This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
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 [ 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 - 2 1 " ! - , ( t ' t "" 'o')u*(t.to) \aSo n-*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(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 [
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 (Y ac.- l)
:2w(n')(Vac.) for m:0. (sz) r*:r'"t"ll/, ttt4 7)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 frfinite. 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
wobei n der Einheitsvektor in der Richtung A ist. Dieses ergibt naph g entwickelt: .(1) l.:
-#\,,w + 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>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
],fmc2fh, 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, an'n' ,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_. - -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(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
( 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.orlzt3
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+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 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 fiblr>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(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"(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.
J.
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
S
MATRIX
IN
QUANTUM
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
.
J
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
.'
MATRIX
IN
ELECTRODYNAMICS
QUANTUM
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
I/JJ
S
MATRIX
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
R.
P.
FEYNMAN
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 -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 ?[ ,
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- xots), 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 .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 ' '
" - 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
.1
\Vriting
Fr1(r- r") : 0(* -.,:"),#
5
rrt (n) dp"io@-*")
(34)
and using the formula e(r-r")
: *"5 41,i, 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 4
q.e.d.
As the functi < " > - 2 L A t a > < " t < o < , t
I
(')
G)
* Ort,"r,")(')- 2
: *
F
A
