advanced book classics

PARTICLES, SOURCES, AND

FIELDS

Volume III

ADVANCED BOOK CLASSICS David Pines, Series Editor

Anderson, P.W., Bask Notions of Condensed Matter Physics Bethe H. and Jackiw, R., Inr Quantum Mechanics, Third Edition

Feynman, R., Photon-Hadron Interactions

Feynmm, R., Quantum Ekctrodynamicr

Feynman, R., S ~ t i ~ t i c d Mechnnics Feynman, R., The Theory of Fundamntnl Processes

Negele, 1. W. and Orland, H., Quanmm Manyeparrick S y s ~ m s

Nozieres, R, Thew of Interacting Fermi System

tical Field Theory

Pines, D., The Many-Body Problem Quigg, C., Gauge T f i e ~ e s of the Strong, Weak, a d Ekctromagnetic Interactions

Schwinger, J . , Particles, Sources, a d Fields, Volume I Schwinger, J., Parn'cks, Sou~ces , and Fields, Volume II Schw inger, J . , Particles, Sources, and Fields, Volume III

ULIAN SCHWINGER late, University of California at Los Angeles

PERSEZIS BOOKS Reding, Massachusetts

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Editor's Foreword

Perseus Books's Frontiers in Physics series has, since 1961, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics-without having to devote the time and energy required to prepare a formal review or mono- graph. Indeed, throughout its nearly forty-year existence, the series has emphasized informality in both style and content, as well as pedagogical clarity. Over time, it was expected that these infomal accounts would be replaced by more formal counterparts-textbooks or monographs--as the cutting-edge topics they treated gradually became integrated into the body of physics knowledge and reader interest dwindled. However, this has not proven to be the case for a number of the volumes in the series: Many works have remained in print on an on-demand basis, while others have such intrinsic value that the physics community has urged us to extend their life span.

The Advanced Book Classics series has been designed to meet this demand. It will keep in print those volumes in Fronriers in Physics or its sister series, L e c m Notes and Suppkmenu in Physics, that continue to provide a unique account of a topic of lasting interest. And through a sizable printing, these classics will be made available at a comparatively modest c a t TO the reader,

n e s e fecturt3 notes by fulian Schwinger, one of the most distinguished theoretical physicists of h is cencuv, provide boek beginning paduate students and experienced researchers with an invaluable introduction to the author's perspective on quantum electrodynamics and high-energy particle physics. Based on lectures delivered during the period 1966 to 1973, in which Schwinger developed a point of view (the physical source concept) and a technique that emphasized the unity of particle physics, electrodynamics, gravita- tional theory, and mmy-body theory, the notes serve as both a textbook on source theory and an infomal historical record of the author's approach to many of the central problems in physics. I am most pieased that Adwanced Bwk Clarsics will make these volumes readily accessible to a new generation of readers.

David Pines Aspen, Colorado

July 1998

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Vita

Jdim Schwinger U ~ v e r ~ t y niwzsity of C&fo&% md Profeswr of Physia at the U~versity S Los hgeles since: 19'72, was born in New York City on Febmw 12, 1918. hofesar his PhD. in physics from &lmbb Udversity in 1939. )3te has o r q dmtarates ixll from four institutions: Putrdue w a r d Univiefsity (1962), Brmdeis Uivasity (19731, and Custaws Adolphus College (1975). In ad&tion to teacbg at the U~versity of Gali fo~a, Professor schvviiager has tau@t at firdue University (1941-431, suld at Hwmd U~versity (2945-72). Dr.

wa a Re~mch Aswiatr: at the U~versity af G&EoMh Berkeley, m& ai Saff Member of the Mmsa~husf=tts Institute of TwboXot%y bdiatian Labaratav. fn IS165 hafessor %hknger b c m e a ca-resigient ( ~ t h 1Richmd

in Physics for work in qumtm A Na~onaf Rese FeUow (1939-40) and&& (1970), Profemr W the rsipient of the 6, L. Maym Nature of Li&t Award (1949); the First &stein P&e Award (1951); a J. W. Gbbs Honarw h t w e r af the h e n e m M a t h m a ~ d Society (19a); the Nationd Medd of % i e m Awmd for Physics (1969; ai Humboidt h a d (1981); the P r e ~ o Citta di Casti&one de Sieifia (1986); the Manie A. Ferst S i p a Xi Award (1986); aed the h e ~ c a n Academy of Ackevement Awwd (1987).

Isaac- Newton used his newly invented method of fluxious (the calculus) to compare the implications of the inverse square law of gravitation with Kepler's empirical laws of planetary motion. Yet, when the time came to write the Principia, he resorted entirely to geometrical demonstrations. Should we conclude that calculus is supeffluous?

Saurce theory--& which the conmpt of rmomalization is foxeip-and renormalized operator field theory have both been found to yield the same answers to electrodynamic problems (which disappoints some people who would prefer that souree t h e o ~ produce new-and wrong-mswers), Should we conclude that source theory is thus superfluous?

Both questions merit the same response: the simpler, mare intuitive fama- tion, is preferable..

This edition of ParticIes, Sources, and Fzei& is more extensive than the original two volumes of 1970 and 1973. It now contains four additional sections that finish the chapter entitled, "Eleetrodynadcs 11.'' These se~tions were written in 1973, but remained in partially t m d fom for fifteen yews. I m ag& indebred ta Mr. Ronald Bob, who managd to d&pha my f a d i ~ ~fibbles and completed the typescript. Particular attention should be directed to Section 5-9, where, in a context mmewhat luger than el~trodyrrt , a dimgwment

and operator fidd theory flndly does appear. their first q u a i n t a c e with source theory should wnsult

the Appendix in Volume I. This Appendix contains suggestions for threading one's sway through the soxnetimm dutterd pages.

h 8 Angeles, Glvnr ia April 1988

Contents

Itvo-Particle Pnterac?im, N~n-relariui$tic Disiseussion Two-Parficle Ijzter~ctiom. Xielatiuistic m e ~ r y f M-Paaicle I~ferac?ions: Relativistic me~ry 11 Phatm Propagation Fmcfion Il P ~ 8 i t r ~ ~ i m . Muonim Str~ng Mapetie Fie[& Eleetr~n Magnetic: M~ment Photon Propagation Fmction 111 Photon Decay of the P~QH. A Confrontation

I f p can "&in 'em, beat "em*

Par 9 Sources,

For some time now we have been occupied with the implications of two-particlie exchange, This leaves several important areas unexplored, however, There is the obvious question of extending the procedures to more elaborate multiparticle exchange mechanisms. And the practical applications of the results have been essentially limited to the idealization of a particle moving in a prescribed field, avoiding the relativistic two-body problem, This chapter is concerned with both types of investigations. But, in order to prevent too heavy a concentration of the often ponderous calculations involved in the higher order multiparticle exchange processes, such discussions will be intersprsed among the two-body considerations, somewhat as dictated by the relevance to comparison with experiment.

5-f TWO-PARTICLE INTERACTIONSc NON-RELATIVISTIC IBISCUSSiON

It is helpful to set the stage for two-particlie relativistic theory by first assuming the simpler nonrelativistic context. Let us consider two kinds af particles, labeled I, and 2 (no confusion with causal labels should occur here). The vacuum amplitude that describes them under conditions of' non-interaction is

(dr) dt (dr') dt' qq*(rt)G(r - r', t - t"q(rVt") 1,

(dr ) dt (dr ') at' g* (rt ) G(r -- r', t - if)? (r"')

To avoid writing out all these space-time coordinates, we shall often convey such an expression by the notation indicated in

The particular term in the expansion of exp[z'W] that represents two particles, one of each type, is

Chap. 5

whkh displays the propqation function of the noninteracting two-gafticle system as the prduct of the in&~duati propagation functions:

Ilftibing the indi~duaf Sfferentid equations [cf. (4-1 l .Q)], which we shail h t e as

one deriva the differential equation for the two-particle propagation function,

The expgcit etxprwsion (61 .4 ) can be recovered from the differential equation by adjoining the retarded b u n d a q conditions that are exhibited in Eq. (4-1 1 .S),

A related vemion of the dlfferentiat equation emerges on introducing the in&~daaJ pbicle fiel&,

n e n , the two-pa~icle field, def i n d under noninteract ion circumst ants by

The basic characte~stic of a mrelativistic theory is the maninglulness of absotute simdtaneity. According-ly, it is naturd to consider the spcialization of the* =&&the fields and propagation functions to the equal time situation. The expficit prop~ation hxnction construction of Eq. (4-1 1.3) can be preented as

&(F - F" t - g') = -- iq(t - t') expf- iT(t - t')] B(r -. F'), (5-1.11)

and thus

-. - q(l -- $7 expf-- i(TI + G)@ - l')] &frl - rf1) 4% - F'%). (&I.f2)

The fmction defind by

Two-particle f neermf onr. Non-rcilatlvistic d fscuasion 3

then obeys

which is a more familiar two-particle generalization of the one-pa&icle Green's function equation of Eq, (&l,(i),

To examine further the re1 at ion between the two t y p s af propagation functions, it is convenient ta adopt a matrix notation with regard to spatial variables, while time variables are made explicit. Thus we present the Eqs, (5-1 '4 ) and (5-1.12, 13) as

and

G1+%(ts i f ) - ig(t - t') expf- i(T1 + I",)(t --- g ' ) ] , (5-1 ,l@)

where the latter notation, Gx,zl emphasizes that tbe equal time version regards particles 1 and 2 as parts of a single system. Suppose, for example, that t, > t2 and t$ > >'%, Then

x expf- iT2(it\ - kf2)"j

Z'G~(tlit t~)G1+2(t%t t'lj)G2(tf1a t ' ~ ) r

which is an example of the general relation (assuming t , => t",)

where, on the right side, the first single particle Green's function refers ta the padiele with the larger (4,) of the time variables t,, t2, while the ather single particle lunction is associated with the particle hwing the lesser (t',) of the time vdues t$l, it2. This is made exeieit. in the constmctions (remember that these are retarded functions)

The physical picture of the multi-time propagation function that (6-1.18) suppEes is quite simple, At time t> ,, one of the particles is created. This single-particle

4 Electrodynamics I1 Chap. 5

situation lasts until time t',, when the other particle is emitted. The two-particle configuration endures until time t,, when one of the particles is detected. And eventually, at time t,, the final particle is also detected. Let us also note that Eq. (5-1.18) includes the original definition of (6-1.13), or

Cl+e(t, t') iG12(tt; t't'), (5-1.20)

since

The equal time two-particle field is correspondingly defined by

On using the following special example of (5-1.18),

this becomes

which, with the definition

reads

The equivalent field differential equation is

(E - TI - T2)$1+2(t)narfnt. 7i+e(t)# (5-1.27)

and this identifies ~ ~ + ~ ( t ) as an equal time two-particle source. The coordinate indices are made explicit in writing the latter equation as

which also illustrates the practice of omitting subscripts when the necessary information is amply evident in the arguments of the functions. With the analogous source definition

S1 Two-particle Entarzctf ono. Non-ralativtotlc dbscurrrtan S

one can prewnt the two-particle vacuum amplitude of (5-1.3) in the fom applicable to a single system:

One should note, however, that and q:12(t) are a complex conjugate pair of functions only when the earlier acting emission source of qlcz and the later acting detection source of q:+p are extended sources, dealing with virtual, rather than real particles. This is reasanable, for these are the conditions for the degree of time locality that permits an effective description by just one time variable. To s@e it in detail, let us write out the sources, using Eq. (5-1 -19) :

where each propagation function can be represented as

m dE exp[-- iE(t -- t')] G(t, t" -, 2n E + ia - T '

Tfie complex conjugate af the first structure in (&1.31) does regroduce the form of the second one, except that, instead of the typical function

one finds the transpsd, complex conjugate, or adjoint function

But the two are equivalent, if the sim of i~ is indevant, that is, if the relation E - T = 0, the condition Ior real parlicle propqatim, i s effectively not satisfied in Grtue of the nature of the sources.

Now let two particles, one of each type, approaGh each other and scatter, in a prirnitivrt interaction act, The nonrelativistic concept of a primitive interaction is an instantaneous proces, which is not localized spatially, in general. The scatlered particles can be described by m effective two-partiele wurce, which is measured by the serength of the excitation--the product of the two in&vidual fields--and by a function F', intrinsic to the mechanism, Thus, we write

6 El-rdponrtcs C l Chap. 5

where, hsides the explicitly stated traslationally invariant dependence upon spatial coordinates, the function V may involve momenta, spins, and other paflticle attributes. The field prduced by the combination of the emission sources and the effective source of (&l '35) is, according to Eq. (&1.9),

where

An impodant proprty of the interaction function V can be inferred from the stnrcture of the addition to W that describes the exchange af a pair of partictes htween the effectiw emission source (6-1 *35) and detection saurces. On, referring to Eq. (&l.$), one sees that the addition, dW, can be obtained as

where 4+(12j reprewnts the interaction-induced field of (k1.36). Thus,

according to the constmctiarrs

The more explicit fom of (6-1.39) is

S u p p e ~ f ? consider circumstances in which the soufces are incapable af erni(ting red pa~icles (E - 2" + 0). Then; [cf. Eq. (4-1 1. l l)]

E(r - F', t - t') rr: exp(i[p (r - r f ) - E($ - l')]) E - 7(pj

(5-1.42)

G(r - r', 6 - f)* = G(rt - r, t ' - g), (&l .G)

which retates more concretely the relation ktween (45-1.33) and (6-1.34). In mnsqaence, each #*(F#) is the complex coajugate of the corresponding $(rt). Under t h e circamstmces, h, the vacuum persistence p b a b ~ t y must relnain

unity, or the quantity W real, That is tme of tbe individual particle contributions, Eq, (6-1. l ) , and it will be true of 6W as well if V(r, - rz) is a real, or more generdly, Hemitian funct ion of its vaGables.

The reptition of the primitive interaction will add further terns to the field {&l ,36), But these effects are easily summa~zed. The complete field $112) is the suprposition of that representing noninteracting particles, #(l)+(2), with the field representing particles coming from their last collisian, as excited by the field generated by all sources, namely $(12). T%us, the replacement, under the integration sic, of #(l7)lJ"(23 by #(l%') prduces an integral equation that describes unlimitd reptitions af the primitive interaction:

$(12) = $(1)#(2) + i dl' d2TG(I1,17)G(2, 2')V(112')+(lf2'). (5-2.M)

The equivalent differential equation is

wKch is dsa ob t~ned directly from (6-1.35) by replacing the field of noninteracting particles with the total field. The detemining differential equation for $(12) is, therefore,

We mite its Green's function solution a5

dl' d2' E(12, 11'2')q(l9).111(2", (6-1 .M)

where

[ (E - T),(E - 2")% - i'Fr{12)]G(l2, 22 ' ) = 6(2, l') &(g, 2') )&l.M)

is a genesailization of Eq. (Gf.6) to interacting parlieles, Since the efiective source af (G1.35) is only oprative at equal timm, the?

single-time, tw-particle sour= defined in Eq, (5-1 35) is particularly sinpk. The delta function extracts the equal time limit of G(t',, l',) which is taken from the side of positive time difference [Eq. (5-1 .%?ill. Thus,

md the %me tine of arwment, now applied to the differential equation (5-1.281,

CE - T, -- I", --. V(r1 - r%Zl#(rlr,~) = v(flrzt)*

The Green" function solution is

where

Chap. 5

(5-1 .sr)

extends (5-1 *14) to the situation of interacting particles. We recognize that the primitive interaction function IV plays the role customarily assieed to potential energy.

It is intuitively evident that the relation (5-1 . f 8) between the twa types af propagation functions should persist in the prexnce of the interaction V, since its action i s relevant only when both particles exist. Nevertheless, let us pmve this directly. The Green" function. integral equation equivalent to (5-1.44) is

Consider the situation with I , > t2, Then, switching to the matrix notation, we csn w i t e (provided t2 > t t l )

G ~ ( ~ l , i " ) exp[-iI"~(gl -t2)]Gl(t2,tfl) = aGZ(tl,12)C1(52,t'l}J (5-1.55)

and (&X ,M) will read

where 1;"" indicates V(rl - rz). Again, if 1" 2 t*,, we have

Aceor&ngly, if we now define G1+% by the (more generally stated) relation

G(tltz* t'lt") - &(I,, t,)G,,%(t,,c, tf,)6(t',, t',f, ( 6 1 ,m) tXxcl;t function obeys the integraf equation

We recognize that

is the equal-time Green" function for the noninteracting system, which obeys the differential equation [Eq. (&1.14)]

Therefore,

which is the differential equation (5-X,B3), in matrix notation:

I t should be emphasized here that this discussion assumes that there is a time interval during which the particles cmxist ( 1 , > g',,). Xf that is not the situation ft, < t",), we have a noninteracting arrangement where the Green" function is simply the product of the single particle functions. The Green" functions for the two domains join continuousfy,

We now proceed to set up action principles that will characte~ze the fufly interacting system, a t feast in its two-particle interaction aspects, Some ingredients are already available - the action expression for noninteracting particles [Eq. (4-X 1 ,lZ)],

and the primitive interaction af Eq, (&f,38),

The action pxlnciple should also involve the two-padicle field #(X2), but this should occur in such a way that nothing of the kind is required when interactions are absent, since the fields $(l) and $(2) would then provide a complete description. That suggests the introduction of the field

OR combining the differential equations (&l '8) and (6-1.46) we infer the folfowing equation for this difference field,

10 El-mdymmfict II Chap. S

[ ( E -- T),(E -- T) , - iV( i2) ]~(12) = iV(12)$(1)#(2). (5-1.67)

It identifies the source of the field X as the first interaction of previously noninteracting pa&icles.

A suitable action p~nciple will now b stated where, for simplicity, no additional source for the X field has been exhibited:

in which

This sCructure vvill be justified by its consequences. The field equation obtained by vaqing %*(l21 is just Eq. (5--1.67), and the variation of X ( 12) provides the analogous equation

Their solutions can h stated with the aid of the Green's function G(12, l"'),

dl' d2' G(12, 122")i'C/(l'2~~fl')ylr(2f),

d l d2 46*(2)$*(l)iV(f 2)G(12, 1%'). (45-1 .?l)

The jatter one uses the alternative presentation of (S1.48) as

and the consistency of this system. is confinned by the fact that W, cm h evalualed in two alternative ways to @ve

The sum of WFim*,. and W, involves the following combination standing betwwn producb of single-particle fields :

which makes successive use of the Eqs. (6-1.48) and (5-f.722) for the Green's function. Qne can also write the latter form as

The explicit expressions aswciated with the first right-hand version of (61.74) and with (6-1.75) are, respectively,

X V(lJ2')#(l7)4t(2') (B- X . 78)

and

After the elimination of X and X * , the action principle still applies to variations of 4 and $*., Thus, the field equation far $(g) derived by using the form (&X.%) .would be

I t is an exampile of a set of nonlinear equations that could be salved by successive iteration, Evidently the right-hand side of (51.78) is at least cubic in the sources (counting bath emission and absorption sources), If it were omitted, the error in evaluating W would be, not quartic, but sextic in the sources owing to the stationary n a t u ~ of the action. Accordingly, if we confine attention to the quadratic and quax-tic source terms in the expansion af the vacuum amplittlh, descriptive of a single particle ox a pair of particles, it suffices to use the noninteracting solutions for the fields $(l) and tlf(2). Then, we have

and

The effect of the quartic term in (6-1.79). which we are now verifying, has been to substitute, for the Green's function of two noninteracting particles, the function G(12, 1'2') that contains the full account of the interaction.

In the situation under discussion, where interactions are instantaneous, the action principle can also be formulated using the equal time field +(r,rd), or rather,

The structure (5-1.68) is maintained, where we might now write

but

The X field equations are [V = V(rl - re)]

and

which are solved by

= ( d ) d ) dt ( ) ( t ) V - ) G ( l , ) (61.86)

The use of either solution presents W, as

Twouparticle inteructlons. Non~retotlvfsttc discussion t 3

The addition of the primitive interaction in traduces the combination (matrix natation is wed)

where the latter form involves the differential equation of (S1.W). The aiternative version of this equation,

completes the elimination of T/" ta give

= (E - T, - T,)[G,,,(k, t') - iG,(t, tt)Gg(l, t')](E' -- T1 - Td. (61.90)

The last form is written out as

As discussed before, we now use fields that obey field equations withoat interxtian. As a result,

where the added term again serves to intmduce the Green's function of the interacting system in the relevant term of the vacuum amplitude [cf. Eq. (5-l.$@)].

As is usual with nonrelativistic syslems, advantageous use can be made of center of m s s and relative eoor&nates,

The associated momenta are

Chap, 5

From the inver* relations ( M mx -+ m%)

m x r + r , rz = ft - -r, M M

p1 =%P+P, M p2 = % P - _ , (5-1.95)

we infer that

m, P l m, P8 I T, = M m + M~*Pfz , T~=------ ---P e p + --, P (5--1.96)

2m1 M 2M M 2%,

wfiich implies the kmiliar decomposition of the total kinetic energy,

lt)2 p2 l " l+ I "z= : -+ -=Tp+T, 2M 2p

(5-1.97)

where

f/p = l lml -I- lfma (Sf.88)

defines the reduced mass p. The independence sf the center of mass and relative motions is conveyed, in Green" function langu;age, by the factorization

We shall verify this, beginning with the Gmenk function equation of (5-1.53) which is now witten as

The introduction sf the Fourier reprewntation

exp{iw * (B - R') - Tp(t - t')]) G(rt, r't') (61.101)

yields tbe foflowing P-independent equation,

G(lrt, F?') = 8(t - t') &(F - F')). That is the eontent of Eq. (5-1 .W), where G(R11, Ktf) is identified as the green*^

Eunctlon of a free particle with mass M. Eigenfunctions, labeM by ener$y E, and other quantum numbers caUeetively

c a d a,

are solutions a f the homogeneous Green 'S function. equation,

that have the or(honarma1it y property (for discretely labled states)

With a knowledge of the Green" function, all the eigenfunctions can be exhibited and, conversely, the Green" flunctjion can be constmcted in terns of them, As for the latter, let the Green's function equation of (&L.IO";Z b multipged by $;#(r) and integrated. In view of the adjoint form of (5-1.104).

that gives

The solution af this Green" ffunction equation is

Use of the completeness properly, as expressed by

then supplies the eigenfunction eonstmction

and, conversely, the completeness property is recovered by eampa~son of (5-1 .l 10) with tbe limiting value deduced from the differential equation and the retarded boundafy condition :

The center of mms motion can be reinstated, in accordance with Eq, (6-1.99), to produce the Green's function expansion

where

These eigenfunctions have the arthonormality property

We can also write

with

Eigenfunction expansions for the multi-time Green" functions will now be cansidered. We reed1 that ( t , > 4")

where

and the matrix notation implies integrations over all spatial coordinates, The introduction of the expansion (lE-1.115) for Gl,,(t,, 6") gives

with

5-1 Two-particfs interactions. Non-relativistic discussion ? 7

The factors of 2 have been introduced so that we shall have

.As the construction (5-1.1 f 9) suggests, the multi-time functions are eigenfunctions of the homogeneous version of the Green's function equation in (61.48). To verify that, consider, fczf example (the quantum number labels are omitted),

- sf (4, - t,)($(r,r,t*) -- 3jr(~l~%t2)). (&l. 123) Now,

and thus the right-hand side of (5-1 .l231 is

In view sf the first relation in Eq. (&l.l22), this is the anticipated equation:

[ (E - T),(E - c1"I2 - 6(tl - 8%) y(r1 - ~2)]#(~1t l r2 t~) 0 8 (G1.126)

and a similar procedure shows that

AnaIogous operations can be exploited to give alternative fom to the eigenfunctions. Thus

from which we infer that (matrix notation, with labels omitted)

18 Elsc$dymomics f l Chop. S

The latter version suppfies a physical intevretation of the mdti-time eigen- functims in terns of. measurements perfomed, after the last interaction, on free particles. Similarfy, vve have

Next, we are going to discuss how the multi-time eigenfunctions are used to express or-t,honsmdity., As a first step in an: empirical investigation of this propedy, consider the prdact of: [matrix notation]

and

Or&nary ortbonomality statements, whether written as (5-1. X 14) or in the quivalent Iom

do not involve time integations. Here, however, there are two time vahaitsfes, 1% and tz, or, afternatively,

which suggmts that an integaition aver the relative time va~able z i s required. We note that

where z is r e q ~ r d to be gasilive and negative, resptively, ia the two forms. Now we see that, after compensating the effmt of the z integration from - aa to oo by a faetor proprtiond to #(E + E" - TTt - T2, the ordinav o&lrro- nom&ty statement is recovered :

This stmcture is suggested more Birectly by considering the eigenfunction equation of (&1.128), where we nrsw write

El = + E,, Ep, 6E= gE - Et' (&l * f 39) with

~ = z ' ( a / a t ) , ~ , = i ~ a / a ~ ) , (5-1.140) That ~ v e s

and, sirnjilarly,

&(E - T-1 T2)% - where the differential operator E has been replaced by energy eigenvdues, since the latter deternine the r e s p n s to a ri@d translation of both the v&abls, which leaves z fixed. The: remnGning vac~ables in Eqs, (6-1.141, 142) are rl, rg, and z. We now procwd conventiondy by cross-multiplying the two equations, which are then subtracted and intevated over all vafiables to produce

The fom of (5-1.138) is hereby recopized. But, to fix the abalute factor of the nomdization statement, one falls back on the prwding development..

