r function Dr. This is also true for brackets with with the f sign, since otherwise it would follow that gauge invariant quantities, which are constructed -rr)+) for xolr lJol|(xo, I bilinearly from the UG>, as for example the F(r,ro):l 0 f - o r ) x o > -. r i ( t S ) charge density, are noncommutable in two points l- Jolr(*or-rr1+1 l o r - r > x & l with a space-like distance.lo The jump from * to - of the function F on The justification for our postulate lies in the the light cone corresponds to the d singularity of fact that measurements at two space points rvith D on this cone. For the following it will be of a space-like distance can never disturb each decisive importance that D vanish in the exterior other, since no signals can be transmitted \,vith -r). of the cone (i.e., for rlxo) velocities greater than that of light. Theories The form of the factor d7k/kois determined by which would make use of the Dr function in the fact that dak/ko is invariant on the hypertheir quantization would be very much different boloid (i) of the four-dimensional motrrentum from the known theories in their consequences. space (k, fr). It is for this reason that, apart At once we are able to draw further conclusions from D, there exists just one more function about the number of derivatives of .D function which is invariant and which satisfies the wave which can occur in the bracket expressior.rs,if we equation (9), namely, take into account the invariance of the theories under the transformations of the restricted 7 cos froro f Lorentz group and if we use the results of the Dtk. xo): _ (t6) | d 3 f te x -p [ ; ( k x ) ] - . preceding section on the class division of the (2r)3 J ko tensors. We assume the quantities [/(o to be For r:0 one finds ordered in such a way that each lield component is composed only of quantities of the same class. 11 Dr(x, ro) (17) * For the canonical quantization formalism this postulate 2r2 f2- rs2 s S e e P , A. N{, Dirac. Proc. Camb. Phil Soc. 30, 150 (1934).

is satisfied implicitly. But this postulate is much more general than the canonical {ormalism. r 0 S e eW . P a u l i , A n n . d e l ' I n s t . H . P o i n c a r 66 , 1 3 7 ( 1 9 3 6 ) , esp. $3.

378

w.

PAULI

We consider especially the bracket expression of a field component [/(") with its own comptex conjugate f U r ' > ( x ' , x n ' ), U * G )( x t ' , x n " ) f . We distinguish no'rv the two cases of half-integral and integral spin. In the former case this expression transforms according to (8) under Lorentz transformations as a tensor of odd rank. In the second case, however, it transforms as a tensor of even rank. Hence we have for halfintegral spin IUG) (x' , xst), U*t') (xtt , xt")l :odd number of derivatives of the function (19a) D(x' *x" , xo'- ro") and similarly for integral spin fUr't(xt, xst), U*t)(xtt, xstt)l :even number of derivatives of the function (19b) D(x'-x", xot -rcott). This must be understood in such a way that on the right-hand side there may occur a complicated sum of expressions of the type indicated. We consider now the following expression, which is symmetrical in the two points t, t)l X : l U t >( x ' , x n t ) , U * { , t ( x t x s t

-flUo)(x", xn"), U*{')(x',xn')f. (20)

Since the D function is even in the space coordinates odd in the time coordinate, which can be seen at once from Eqs. (11) or (15), it follows number from the symmetry of X that X:even of space-like times odd numbers of timelike derivatives of D(x' -x" , xyt - K1t'). This is fully consistent with the postulate (19a) for halfintegral spin, but in contradiction with (19b) for integral spin unless X vanishes. We have therefore the result lor inlegral spin t)f ) l(l(,) (x', xn'), U*t (x", xo' l l U t > ( x t t , x o t t ) ,U * k ( x ' , r i ) ] : 0 .

(21)

So far we have not distinguished between the two cases of Bose statistics and the exclusion principle. In the former case, one has the ordinary bracket with the - sign, in the latter case,

722

according to Jordan and Wigner, the bracket

lA, s1:a313a with the * sign. By insertiag the brackets with the ! sign into (20) tae haztean algebraic contrad'iction, since the left-hand side is essentially positive for x':x" and cannot vanish unless both Uc) and, I/x(') vanish.* Hence we come to the result: For integral' spin the quantization according to lhe excl'usion princ'iple is not possible. For lhis result,it'is essential, thatr the use oJ the DlJunction in place of the D Junct'ion be, for general reasons, d'iscarded. On the other hand, it is formally possible to quantize the theory for half-integral spins according to Einstein-Bose-statistics, but according to the general result of the preceding section the energy of the system would not be posiLiae. Since for physical reasons it is necessaiy to postulate this, we must apply the exclusion principle in connection with Dirac's hole theory. For the positive proof that a theory with a positive total energy is possible by quantization according lo Bose-statistics(exclusionprinciple) for integral (half-integral) spins, we must refer to the already mentioned paper by Fierz. In another paper by Fierz and Paulill the case of an external electromagnetic field and also the connection between the special case of spin 2 and the gravitational theory of Einstein has been discussed. In conclusion we wish to state, that according to our opinion the connection between spin and statistics is one of the most important applications of the special relativity

theory.

* This contradiction may be seen also by resolving I/t": into eigenvibrations according to u*c) (x, f,0): v-, 2 r I u +* (k) exp [t J - (kT) +[or,o | ]. * U-(F) exP Li { (kx) - }oro I J I L I c ) ( x , x 0 ): V - t > k l'U + ( h ) e x p [ i l ( k x ) - ] o x o ] l

417-*(llexo[;{ -(kxt+froro}]}.

The equation (21) leads then, among others, to the relation

I I/**([), U+(k)f+ lu -(b), u-t(fr) ] = 0, a relation, which is not possible for brackets with the * sign unless U+(k) and, U+*(E) vanish. lM. Fierz and W. Pauli, Proc. Roy. Sc. Ar73, 211 (1939).

-452

PHYSICS: J. SCHWINGER

PROC. N. A. S.

ON THE GREEN'S FUNCTIONS OF QUANTIZED FIELDS. I By JULIAN SCHWINGER HARVARD UNIVERSITY

Communicated May 22, 1951

The temporal development of quantized fields, in its particle aspect, is described by propagation functions, or Green's functions. The construction of these functions for coupled fields is usually considered from the viewpoint of perturbation theory. Although the latter may be resorted to for detailed calculations, it is desirable to avoid founding the formal theory of the Green's functions on the restricted basis provided by the assumption of expandability in powers of coupling constants. These notes are a preliminary -account of a general theory of Green's functions, in which the defining property is taken to be the representation of the fields of prescribed sources. We employ a quantum dynamical principle for fields which has been described elsewhere.1 This principle is a differential characterization of the function that produces a transformation from eigenvalues of a complete set of commuting operators on one space-like surface to eigenvalues of another set on a different surface,2

(rl', (T1jr2, 02)

i(r1' 711 afUl (dX) -CI 2,p 0'2)

(1) Here £ is the Lagrange function operator of the system. For the example of coupled Dirac and Maxwell fields, with external sources for each field, the Lagrange function may be taken as £ = -..1/4[P, 'Y;(-ip - eA>)P + m+/] + 1/2[4, 'i] + Herm. conj. + 1/4F,P2 - 1/4{Fp,, )A, - 6A} + J,A,X, (2) which implies the equations of motion 'Y;&(-ib - eA,u)# + mi = 71. = J, +ji,, F,, = 6g.A4 - ,A,;, (3) =

where

j;&

e'/2[l, y4]. (4) With regard to commutation relations, we need only note the anticommutativity of the source spinors with the Dirac field components. We shall restrict our attention to changes in the transformation function that arise from variations of the external sources. In terms of the notation =

(r1', 'l Ir2 , '2)

(

= exp iW,

al'I!F(x) |2, '2)/(r1', '711

2,

'2)

=

(F(x)),

(5)

PHYSICS: J. SCHWINGER

VOL. 37, 1951

453

the dynamical principle can then be written bW = j;`°

(dx)(bc(x)),

(6)

where = (k(X))5ii(X) + "(x)(#(x)) + (A,(x))6Jp(x). The effect of a second, independent variation is described by

l(b2(x))

=

i .J '

(7)

(dx') [((5 e(x)6' e(x')) +) - (5(x))(5'.e(x'))], (8)

in which the notation ( )+ indicates temporal ordering of the operators. As examples we have

6v(+(*

))=

02j, (dx') [((#(x)7;(x')5n(x')) +)-(t(x))(;(x')5v(x'))], (9)

and

bj(o(x)) = i J:" (dx')[(4,(x)A,(x'))+)

-

(4/(x))(A;(x'))]5J,(x'). (10)

The latter result can be expressed in the notation

although one may supplement the right side with an arbitrary gradient. This consequence of the charge conservation condition, 6AJ;, = 0, corresponds to the gauge invariance of the theory. A Green's function for the Dirac field, in the absence of an actual spinor souree, is defined by = (dx') G(x, x')56(x'). J,0

(12)

According to (9), and the anticommutativity of 65(x') with 4'(x), we have G(x, x') = i((4,/(x)i(x'))+)E(x, x'), (13) . On combining the differential where E(x, x') = (xo - xo')/ xo equation for (y6(x)) with (11), we obtain the functional differential equation -e(A,.(x)) + ieb/,Js(x)) + m]G(x, x') = 6(x - x'). (14) An accompanying equation for (A,(x)) is obtained by noting that (15) (j,(x)) = ie tr 'y,sG(x, x')x' x, in which the trace refers to the spinor indices, and an average is to be taken of the forms obtained with xo' -- xo h 0. Thus, with the special choice of gauge, b6(Av(x)) = 0, we have -62(A (x)) = J,(x) + ie tr y;,G(x, x). (16) The simultaneous equations (14) and (16) provide a rigorous description of G(x, x') and (A,(x)).

xo'l

PHYSICS: J. SCHWINGER

454

PROC. N. A. S.

A Maxwell field Green's function is defined by &Pv(x, x') = (8/bJ(x'))(Ap(x)) = (515J=(x))(A(x'))

i[((A,(x)A (x'))+)

-

(A;(x))(A ,(x'))]. (17)

The differential equations obtained from (16) and the gauge condition are + ie tr 'y(6/5J,(x'))G(x, x), -b)2S;,(x, x') = (x-x) bA9;v(x, x') = 0 (= 6'x). (18) More complicated Green's functions can be discussed in an analogous manner. The Dirac field Green's function defined by

5,72((jt(XI) t(X2)) +) e (XI X2),v

= 0

=

J91((dxl) ,20/" (dx2')G(xl, X2; XI', X21)5V(XI')5j((X2%) (19) may be called a "two-particle" Green's function, as distinguished from the "one-particle" G(x, x'). It is given explicitly by

G(xi, x2; xI', x2') = ((4(x1)4#(x2){(x1')l(x2'))+) E, e(xI, X2)E(XI', X2')E(Xl, xI')E(xI, x2')e(x2, Xi')e(X2, x2') (20) This function is antisymmetrical with respect to the interchange of xi and X2, and of xi' and x2' (including the suppressed spinor indices). It obeys the differential equation W G(x1, x2; xl', X2') = 6(x - xi')G(x2, X2') - (xl -x2')G(x2, x1'), (21) where 0 is the functional differential operator of (14). More symmetrically written, this equation reads e

=

i 1a2G(xi, x2; xi', x2')

=

5(xi - xl')(x2 - x2')-

6(xi - x2')6(x2- xi'), (22) in which the two differential operators are commutative. The replacement of the Dirac field by a Kemmer field involves alterations beyond those implied by the change in statistics. Not all components of the Kemmer field are dynamically independent. Thus, if 0 refers to some arbitrary time-like direction, we have m(l - #02)4, = (1 - #02)rt - Pkk(-i2) - eAk) #o2#, k= 1,2,3, (23) which is an equation of constraint expressing (1 - #o2)4, in terms of the independent field components 13o2#, and of the external source. Accordingly, in computing 5,(4,(x)) we must take into account the change induced in (1 -,o2), (x), whence G(x, x') = i((4,(x)1(x'))+) + (1/m)(1 - #02)5(x - x'). (24) The temporal ordering is with respect to the arbitrary time-like direction.

PHYSICS: J. SCHWINGER

VOL. 37, 1951

455

The Green's function is independent of this direction, however, and satisfies equations which are of the same form as (14) and (16), save for a sign change in the last term of the latter equation which arises from the different statistics associated with the integral spin field. 1 Schwinger, J., Phys. Rev., June 15, 1951 issue. 2 We employ units in which h = c = 1.

ON THE GREEN'S FUNCTIONS OF QUANTIZED FIELDS. II By JULIAN SCHWINGER HARVARD UNIVERSITY

Communicated May 22, 1951

In all of the work of the preceding note there has been no explicit reference to the particular states on 01 and 01 that enter in the definitions of the Green's functions. This information must be contained in boundary conditions that supplement the differential equations. We shall determine these boundary conditions for the Green's functions associated with vacuum states on both o1 and a2. The vacuum, as the lowest energy state of the system, can be defined only if, in the neighborhood of a1 and U2, the actual external electromagnetic field is constant in some time-like direction (which need not be the same for a1 and a2). In the Dirac one-

particle Green's function, for example, G(x, x') = i(+i(x);(x')), xo > xo', = -i(4(x') A(x)), xo < xo', (25) the temporal variation of +1(x) in the vicinity of o1 can then be represented by

(26) exp [iPo(xo - Xo)]4I(X) exp [-iPo(xo - Xo)], where Po is the energy operator and X is some fixed point. Therefore, x -- or: G(x, x') = i(4/(X) exp [-i(Po - Povac)(xo - Xo)];(x')), (27)

O6(x)

=

in which Povac is the vacuum energy eigenvalue. Now PO -Povac has no negative eigenvalues, and accordingly G(x, x'), as a function of xo in the vicinity of a,, contains only positive frequencies, which are energy values for states of unit positive charge. The statement is true of every timelike direction, if the external field vanishes in this neighborhood. A representation similar to (26) for the vicinity of 01 yields X --

02: G(x, x')

=

-i( (x') exp [i(Po - PoV")(xo - Xo)]1(X)), (28)

456

456 PHYSICS: J. SCHWINGER

PROC. N. A. S.

which contains only negative frequencies. In absolute value, these are the energies of unit negative charge states. We thus encounter Green's functions that obey the temporal analog of the boundary condition characteristic of a source radiating into space.' In keeping with this analogy, such Green's functions can be derived from a retarded proper time Green's function by a Fourier decomposition with respect to the mass. The boundary condition that characterizes the Green's functions associated with vacuum states on a, and a2 involves these -surfaces only to the extent that they must be in the region of outgoing waves. Accordingly, the domain of these functions may conveniently be taken as the entire four-dimensional space. Thus, if the Green's function G+(x, x'), defined by (14), (16), and the outgoing wave boundary condition, is represented by the integro-differential equation, yA(-ib - eA+,u(x))G+(x, x') + (29) f(dx')M(x, x")G+(x', x') = 6(x - x'), the integration is to be extended over all space-time. This equation can be more compactly written as ['y(p - eA +) + M]G+ = 1, (30) by regarding the space-time coordinates as matrix indices. The mass operator M is then symbolically defined by MG+ = mG+ + iery(S/8J)G+. (31) In these formulae, A + and 8/1J are considered to be diagonal matrices, (xl A +,, x') = 6(x - x')A4+4(x). (32) There is some advantage, however, in introducing "photon coordinates" explicitly (while continuing to employ matrix notation for the "particle coordinates"). Thus

-jA + J(d{),yQ)A +(t),

(33)

where -y() is defined by

(x,Yr(;)jx')

=

yp5(x -

)(x - x').

(34)

The differential equation for A +(t) can then be written -

t2A +(t)

=

J(Q)

+ ie Tr [y(t)G+],

(35)

where Tr denotes diagonal summation with respect to spinor indices and particle coordinates. The associated photon Green's function differential equation is

-at2q+(t, {')

=

(- ') + ie Tr [-y()(5/6J(t'))G+]-

(36)

PHYSICS: J. SCHWINGER

VOL. 37, 1951

457

To express the variational derivatives that occur in (31) and (36) we introduce an auxiliary quantity defined by

r(a)

=

-(61beA +Q))G+-I - (5/beA+(Q))M.

(37)

ef (d{')G+F(t')G+S+(t' t),

(38)

a()

=

Thus

(6/5J(Q))G+

=

from which we obtain

M= m+

ielf(d))(dt'),yQ)G+rw)9+Q1, 0,

(39)

and

-aZ29+Q, {')

+

f(d t)P( , `)

+(t` ') = 6( P(t, 0') = -je2 Tr ['(y)G+r(Q')G+] (40)

With the introduction of matrix notation for the photon coordinates, this Green's function equation becomes

(k2 + P)9+

= 1,

[,, k^] =

(41)

i

and the polarization operator P is given by P = -ie2 Tr [yG+rG+].