The multi-time Green" function is now b o r n exglicitlly for the two Gsjoint time r eans t, > t\ and t, < g',. In the first r+an the two pmicles coexkt for a finite time intewd ; the Green" function can be represent& by the eigenfunction expansion (Fi--3,139), The other time redon is such that the padicles do not coexist, and therefore do not interact:

It would be desirable to obtain thew two foms in a unified way by prwwding from a single expression. We shall do this by amlying the integral equations that are equivalent to the differentfat equations of (b1.48) and (&1.72), namely

and

G12 = GIG% "'t 612zF"(12)GlGz,

which are written in a four-dimension& matrix notation [cf. Eq, (Slf2.2X)3, The combination of the two gives

which is also effectively contained in Eq, (&-l ,744). We then s ~ t c h to three- dimension& matrix notation and write out this equation as

in which it has been recognized that the last term involves the equd-time Green's function

The +(g) a p p r i n g here are the eigenfunctions (5-1 . 3 38) with the labeh omittd, for simplicity, We now observe that

V7/ft - $7 2 $(t)$*($') = qft -- &')(E - - Tp) 2 $(t)#*(tt)

= (E -- T1 -- T,)q($ - t') 2 #(q+*(tf) - i b(t - C'),

(45-1.161)

which uses the homogeneous equation obeyed by the eipnfunctions,

( E - r1 - "irz --. v)+(t) = O, E = i (a/at) , (s--relsz)

and the expression of completeness,

$(g)#*ttl = 1 - (61.153)

The additional deltar, function term obtained in this way cancels the finear V term of (b1.149). Furthermore, we recall that

which gives the reduced form

q@ - 8 7 2 $(t)$*(t6)vG,(tp, t f1)G2(t f , j'gb (5-1 ,rsa)

We praceed analogously ta complete the elimination of V, in its expticit manif eslation,

This combines with the relation

to give

Chap, 5

"" [Gl (gx, t") - iG,(E,, t,)Gl(t,, t")Jfcz(tzt t") - 2"G2(82, tl)Gft(tlP t i l l

In e t i n g the latter form we have useta the vanishing of the prdact Gl(tl, t2)G2(Ee, tl), noted that the double tirne intepation assips to t and t-he values I, and if,, resptively, and recopized the constmctions (5-1.120, 121) of the multi-time eigenfunctions, Concerning the combination of free padicle Green" functions that appears here, we recall that

whereas the prduct on the fight-hand side vanishes if the tirne variables are not in the indicated wquence. Accordingly, the free padicle tern of (6X.159) disappars if tl > be 2 ifl, or if 12 > tl > tf2, which are summanized by t, > t", while, in the opposite situation, 1, < t",, the prducts of two Green's functions refer;ing to the =me padicle are zero, The result is the anticipated one,

The states of the two-particle system fall into two distinct categories: those with E > 0, which constitute scattehg situations, and those with E < 0, the bound states, Each example of the latter constitutes a czomposite particle which, in the present simpfified description, appeam as a stable particle. We must check a consistency aspect of our theory - the composite nature of a particle should be irrelevant to its phenornenological description, Let us return to Eq. (61.79) and pick out, in the quastic source term af W, the contribution of a partiGular bound state to G(12, 112')), using the construction of (5-1.148, 149) for this purpse,

ed notation, that give

where, accorhg to (&1.98) and (&X.110),

is the eigenfunction of the spwifie bound state under consideration, Isolating the motion, of the compsite partiicle as a whole then gives

(dr) dL' d2' q*(2')q*(If))Gz(-t', rlt)GS?(2', r$) V(r)Zt(r(rt), (6-I.166)

and [Eq. (5-2.9Ei)l

The fom of (6-1 .165) is c~nec t [cf. Eq. (5-1. l)], but the complete phenomenolag- icaf stmcture is attained only if q(W) and qe(lttl) are indeed complex canjug;ate quantities, That will. be true if each of the single-particle Green's h n e t i m ~ effective1 y obeys

As in the discussion falfowirrg Eq, (&1.30), and also (5-1 .&X), the eon&tion far this is that no real single-particle propagation shaE occur under the circumstancers that characterize the functioning of the composite particle sources, which is surely satisfied if neither single-pafiicle source is capable of emitting, or absorbing, real padiclm. Let us dso Bispfay the sources for a particular composite particle,

and

2.9 Elmrdynamfcs t l Chap. S

where Ifia(rlrzt) is the eigenfunction of (6-1 .f 18). We recognize here the structure of the multi-time eipnfunctians in Eqs. (s1.129, 130) [remember that the latter use matrix notation], and thus

These multi-time constructions are also in evidence when the eigenfunction expansion of Eq, (5-1 . l 19) is inserted in (5-1.79).

No specific reference to the I3.E. or F.D. nature of the murces has occumed in this seelion. But we should comment on the statistics of a composite particle in relation to those of its constituents [cf. Section 3-9, p. 2521. The prducts of two commuting numbers or of two anticornmuting numbers are eompfetefy commuting objects : if the tvvo constituents have the same statistics, the composite is a B.E. particle. The product of a commuting number with an anticommuting one is an anticommuting object : constituents of opposite statistics prduce a composite F-D. partick.

There is an interesting way of exhibiting symbolically the solution of the multi-time Green's function equation of (6-1.48). Zt is suggested by the first two terns of the construction (&1.148), which are also the initial terms of an iterative sofutian,

Let us return to the single-particle Green" function equation, and introduce an arbitrary potentid energy term, a function of space and time :

The effect of an infinitesimal vafiation in V(l) is given by

md the solution of the differential equation is

J

A functional derivative notation will be used to convey this differential expression,

5-1 Two-particle Intttractions. Non*rttlaiQlvisffc discussion 25

If the auxiliary function V(!) is set equal to zero after the differentiation of (&1.177), we encounter just the product of two free-particle Green's functions that occurs (for each particle) in Eq. (5-f ,173). Accordingly, we can write the latter as

It is a natural presumption that: the effect of the indefinite repetition of the internetion is expressed. by the expnential operator that has its first terms exhibited in (5-1 .178) :

S 6 G(12, f2') = exp dT d5 - V ( n j - &V(T) SV(Z)

We shall verify this. [For a; related quantum mechanical discussion using the action principle, see Q%asztu-m Kznematzcs and Dynamics, Section 7.9.3 According to the equations illustrated in (&1.174), we have

(E -- II"),(E - T)%G(12, f 3')

= l (d(1, 1') + p(f)GV(l, 1'))(4(2,2') + V(2)GV(2. 2' ) ) /vm0

= d(f , 1') 6(2,2') + exp[ ] V(1)V(2)GV(l, 1')GV(2, 2') (&X. 180)

where the bracket indicates the functional differential operator of (&1.179), and simpEifieations associated with terms that do not contain both V(1) and V(2) have k n inserted. Now observe that

according to the functional derivative relation

After V(1) has been moved to the left of all functional derivatives, it i s set equal to zero. The first stage in prforming the same service for V(2) involves

Chap. S

Naw the use of tbe relation analogous to ($1,181) gives

[(E: -- T),(E - T), -- .iT/T(12)J6(12,1'2') -- &(l, 1') 6(2,2')

The cfouble functional derivative appearing here is evaluated as [Eq. (&1.177)]

At this point, the instantaneaus nature of the interaction and the retarded character of the Green" functions become decisive, The time delta functions in V demand that

Z g = t X I f l = l p (5-1 .M?)

Hence, the Green's function product in (5-1.186) contains the factor

since

apad from an isolated p i n t at t , - tz =.. 0, which does not yield a nonzero time inteeal. That completes the verification of Eq, (6-1 .l 79).

The instantaneous character of the interaction i s made explicit an writing fLli.179) as

One can also specialize to the equal-time Green" function :

(kr-rsx) where

For a direct derivation of the differential equation in (%-1.63), we apply the preceding equation for GYtss,

and then use (5-1.181) to present the right-hand side as

X Gbl*(rlrzlt rilrfZt') (5-1 . 194)

Next, W consider the two-particle analogue of Eq. (5-X.l76),

+ BVp(i,l)]~Vg**(il ?,E, r',r',tl), (&1.19&]

which is conveyed by the functional derivatives

What is required in. (kf.194) is the evaluation of these functional derivatives at 8 =;: t, wkicb means Phe equally weighted average of the two limits E -+ t f 0, Recalling that

G ( ~ , r t , Exid $. 0) =e 0,

G(rIr2t, 5%t -- 0) = (I j i ) S(rl - i $(r2 - E%), (&L. 198)

we have

Lim i6(rlr&, 5,fZ-Z) ~ = l itlE(rl - 2,) &(rz - g2), (6-1.1@9) E-+#

The outcome far the ~ght-hand side of (6-1.193) is

28 tilerodynamfcs ll Chap. 5

The discussion thus far has been concerned exclusively with the interaction of two different particles. Some words are in order concerning the modifications needed when the two particles are identical, 1Efiee;inning with

the quadratic term in the expansion of exp[zWnoni,,J is

where the integers no longer have the context of different particle types, This is rewritten as

in which

makes explicit reference to the statistics of the particles under consideration (+ , B.E. ; - , F ,D,), Correspondingly the two-particle Green's function has definite symmetry properties (in general, not only under noninteracting eir- cumstances)

The factors of 4 in the differential volume elements are thereby understod as avoiding repetitiaus counting of the identical particles. The Green's function (hf.205) obeys the differential equation

Xt should now be sufficiently clear that, as a general rule, a11 previous results are translated into the identical particle situation be replacing delta functions with the appropriately symmetrized combinations itlustrated above, and by avoiding duplicate counting in all inteeations,

S 2 TWOmPARTlCLE IfJTEPtA~fOPJS, RELATIVISTIC THEORY l

Before enlbaking on the first stages of zt relativistic theory of ekectronrragnetieaIly interacting particles, let us review some aspets of skeletal interaction theory, as discussed in Section 3-12. We arc? going to be interested in the multi-photon annihilation of a spin 4 particle-antiparticle pair, and in the inverse amantiyement.

These processes are described by the skeletal interaction terns of Eq, (3-12.17), as detailed in Eg. (SX2.24). The first examples of the latter can be written as the vacuum amplitude

where the effective photon sourGes are

and

Since these structures o~ginate in the expansion of the interaction expression

they are given a more compact and unified presentation in the notation of functional derivatives :

1 8 eff. = 7 8AL(x) iW ,'*. I,,.

We see here the sense in wlkich (Xli) (rj/6AU(n) plays a symbolic role as the source of the nzulti-photon emission, or absorption. AI1 expressions of this type are conngriwd in the functional form of a Tayfor series expansion,

It Etas also been noted in Section 3-12! that the question of photon radiation from the charged particle sources can be avoided by using the phton propagation iunctions af a certain class of gauges, Tbat is expresse-d here by writing

30 ElWMynamla II Chap, 5

where D + R V is of the f o m descnbd in Eqs. (%12.8,9). This differs from the simpler vemion, giuvR+(x - g'), by gauge terms that are associated with one, or both, of the vector indices p and v. Spcific details will h re~alled later,

The system of interest in tkis section contains two spin g charged partieltjs , which are labeled L and 2. The vacuum amplitude that describes

them in the abwnce of interaction, is

wkch we express by means of a twepadicle Green" function,

G+(XIX~F d~x'e)d,t, a; G+(RI - ~'I)G+ (2% -- g",). (5-2 .Q)

'Z'be effect of interactions can be variously iIltroduced by considering different cawd mangements. VV4? have prepared the way far the following one, Particle and antipadieke of typrt 2 annihilate into an arbitrav number af photons, which submqueatly recombine to fom the pa&icle and antiparticle of type f , or vice versa. The skeletal desc~ption of these processes is dven in (62.6) ; it is the omission crl[ f om factors for the vadous acts that constitutes the skeletal nature of the dmription. And, in the characterization of the exchanged @otons by the simple propatgation funetion D+ (p@ questions aide) we aiso employ a skeletd description. The unlimited exchange of noninteracting photons is expresmd symbalically by the vacuum amplitude factor

acting u p the particle past of the vacuum mpfitude. The latter is (&2.8), with the propwation func~ons replaced by those that contain the effects of the electrw magnetic fields At,,, as described by

What emerges is a symbolic expression for the two-particle propagation fundion that incoyrates the sXreXetm intermions b h g considered :

where the; space-time presntation has removed the reference ta tb:e initial caasal amangensent ,

There is an evident resemblance here to the nonrelativistic canstnretian in Eq. (&X. 1179). VVe can proceed analogously in the first s tqes af deriving a differential equation

The next step involves the reanangement

which, ~ t h a similar statement refer~ng to Az9, lea&. to

wbere

1[1Y(.1.2) = - i(q3tUI1 D+@I -- %)m ( v W ) z * (6-2.16)

The differentid rebtion inferred from Eq. (&2,11),

(?U + m)bG,"(x, x') = w 4 A (x)GtA(x, x')), (6-2.17)

is solved by

Chap. 5

and this result is convey& by the functional derivative

[An example of this relation is the equivdenee of the first statement in Eq, (6-2.5) with ($2.2) ,] Accordingly, the right-hand side of (62.15) becomes

in adaw with (G1.185, X86). But here the resmblance to the mnrelativistic discussion ceases, The photon

propagation function does not transmit an instantaneous interaction, and the paseide propagation functions da not obey retard& boundary conditions. The apparance of four padicle propqation, functions in (62.20) means that new classes of propagation fvnctims are: being introduced in the process of finding G,(x,xz, X'~X'~) . Thus, the two-paPticb equation of nonrelativistic theory has no strict eaunteqart in the relativistic domain, except in the inexact sense of approximation =hemm that relate (g2.20) directly to the two-particle Green" function. An illustration of this erne- on distinguishing the propqation function factor C$(t, x',)G$([', g',) from C$(xl. 6)E$(xp. E''). The former describes the propagation of the p&icles from their initial ereation re@on up to the domain of the two-photon exchmge process considered in (&2.20), while the latter represents the p&icles i5u~ng the interaction process, It is plausible that circumstances &odd exist: where the additianaf interactions between the particles (symbolized by exp[ 3) during tbe tm-photon exchange process wuld be relatively negligible, whereas they certainly cannot be ignored throughout the previous history of the p d & s . Accepting this arpment gives the fol io~ng appraxirnate evaluation for (&2.20)8

in which:

IcB)(xz.28 %l$%) -- - (W~)~(~Q"Y~)% -- g,),, D+(xz - f ,),, - %l)G+lx% -- ~,)(W),(W"Z,

and leads to the spboLicaXly presented two-p&icb equation

ffrP -r- ~)I(Y@ + -- ~xdG,z = 1,

where

I,, = + I;;) + * * * .

This discussion has maintained the generality af

I)+(% - zf),, = g,,D,(n; - X ' ) f gauge terms, (5-2.26)

But clearly our principal concern is with the physical implications of the formalism, which mmt be indepndent af the sgeeific choice of gauge terms, To examine what the latter influence, consider the result of changing D,({ ---. E')"" b y a p- dependent gauge transformation. This alters (5-2.10) by a factor of the form

where A ( f ) is also a linear functional of &/&A2. The efkct of the operator (5-2.26) is to prduce a translatian of A U :

This is a gauge transformation, and its consequence for G $ ( X ~ , x t l ) is given by

The thing to appreciate is that the alteration involves the terminal points of the Green's function, which is not surprising when one recalls that the gauge terms appear as an alternative way of representing the electromagnetic m d e l of the source, and the source radiation that it characterizes. Such aspects of the Green's function are generally not of physical interest, and we must learn to separate them from the information that is desired. This situation is not new, af course, Zt is encountered in any scattering arrangement, but in such circumstances there are intuitivdy evident theoretical coun terparts for the experimental shielding that absorbs direct electromagnetic radiation from the particle sources,

Our concern in this wction is with energy spctra. Ta illustrate the problem in a vefy simple context, we consider a limit in which the particles are very massive and remain relatively at rest. In these circumstances, the particles should be describable by the pfioton source formalism, This i s evident in the reduced form the Green's function equation acquires when all reference to spatial momentum is deleted :

for (xQ > .P)

exhibits the properties of the charge cq, which is located at the point X during the time interval from xO' to xO. Whether we use Eq. (5-2.12) or apply the source description directly, the interaction between the particles is expressed by the factor

In the radiation gauge, where [Eq. (3-16.61)]

AO(xt) = (dx')g(x - x8)JO(x8t), I - we have

D+(x, - X*, t , - 4)* - d(tl - t.) 9(xl - x2). (8-2.33)

Then (5-2.31) redtlces to

in which T is the duration of the interval that the particles coexist, and

is the anticipated Coulomb interaction energy of the charges. Now let us compare this elementary result with that obtained by omitting

all gauge terms and working directly with c v D + :

In carrying out the time integrations of (5-2.31), it is helpful to use the differential equation

i (a,' + lkll- =p(- ilk1 - F I ) = &(+Q - F ) .

2lk1 (62.37)

Thus,

(&2*38)

where 7" is again the coexistence interval for the two particles. For the equal-time situation represented by

this reduces to

(62.41)

and the factor (S2.31) becomes

(5-2.42)

where E retains the meaning given in ($2.35). 'Dais is a pnerating function for an energy spectrum. The notation

36 IECatdynmtcs C l Chap. 5

We recopize that the system has the mound state energy E, and excited states in which an arbitrav number of photons are presr?nt, Tbe latter are an artifact of the parlicdar way that the two-padicle system has k e n created (a nonphysical one). The only physical infomation, contained in the generating function (G2.42) is the enerm of the system d thou t photons, the Coulomb energy E. One will ask how this single bit of meanin,@ul infomation might have k n identified, without

nowing it. The answer is found on minimizing the i~e levant teminal nsidtefing a very tong time ervd. The osciUlatory character of ensures that only values of g 1lfl" contribute to that po&ion af

the momentum irrtwal in (&2,42), which sexves as an effective infrared cut-off to the T-independent part of the inteeal. Thus, the mmptotic form of (S2.42) is

where, rough1 y,

The factor exp(- AT) has the appearance of the (infrared sensitive) probability that no photon will be emitted dur;ing the creation process, but even this is not physical infomation since exp(- A,) exceeds unity for apposite s i p s af the charges. What remains is the enerw E.

The rdiation gauge has qualified far further consideration, at least in pre- dominantly nonrelativistic situations, by shourl'ng two advantages, f t simplifies the problem of extracting phpically significant information, and. it improves the convergence of the sr ies (g2.24). Bath propdies stem from the presumed dominance of the instantaneous component of the prapqation function tenwr, &splayed in Eq. (&%.33j, The remstining; components can be extracted from the complete constmetion given in Eqs, (s12.8) and (3-15,48), as cornbind in

ntly, from the transverse field equations of (3-f6-,62,63), The constmctian of the divergencelas part af J(x) that is given in the latter equation can be p r e n t e d spboticafly as

and then

This exhibits the spatial components of the propagalion function tensor,

which is also the content of the 1st t e rn in Eq, (&2,48), cansidered in the coordinate system where n, coincides with the time axis.

I t is useful to extract an instantaneous part from (S2.51) a well, This is accomplished by the rearrmgement indicate$, in

Thus, the instantaneous part is given by

with a noninstantaneous remainder of

To give an explicit spatial form to the instantanmus function, we note that

according to (k15.48), where the possibility of an added canstant is without interest since we are only concerned with

This yields

'.W

C

0

0-".

tct" . 4-J g

l

C"

3

E .9

0 2, G

M '*

2

% II

80

,: g 5

4 a-=. 6.1 ;iia

y

= 8 $

82

3

z .S $j

c,

g *G

c:

95

P

3

U)

"C

"

1 $

2

3; cl(

8 C

0

0".

CV3 CO

3 W Q;; @

: Y

t? .- a-=.

U

N

k

*rl i-i

4 H

U

d? U

W

Fi

F(

2 ll #--+ U

bV

-5 $4

V

6)

Z W 0.I C

S

Let us begin with a discussion of the two-particle equation of (5;-2.23), in which only the instantaneous part of 1% is retained. We write

4S(xix2)- = i 8(xi0 - xe")ri~g0V(~i - ~ 2 ) ~ (5-2.59)

with

where we have returned to the use of the Hennitian matrices

The equivalent integral equation is

GglL = GIG, 4- GlG24V*.

We shall work with the equal-time functions

and

The corresponding specialization of the integral equation is

We proceed to convert this into a differential equation by using the following form of. the Dirac equation obeyed by G+(% - X'),

(- i a,, + H)G+(x - X') = yO b(# - fi) b(x - X'), (5-2.66)

where

Thus, we have

(iat - HI -- He)Gi+e(~lxd, x'lx8d')naatnt.


4S(xix2)- = i 8(xi0 - xe")ri~g0V(~i - ~ 2 ) ~ (5-2.59)

with





and



(- i a,, + H)G+(x - X') = yO b(# - fi) b(x - X'), (5-2.66)

where

Thus, we have



4S(xix2)- = i 8(xi0 - xe")ri~g0V(~i - ~ 2 ) ~ (5-2.59)

with





and



(- i a,, + H)G+(x - X') = yO b(# - fi) b(x - X'), (5-2.66)

where

Thus, we have


The value of each G,(lrh, x2t) i s computed as an average of the two limits, t - 6'- & 0 :

The latter is a symbolic way of presenting the result, using the notation

W = (p2 + m2)lj2. (62.71)

f t puts the differential equation af (5-2.m) into the form

(iat ffl f i 1 2 ) G ~ + 2 ( ~ 1 ~ 2 ~ ~ ~~l~'"tt ' )noniat .

whic2s. differs from the nonrelativistic version in (5-1.14) by the presence of the factor involving the Hf W.

Vfre observe that

which ascribes the eigenvalues & 1 to the Iliemitittn quantity H/W, Accordingly, the additional factor or the right side of ($2.79 has the eigenrtalues I, 0, - 1. That i s made explicit on w ~ t i n g

The related solution sf tbe differential equation (k2.72) is

40 &lsrtrodynamfu I t Chap, S

where G(x1xZ, x * ~ x ' ~ t - 1') is the retarded Green" function that ohys

Qf course, this result is obtained directjy by muftifiying the equal-time forms of the two single-padiele Green's functions,

8 2 X@' : E+ ( X -- x')y@

The structure of the differential equation (5-2.72) can be simplified by noting that

where

p\lW, m gb = mlu.", (5-2.7%)

is a unitary matrix, Thus, far the noninteraction situation,

Cl+% = U1&61+2U1-EUZ-x (g2.80)

o b y s the equation

[''g - (Y'W) X -- (PW),ICl +,(xlx,t. ~'~~'~~')~i.t.

d(t - t")a(ylo t-. yzOf 6(x1 -- X",) &(X, .--. X"), (g2.81)

In a representation where both yla and yso are diagonal matrices, with the e i p - vdues 1, only two possibilities aplpear,

and

They are united in the constm&ian

which, naturally, is the transform& version of Eq. (62.75). The differential equation derived from (6-2.66) is

( i d t - .Hl - HZ)G1+2(~1~Zdr X ' ~ X ~ ~ ~ ' )

Xt is transformed, according to (5-2.&0), into

where

The presence of the factor %(yIQ + y2@), padieularly in the inhomogeneaus t e r n of the equation, ikplies that only y,@" no' need be considered in the row and column labels of the matrix G,,,. Mowevm, the introrlwtion of the matrix p removes the diagonal nature of G,,. Using the symbols + and - to indicate the common value of y,@" yza' in row and column indices, and employing three- dimensional coordinate matrix notation, we write out (5-286) as the twa pairs of equations

and

( ;at t- trJ, -+- + P--)G"-- + P-+G,- == - d(t --.. t f ) ,

(aa, - - W , - P+,)G+- --. ~r,-i=-- m 0. ( k 2 -891

Before continuing we must note that

v++ = v-- = V &

and that it is possible t o anan@ matters so that

The= are comments about the stmcture of 1V fEq, (&2.69)] in its depndence on the matrices yO and the complementary matrices y,, The submatrices a,, and v-- contain no y5 matrices, and an individual term may .have no y@ matrices, or the factor yl@ygO. Single yO matrices are absent, for the origin of any y@ is in the matrix

y = iy0y,cr,

as distinguished from

and no single y5 matrix survives. Since only yl@yza -+ 1 sccurs, there is no distinc- tton between P,+ and P--, as claimed. The submatrices P,- and P-, come from the part of P that is proportional to Y ~ ~ Y ~ ~ , where any factor y1@yZ0 --+ 1, as before.

that can enter the elements of the Hermitian matrix y 5 1 ~ 5 2 could always be adjusted to make this matrix symmetrical, which is the content of (&2*9f),

Using the notation

we now convey the Eqs. (k2.88, 89) by the sets

and

(iat + H,)G-- -t- F',C,- -.-. - 4 6 - E') ,

The non-diagonal elements of G can then be found with the aid of the retarded and advanced Green" functions that obey

(%"a, -4- Ho)Gad,,(t - t" = - 4(t - E", (&2 .97)

together with

This canstmction is even symbc3licalfy by

5-2 T w ~ i c k interactions. Relativistic theory 1 43

The retarded and advanced functions are used again in recasting the remaining equations in integral form:

The time symmetry of this system indicates the additional relation,

G--(t - t') S G++(t' - t ) . (5-2.101)

However, these functions individually are not of the retarded or advanced type, but satisfy in a more general way the boundary conditions of the G+ class of Green's functions.

We shall be content with the approximate solution of Eq. (5-2.100) that is produced by one iteration,

The time variables are made explicit in

coo coo

To see that' G+ time boundary conditions are satisfied, it suffices to consider individual time exponentials in the construction of 'the various Green's functions. Thus, for t - t' > 0 we have, say,

l 0 dtdr'q(t + t' - (t - t')) exp(- iEr) exp[- iE'(t + t' - (1 - t'))] exp(- iE"r'),

which, using the variables

becomes the integral

ds exp[- iE(?T + S)] exp[- iE'(T - (t - t'))] exp[- i E (+T - S)]

i dT- [exp(- iET) - exp(- iE"T)] exp[- iE'(T - (t - t'))]

6-t' E -E"

44 Elearodynamfclr I t Chap. 5

Here are the required psitive frequencies far positive t - t'. W e n t - 6" < 0, on the other hand, the advanced Green" function imposes no further restdctian on z and s', which immediately supplies the time dependence

negative frequencies alppar for 1 - thrxegative, These general characteristics also apply to C;--(& - 6')).

The spctrum af the energy oprator HQ [Eq. (62.94)J will be ctjiscussedl in an essentially non-relativistic context, That is to say, on1 y first deviations from non-relativistic behavi~r will be considered conesponding, far example, to retaining only the indicated terms of the expansion

The unitary matrix [Eq. (62.78)J

is therefore simplified to

The con1 binations af interest are

There are two kinds of terms in V , One is proportional to the unit m a t ~ x ,

and the ather contains products of Dirac matrices. This W write as ylijlyaeVb, with

In evaluating the submatrix Vo, all terms containing a ys matrix of either particle are rejected. Accordingly,

which uses the relation

a*pTi,a*p = +{p2, V,) -4- $v2V, -4- @*W@ X P, (5-2.116)

and

For the latter caIculation, where

V@ = al*A*agl

the origin of this combination in the transverse propagation function (5-2.64) makes symmetrized multiplication unnecessary (V * A = 0) in such combinations as p, * 4. p,:

(a1 PI. {o~*.Pz. v,)) = PI* A * PZ + 2 ~ 1 X A * pg

+2az*Vz X Aepl + (al X V1)(aZ X ' f l e ) : I I ;

(62.2 11 8)

the last term is a way of writing the scalar product of the dyadic h with the two vectors, When the transverse structure is made explicit,

we see that

VxA=PT71V,x.