(42)

In this notation, the mass operator expression reads

M = m + ie2 Tp [yG+rS+],

(43)

where Tp denotes diagonal summation with respect to the photon coordinates, including the vector indices. The two-particle Green's function

G+(xi, x2; xl', x2')

=

(xi, x2| G121 xl', x2'),

(44)

can be represented by the integro-differential equation

[(Ylr + M)1(77r + M)2

= 112, (45) p - eA +, thereby introducing the interaction operator 112. The unit operator 112 is defined by the matrix representation 7r =

(X1, X21 1121 XI 1, X2) = (X1 - X1')6(X2

-

X2')

-

6(xl - X2')5(X2

-

x'). (46)

On comparison with (21) we find that the interaction operator can be characterized symbolically by

458

PROC. N. A. S.

PHYSICS: J. SCHWINGER

12G12 = -ie2 Tp[LYP2S+]GG12 -ie2 Tp[y1G,6/6J]1(112G12) - -ie2 Tp[Y2riFi+]G12 - ie2 Tp[y2G2i5/J] (I12G12), (47) where G1 and G2 are the one-particle Green's functions of the indicated particle coordinates. The various operators that enter in the Green's function equations, the mass operator M, the polarization operator P, the interaction operator 112, can be constructed by successive approximation. Thus, in the first approximation, M(x, x') = mb(x - x') + ie2ey,GG+(x, x'),y,DD+(x, x'), PMV(R, i') = -ie2 tr[-yMG+(Q, i')-yvG+(t', c)], I(xb, x2; X1', X2') = -ie2y,y2,.D+(xl, X2) (X1, x21 1121 Xl', X2'), (48) where

9;AV Q, 0'

=

6,D +(,i)

(49)

and the Green's functions that appear in these formulae refer to the 0th approximation (M = m, P = 0). We also have, in the first approximation,

FJ(t; x, x') = 'yJA(t - x)6(x - x') -ie2y,,G+(x, t),yG+Q, x')'ypD+(x, x')

(50)

Perturbation theory, as applied in this manner, must not be confused with the expansion of the Green's functions in powers of the charge. The latter procedure is restricted to the treatment of scattering problems. The solutions of the homogeneous Green's function equations constitute the wave functions that describe the various states of the system. Thus, we have the one-particle wave equation

(51)

(,rr + M)# = 0, and the two particle wave equation

[Q(y7r + M)y(77r + M)2 - 112h'12

=

0,

(52)

which are applicable equally to the discussion of scattering and to the properties of bound states. In particular, the total energy and momentum eigenfunctions of two particles in isolated interaction are obtained as the solutions of (52) which are eigenfunctions for a common displacement of the two space-time coordinates. It is necessary to recognize, however, that the mass operator, for example, can be largely represented in its effect by an alteration in the mass constant and by a scale change of the Green's function. Similarly, the major effect of the polarization operator is to multiply the photon Green's function by a factor, which everywhere appears associated with the charge. It is only after these renormaliza-

VOL. 37, 1951

ZOOLOG Y: ENGSTROM A ND R UCH

459

tions have been performed that we deal with wave equations that involve the empirical mass and charge, and are thus of immediate physical applicability. The details of this theory will be published elsewhere, in a series of articles entitled "The Theory of Quantized Fields." 1 Green's functions of this variety have been discussed by Stueckelberg, E. C. G., Helv. Phys. Acta, 19, 242 (1946), and by Feynman, R. P., Phys. Rev., 76, 749 (1949).

DISTRIBUTION OF MASS IN SALIVARY GLAND CHROMOSOMES By A. ENGSTROM* AND F. RuCHt DEPARTMENT FOR CELL RESEARCH, KAROLINSKA INSTITUTET STOCKHOLM

Communicated by C. W. Metz, May 15, 1951

The measurement of the absorption of soft x-rays, 8 to 12 A in wavelength, in biological structures makes it possible to determine the total mass (dry weight) per unit area of cytologically defined areas in a biological sample. Knowing the thickness of the sample or structure being analyzed the percentage of dry substance can be estimated. For theoretical and technical details see Engstrom' 1950. Dry substance is an accurate basis upon which to express the results obtained with other cytochemical techniques, e.g., the amount of specifically absorbing, ultra-violet or visible, substances. The present investigation is an attempt to determine the- dry weight (mass) of the different bands in the giant chromosomes from the cells in the larval salivary glands of the fly Chironomus. The structures to be observed, however, are just on the border of the resolving power of the x-ray technique for the determination of mass. The results reported, therefore, must be interpreted with care. The specimen intended for x-ray investigation is mounted on a collodion film circa 0.5 micron thick. This film supports the object in the sample holder, a brass disk with a slit about 6 mm. long and 0.5 mm. wide. In the first experiments salivary glands from Chironomus were isolated on a microscope slide and the chromosomes transferred to the thin carrier membrane on the sample holder. When examining the x-ray picture of these chromosomes no details at all could be seen due to shrinkage effects when the chromosomes were dried. For the x-ray determination of mass the specimens must be dried before they are introduced to the high vacuum of the x-ray tube. The water must also be taken away for another reason: The high absorption of soft

387

P o p e r3 2

Electrodynamic Displacement of Atomic Energy Levels. IIL The Hyperfine Structure of Positronium RoBERT KARPf,us etp Aerlsru Krr:rx H truud Unitersily, Caubrid,ge, .ll ussltchilset!s (Received NIay 13, 1952) integro-difierential equation for the electronA functional nositron Green's function is derived from a consideration of the effect of sources of the Dirac field. This equation contains aD electron-positron interaction operator from which functional deprocedure. The by an iteration rivatives may be eliminated ooerator is evaluated so as to include the efiects of one and t*'o virtual quanta, It contains an interaction resulting from quantum exchange as well as one resulting lrom virtual annihilation of the pair. The wave functions of the electron-positron system are the solutions of the homogeneous equation related to the Green's function equation. The eigenvalues of the total energy of the

I. INTRODUCTION

system may be found b). a four-dimensional perturbatiol teclL nique. The s1'stem bound bv the Coulomb interactjon is here treated as the unperturbed situation. Numerical values for tbe spin-dependent change of the energy from the Coulomb value in the ground state are finally obtained accurate to order a relative to the hyperfine structure d2 Rr'. The result for the singlet-triplet energy difference is LW n:

la2 Ry-17 /3-

(32/9 12 ln2)a/ ol:

2.0337X 105 Nfc/scc.

Theory and experiment are in agreement.

investigation to be describedin this paper I was suggestedby the current theoreticalinterest in the quantum-mechanical two-body probleml-3 and the recent accurate measurement of the ground state hyperfine structure of positronium.a b The system compbsed o{ one electron and one positron in interaction is the simplest accessible to calculation because it is purely electrodynamic in nature. Moreover, the success of quantum electrodynamics in predicting with great accuracy the properties of a singleparticle in an external field indicates the absence of fundamental difficulties from the theory in the range of energies that are significant in positronium. The discussionof the bound states'of the electrolpositron system is based upon a rigorous functional difierential equation {or the Green's function of that system, derived in Sec. II by the method described by Schwinger.I In order to obtain a useful approximate form of this equation (and of the associatedhomogeneous equation) we have iterated the implicitly defined interaction operator) in this way automatically generating to any required order the interaction kernel obtained from scattering considerations by Bethe and Salpeter.3 In the present case we have included all interaction terms involving the emission and absorption of one or two quanta. The latter include self-energyand vacuum polarization corrections to one-photon exchange processes as rvell as trvo-photon exchange terms. The particle-antiparticle relationship of electron and positron is represented by terms describing one- and twophoton virtual annihilation of the pair.FE In contrast

to the caseof scattering,only the irreduciblesinteractions appear explicitly. Our subsequentconcern is llith'the solution of the associatedhomogeneous equation.It should be enphasized at the outset that we shall be silent (out of ignorance)on the questionof the fundamentalinterpretation of a rvave function rvhich refers to individual timesfor eachof the particles.The possibility,nevertheless, of obtaining a solution to our problem entirely s'ithin the framework of the present formalism dependson two conditions.The first of theseis that most of the binding is accountedfor by the instantaneous Coulombinteraction.Salpetere has shownthat when the interaction is instantaneous,the rvaYeequation can be rigorouslyreducedto one involving only equal times for the trvo particles. trforeover, the rvave function for arbitrary individual time coordinates can be expresseci in terms of that for equal times. This last circumstance can alsobe exploitedin the der-elopmentof a perturbation theory x'hich yields the contribution to the energ)' levels of a small non-instantaneousinteraction.eThe relevant resultsof this treatment are siven in Sec.IIL The secondcondit ion is thaI I he freJpa rt icleapproximation for all intermediatestatesshall be an adequate one. Ihe essentialpoint here is that \yhether one derives an expiicit interaction operator by the iteration procedureadoptedin the presentpaper (tantamount to an expansion of the intrinsic nonlinearity in terms of free particle properties)or by a partial summationof a scattering kernel, the propagation rvhicl.r naturally enters in intermediate states is that of free particles. In the treatment of fine-structure effects, the contribu-

I L Schwinger,Proc.Nat. Acad. Sci. US 37,452,455 (1951). ,-lr. cell-Mann and F. Low. Phys. Rev.84,350 (1950. 3 H. A. Betheand E. E. Salpeter,Phys-Rev. 84,1232(1951). {M.DeutschandS.C.Brou'n,Phys.Rev.85,1047(1952). 6 M. Deutsch, latest result reported at the Washington Me€ting oI the American Physical Society, May, 1952. ?hys. Rev. 87, 212(T\ (1952). u j. lir"nn6, Arch. sci. phys. et nat. 28, 233 (1946);29, 121, 2O7,and 265 (1947).

? V. B. Berestetskiand L. D. Landau, Exptl. Theoret.Phys. J. (U.S.S.R.) (1949).Seealso V. d. Be;estetski,J. blxitl. 'fheoret. 19, 673 Phys. (U.S.S.R.)19, 1i30 (1949). 3 R . A . F e r r e l l ,P h y s , R e v . 8 4 , 8 5 8 ( 1 9 5 1 )a n d P h . D . t i r e s i s (Princeton, 1951). Dr. Ferrell kindly sent us a copy of his thesis. 'gE. E. Salpeter, Phys. Rev. 87, 328 (1952). We are indebted to Dr. Salpeter for making available to us a copy of his paper prior to publication. \\re hive found his ideas very helpful i'n iut work.

'fHE

848

388 I C E L E C TRO D Y NA N'1

849

D I SP LA C E M E N T

tion of nonreiativistic intermediate states, lvhere the and satisfy the difierential equations Coulomb binding cannot be ignored, must then be eA1,@)*ieal6J rQ))*ml obtainedin a mannerreminiscentof the first treatments ly r(-i0 uof the Lamb shift.'g This rvill not be necessary in the XG-(x' r'): a1a- ,"'7 (2'aa) present paper since rve shall be concernedwith the and hyperfine (spin-spin)type of interaction to tvhich only i 0,1 eA+ r@) - ie6/ 6l r@))I mf relativistic intermediate states contribute to the re- ly,(quired precision.ro y6+ (t, xt): 6(x- n'), (,2.4b) The practical goal of this v'ork is to obtain the splitting of the singlet-tripletground-statedoublet of posi- with the outgoing wave boundary condition. They are, tronium correctto order a3 Ry. Previouscalculations,6-8 of course,relatedby the matrlr C: accurate to order a2Ry, have included the lowest (2.5) G " B + @ , x ' ) : - ( " " , C t B B , G p , " ' - ( rt )' ,. order contributions of the ordinary spin-spin coupling arisingfrom the Breitlr interaction (the analogof rvhich We shall now introduce matrlr notation for the in hydrogen is responsible for its hyperfrne structure) combinedparticle coordinatesand spinor indices,and and of the one-photonvirtual annihilation force, char- the combined photon coordinatesand vector indices. acteristicof the systemof particle-antiparticle.The ex- Becausethe formulaswill get quite involved, the matrix pressionfor the energyshift given in Sec.III, Eq. (3.6) indices will be erpressedas arguments, by numbers for yieids these again in lowest approximation and contains l h e p a r t i r l e sa n d b y ! . f ' , " ' f o r t h e p h o l o n sa. n d t h e as well the matrix elementsof all interactions which summation convention n'ill be understood. Functions can contribute to the required accuracy. ol one coordinateare to be diagonalmatrices; quanti SectionIV is devotedto the detailedevaluatiorrof all ties alfixed with only one matrix index are to be vectors the matri-r eiementsthat may be looked upon as general- with respect to that index. The arguments of the Dirac ized Breit interactionsbecausethey dependpurely on matriceswill refer only to the vector and spinor indices the exchangeof photons between the two particles. of these quantities; they will be unit matrices in the In Sec. V we consider the annihilation interaction coordinates. Similarly, functions of the coordinates peculiar to the electron-positronsystem. Fiually, the alone must be understood as multiples of the l)irac comparisonlvith experimentis given in Sec.\iI. unit matrix. As an example, Eqs. (2.4) and (2.5) rvill be tranII. THE WAVE EQUATION scribed rvith the symbols I and S+ standing for the A discussionof the one-particleelectronand positron functional differentialoperatorsin Eq. (2.4): Green'sfunction associatedr.viththe vacuum state will a(r,;); serueas an introduction to this section.If the notation Q.a'a') 3-(rz)G-(23): of reference1 is extendedto include the positron field (2.4'b) ; [+(rz)c+(23):6(13) variablesQ'@), {' (") , and their sourcesthat are related to the electron variablest@),0@), and their sources G+(t2): _C(Lt')C L(22,)G_(2,1,). (2.5') by the usual chargeconjugatingmatrix C, II the mass operator M(12) is defined in the usual way, cic-t: -t. c: -4. c'c:1, () 1t (2.6) M+(lDGrQs):1n+Q2)C;t(2j), ,tr':C,l',,tt':C',1r. n':Crt, r'-C )n. operator where !J? is the functional difierential the Green's functions are defined by the vacuum exDlt(r2): m6(12)+ie7c, 12)6/6t (0 , Q.7) pectation values ^ o\

6l/(r))ol-o:

I

d a x ' G( x , r ' ) 6 4 ( r ' )

,l oz

and fo'

( 2 . 2 a ) then the Green'sfunction equations(2.4) can be rvritten in terms of integro-difierential operators F that are obtained from the 3 by the replacementof \)l.by M. A v e r l e v o n e r a t o rf { } - l 2 ) m u s t n o w b e d e f i n e df o r

60,(f'(r))01,,:o:1 dax'G'(x, x')bl(:'), (2.2b) J62

ecch

(lrcpn'c

frrnefinn

r+ (t,t2) : (6/ 6eA {))

where d4 and 0q/are arbitrary variations of the electron and positron sources,respectively.The Green'sfunctions can be qxpressedin terms of expectation values by and ,

G-(*, r'):i((!(x){,(x'))*)0.(r,

r')

(2.sa)

a.nd

6+(s, t'): i((!' (x){'(r'))*)0,(r, *')

(2.3b)

10R. Karplus and A. K)ein, Phys. Rev. 85,972 (1952). 11G. Breit, Phys. Rev. 34, 553 (1929);36, 383 (1930);39, 616 (1932).

(G+(r2) )-1 : (6/ 6eA+(0)F+(r2)

I-(f , 12) : - $ / 6eA aQ)) (G- (12))-l : - $/aeAa({)F-(r2).

(2.8a)

(2.8b)

In the absenceoI an external field thesetrvo quantities becomeequal becausethen the charge occurs always

389 R.

KARPLUS

to an even power only, and the two difier just in the sign of the charge. We now proceed to the two-particle system. The electron-positron Green's function for the vacuum state is defined by the relation d,6,,(Qt(x),1,,(rz) )*)oI o,,:o.rr,, :,

?ot Joc

A.

KLEIN

850

Finally, the equation may be written in the form

lF- (rt')F+(22')- r (12,r' z')fc--+(r'2" 34) :6(13)6(24), (2.16) r.here the interaction operator 1(1234)is definedby

",.r

I (12, l' 2')G-+(l',z',,34)

fol

anri I

AND

dnh,G-(rfiz,rih,)

: - F+(22')lyn-Qt')-

Jaz

M-(r|'))G--+(L'2" 34)

XEq(x1')6n'(x2'). (2.9)

-L + ie^/(8, | 3/) c (s'2)c (44,) (6/ 6J (0)G (4,s),

Evaluation of the variations with the help of Eq. (9), reference1, leadsto the explicit er?ression

: - F-(1 1')[IJt+ (zi',) - M+ (22')]G--+(r' 2" 34)

G+(rp2, *1'12')

- iey 22')C(2'1)C-'(33')(6/6"r( Q, t))G+(3'4). (2.17)

The secondexpressionarisesrvhen $+ and then F- are appliedto the Green'sfunction. Theseerpressionsmust norv be rearrangedso as to yield the interaction operator e x p l i c i t l y a s a n i n t e g r a l o p e r a r o ru p t o t h e d e s i r e d x2'). (2.10) order of accuracy.In other rvords,the functionalderivax((0@)0' @z'))a)ne(x1', As might be expected,this Green'sfunction is related tives may occuronly in terms that contributenegligibly to a charge conjugate of the two-electron Green's to the effect that is being investigated.The subsequent function with arguments interchanged properly, by operationss'ill be directedat finding an expressionthat is suitable for the purposesof this paper. (For other E q s . ( 1 3 ,2 0 ) , r e f e r e n c e1 : effects,such as the Lamb shift in positronium,a difierG"Br6+(xg2, lc1'r2') e n t f o r m o f t h e i n l e r a c t i o no p e r a t o ri s n e c e s s a r ) . ) With the help of the definitiono{ the vertex operator, = -L 6F'L '66uob'as' (Jlr2, f,rf ?, Eq. (2.8),the lorvestorder interactionmay be separated - C pp,C-t ay G p,- (ayx2)G5, (x2'x as follows: 1'). (2.11) ; : (@ (",) *, () 0 (x| ) 0, (r z,)) +)oe ") - ((Un,){' (rr))+)oc(rr rz)

"

'I'he

antisvmmetry of the two-electron Green's function itssures that both direct and exchange processes are contained in the electron-positron Green's Iunction; the second term merely corrects for the fact that the uncoupled electron-positron system cannot undergo an exchange process. In this case, G. y, 6,- (r 6 2', t 1'* r) -4,

( 2 . 12 )

wnence G'y5+(rp2, -

34)

: i e'? t (t, 11')9+(f, €')t' +1.v,22')G' + (I' 2', jl) _ _ ItJn- ( r, ) = M (1r' )lF + (22,) G-+ (1'2" 34) *i.e'7(1, 13')C(3'2)g+(t, t')C-t(2' 4') xt-(t"

r- 7y,ta t' ) G y p,- (x z'x z) G " e, - ( * t - r r ) G r , r ( r r ' r r ' ) ,

I(12,l'2')G+(1'2"

+' l')G (1',3)G+(2'+).

(2.18)

The secondterm in Eq. (2.18)can be simplifiedb-"*the use of Eqs. (2.16)and (2.6), rvhenceit becomes _ i", t (8, 11, (t, t,, )G_ )16/ 6eJ())

r1'r2') - Cp o,C-t t yG.; : G

(r$t')G y a,- (rz' rz) (xp1')G 96+(rrr2'), ";

xI(1"2, s'4')G-+(3'4" 3+). (2.t9) (2.7j)

the proper description for noninteracting particles. The difierential equation for G+ may be obtained rvith the help of that for G--, Eq. (21), reference 1, and of Eq. (2.4'). They yield

3- Qr' )G+ (1'2, 34): 6(13)cf(24) I i e1 ft , t I' )CQ'2')C-| (M') xG+(22')(6/6J(0)G-(4'3) (2.14)

and F+(22')3-01')G+ (1,2,, 31): 6(13) 6(24)

* i et G, Ir' )C(1'2)c-t (44') (6/ 3J(t) )c (4'3). (2.1s)

The last term, finally, is brought into more useful form rvith the help of the identity g*(E, l')C-t (2'4')r- (t', 4' t')c- (r' 3)G+(2'4) : D+(E, E')C-t(2'4')7(1', 1't')6+(1'2' , 34),

(2.20)

rvhich may be verified by iteration of both sides.The interactionoperator thereforeis given by I (r2, 31): ie'zYc,13)9+(€,t',)r+(t',,24) * ie't (t, lr')C (1'2)D+(t, t')c-1(41'h (t', 4'3) - i e \ ( t , n ' ) G ( r ' 1 " ) l ( 6 6 e J( ) ) I ( 1 "2 , s ' a ' ) /

y6-'+(3,4" s"4")lrc- +(3,,+,,,34)l-t. (2.2r)

390 851

ELEC

TRO DYNA

M I C

This, and a correspondingexpressionobtained from the alternative form of Eq. (2.17) correspondto Eq. (47), reference1; the only difierencelies in the secondterm above, rvhich represents the interaction due to the virtual annihilation of the electron-positronpair. The last term contains the effects of higher order electrodynamic processes involving more than one virtual photon, such as multiple photon exchanges and the corrections that symmetrize the first term in the interaction so that it dependson the vertex operatorof both the electronand the positron. We are interested in the effects of one and two virtual quanta, terms of order I in the interaction- For this reason, the functional derivative in Eq. (2.21) needsbe evaluated only to the lowest order, (s'4" 3" 4")l l(6 / 6cJ(t))r (1" 2, 3' 4')G--+ XIG- +(3"4" ,34)l-'=-

I(7"2, s'4')

XG-+(3,4,, j,,4,,)16/6eJ(l)f XIF- (3" 3)F+ (4'' 4)l:

- ie,l^y(E, 7''s')

Xt G', 24')l t (E,7" 2')C(2'2)C-r (4" 2") x7 (8" 2" 3',)fD+(t,E )G- (3'3")G+(4'4")

D I SPLA CE M ENT

interested,the operatorF(12) is a multiple o{ the Dirac operator F(72) that depends on the experimental mass m oI the electron, Ft(t21:(-aB/2r)

tFt\12),

(2.25)

rvith l-*(x, *'): 6(r.- r')17 u(- i0,'teA+ u@))*mf. (2.20) We may now introduce the interaction

Iqrz,s+7:e-aB/r)r(r2,3e,e.27) which enters the equation of the usual form for the wave function,

lF- (l t )F+(22, ) - I (12,r'2,)1,t, Q,2,) : O. (2.28) To find the energy levels of the system, we seek solutions of the form {(rp2):

4rx tu(r) ; X :i@ft

rz), x: rrxz,

(2.29)

that are eigenfunctionsof the total momentum operator with eigenvalue K. This eigenvalue is the goal of the calculation. fn the absenceof an external field, the interaction operator conservesthe total momentum, so that it is possible to write an equation for the function 96(r) of the relative coordinate *,

X D +(t, t' )l- F+ (4" 4) ^tG', 3" 3)

lF x(nc')- I r(x, *')lea@):s,

(2.30)

t F-(s"3)yQ',a"4)1. (2.22) rvhere Whenthis expression is multipliedout, the first of the e ; R X I Fs ( r x ' ) f . B 1 t four termsis conveniently includedin a symmetrical lowestorderinteraction, and the (.t) superscripts can r-71x,X' + - t"x') : I F ix') F'pE\X- ix, X' be droppedin the limit of vanishingexternalfield. ",(X This form of the approximateinteractionoperator, xeiK'x'd4xt,

I (12,3+)=ie'zr(t, 13)9+({,t')r (t', 24)

(2.31)

anrJ,I6(r, r') is similarly related to I(1234). The Dirac indices in Eq. (2.30) are summed in the same way as 7")t (E,I" 3h (E',24') those in Eq. (2.24); ,px stillhas two sets of Dirac indices I (ie')'t G, 11')G(1' even through it has but one four-vector argument. To ({', 4" D+(t, XG(4'4")7 4 {) D*(E,E') avoid complications in the notation, this matrix notation will be continued; where necessary, superscripts I Qe')\ (t, 11')G(l'1")y Q, 1"2')C(2'2)D+(tt') 7 and 2 will distinguish Dirac matrices that operate, 4")7 Q', 4"4) xD+(t, E')lC-1(i3')tG',3'4')G(4' respectively, on the first and second particle index of the wave function gs(r). + C-' (M')t (E',4'3')G(3'3")1 (l', 3"3)1, Q.23). Before we proceed to solve Eq. (2.30),_weshall decan be easilyunderstood in termsof the equivalent composethe first two contributions to 1(1234), Eqs. (2.23) and (2.27). W\th the help of the expressions!2 Fp'nman rlleorcm The wave functions {(12) of the electron-positron I,(t, 13): z"(f, 13)(11 aB / 2r) systemare solutionsof the homogeneous equation,

* ie't G, ll')C (1'2)D+(t, l')C-\ (44')7Q', 4'3)

+ (2,32) +^A(r(1_€' f-3) l F - ( l 1 ,) F ( 2 2), - I ( t 2 , 7 '2 ') ) , 1(,t ' 2 ') : 0 , ( 2 . 2 4 ) andr2,13 related to Eq. (2.16).It is important to realizethat the operators-F(12)alsocontainelectrodynamiccorrections. g+,"(t, t): Ql qA/2r)D+(E, t',)6Thesemay be obtainedfrom the corrcctionsto the one*D+Q(t, E')iF", Q.33) particle Green'sfunction G(12), ol rvhich Ii(12) is the inverse.IzFor the nonrelativisticstatesin which rve are 13Note that r2R. Karplus and N. lI. Kroll, Phys. Rev. 77, 536 (1950).

,r:|i,Dr,,

Da=!iDy, D+@:+iDFo.

391 R.

KARPLIIS

AND

,{.

8.52

KLE]N

they become

then given to a sufficient approximation bye

4ni.a7u(f, l3)D+(t, t'h,(t' , 24) * ie\ r(t, 11')C(r'2)D+(tt')fl (43')y,(l',3'j' X (1- aB/ r) * 4ria1,(9,1i) U (t, t')

tn: _ t. an*an*'p"(r) f X

X t,,tzt12- 1',t' - 4)+ 4r i o^,e,(l - {, E-.i) X D+(t, t') t,G', 24)! 4ria7 uQ,13) D,(D(tt).y,G,, 24), (2.34)

l",tr, {

*/rrat"(x,

(x, x,) r')l I y21(t,x')l I a2po)

t')l

f I d a x ' t d a t ' t ' Ix r ( x , x " )

up to termsinvolvingtwo virtual photons.The experiXfF6c(x", x"'))-'I *,(x"',"1],P"1*1, 1.t.0; mentalvalueof the fine structure--censtant a hasbeen l writtento absorbthe chargerenormalization factorin measured in the reference frame in which the total r'.q.(2.33),12 4ra: e2(ll oA/2r):4o11i7.O,' . . .

spatial momentum vanishes,

(2..35)

III. PERTURBATIONTHEORY Salpetere has discussed a method for finding the eigenvaluesof the total energy of a two-particle system describedby an equation like Eq. (2.30) if the interaction function does not differ greatly from a local instantaneousinteraction of the form 6(r-r')6(l)/(r)

(x,:r, t;

i:1,2,3).

K!:

( 0 ,K o ) .

(3.7)

The function 9g@) is the relativistic Coulomb wave function that is a good approximation to the actual wave function of the state whoseenergy level is sought. It is a solution of lFac(r,r')*Ic(r,x')),pg(x'):0,

(3.8)

whence AE: Ko- Koc.

(.3.1) 'Ihe

(3.e)

expressionEq. (3.6) is accurate to order c relative to the {ine structure contribution 161 and further presupposesthat the intermediate states in the secondorder perturbation term, the last in Eq. (3.6), can be replaced by free particle states. This is the casefor the IC (r, r')l I 6y(r, x'): Ic (r, *') spin-spin interaction under investigation. Before closing this section, we must briefly discuss I I xn(x, r')! I y1a(x,r,), (3.2) the wave function pc(r) that enters into'Eq. (3.6). where As is the case with the electrodynamic corrections to Jc(n, r'): -i.a6(x-r')1nt7nz6(t)/r, (3.3) the magnetic interactions in hydrogen, the contributions to A-E come mostly from the vicinity of the relative the Coulomb interaction,and coordinate origin. The two-photon contributions, therefore, will be at most of the order a,196(0)1,,where I nn:2ia(2r)-36(xr') po(r) is the Pauli wave function for the ground state 'Y0t1o2ko21 of positronium. Since this is the smallest magnitude f I T''T' that is being considered, contributions to these terms \ | d'*ett'1 \-"/ ,1 L ku' hi2ktz l' that are proportional to the relative momentum can be neglected. ft therefore sufices to approximate I a1a: iez(7uC)6(x)6(x')(C-\ ,)(7- aB/ r) / K,z. (3.5) pc(")l lp"@') by the product of I esQ)l': (lam)s/r These include the Breitlt interaction, retardation efiects, and the appropriate spin matrix element, which will be denoted by ( ). In calculating the effect of 161, which and the virtual annihilation exchange interaction. All the contributionsderivablefrom Eqs. (2.23) and,(2.3a) contains contributions of order olpo(0)1, due to the that are not included in Eqs. (3.2-5) depend on the exchangeof one virtual photon, the relative momentum appearanceof two virtual quanta. The two-quantum can no longer be neglected. fndeed, corrections of terms that are includedin Eq. (2.34)will be denotedby relative order a that arise from the larse momentum 162s(r),while those that are explicit in Eq. (2.23) will c o m p o n e n tos f t h e w a v e f u n c t i o nm u s t n o t b e o m i t t e d . be denotedby I urn(r, rt) or I 6s{2)(a, r,) dependingon As Salpeterehas pointed out, an improvementover the Pauli wave functions is obtainecl when the intesral whether they are exchangeor direct interactions. 'l'he change in energy levels produced by the per- equation, turbations 161 and 1yq2acting on the electron-positron vc(x): -io I lF*c(x, *')l-t,pe(r',0)dr'fr', (3.10) system bound by the Coulomb interaction Eq. (3.3) is JS-ucha term can indeed be separatedfrom the centcrof-mass transform of the first two contributions o{ Eq. (na), which may be written

392 853

DISPLACEMENT

ELECTRODYNAMIC

The following observationscau now be made about that part of the energy change which depends on the spin of both particles. Only large contributions of magnitude f a 2 and a-r will be important in the integral. It can be (3.1 y')l-tqs(r')dr' r'. 1) ,ps(r)-- ia | [f'^c(x, / seenthat only small valuesof the momenta k', k"Sam make such large contributions. The important region IV. THE DIRECT INTERACTION of integration, therefore, extends over small values of either or of both these momenta. When both momenta We turn now to the evaluation of the matrix elements are large, k' and k" -m, the integral becomesnegligible for the energy shift that was obtained in the previous for the purposes of the present calculation. A term section. We shall consider first the contributions A.Er proportional to k'2 and k"2 in the spin matrlr element, from interaction, arise direct which of those terms for instance, is negligible becausein its evaluation one namely, those in which an electron-positron pair is may neglect (arz)2 compared to h'2 and *"?, so that the present in each intermediate state. According to Eq. integral in Eq. (a.3) becomes effectively independent (3.6) and the definition precedingthis equation, of a.Ia One may now see that the spin-dependent contribution of the retarded Coulomb interaction involves i , d 4r c dx4' , p c ( r c ) I r r r ( x x, ' ) 9 6 ( x ' ) A[. B: one of the cl.k'c2.k' terms of both F(l) operatorsand I is therefore a negligible large momentum effect. The Breit interaction, of course,is important and contributes t/ -il eo(o)i2 dnilA.v'\lK2Bt2)Q(, r') in conjunction rvith only one factor cr'k/cz'k'. Since J corrections that involve an additional factor fr"2 are too small, one may use an approximateexpression

is used Ior au iteration procedure based on the Pauli wave function,

+

J

dLtc"d4x"'I Ktn(x,x")

F*(r)ry+(1+ a1.k / 2m)(l - ar. k / 2m) i( z+n) I t!) tt Xl(m / E) (e-Nn-n) | ! s-

, r')> XIF vc(x", :c")l-Ll KIBQ"'I r -il p0(0)| rJ daxdar'(I *'17. orutr>,1,

l(e-&a

(4'1)

The one-photonpart of the interaction, AEBL: - i

J

to evaluate AErr. The spin matrix simple,

has now

quite

become

X c r . c r ( l * c , . k , ,/ 2 m ) ( l * a z . k ,/, 2 m ) l +(at . o2k2- o\ .kc, . k)+3(or . cr)[t,

2qr (2r)3J

(4.4)

( 0 | . a t . k '/ 2 r n ) ( 1 - a 2. k ' / 2 m )

d'a x,Ja x' Pc(x)I x rs(*. x' t,p6(*')

:-

element

nttt -p-i(f In)ltl)]

| d4xd4k\oc(x)eik.