In part;icular, this @ves

where the last tern isolates the result of a spatial. rotational averaging process, The outcome of this prwedure is the energy operator

where

and, for simplicity, we have written el ,, in place of (eq)l,z. Of principal concern is the spectnrm of hFo in the rest frame of the two-paticte system-the internal energy, Nevertheless, it is of some interest to see how the anticipated dependence of the energy an the total momentum of the system emerges under these eircum- stances of small. relativistic deviations from non-relativistic behavior. Let us insert the momentum relations of (Ei-1.95) and extract the terms of Ha that involve P :

although we have omitted expressions suck as P (l/r)p ,and 61,g X F P, which will not contri'fiute to expectation values in a state of definite internal parity,

We note the appearance in Eq. (G2.124) of the non-relativistic internal enerm operator

which enters the expectation value of (&2.l%) through its eigesvalue Eiat,,

The first two terms are the expected ones, where the total mass af the system is recognized to be hl + E,,,, , with E,,, << N. It is neeessav, therefore, that the last term vanish:

If we consider the diagonal sum of this dyadic relation,

we encounter the familiar virial theorem connection between average intern& kinetic and potential enesies, in the form apprerpriate to the Coulomb fietd, To be verified is the dyadic e;~tneralization of the viriail theorem stated in (&g, 127).

The proof folfows directly from a simple generalization of the scale trans- forrnatio1.1 that implies the usual virial theorem. Consider the infinitesimat unitary transformation described by

where &K is an infinitesimal dyadic that is real and symmetrical. The transformation is [cf. Eqs. (1-1.18, 19)j

with

l 8r = fr, G'j B ~ K : . F,

1. S

BP = 7 [P, G] = -- BK * p. (&2.1$1)

The induced change in the operator H,

f &H = 7 [H, C],

is given by

48 El~cecrodynrrrmfrrr It Chap. 5

But, since the expectation value of H is stationary with respect to variations of t he wave function or, more directly, in consequence of the commutator stmcture of &H, the expectation value of G f f vanishes. That is the content of Eq, (5-2.l27).

The specialization of the energy operator (6-2-122) to the rest frame is

The last terns, which contain both spins, will be recognized as the hyperfine structure interaction for particles with magnetic moments characterized by 48 == 1, as the clloice of primitive interaction implies, Let us ignore all terms containing 02, say, thereby omitting hyperfine stmeture, and consider the situation, appropriate to the hydrogen atom and muonium (p-+ + e-1, where

When second and higher powers of nzlfil are neglected, the energy operator WO reduces to [e1e2/4n = - a]

The combination in braces will be recognized as the approximate transformed version of the Dirac energy operator far a particle in a Coulomb field:

Tbat is what one expects to find as m/M -+ 0, The term with 1/&1 as a factor thus supplies a first correction to the idealized treatment of the more massive: body its a fixed source.

Elemelttary perturbation theory indicates the cor~c ted energy value

where E,,. i s a typical energy value associated with the transformed Dime enexrgy operator :

A useful result is obtained by applying an isotropic scale transforrnrttion to this operator,

The comparison of the last two equations supplies the information that

An evaluation for the remaining structure is produced by exploiting a modification of the scale transformation. It is generated by

where symmetrized multiplication is understood. This translomation is

Since the rellevant term in (b2.138) earties the additional factor of a, it suffices to consider the non-relativistic energy operator in applying this transformation:

where the full symmetrization required in p * ( I m p is m d e explicit in

Alternative1 y, one can write

5 0 Llectrodytsamfcs l # Chap. 5

and get

The vanishing expectation value of (k2.144) then mserts that

which combines with (S3.141) to give

In writing the last expression we uxd the non-relativistic symbols

For 3 state of non-relativistic energy E,,,-,,,, one has

(3T2 -f- % P - 3 Yf + V 2 ) = Enon-,.(E,,,,,. + 2(T))

= - (m - E,,jaS ($2.151)

which employs the nan-relativistic fom of the virial theorem (&2.141),

Thus, we have the following first indication of the maa dependence in the spc tmm of the two-body system, with M T> m,

We have not been concerned with fine stmcture before now [except; for the indirect reference t s the fact that simple fine structure theory does not remove the degeneracy of certain levels, which accompanied Eq, (4-1 3.109)], and lack an explicit expression for Em,.. But we have only to intrduce the known values for the various terms in the expectation value of Eq, (k2.139). Thus, fox I # 0,

5-2 Two-portfcla iinteraceiona. Ftslativfstic theory I S 1

accor&ng to Eqs, (4-11.104, 1071, and

which uses (4-1 l, 106) and tfie virial theorem statement, equivalent to (5-2. X 521,

On remarking that

we get the desired result :

which continues to hold for I -- 0, j = 9, as one s e s by replacing the actually vanishing last te rn of (5-2,154), ma4/2n3, with the delta function term

The reXativisti~ correction exhibited in Eq, (S2.158) describes the splitting of the 2e2 degenerate levels of quantum number .rz into n distinct levels that are labeled by the total angular momentum quantum number j = &, #,. . ., n - 4. For each j' n --- f , two different values of the orbital angular momentum can poduce the given j and the multiplicity of the level is, therefore, 2(2j + 1). The exception is j = n - 9, where the multiplicity is 2;j + 1 2%. The first point to notice about tfie mass dependence given in (5-2.163) is that it is a function of E&,(%, i ) and hence intrduces no new splitting of the still degenerde levels. For a more quantitative expression, let us write

E = m ( g * g = (&2,1 SO)

and get

To the extent that

Chap, S

the reduced mass, the latter is the mass parameter Chat enters both the floss structure and the fine structure splittings :

But we shall see that this simple mass characterization of the fine structure ceases to toe valid when the remainder of the theory is consulted.

Before embarking on this task, however, we shall add a comment to the calculation just concluded. Observe that the last tern of (62.138) is the approximate, transformed version of

which form would be obtained directly by applying the unitary transformation only to the heavy particle. Let us ask what emerges if we evaluate this expectation value as i t stands. Now we apply the transfermation (5-2.143) to the Dirac energy operator :

and get t f lcr expctation value implication

With this substitution, the expression (5-2.164) becomes

The calculation is completed by applying an orOLinary scale transformation to the energy opefator, which yields

This result

gives an expression, Ior (k2.1CS4) that is identical with what is displayed in Eq. (&2.15.3).

There is an implicit difference, of course. In the last calculation, E,. refers to the exact eigenvdua of the Dirac equation, while in (62.163) signifies the terns displayed in the expansion of Eq. (S2.168). I t is worth verifying that no discrepancy arises to the order of accuracy that has b e n retained in the appro&mate treatment, which is such that the fine stmcture, or its m- dependence, is regarded m small, of order @, relative ta the moss structure. We shall

the sc;cond order fom of the homogeneous Dirae equation,

Spcidized to the chage assignment required fer a b u n d state in the Coulomb field of a nudear charge 25 , the equation for the energy eivnvalue E reads

We write the decomposition of p8 into radial and anmlair parts as

That gives

which we compare with the non-relativistic equation appropriate to the quantum numb= n and I ,

where

Xmmediate-Xy evidtsnt are the eomespndence

bat what comesponds to the non-relativistic 41 + l) still appeaxs as an oprattor. We need its eigenvalum,

S4 Elmrodynamlcs l l Chap. 5

Concerning the angular momentum propedies of spin 4 particles, we know [Section 2-71 that a given total axlmlar momentum quantum numkr j can be r e a l i ~ d with I j + or j - 3, which are states of apposite orbital parity. We dsa know that the effect of multiplication by z ' y p * (rfr), where iy, reverses intrjinsic parity (F), is to interchange the two kinds of spin-orbit functions wwciated with a even j. Aceodingly, the coefficient af 1fr2 in (k2.173) can be represented by a two-dimensional matrix, with row and column Iabeled. by I = = j + + , j - - 8 :

The eigenvdues of this matrix are of the form I'((t'-t- f ) , with

which, af 2% -.. 0, reduce to j -+ 9, j - 3, respectively. The comespondences (k2.1'76) are thus completed by

The result: obtained for the relativistic energy spctrum by applying these relations is

(5-2. f SO)

or, using either value of I' and identifying n aceor&ngly,

As a by-product, we obwrve that the anaXsgaus equation for a spin 0 particle lacks the spin term of (62.171). Hence, the coefficient of l frg in Eq, (S2.173) becomes

1(l + l) - ( 2 ~ ) ~ X i!'(I' + l), (6-2.182)

and

t" E [ ( I + - (2a)T1ig --- +, (45-2.183)

The camespnding enerw fomula i s again (5-2.181), but with the integer + 3 quantity, j replaced by the integer I. In the spciaf. circumstance n = 1, j = 9, the formula (62.181) reduces to

which has already been encountered in Eq, (4-17.31). A similar result holds for all n = j + 4,

and the expanded version agrees with Eq, (B-2.f58), where 2 .=. f , as far as the latter is stated. The same agreement appears when the general formula of (5-2.181) is expanded,

With regard to tl-te mass dependence exhibited in Eq. (&2.153), when the result of (62,181) is used for E,,,, the fact that the latter is an even function of a implies that no term appars of the form

m2 m a5 - -- or (fine structure),

M M

The next section is devoted to evaluating effects of this type, which are contain4 in the full theory,

5-3 TWO-PAWT1CLE I NTERACTIQ NS. RELATIVISTIC T W EORV IE

We begin thissection by demonstrating that there are energy shiftsof the magnitude indicated in Eq. (5-2.187) ; effects of order a, rather than a2, relative to the mriss dependence (m/M) of the fine structure. Although it is not the most important one numerically, the simplest example of such an effect appears in the comparison between the calculation just concluded, employing the Dirac equation -with the instantaneous interaction, and the one bmed on the equation system of (5-2,94,85). Since the entire transveme part of the interaction is advantageausly handled in another way, on1 y the instantaneous Coulomb interaction, will now be considered. To put the two approaches on the same footing, the Eqs, (5-2.94, 95) should be simplified, in the: sense of retaining anly the: first power of m/M, which characteri~s our present limited objective. Thus we have

where Ci" refers only to the particle of mass m, and the (+) subsedpts correspondingly indicate only the "igenvalue of this particle, The latter notation is used

54 Elerdymarnicb II Chap. S

to distinguish such labels from the ones in (5-2.95) where +, for example, indicates the common eigenvalue of yl@ and y;.

First, fet us recognize that in the equation of (5--2.95), combined as

the term invofving V1 is of relatiw order (%/M)" and will not be retained. A rough indication of the mqnitude of V, is given by

and the denominator appearing in the V , term can be approximated by 2M. The inference that the term in question has three powers of M in the denominator is incorrect, however, as one zecognizes f r m a. mmentum transcription of its expectation value in which the momentum associated with the wave function i s neglected (this is, equivalently, the use of the wave function at tbe origin of the relative coordinates)

1 vlrl -- i a t -4" Hi,

The final step recognizes t bat the non-relativistic evaluations, such as (5-3.31, hcome incorrect and give an overestimate at momentum transfers of magnitude

$(@)l2 (am)3, this effect is of the order of (mjM)2a(cr4m), and is omitted.

The comparison of interest is that between the eigenvalues of the energy opra tar

and those of

Consider, then, the Green's function equation associated with N or, rather, with the transformed version of H,

(ia, - RIG = 8(t - 1').

This equation is now decomposed in accordance with the eigenvatues of y@:

giving

It will suffice to ignore the interaction term in the denominator of (&3.9),

As another simplification, the energy af the state of interest, which replaces ;a, in the denominator, can be approximated by M -+ m, omitting the binding energy. The expctatian value, in an eigenstate of HQ, of the effective energy operator obtained in this way, gives an evaluation for the corresponding eigenvalue of H,

1 E = E , + R(+lf-k m + W (p) -- /pP/2 (k3.f I)

This determines the E. spectru~n from the known E spectrum, The eigenftxnctions of H. [cl, Eq, (5-2.1 36)] are approximately those of the

non-relativistic system with the reduced mass p fEq. (5-2.1 62)], In the application to (5-3.llf, which is dominated by relativistic energies m [rather than - M, as in. [&3.4)], the momentum afisociated with the wave function is negligible, That reduces (5-3.11) ta

in which the matrix elements of B are now &omenturn transfoms, which take into account that U reduces to unity for p == Q. Thus, extracting the coefficient of iy5, we have

where, illustrating the discussion that accompanied Eq. (g3.41, the factor [(W - m)/2W]1/2 behaves as lp 12% non-reIativistieii.I.ly and approaches a constant for W 3> m. This gives

58 Elarwmmicco l@ Chap, S

or, implified by m/M < I ,

Introducing- a new integration variabte 8,

$ == m sinh 8, W = m cosh 8, (S3.16)

we evaluate the two integals:

a0 f 1 P d' W(W + m)%= (cosh @ +T m W

W -- m cwhIf--1 X dp W(W + m)% = (cosh 0 + 11% = G (&3*k7)

The rwult, stated for an s-state of p~neipa l quantum number n, is

Here is an example of an energy shift of order a(m/Mj(a%), The ather impgcation of this calcufation i s somewhat disconcerting, however.

Xn the &fit m / M -+ 0, the &fference of the two energies does not vanish. That is, the spectmm of H. in this limit is not that of a charge in the Coulomb field of a static source. The apparent ttis~repancy is a rexninhr that another aspect of the instmtaneaw interaction remains to be considered-the interaction I(" oaf Eqs. (62.22224). When only the instantaneous Coulomb part is retalneb, the latter Becomes

To draw the con%quenees af this non-lwal interaction, we need a further devefop- ment of the Green's ffunction equation for a laeal interaction, since the description of the equal-tim situation no longer sufZices.

Consider, then, the Green's function equation

and its equivalent irzteeaE farm

5-3 Two-paeicte interactions, Ralatfvistlc theory 11 39

Let us set xI0 = x2@ =I= I in G(x1z2, X ' ~ X ' ~ ) , as the interaction term requires, but leave X,@' and xz@Yree. We need the differential equation obeyed by GIGz under these cireutmstances, It is conveyed by

The implied differential equation for G, gresnted in symbolic notation, is

which reduces to (h2.85) when one sets xI0' = xZa" t', In the transform& version of this equation one has

where the factor in brackets can also be written as

in which

As with any Green's function equation, there is another, transposed, form for which the second set of variables is involved in the differential aspects of the equation. This counterpart: of (s3.23) is

where we have wrltten a' to indicate differentiation to the liefit, with an additionat minus sign.

Now let us add to the instantaneous interaction of (S3.20) a non-local intexac- tian, written as

Chap. S

which replaces the integral equation (Ei-3-21) with

Then the differential equation (5-3.23) becomes

In virtue of the non-local nature of v , the Green's function G in the last term does not have equal time values far its two left-hand coordinates. An approximate description of the needed relative time dependence can be obtained from (&3,28), where it i s tfze right-hand time variables that are equal. The equal-time Green's function associated with the instantaneous interaction obeys the equations

With its aid, a sufficiently accurate solution of (&3*28) is exhibited as

which we use in the approximate conversion, of (5-3.31) into the equal-time equation

The transformed version of this equation, in the anticipated approximation that retains only the + compsnents, reads [hopefully, the trmsitional mixture of three- and hur-dimensional matrix notations for coordinates is not too unsettling]

where

and

The tirne variables that have been suppressed in (5-3.36) are written out m

fn the application to the non-local interaction (&3.19), which has the form

va(tltzl ttltt2) == 8(tl - f 2 ) 8(t2 - ttX) - t2), (G$. 39)

the terms in (5-3.38) tha"tnvo1ve the Green's functions of different particks do not survive, as illustrated by

c l,,. ft - tl)~,,,.(t, - t ) = 0,

This reduces (63.38) to

If &F', were an instantaneous interaction, its expectation value, a three- dimensional coordinate integration, would estimate the energy displacement that it induces. But the actual structure involves a tirne inteeation,

and one must recognize that, when a particular state of energy E is consi&redl, G,,(f, t8) effectively differs from G,,($, t') by the phase factor

Accardingl y, the energy shift i s

Whm the stntcture (5-8.41) is introduced, with

Chap, 5

(&3,4fj) this becomes

As y e shall we, onl y relativistic energies contfibute to the efhct under consideration. That enables the energy of the state to be approximated, E g M -+ m; the internal wave function is replaced by It(@), which reduces the unitary transformation U I U z to unity, and the retarded Green" sunetions refer to the rest mass of the associated particle. The energetic simplificsttions reduce (5-3.46) to

Xn view of the + character of the wave hnction (yI0' = yyZQ'= =.E l), the functions v(-& t) obta-ined from Eqs. (5-3*19, 29, 31)) are

Let us recognize immediately that the energy shift associlcted with the; V ( T ) t e r n is of relative order (rn/M)g, and is therefore omitted, fn a non-relativistic treatment of the heavy particle, the z dependence of that t e rn is - @xg[-- i2Mz], which contlributes a factor 1JM on performing the z integration. We &so have in (W, - IM)/W, a factor - p22/1M2, AS in the discussion of Eq, (&8.4), the: apparent IfW dependence is reduced to l j M 2 on examining the momentum depndence of the inteeal.

The expetation value of (5L3.47) involves internations over both particle cmrdinates, as restricted by the wave function [cf. EQ. (6-1. l f 3)]

which decr_ibes the factorization of the esrxtially non-relativistic motion into relative and center of m w motion. Since we are concerned with the system in its center of m a s frame, and nqlect relative momentum, the first set of in ts ra- tians is

Two-particle interactions, Relativistic thwry l l 63

where

and we have proceeded cautiously with the center of mass wave function to avoid obscuring the application of the normalization condition

Since (5-3.51) only involves felative coordinates, the remaining inteuations directly give

( V ( - t)) = - 1#(0)/2(4na)Z exp{ - i [W + M f (pg/2M)]~].

where we have used the non-relativistic form for the heavy padicle enerw, and omitted the subscript on Wl. Carrying out the relative time integration in (g3.47) now eves

which is very closely related to (5-3. f 4). Indeed, on neglecting pg/2M the two expressions cancel, which is as it should be. Also, the terns af relative order m[M in (5-3.14) and (g3.54) are equal, Hence, the net energy shift obtained from the instantaneous Coulomb interaction is

the last form refers to an ns state, in which the distindion between p and m has not been retained. For the important example of 1.r -. 2, this r e d (the numerical value refers to hydrogen)

Before embarking on the relativistic discussion of the transverse interaction, let us return to the non-refativistic considerations of Smtion 4-11. The causal, vacuum amplitude for the exeXlange of a photon and a particle is written in Eq.

64 Elactrodynamlclt 41 Chap, 5

(4-1 1 .S), where the choice of particle propagation function descriks the dynamical nature; of the system. For a composite of two particles with charge and m a s assignments given by -- e, m and e, M, considered in the center of mass frame, one has only to make the substitution

while indicating, through the coordinate dependence af the phaton propagation function, which particles are involved in the emission and absorption acts. [See the related &scussion in Section 3-15, pp, 357-358 where, llowewer, the symbol M indicates the mass of the composite particle,] The additional coupling Between different particles has an explicit f [Af factor, Accordingly, the remaining calculation, is performed as though m/M = 0, This enables the heavy particle to be stationed at the origin, while the light particle is assigned the coordinate vector r. The two vacuum amplitude terns obtained in this way are

The presence of the phaton pmpagation function jEq, f4-.11.%)],

ad& the energy kl" to the compasite padicle energy, as before. When the space- time extrapoltati~n is prformed, t he resulting addition to cSY f Eq, (4-1 1,58)3 is

The point of the latter decomposition appears on noting that

the corresponding tern is the non-relativistic version of the instantaneous part of the transverse interaction, I t has already h e n considered, in its relativistic form, and should be removed from (5-3.61).

The presnce of the exponential factor exp(ik r) in the significant tern of (6-3.61) introduces a characteristic photon enerw K, a small numerical multiple of (ao)-" gm, say

For kO 2 X, the exponential factor is effectively unity, and the perfsmance of the integration over alI photon directions gives

One will recognize, multiplied by 2m/M, a more compact f o m of the structure of Eq, (4-11.59). Underlying this simple result is the obsewation, based on (5-3.571, that the two particles emit and absorb long wavelength photons in the same way as the single particle, but with the coupling constant alteration elm -.. elp. The additional energy displacement for a 2s level is obtained from (4-11.91, 5321, with Z == l, as

For the f?.P level, according to (4-1 f .97), we have

Wh'en R@ > K, which is an order of magnitude larger than the atomic binding energies, one can neglect E - H relative to k@ in the denominator of (g3.61) :

For the expectation vdue of this operator in an eigenstate of H with ei~nvalue E, one can modify (k3.67) according ta

(E - W)p -.+[E - H , p] = - iVV,

PIE - H) 3 [p, E - W ] = iVY,

W ~ C ~ cotnve~s it into

Chap, f

(G3.68)

: explila. r5 VVV.

It is shpler to use (6-3,692 with a real wazre function choice, givina;

As in Eqs, (4-1 1.76, 761, we shdl employ dimensionless variables,

together with

On introducing the cosine of the mgle htween r and K,

we find that

(W.76)

Aecordhg ta the generating h c G ~ n [Eq, (4-1 [1 ,M)]

which, incidentagy, is v e ~ f i d as a solatian of the afferential equation

we have

and therefore

One now encounters the successive integals

where the last step refers to the nature of the subsequent y integration ; then,

P

A "'" log - %@&<C X . ;

X IC kg + ~~p~ naaKp "

and finally,

The resulting generating function is

in whkh the coefficient of tn-ltk-', divided by PP, is the desired expctation value in the ;ns state. For the important example of n = 2, this recipe gives

When (k3.65) is added, we get

We have only had occasion to exhibit the wave functions of S-states, based on the generating function (g3.76). The relation between the states of common 111 but diiferent I quantum numbers can, bet d e x ~ b e d with the aid of the axial vector

]I" A, = - + a,+(& x p - g x L).

I (k3.86)

68 Blccrttody~amfn t l Chap. S

It is directly verified to be a constant of the motion as a consequence of the equ* tions of motion,

on employing the symmet~zed multiplication that is justified by the quadratic dependence of the energy operator on the momentum [cf. Eq. (l-2.51)]. As a constant of the motion, the action of A on a wave function of given energy must prduee another such wave function. According to a commutation relation that characterizes p as a vector,

the effect an an s-state wave function =: 01 is

which, as a vector, constitutes a $-state wave function, For n = ]I, where only an S-state exists, we conclude that

which implies the simple expnential function known ta represent this state, The radial depndence of the 2s wave function [Eq. (4-1 1 .W)],

produces the three 2)p wave functions, which are exhibitd as c~mponents of a,

vector :

i t has been supplied vvlth the proper nomalization constant, Instead of introducing one of the p wave functions into f&3,71), we shall

employ their iverage,

where the latter form u ~ s the &mensionless variables for llz- 2. The application of ($3.76) tben gives

If we write

we can make use af the generating function (5-3.&3), without the factor (1 -- x (]l - E'))-... This yields

which is combined with (&3,66) in

The magnitude of the effects we have been discussing are indicated by

which can also be realized through the exchange of tvvo photons betwen the two particles, Indeed, we reeopize in acJm and orfM the amplitudes for low energy photon scattering by the respective particles [recall the Thommn cross section], which also govern the emission and absorption af two law energy photons. We shall evaluate this effect for the non-relativistic regime. Suppbrnenting the action term of (4-L1.30) is

which is extended to the two-particle situation in

Xt is the latter process, rather than two successive acts ~ v e r n e d by the two- particle extension of (&l1.30), that dominates the non-refati~stic situation,

7Q Elarodyomfcs II Chap, 5

Thus, the effective source for the emission ad two photons and the c6mpsite particle by an extended warce of composite particles is

with a similar expression involving Ift*(r,rzt) for the three-particle absorption process. The vacuum ampEtude des~ribing the three-particle exchange b t w e n different component partides is

Nate that the tensors deafibing the transverse character of the photon. propagatian fundion are cornbind in

whicfi is reminiaent of the angle dependence in low energy photon scattering [Eq, (%18,1X8), for example]. We ajso obsewe that the presence of the factor 1JM enables us to station the heavy partide at the orien, and then infer an addition to the action that has the farm (&11,BS), with the interaction

I'he expectation value of QYYn a, nutH eigenstate of E - IE is

Consider first an approximtian in which the particle momentum is negIected, commpnding to the replacement of $(F) by $(O). This gives

The photon energy inteeation apparing here will be t emina td by relati.vistic effects at kO m, and by the finite momentum assaciated with the bomd state at k@ PW Ern, Hence,

and we get the rou& estimate, for n = 2,

The effect is approximately ) of 65-3-85), and of opposite sign, Now let us improve this estimate at the low energy end. Accordiing to the

generating function fw s-states, Eq, (5-3.761, the coordinate in tegratian of (5-3. IQfi), exprwed in BimensianXess variables, involves

(dx) exp[z"(~ + K') X] exp(- h) = 4% dx exp(- h)

The representation

then presents (63.109) as

Using the term in the same wnse as in tfie context of Eq. (5-3+83), the gentrrating Iunction associated with 663.104) is given by

where r and K' indicate the magnitudes of the respective vectors and p is the eosine of the angle between them, It is convenient to intxoduce a change of variables :

The upper limit 2" expresses the necessity of stopping the photcm energy integration before one enters the relativistic region, With tllesse variables, (5-3.112) reads

where

We shall treat the two terms appearing in the rearrangement

in different ways. For the eonstant term, the integrations over p, and then S,

are p-erfomed first :

where the latter fom is produced by partial intepation. The logarithmic depndence an, Y is a Consequence of the vy combination in (G3.fX.7) far small values of v. Thus, it refen to the situation K - K' - 2-1-f/sy. In eonsequenee,

where L is an upper limit to photon energies which, in anticipation of a junction with a relativistic calculation, is chosen intermediate between nonrelativistic momenta (m %m) and relativistic ones (W m), say,

The fairly large value of V - a-f/% permits the simplification of (6-3.118) to

wlrich employs the integral

log t dt - = Q.