''T' '

f T' \'

'to''vo'ko'1 "Yo'll'ko'

;_ 8,," L k,,

* lpc\x), \+.2) k;nk,,

R;'ft"'J

(4.5)

and the since the 0-function implies that k'-k":k, integrand has the necessary spherical symmetry. The &6 integration with the usual treatment of the poles yields

presents the greatest complication becauseit contains the lowest order hyperline structure as leading term. When the approximate Coulomb wave function evaluated in the appendix is inserted here, one obtains a spin matrix element and multiple momentum integral which is multiplied by the explicit factor c3l 96(0)1'?:

e ikotdk|(kr-k02-it)-t-Tik- t.-;tlt (6)0). (4.6)

I

The function of time in Eq. (a.3) is therefore even) so that the time integration may be carried out only over positive values if a factor of two is supplied' I'he integrals encountered are of the form

Sarl ,po(O) l'? r AEBr:

(2n)am2 J nt2

( P 'zj

I

dle

,to't^:t"-ikte

- i(k+ L' + E" +m*m)-t,

nf

-6(k-k,+k,)

|2 ! a 2n 2 ) 2 ( k t + I a , m 2 ) 2

:!r!!!f,.,a>)(43) " (,"eolff

i\n"xmtl

since

the

denominator

never

vanishes.

'fhe

(4.7) energy

)a Detailed examinalion shows that the integral actually is pro portional to loga in this case. This rJependenie, however, is still negligible for our purposes.

393 R.

KARPLUS

z X f ,Jk,/k',1k"6(k-k'ali'11p': FLa2mzl1(E'fm)(E"!m) ' ltnqz/ trzl-'?1 h.

mz-E'E"

k X,

2E'8"

. (E'-m)(E"-nr)

at d l x d o xe' - i K " \ x- x ' ) d X

kaE'1p"

k I -l. kIE'*D"_l2ml

4EE"

854

: - i l,ps(0)l' (4nia)' LE Bz(2)

4E'8"

h*E'lE"-2m

KLE]N

of this treatment lies in thc fact that the sum of the direct interactions is independent of the cutoff; that a cutofi need not have been introducedat all if the terms had beengroupedproperly accordingto the photon momclltum that rnakes the contributiorr rather tltan lccording to the physicrl processthal is represenlecl. 'fhe evaluation of the remainderof Eq. (4.1) is relatively simple.The secondline contributes

A,Epl : *(c1. o2)aa(2r)-3 | pn(O)l, J

I

A.

'l'he justification

change has now been reduced to

X(k"2

AND

(+.8)

X {((r*1G1(X*1x, X' jlr')y"t) X (t

"'G'

(X - ir, X' * L*') t u,))

X D+(X- X' + +(r- r'))D.,(X'- X t i@l x'))

As it stands, the integral in Eq. (4.8) is quite difficult to carry out. We must remember, however, that at f least one of the two variables k', k" must be small comt f (2t) l d1kd4k'ei+tpik'!'f kt2kF'2 pared to m, a fact which permits replacement of the corresponding kinetic energy by the rest energy, Furthermore, the occurrence of a factor (E'-m) im' X((T'. r'- yo'r o'(ko2/k;2))Gt(x*lr, X'llx') plies that the particular term contributes only for XG'(X - lr' X'-\x'11rt . rz- 1otyo,(ho2/ kir))), large kt-m, whence &" must be small, and aice uersa. (4.10) In such a case, the small momentum may also be neglected in the argument of the d-function. The an expression derived from Eqs. (2.23), (2.31), and remaining integrationcan then be carriedout: (3.4). When Fourier transforms are introduced for the Green'sfunctions, the energymay be written LE 31: -(ot . v2)atQ") -"1 po/n)|, 4-2 Afu"r:'. . . . l e . ( o )i ' l , l o k 1 h , ' 7 ' '

"
| X (h'2+ +q2n 2)-2(kt 2+ I d2nt')-2 { ((k - E' ) / 2mE kk'') 6(k-k') (k'' 2+ ia2m2)-2

"\ x(,",!s!:?-' \ F.,-(n-ko)r

* ((h - E" ) / 2mE" kk"'z)6(k+k") (k' 2+ !azrnzl-21

:

2r

a

|

. ( u , . o .i )l t "' l e o ( u ) Il !l -l J

4a r

-

2q r

J

\ZT)'

x I dkdk'dk" | (m,/ ti' E')6(k+k" -k )

"')

G !" - ! -l (T' "! T t - ? u ' ? u 'P u I '1r,"r't E2- (n- ko)2

tnl

l n' 2_k .^-l 1 . ( + . , )

In the 6rst term both k' and k" are of the order am, in the secondk'-m, and in the third k"-m. A cutoff ft,, has here been introduced as a lower limit on the final momentum integration. Its presenceshows that some contributions of order a2| ,pr(0)] , to AErr do arise from small values of momentum, contrary to expectation. It will be seen, lrowever, that the direct interaction I xzxQ)'and the second-orr'lereffect of 1{r, also cotrtain c o n t r i b u t i o n sf r o m s m a l l v a l u e so f t h c m o n t e n t u m : r s represented by the appearance oI ln(m/2k,,). Just as here, these are being treated incorrectly because of the assumption of free intermediate states that is implicit in the derivation of the interaction operator.

t'(iKC lk)-n [r-

( T ' ' ' ( ' - " Y o: '' o ' k o/ 'k : )

@+ W

\ ),

(4'l | )

where, as before, Iiz:k2lm2-ie

(e)0),

};Kna>m

(1.12a,)

it nrl hu't- Pz-Put-1,

(4.121t)

define the treatment of the poles. Explicit display of the spin matrix element and spherical averaging wisely precede the momentum

394 855

ELECTRODYNAMI

C DISPLACtrMENT

integration,

only anticipate the result [see Appendix, Eq. (A.3)] that the latter will contribute a term that renormalizes the charge occurring in the first-order virtual annihilant r' tion from its uncorrected value d to the measured d h o ( k p z ) z h , d h , xf value 4ra [see Eq. (2.35)]; to the order consideredin Jo J-6 this paper, therefore, all quantities depend on a from - ? k')lE2 - (/n - h0)21-2 X { (ko'z nere on, The first one and last two terms in Eq. (5.1) present * ik o'lEz- (m.- k o)zf-tlE'z- (rn+46)'?1-') ; (4.13) some complications since the quantity B is actually a it depends on the identities divergent integral.r2 We expect that other divergent integrals will make the complete result finite, but we : (outo 6iiia;i; (4.14) 2(o'' o'). ni\ 7;7 i: must exercise great care to obtain the correct finite The evaluation of the integrals is straightforward, result. For purposes oI orientation it is instructive to except that the same cutofi fr- for small momentum consider briefly the matrix element in Eq. (5.1) for values must be introduced. The result is noninteracting, nonrelativistic initial and final states, because the high energy divergencesmay be expected (5q 2a n 'l 2r q : - -(c'. c') | es(0)l, - - *aE B2t2) 1n^- l, (4.15) to be the same in this simpler caseas in the positronium I J m' tZT ZR^l T atom. The wave function ,pg(r) then represents the and gives the total efiect independent of ,t- of processes initial state plus a co'rrection due to one Coulomb where all quanta are exchanged between the two scattering, while { lF (r, xt )f-t I x B (r', r" ) s s(0)d4*' d4*" represents the correction to the initial state due to the particles, Breit interaction and retardation effects. The three 2r a 3al I A E a t l L E e z ? ) : - ( o t . " r ) l e o ( O ) l ' { 1 - - f . ( 4 . 1 6 ) terms we are now considering, therefore, comprise the 3m2 | 2rl matrlr element of the virtual annihilation in the initial trEp2t2): (8a2fr) I ,ro(O;| '1",."t;

The perturbatio\ AEB1G)includes efiects of vacuum fluctuations on the exchangeof a single quantum. The spin-dependent corrections to the vertex operator are contained in the anomalous-magnetic moment (a/2r)(e/2m) of each particle while the vacuum polarization has no efiect on the singlet-triplet separation. The added contribution is therefore 2ra AEB1(D: - -(a1.r,)le6(0)1,{a/nl. 3 rnz

t!

Y

Y

lr

ir

|

\7 Y ir

Itr I b

,. Feynman diagrams for virtual annihilation electron positron scattering. "r"-

(4.17)

state plus a correction due to the four-dimensional interaction representedby one quantum exchange.The V. EXCHANGE INTERACTION Feynman diagrams for theseprocesses,Fig. 1, show that In this section we shall evaluate the matrix elements the electrodynamiccorrections,Fig. 1b, 1c, are just of the exchange energy, embracing all processes in the correction to the vertex operator, and therefore which there is an intermediate state with no pairs contain each a contribution (q/2r)B multiplying the To our order of accuracy, basic interaction Fig. 1a.r?,16 present. The energy change,according to Eq. (3.6), is the divergent integrals disappear. f With this understandingwe can attempt to evaluate A E n : - ; I d a x d a xe's ( x ) l y ' ; ( r , r ' ) e 6 ( r ' ) ( 1 - a B / r ) the actual matrlx element in Eq. (5.1). In order to keep track of the inf,nite quantities, it is very conf venient to regulate the interaction brought about by -tl eo(0)l' dardax'(I6ra(x.x')) I photon 11 in Fig. 1 with a heavy photon of mass .4..r8 The integral B can be evaluatedto B^,l'? f -tl eo(O) d4)cd4#td4r"d4r'tl frf l' t J B^:(ir'?\ t I uduldak Jo

X(I rtn(r, r")lF ac(x" , x"')l-11 s1a(x"', r') t I rra(x, *")lF 6c(rt' , x"')f-tI yla(x"' , r') t I ru(r, r" )lF xc (x".,x"' )f-' I x t n(x"', r')).

(.5.1)

Considerationof the virtual two-quantum annihilation I xzr and of the second-ordersingle-quantumannihilation will be postponed to the end of this section. We

J

2 X l(kzf m2u2)2- (k2J m2u2 | L2(l - tr)) - 4m2(1- u - lu2) (h2+ m2u2)-3l :ln(tr/m)!1a*1n(m/2k,,),

(.s.2)

* J.C'.lVr.a,Phys.Rcv.78,182(1950). 16R. P. Feynman, W. PauliandF. Phys.Rev.74,1430(1948); Villars,Revs.ModernPhys.21,434(1949).

395 R.

KARPLUS

AND

where quantities depending inversely on ,4.have been omitted and the low energy cutofi A- has been introduced [see text following Eq. (a.9)]. The structure of the exchange interaction 161a implies that the energy change corresponding to Fig. 1 can be written

A.

Tr[C-17;e1(0)] :ll-

(a/2r)B tl Tr[C-12;i,o(0))(- a/rm,) f*.

LE a1: - v o*-2 o t(O) "aQ iC)"p

o'opt(o)

"'

h2dk{-?n2E-t(4k2/3} 2n2)

X (kz+ +q2rn2)-2t I (E: r - P- t1t, *z P- s -h2(h2+ l\2- L4/4rn )-rl_ (rn2_I.le) t X E (4k,/ 3* 2m2)h-4+ + (E-t - E - 1)

(5.3)

where (C-\)

-1f1. + (tnz- + L2)k-2E

p'=TrlC-tttpt(O)l Here

E _(h2+L2)r,

=lr- (a/zr)B Trll d.k(rn,/ E) ^12"(2")-, I X (h2+ iq2nt2)-2| (l - r. k / Zm) t XC-tt i0* r.k/ 2m)l C-t^yft2/ 4*rl

eo0) l

+ (i/zr)

f

J

(5.5) (5.6)

in the second set of terms, which came from the regulating expression. One can observe that these reduce to the first set when h:O il am there is neglected with respect to fr. The integrations are similar to the ones encountered in connection with Eq. (4.9) but made more complicated by the regulator. If one erpands the result in porvers of (m/L)'and keeps only the leading fprm

nne

nhtqinc

Tr[C-r7;e1(0)]

dak(i ;@- i (],KC* h))

:ll-

XC-ty 1(tn- 7 (! KC- h)h ;- (ks2 / k;2) Xto(m- 7(iK"*k))C-tt;(m-

856

integral, Eq. (5.a) becomes

X I

X(C-\ ) y "',p{o)",0,,

KLEIN

(d/2r)B ^l Tr[C-\2;,po(0)]

x U -f (q/ 2r)Lln(rt/ m)l- +* L- In(m/ 2k-)l

t(iK" - h))tol

: (l- 2a/ r) Tr[C-12;po(0)], (5.7)

(5.2), whence Xlk p'zl-tlE'- (tn+ ko)'f-Lla, - (m- ho)rl-L,po(o) with Br given by Eq. LE a1: - (r a/ mr) (1- 4a/ r)

f

- (i/ 2r) dah{i,@- i G KC* k)) J

XTr[po(0)zF]

XC-t y1(m.- 1 (l KC- k)), Fllh t2+ L'zl-\

Tr[C-r'y;e6(0)],

(5.8)

because7rC is a symmetrical matrlx. When the usual representation of the charge conjugating matrix

- (n- fr0),1,e0(0)1. (s.4) XlEz- (m I ho)zl-'lE2 I In writing the contribution of the regulating term, the last in Eq. (5.a), we have taken advantage of the fact that a very short range potential has no bound state so that the scattering picture describedby Fig. 1 is applicable. The total energy has been approximated by 2m everywhere except in the correction to the Coulomb wave function, rvhich comesfrom Eq. (A.9) evaluated at the origin. Only a spaceJikepair-producing Dirac matrix need be taken in Eq. (5.3). The trace is evaluatedrvith the help of the facts that the Pauli wave function has only largecomponentsan<1that the chargcconjugating matrix C is an odd Dirac matrix. After integrating over fro with the usual treatment of the poles and after spherical averaging of the momentum

C:C-L:

(5.9)

"to"yz

is inserted, the direct product of the Dirac matrices in Eq. (5.8) can be relabeledso that the operatorsrefer to the spins of the individual particles:F8 | Oo(O)" eQ {) e"(C- y ) p'"' p o(0)"' p'--

po*(0) "B

a po,l,po(0)y X [86"",68p,+]o """ "'

:-

Ieo(0)l'(s'),(s.10)

where S is the total spin of the system, S:*(a'f

a'z).

( s .1 )

We then obtain the known effect of the virtual annihilationrs plus a large correctionof relative order a, 6 E " ' : ( r a / m 2 ) \ S ' z ) l , p o @ ) l ' 0a- a / r l .

(5.12)

396 857

ELECTRODYNAMIC

DISPLACEMENT

We norv turn to the contribution of the second-order prevents them from contributing. Since the wave funcsinglequantum annihilation, tions in which the spin matrix elements are evaluated have only large components,Eq. (5.18) can be simpliaE ^2(\, iTzd2m-,TrIpo(0)roC](C-tr) Iied to X[Frc(0, 0)]-'(z,c) Tr[C-I7;e6(0)],

(5.13)

rvith the spin sums as inferred from the derivation of this expression. In the appendix this efrect is interpreted in terms of the polarization of the vacuum by the photon produced in the virtual annihilation. The evaluation given there together with Eq. (5.10) shorvs that the effect on the singlet-triplet splitting is8 L E A 2 ( D : ( r a / n ) ( S r ) |e o ( 0 ) l , | - 8 q / 9 r ) ,

- 3k,'z\Q orsc)(C- lvorr)) : 3}"'!(("vzro)(rzvo)), (5.19) where

ia2 A E d r ( r ) : - l 9 o ( 0 )l t

The momentum integration,

J

? I d[hk!21k"2+L2l-a:ir2/3L2,

ar

xl Jo

x((t rltGKc-k)-m)y"C)

| 9r(0) l,

FL

Nt | "a*7*12-y1'-4(r-y)-ief-r .ro

: - (a' / nf)(2- S'z)l,po$)l'z(2- 2 ln2+ il).

(i KC - k) mft

" - k) - nl1,\,

(S'rs)

u,herr lourier representationsare iutroduced for the Green'sfunctions.The spin matrix elementsof the tso parenthesesis to be taken as trace with the final and initial state n'ave functions, as in Eqs. (5.3) and (5.a). The momentum integration is simplified by the usual procedure of combining the three distinct denominators accordingto the formula

(5.22)

brings the energy perturbation into the form LEp{z): -6oz*-r(2-5r)

I C-\,6GKC

(5.21)

3k,z(z-521.