0 X + t 2

fn the second piece of I,

the vanishing of 1 -- p2 for p =. - 1, which is the high energy situation according to (S-3.16fS), enables one to replace -V by m. Here we first perform the _v integration, and then the s and p integrals:

l

-1

x exp f -- 2%) C1 + VS + p(1 - VS)] --W2

To aid in the p integration, a partial integration is used,

and, finally,

The two parts of f are united in

74 E l ~ t r d y m m f c s tl Chap. S

The generating function (63.114) is thus found to be

and its eonsquenctl: far n = 2 it;

1 m (6v'"22, -- - - -2L 37 4 a;n: M log --- -- - f -$ log 2 m 12

As in the discussion of (&3.96), the result for the 2p level is obtained by applying the operator + (d/dA)g t o the generating function (5-3.128). without the factor (1 - E)-%(l - t"-2, This gives

The principal remaining task is the calculation of the relativistic process that joins with (69.129) to remove the dependence upon the parameter L. Let us return to the construction of Eq. (s2.12) *ere, in the radiation gauge, D,(n - S"),,

has the two types of components restated by

and

Mle have dealt with the instantaneous interation by devislng a differential equation for the two-particle Green's function. The transverse interaction will be handled in a more elementary way, by using a power series expansion far that part of the functional diff%rentid agerator :

mese functional CIerivatives can be applied directly to the individual particle Green's functions, in accordance with

and

This is unnecessarily complicated for the heavy particle, however, Since only effects of order X/IM wilt be retained, it is simpler to replace the non-relativistic: reduction of these foms by a direct derivation from the non-relativistic version of the Green3 function, A transitional form of the latter that foHows from the constmktion

referring to a particle of charge e. Inasmuch as W are interested in the dependence on A, which dready involves the smallness parameter X/M, the kinetic energy p2/2M can be neglected to give

which is a generdizatian of Ep. (S2.30). The functional derivatives obtained in this way are

and

76 Elsctrodynanfc;~ lE Chap, S

for the situation where the times p" and p are intermediate between xO and X@' ; otherwise, the functional derivatives vanish,

f t may be helpful to interpose here an illustration of the technique for extracting an energy &ift when the Green" ifunction is given by such an expansion, For simplicity we consider a single padicle, the Green" function of which has an expansion with the leading terms

The initial Green" function has the eigenfunction representation

with positive frequencies, under the indicated time circumstances, while negative hquencies occur far .rO < xU'. Associated with a pa~icular eigenfunction in (5-3.141), as the factor of i#(x)4*(x'))y0, is (fl - X @ ' = T )

where

and we have only exhibited the contribution associated with the time domain

The reason for that %ecomes apparent on giiicing the i n t e s d of (5-3.143) an asymptotic evaluation in which the microscopic time variable t = yO - yO' effectkely ranges from -- m to m, while the remaining t i m variaMe yO 2: yO" covers the interval of duration II". This gives for (8-3.143) :

exp(-- zET) [1 - 2 SET] g expf --- i ( E -+ SE)T], (5-3.146)

with

In contrast to the secular variation exhibited in (!5-3,146), the significant time intervals in the regionsyB > x0 and y 0 2 <' are microscopic and do not contribute

to the energy displacement fomula (6-3.147), which is the counterpat-l of Eq. (k3.M).

Another usefd obsenratian follows from the remark that

1 --. %[G,A(x, n)y.G+d(2, X')] f8(% --- X' ) ---- S(X - g)]G+&(x, n'), (b-3.148) 2

which is a version of the divergence equation [cf. Eq, (W,48)]

a, C.lf;#(x)?@~"e9#(x)l m i$f ~)$~11 (%) * (5-3.149)

Consider the situation with xQ > X@', and let P be such that

9>@>#. (5-3.150)

Now prform a space-time integration over the wmi-infinite region with time values less than 3. This gives

which i s a multiplicative composition property for the function (I /i)G+.'yO. This is an elementary statement for free particles; here is a e;eneraEzation to arbitrary ebctrornapetic fields, Note, incidentally, that if both inequalities in (5-3. I 50) are reversed, a minus sign appears on the fight side of (S3.1B1). 15 only one inequality is reversed, the integraf of (63,162) vanishes.

The factorization of the heavy padicle Green" function that is exhibit& in Eqs, (63,139, 1M) enables one to consider an effective single pa&icle Green's function far the light particle: This elimination of the two-particle aspct also involves the restriction to relative motion, accor&ng to which the heavy particle momentum in (63.139) is replaced by the negative of the light par;ticle momentum at the same time. To do that WC? make exphcit the heavy particle emission or absorption time, using the analysis of Eq. (63.151). With the position of the heavy paftiele adopted as spatial o ~ @ n , the effective change of the light particle Crmn's function that is associated with single photon exchange is

h e r e exp[ 3 indicates the instantawous interaction part of the functiond oprictor in (k2.12). The effect of the Eattftr is most simply desc~bed if one lets

78 EtWrodpamlcs ll Chap. 5

the time span of the h e a v particle, the range of time internation in the last factor of (H, 1521, be large compred to xO -- xQ" - If while campletely including this interval, With tbis eEmination of end effects, the functional operator simply replaces q A Q in G, by the static Coulomb potential. Comparison with the structure (&a, 141) then eves the energy shift formula (somewhat different notation is used, and D,,, is made explicit)

Now if we introdace the unitary transfomation of (S2.79) and exploit the essentially non-relativistic nature of the system, according to

and [cf. Eqs. (&3,137, 138)]

the resdting energy shift fomula, is just that implied by the consequence (5-3.W) of the cornplettsly non-relativistic dixnssion.

Two-photon exchange prduces the foIlowintg change in the propagation function :

There i s an in te~at ion here aver the common time variable p' = p". If we confine oar attention to the relativistic domain where the momentum associated with the wave function is negligible, and the function G+(t, 6') can be approximated as the translationally invariant free partklt: function, we then effectively encounter the four-&mmsional intepal

Tbe energy shift fomula i s (E 2 m)

5-3 Two-particle interactions. Ral8tivittic theory 11 79

where

Since $(O) is an eigenvector of yO with yO' = + I, the expectation value of any odd power of y vanishes, so that y k can be omitted, while yO in (5-3.169). which acts on y&(O), can be set equal to - 1. That reduces (b3.158) to

The frequency integral appearing here can be evaluated by contour integration,

In order to join with the non-relativistic calculation, the subsequent three- dimensional momentum integral will be stopped at the lower limit Ikl = L << m. This gives the partial energy shift

which unites with the generating function (5-3.128) or with the explicit result (5-3.129), for n = 2, to effectively replace L by bm. Thus,

The several contributions to the energy shift of the 2s level, listed in Eqs. (6-3.50,86;, 163) are combined in

while the 2p displacements listed in Eqs. (543.97, 130) give

The unit that appears in these results is, for hydrogen,

and

The additionai relative displacement of 0.36 MHz adds to the last theoretid estimate of (4-1'7.132) to produce the new value :

This time we have somewhat overshot the nominal experimental resalt.

H PHOTON PROPAGATION FUNCT1ON I1

The treatment of the modified photon propagation function @ven in Section 4-43 involved the exchange of a non-interacting pair of oppositely charged padicls between extended photon sources that are in cawal anay, We shall now consider the next dynamical level for this process vvhich, for example, recognizes the possibility of interaction between the pa&icles as p r d u c d by the exchttnge of a virtual photon. Qther mechanisms must also bte taken into account at this dynamical level, as one can see from several points of view. Thus, any process employing a virtual photon can have a counte~ar t employing a real photon. In. this causal arrangement it arises through the possibility of emitting, not a pair of red particles, but a real. and a virtual particle. The latter ra&atw a photon ta produce a three-particle emission act, The correspnding a b w ~ t i o n p rwas takes place by having the photon combine tlvittz. a charged p&icle Lo produce a vidual particle, where the latter, together .tlvith the other real pa&iele, is sub- sequently detected by the extended photon source, There are two ps ibGit ia here, however, since the photon can either h absorbed by the amet pa&icle that earlier emitted it, or by the other particle. It is the scond p s i b s t y , invoivhg a photon exchanged htween the oppsitely charged pa~icltes, that is the counterpart of the seattedng, or virtual photon exchange, procas, Alternatively, we can remark that mechanisms must exist to introduce the approp~ate mdf iea t i~ns in, the various elements of the two-particle exchange act. Thew are: the p ~ ~ t i v e interaction that interconverts a virtud photon with a pa&icle pair, and the padicle propagation functions. We recopize, in the particulla three-padicle p rwes where one of the particles does not share in the exehmge of the photon, just the mechanism for the modification in the paAiele propwation fanclians, And the scattering process is the one that introduces the f o m factor mdfication of the primitive interaction, Since the latter occuw at both the enrission and abmqtian en& of the prwas, the scattering mechanism, with approp~ate e a u 4 controh, must %we both functions,

5-4 Photon propagatfon function t E 8 t

The three-particle exchange pracess is a new feature of such problems, and we

discuss i t first. As an introduction, let us consider the three-particle kinematical integral that is the ~;~tneralization of (4-1.231,

One elementary procedure first performs the integrations that unite two particles into a composite system of variable mass M', and then deals with the effective two-particle system that remains. This point of view is conveyed by writing

- P , - P, - P,) = &f' -- P' - p,) dmpt $MtP (2~1% B(P' - p, - p,). (5-4.2)

Carrying out the successive integrations for two particle systems gives

X(M, m,, m,, m,) =c dM 'B 2 (M' , mm,, m,) ( 2 ~ ) " dm,, dWc (j(P - P' - p,)

dM* 2r(Mf, m,, 111-,)IfM, M'$ m,). (54.3)

The simplest example of the final integration over M' occurs when the in&ivictual particle masses are zero or, equivalen.tly, under the circumstances M >=. m,, m,, m,. Then ( M . 3 ) reduces ta

Next in simplicity is the situation w h e ~ only one mass diffess from zero:

For the system of present interest, where m. = m, = m, m, = 0, we have

82 ttarobynarmllc?r t l Chap, S

which evaluation uses the variable

with

The asymptotic khavior for M2 >r (2m)2, ZIO l, is indeed given by (W.41, while near the threshold, M2 2 (2mI2, va <<I l, we find

A treatment that is more symmetrical among the particles can be srzpplid by using an infinite momentum frame, generalizing the discmian that led to Eq. (4-1.32). We note the invariant momentum space element (&1.28), and the delta function expression that is the tliree-padicle counterpad of (41.30) :

In a temporarily unsymmetrical pr fomance of the transverw mornenta integrations, we use the relevant delta function of (M.10) to eliminate pcT, thereby prdueing in the energy delta function the quadratic form

We have also indicatell here the possibility of diagonalizing the quadratic form by an oxtXlaganaX transfomation, which is such that

The latter f o m exploits the rotatimd symmetries of the infegrand, which now appars as

where it is understood, that the u-parameters are positive.. The spectral restrictions are evident here, in parametric form:

~ ~ 1 2 9 , 1. g,

The anticipated threshold mass

M@ = 2 mm,

can be used to rewrite (M.14) as

and the possibility of attajning the indicated lower limit, with

(W. 14)

(M. 1 5)

confirms the significance of MO. As the genhality of the notation indicates, these considerations hold for any number of particles.

The r e m a i n i ~ momentum integrations in (54.13) are perfomed with appropriate variable changes,

under the circumstances of (M.14) . The product A l l z is the determinant of the quadratic f o m ( H - I l ) ,

This gives

I(2M-* m,, m,, m,)

1 m mb2 m------- d%, d%& &ge fl(1 - "U, - 2aCb - H,) - - -

( 4 ~ 1 ~ M, M@

where the integration doxnain is restricted by the delta function and the requirement of (5-4.14). The high energy limit ( M . 4 ) is obtained directly:

84 EIutmdpunia I1 Chap. 5

For the situation of two vanishing masses we omit the final integration of (64.21),

The resulting single parameter integral,

is equivalent to (64 .5) . Turning to the mass assignments m, = m, = m, m, = 0, we eliminate one parameter to get

It is convenient now to introduce new variables:

, = U + V , U , = ~ a ( i - V ) , dub = du W, (54.26)

which present (4n)'I(M, m, m, 0) as

Both variables here range between 0 and 1 , subject to the restriction

on introducing the definition (5-4.8). Since U cannot exceed unity, it is necessary that

Carrying out the U integration then yields

which is reminiscent of the parametric integral in (64.6), but differs from it in

5-4 Photon propagation luncfllon II 8S

detaiait. f n fact, the two versions are equivalent, but (W.30) leads somewhat more directly to the high and low energy limiting form.

The p e s s of emitting two particles and a photon Irom an extended photon source is repsented by a coupling among tvvo particle saurees and two pholon sources. As such, it is another application of the interaction Wzz which also describes pboton-particle scattering [cf. Eq. (3-12.29)] :

(dx) (dxP)#(r) C2eqe.A ( X ) A,(% - x') 2eqp.A (X ' ) -- 6(n -- x') eZAz(z)]+(n') . (M.31)

The part of the vacuum amplitude iWz2 that is linear in the field of the extended source, A,, and in the field of the emitted photon, i s compared with the equivaIent three-particle vacuum amplitude

to give the effective emission source

The momentam version is

- K2i$)Kz(p'l/,.t(k) l,, = 2eZyzrvA %*(K). (M*%)

where

has been simplified by introducing the reaX particle properties

is the total momentum emitted by the source. We note the con~rvation and gauge invariance statements

86 Blmrodynomlers )l Chap, S

- &.(-- P')K,(-- $)l,"- 4 I,. = 8ePA g,(-- K ) yA, (M.39)

in which

Incidentally, the unit matrix in the charge space is implicit in the latter objeds. The vacuum amplitude for the three-particle exchange process is

where the trace acts in charge space, and supplies a factor of 2 in the resulting expression

which collects all the internal workinp in the tensor

dw, do,. dw, ( 2 ~ ) ' &(K - p - p' - ~)VI,AVE% ((5-4.43)

According to the relations (5-4.38, 401, I,, is symmetrical in p and v, and obeys the condition of gauge inva~ance

This spcilies the tensor stnxcture:

where the scab I(M7 is

The tensor VP can be exhibited as follows:

There are three sets of terns here, each of which vanishes on multiplication by kA.

Consequently, the one Iraving khas a factor does not contribute to the required produe t ,

W recognize the rnultipticative structure that dominates soft photon emission. I t has often been encountered in describing the deflection of a particle. Were it refers to the creation of an oppositely efiarged pair of particles, One can use the relation

to combine the first two terns on the right side of f5.tim48j,

We shall cany aut the intqration in the manner of Eq. ( U . 3 ) , first grouping the padicles into a, composite of mass M':

The integation over the padicles, in (W,46), now p r d u c s the scalar fnnetion

dw, dw,. ( 2 n ) q ( P - p - P') v ~ v ~ , , (M.58)

and the remaining kinematical integral then gives

although, as M" M# we must modify the indicated kinematical tactor to take account of the fictitious photon mass p,

88 t l ~ n u n f c r II Chap. 5

M q 1 1 - 4 - 2 [ (W - M'?' - 4paM9ln g - [(M - M')' - /8j". (-.m) W W M

The integration of (M.53), as expressed by

is performed in the rest frame of P. Some invariant expressions for quantities in this coordinate system are :

P = W - M * 1 2M' Ikl = m [(Ma - M?' - 4p4M'Tln,

and, there is also the invariant

The infra-red sensitive integrals are

and

We shall only exhibit the latter integral in the two domains that were finally i n t d u c d in (64.59). For the first one, with M - M')> p, the integral is variously expressed as

5-4 Photon propagation function II U9

In the region M - M' I p it becomes

I " (M - M')' - ' [ l - (4m2/M7][(M - M')' - p7

1 M2 (M - Mt)[(M - M')' - ~ $ 3 ' ~

Jf - M' + [ l - (4m2/W)11"[(M - M')' - p$31fl X log- M - M' - [l - (4m2/W)l1a[(M - M')" p711' (64.62)

The remaining integral is

which is evaluated as

Before combining these structures, let us be explicit about the quantity of actual interest. It is the weight function a ( M 3 , displayed in the action expression of Eq. (4-3.70) and in the implied propagation function formula (4-3.81). The calculation of Section 4-3 gave the two-particle exchange contribution to a(M4) :

We are now finding the three-partide exchange contribution U(~)(W), as another term in the action, of similar structure, and thus appearing additively in a(@). The value inferred by comparing the coupling (6-4.42, 54.56) with (4-3.34, 37) is

dM" h 2 ' l 2 wa(a)(m=X-I 12* ~t ' ( l ) ' ( - ) <v~v*av), (-*m)

where the 'quotation marks' recall the necessity of using the version in (54.55) forM - M ' N ~ .

A function of frequent occurrence in these integrations is

With its aid, we convey the form of (vPv~,,) that is appropriate to M - M' p 83

Chap. S

which U= the vaxciabXes

The Xatter also Bc: employ& to write

m& then the cmfficient of ary12~ in (M.66) reads

The internation here mges from M" 22m to M" M - --M, where

p<<8iM<<m, (M. 72)

or, from v' = 0 Oto v' = v - -.v, with

Other terns da appar in addition to (-.7X), namely

but: this combjstation vanishes. That is verifid by explicit intcl~ation, of

and

In the re@on M - M" p, we have

only the infra-red singular terms are retain&. For this region, the integral of (M.68) becomes

(l - v2)v

m2 d(M' -- M)[(M M')" p/172(~F~~~r). (M. 78)

where the integration over M - M' ranges from p to &M. The basic integral encountered here is [cf. Eq, (&,97)]

and the resulting fom of (M.78) is

The depndenee on &M will &sappar when (5-4.71) is added. To remove the fietitiaus photon m=, we mast consider the second effect, the mo&fieation in the two-padiele exchange process,

We know that the modifications in the individud particle-pair emimion and absorption acts are described by the farm factor

But we must appreciate the causal situation before applying (3-Q.81). In the initid tvvo-pa&iele exchange process, there is a causal control over the emission and absorption reg;ions. That ceases to be true, in general, when the form faetor is intrduced into the desc~iglion of the individual emission and absorption acts, since there is complete non-locality (propagation) under the energetic circumstances expressed by the vanishing of the denominatar in (M.81). The situation is similar to that encountered in describing unstable particles where simple sources

cannot h considered if a causal control is to be exerted. As the andope of the extended source employ4 in the latter discussion, we must exclude, for each choice of v'iin (W,8t), those saurces for which o is in the immediate neighboxhod of V' , where

(M. 82)

And, ats in the unstable particle considerations, a final limiting a principal value integal, mus, (W.81) is effectivefy replaced by

with

This repreents the madif ying effect of those gartieh interactions that are s~tably localized near the emitting or absorbing source. The net effect on the causal two-particle exchange process is conveyed by the factor

The resulting change in a")[(1M2) is then gJven by

To exhibit the photon Fass dependence in the above equation, we deconnpse f (v) fi

vl% 1 (1 + v18) log -

1 - 9 (H,&?)

Then, using the fact that

we have

The resulting coefficient of a2/f2$ in (W.86) cancels the photon mass te rn of Eq* (-.W).

The remaining integrals in Eqs. (5-4.71, W, 87) can be p d o m e d in terns of one type of transcendental function, whieb will be descdkd later, bat the resulting expression is not very ifluminaiting, Rather, we now prop% to use these integrals, as they appear, to extract a numerical corrwquenee of the proces under consideration, f t is the rnsdification in the vacuum polafization edcdatian of Section 4-43, where it was recognbed that the significant quantity is the zero momentum limit of (FD+(K). According to the constmction [Eq. (&3.81)]

this quantity is

The two-part-iele exchange contributisn, to "fe integral is

The desired supplement to it is given by the v2 integral of the sum of ( M . 7 4 ) and (54.80;), multipfied by a2/12n2, and of (54,8(3).

Let us begin with (M.71) , first integrating over vs from v" + &'a to 1, The basic integral here is

1 1, 1 - vf2 dv2 - = log - = log

0p2+gy#z 81f6 -, ~~8 8 ~ ' s

One then verifies inductively, by differentiation with rmpect to v'%, that

Using these results, we find that the integral of Eq. (W.71) becomes (dropping the pl-irne on the remaining integration variable)

Turning to the integral of (-,SO), we first observe, through pa&iaI intwatian, that

Chap. 5

The sum of these two contfibutions, which cancels dM, is

To this is added [Eqs. (M,86, 8711

where we have wed the pdxxciprrzl value intepd

When we add (5-4.97) and (M.98). all non-physical parameters disappear, to give

v", dar ($ + - vOg(u))(l f vq lag - (M.IW) X - v"

For the remaining integrations, we perform various partial integrations, as illwtrated by

we exmples of the inteuab

(W. 10f)

md note the spif: i :e reu1.t

The outcome is expressed by

This increse in the vacuum plla~zat ian effect is raughlly one prcent, A quantitative statement wilt, be reserved for the more experimentally relevant spin + discussion.

The integals that must be pedormd in order to exhibit a ( M 3 haw stmetares eontaininf: a denominator and a logarithm which are different linear functbns of one vadabb. A standard function of this type is

" dt It a2 X"

O < % < X : I(%) = -log- = 2- f, t l - t .=l nB'

variously called Euler's dilogarithm and the Spence function. As one recognizes through the substihtim t -+ It -- t , followed by partial integration, this funclion obeys

which incovorates the fact that

(H. f 07)

Andogo- functions defined far other ranges of x are simply related to l ( x f , Thus, for x > 1 we consider

and n&f: that t + L l f prdaces

The function that effectively appears on changing the sign of X is

and the analogous relation for x > I reads

These are all aspects of one function, of course, but for numerical purposes we prefer to use l(%) as the standard function.

Other relations appear on making the substitution C 4 1 + t in (5-4.108), yielding

- log t = log x log(% - 1) - +t ) . (64.113)

This is given different forms depending upon whether x - 1 is greater or less than 1. In the latter situation we can apply (5-4.1 11) to get

1 < X < 2 : t(n) = log X log@ - 1) - I(x - 1) + +J((X - I )%) , (84.1 14)

whereas (64.112), together with the integral

is used to produce

With the aid of (64.110). these results are transformed into

and

which are interconnected by the statement of Eq. (64.106). A particular consequence is reached by parametrizing x as +(l f v) in the respective domains of Eqs. (64.117, 118). and subtracting the latter:

To illustrate the use of the dilogarithmic function, consider the second integral of Eq. (54.87). which can be resolved into individual integrals containing either v - v' o; v + v' as denominator, and a logarithm of v', 1 + v', or 1 - v'. We first observe that

p[ N v - V' log V' ,,V-v'

The substitution v' = vt then brings these integrals to the form

Another consequence, produced by the substitution v - 1 -- v, v' -, 1 - v', is

and, incidentally,

according to (64.106). Changing the denominator in (64.120). we have

Chap. 5

fog v" -

38 X =g - - ---

6 2 (W. 12q

Also required i s

Now it i s the transfomactian I + v'= (1 + w ) t that produe@ the fsm

dt 1 (%+@l & - log - 4- Flog(t - l )

,(l+a) 6 1

Fadhemore, we have /4l+of &

- lag(1 - v" I

- log - Q t 1 - t

w ~ c h uses the transfomation v ' = l - (1 + v)$, arid, with the transfamation v" - 1 + ((1 -- v)$,

l + " [ ( 1 ; v T -;( 1 ; ~ ) ~ I- -*+- log- log -

28

l + v - 2log2log- + #(v) - l - v l(* - l ( ~ ) + l(?)] +2log2.

Note that the relation (8-4.119) could be used to give this another form. Without going into further details about the integration, we state the result

for a(Mt) :

- ( ~ 3 = ~ # + - 12z 1238 { @(l+* [T+lOp-i-lw- " I + v I + v l - v

This elaborate structure can better be comprehended in the high energy (v -c 1) and low energy (v -c 0) limits. Thus,

where the contribution of order as comes entirely from the last term in the braces of (64.132). and

here the aa term arises from the first bracket in the braces of (M.132) and can be traced back to the partial form factor integral (8-4.126). The latter result is particularly interesting since the threshold behavior has been changed. This can be understood from familiar non-relativistic considerations. The effect of the Coulomb attraction between charges that are produced with relative speed v , increases the probability of establishing the state by the factor

100 Elrlrctdynrrmiccsll Chap, 5

whert: the approximation refers to the circumstances 1 >r umi' >> a, \vhich vadate the treatment of the Coulomb interation as a weak, non-rdativistic effect. Accord- ing to the non-relativistic relation

we have

which ixlded identifies (-.l%) with the mscfiification factar of (M.13.4). Incihntally, the elastic fom factor itself, in the non-refativistic limit, is emntiaHy identicd with the wave function for relative rmtcr-tion in the Coulomb field, evduated at the origin, with the nomalization set by the unit amplitude of the asymgtotic plane wave.

We could repat the vacuum pladzation computation given in Eq- (M,XM), using the explicit expression for M%(Mz). Instead of Boing that, let us make the falbwing approximate ohemation. A simple, but slightly contfived, dormula that intergolakes between the two limiting foms of E*, (W-133, 134) is

The result of perfoming the .uZ inteeation of this function appmximates the numctrical coefficient exhibited in (M.IM),

Now let us go through the analogous calculations for spin f charged particles. To describe the three-particle exchange process. we begin with [cf. Eq. (3-12.24)]

Comparing the appropriate part of the vacuum amplitude iWee with the eq~valent ampfitude

5-4 Photon propagation function 11 101

we infer the effective source

- r l t ( ~ ' J A l . P 4'4% - €)yAG+(% - x')YAz(x')

+yA2(x)G+(x-x' )yAd(x' - €)l. (64 .144)

The momenttun version is

- 72@)tlr(flPJ%"k) b

and the analogous absorption process is represented by

- 71 (- P ' l ~ i (- P)PJIA(- k ) IM.