T.

KC - h) \'z+ m'?l' l- L X Ihe2(K c - k) "21(+

(5'20)

Rearrangement of indices according to Eq. (5.10) finally produces the ordinary spin matrix element of a function of the total spin, Eq. (5.11),

(5.t4)

since the renormalization constant ,4, Eq. (A.3) has been incorporated already. The final item to be discussedin this section is the energy shift associated with tivo-quantum virtual annihilation given by

x (- C-' t,lt

7s:'Yt"t2"l3"fot^ls2=- l.

(s.23)

The real part of this expression corresponds to the energy change of the level while the imaginary part correspondsto the well-knownrTdecay rate of the singlet s t a t eb y t w o - p h o t o na n n i h i l a t i o n , r-1:a3 Ry-:0.804X10ro sec t.

(5.24)

'I'he

total contribution of the virtual annihilation interaction may be collectedfrom Eqs. (5.12), (5.13), ano ().rJr, 6B o: (r a/ m2)l,po(0)l,{(S'!)(1- aa/ r - 8a/9r)

f 2(s'- 2)(1-ln2)).(s.2s)

- h)^t* *,1,l-, , d 4 k ath2 ( K C k ) , r l ( i K "

VI. SUMMARY

:6

'I'he

ftftf

J^

dependenceof the 1tS and 13S stales in positronium on the spin of the system is obtained by the addition of Eqs. (,1.16),(a.17),and (5.25):

t'dx tuJ a,n J^ vn

X llk - i Kc (xy -t 2(1 - *))1'?

tE:

{rm2(xy2-4(1- x)(l- y))- i.el-a. (5.16) T h e d i s p l a c e m e n tk s - h s f n ( y + 2 ( 1 - r ) ) denominator into the form

h2| rm2lx(2*y)'- aQ-il

(2r o/ m'z)l,po(0)l,{t(ar. c')lr-

}a/ rf

+ Xsl[l - (26le| 2 tn2)a/r)1.

brings the

By taking the difierence of the value of these operators in the singlet and triplet states, one arrives at the ( s . 1 7 )hyperfine splitting

and leavcsthe numerator proportional to (- c-''v,'r o'r,* C-'r,r,r u)) (5.18) "t'c) after hyperspherical averaging and the discarding of some terms whose deoendenceon the Dirac matrices

LIV 6: (2raf m2)l *oo(0)1'z{7 /3-(32/9+21n2)s/tl :la'zRy-17

tk x'((t n

-lJ.

/3- (32/9+2 tn2)a/rl :2.0337X106 Mc/sec.

A5h..t".,Ann. N. Y. Acad.Sci. 48,21g(1946).

397 R,

IiARPLUS

AND

The singlet state is the lower one. It can be seenthat most of the rather large negative electrodynamiccorrection comesfrom the virtual annihilation interaction. When the experiment of Deutsch and Browna is interpreted on the basis of a Zeeman efiect that depends on the totar magnetic ^. ;;;"G;i;;;,jfi]"7i'"1'"i each particle, the value of the separationobtained by them isb

A.

The center-of-masstransform of the noninteracting Green'sfunction appearsin tire integralequation (3.10). Its Fourier representation is ,. [j?xc(r'r')]-t:-)

APPENDIX The operator [tr.(13)F+(24)]-1:G-(13)G+(24), the n o n i n t e r a c t i n gl w o - p a r t i c l eG r e e n ' s f u n c t i o n ,a n d i t s F o u r i e r t r a n s { o r m sa p p e a r s o f r e q u e n t l y t l r a t r h i s appendlr will be devoted to a discussionof someof its properties. fn connection with the second-ordereffect of the virtual annihilation, there appears the tensor

_ t

r

dok"ou o'

-. /.{.4\ l m - t t ( i K-c l h ) ) L n' -t-t G K c - h-) f x- ki'?) (+Krc t E2- (LKoctha)zllE'z-

LWh: (2.035+0.003) 105Mc/sec. Theory and experiment are thus in satisfactory agreement. We are grateful to V. F. Weisskopffor calling this p r o b l e mt o o u r a t l e n l i o n .T h e a u t h o r s a r ea l s o lne l : db l , e d t o t h e m e m b e r so f t h e I n s l i t u , t . e , I o r . A d v a n c S ludy Princeton,for an informative discussion'

858

KLEIN

when the center of mass is at rest. The quantity -R is defined by (A'5) E-h'zlnf-ie,. e)0, 5 i n e eF . o ( A 4 ) r e n r e s e n r st h e n o n i n t e r a c t i n gt w o plt'ir"tr.."'='r""Iti"" i " t o u t g o i n gr v a ' e s .T i e i n r e lration over the fourth component of the momentum

i;Ji.IfTJ;.,ilJ.ilIi:,.T]''Jil':,Y; iltimecoordinafts , lFxc(x,r'1f

':i(2r)-3

f dkr;t r'-"r J| X.(hrllqrmr)Filt-t'),

(A.6)

where

(C-1tl)lFxc(0,0)l-'(ry,C) :, t

F o Q ) : L 7 s - o t u _ m ) )+t e _ i ( E + n ) t t t )

f2E d 1x ' e - i K ( x- x ' ) ( c - L ? ; ) " " . G " , s , - ( x ,x ' )

f/

k'l

o,.k\/ o,.k\ |/ Xl I l+l l t * 2* m I 2m / \ [\ /

t h a t a p p e a r si r r t h e v a c u u mp o l a r i z r t i o nt e n s o r . rh2 i s e q u a tl o

(4o;tt"""

otk\

t)l tt I !le-ttz-^> t - ri(E+m) 1

x T r t l p - ; . x , ) t , G - ( x , , x ) 1 ,( A . 1 ) l yeq u r n t i t y b y E q .( 2 . 5 )T. h i si s .h o w e l epr r, e c i s er h

i --l^..lKc\2-K.cK

o'k\/

xlll+-ll1--l+-l 2 m' l 4 m 2 J 2 m/ \ L\

r X G " p + ( X , X ' ) ( t , C t ul d, u n :x ' r - ' n ' * - * ' '

k, /

;,\t*

c\

'

I cr.k-a2.k\l

l't

u

)l

(A'7)

und the total energyo{ the 1,5state has been inserted,

v,(1- +vr)(Kc),I

xlza+["'dv m2lf,(KC)'(1_V2) ].

Koc:2m-ta2m.

(A.2)

In the frame where Knc:6, (KC)2--4m2, the tensor becomes

'#','(;-^-::) (A.3)

(4.8)

The wave function derived from Eq. (3.11) with the help of the operator just obtained is

eg\):

(2a/(2n)2)

f | dketk'n

X (h2!la2m2)-2F{t) eo(O). (A.9)

P o p e r3 3

OF ONTHEMAGNITUDE CONSTANTS THE RENORMALIZATIOI\ EI-.ECTRODYNAMICS IN QUANTUM BY

GU){NAR KII,LI'N \I lith the aid of an exacl forntulation of the renoimalization method in guantum electrodynamics which has been developed earlier,it is shown that not VV all of the renormalization constants can be finite quantities' It must be stressed theory. that this statement is here made rrithout any reference to perturbation

Introduction. In a previous paperl, the author has given a formulation of quantum electrodynamics in terms of the renormalized Heisenberg operators and the experimental mass and charge of the electron. The cor^sistency of the renormalization method was there shown to depend upon the behaviour of certain functions (II (p'), Dr(p') and Xr(p'z)) for large, negative values of the argument p'. If the integrals

(rr (- o\ \)a)a

de,

f3, (- o) \

"

do

(i:

1,2)

(1)

converge, quantum electrodynamics is a completely consistent theory, and the renormalization constants themselves are finite quantities. This would seem to contradict what has appeared to be a well-established fact for more than trventy years, but it rnust be remembered that all calculations of self-energies etc. have been made with the aid of expansions in the coupling constant e. Thus what we knorv is really only that, for example, the selfenergy of the electron, considered as a function of e, is not analytic al the origin. It has even been suggested2 that a different scheme of approximation may drastically alter the results obtained with the aid of a straightforward application of perturbation any theory. It is the aim of the present paper to show-without attempt at extreme mathematical rigour-that this is actually not the .case in present quantum electrodynamics. The best v'e can 1 G. KAr,r-6Nn Helv. Phys. Acta 25, 417 (f952), here quoted as I. ' Cl., e.g., W. TurnnrrC, Z.f. Naturf .6a 462 (1951). N. Hu, Phys. Rev. 80, 1109 (1950). 1*

399 4

Nr. 12

hope for is that the renormalized words, that the integrals

theory is finite or, in other

n'rr(- o). f},(-") da, \.. V

o0,

(2)

Jq',)a"

appearing in the renormalized operators, do converge.No discussion of this point, hov.ever, r'ill be given here.

General Outline of the Method. We start our investigation with the assumption that all the quantitiesK, (l -l)-1

and *
that the integrals (1) converge. This will be shown to lead to a lower bound for I/(p,) u,'hich has a finite limit for - prn *, thus contradicting our assumption. In this way it is proved that not all of the three quantities above can be finite.'Our lower bound for II (pz) is obtained from the formula (cf. I, Eqs. (82) and (32 a)) r/ \- r,

Ir (p') : :+ 1- 1;wt').r; ) , |
(3)

p\'): p

It was sho*'n in I that, in spite of the signs appearing in (3), the sum for II (p') could be written as a sum over only positive terms. Thus we get a lorver bound for II (pz), if we consider the following expression

Ir (p')r --+; )l'l aol,i"lq, q')l'. "r

(4)

ofrp

In Eq. (4), (0 ljrlq, g') denotes a matrix element of the current (defined in I, Bq. (3)) between the vacuum and a state with one electron-positron pair (for no>- co). The energy-momentum vector of the electron is equal to g and of the positron is equal to q'. The sum is to be extended over all states for which q + q' : p. We can note here that, if we develop the function Z (pr) in powers of e2 and consider just the first term in this expansion, only the states included in (a) will give a contribution. For this case, the sum is easily computed, e. g. in the following way:

) r ' l ( 0t i , 1 " , 1: '

Z

( i : r , , r t 1 1 ; ;"';- t ( 0 t L4t l ' )

400 Nr. 12

(5)

'lhe

for large function Z(0) (p'z) has lhe constant limit ,{n, 'of values Oz. This corresponds, of course', to the rvell-knorvn for the first-order charge-renormalization.We shall see, divergence however, that with the assumptions we have made here the Iower bound for the complete If (p'), obtained from (4), is rather similar to IIo (p').

An Exact Expression

for the Matrix

Element

of the Current.

Our next problem is to obtain a formula for (0 lirlq'q'> rvith which we can estimate the matrix element for large values of - (S * q')'. ,For this purpope we first compute

= -"!!Jt 3)lip(*),f (3)ldr"' lir@),rtor(r')l

(6)

- r" rp(B)idar"' . I;,!,i ) vnlirt*), (Cf. I, Eq. (5a).) The last commutator can be computed without difficulty if we,introduce the following formula fot ir(*) ieNz

i r @ ) : =E,slr)

L

0 2A " . @ )

+ r _ Ltaiffi-

rvith €6:

6pit- L6r+6i,*

L6p+lAn(r) (7) (7 a)

and

: (r)). s7@) f;fO<"1,yrp

(7 b)

The expression (7) is rvritten in such a way that the second timederivatives of all the Ar's drop out. With the aid of I' Eqs. (4)(7) we norv get

401 6

Nr. t2

(r), Lir(,rl,vG11*-,-*: ,'?, €utlrt rp(3)l I : -

-i"') -=€r7,TaT1"V(r)d(F

j

(8)

l

It thus follo*'s that /.x

t i r @ ) , t t o r ( r ' ) J: - N ) j . ( 1 3 ) [ . r p ( ' ) ,f ( 3 ) ] d -

pIV

t _ tfulS(t

(e)

t)y;rp(x).

We then proceed bv computing

Q l {fj r(r) , ,ro)(r')), Ttor(r")} | o ) ieAf" (1e (* i_ Lfui.S ) zr S 2) N ) S-(tl) de"" (r), {?(o) x [(0lf,rp (2),/(3)}]lo>-
:

If this expression is considered as an identity in r, an I r nI d fr' :t:' c" ]t will obviouslv give us a formula for (0 lirlU, A, anr dd fo for rilightthl rie Q l i p I q ' ) . ( C f . I , E q s . ( 6 8 ) a n d ( 7 7 ) . ) \ \ / e t r a n s f o r m the iht hand side of (10) in the follo'wing u.ay: l''"' (2),/(B)): r)4
9,,7a 1e;+ Iili l si ( 3 2 )

(11)

and, hence,

( o l l i r @ 1 ,{ 7 ( 0 ) ( 2/ () 3 , ) } ll o l :

# r ^ t ( 3 2 ) ( 0l t i r { O , A i ( 3 ) l 0 )

+ u'Iljf'u(ol.jp(*),{/(e),7<+>>ll 0)s (42).

(12)

The last terrn in (10) can be treated in a similar \vay: f"

oUl

(r}l L i , , @ ) , ? t o r ( 2: ) Jn - ) j 7 , ( r ) , f ( 4 ) l s ( 1 2 ) d r l v 1 , L _r O G ) y t s ( * z ) E ^ p and

t i ! -. < . t 3 ) d x " ' ( o 1 ( F/((' ): ), ) l o 1: - ( 0 1 { o( r ) , t p ({ r, t' ) * 5 " ^ :B" -) ty n , p ( r ) d . r " ' ): l or > s(r ) -*,] [t

I tt*' j

Nr. 12 Collecting (12), (13) and (14) we get ( 0 | {[J& (0), ,p(o) (r')), y{o)(r")) | 0 )

:

i+]+2(N-1)l6r1s(1 r)vf

(r2)

pI

-,

\ s _ 1 r s7; 7 . s ( 3 2d)r " ' ( 0 l l i r { * ) , A i ( 3 ) l | 0 )

(15)

a-h ,.J

-"')jf

FI"'

( / ( B )f, < + l )l lo ) s ( + z ) U f ' " s ( 1 s )< o l l i p ( r ) ,

- N ' ) j j " ' ! j f ' " ( 1 3 ) ( o| { f ( 3 ) ,l iu @ ) , / ( 4 ) l| 0 } )s ( + 2 ) . 'Ihe

second term in (15) can be rewritten lvith the aid of the frrnctions II (p') ana n (p2).

(0 | li

: 0) dr lv r , @) ,A ,.(3| 0 )l) ta o {a +l1 oI h1,@) ,i 7@\l : ap,'ot''u(p)lptpt- oz6rtJ4\P\ -;t\

(16)

i

We are, holvever, more interested in the expression

t''"1'[ L' dPPin,*\w'') e ( r 3 ) l ( o l t T r ( r ) ,A i ( 3 ) ll o l : 5tr+ e,r). )

(1i)

-t ine(p)n (p,)l+ tr * e(r Dlu#::}, | lvhere

@(*) : Obviously, rve have

,, \-'"/ "fa\a l

dpeip' u( p)n q') .

(18a)

@(3r):0

ryP:

( 1 7a )

-in(o)d(t-;"')

(18b)

for rld' : ro. It thus follows

a#+9 e '( * B ) o e u o f r l - T - f r l ' [ e ( r B ) @ ( B r+) ]2 i n ( 0 ) d u * 6 7 . * d ( r 3 )( .t e ) o r:,^r o

403 8

Nr. 12

Using the equation

u u ' ;t ( l 3 ) z l s ( s z ) : o '

weger (ol

-'

: -

(20)

a

- f e(r B )]s(1 s)y7 s(8 2 )(o I t7r ( r ) ,a,( 3) Jlo>dr "' \ ; [1 J_-

I ine(p)rI @')l d;f\o*"'too"o*u'(l')7"s(zz)lnQ'z)

(21

t i6r+f-sQ *)yns(r2). Introclucing(21) into (1b) we obtain ( O| {[7, (r), yt(o) (ct')], 1p@ (x:,,)>| 0> nn t

: re rr"' r - d p i ^ , y e r 3)yps(82)i - rr er) \a ' u \ (iLn'nr'or*",s(l +n@)-ine(p)n(pr)l

-"'

c.r

\

al-

Pr"'

o * " ' d * r v s ( r s () o l tj r { * ) , ( / ( 3 ) ,f f + > >0l l) s ( 4 2 ) \ * a.l- o

(r.r\

- r ' \ 0 r , , , \ d . r r v s ( 1 (30) l { f G ) , [ i r G ) , f f + 1 1 1 , 1 0 ) s ( 4 2 )

'-a'*z"g7fr1s(1

r)y,s(*2). The first ter-m in (22) describes the vacuum polarization and is quite similar to the corresponding expression for a weak external field (cf. I, Appendix). The remaining terms contain the ano_ malous magnetic moment, the main contribution to the Lamb shift etc. Introducing the notation

- N z0(rB ) 0(8 4 )(0 l l i u @),{f (B) ,lf + .r }Ilo> - LN 20(r 3 )g(r 4 )(0 l {/(3 ), l i u @ ) ,/ta>l}I o> _ 2 _ ! ( N _ 1 ) , "L 6 1 , n r n. ,(6r 3 n) d(34) t:

ie

nn


arar'tto'',r'+i,(x4tAp(p', p)

(23)

404 9

Nr. 12

o(*):]tt+u(n)1,

( 2 3 a)

rve obtain from (22)

(ol;r,lq,q,> | _ - ( 0 l 7 ; l' rq , q ' ) - Z < < n- t q ' ) ' ) +_z ( o-)i n l r ( ( q * q ' ) ' ) lt *, i _-l I * ie( 0 l,t@l,t') Ar (- q',q)<01 v(o) | q) . |

-

I

(24)

L)

This is the desired formula for the matrix element of the current.