The three-particle exchange vacuum amplitude is then derived from

X '12@)tl2(P')Yqlrr(k) Itff. (- m - YP')I. which is a rearranged version of

It can again be written its

where now

and tr, inacatef the trace, so nomalized that

trs 1 = 1" (M. xar The gauge invariance of the coupling (M.149), which implies the tensor stmcture

can be verified directly. What we must calculate i s the scalar function

h, do,. do, ( 2 3 ~ ) ~ b(II: --- p --. p' - k) trB f

r(P + k) 3- yd

l 1 + ~ " y y ( ~ ' + k ) + n r +&) + m Y v

1 + R) + m (W.1 m)

The matdx factors in brackets are reduced ~ t h the aid of the projection nnatriw in (W.153). Thus,

The matttrix prduGt af tM.153) then becomes

We specifically note the appearance of the term

which is the expected, infra-red sensitive, radiative modification of the two- particle exchange mecfianism,

There are two t y p of terms in (54,158) that involve a pair af yk factors. One is

yi lykYV(~ - yp)ylykyv(-- m - ?P')

= r?hk[zvlvkvP + (m + rP)yYy iy~yy l (- m - yp') = 0. (5-4.158)

which holds since

yA?/kyAyk = 2(yk)% =t 0,

while

zvv~Avkyt = 4 k

also produces the null structure ($)? The other,

since these axe equivalent with respect to the trace, and the latter is further ~ d u c e d to

the last step records the result of the trace operation. An example of a te rn in (H.1Slij with one yk factor is

which has already exploited the cyclic property of the trace. If we write

the =pression in ( 5 4 . M3) decomposes into

Chap. 5

where the projection matrices have been used to simplify the stmcture, Now, the trace of the product of an add number of y-matrices is zero. The proof is an immediate generalization of that for one y-matrix, based an anticommutativity with y,, which is @ven in Eq. (M.79). Hence the trace of (54 .1 65) mduces to

since

tr, yAyB = - AB,

while, in (54 . l66) , we encounter

(M, 167)

tfn(m - yfi)yk(m - cyp" = - trPI(y&k -t- Y ~ Y * ~ )

-- m(Pk 3- P'& (H. 169)

and

tr,(m - yfi) (m - ?p6) = m2 - pp" --Mfi2. (M, 170)

The immediate expression for the traee of the matrix in (H.153) is

This can be rearranged as

where one will recognize much of the spin O structure displayed in Eq. (M.51).

5-4 Photon propagatfon function I # 1485

Indeed, no new integrals are encountered in evaluating the expectation value of this function, as required for the analogue of Eq, (54.86) :

Following the spin 0 procedure, we first consider the domain M - M' > p, where

This gives the following contribution to the coefficient of cra/3?t2 in (54.173):

*ere the integration domain is that described in the context of Eq. ( 5 4 . 7 1 ) . UnIike the latter equation, the infra-red insensitive terms of (W.175) do not disappear on integration, and have been left intact,

The behavior in the region M - M' -o (u is the same as with spin 0, except for the factor that expresses the different form of acZ)[(M2f, M" 4 d -- N3 + 29133, where the additional factor of 4 is used to replaw ~ ~ / 1 2 n ~ by or2/3na. Thus, with the multiplicative substitutioll in Eq, (5-4.80) of vZ -- 4(3 - v%), we get the following addition to (M. 1751,

It will cancel the parameter 6M. The form factor effect is a little more elaborate with spin 4 particles since the

additional magnetic moment coupling comes into play,

The consequence for the trace calculation of Eqs, (L3.20, 22) is indicated in

106 Electrodynamia II Chap. 5

which has exploited the conservation property of the stmcture to omit terns containing k g . We again use the algebraic basis for this property, the projection matrices, in reducing the magnetic mornenl coupling. Th& is described by

and, similarly,

where the resurling yfi combination can then be replaced by - m, This gives, for (W.]L78),

3cx Fx2(M2 + 2mg) + - F2M2, 2n:

(M. 182)

where only effects of order a have been retained. The farm factors that appear here are

a Ft(v) m 1 - -!x(vI8 (W. 183)

2%

with [cf, Eqs, ( M . 6 8 , 77)]

Photon propagation function 11 107

The consequent change in

WU(~(W) = f iv t (3 - fl (M. 186) 3n

is given by

As in the spin 0 discussion, we shall first evaluate the integral, drP M.a(M'), which measures the vacuum polarization displacement of atomic energy levels. The v' integral of (B4.176), produced by appropriate modification of (5-4.95). is

where the added constant, - 8, gives the integrated value of the non-singular terms in (6-4.176) :

For the integral of (8-4.176), we observe that

The sum of the integrals of (6-4.176) and (M.176). from which $M cancels, is then

(64.191)

where we have introduced the numerical values of all the integrals of type ( M . 102),

dv vL~(v), n 2 1. As for the integral of (64.187), its contribution to the coefficient of aa/3# is

(84. 192)

which uses the principal value integral

Vt(3 - Vt) 8 --v*- V*@ -- vY)x(v'). v2 - v'% 3

The sum of (64.191) and (M.192), from which the fictitious photon mass finally cancels, is

Some significant combinations for this evaluation are

and

The result is expressed by

This fractional increase is somewhat smaller than in the spin 0 situation, bat it is still roughly one percent. The effect on the added constants of the energy displacement calculation is given by

where the unit [Eq. (4-1 1.1 14)J is 136.6 MHz. This represents a decrease in the 2s-level splitting of 0.24 MHz. I t alters the last estimate, of Eq. (6-3.168). to

W Photon propagattan funetfan I t l09

H : -- E2 ,,,, = 1067.93 MHz, (M. 199)

which is strikingly close to the nominal experimental value of 1057.90 & 0.10 MHz, The usual caveat a b u t still unconsidered effects continues to apply, however.

The integrations required to exhibit &(Mz) are very similar to those of the spin O situation, Such a relationship also agpears in the results, far the substitution v" +(3 -- V%), performed in all the terms of (5-4.132) that have such a factor, yields the precise spin 4 munterpads, as dispfayed below:

a l i + v I + @ W g ( M % ) = - v4(3 - V%) + -- + log -yj"-- log --

3 z 6 X.-v

~ - - - v " + v $-V' + 6a- 2

log- - 4%- 2 2

log v + Sv(6 - 3vq

The limiting behaviors here are

where the a% contribution again comes entirely from the last term in the brace, and

in which the a8 tern continues to s p ~ n g from the first bracket of the brace, with its origin in the fom factor: Indeed, as was to be expected, the multiplicative faetor of (Mew) is the s m e as with spin O [Eq, (H.136)]. A simpfe interpolalion fomula, which is wei&ted somewhat differentl;~ than for spin O, is

The reson far this shift in weihlfit a p p r s on comparing the two braces of Eqs, (M. X32) and (54.2W) in the following way :

l + v 3 3 3 - v % X log - + -v(& - 3 ~ 2 ) - --

l . - v S 2 2v (1 + v2)

When the inteqolation fomulas are ugd, with the weight factor Q symbolized by A for the mornent, the above combination &comes

- - # v E ~ ( Q ~ E ' - - ~ ) - - * ( * z ' - - - ~ ) ] , @ < I * (M.205)

The identifiation of the two expressions, for v <( 1, then gives

which, for simplicity, has twjen replaced with the nearby fraGtion 8. m e n the intevlalion formula (M.203) is used in the calcdation of (5-4,19?), the coefficient of aa/n%is fomd to b

as compared dth, the exact answer,

Warold has a qustion.

H. : Perhap 1 am overlooking a point, but shouldn't there be some mention of the annihilation scattering mechanism which accompanies the Coulomb scatte~ng proces that YOU have consictered, in campating the vacuum polarization energ-y shift ?

S. : Let me restate the question and, thereby, jog your memory. The modified photon projp~alion function has been exhibited in two forms, One [cf. ]Eq, (48.81)3 i s

md the other [Eq, (&$.83)] b given by

where the eonnwtion between them [Eq. (&3.85)] is repeated as

The weight function a(M9 characterizes an irredudble interaction process, the indefinite repetition of which is dexribed by the denominator stmcture of (H.209). To the accuracy with which we ].rave worked in this section, it suffiw to expand the denominator factor :

It is the last tern here that represents the annihilation interaction, the repetition of the two-particle exchange process. As we see. it does not contribute for k = 0, which is the approximate situation in the energy shift cdculation. Now. one might ask how the %me conclusion emerges on using the form (5-4.210). where the required quantity is the integxal

since A (M8). as given by (5-4.211). certainly incorporates the repetition of the basic interaction process. Let us just note that, to the required order,

and, inded,

Incidentally, I should draw attention to the relation (M.2141, written as

since it is analogous to the use already made of form factors, in improving the two partide exchanp contribution. The form factor o c c a ~ n g here is the one that multiplies B,(k) to eve B,(k), evaluated at R2 =: --- Map

The relation (5-4.216) is an approximate one which, according to Eq. (6-P.211), is precisely stated as

I t 2 Eleetrdynamlcs l t Chap. 5

This prescription is physically sensible since, as a probability measure, the vveight function A(M2) can be constructed from the absolute squares af emission probability amplitudes.

Electrdynamics, in its narrow wnse, is concerned with the proprties of those few particles whose darninant interaction mechanism are electromagnetic in character. These are : the photon, the electron (positron), and the muon (psitive- negative), There are also two kinds of unstable composite particles that have become accesible experimentally: positronium (e+e-) and muonium (p+@-). This section is mainly focwed on lpositronium. I t is the purest of electrodynamic systems, These atoms have fine and hyperfine structures that reflect completely known electromi~petic interactions, and their instabitit y only involves decay into photons. In contrast, nnaoniurn invokes the weak interactions, which introduces the neutrino: p+&- -.. e+ + e- + 23".

The positroniunr structures are essentidly non-relativistic, with a grass energy spctrurn dven by the Bohr formula that is appropriate to the reducd mass of +m. These binding energies are

X 6.8029 =2 -RY =-

2n2 ev.

P??

me states af given principal quantum number $2 =.: f , 2, 3,. . . can be further lahled by the quantum numkr L -- 0, 1,2, . . . of relative orbital angular momentum, the spin quantum number S == 0, I, and the total angular momentum quantum number ] = 0, 1, 2,. . . . A particular state is designated as n'S+lL,. Relativistic effects and electromagnetic i~lteractions other than the Coulomb attraction induce a fine stmcture splitting and a byperfSne stmeture splitting, Unlike hydrogen, with its Izge mass ratio, the fine and hyperfine stwetures in positronium are of the same order of mwitude . Particularly interesting is the hyperfine stmcture of the punc-l state, the splitting between the X and 1 ISo itevels. Positroniurn atoms famed in excited states wiH radiatively decay down to one of the hyperfine levels of the ground state. These atoms eventudy annihilate completely into photons, We be@n with a dixussion of the annihilation mechanism.

The nature of the photon decay of psitronium i s governed by a selection rule mociated with the concept of charge reflection, In general, charge reflection (Q -. - Q) converts a 8ven state into a different one. But, far eltectrically neutrd systems, another state: of the same kind is produced and one can introduce the eigenvedars of the charge reflection operation. With two particles of opposite charge, as in posifronium, there is a symmetrical and an antisymmett-ical cambination of the two ehwge amignments, corraponding to

Now, the effect of interchanging all a t t ~ b u t e s of the tvvo padicles is controlled by the statistics of the particles, which, for F.D. particles, demands a net sign ehmge. When the spatial coordinates are interchanged in a state of orbital quantum number L, the spherical harmonic pveming the ang!e depenknce resgonds with the factor ( - 1)" [cf. Eq. ( s7 .21 l]. As for the spin functions, tfiplet and singlet states are, respectively, symmetx*ieaf and antisymmet~eal, as symbolized by the factor --- (-- l)$. Thus, the fulE expression of F.B. statistics for the electron-positron system is contained in

Accordingly, the state is charge symmetric (r, = + l ) , and the BSI state is charge antisymmetric (P, ==: - X ) .

The state of a system of n photons is represented by the product of the sources, n in number, that emit or absorb these particles. Since evefy photon, murce, as an electric cument, reverses sign under charge reflection, the charge p a ~ t y of an n-photon state is

Hence, if charge parity is to be maintained in time, the I. ISo state, with r, = $- 1, can anfy decay into an even number of photons, rnost probably lpc 2, while the f %SE state decay is restricted to an d d number of photons, rnost probably rt 3, since a single real photon is excluded, This inhibition in the decay mechanism of the 8.5, state wilX result in a considerably slower rate of decay, compared to that of 1 So positronium.

There is mother reflection asyzect of thew states that desewes mention, f l: refers to space parity. The space reflection m a t ~ x is [Eq. (2-6,39)]

which inlplies that the intrinsic paxity, charactenlzing a particle a t rest with p'== + 1, is i , This value of the intrinsic parity (which could equally well be -- i) is independent, of the electric charge value. Any arbitrariness in definition disappars for the two-particle pasitronium states, where the inlfinsic parity becomes (-& iI2 = - l, That is suprimposed on the orbital p&ty which, for a state of angular momentum quantum number L, is f -- If Accordingly, the complete space parity is

Chap. S

For consistency with conventional notation, we shdl then desipate the charge parity as C :

and note that

To the extent that C and P, or at least the product CP, are exact quantum numbers, the distinction btween singlet and triplet spin states is precisely .maintained. The singlet and triplet classes of psitronium are sometimes refemed to as para and odho psitronium, respctively.

The S-levels that constitute the ground state of the cos structure have 6" == - I, which asseds the intrinsic parity of the twa-particle system. Hence, the 'S, padicle, with zero total angufar rnontentunn and odd parity, would be described by a peudoscalw field (41, while the %Sx particle, a system with unit anplar momentum and odd parity, is characterized by a vstor field (4,). A phenornenolo@caf description of the two-pfioton decay of ISo psitronium is provided by the gaue invadmt coupgng

and, indeed, this pseudwcalar type of coupling has $ready been exhibited in Eqs. (S13.76, 76). As noted in that context, it implies that the two photons are orthogonauy polarized. There: are two pomible gauge invariant combinations for the unit spin system, namely,

and

comespnding to the two ways in which, a four-dimensional rotationally invadant combination for a pair of photons can be famed, We shall later d e ~ v e the precise combination that is appEeable to oxtho psitronium, and also point out an essentiaf &fference in the nature of the two-photon and three-phobn couplings.

The decay rate of pwa psitronium can be quicuy obtained from the annihilation cross section of free padicles given in Eq, (3i-18.74). In order to deal d t h annihgation in shliyXet stales, the factor of that was introduced in aver@ng owr all spin o~entations must be removed. Then the rate of annihifation is produced by multiplying 4a by the relative p-iele flux v where y% is the

non-relativistic wave function for relative motion. I t is evaluated at the origin to represent the conditions of the relativistic annihilation process. Accordingly, the decay rate is

since

in which the positronium Bohr radius is given by

The implied lifetime for the ground level of para positronium is

where it has been helpful to note that

= or" Ry = 3n X 136.6(2)1 X l@) secse, (5-5.17)

according to the energy unit of Eq. (4-1 1.1 13). Now, let us repeat this derivation in the spirit of the phenomenological coupling

(W. 10). as derived from (3-1 3.76). The field product that appears here, a specialization of $(X)$($), is to be replaced by the two-particle field of the interacting system, )(%X'). The field of a given para positronium atom paums a rather simple* non-relativistic character. For the two spinor indices we have, effectively, yO' = + 1 ; the charge labels in this C = 1 state aie combined in the symmetrical, normalized function

and the normalized spin function of the antisymmetrical singlet state is

2-% 6u,4*. (M. 19)

There is also a normalized (equal-time) wave function for relative motion, $(r), and a wave function for center of mass motion [cf. Eqs. ( 6 1 .l 13, 116)] which in the rest frame is

1 16 Elardyntmrfcs ll Chap. S

since the m- of psitranium is very close to 2%. We mwt f indy remmk that the normalization condition for two identical padicles cont&ij,s tbe factor + to avoid repetitious counting. Thus, the field iated with. a s p i f i c atom is indicatd by

where the yO" + f restfiCtion is implicit. Now, the a n t i s p m e t ~ c d matrix yOys anticommute with yO, but connmutes with the charge a d spin matrices, Accordingly, it connects equal values of yO and oppsite vdua of chwge and spin quantum numbers, since dIt the asmciated matrices we antispmetricd. fll"his is illustrated by wy6q$ = w y S # = (--. q+pyfJ$.] fn the yO' = $- L subspace, the an t i s~met r icd f y f J ma.trix effectively reduce to cr l(f,,,#, apa& from a p h w factor, as exhibit4 in Eq. (%13.72), where heficity fabls me used, This resdts in the equivalence

where the pbenomenolagical para positronium field # ( X ) has been introduced to charae-te~u: the center of mass motion. The interaction tern of Eq. (3-13.76) is thereby repheed by

Its preaction for the decay rate is

which, naturalty, coincidm with (H.f3) , The vacuum amplitude that descxiberj three-photon decay is @I. Eq. (%X%%)]

We exhibit the coefficient of iGi];ld~:,l,t, With the phcfcf field taken as # e x p [ i p ~ , g $ $, mB =;; 0:

(dx) expfi(2fio - k - k" - R"")] (dwk dwh. dwL. .)lBi)wq

which &splays only one af the six ways of assigning the three photons. In the approximation of free-padicle motion we have

and we shall also adopt the gauge

PO@ = 0,

for at1 photons. This leads to the substitutions

WY@C~ ---- yfk ---. Pa)l = Wfm --- yp,ty@ -- thpyeya

= " I p y ? k ~ ~ ~ , k , (g6.29)

and

where the mqnetic moment interaction wit h the efeetromagnetie fidd has become explicit, The latter will be expressed in three-dimensional, form,

which uses the relation

" P E == - y5@E,

or, alternatively,

and applie the gauge condition (G5.28) in the rest frame of ea. The braced expression of Eq, (66.26) can now be written out as

where we make explicit the result of interchanging k , e and k", e". The v a ~ o w spin grctdltcts can bt3 reduced with the &d af the relations

1 18 ECerodymamfca II Chap. 5

The latter combination appars muttiflied by yti. Tht: resulting Dirac field structure, we^+, does not contribute under the assumed conditions, however. That fol2ows by combining the equations of (65,f;) lq into

in which the last vemion is the p. rest frame evduation. The form that emerges for (65.38) is

-- e x k * e" X kE'y e'ko" k@ko"ko"""(e a'y * B" -+ -t-' ef "y. e

- e * e""y @')l + cycl. perm. (s5.38)

The effect of adding the three cyclic permutations is particularly simple for the second set of t e m s in (5-5.38), where it produces the symmetrical combination

In the elements e X k and ek@ we recagnke the veetorial aspects of the magnetic and electric fields that are associated with a given photon, Indeed, in (5-5.38) we have a combination of individual photon fields that is produced by the total field stmcture

With the proportional identification of +wqykJI with - a,+o, we encounter a pa&icular linear combination of the two stmctures given in Eqs. (5-5.1 1, 12). But what is not antiGipated in those phenomenological forms is the addition& factor of (kp&'fikH'po)-l that is exhibited in ftr-5.38). This constitutes a f o m factor, coupling the photon fields non-locally to the ortho psitmnium field. In retrospect, W recaeize that the two-photon probabilily amplitude, expressed in terms of field strengths, afso has the factor (&,ktpO)-" but the latter is eompletdy fixed by the kinematics. We are being reminded that the intuition usually awciated with phenomenological couplings refers to a system with inverse dimensions that are large compared with the momenta of the excitatians that it emits or absorbs. Here we have the opposite extreme since [cf. Eq, (66,lfi)j I/a is much smaller than m, the characteristic unit of the annihilation process. Accordingly, there is non-locality on the latter scale.

The irnpartance of the form factor is emphasizd by considering the limit as one photon enemy approaches zero, thereby simulating a physical process in which the presence of a homogeneous magnetic field induces an ortbo psitronium

atom to decay by tw-photon emission. The singular limit of zero photon enerw indicates a seeular gowth with time of the probability mplitude, which is actuatfy limited by the finite splitting between the ortho and pars pound levek, and constitutes the bmis for measuring this quantity, If this hsmogeneous field is designated by B@, the field structure infemed from (s5.N) is

The last term does not contribute in two-hdy decay, where the photons have equal! and opposite momenta, since

e x ( e b kk") + a ' x ((e x k) = - (Er + kf))e * e' 2;: 0. (k6.43)

What remains is just the pseudo-scalar coupling that characterizes the two- photon decay of para positronium. This stimulated decay takes place in ortho positronium atoms that are plarized parage1 to the magnetic field, or, equivalently, have zero magnetic quantum number reli-zlive to the mqnetic field direction. We have arrived a t a deset-iption of the mametic field induced mixing of para and ortho pasitxonium, which is customadjy handled by atomic perturbation theory. Some details of the latter, including the removal of the limitation ta weak m w e t i c fields, wiXE be given later,

To proceed with the cakufalion of the three-photon decay rate, we consider the particular state af the ?SI, C = - l, system that has zero mapetic quantum number, as above, relative to an arbitrary dimetion with unit vector v. The associated field, analogous to f&5,2X), i s

2'i2[2-1iq 8e,-9p] [2-1/1 fi,,-,.]+(r) (&m dcu,)llz =p@&), ( 6 6 . ~ )

and

4NxfPqv * y+(x) -+ *'*fi24L(Ofv +(X). ( ~ 5 . 4 5 )

The decay rate is then inferred as

where, intrdueing unit vectors n, along the photon propagation directions, we have

! X , E l e d y m m l c r I t Chap. S

and the summation in (66.46) is extended over at1 posible psfakatians. The factor of -Q then: removes the reptitious eoun"tn:g of the photons, She@ the final rmdt is independent of the vector v, it is convenient to averwe aver that direction, fimt, The polarization mmma-lions are perfamed using. the dya&e refation

which expfesa the completeness of the two e vectors and n. These summations are illustrated by

= 2 x [l -- (n' B X a)' + 1 - (a'* e)$ - 2n n'] = 4(1 -- n R')" ((M.49) Q

= 2 2 [ ( l - n * n')P(e @')P + (e - n8)'(e' . n)' + 2(1 -- n * nf)e e' B * n' et n] 61)"

= 2 2 [ ( l - n n'18(1 -- (n' a)$) + (l -- (n n')2)(n' * e)' -- 2(1 -- n m n')n n'(n' 0)7 6

= 4(1- n m")", (&6,50)

X t turns out that all the other types sf terms combine to caneel, thus giving

w ~ c h him been written: in the unsynnmetrieaI form pmi t ted by the equivdence of a& the photons with wspct to the integration, This stwe of the calculation is

Qne @vks invharrt fom ta 1 - n * a i k

kk" I - n * n L - h a T --&g% kk"

hPk P (kk' + kRU)(kk' + k'k") ' (H*=)

We shaaf p u p two of the photons into a system of m m M, as in&catd by

where

and

(fZm)Z = - ( K + ktl)% = M2 - 2Kkt'. (5-5'56)

In the rest frame of K , we have

(66.57)

where a is the cosine of the angle betvveen R == - kband h". Performing the z integration gives

Then, since

1 dcu, dwkPe (%E)% 8(K + kf r - PP) ;=; - (65.69)

(4%)"

the remaining integration is proporPianal to

in which

21 t;: M2/4m2,

Successive padial integrations reduce the latter integal ta

which evduation makes use of yet another partial intepation to get

f 1 dzc - log -- =

3%

@ f + @ =X"

Chap, 5

(5-5.62)

as an application of Eq. (64. lf5) . Putting things together we find the decay rate to be

and the lifetime for the eound level of oflho positroniurn is

The energy spectrum of positmnium is first approached by applying the results of Section 5-2, spcifically the rest frame enerw oprator of Eq, (&2,XM), where we now have

1121 -- m% = m, y =. Qm, el = - ez = e. (65.W)

This gives

The simplest application is to the singlet levels of para psitronium where, effectively,

thus reducing (b5.67) to

5-5 Positronlum. Muonium 123

In order to find the first deviations from the gross structure, the spectrum of the non-relativistic energy operator

we apply the result of Eq. (5-2.148), with m replaced by fm in accordance with its origin in the non-relativistic energy operator [Eq. (5-2.144)]. The consequence, expressed as

enables us to present the expectation value of (54.69) in the form

The elimination of the potential energy gives

since, in the Coulomb field [Eq. (5-2.152)],

We also know that, in the state of orbital quantum number L [Eq. (5-2.155), with t -,L, m -, +],

The outcome is an expression for the first terms in a power series expansion of the para positronium fine structure :

Notice that the reduced mass formula of Eq. (6-2.163), with M = m, p = fm, j = L, reproduces this result, except for the numerical coefficient 11/16. [That too would be right if, consistent with its reference to M )> m, the last term of (62.163) were replaced by - p%r4/8(M + m)n4.]

The spectrum of ortho positronium is considerably more elaborate. We must take into account the spin-orbit term of Eq. (&6.67),

124 El=trodpanrTcs I f Chap, 5

and the tensor spin-spin coupling also exhibited there:

For a given total anefsr momentum quantum numhr J , which values generally are ] = I, + I, L, L - l, one has only to take the expectation value of (68.77) in the state ntJ, according to

The factors appearing here arc:

and [Eq. (4-Xl,tOl), with aa --, 2aQ]

For the particular example J = L = 1,2, . . , , the spin-orbit errerw shift i s

The tensor interaction (6-6.78) is more complicated than the spin-orbit coupling, fox it can change the orbital anplar momentum while maintaining the orbital parity ( - l)&, thus mixing the two t w s of states with L -- J -& f . Since there is no mixing for cJ == L, the spin-angle. factor

must have an eigenvalue in that kind of state, There are just two ctigenvalues for this combinatian, corresponding to the unit spin gossibilities

which yields

(3(8 * n)" 8%)" I* - 2.

To learn which of the* is the comwt elgenvdue for J =;; L, it suffices to me quaiitative arguments .that are w ~ p t o t i c d y accurate for large L. Tbe eigenvdues of (W.%) disltin@sh two situations that can be dwMiM as S b h g parallef (antiparallel) to n, ar orth~gond to n, rspctively. Since the unit radial

vector n is orthogonal to I;, the first of the two situations detailed in Eqs, f5--5*84, 85) can be characterized as one in which L and S are odhogonal. On inspecting the eigenvalues of L * S exhibited in (5-5.80), we recognize that (L* S)'/IL, for L > 1, is 1,0, - 1, ewresponding to J ==: L + 1, L, L, - 1, rapectively. Accord- ingly, the asymptotic situation L = S - 0, in which (k5.83) has the eigenvdue 1, occurs for J = L, Naturally, the unit eigenvalue that appears for J = L can be derived in a more formal way, which i s not very lengthy, tout such an approach gives no understanding of why that particular eigenvalue appears. The energy shift now deduced from (5-6.W) is

On adding (L5.82) and (&5.86), we get the enerw displacement of the non-S ortho positraniurn $JJ states relative to the para positronium 'lJ levels:

The first examples of mixed ortho positranium states are: + WL, %P2 + where both orbital states must klong to the same gross stmcture levef , if the mixing is to be appreciable. Hence, for n ==: 1, where only L =;.. O occurs, and n = 2 with L =.: 0, I , no such mixing can appear. Since availaibb experimental data are limited ta n = 11, we give no further details about the mixing of levels. The ortho- para splitting of the ground S-level corn= entirely from the last term of (65.67). Recalling that

we get

which is also produced by the hypedine stnxcture formula of Eq. (4-17.16) on placing &l, = m, Z = 1, S = f, g, = 2. Even a t the present level of accuracy, this is not the complete story, however.

Qrtho positmnium, with C = -- 1, decays into three photons because a sin@@ real photon is excluded kinematically. But a single virtuaf photon can be emitted and reabsorbed, 'which leads to an additional energy displacement relative to para psitraniurn. The exchange of a virtual photon is dwFibed by the interaction

116 Efarodyrrzmfcr ll Chap. S

where, in consequence of the covariantly stated correspondence of Eq. (&5.45),

Since the annikiiatian mechanism involves the exchange of the mass %m, we have, effectively,

and the annihilation coapEng &comes

the fast exprasion states the phenamenolo@cal interpretation of this term, as a m a s displacement. Hence,

and the complete: statement that replaces (65.89) is

A recent expe~mentaf value is

The agreement ta within percent is very gmtifying, particularly in view of the anticipated presnce of: theoretical modifications of relative order a.

But before considering the latter, let us fulfil1 the promise to discuss the ef feet of a m w e t i e field in mixing oxtho and para positroniurn. In the S-states of relative motion, the coupling vvith a magnetic field comes entirely from the spin magnetic mamlents. This interaction enere , with a particular assignment of the charge labis, is

The a n t b y m e t r y ia the two spins implies a vanishing expectation value in the singlet and triplet stales. Indeed, the sole effect of (k6.97) is the expected one, of mixing the l&, C = X, state with the %S,, C =: - 1, state, Only the mqnetic level m = O of the %S1 state is coupled with the slate since the an@= momentm

about the z-=is, the mqnetic field direction, is still consc;med, The matrix element can be inferred by noting that

since al + oz vanishes in the singlet state. Accordingly, with a pmissiblt3 choice of phase?, the submatriar of the energy operator for this pair of levels is

whik the amplitudes of the two states, as determined by the eigenvector equations and the normalization condition, are given by

These amplitudes enable one to compute the decay rate of a mixed state :

r = I+.I.IZYPPR f ~ti-s.xo3)

For weak magnetic fields, as &fined by

@H/m << 9 AE, (&6* l@&)

where

,?!+!E = Eortbo - Em. (M. 106)

the energy eigenvdaes are

Corresponding to these two alternatives, which describe perturbed ortho and pas& levds, respectively, we have

(M. t 07)

128 E l a r d y n a m l a I1

and

Chap. 5

The wwiated decay rates are

the first: of which describes the incremd rate for ortho pitronium decay owing to the induced process of two-photon emksion. Here is the mechanism that was earlier discussed qualitatively as an application af the f o m factor associated vvith three-photon decay. The general situation of arbitrary mwnetic field strength is described by the perturbation theory results just obtained. In particular, the strong field limit, where the inequiility af (g6.10.1C) is reve +wra = F #ar&, and, thus, both decay rates tend to a common limit, the equally weighted average + y,,,).

As a first step toward evaluating the mocfification of order a, in the odho-para splitting energy, we consider the single photon exchange of the annihilation mechanism. The two elements that have been combined in the calculation are: the primitive interaction descnibing the interconversion of photon and electron- positron pair; the photon propagation function- For the latter we must now use the modified propagation function

while the primitive interaction is altered in accordance with the form factor generagzation

a 11 Y A ( R ) W F,(k)yA (4 -l- 5 S pg(k)flF(k) e

(&G. 1 12)

The propagation function calcuEation is immediate, The evduation for k2 = - 4mg @ves

1 D(R) - - 4mg

(&&. 1 13)

which constitutes a deerew in the annihilation contribution to the ortho-parit

splitting by the factor

Twning to the additional magnetic moment coupling that appears in (b8.112), let us note that

In the rest frame creation process, for example,

and

which means that the primitive coupling yA(k) is multiplied by the effective form factor

The components of this form factor, of which only the real part is significant, are given in Eqs. (M.183, 184, 185) as

Ft = - (1 - *%(v)*

which leads to

In the non-relativistic situation of interest (v < l), %(v) reduces to unity, and [QS. (5-4.87,89)1

according to the small v limit of (64.131), which uses the properties

Notiee that the reference to the photon mass, the sign of inf-ra-red sensitivity, has disappeared at this level of accuracy. The implication fur the modification of the single photon exchange process is given by the factor

We recognize in X f (na/2v) the function that gives the non-relativistic evaluation for free particles [cf. Eqs. (8-4.135, 138)1. The replacement of this

evaluation by the one appropriate to the bound psitronium atom i s already incorporated in the initial calculation. Accordingly, the actual modification of that calculation is @ven by the other factor of (&6.128),

(W. X 24)

Tbe complete ma&fication of the singge-lphoton annihilation part of the or-thu- para splitting is, therefore, provided by the factor

With the level of daription altered by the ad&tional factor of a, one: must now also consider two-photon proeaes, such as the annihilation mtlchanism of par& positroaium. The effective two-photon source "eere is

and the vacuum amgIjlude desGb.ing a twephoton exchange is obtained from

We have the option af pdoming a causal or a nan-eausal evaluation of this coupgng. The Iattctr faciIitata simpllifieatiuns basd on the spcid natare of the % sate, and we introduce it by replacing (M, t 27) with the momentum vctrsion

'The spaicetime stmcture of the p&icle fields is sufficiently. indicated by

rlrl(x) = exp(- i#$tx)tC", #%(..l = exp(i4132x) Ib (5--.5.129)

where, in the positronium rest frame,

PI0 == Pz@ 2 2 n ~ . (6-5.130)

Accordingly,

and

In view of the momentum restrictions expressed by the delta functions, only one four-dimensional rnomentum integral occurs. We shall express this through the: change of variables (P1 = Pz = p)

--+ # p 4- k , k' -+ &p -- K, (5-6.133)

and thereby present (65.128) as [- ie is understood]

The nature of the 1s state is such that only pseudoscalar and pwudoveetoir combinations of the particle fiel& can be farmed, The products of three y- matrices in (6-5.134) meet this requirement, since

where the dots indieate other terms involving a single pmatrix. The introduction of ihis simplification in (5-5.134) reduces the latter to

which. uses the relation

Let us note here that, in accordance with the spin Q character of the system, the psudovector combination is proportion4 to the padient of the p a d a a l a r ; or

This is verified in the rest frame of the vector p, where the prapnty Yoqt = #, and the antisymmetry of p, implies the vanishing of the field stmcture eontdnhg ykye, which commutes with yO. The andogous relation

makes similar reference to the property = $*" A partialar consequence is @ven by

To evaluate the momentum intepal of (&6,t36), we use the represntation

ds, dsz dss sl, expf - is,(k -I- +$)T expf - iiiz(R - ifi)%expf- is8(kS + m%)]

OR introducing the! praprty

p% :, - &pp,

the caefficient of - i s in the expnent hconxes

The basic structure of the momentnm intwal is, then,

i 2s -- - g,v 3- *%2Vgfi,Pv

according ta the momentum integrals of (&8,57) and (&lQ.ci"fi), More spcificafly, we have

The combination pgg,, -- ;b,p, will not contribute because af the null curl fmm of the vector (65.138). This gives the effective evaluation

f L = g-- #g,, - I ,

(4~1% m2 (w.r&)

where

On employing ( 6 5 . X 401, and in trducing the pseudoscalar field camespndence of Eq. (5-15.221, the vacuum amplitude (g5.136) bcornes

from which we infer the action contribution

This has the stmcture af a mass tern and identifies a (mass)Wiqfacement, which is complex;, co~espnd ing to the instabgity of the pa&icXe. n m ,

and (m, g 2m)

For a consistency check k t h the earliea calcutilti~n that is record4 in Eq, (&6.24), we note that

where the final evaluation employs the substilulion

The result infened from (&6,151) does indeed aeee with (Ei-5.24). The evaluation of the real part of I,

proceeds by separating the two regions, u < .i) and ac > 4. In the fint of these we have

and the introduction of the transfomation

1 - 2% == 29

brine this contribution to the fom

For the re@on u > we replace (SB.fEi5) with

Using the transfsmatioa of (&6.163), this cont~bution t-o Re I is found to b

S 5

and

Re I = g(1 - log 2).

This Efives, as the two-photon annihilation contribution,

ara amwra = - 4(f - log 2) - z = - X - - l o g 2 a a ~ y . (5-5.161) m% 7t

Xn addition to the annihilation processes characteristic ol psitroniurn, there are conventional interaction mechanisms in which the two partides maintain their existence. We shall follow tbe clasificrrtion of Section 5-3, where the exchange of transverse photons was superimposed on an initial desc~ption employing the instantaneous Coulomb interaction. But, in view of the high energy nature of the process now under consideration, we prefer, from the hginning, to regard the particles as essentially free during the photon exchange acts. Besides the exchange of two transverse photons, we must consider thosct effects of the Coulomb interactian on single transverse photon exchange that are not summarized by the use af the wave function $(Of, in conjunction with a static s@n interaction. There is, for example, the pssibility of an additional instantaneous Coulomb interaction while the transvers photon is in flight. And, we must ceaw to ignore completely the momentum asociated with the relative motion of the particles [this is the useof #(@)l+ h we have often explaited in single-pa&icle contexts, the daired short &tance bhavior is introduced by a first iteration of the Coulomb interaction on the wave function +(Q). The ptocesses we have just enumerated constitute all pssibfe ways in which a transverse photon can be combined with an instantaneow Coulomb interaction. In effect, then, we are interest& in the totdity of two-photon exchanges. [The inclmion of a repeated Coulomb interac-

, since it contains no spin-spin interaction,] Xt is then slightly simpler not to use the decompsition into instantaneous. and t r ansve r~ interactions of Eqs. fk3.131, 1321, but to work with the cova~an t propwatian function

With the introduction of causal labefs for the pa&icle fields, the elflective source of (66.1126) &comes

The two padieles will be &theshed by s u h e ~ p t s a and b, so that the vacaunr amplitucfe for two-photon exehmge is in&ieatd by

td;b E l-rodynmicr E l Chop. 5

It suffices wain to use the simple form of the pa&icf e fields @ven in Eq. (k6.129). As a conwquence, we have k" - k, and the vacuum amplitude reduces to

When space and time carnponents acre exhibited sparately, arid the tern with only t h e compnents dixarded, this expression becomes

In view of: the y0 eigenvectof pmpdks of # and JI* in the rest f r am of p, snly ant even numhr of y matrices can survive in the individud terns of (&6.166), The resdting sirnp5fications are illustrated by

and

in which only the dmircjid spin structure has k n retaind. 'The spa~e vector of the latter result can then be ro ta t iody avawd,

What emerges for the two terms of the vacuum amplitude (66.166) is

where, written covariantly,

(dk) I (PR)' 1 11 =Iw(p) -r(Kp+&Y

and

The sum of the two terms is

(M. 172)

(M. 173)

and the two contributions that appear under the integral sign can be identified with the exchange of two transverse photons and one transverse photon, respectively.

The integrals are evaluated with the aid of the representation

where

P - p k v = (R- &pu)r + m W (66.176)

After the redefinition 4 - 4pu -, K, the basic momentum integral encountered in I is

(M. X 77)

Now note that

which eonve&s (&G. 197) into

H, : Surely something is wrong here I That last integml daesn? texist I

S.: Indeed. But X was about to recall the spctcid calculatianal rule that accompaniw the technique for intrdaCing the shod distance bhavior sf the wave function. It is stated in Eq. (415.45) and implies that the particle propagation function (p' + mP - ie)-l, evaluated for fie = m, is to be replaced by (ps - @)-l, E -r 0, which enten integrals as a Cauchy principal value. That is what we are encountering here, with the quantity m%% playing the role of p'. As physical evidence for this identification. note that the first term on the right side of (66.173), which is contributed by double transverse photon exchange, does not prduce this kind of integral:

I t is the second tern of (66.173), which combines one transverse photon with the instantaneous Coulomb interaction, that is responsible for the singular integral of (66.179). I t is therefore correct to invoke the prindpal value rule.

One could have i n c o ~ r a t e d it explicitly earlier in the calculation, but it seemed simpler to wait until the need tzecame evident. The inteeal. that should appear in (gEi.179) is, then,

and

The vacuum amplitude (k5.170) has now been evaluated as

We r e c o p k in expP(iPzx) the f ree-pndicle form of the two-part,icle field $(X%) that is msociated with an emitted positroniurn atom. The field X

e q ( - refers similarly to a detected atom. What replaces them, in describing the 'boand system, factors into a normalized center of mass function, which is removed by the spatial integration of (&5.183), and a wave function far relative motion that is evaluated at the origin : #(Q). The resulting coefficient of -- i dro in (66.183) is the desired energy shift,

with

Q," CTb = para: -- 3 '

This contribution to the o&-tfio-parn splitting is, therefore,

There is one other effect to be considered at this level of description. It is the a/2a nnodjifimtion of the magnetic moment, which nruitiplieies (65.89) by

The v a ~ o w eont~butions to the ortho-para splitting of order a8 Ry, as contained in Eqs, (G6.125, f 61, 186, 187) are ma& explicit in

and the resulting modification of Eq. {&5,95) is

This represents ai decrease that is close ta Q percent, reducing the numefical value of (k5.95) to

E,,,, -- E,, = 2.0338 X l@ MHz, (6-6. f W )

which greatly improves the comparison with the experimental value even in. Eq, (5-5.96).

H[. : X see that you have reprduced the old KarpXus-Klein result [cf. Salelrtea! Pa$ers on Qzllarctgm Ekctmdynamics, Dover, 19581. But, with no reference to such irrelevancies as divergences, heavy photons and inft-a-red cutoffs, there is quite a conceptual improvement, Also, since elementav arpments replace the machinery of the two-particle equation, the calculation is ~ e a t l y simplified, Presumably one could, now go on to the next level of descfiption in much the same way I

S. : f should think so, at least with regard to effects of relative order as log Ila, but we are not yet prepared for an or2 computation. However, 1 would prefer naw to discuss the closely related hyperfine splitting of muonium.

For most purposs, rnuonium behaves like hydroen with a lighter nucleus jm,lm. = 206.77, m,,/m. = 1936.11. In particular, the hypedine splitting of the gmund state, which is the quantity acealbie to memurement, shodd be desc~babte in large part by the tfxeov developed in Section 4-17 for m immab2e nucleus. But there are dynamical modifications involving the m m ratio m,/m,, and we now proceed to evaiuate them in the spirit of the precdn-g psitronium discussion. There is no counterpart to the annihilation mechanism, of eau=, and our attention moves directly to the &scumion of two-photon internetion prmwses as described, for pasitronium, in Eq. (M.1615). The only changes that mwt be introduced refer to the occurrence of unequal masws, as indicated by

where

The earlier discussion can be followed to Eq, (&5,170), but Raw f l\m2)(1, -+ I z ) is replaced by

The reprerzentatfons

are combined in

with

The integral (5-5,176) continues to apply, with 4m2 -+ M" v 4 V , and the structure of (X/m2)X, as inferred from (&5,177), is replaced by (-- ie i s omitted)

Now we notice that

md a partial ixrtegration converts (&5,l97) into

the combination in brackets can also be presented as

The v integation i s performed first. The computation rule af Eq, (66.181) is intrduced by witing

Ii 1 -Bv- m I

2 T/13 1 + [ (m, - m , ) / 2 m (v, -j v,)

wherc4 the firnits of V i n t e~a t ion are 3 + [(m, - m,)EWv+ and - (1 i- [(m, --. -- m,)/Mjv-). This @-;v=

The other neded inlegal, apparing in

They are combined in

P 1 [(m, - ms)/2W (V+ -- "U-) -g&[ ] = - 1 + [(m, - m.)lWv+ 2 [l + [(m, - n,)/&Ml(u, + v-)l' log 1 + [(m, - m,)lMlv-

where the brwket OR the left side represents tbe combination of Eq. (M.2W). NW we have only to abseme that the fight-hand side of (5-5.205) is reproduced by p d o m i n g the differentiations in

(b6.206)

This immediate1 y yields

mid (M, ]Em) becomes

i 2 m# log -- * (&5*2013) ( 4 ~ ) ~ mBS - m, m,

The muanium energy c2ispfacement can be inferred from the positmnium result of Eq, (S6.f M) by the substitution

together with the appropriate form of

This contfibution to the muonit-rm hyprfine splitting is, therefore,

On comparison with the energy splitting of the elementary theory, as derived from (417.16) by the substitutions M = M, +m,, m - me, Z = 1, S = f . g& = 2:

we em chmade~ze (&6,211) as a fractional modification given by

both at which are q u a t d only with sufficient accuracy for our presenl p u p @ . Most of the discrepancy of 8.6 MHz is removed by incorporating the a/2n modifies- thn of the two m w & c moments, which increases (M.214) to

The &dual h m p n c y , which is now in the opposite sense, is 0.9 MHa. As

144 Eiec)todynamiu II Chop. S

described in Eq. (4-17.1 10). the effects discussed in Section 4-17 imply a decrease in the theoretical value of

thereby reducing the discrepancy to 0.8 MHz. But this is just what is produced by the mass effect described in Eq. (66.213).

- [1.8 X 104][4.46 X I@ MHz] = - 0.8 MHz. (66.218)

Thus, in contrast with the hydrogen hyperfine stmctm, purely electrodynamic mechanisms snf f ice to give excellent agreement wit h experiment in the posit ronium and muonium systems.

H. : I am disturbed by one thing in this comparison with experiment. You have taken into account effects of order as that arise from the interaction between the particles. as in (56.217). but not modifications of the same order of magnitude in individual particle properties- the magnetic moments. Isn't this inconsistent ?

S.: You are right, in principle. But, in practice, the numerical coefficients in the effects you mention are suf£iciently small that our limited comparison with qmkent, a t the lwd of some tens of parts per million, is not significantly affected. Neverthelees, direct meammmmts of an accuracy to detect the 2 modifications in the electron and muon magnetic moments do exist, as we have already noted in Section 4-3, and one of our next tasks will be to develop the comqmding theory of the electron magnetic moment. But, although it is greatly simplified by the use of source theory, this is still a rather lengthy calculation. Neverthele~s, it might be helpful to pause and fill a gap in the treatment of quantum electrodynamic effecta of order a. As in the forthcoming dliscuseion of the electron magnetic moment, this topic refers to the effect of external electromagnetic fields. In the next seetion (Section &Q, the foeus is on strong fields This g e a d treatment wil l be useful in understanding Section 5.9.

5-6 STRONG MAGNETIC FIELDS

A major objective of these ccmcluding s d 0 a 8 on electdynamics is an improved treatment of the electron magnetic moment, a quantity that is defined in weak magnetic fields. But, firet, we shall explore same effecta of strong magnetic fields. Them include the etmng-field modification of the a/2r moment, the existence of an induced mament-a magnetic polarizability--and the proprties of the photon radiation emitted while moving in the magnetic field. In addition, we shall develop and apply a useful variant of the non-causal computational method.

TheIltartingpaintiatheactimtesanaeaociatedwiththeentchangeofasinde photon aumnpanying the particle,

a umtnhtion that a l h s the spin- Gmedt+functim equation into

( y ~ + m+M)F,= 1.

Here [d. Eq* (4-16.1)3,

where contact teanrs are left implicit. The exponmtial repmentation used in Eq. (4- 162),

andthe&devi08ofSectiondl4(with(= (+or,) = IQ = O)), conve& M into the expmJsiCxl

We are going to exploit the analogy between e-'.X and the unitary operator that d88Qibee the development during the time interval S, under the action of the energy operator X. Thus, with definitions such as

146 EIoctrodymonricr II Chop. S

The procedure membles that of Section 4-8, but is here applied to the system of charged particle and photon. Saw of the equatiam of motion are

and

The latter usee the cammutator IEq. (4-8.4411

[H, nl - @F, andaaeumeetheamstancyofthefiald F. I t i s t h e l a s t ~ o e t h a t d l ~ us to solve them equatione of motion, which then form a linear syetan. Thus, (5-6.9) is solved by the matrix statement [it is Eq. (4-8.48), with us replacing S J

n (s ) - k - e2-qR(n - k), (S-6.11)

and then (6-6.8) yields

Weccmalsopreeeartthela~m

eqqt(8) - t) = - m, (S-6.13)

where

As a firet indicatian of the method to be follwed here, let us relate the expectation value (e-'.xk), appearing in (5-66), to the basic expectation value (e-'.X). For that, we employ the statement of time evolution [Eq. (5-6.6)]

and the null dgenvalue (4 - 0) referehce of the expectation d u e , to deduce that

5-6 Strong magnetic fields 147

which gives

Another uaehrl relation appeam on employing two cammutabm,

The commutator appearing hem is evaluated in convenient form as

on applying the commutation relation

and introducing the transpoeed D-matrix. The latter is obtained from D by reversing the sign of F, according to the antisymmetry of F,,. The statement in (5-6.18) now becoma

and then, applying (S-6.17),

W t e the unsymmetrical appearance of the right-hand side in (5-6=), this structure is indeed symmetrical in p and v. The necessary algebraic property of the matricas is

It is confirmed, first by noting that

( A + l)( A= + 1) = e2UqF8e-2uqn = 1

1 Elutrodynamicr I1 Chap. S

and then by applying the an- of D - A,

A + A ~ = D + D ~ .

The main problem, the evaluation of (e-h), is now solved by devising a differential equation:

a i- ( e - ~ x ) = ( e - k x ) a8 - (c-'.x)ufl- (c-'.xk)2ull

+ (c-'.xk2) + (e-'.x)u(m2 - e q u ~ ) . With the aid of Eqs. (S-6.17,22), it immediately follows that

where the prime an the tram is a reaninder that only the vector indic88 are involved. In order to have a symmetrical matrix in the II quadratic form, we d t e the right side of (6-6B), apart from the last term, as

which applia the commutator (b6.10). Now, if we use the relation (5-6B), in which A and AT are commutative, we get

where the last d a n incorporates the propediea

5-6 Strong magnetic fidds 149

Thia puts (5 -6s) into the form

and the d t i n g integral of (S-- is

( 2:F)-1fie-". (eux) - C det' -

In introducing the determinant we have made use of the differential property IEq. (4-83311

8 logdet'X - t f (X- lbX) = 8 tr'(1og X), ( 6 6 s )

and provided a multiplicative factor to simplify the farm of the integration conatant C.

To evaluate C, we amsider the limit of amall S, where

Then, (S-6.33) exhiits the dominant behavim

1 ( c - h ) - C-

S~ '

since the d i m d d t y of the determinant is 4. The eingularity at s - 0 arisee frwn the incregsingly large value3 of k that are demanded, as s -, 0, by comple- mentarity with 6 - 0. Accaadingly, the limiting structure is given by the elementary i-lim (4-8JjVI'

and

150 Eloctrodynomics II Chap. S

The complete d t can be presented as

The known mm-field fonn [cf. Eq& (4-16.9, lO)] emerge8 on using the expansions of (S-6.36), since the detenminant redum to unity, and

To facilitate the next step, which is concerned with the Dirac matrices, we write

thereby isolating the spin matrices, and preeent our results to this point as

whetre the umtact ternrs (ct.) must still be made explicit. Now we again examine equations of motion, this time of the matrix quantities

Them equations are

d - d s ) = 2ueqFr(s), ds

sin-

[Y,@FI - -2iFy,

and the solution is

5-6 Strong magnetic fields 1 S1

which is then combined wi th the rearrangeanent

In doing this, one encounters

y(1 + AT)y = -4 - tr'A + 2bA,

where

The ambination is

In the abeence of the hdmogeneous fidd, the latter reducm to

-4["1 + (1 - u)yn] + 2(1- u ) y n - -2[2m + (1 - u)ylTJ, (6-6as)

and (S-6.43) beunnm

The contact team is chcsen to make K, and its first derivative with respect to yIf, vanish at yIf + m = 0. These are the normalization conditions. Accordingly,

cot.- -mc - tc(yn + m), (6-666)

where

Chop. S

and

Of course, only the combination of the two parts of K, and of M, is physically eignificant. Writing out the contact tesms mprately is a fiction, which is given mathematical meaning by stopping all the S-integrals at some arbitrarily small lower limit. (The photon mawr is another such fiction, used in connection wi th the U-integral.)

Now let us specialize to a pure magnetic field:

In this situation the matrix F hrra the dgenvalu~~ iH, - iH, 0,0, as foUm from the components

One can then evaluate the determinant of (6-6.43)-or, rather, its inverse-as

where the p a r t i e sign of q - * l is irrelevant. On introducing the variable

which quantity varie~ from 1, at X - 0, to (l - u12, at X - 00. The related


combination appearing in (S-6.52), 2(1 - u)egFs/D, has the twofold eigenvalue 1 - U, associated with the 03 plane, and the conjugate pair of eigenvalues

The various potmibilitiee are united in

where IT, is the projection of IT onto the plane (12) defined by the magnetic field. Alternatively, one can remark that

according to the algebraic properties of ag = iyly2,

%Yl = , %Y2 - - iY, P

which enablee us to w t (5-6.64) as

where

S = m*

The analogous combination of (5-652), with the additional factor

1 + AT e-2ue~F* (5-6.69)

1W Eloctrodynannics II Chap. S

ale0 hm the twofold eigenvalue 1 - U, while the 0 t h ~ e i g m v a l ~ ~ ~ are

and aA = f'sin2x, (5-6.73)

which occur as

-4 - t f A + 2iaA = - 4 c a 2 x + 2icsin2x

Uti;lizing thew mmhs, we psleeent the brace of (6-652) tm

S-6 Strong magnetic fields 155

The spin factor standing to the left of the brace in (5-6.52) can be recombined with @l to form 0 (Eq. (&6.42)], which we exhibit as

The last term wmhhes for F- 0, according to (5-6.36), and therefore only involves the nH components The quantity in brackets has a unique value in that mbspace, namely

where the angle /3 is specified by

Note that j? begins as (1 - u)x, for small X, and approaches x for large valuee of X.