Analysis of the Function Ap(p',p). We nov' v'ant to investigate the function Ap(p',p) in some detail, especially studying its behaviour for large values of - (q + q')' in (24). For simpliciff, we put p' : k * 4 and study i eA e ( p ' , p ) :

r'r' I (3t'1-ip(r4\ 1 s 2 0{ ( r 3 ) 0 ( . r a ) ( 0 1{ / ( 3 ) , d*" r'ip' ! \d*"' [,ru,

U o @ ) , 1 ( 4) )l o l 1 - 0 ( r 3 ) 0 ( 3 4 ) ( o tl " r r ( " ) , { f ( s f) (, 4 ) i l | 0 ) } '

i

We treat the two terms in (25) separately. The first vacuum expectation value can be transformed to momentum space with the aid of the functions

t \ i ' )@ ' , p ): v , : Z ( 0 1 / zl ') ( z ' l j o l r ) ( r l f l o )

(26)

t \ , - () p ' , p ): v ' ) . < 0 l i l t ' ) < r ' l i * l r ) < r l l 1 0 )

(27)

n l !@ ' , 0->- , ' ) i : (0 l / l z ' )( z ' l i l ' >< z l i k l D

(28)

a ? \Q ' , p ): v ' 2 . ( o l , rzo')l ( z 'l f l r >< z l f l l) .

(2e)

ii:,,=i,

It then follows that ( 0 | {/(3), lt^(r), f(4)l ) | 0 ) :

u1,;( ffi

(3r)+ipr'+}Af) rip' @" p)

.. -'iP'(34) nf)(P" P) :,:f.',:( :!.J^i!,' r';,,.r';;-"

(30)

405

Nr. 12

10

Our discussion started with the assumption that all the renormalization constants and, of course, all the matrix elements of and /(r) are finite. As this'is a condition on the operators jr(r) the behaviour of, for example, the function II (p') for large values of -p2, and as this function is defined as a sum of matrix elements, it is clear tirat we also have a condition on the matrix elcments themselves, i. e. on the functions A and B defined in p ' ) d .T o g e t and -(p(26)-(29) for large values of -p2,-p'2 more detailed information on this point we consider the expressi

(31)

u'ith

Frr(r-r")

: 0( " - * " ) < t l l i p ( * ) , i r ( * " ) ) | " )

(32)

(cf. I, Eq. (A. 8) and the equation of motion fol Au (r)). Supposing, for simplicity, that I z ) does not contain a photon u'ith energy-momentum vector 1c, we have

t

("ljp(")lz, k)

t'-*r*,(

(') | * ) + i \ dt:"Fp,(*-* ,,)(olnf o lAto) {"")l*>.1

\Vriting

Fr1(r- r") : 0(* -.,:"),#

5

rrt (n) dp"io@-*")

(34)

and using the formula e(r-r")

: *"5 41,i,<*^-*"')

(35)

dpein'-t"){FpQ) + inFri.@))

(36)

we get

iFrtt(r- *") :,#

\

.J

with

F6@) : e \ * r a ( p , p o *i .

(37)

(33)

406

Nr. 12

1l

We further note that from (34) it follo'ws that

F r 1 f u :) v : ( z l i t l z ') ( r ' l iu l z ) - I ' f ( r i p l r ' ) ( z ' l i , . l z ) .( 3 4 ) p,- ,:

t)\- t:

I)'-,+ p

p'-t-D

If every expression appearing in our formalism is finite, the integral in (37) must converge. This means thatr)

I::.It(P'Po): Putting p :

A:

o'

( 3e)

fr rve then get frorn (38) and

( 3e)

1i o l " ' ) l ' ( - 1 ) N ! :+r x i : ' r: 0

lim )

|(

lirn )

| ( 1 1 io l " ' > l ' ( - 1 ) N l " ) + N l :" i 0 . -P P(t)

"

(.i0 a)

po:>a Dt:'):p(.)+ p

and Po )

6

Pl''):

(40 b)

If we first consider a state I z ) rvith no scalar or longitudinal photons, it can be shorvn with the aid of the gauge-invariance of the current operator (cf. I, p. 126. Eq. @Z) there can be verified explicitly rvith the aid of (32) and (33) above) that only states I z') rvith transversal photons will give a non-vanishing contribution to (a0 a) and (40 b), and these contributions are all positive. \Ve thus obtain the result

('ljolz')l' : 0 - l i m' t|+ a

(41)

tP\-'-P\o'

if none of the states lz) and lz' ) contains a scalar or a lorrgitudinal photon. Because of Lorenlz invariance which requires that Eq. ( t) is valid in every coordinate system, it follows, horvever, that (41) must be valid for all kinds of states. If lve make a Lotentz transformation, the "transversal" states in the new coordinate system will in general be a mixture of all kinds of states in the old system. If (41) were not valid also for the scalar and longitudinal states in the old system, it could not hold for the transversal states in the nerv system. 1) The case in which the integrals converge without will be discussed in the Appendix.

the functions vanishing

407 Nr.12

L2 From equation (41) rve conclude that

lim Al+)(p,,p) : o

@2 a)

-(p-P')"+a

lim B[+)(p,, p) :

o

(42b)

__pr_>a

rim B[-r (p,,p) : o. t'2 .>

(a2 c)

a

It is, of course, not immediately clear that the sum over all the must vanish because every term vanishes. terms in (26)-(29) What really follows from (40) is, however, that the sum of all must vanish. If the limits in the absolute values of (zlirlz') performed in such a way that p2 and p'2 are A and B are then one of the p2's are kept fixed kept fixed for ,4 and (p-p')'and for the B's, equations (42) will follow. To summarize the argument so far, we have shown that if we write

( 0 I {f(B), lio@), l(4)l} | o I : we have

(tP

rz 116\ \dpap'

tim F1(1,, P ):

p) (43) r'o'Gr)+ip(r4)Fk(p',

o.

(44)

Pa. ,_rr,o) Fu(p',p) : \:; FkQ

(a5 a)

-(p*p')'>

Introducing

t

a

the notations

ar

and

(45 b)

\(p',il:\+r.(p',pter)

(e is a "vector" with the components er : 0 for 1t * 4 and eo : 1) we find from (44) and the assumption that the integrals in (45) converge that lim F* (p' , p) : -(p-p')'>

(cf. the Appendix). now rvrite

*

With

lim Fo(p',p) :

-(p*p')'>

o

(46)

q

the aid of the notations (a5) we can

408

Nr.12

t3

0 ( r c 3 )a G a ) ( 0 J{ / ( 3 ) , l j o @ ) ,l ( 4 ) l ) | 0 > -r :,r1'"\

fi' ip(r4) apap,.'n'(3r)l F - * @ ,p, l \ tl

(47)

al a)

- n' Fo(p', p) * in (Fo@' p) + F,,(l',1))J . , In quite a similar r,vay it can be sholvn that the second term in (25) can be written in a form analogous b @7) with the aid of a function Gn(p',p) rvhich also has the pr-operties (44) and (46). It thus follows

lim tIo(p', p) :

(48)

o.

-(p-p')'->@

It must be stressed that this property of the function Ax(p,, p) is a consequence of (41) and thus essentially rests on the assumption that all the renormalization constants are finite quantities. It is clear from (24) that the function ,4, transforms as the matrix /p under a Lorentz transformation. The explicit verification of this from (23) is somewhat involvcd but can be carried through with the aid of the identity

0 ( r 3 ) 0 @ a{)f ( 3 ) , l i r @ ) , 1 ( 4 -) 0 l )( * 3 ) 0 ( 3 4 )l i r @ ) , { f ( 3 )f,( 4 ) } l | , . . ) (49) : 0@a )0@3 ){ f ( 4 ),l l r@),r(3 )l } 0(r a )0 G3)lj p( ,) , { f( +) ,/( 3) ) l and the canonical commutators. Eq. (4g) can aiso be usecl to prove the formula

- c-l Ar(- q', q) c : t[(-

t, u')

(50)

which is, hot'ever, also evident from (24) and the charge invariance of the formalism. From the Lorentz invariance it follows that we can r,vrite

Ar(P' ,p)

I

* pt,Ga'a Gr n'* ^)a' lypFa'a t p*gt'el Qyp * nr)e (;t)

:f g' : 0,1 p: 0,1

lvhere the functions .F., G and ,FI are uniquely defined and depending only on pr,p'r, (p-p'), and the signs e(p), e(p,) and e(p-p').From (50) it then follows

409

t4

Nr.12 pes'(- p,p) : Fe'p(- p,, p)

(52 a)

Gaa'(- p, p) : Ha'aG- p, , p).

(52 b)

Utilizing (51) and (52) tlv-eget

q ) : (0 l#) lq,q') R( ( q+ qj) ' ) ie( o l P < ol qi ') A r ,(- s',q )(0 | e to rl (53) + , ^ s ( ( q + u )\ (o r- e ') (0 | ? (0I )s' ) ( o Ip' o) | s) 'where, in vierv of (a8), I i m R ( ( q * q ' ) r ) : l i m s ( ( s * q ' ) z ): 0 .

- @ + q ' F+ r

* ( q + q ' ) '+ o

(54)

The equations (53) and (54) are the desired result of this paragraph.

Completion

of the Proof.

We are nor*' nearly at the end of our discussion. From the assumptions made about II (p') (and its consequences for fr (p'), cf. the Appendix), Eqs. (53) and (54), the limit of Eq. (24) reduces to

liry.( {lIi,l q,q') -: ( oITtor I q,q')jI + n( 0)+ 2'I-r r r -Ll I I

-(q-q')'+q

(55)

:(olffrlq'q')E Oul inequalitl' (a) nov' gives

n Q\)+

>'i
q+q : p

' .'*' ZI
: n(D( pz) (#) Except e2 and

for the possibility mt\ ql

Lt- l

-

"r"

(56)

(T=-;f

of l/ being exactly of | (irra"p"naent 'we have then proved that, if all the renormaliza-

4to Nr. 12

t5

1 I and are finite, the function (1 _ Z) N oo. This is an obvious II(p') cannot approach zero for -p,contradiction and the only remaining possibility is that at least one (and probably all) of the renormalization constants is inIinite. ,1 The case ,A/: ; is rather too special to be considered seriously. tion

constants K,

We can note, hoiever, that N must approach 1 fbr e -> 0 and that one of the integrals in I Eq. (75) will diverge at the lower limit for p --."0, independent of the value of e. The constant N could thus at the utmost be equal to

1

i

for some special combina-

tion (or combinations). p zof e2 and 4.

O, pr is an arbitrarily

small quantity it is hardly possible to ascribe any physical significance to such a solution, even if it does exist. The proof presented here makes no pretence at being satisfactory from a rigorous, mathematical point of vierv. It contains, for example, a large number of interchanges of orders of integrations, limiting processes and so on. From a strictly logical point of view we cannot exclude the possibility that a more singular solution exists lvhere such formal operations are not allowed. It would, hov'ever, be rather hard to understand how the excellent agreement betrveen experimental results and lowest order perturbation theory calculations could be explained on the basis of such a solution.

Appendix. It has been stated and used above that: if

f("): where /(r) and fulfills

,\:H-

(/(0) :

0)

(A.1)

is bounded and continuous for all finite values of c

lf (*+u)-

f (r)l(Mlc I

for all c

(^.2)

4il

l6

Nr.l2

anrl if the integral converges,both /(r) and l(r) u.ill vanish for large values of the argument. This is not strictly true, and in this appendix we uill study that point in.some detail. We begin by proving that if the integral in (A. 1) converges absolutely and if

,

lim logrl/(r)l:0

(A.3)

I->A

it follows that

lim l(c) : 0.

(A.4)

c->t€

o" (Note that the integral \ =+ is nof convergent ancl that the n-tog_ e J vanishing of f (r) is already implicit in (A. B).) To get an upper bound for f(c) v'hen c ) 0 we rvrite

:"5,9,, : (J:"ffn1y,.,, (A r) r(*) ('n

",..,

/f

t'

(? r'

(The limit o is simpler and need not be discussed explicitly.) The absolute value of the first term in (A. b) is obviously less than , ('l'

, f"l'

i\trrolldv(const 1\,ffr.*0. uo ll

(A.6)

Uo

The last term can be treated in a similar way and vields the result

It'--,.

I o?-..'

I Ji,n

I uetp

l\#qoul=\l/(Yrlds-0. "l:l rru_r UIB

(A.7)

,\P,.,1: c)-.r(r-r,,1 \"*,,"*

The remaining term can be written

.Fl'"*y)-,I.5"f r(, l,o.,, |.!:+1,,,-,, (A. {t)

412

Nr.12

t7

In view of (A. 2) and (A. 3), the three terms in (A. 8) vanish separately for large values of c. It thus follows lim l(c) :

g

t>4

q.e.d.

As the functi
f(') :

ds;f Q) : o for yS o

+q -'-

"\. ) 0 "

(A.e)

will follow

(A.10)

r(r):-#"\Pr' where both /(c) Note that

and f(c)

are finite.

r "fl---er)(z-c)

n2'

:

#\

-,o-, I

t

)!z--

utuz a',"* * '' tz' "-iw'lx-iw11t

l-t*rl

(A.1t)

.P :

;\

dwrsi"(u-'):

d(y-c).

It then follows ,r1", ,n" integral

Dt'a,- [!14(')J a, fT-qtu.r< llclc

AU

is divergent, because the second term is convergent in view of (A. 10). This is everything that is needed for the proof. Drn.Xrt.Fyr.f,edd.27,N.l2,

2

413 18

Nr. 12

It is, of course, possible to construct functions f(r) where (A. 10) does not follow from (A. 9). In that case we are not allowed to interchange the order of the integrations in (A. 11); but we have already excluded such cases from our discussion. For simplicity, the statement that the functions ..vanish,' for large values of the variables has been used in the text. If a more careful argument is wanted the phrase "the functions have the property that the integral (.*^ , .

t /(r.) ' \'"'

,U

converges" should be substituted for the word "vanish" places.

in many

The author wishes to express his gratitude to professor Nrrr,s BoHn, Professor C. Morr,rn, and professor T. Gusra.soN for their kind interest. He is also indebted to professor M. Rrusz for an interesting discussion of the problems treated in the appendix. CERN (European Council lor Nuclear Research\ Theoretical Studg Group at the Institute lor Theoretical Phgsics, uniuersitg ol Copenhagen, and D_ep.artm.entol Mechqnics and. Mathematical phgsics, Uniuersitg ol Lund..

Indleveret I.'erdig

til selskabet den 28. februar 1953. fra t['kkeriet den 27. maj lgb3.

414

P o p e r3 4

On the Self-Energy of a Bound Electron* Nonu,llr M. Knoll** AND WrLLrs E. Leuo, Jn. Columbia University, Nm York, New York (Received October 7, 1948) The electromagnetic shift of the energy levels of a bound electron has been calculafed on the basis of the usual formulation of relativistic quantum electrodynamics and positron theory. The theory gives a linite result of 105,1megacycles per second for the shift 2'S1-2'P1 in hydrogen, in close agreement with the non-relativistic calculation by Bethe.