The material for the general canstntction of M is now available. For most applications, however, it to use the fact that the particle field, to a good first approximation, obeys the equations

and thereby to simplify the structure of M as it contributes to the action. Doing this, we find that

1 Ol.Ctrodyromk8 11 Chop. S

in which the amtact term has now bem made explicit (as simplified by ylI + m - 0). O h e that in the h t x of the magnetic field, when x - 0 and A = 1, the yIIn team disappeum and M vanisha as it should. A further simplification will result from the remark that

effecthmly vanishes w h set betwmn +fields obeying (6-6.80). Under ckumtmc88, where q&dependmt eqmentiab of the form

multiply ynn, ths l-term can be mitt&, which is equivalent to replacing the exponential functitm by the average of its d u e for - * 1. But, before we can apply this obmrvation to (&6.81), in which an exponential function of D; .Is0 appeera, it is nemmry to study the energy epectnrm and the mmciated edgm- functions, as they are implied by the field equations of (S-680).

It is coavdent, and involvee no loee in g e n d t y , to specialize the coodinate aystem by choosing Il,, the component of mamentum along the magnetic field direction, to be zero. The field JI will be pmjected onto eubepc##e of intrinsic parity, labeled by the eigearvalua 4' - f l. Since y0 and iy, anticommute, the matxi-

only couple different Accordingly, the Dirac equation referring to an energy eigenvalue p' - E, when preseated as

d80ampoaa into the pair of equatias (D8 - 0)

On dimhating fields between them equatiane, we infez the systean

which would abo fo1101~ directly fraan the m c o n d d e form of the Dirac equation w t e d in (6-6.8)). Evidently the energy eigmvaluas are obtained by


assigning to

S - Q%,

an eigenvalue, S' = f l, and, independently, introducing an eigenvalue for

The familiar one-dimdmal oeciliator problem provides the latter spactnrm,

and one infm the energy values

Note that the ground state of the syetem, with energy E = m, ia uniquely characterid by the quantum numbers

M other cnergy levels are doubly degenerate with the same energy, (m2 + 2n'&rfl, being realized by the two sets of quantum numbers

n-n', r = + l and n = n t - l , c--l. No distinction has been drawn in this account between the quantum numbers

assigned to JI+ and to +,. But, since a, anticammutes with

the eigenvalua BlSgiared to S in the two subepaea, for a state of given eneqy, must be of oppaeite sign, with ameqxmding differences in the eigenvaluee of nH2. Thus, a more precise d d p t i o n of the eigenvalues mmciated with the energy (5-6.91) is given by

which are eummarieed in the following chara&rbtica of the complete +field:

1 8 floctredynamlca II Chap. S

That yOl providee an CaaCr quantum number can be men directly from the Dirac equation of (tb6.85). For the ground state, n = 0, 5' - + 1, the negative value that would.be uaignd to lIH2 in the yO' - - 1 subepace shorn that 4- vanishe& The special propedits of the ground state are amveyed by the eigenvector Btateanmts

Far excited states, yRH doee not have a d a t e eigenvalue. But what is req\tired is a kind of expectation value

(m,) - ((r0p8 - m)) - (YO)E - m, (6-698)

d e m i n g to the field structure that appears in the action [Eq. (Ml)]. To svaluate (TO) we use m as a variable parameter, deducing that

and themby

Accding to the energy expremim (5-6.91)'

and thu8

which yielda

Hardd looks bewildered H.: How can the expectatian d u e of yO, which has eigenvalum of unit

magnitude, be greater than unity? S.: I remind you of the y0 factor that appears m the action. If we make the

d g of (yO) q i d t for the c011tn'bution of a particular eignffunction 4 and

Strong magnetic fields 159

its complex conjugate JI*, it reads

which is ceFtainly greater than one, in g m d When we introduce the intrinsic parity decomposition, thie expectation value becomes

where, according to (S-6.86,87),

and, indeed,

While we are about it, let us note another useful expectation value, that of 1. We have only to use the eigenvalue stated for yOf in Eq. (5-6.96),

Let us also inquire about the energy spectrum when the a/2s magnetic moment is introduced into the Dirac ecluation:

The equivalent pair of equatiam is (l?, = 0)

where f' replacet qyoq. Since these equatians only differ from the aet (5-6.86) in

160 Elodrodynamiu li Chop. S

the subetitutim

Far the gnnmd state, in particular,

With inmiming magnetic field m g t h the total energy demem mondca l ly and, if this formula ccmtinued to apply, would vanish at the field strength

While edmordinarily large, this magnitude might be appmdmd under the astrophysical chmstmcee encountered in neutroa stars The actual situation a m c m h g the formula of (5-6.113) is quite different, howevea, as we now proceed to explain.

We exhii t M IEq. (6-Wl)] in the ground state, where

which makeo me of (6462). We give thia exprdon another fonn by the


tramformation

which yields

The mbetitution (6-6.118) is a rotation of the integration path to. the lower imaginary axis. Its justification involvee the absence of a singularity at the origin, and throughout the quadrant

In particular, a zero of the denominator in (5-6.117) would require that

which obviously cannot be satisfied for t) > 0. The reality of M thus made explicit was to be expected-the ground state is stable against radiative decay.

For weak magnetic fields, which are characterhd by

only correspandingly small valum of y contribute in (S-6.119), provided U > eH/m2. The initial tenn in the y-expadon of the brace in (5-6.119) is

and

which, as the weak-field magnetic moment term for - +l, re&atee (5-6.113). The nerrt power in the expansion of the brace is displayed in

Chop. S

a n d h e a r , w e n r s e t a n ~ - ~ I # o b l ~ B u t t h e ~ ~ t i d ~ o f t h i s 'problem' is already d e ~ ; the eqmmicm (5-6.126) can only be used for d u m of ueuchthat

To deal with the mmahing interval, U < U,,, the brace is w d e d in paw- of U:

where d y the tmn linear in U, which ccmnect8 with the logarithm of (6-6.126), need be retained. Thia umtribution to M is


The term linear in H is needed to restore the piece mixing from the integral of 1 - U [cf. Eq. (S-6.1!24)] because one has now stopped that evaluation at the lower limit U,. On adding the rest to (5-6.126), the logarithmic dependence on U,

disappears, yielding

Here is an indication that, with increasing magnetic field strength, the energy of the ground state does not continue to decreese below m at the rate suggastsd by the weak-field moment. The precedbg calculation referred to a definite spin orientation, l' = + 1, which prevents any further physical identification of individual terms. We shall soon see, however, that the tenn quadratic in H (apart from a logarithmic dependence) is actually spin-independent. It therefore represents an induced magnetic moment, a magnetic polarization of the particle, which, being oppoeed to the direction of the field, is diamagnetic in character.

The question now naturally arisae about the strong-field behavior of M, where the inequality of (5-6.122) is reversed,

To ammer it, we divide the y-integration domain in (6-6.119) at a, where

The contribution to M fram y < a is independent of H. For y > a, we can simplify the double integral of (5-6.119) to

whichisdominated byvahrtsof y - e ~ / m 2 > 1, and of 1 - U - m 2 / d * 1.

1 QIwtrody~onricr II Chop. S

The performance of the tbinbgd, undcir theee cimmm&mtm, givee

Thk leading mymptotic term is quite d&ieant to indicate that, far h mdah- ing at the magnetic field atnngth of (5-6.114), the energy of the ground state in very strong fields:

has increnmd above m. The two limiting fannr, indicate that, at a value of H in the neighborhood of the chm&d&ic value

thetohlenergyrea6baminixnumvalua, whichisonly lernrthan m by a fiactianal amount of the ordm a Incidentally, the l a m obmmatian is emential to justify this treatment of the strong-field situation, since it is still based on the eimplificatioas of Eq. (5-6.80).

In order to f d t a t e writing the g e n d expmuion for M that refem to a state dth quantum numbera n and S', we introduce the symbols

Accodngly, we have [Eq. (6-6.62)]

while the d c i e m t of ynH in (5-6.81) u c q u h the fom pmmted by

and the ground-state combination of (S-6.116) mds

5-6 Shong magnetic fields 165

For the -era1 situation wheite IEq. (6-69Q]

the structure appearing in (66.81) becom8~

Thaq if one raer the projection matricee :(l f 5) to erpr~as the (dependace of functions, and recalls that the function of 1 multip1ying y& is to be averaged wer ib ( = *l values, we infer the following effective form of M in a state of energy quantum number n':

Now, thetradomatian X-+-&cannot beuaed, asexpected fromthe radiative instability of all the levels above the ground state. Nevertheleeg, for weak magnetic fields and U > U,,, small valuee of x should still dominate. We shall first proceed to the eame accuracy as in the ground-state discusdon, retaining only terms quadratic in X. W1th that limitation,

is replaced by unity, which appears to remove this part of the n'dependenca But clearly a d c t i o n an n' is implied, such that

166 Hectrodynamicr II Chop. 5

which excludes E2 - m2 being large in eornparhn with m2. This is an eamtially non-relativistic situation. To the r e q u i d H2 accuracy, the term explicitly linear in H do- not oontn'bute, and the expansion

1 + ue* - l + U * iu(1- u)x - U($ - !U + u2)x2 + (5-6.149) D*

produces the following U > U,, ccmtri'buton to M:

Note that this expremioa is real, and that it coincides with the comsponding ground-state d t on placing f = + 1.

For u < U,, we expand in powers of U:

1 + ue* 2h = l +

D*

and evaluate the explicit H-term at U - 0. The d t i n g contriiutim to M is

On placing n' - 0, - S' = + 1, this redmm to the already evaluated ground-state expression, where x can be replaced by -iy m. (5-6.130)]. Accordingly, we remove the ground-state farm to get the additional terms

Since this is the entire mume of the haghwy part of M, we have

Apart from wrnishing delta-function integra4 the three c l d y related in- grals encountered here are (X > 0)

& X r &*&ask- x2 - -(l 2 - tA)()(2 - A),

whem ~ ( x ) is the Heaviside unit step function

Thus, all the inbgmb that constitute ImM vanish for U > 2eH/m9. This promy is clarified on noting that, in such circurnstana the substitution X -, - iy ia permissible in (6-6.153), aa evidenced by the existence of the d t i n g y-integd, which is then explicitly real. To the limited accuracy that we are working at, no distinction need be made here between S' and l. The immediate outcame is

which is indepmdmt of S'. With the notation

Chop. S

The imagbry part of M effectively produces the replacement of m by m - iiy,,. What this impliee for the energy of the eystan is contained in the differential relation (6-6.101), namely

with

Y = (m/E)Yo.

The latter quantity is identified as the decay constant of the system, the inverse of the mean lifetime, as expressed by the probability time factor

For non-relativistic 8tat.m of motion, y S y,. It is in theee circumstances that an elementary semi-classical calculation of y can be performed, with concordant results. The classical formula for the power radiated by an accelerated electron is

where, according to the c M c a l equation of motion,

Considering motion in the plane perpendicular to the magnetic field, we have

Now, the quantum expmsion for the non-relativistic energy #mv2, as implied by (5-6.90), is

in which o is the chmical orbital rotational frequency. To produce the quantum tramaiption of this semiclaasical redt , one divides the radiated power by the

5-6 Strong mclgnetic fields 169

energy of a quantum, which is U, to get the emission probability per unit time:

and alao r e p k the total non-relatiwc energy by the excitation eaergy above the ground state, in &ex to incorprate the stability of the latter. The result for y S y , is just (S-6.159).

This limited treatment is completed by evahrating the real part of the h (5-6.1S3):

Here we meet the integrals

which are mbined in the partial integration evaluation of

[Note tao that the A-derivative of the last statement reproducxs the first entry of (5-&l@).] Them is also the elementary integral

At this stage, (66.168) can be v t a d as

170 floctrodynomics II Chap. S

where

and the integrals have been so arranged that, with negligiile error, infinity can replace the actual upper limit at

We have also subetituted for 1 in the quadratic H-term. The three integrals that appear here are evaluated as real parts of cormpond-

ing complex in-

- Ref log- ;: ; 1; - 0.

and

All that remaine, then, is the tenn linear in H and proportianal to U,,, which m&oree the piece -wed frr#n the related integral that had been stopped at the lower limit U,,. The net reeult for the real part of M is to replace the logarithmic lswar limit U, in (66.160) by the same value that appeared in the pound-state calculation. The complete statement of M, to the present accuracy, is therefore


confirming the earlier remark about the generality of the diamagnetic term. One should observe how simply the diamagnetic and radiation damping terms are related, as expmmd by the combination

The &mce of dependence in the decay constant (S-6.1S9) is to be expe&d, according to an elementary non-relatiac calculation of the magnetic dipoleradiationamociatedwiththespintransition S - -1 -, +l.Itisbasedon the electric dipole formula of Eq. (3-15.69), which is converted by the subetitution

into the decayanstant equeadon

Hem, we have used the matrix property

and recoeplized that the spin transition frequency o equals that of an orbital transition n -, n - l (recall 2n + 1 - l'), which is the classical rotatian frequency of (6-6.166). Being cubic in the magnetic field strength, this pmesss has escaped a treatment that halts at the second power of an expansion. We shall now proceed to include thew cubic term&

We first attempt to repduce the decay constant of (6-6.180) by anmining a state that decays entirely by magnetic dipole radiation. This is the fir& excited lwel with n - 0, S' - - 1. The starting point, then, is the n' - 1 form of M:

l72 Hoctrodynamics I1 Chap. S

Since we are intemted here only in the imaghuy part of M, it is well to recognize that some of them temm do pennit the X -, - iy transformation that leads to a purely real expremion. They are the on- involving only D,, where

and the mee containing

where we have applied the contact tenn in individual expmmians to remove any singularity at X - 0.

It will be seen that the transformation x -, - iy, with its implied reality, is permitted for U > 2eH/m2. Accordingly, ady Sman values of U contribute to Im M, and we employ an expansion. Far the second of the two s t r u m that corn- (5-6.18Q9 the v c e of U a eH/m2 as a factor implies that no more than the linear term in U of an expansion is needed,

A similar remark applies to the find tenn, and it is strengthened by the obwma- tim that

it suffica to d u a t e its factor at U = 0. Introducing the expectatin value

we get the following eqmaaion far Im M:

The two X-integrals that appear h are of the forma

ao& j - [ r i n A x - i n ( A - 2 ) r ] - 4 2 - A ) 0 X

(6-6.191)

and

m & ain X - [ A - - ( - 1 1 - ( 2 - A). (6-6.192) 0 X X

For the umtnWcm with only m e U or H factor, ane must be careful to note that

The immediate outcome is

174 Electrodynomicr II Chop. S

and

aQIXPBCted* Now let us peaform the expansion up to cubic terms in H for the g e n d

&on. We begin with the x-erpadon for U > q,:

1 + ue* - l + U * &(l - u)x - U($ - f u + u2)x2 D, (6-6.196)

These are combined into the following expanded form of the brace in Eq. (5- 6.145):

A f b p e r f d g the X-integration, we have

and then

eH 1 1 l -!2t(2)3(G - $log- U, + !) 6 - ! n ' ( $ ) 3 ( ~ U, - 2log- U, + L ) ) . 4 (5-6.199)


Since no distinction need be made between S' and in H3 terms, one can alao pmmt the last two terms of the brace as

In writing the last fanas, we have not troubled to keep poaitive p a s of U,, which we know wi l l be cancelled eventually by related terms in M(u < U,,). But we have tempcmdy retained another type of uodepeadence, the H3/u0 structure. To me how these terms are cancelled, consider the special example of the ground-state evaluation of the U < U,, contribution, as preeented in Eq. (5-6.130). Carrying the expan8ion in eH/m2u, one step beyond that given in (5-6.131) aupplie~ the additional team

which doee indeed cancel the similar tenn in (6-6.199) when the latter is evaluated for n' - 0, f - + 1. In short, the H3/u , terms are a d d u e of the H2 calculation, and wil l not be cansidered further.

Now we proceed to the u-expadon for U < U,,. The ingredients are:

& X sinx +u2(7e*2k - 2-."+l- X -e"k+e"k) & X X + ...

and

+ 2 ~ ( = sinx + * g o . (5-6203) X X

176 Eledrodynomio II Chap. S

We shall record only the u2 and Hu terms in the structure of M(u < 4):

The following symbols have been introduced for various functions of X:

sin2x sin 2x sinx &=X' coe2x-- X + 1 - - X coe3x + coe2x,

and

also

sinx 8inx Cl - - coe3X-2-cosx+l,

X X

We have already seen, in the t3immdon of the portion of ReM, that no contribution appears for U < uo unlm there is a related term in the U > u, structure. Specifically, this expmsea the vanishing of the terms linear in n' and [Eq& (5-6.168-176)], which do not appear in (5-6.150). A similar situation occurs in the H3 calculation. The U > u, form of (5-6.199) only has H3 tenns propor-


tiond to 1 and n'. Carreepondingly, there are vanishing contributions in (5-6.204) for the terms with coefficients nn, n'l, as well as the one lacking any quantum number dependence. The integrals involved in demonstrating this are of the general types already encountered in the H2 diseuasion, and we refrain from giving further details. The remaining structure in (5-6204) can be preeented as (c S l)

where 1

~ , + & - ~ d n 2 ~ - 2 x e o s ~ x - - a i $ 2 ~ . 2x (5-6m)

The X-integrals appearing here are effectively given by

and

The subt3equmt integrals wer the variable

Chap. S

in which poeitive pwem of X, - U, are d t t ed , appear as

and

This giva the following d t for (5-6204):

which, added to (5-6.l99, 200), together with the analogous H2 tarn [Eq. (&&18l)l, yields

We me h the first thong-field modifications of the a/2s spin m0m-t and of the magnetic polmhbility, which work in o m t e direetions cm them two -88.

A bit of A m w camplicatad-appearing but equivalent reeult was produced aame time ago by R Newton, Phya Rev. 96,523 (1954). His method was a related one, in its uae of the mnwr opemtor M, but was sufficiently more cum-e that only them initial teame of an empadon in H were erhibited.

One an, of came, compute the H3 terms in ImM h, but we shall not trouble to do m. The physical that would be included here involve relativistic modifications in the =dipole proeas and electric quadrupole amtributiona Themm addi t id effects must remain small, however, for weak

5-6 Strong mognetic fields l79

fields and low energia The situation ia quite different at ultra-relativistic ener- where very high multiple moments dominate, and we proceed directly to

that calculation. Let us return to the x+on of (5-6.146) and remove the n' d c t i o n of

(6-6.147, 148). Now, we write

aa an approximate +on of the regime

The integral formula for M, Eq. (&6.146), wi l l be dominated by the two expanential functions with large coefficients in the argument, (5-6.217) and 4 - i ( m 2 / e H ) u x ] . The important range of X ocerrm where the two arguments are mghly O J l e ,

Under the highmergy circumstanw expmmd by

them dominant value^ of X,

will indeed be small compared wi th unity, apart from a namrw range of U near one. Accordingly, we retain only the leading terms in an x-expadon of the bracket in (5-6.146),

Having in mind the eatbate of (5-622l), we introdu- a new in-tim variable y, such that

l#) Eluirodynamicr I1

This amverts the bracket of (6-6XB) into

Chop. S

where we have also replaced f (m/E) by its effective value S', and the product of the two expanential functioas become6

=P[ - it€( Y + V)] (&6=)

in which

Since our concern h is with Im M, we write, directly,

The basic integral that appears is

4- °D- !€(Y + 49) = 3-1flKl/a(C), (8-6=)

which trsee an emmple of the Bsseel fuuctian of imaginary argument,

(Fa all needed infurmation on snch functions, the chmic refeatace, G. N. Watam, -1 Jhdons, Cambridge Univdty Press, can be canaultw) On &g the property

we infm that

5-6

and the^

Strong magnetic fields 181

The integal that produced +r was identified through the limit of large [, where y IKJCom- ==P=W~Y 6

Combining the d t of a partial integration,

! € ~ Q ( Y + y " ) h ? € ( Y + $9) - 3- lDK1fl(€) , ( 5 - 6 - 9

with the remrmnce relation

we also deduce that

4- &Y Y + $y3) 3-lnK2/a( t) ( G e m )

A final property of interest, which exploits mother mcurrence relation,

The various integral evaluations are inserted in (6-6227) to give

Chap. S

For all pawe~tly attainable artificial magnetic field strengths and electron emer- giee,

is very d compared to unity. Since the hctiam K,([) of direct amum decrease expanemtially far 1, the important values of U are such that [cf. (6626) l

Accordingly, a lending appxdmation to y is given by

which evaluation an example of the integral

p - v p + v L* d€ tp-lK.( €1 - T ) 4 T ) The radiation emitted under the cirmmtacm being conaidered, as exp.mwd

by (&M), has a special charactea-it is damical radiatim. To appreciate this, it may be simple& to turn matters about and inquire c o n d g the clRncrical radiation that d y ia emitted by an e l m moving in a macmcopic orbit under the control of a ma&c mqnetic field; this is the experimentally well-kaown s y n m radiation. In this damid limit the qmnestial function 4 - iex(u)] of Eq. (6-64 wil l be rapidly oecillatory, and the major contribution


to the k-integral is umcentrated in the neighborhood of the stationary phase point,

This is not the momentum of a red photon

-h2= -UZIIIsu2m2,

but beumes so if U is 8ufficiently rmmll, Aamdbgly, to an accuracy that neglects u2, we idemtify the energy of the emitted photon, W - kO, with

and we mm&e in the clamid criteria,

u (: E ,

8 comhte~t d c t i m t0 4 d u e s of U. We can now present (6-6.243) as the spectral integral

and pick out of the integrand the dnnnical power spectnrm,

where

With this identification, the total radiated power is

184 E l ~ o d ~ a m i u II Chap. S

which appliee the integral [(6-6.!246)]

To vedfy that (5-6- is indeed the d a m i d expredon for radiated powes,, at high energies, we return to the clasaid, non-relativistic formula, Eq. (5-6.163), and remove the low-energy limitation. To do that we ramark that p e r , the coefficient in the linear relation between emitted energy and elapBed time, must be a relativM5c invariant. We therefore replace time by proper time, and non-relativistic momentum p - mv by the four-vector of momentum, to produce an invariant:

wheae [cf. Eq. (1-&76)]

Since the claeaid equations of motian read [they are the first line of Eq. (1-3.77), written in tbreedmdonal notation]

we have, for motion in the plane perpendicular to the field,

At ultra-relativistic aaeqjics, where v' r 1, this is indeed the same as (5-6.253). The nature of the spectrum, in which the important frequencies [I - l] are of

the ordea

is alao a c?lnrwical dt. The rotational frequency of the electron in the magnetic fie& as inferred from (6-6357') and the relation p = E v, is


This is the fundamental £requency of the cWcal radiation, generalizing the non-relatiMc result deduced from (5-6.164). But, unlike the non-relativistic situation, most of the radiation appears at very high harmonics of the fundamental. There are two reaeons for this. Firat, owing to the high energy of the particle, the radiation is concentrated neat the instantaneous direction of motion of the particle, appearing in a narrow cone with an opening angle of the order

Only that fraction of the orbit is actually effective in directing radiation toward an o h e r , and on this account the important harmonic numbers would be - E / m The second point is that, in consequence of the Doppler effect, the detected frequency differs markedly f'mn the emitted £requency. A signal generated at the point r(t), at the time t, is received, at the point r', at the time

where 8 is the angle between the &on direction and the instantaneous velocity. Then, since

we have

and the significant detected frequenciee are of the order

which is the content of (6-6.259). In the strict classical limit, where U (= 1 is d d e r e d negligiile in comparistm

with unity, all reference to the spin naturally disappears. It is interesting, then, to proceed to the level of first quantum corrections, where the quantum number S' does make an appearance. For that purpoee, we return to the expression for y, Eq.

Chop. S

(S-W), and appmrhate it m

whem, in the leading temm,

and we have now mtrodugd the chmctmWc classical frequency

It sufiicee to nite 6 - o/oC in the terms that have an explicit u-faeta. Cancemn- ing the quantum d o n terms that are independent of S', we note that

.r a e ~ a r s g u r ~ g of the integral evaluati- deduced fmm (S-624S). One aspect of thiafactreferstounpolaridpnrtidas,whare~- +l and -1occurwithequal pdmbilitim. Then the qectral denaity of y, which is o-lP(o), has the classical form, but wi th the subetituticm

All first quantum carmcti~~ll m y are made explicit by writing

where^ E now revatr to its clmic8.l farm, U/%. The mrsted decay rate is

S-6 Stroms magnetic fields 187

Introducing an additional factor of o in (5-6272) then gives the radiation power,

Harold looks alreptical. H.: That last calculation, of the first quantum carrection to radiated power, is

not very convincing. Suppaw you had not recognized that two of the tenns in y cancel [Eq. (5-6270)], and had hx&d the additional factor of o in their integrands as well, That would give a different answer for P.

S.: You are quite right. Put mare generally, one cannot infer a unique in& grand, o-'P(@), given only the integral expreslsioon for y. For that, an additional argument is needed, which we shall now develop.

Let us return to the @on for M, Eq. (5-6.3) or suhKquent forms, and insest a unit factor,

When applied to I . M, where only d proceeees 001ltn7,ute, the inferred spectral distribution in o will be that of the radiated photon energy, thus supplying the d& photon specbmn without ambiguity. The k-integration symbolized by ( ) is now modified by the preeence of the factor

This effectively inducee the following subetitutian in x(u) @3q. (6-6.4):

The aUbt3eJqu-t K-warmation

togethe6 wi th the hat two Xdepsadent terms of (5-6.277), m&- Eq. (S-64, for example, by the additional factor (this procedure depends upon the circum- Stancex-0)

1 Hoctrodynomicr II Chap. S

and by the substitution

The additional term that the latter produ~e~ in the brace of (6-6.S2), or its rearrangement in (5-6.76), is

In the last vdcm, we have also made explicit the factor of -2m that is removed in producing the bracket of (56.81). Thus, the additional tenn appears in the related bracket of (5-6.146) as

which also involvae the effective subetitution of Eq* (5-6.102). One &odd not werlook the n d t y for a related supplement to the -tact term -(l + U).

Recall that the latter is deigned to produce a null d t for zero field strength, where X - eHw, but not x/H, vanhihe& Hence this additional contact term is

Leaving them extra tearrm aside for the maneant, we see that the explicitly rdependent factars of (6-6275) and (6-6.279) combine into

whare we have introdwed the variable y of (5-6223)' and 6 CEq. (5-6226)]. The omhdtm of the Gauaaian function of r would instantly reduce this integral to 8(0 - &), which ia the chmical identification of (5-6.248). Since the important rangee of the several variablee are y - 1, - 1, ruE - 1, the Gauseian function is indeed cl- to unity uxtder ultra-relativistic t h x m s h n ~ ~ ~ . If that were the whole story, we should arrive at (6-6.267), with the additional factor

5-7 Electron mognetic moment 189

serving to effectively replaw U with w/E, and the terms d i a c w d in (66.270) would be retained in the spectral distribution.