I. INTRODUCTION

1927-1934 formulation of quantum electrodynamics due to Dirac, Heisenberg, Pauli, and PETHET has recently discussedthe anomalous L) fine structure2in hydrogen on the basis of Weisskopf. It will appear from this that the relativistic invariance of the present non-relativistic quantum electrodynamics. His formal theory is to some degree illusory in that all selfresult for the 2254-22Pqdisplacementwas energies diverge logarithmically, so that the difference of two energies such as W(2'S) and AW:W(2'S) -W(z'P\) : (aiRy/3r) log(K /z), (1) W(z'P), although finite, is not necessarily unique. The method we have used has a certain the fine structure con- simplicity in its motivation, however, and the where a:e2/hc-l/137 stant, Ry the Rydberg energy a2mc2f2, and e results are surprisingly plausiblein their mathean average excitation energy of the atom, calcu- matical appearance.In any case,the calculations lated to be 17.8Ry. As Bethe's calculation di- may serve as an illustration of the extent to verged logarithmically, it was necessaryfor him which physical results may be derived from a to introduce a cut-off energy K for the light divergent field theory. quanta which could be emitted and reabsorbed The calculation is incomplete in several well by the atom. On the basis of speculations as to defined respects. It is only made to order a in the improved convergenceof a relativistic calcu- the coupling between the electron and the electrolation which included positron theoretic effects, magnetic field, and to fourth order in the ratio Bethe took K equal to mc2.This led to a value of of the velocity of the atomic electron to the LW/h:1040 megacyclesper second,which was velocity of light. It is expected that these dein very good agreement with the then available ficiencieswill be made up elsewhere.We will observationsof 1000 Mc/sec. make no effort to improve on the low frequency The purpose of this paper is to show that a part of the calculation as done by Bethe, for this relativistic calculation of AIl does,in fact, give is essentially a non-relativistic problem. a convergent answer, and to present the results II. DERIVATIONOF EQUATIONSFOR and some details of a calculation based on the SELF-ENERGY * Work suooorted bv the Sisnal Corps. +* Now Niiional RisearchFellow ;t The Institute foi Advanced Studv. I H. A. Bethe, Phys. Rev. 72,339 (1947). It may be of sme interest to obserue that if the non-relativistic theory is taken seriouslv to such an extent that retardation and tJre recoil enercy- ir the enerqy denominators are retained, the dynamic dli-energy diverges only logarithmically, and tlre S-Pr level shift conilerges,and, in fact, with K determined to be K -2mc2, The iesulting shift of 1134 Mc is in disagreement with the ob*rations. z W. E. Lamb, Jr. and R. C. Retherford, Phys. Rev. 72,

We start from the Hamiltonian for a system of iy' electrons moving in an external static electric potential energy field Tand interacting with the radiation field. After elimination of the longitudinal and scalar photons in the usual vr'ay, we obtain the Hamiltonian

24r(re47).

. A lat6r tentative value reported at the April 1948 Society meeting was 1065+20 Washington Physiel Mc/ec.

388

H:H,^alH*"r*I1i'r,

(2)

u"^:Iax'LNtxhclkl,

(3)

415 389

SELF_ENERGY OF A BOUND l1-*-I

lcs.;.pti0;mc2+V(rc)1,

e)

1.:1

n/

Hn": -L

nr'nr'

e s ,A . (r)ti

t L

"r/,o;.

(5)

Here e is the (negative) charge on the electron; s, A are the'Dirac matrices in the form

":(:

,:(;

;)

-:)

(6)

where c are the usual two-component Pauli matrices. The vector potential of the radiation field is expanded in plane ivaves normalized in the continuous spectrum as 2

A(r):- rtP*1fat | tr:l

(hc/k)rbw.eux X exp(lk.r) f conj., (7)

where bp1+, b11 are the creation and destruction operators for a light quantum of wave vector k and po,larization type I:1, 2. In positron theory, because of the indefiniteness of the number of electrons, it is convenient to use second quantization for the electrons as well as for the light quanta. Then II-u,+

| a x r t r * 1 x ;{ c a . p l B n c 2 l

lz(x) lrl:(x),

(8)

f

H inr+ - | dxq+1x)eo.A(x)t(x) ff

* | | dxdx',1+6),1,9)(e'/lx-x'l) JJ Xt+(x')t(x') :Hr*Hc,

ELECTRON

in state a plus the vacuum electrons" minus "self-energy of the vacuum electrons alone." The highly divergent interaction of the extra electron with the infinite charge density of the vacuum electrons must still be removed. This is done by the process of symmetrization 6 in which the calculation is also made using the equally justified picture that all the electrons in existence are positively charged, so that the observance of a negatively charged electron in state a corresponds to a vacancy in the sea of negative energy states othenvise filled with positively charged particles. Then the results of the two methods of calculation are averaged. Since in the first picture there is one particle present in addition to the vacuum particles, and in the second picture one partiile fewer, the self-term i:j in the electrostatic energy cancels out as does the direct Coulomb interaction between the bound electron and the vacuum electrons, insofar as the latter are not polarized by an external electric field. The result is an avoidance of all singularities worse than logarithmic, and these may be plausibly discarded by renormalization of charge6 and mass. The self-energy to order a consists of the first-order Coulomb self-energy Wc and the second-order electrodynamic self-energy Wo. The former will split naturally into a direct or vacuum polarization term Wp and a static exchange term Ws. The static and dynamic terms trZs and Wo were first calculated by Weisskopf6 in 1934 for the case of a free electron. To calculate Wc we need the expectation values of the operator

(9)

I cr

where {+(r) and r[(r) are, respectively, creation H":- | | dxdx'r!+(x)r!(x) 2J J and destruction operators for an electron. We will expand r!(r) in terms of the eigenfunctions X(e'/ lx-x' l)q+(x/){(x/) (11) u"(x) of the potential field Iz

:t

t(x):I

a"u"(x),

(10)

where the coefficients a, are operators corresponding to the destruction of an electron in state n, etc. We are concerned with the self-energy of a "single" electron bound in some stationary state u"(r) in the potential field Z. In positron theory, this is taken to mean "self-energy of one electron

I ZEA"uda"+opafo'6,

aFt6

where 1rr A.tut:- | | dxdx'u"*(x)u6@) 2J J

X (e'/ I,x-x'l)ur*

(x')u6(r'),

'W. Hei*nberg, Zeits.f. Physik 90,209 (1934). 6V. F. Weisskopf,Zeits. f. Physik 90, 817 (1934). I P. A, M. Dimc, Solvay Congress,1933.

(12)

416 N. for the states represented functionals

M.

KROLL

AND

by the Schrtidinger

O ( 0 1 p ; 0 1 " ; 1 1 ' ,i1D) (, 0 ( p ) l c ) ) , s e c o n d p i c t u r e .

E.

390

JR.

L A o ' >< , ' > a ' ) G+' ) I

D A tr>

(r) (r';

(13a)

A<,,>ato't<,'t,

*D (r')

(13b)

Here r denotes any positive energy state, while a prime indicates the exclusion of the state a occupied by the bound electron. The letters p, o denote any negative energy state, while the indices z, d, P, y, and 6 are to be used for a complete set of states of any energy whatever. In the alternate picture, a positive energy state of a positive particle is represented by (p) and a negative energy state bV (r), (s), The state whose vacancy constitutes our electron is denoted by (a). Consider the expectation value a " + a B a r + a 5 A . p rO , (1"0,,1,).

a8+6

The vacuum term is I

(r) (p)

and the difference , 4 r o l r , t c , t r-" r L A o a t t o > < " > - 2 L A t a > < " t < o < , t

I

(')

G)

* Ort,"r,")(')- 2

: *

F

A <"tt"t<,tt,t,

represents the self-energy of the electron in state a as calculated on the basis of the positive particle picture. The wave functions un and uh, are identical for physical reasons, so that we may now drop the parentheses. Averaging the two results. we obtain

L +A""""+L +A"""":Ws*Wp. The static term

Aoo,o+L A""oo

I A c>t'>,

E lot,rr"rro *E

G) (r)

Using the matrix elementsbfor the destruction and creation operators, we obtain L 4",,"+L

LAMB,

and obtain L

o(1.0",1r),6(0,1,), firstpicture,

O*(1"0,,1,) |

w.

(14)

/v^9rs

rP

r I\ -L \ - ,L .rl 'd d o p

- ILLF \ -L ./L .p .f . f . o .

Ws:L

P r'

rf

=+ ? *; J dxdx'u"*(x)u"(x) X (e,/ lx-x' l)u**(x')u"(x'),(15)

Subtracting ,tr" *ru.unrri,"r-. \L -L .\--d /d p p t L L / rL p\ f-r \r -t / poPf

the self-energy of the electron in state a on the basis of the negative particle picture is \L l-t a/f r a

r

r |r \! - L/

lrdapp

|

-\-/ L

*o"* nF

I

by

"*2EA""oo,

use of a Fourier

representation

for

l/lx-x'l _: lx-x'l

where the upper or lower sign is to be taken for a positive or negative energy state, respectively. The first term represents an exchange term and diverges only logarithmically. The last term is the direct Coulomb energy of the electron in state a interacting with the sea of negative energy electrons and diverges quadratically. As mentioned above, the worst part of this diy'ergence is removed by the process of symmetrization. On the basis of the alternate picture, we therefore calculate the expectation value O*(010"1,,)

and

tf zLapta

' :L

+4""""

(1/zr'z) t dk exp(zk.(x-x'))/k2, r

(16)

may be written as

w s: e,/ao,) f

d!

T + [ an.*rxl

X exp(ik'x)2,(x) d'x'u^*(x') f XexP(- ik' x')u"(x'). (17) The polarization

tetm WP(a)

w,(o):T-o*-:;

a61+a61a61+a1rl.4t.lrplr"lcrl

(a) G) (z) (;)

(D(0aoy01"111",y),

xl

r

r'

ezr

J

axlu"lx)lz

dx'

_I+lz"(x')1'?, Ix-xln

(18)

417 OF

SELF-ENERGY

39T may be written

A

as

BOUND

ELECTRON

gres ls ?

Wptu): I dxlu"(x)l'ett(x),

(1e)

- l df 2k r

wherethe functionl,*, ,. ,n. potentialdue to a

- E")) ( | (/k IH, Io) l' / (8,1-hck

[t

1 l( o kl H , l o )l ' / ( E " * h c k * E " ) )

-5-

charge densityT-e

p ( x ) : ( e / 2 )L + l u " ( x ) | " ' z ,

*2 I ((aIH' I ak)(pkiEtlp)/tuk)f,

(20) which

induced in the vacuum by the external electrostatic 6eld. The energy trZp vanishes for a free electron. The second-order electrodynamic self-energy Wn@) of the electron in state o, according to the electron picture, is given by the difference of the energy Wo(1"0,,1r) for the electron in state o plus the vacuum electrons and the energy Wo(0,1,) of the vacuum electrons alone. The vacuum energy Wo(0,1r) is given by second-order perturbation theory, and involves the virtual emission and re-absorption of a light quantum of wave vector k and polarization type tr. There are two types of terms, representedby the following transition schemes: ,/o+r*k1 I landl \rlk-p/

may

be written

as

fz

- | dk E tE (+ | (nkllJtla)l'/ J

r-r

( l E " l + h c k + E " ) ) + 2Z , ( ( a l I { r l a k ) x(pkllJrld/hck)1. affects only

Symmetrization and gives

the second term,

f2

wo@):-ld.kLL J

x:t"

v (+ | (zkIr{r | l' / (lE"l *hek+ E")) ") f2

/p-p+k\ \ofk+o./

1'

+fdktE Jr-tn

X (+ (alHr lak)(nkllJrlnt/hch). In the caseof the energy WDQ"0,'1), there are added which the transitions additional some The last term can be written as electron can make, and some of the previously presence prevented by the are allowed transitions f2 (dk/h,)u L of the atomic electron in state o. One has then | J r:rn the following types of transitions: ,zo+r'tk1

I

\r, |-k-p/

t,|

1a+r*k\ \r1-k-a/

L

/o+o*k1 /n*n-|k1 I t,| L \a*k-a/ \o*k*o,/

r1a+ai-k\ l \Pfk+P,'/ The difference of the two corresponding ener? See reference a, Eq. (40). 8 E. A. Uehlins, Phys. Rev' 48, 55 (1935), W. Pauli and M. Rose. Phvs. Rev. 49,462 (1936), and V' F. Weisskopf, Ksl. Danske Vid. Sels. Math.-Fys' Medd 14, No. 6 (1936). i R. Serber, Phys. Rev. 48, 49 (1935).

+

ff

J J

(21)

dxdx'u"*(r)o'errz.(x)

X exp(ik' (x - x'))2"*(x')a'erran(x'), 'rvhichwill be zero if the polarization currentT-e j(x'):el

+u"*(r')uu"(x')

(22)

is zero, In the absenceof an external vector potential, this current is in fact zero' so that the last term in Eq. (21) will henceforthbe dropped. It should be noted that two physically different k-spacesare involved in the expressionsEq. (17) and Eq. (21) for Ws and Wo'

418 N,

M.

KROLL

AN D W.

III. COMPUTATION OF THE SELF-ENERGIES We turn now to the evaluation of the expressions trZs, Wo, Wp for the self-energy. We shall pay particular attention to the static and dynamic terms Ws and Wo, as the polarization (Uehling) term Wp is directly related to the polarization charge density which has been cdmputed by others.T-e In the calculations to follow, relativistic units will be used throughout, in which h, m, and c are taken equal to unity. Our main interest, of course, is in the case of an electron moving in a Coulomb field for which V : - e2/r. The integrals like

E.

LAMB,

392

JR.

The sums over n can be performed, at least formally, by making use of the completeness of the solutions of the Dirac equation. Thus I,I/s can be written in the form ff W s: (e2/4r2) | (dk/h\ J

| dxu"* (x)

| NJ

xexp(ik x)(H/ | u |) u"1x7 dx'u (x') "* J Xexp(-

ik.x')u"(x'),

(23)

where H:a.p*AlV

Q4)

is the Hamiltonian of the unperturbed electronic motion, and I 111 the absolute value of the Hamil(nklr.ll a) : - (ie/ zrntl d.xu^*(x)a. ey1 tonian, by which we mean an operator having I the same eigenstates and spectrum as the HamilXexp(-zt.x)2"(x) tonian, except that its eigenvalues are taken to occur in the theory of the relativistic photo- be positive. It can most conveniently be comelectric effect and have been studied extensively puted by representing it as +(11,);. The equivaby Hall.10Becauseof their complexity, it seems lence of Eqs. (17) and (23) follows from the fact is f 1 when operating on a positive hardly likely that we could perform the neces- that H/lHl energy state and - 1 when operating on a negasary further operations on them to evaluate such tive energy state. Using the completeness of the expressionsas Wp. Even more must such a direct u"(x), we now find attack be ruled out for the case of an electron

moving in a general potential field I/(x), for rf which the relativistic eigenfunctionsu^(x) are W 5 : (e2/ 4r2) t| , (dk,/ hr) | dxu"* (x) not known. The only remaining method of apxexp(i.k. x) (H / | -r1l) exp( -lk . x)u"(x), (25) proach seems to be to make an expansion of some kind. We observethat if the electron is free, so that the problem of computing IZs is reduced the evaluation of the sums is a comparatively to that of finding the expectation value of the simple matter. Thus, if u"(x) is a plane wave of operator momentum p, then

r.

I dxu"*u' e exp( ik. x)zr,

(e2/4r2)

| (dk/k'z)

Xexp(-ik.x)(H/ lHl) exp(f zt.x) is different from zero only if the momentum of the state a is kfp, and there are only four such for the state o. We first note that for anv.oolvstates.In the caseof a "weakly" bound electron, nomial function f(p, V) i.e.,for l@, V) exp(fik.x)2,(x)

h>>(al lpl la),

one might expect that the matrix element above would have an appreciable value only when lE" I is of the order Ep:*(1+p)1. We take advantage of this fact in the method of calculation used. r0 H. Hall, Rev. Mod. Phys. 8, 358 (1936).

: exp(f ik. x)/(k-fp, V )u"(x). This theorem may then be usedfor any function J6, V) such as H/IHI, for which a seriesexpansion in p and tr/ is valid. We therefore write (H/|Hl)

exp(tu.x)u"(x) : exp(tu. x) (H / | H | )y+pu "(x),

419 393

SELF_ENERGY

OF

A

where the notation

BOUND

ELECTRON

where

(

Eh: (r+k')\' no: lHl -E*,

)r_n

means that the operator p is to be replaced by kfp everywhereit appearswithin the brackets. One then has

Do+:BrSP-t, w":E"-1.

(31) (s2) (33) (34)

is, of course, also necessaryto evaluate A* ws: @'/a,,\("ll #r, t"rl,*1,).
t lr

wo: -Q,/4r')lal I tav.rDE o.eo^ ^:r \ lr'

r/ H x{l-*1

\

\ : IH I pay- E1,: (Ey, | 6y)r- E* :i6t-t(6v" / E) { }6(6v3 / E*') +,'' " 'lvhere 6r : 2k. p-|p'* 2 V (c.kl- s. p -f 9) le'aVlV2.

/ l/(lHl+h-E')

(3s)

(36)

t\lr1l //

All expansions indicated are to be carried to sufficiently high order so as to include all terms which are effectively of the fourth or lower order in vfc. The operator p is obviously of 6rst order The evaluation of the terms trZs and JZo thus in zt/c, V is of second order becauseof the virial theorem, while k, 0 are of zeroth order. Since hinges on the expressionof the operators e.,2:sr2:su2:1, a must be regardedas a zeto' (r/lijl)*p and (r/ (lHl lk+E"))v+p order quantity until the expansionhas beenfully out. worked whose expectation values operators of in terms The expansionof the operatorsin the manner can be readily obtained. indicated and the summation over the polatizaTurning now to this task, we write tion direction I:1, 2 is a straightforward but ( lfll)r+r: ((/1'zf)rap: ((("'p*0* 7)')i)t+o lengthy matter. This being completed, one is : ((1*p'*"'p V I Va'Pl2tsV* tr/'z),)11o left with a sum of expectationvalues of various : (l + k' +2k -p *p'z-l 2 V(a' k|_o' P+B) operators (28) la'rV*V2)\, 1 , 0 , a . p , 0 o ' p ,p ' , A p ' ,V , A V , where rr denotes an operator p which operates t' nV Xp, a' nV, 8s' *'V, d2V, Vp2,p2V' nV'p' only on the quantity immediately following it gn,Bpn,V2,FV2, Va.p, tsVc'9, ( e . g . c, ' p 7 : V t ' 9 * e ' t V ) . a . p p zw , : E - 1: c . p * F * V - 1 , w 2 ,B w , It is clear that the ratio of (1111)r+pto combination of some fifty Q+n'1t approachesunity as & becomeslarge, each multiplied by a which correspondsto'our previous statement elementary integrals over k. The result of this regarding the relationship between the magni- calculation is given in Eq. (73) below. Before tude of the matrix elements and the energy of coming to it, we shall first discuss briefly the validity of the expansion used and the form in the state z. Thus we exPand which the sdlf-energyis expressed' (r/l/Jl)t+n and (t/ lHl lk+E")t+p Assuming for the moment that p and tr/ can be regarded as numbers less than unity (in as follows: relativistic units), then the expansionsof

r,r .(#,-') / tH|+k+E") |"*;."-^1,)

(r/ lHl)*r:r/(E**^)

- - L / E * - A * / E n ' * L f/ E r x - ' ' ' , ( 2 9 )

(r/(lHl +ft+8,))r+p :l/(Dk++Lk+w"):1/Do* -(a*+w")/(D.+)',*' ' ',

(r/lEl)t+o

and (r/(lHl+k+E"))k+,'

(if carried far enough) are valid for all values of ft, sinceEr approachesunity and D,'- approaches (30) two as & goes to zero. On the other hand, Dl+

420 N.

M.

KROLL

AND

approaches zero in this limit, so that one should examine the low ft behavior for this case. It turns out that that part of A*-w"' which does not approach zero as & approaches zero is of the order (u/c)2.Therefore, in the term involving Df we shall carry our integrals down wave number &r, only to some intermediate which, for convenience, v/e take to be of order aB. This term must then be given a separate treatment for the low & region O
(32)

".(:),

V ) ) " . p * Z * 1- E ) 4 :

@ : ( r/ ( 1 + E - V ) ) " . p 0 ,

E.

LAMB,

394

JR.

ord,er (a/c)2 is just a non-relativistic two-component Pauli-Schrcidinger wave function. One can'then see, for example, that

\ a l p '- 0 p ' l a :) 2 | d x \ (1/ ( 1+ E - V )) I

Xo' p6)ap'z(r / (1+-E - V) )". pQ" tf

__ | dx6"*p,e" 2J

-i@lp"la),

(40)

since l-E and V are of order (v/c)2, and f dx+"pn6" and (a lpa lo) differ only by a quantity of order (a/c)6. One can therefore simplify the final result Eq. (73) by expressing all operators in terms of certain arbitrarily we have taken to be

chosen ones which

0 , u ' p , V , d 2 V ,B a ' n V , Vp2,pa, and V2. Our reduction is obtained by using the following relations betrveen expectation values

1+Bf |c.p-$pa{f,pa.dV,

(41\

p'-,r.p**pn*ttsa.zrV,

( a?\

. p p z+ c . p f

|Pc.nI/,

(43)

BV+V-lVp'z-i]r.nV,

(44\

0c'P-0,

//.c\

p2V+Vp2,

(46)

dV'p+-|22V,

(47)

s.rV+0,

148)

'n.-VXp-!a."1't!r2V, \/s.pnVp2{,ils.*V, B V a . p +- l B a . n I ; , a'PP'+P',

(49)

(s0) (s1) (s2)

0pn*pn,

where{ and o are two-componentv,/avefunctions satisfying ( a ' p ( r /( r - l E -

rv.

s,

133;

(3e)

so that for a positive energy state o is of order v/c with respect to @, rvhich apart from terms of

rq4)

BV2+V2, w : H - t_+tc.p+ 7++p'-+Ba. rV,

(55)

- i Vp' - *9a. xV, Bza+lu. p I V - L"pn

(56)

ze,r-lpr* Vp"+Vr.

(57)

From these relations one finds that the total

421 SELF-ENERGY

395

OF

A

BOUN D

ELECTRON

contribution of the static and dynamic terms of the self-energy, apart from the low & contribution of the term involving D1+, is

is the induced charge density calculated by various authors. To the order required, this is found to be

(ws* wo)': ("/d (.1(:f,- o,ru;++)s

pqx):(e/6n,)v,Vl

/f\ i+ | G,dk/Ek') | \.-rt l(e/60r')vaV,

* * c . p * 3 c . p f t ; ! f , P an. V - ( $ l o g ( 1 / E )

(63)

from which one finds

- ! tog2 | 1r/ 72)",v1"). (s8)

lf\ - (2e'z/3r)l l*

Wp(a): In order to compute the low i contribution of the term involving Dr+, it is convenient to take advantage of the essentially non-relativistic nature of this region and to make use of the previously discussed large and small component reduction Eq. (37). One then readily finds that the resultant expression has, to the order required, just the form of the non-relativistic self-energy, so that Bethe'st calculaticlr may be used up to the frequency k;. The contribution is

| &'dh/Ef)

|

\J/

x (al v' la) - 1e/ ts"11alv,v la).

(64)

The prime appearing on tr/' is used to indicate that the gaugeof Z'has beendeterminedby the fact that it arises from an expressionof the form F

-1/4n I dx'vzV(x')/lx-x'1. I

(65)

We shouldlike to point out that the expression for p(x') can be readily and neatly calculatedby - g(losk;/z)n'zV : (a/r)(al - frp'&; la). (59) methods very similar to those used above in the caseof the static and dynamic terms in the selfAdding the t\4'o, and observing that (alp'?lo) is energy. To show this we first evaluate (au.a)*
the same as (alc.plo) to the order required, we find for the total contribution of Ws and Wo

p(x',x") : - (e/Z) | : * (e/2)

us++)a wstwo: @/d ("1(;{,- row *(".p/6)*i,6a.rV-

u"*(x')(Hi lHt)u"(x")

|

44

:-(e/2) I t t @/lgl),,

/l | * log\E

- ! tog2 rrl /7r)"r1"),

*u"* (x')u"(x")

n

r:l

Y:l

\,u^r*(x')u*,(x"),

(66)

where the uny, F: l, 2, 3, 4 are the components of are to be taken and all operators in H/lHl with.respect to x". Making use of the completeness relation

(60) un,

and we note that the result is independentof the joining frequencyfr;. As previously mentioned,the direct Coulomb energy term can be expressed in terms of a polarization charge as follows:

E u"u*?')u",(x"):6u,6(r'-x"),

(67)

we obtain

p(r',x"): - (e/2)L g/ lHl)F6(x'-*")

Wp(a):e I dxla"(x) 'z

p

,

: - ( e / 2 ) ( S p u t H / IH I ) " , , 6 ( x ' - x " ) .

(68)

X I d x ' p ( x ' ) / l x - x ' 1 , ( 6 1 ) To evaluate this, we Fourier-analyze the deltafunction

where p(x'): (e/2) t + l u " ( x ' ) P

I

( 6 2 ) 6(x'-r" 1: Q /8r3) | dk exp(ik. (r" - x')), (69)

422 N.

M

KROLL

AND

TesI-p I.

2tsv,

Ouantity

VY iPs.VV

-1t)

State 2tPVt

22P./,

0 r/6

0 -r/12

and find

p(x',x"):(l/8r3)

f

J

dk exp(-lk'x')

X (Spurfl/ lfll) exp(ik.x"), f

: (t /8r3) dk exp(lk.(r"-x,)) J X (Spur//l1/l )r+n.1(x"), (70) when1(r") is a constantequalto one.Since

p(x'):p(x,,x,),

E.

LAMB,

396

JR

where we have made use of the fact that a: -iY . We obserlre first of all that the worst divergence is logarithmic, and that the expression is invariant to the gauge of I/. We next see that the difference of the selfenergies of the states 225; and 2,P1 of the hydrogen atom does converge, since the expectation values of B, V', (and, therefore, c.p) are, respectively, equal for the two states. (These statements follow from the observation that neither a small change of charge nor mass of the electron will remove the degeneracy.) In order to calculate the numerical difference of W(a) for the two states, we need the values of the expectation values of the remaining operators in Eq. (73). These are given in Table Itr in units of a2Ry. The energy difference is then A,W : (a'Ry /3r)Uog(r

(7r)

/e) -log2

+(23/24)-+7'(74)

which, using Bethe's revised valuesrz for the con-

we obtain, finally,

Yf,,E9#/i;],ii:i'l#ii;:;,i,".-,',J

p(x,):_ (e/,6rs) X f akg ' p' u r 1 l l . 1 7 l ) g 1 r . 1 ( r , ) . J

W.

(72)

The expression can now be readily reduced to the form Eq. (63) bV expanding

and thus differs from the original guess by only a small amount. It should be admitted, however, that one cannot regard this energy difference as uniquely determined, since one is taking the difference of two infinite quantities. With respect to the determination of the abso-

(Spurf/|f1|)tlolutevalueoftheself-energyforastate,itis convenient to attemDt a phvsical interoretation in the manner used for the evaluation of the ofthetermsinvolvei. In thiscontext,ltshould static and dynamic terms. be observed that even if the coefficients of the Z'

rv.rNrERpRErArroN oFRESUr.rs il1'tt'JTff:*T;?.'i-":f::l'::$::#1T: fact that

The total expression for the self-energy is

w(a)

=@/")("1(: I aurcsnt)u -(? awnta,,r++)2, [ * @ . p / 6 ) - ( i / 4 ) 0 s . vv

+ (* rosfr/O- ] log2

rsl v + ( r t/ 72 )-rrU )v, l a ),

( / J,

they would manifest

themselves as a

u It should be mentioned here that the exoected value of V, tr/really diverges for the S stares of the Coulomb field, since it then is equal to the square of the absolute value of the wave functioi at the origiri. Since, however, our evaluation is being carried only to order (u/c)n, one should be able to us the spatial dependence of the Schr
423 397

SELF-ENERGY

OF A BOUND ELECTRON

modification in the real charge and mass of the our self-energy expressionmust be found to give electron, and thus be included in the observed a covariant expressionfor the free electron. We charge and mass. We shall assume that these proceed by subtracting some free electron operterms have been so included in the observed ator from the operators contained in our selfcharge and mass and drop them from the self- energy expressionsuch that the self-energy of a energy expression.The term in dpeu'vlzis just free electron is zero, thereby regarding the total of the form of the interaction of a Pauli-type self-energy as contained in the observed mass. Such a procedureis, of course,not unique: we intrinsic magnetic moment with a static potenmake the simplest subtraction, examine the shall as implying interpreted be thus tial I/ and can an additional electronic magnetic moment of resultant expression, and then investigate the a/2r-Bohr magentons, while the term V27 im- nature of the lack of uniqueness.Thus if one plies a correction to the external potential, or, simply drops the (a/&r)o'p term from the selfmore specifically, an additional short-range inter- energy,one obtains action between the electron and a point charge. 2e) : The term in a 'p is not subject to a direct physical W @) (d/ r) (al iA"' v lry4 + (+ log(1/ interpretation, and, in fact, must be regardedas (7s) + ( 1 1/ 7 2 -t ( r / r s ) having no physical significance. Thus if one ap) v v l a ) ' plies the self-energy expression (73) (with the This expression can be interpreted as arising B- and V' terms omitted as explained above) to in the magnetic moment of the a free electron of momentum p, only the term from an increase 2r'Bohr magnetons and an addiaf of. electron yielding for self-energy the in c.p contributes, given by (a/.6tr)lp'z/(l-lp')il. Now if the electron is to be tional interaction potential / \ relativistic connection particle, the regarded as a -(r/1s) tvv. (76) 6v.ff: - I + los(l/2€)+(1r/72) between the momentum and energy of a particle / \must be retained, so that the self-energy should contribute 68 and 983 Mc/sec., respechave the momentum dependence appropriate These to the level shift. tively, -l/(l*p')' isrs that to a mass correction, with our subtraction prescripIn accordance corresponding to the term in p already subhowever, add any linear combinawe could, tion non-covariant term of the tracted. The presence operators of order up to (a/6tr)lp'/(l-lnz;+], which is reminiscentof the tion of free electron (u/c)a whose expectation value is zero for the free be can self-energy, stress terms in the classical electron. There are seven such operators,\6 uiz., traced to the fact that the total self-energy is l, P, p', 9p', a'p, p', Bpn. The condition that a of the in case the infinite. and can be avoided linear combination gives zero to order constitutes free electron by paying proper attention to the constraints, so that there should be four three k-spaces.ra in the various domains of integration independent combinations giving zero linearly That is, in order to keep the total self-energy free electron. A possible choice for these for the over a finite integrate finite it is necessary to is the following: region of the light quantum space and the elec(76a) a integrates over If one st":l - 9-tp'-c'P*6P'*8Pa, tron momentum space. region which would be spherical for an electron (76b) %:pr-Bp,-*Pa, at rest, a covariant result is obtained. One can(76c) sl":pa-Fpa, not, however, apply this prescription to a bound electron, so that some other means of modifying -0P2. (76d) Oa: a.p 13The enersv of a particle of mass z and morentum p is (mzlpz)t. If z,is irodified by a quantity 62, then the energy to nrst oroer rn ou rs (m2! P2)r! 6m / (mz! Pz)|, as is appropriate for and the correction term with n:1, the electron, is of the form given. nA. p"is. Verh. d. K. Ned. Akad. v. Wet, Section 1, 19, No. 1 (1947).

The expectation values of phe above combinations are all zero for the lfree electron. Their effect upon the self-etergy of a bound electron depends upon their expectation values for a -lfooFtor"

of odd order in t/c have been ignored, as there ire all rero for the bound electron'

424 N. M. KROLL.{ND

\\:. E.

L.{MB,

lR.

398

bound electron. Those for go, Oa,and O" are zero, so that their subtraction would have no physical On the other hand, consequences.

electron has been made by Kusch and Foley,l0 who obtain a value in good agreement with the value a/2zr-Bohr magnetons theoretically computed by Schwinger.lT If we adopt this experi(77) @laala):-i(al\s'vVla)/2, mental and theoretical result, the 2254-22P1 which is precisely the form of interaction of a separation Liecomes uniquely determined to be magnetic moment with a static potential Z. just the value 1051 Mc/sec. obtained above by a Thus, the lack of uniquenessof the subtraction direct subtraction (74) of thd self-energies for prescription is just such as to make the magnetic the two states.

moment correction indeterminate, while-the correction to the potential is left uniquely determined. Now a purely magnetic measurement of the correction to the magnetic moment of the

16P. Kusch and H. M. Foley, Phys. Rev. 74, 250 (1948), also J. E. Nafe and E. B. Nelson, Phys. Rev. 73,718 (1948). ItJ. Schwinger, Phys. Rev. 73,416 (1948) and Pocono Conference. 1948.