But there are the additional contributions of (5-6.282) and (5-62283). Keeping only the leading term in the x-exptdon replac88 the bracket of (5-6282) by unity, and pfOduc88 the following added term in y:

A partial integration in U, followed by one in y, replace3 this by

which precisely cancels thoee spin-independent quantum d o n tenm of (6-6.267) that had been discarded in (5-6.270). We conclude that the identifica- ticm of the spectral distribution leading to the power calculation of Eq. (6-6274) iscorrect,

This already wer10ng section wi l l be c l d hem. While additional topics mnah to be explored, in the areas of very strong fields, and high-energy radiation proceesee, they are sufficiently tied in with other considerations, of astrophyaica, and accelerator deeign, that furthm discuesion would lead us too far from the main line of development.

5-7 ELECTRON MAGNETIC MOMENT

It is our intention n w to 6nd the d d m t of (a/2n)' in a poaerdes eapanaion of the el- magnetic moment,

190 E(octr0dynomicsII Chap. S

where it is known that c, - 1. One u m t n i o n to c, has been available for aome time, and we begin by evaluating it. It is the effect of modifjrhg the photm propagation function, associated with electron which was dbcmsed in Section 4-3 in connection with the difference between the e l m and mum rnamenta. The relevant farm is (4-3.107), with m' = m aa noted in the text. Since this is a multiplicative comction to a/2r, the vacuum polarization comedon to %is

The d t of performing the heintegration is i s y given in Eqa (4-3.112,113), whem now

and we pmmt it as

Partial integration and the ~~b&itution 1 - U - t reduce the integrd to

according to (S-4.107), and

This is a rather d ctmtribution, if one anticipatee that c, should be of &er unity.

The formula of Eq. (4-3.10'7) was derived in a causal manner* aa a modification of the technique developed in Section 4-2. Before continuing, let us note the ameqmnding non-causal derivation, as a modification of the work in Section 4-16. For that, we replace the null photon mass by the variable mass M, with

the d is tn ian of which is dedbed by the weight fsctor CEq. (4-3.105)l

The replaceme~t of kg by k2 + M' altas the function ~ ( u ) o f (4-16.3) by adding

This iafluenc88 the right-hand side o f Eq. (4-16.17), which ie changed to

The consequence far Eq. (6-1618) ia indicated by the mbetitution

When the latter ia multiplied by (67.8) and integrated with mpect to v, the d t i n g coefficient of ( a / 2 ~ ) ~ is

the substitutian U 4 1 - U amfirms the equivalence with ( 6 7 2 ) . Paralleling the me of the modified photon propagation function in Eq. (4-161)

is the introduction of the modified electran propagation function. The dructure of E+ appropriate to a weak, homogeneous eletAmmqpetic field is contained in Eqa (4-2.31, 40). Employing the form given in Eq. (4-2.44) for the explicit field depeadmce of M(F), and the expandon

which is ad5de~tly accurate for our we get

2mM 1 - l +

(M - m)' (M + m)2 + 2mM 1, (5-7.14) yn - M

192 ~octrodynomiu II Chap. S

where qmmetxhtion of the factors multiplying aF is undesstwl in the third term of G+. The 1- limit of integration in the last term is a reminder of an infra-red singularity, which is non-physical. Indeed, this term will be cancelled campletaly by another contn'bution to be intraduced later, and we set it aside to d e the implications of the explicitly field-dependent parts of (5-7.14).

The non-causal method of Section 4-16 will be used. The vacuum amplitude of (4-16.1) is modified to

The tsms of i n M in 0, then produce matrix combinations of the type

where m' may be m, M, or -M, and a symmetrization between m and m' is applied. On using such relations as

and the familiar property

r'eFyc = 0,

we find that (5-7.16), with its implicit qmmtrhtion, redutm to

W e them e~counter momentum integrals of the form (II 4 p)

5-7 Electron magnetic moment 103

in which a photon mass p has alsa becm introduced. The redefinition k - up + k, and the real-particle property -p' - m2, convert this into

where

W - *(l - v) . O h e that an infra-red singularity do- appear for m' = m.

With the aid of these d t s we find that the second term of (6-7-14), d d b i n g the additional magnetic moment of the electron, producea the following contribution to the vacuum amplitude (6-7.15):

Thie &&its another piece~ of q,

Concerning the third term of (S-7.14), we resnatk that the two parts aasociated with M and -M give equal contn'butions, leading to the vacuum amplitude

Chap. S

with

The last version was produced by a partial integration on the variable

x - M2/m2.

Now

and, using the subetitution 1 - U - t, we get

1 1 (4:- -~ld(;1twG - l) - -(; - l), (5-7B)

again invoking (6-4.107). The complete contribution associated wi th the explicitly fielddepeadent tenns in the modified particle propagation function is therefore

We have begun the dimmioa of the magnetic moment problem by computing eame obvious contributions. Now we must examine the whole picture. The initial a d arrangement referred to the exchange of an electron and a photon in a homogeneous magnetic field. At the next level of d d p t i o n , the characterization of the two-particle prams is modified, and three-particle exchange takes place. The d o n s to the two-particle mechanhm involve the introduction of modified propagation functiam for the photon and electron (effects that have already been considered) and, assa5ated with the interaction of the electran and photon, of form factors for the two-particle d o n and abmption acta The three-partide pmmscs are brought in by d d e r i n g the emission and abeorption, not of

5-7 flwtron magnetic moment 195

two real particlee, but of one real and one virtual particle. A virtual electron decays into a real electron and a real photon. The subtwquent recombination of these particles is the mecbanien for producing G+, which has already been discussed. But them is a second pu3siity in which it is the initially emitted photon that later combbee wi th the electron to produce the virtual particle that is detected, along with the other photon. [Aside to the reader: Draw the causal diagram! It can be w t e d as a rectangle with heavy, virtual-electron lines constituting the narrow top and bottom, while wavy, thin, real-photon lines fom the side& A d-electron line t ram one diagonal, and the initial and final virtual-electron linee are attached at the other two verticxs.1 Similarly, a virtual photon decays into two real electrom, of oppoeite charge. The recombination of these partielea generates D+, which effect has already been computed. But the exchange of the roles of the two like thug- at the absorption end producee a new promxt. me causal dhgmm here is also a rectangle, with virtual-photon Knee forming the top and bottom, and real-el- lines the side& A real-electnwr line occupiee one diagonal, and the initial and final virtual-electron lines are placed at the other vertices.]

Before continuing, let ua review the machinery that introduces form factors for the two-particle process It is a comequence of an interaction that contributes to Compton acattaing. The latter is produced in either of two ways that are related by photon cmahg symmetry. The first one involves the recombination of the initial electron and photon to form a virtual electron that decays into the final electron and photon. We do not consider this mechanism explicitly, since it is an iteration of the two-particle exchange that produces the modified particle propagation function. The action principle handles it automatically. (Compare the d.iscudon with Harold at the end of Section 5-4.) In the second Wbility, the rolee of the initial and final photon are interchanged. The initial electron emits the final photon to become a virtual particle which, on absorbing the initial photon, producee the final electron. [A causal diagram can be drawn in lozenge (diamond) shape, with real-photon lines forming me set of parallel lines and real-particle lines the other set. A horizontal virtual-electron line connects two vertices, and the initial and final virtual-electron linee are tied to the other vertices.]

The &istic read= is now in a poeition to mmgnb that the topologiee of theee three c a d diag.lrms are the same. That is, with the distiuction between real and virtual particlee ignored, and on performing deformations that maintain the connectivity of the line4 the three diagrams can be made identical. (The multing non-causal diagram is what is known as a Feynrnan diagnun. A simple version of it is produced by drawing out the electron pictorial r e p ~ ~ t i o n s into a single straight line, with the two photon graphic symboIs traced as intersecting arc&)

196 aechodycramicr l 4 Chap. S

Thus, we have the option of evaluating three distinct c a d m or of performing one nm-cad calculation. It is the latter strategy that will be adopted her&

What has just been d88~1'bed can be denmrmhted analytically, of course. Howevez, we shall not trouble to consider all three causal arrangements (while urging the reader to do so), but just select me to produce the common space-time form of the coupling. For the two-particle procees, the vacuum amplitude representing the partial Coanpton is

where the fields are those of the real particlee that enter and leave the collision, and the pmwnce of the homogeneou8 magnetic field influen~e~ the form of the electron propagation function. The 80- of th- particles are

and

iJ:(r')n(x)rOl,- ~l(~)Y~sqY"~(x - +?P (5-733)

in which +&,(X) here &er to the extended particle anncar Putting t h w elements tagether givee the deeired vacuum amplitude, apart from contact terms,

is the m ~ w r operator amtnian characterizing this two-photon exchange p C8g&


Them is a cl- r a l a t i d p with the structure of M [Eq. (6-6.3)], which dBBQibts single photon exchange,

whem the oaatact tenns appearing here,

c.t.= -mc - fc(yn + m),

are epecified in Eq& (6-6.S6, 67). This example illustratee the completely local nature of contact terms, in cantrast with the non-locality that is chars-c of a multi-particle exchange praces& Now consider the effect of an infinitmbd alteration of the vector field A in (6-7.37),

On writing

and

("') e W x ~ ( k t ) 8iU-j-- (24' h')

81M &4(k3 8A( k') '

we can preesat (6-7.39) aB

4M (W 1 1 1 M , ( L ~ m iesImf~ fin - L ) + m w y v y ( n - L - + m Yp + c ~ Y ~ ,

(5-7.42)

or, interchanging k and K', m

v4 - i e s / - ("9 , l - 1 1

&Ap( -C) ( 2 r ) 4 y kn f i l l - k - k 3 + mQPICfil l - i t ) + m Y. + S . Y " *

198 Elutrodynamitr II Chop. S

Both of them i n b g d structum can be recognized in (5-7.36). But, before introducing them as component parts of M@, we must understand better the role of the cantact terms.

The y" y,, structure in M@ mpmmts the exchange of a photon with momentum R, in which the systean is probed by a photon of momentum K'. This is d d b e d by the combination ( S - 7 4 , which n k t a t e e the following contact tenn in M(2):

(Nde that contact temil, added to Mm as a whole are without effect in this non-causal situation, eince 3/, and $3 do not overlap.) And the y' y, structure in M@) repreeents the exchange of a photan with momentum kg, in which the system is probed by a photon of mome~tum k. That is descri'bed by the combination (5-7&), which requires the following contact term in MQ:

The emmtiaf point is that, since the two photon exchanges are independent, both of them contact terms are needed, and the complete contact term structure in (5-7.36) is

cat. = - 2 S 3 . (6-7.46)

In view of the importance of this conclusion, we add mother consideration in which the effects of the two photon exchange proceesee are more clearly separated. Let us eramine haw M(2) mqmnds to an arbitrary infinitdmd variation of the el-etic field. We write this reepanse, without contact terms, as

+fG+(n - k)y'8G+(n - k - k') @+(l? - k')~.

Wdhgthefirstdthsathree-wesarthatthev - * - y,structurenon involves the differential action of two field& An hspeetion of (S-7.38) shows that no contact team appears for such But the y' y, combination in the same first term has the form of (5-7.43), which demands the contact tenn

5-7 Electron mognetic moment 1 W

exhibited them. The addition thm implied to (5-7-47') is

For the aseond of the three terms, both of the singlephoton exchange s t r u w involve two diffemntiations with rrspect to fields, with the carmquent * C B of contact terms The discwsion of the last term in (k7.47) is analogous to that of the first one, leading to another amtact tenn exactly equal to (67.48). In this way, we recogiza again, and more explicitly, the existence of the ct. (5-7.46).

Fktuming to the structure of [Ecp (5-736,4611 we now o h e that

h08 the Single M structure d y in-tee one of the two epual amtact m Concerning the reeidual me, we note the form of f, as the sum of two contributions, given in (5-6.57):

Se=t'++, with

and (introducing a photon nuus)

Hence the ccmtn'bution to e, that is ammiated with the r-component of the contact term,

removas the fictitious photan-m dependence exhiited in (S-7.30),

200 E l ~ o d y c r o m i ~ II Chop. S

Another compemation, which was anticipated m the discusdon following Eq. (5-7.14), can now be made explicit. The last term of the Bxpreason for c+ given in that equation is

with

a 1 ). ( M T m)'

It supplies the following conbition to the vacuum amplitude of (5-7.15):

Now the mass operator M, referring to an arbitrary elechmagnetic field, can be decampatid in this way:

w h w M, depends explicitly on el-(ie field strengths, and MO is the gaugecovariant form that appliee in the absence of electromagnetic fields. As exhibited in the first term of (4-2.31), the latter is

( M - m)' - lm("+(")[ ,n + M - ( M - r n ) + a + m I

( M + m)' + A - ( M ) [ - M + ( M + m) + yn + m

1 1 + A - ( M ) [ ( M + m)' y ( n - k ) - M w y ~ ~ - e y p ] ) * ( 5 - 7 * B O )

5-7 flectron magnetic moment 201

and then proceed to simplify the part of the vacuum amplitude inferred from Eqs. (5-735 49) by performing the reduction yn + in -, 0 on the right side of (&7.60), as effectively e x p d by

yields a vacuum-amplitude contniution that precisely cancels (5-767). Our attention is now concentrated on the remaining part of MO,

Accordingly, we need an expreesion for

8rMl 4-M &?MO -m---

&A,( K') 8A,( k') BA,( K') = -W~l"(k')

that is accurate to terms linear in the homogeneous field. We shall use an equivalent form of the construction for M given in (5-6.5),

where the -tact tesrns need not be added, since they will cancel between M and MO*

We m construct M, in this way, by d d e r i n g the field-free situation. With the now permislrible transfomtion k-un -, K, and the evaluation [Eq. (5-6.38)]

262 Hectrodynomicr I1

we immediately get (without contact terms)

Chop. S

w h w

This reproducee (6654). of course. What is needed, howwer, is the differential form produced by varying A in M,,, nor applied to an arbitrary eledmmagnetic field:

in which

Ihpmmhg this by a functimal derivative, as nqoirsd for (S-7.64), will indutx the substitution

for any term standing on the right aide of 8A. That giva

whee the symbol X' indicates that the subetituticm (67.71) hrs been made in


X. In writing this form we have alao introduced a simplification associated with its eventual use: where yn + m and S stand entirely on the left, they have been replaced by zero.

Let ~8 begin the discmaim of M by first performing the functional differentiation with reepect to A. From the differential form

one infers the functional derivative as

l - k 2 ~ 8 d ~ , ~ ( ~ ~ e - h ~ ) ~ ~ --- eq &AV( h') J

do l + o - e 2 ~ & 8 2 * T y p ( [ m - fin - k)]m[-bTx]

According to the structure of X [Eq. (5-6.4)], the functional derivative that appears on the right side is

1 4 x --p - ~ [ ( n - h)' + (m - k)' + ik!aAv]. (5-7.75) eq &AV(k')

Since our need is only for an evaluation to the first power in the homogeneous field, we shall adopt more elementary methods than t h m elaborated in the preceding section, for example. Thus, in order to combine the two exponential factors in the second term of (5-7.74), we move intervening factors away by using the approximate e x p d o n s

204 Eloctrodynamia II Chop. S

and

The unification of the exponential factors in turn employs an approximation, which is based on the following theorem applicable to operatom A and B such that the cammutator

[ A , B] - iC (5-7.78)

is commutative with A and B:

A proof is immediately supplied by compnring the evaluation

wi th the alternative ows in which ea is used to effect such trandormatiiona [The same procedure also suppliee a short derivation of Eq. (2-1.!21), which is a generalization of (6-7.79).] The relevance of this thin our situation steam h m the remark that the commutator [X, X'], being explicitly linear in the h 0 m 0 g ~ 8 0 ~ ~ field, is effectively inoperative in forming additional commutators, since they would contain higher powers of the field. The same &ction to the firet power of the homogeneous field h permits a simplified application of (5-7.79),

The commutator is emluated as

Perhape it is not too man to introduce another simplification of which we shall make repeated uae. The combination

appearing in (6-7.89, fop example, wil l through to the final calculation,


where it can only appear in the form

since the alternative

nFn = :icqPVFp,

is negligiile. But the commutator appearing on the right side of (5-7.84) do- not contribute in the application to particle fields that obey (yn + m)$ = 0. Accord- ingly, the IIF stnrcture will be systematically omitted as the calculation proceeda It is in this sense that the commutator of (5-7.82) is replaced by an equivalent statement,

Incidentally, the combination of exponents on the right side of (5-781) is

which employs the relation [cf. Eq. (6-7.71)]

Whenewer can be replaced by -m2 and the OF term d t t a d , owing to the preeence elsewhere of an electmmqpetic field factor, this combination wil l appear as

l + o 1 - 0 - X + -X' + + D',

2 2

where

206 fl0CtrOdyRCrrni~rfiI Chop. S

and

It is ueeful, in any fairly edaborate calculation, to have same independent t h d m on the algebra One such check is applied by the mquhment of gauge i n v h c e . By ddinitioa, M, depends qlicitly on field strengths. There are two field typee of intemt here, the weak homogeneous field F, and the Wt#rimnl arbitrary fidd, say f , that is requhd for the functional derivative of (5-7.64). Theee are indicated, adequately for our purpoee, by the initial terms of an -on,

which contains two linear and one f i e a t +on in the various fields. Now amaider the gauge ~ o a m a t i i o n

m d evaluate

which we sapaeer, m

--- I W, =k;y:(k'). ieq bX( k')

Since the field stmngtbs am unaltered by the gauge tramformation, and f is set equal to zero after the functional differentiation, only the first of the three teams indicated in (5-7.92) c o l l t r i a It is given by CEQ. (4-2.3l)J


where the dot recalk, the necesgity for symmetrhed multiplication, and

The reepanee of this stnrcture to the gauge trandorm8tion is given by

in which

and 8-ly D[, -ts a s p d a h t i o n of (5-7.90) to the situation +(l - v ) = 1. On noticing that

and availing on& of the aimplificati~~) asmchted with the use of %"(h') in (5-7.63), this reduces to

which will provide a umtrol on the direct calculation of y,'(k'). Harold speaks up. H.: It seems that almoet every paper touching on electrodynamics that has

appeared recently rnakee some reference to Ward's identity. Is it dated to your last remarks, and what is it?

S.: The -er to your first queetitm is ye& It follows from tbis affirmative response that "Ward's identity* must also be an expmsion of gauge invariance. Indeed, consider M, 4, or M,, which is to say, any object that contains

Eloctrodynclmicr I1 Chop. S

and field streagtha AB m&, it is invariant under the combined gauge transformation

One can expram this, using M(A) as an emmple, by the statement

or, in infiniteeimal form, as

The momentum v d o n of the latter is

where, again, the prime on M indicates the subetitutioa of lI' for lI . This is a fann o f Ward's identity* AS we have xwmded, the same formula will apply to M,, which statemeat can be written as

We M themby chde~getd to rbm the equivalence with (67.101), fa example, which does f o l k h m the rsductioas of M, and M[, as they are inferred from (5-7.96) :

(6-7.108) a 1

M[ -r - - / '&u( l - u)eqoF* [mu - (2 - u)yk8]- 2~ 0 D{

The list of ccmtnWo118 to $(h') lads off with the explicitly fielddepadent tarns produced by the rearrangements o f (6-7.76, 77),

X ([2m + r( lI - k ) ] [8u2(1 + O) ep( m)'

5-7

The transformation

Electron magnetic moment 209

combined with the basic integrals (4-14-78, (4-8.531

then yields

In connection wi th the teat provided by (5-7.101), we aleo record that

The next mntn'butim involvee the commutator introduced in (6781) by combining the exponential functions, which commutator is effectively evaluated in (67.86). Setting aside the spin term of (5-7-75) for later consideration, this gives

210 ~ ~ o d y n c m l u iI

and the trandormati011(6-7.110) effectively 0 0 1 1 ~ it into

Chop. S

xcq(PK)' + [(l - u)(2n - k') - uvk']veqyRk' l

Again, we nute the cmtrhtim to the product k:y:(k'):

which, M it h a m 2#ecisely cancels (6-7.114). More embarking on our major taalq the comput~tiem of the llleamurged form

of the^ m d term in (6-7.74)' we amaider the firet tenn of that expreesion:

5-7 Electron magnetic moment 21 1

Hare, we have introduced the symbol (omitting the prime)

X - u(1- u ) ( P + m2), (6-7.120)

a distingukhed from

X- ~ ( 1 - U) ( .@ + m2 - e q t r ~ ) . (6-7.l21)

Using the notation [it is (S-7.91), with $(l - v ) - l ] - k - k - un', (6-7.122)

we employ the spin expansion

e-w -[ - k(g2 + X' + m2u2)] ( 1 + ieqtrF)r (6-7.123)

The obeervations that

y - 2 y'yCoFy,--2uFyC (6-7.124)

The w e n t i a l function involving the sum %' + X.' am be decompomd into a product of exponentials with a compensating commutator term, in two different ways [cf. Eq. (5-7.79)]. To the Fequired accuracy, limited to the power of F, an average of the two forms wi l l cancel the additional comutatars,

Then, as we have remarked before [cf. the dimmion preceding (4-16.13)], the distinction between and k in the reprdred integration can only appear in terms quadratic in F, which giva for (5-7.125)

In the latter version we have reintroduced X', with the appropriate aF anmctian

212 Noctrodymomicr II Chap. S

team, and where the aF factor is already pment, ignored the dbthction between S' and #". The point of this is to recognize that the initial term of (5-7.127), the one that doee not have an explicit OF factor, is cancelled by a piece of (5-7.72), refenkg to MO, namely, the firat of the two terms produced by the factor 1 - U.

Accordingly, we are left with this umtnlbutim to 7:(k3:

a 1 - - 1 1 ~ U 4 V ~ ~ p m 2 U 2 2~ o +*,S (5-7.128)

and [Eq. (5-7.99)]

With the qh tQam of (6-7.75) still set aside, and conscim of the rearrange- men* already introduced, we find that the &dual form of the second tenn in (5-7.74) lmdf3

l + v 1 - 0 X ([m - r(n - &)]((U - k)'.Ip[-i.(TX + TXt)]

+m[-i.(Fx + (D' -

First, we &it two explicit itpin tenna, one of which is evident in (S-7.131), while the other appeam on moving F + m2 to the left, as can be done without effective

5-7 Elutron mc~gnetic moment 21 3

change, there to be replaced by egaF. This givts

du l + v y:( 1') 1, - - 2ie2/ c& 8' cl^ u2(l - U ) - q a F

2

tib - 2ie2/ c&88 & - u2eqirF (l( - k ) [ 2 ( n - k ) - k'] ve-bx') c - ~ ~ . 2

The use of the tramformation (S-7.110), and of the integah, (67.111) tha yields, after the 8-inbgmtian,

1 - 0 m(1 + U ) + U-+') [ ( l - u) (2n - k') - U V ~ ' ] ~ - + yp-

2 D" D'

As in (5-7.118), the product with K; contains

and we obeerve that

1 - 0 cEo a m(1- U) - U- yk'

+ f / & z u e y a ~ - U av D' 2 , (5-7.136)

Chop. S

or, with a partial integration,

the end-paint 8VBJuatiom have introdwed

Aftes reamwing the spin terme fn#n (6-7.130) in the mamex d e e c n i we have an effective reduction of the structure of the exponential function, as d d b e d by

where

i s tobedis t in-h D', w h i c h i s m by useof P. Inview ofthe a b c e of spin matricee in this eqmdon, the implied form of (6-7.130) can be simplified to

+-[-it(z2 + v] (n - c)')), (5-7.140)

which has already invoked the poaa i ty of omitting yII + m when it stands entirely to the left. Cancaaing the integration symbolized by ( ), we first note that, as in the dimmio11 based an (6-7.126), we have

5-7 Electron mognetic moment 21s

Nest, we need to evaluate

( L eq[-*I2 + V)]) r e-bw* (5-7.142)

Far that, we begin with

1 0 28 )

and then use the appmximate rearrangements

l - W , S ru'n + is- [ k2, n ] ~ ~ ~ ' , ( 6 7 . 1 ~ )

2

But the latter commutator term disappears after the ~~inbgmticm, from which we learn that

Finally, we require an evaluaticm for

(Q, exp[ - YK + V)] ) a (k , ,k ,~-~ ' ) (5-7.147)

and theadare amside?

The preaeding discumim can again be applied to the last integral, with the

216 Nectrodynomics II Chop. S

additional k-factar, which yields

and then

Thus, all is as in the a c e of the homogeneous field, except for the symmetrization required in the last tenn above.

In dealing with the double symmetrizations that are occasionally required, it is well to keep in mind the identity

which, through the appearance of a double commutator, implia that all such double symmetrizatiom are equivalent, to the required accuracy. Then using the reduction yII + m -, 0, when this combination stands on the left-hand side, we get the following for (5-7.140):

But let us quickly detach the last tmn, and carry out a partial integration,

where


Apart fmm the distinction between D; and D;, the first term on the right-hand side o f (6-7.1s) cancels the second piece of (5-7.72), the one arising from the last half of the factor 1 - U. The spin term that aurvive~ £ram this incomplete cancellation is

We now pesform the 8-integratiaae in the four eqmdons that have appeared: (&7.156), the lest team of (5-7.163), the firat part of (5-7.152), and, setting aside the explicit spin term, the latter part of (5-7.72). This give8

In working out this umtri ion to k,'y:(k'), we encounter

~ ( 1 - u)k8(211 - k8) 4 -u(l - u)[ ( l I - k?' + rn2 - e q o ~ ] - -M', (67.157)

and then iind that

The result o f combining (5-7.129), (&7.136), and (5-7.168) [recaUing the cancella-

Chap. S

tian of (5-7.114) and (5-7.118)] is

which m plrecieely with the anticipated eXpreeaian (6-7.101). This may occa- mon some surprh, since the calculation of y;(k') is not yet complete. The clrvitying obaervatian is that all terms not yet d d d involve the spin sbucture kid', which vanishee identically on multiplication with k,'.

To complete this first stage of our program, we return to Eqs. (&7.74,76) , and isolate the spin term that has thus far been set aside:

l + v x ([m - dn - k)] q[-c-p] .+~pq[-.qX'])~~. (5-7.1tm)

and begin by picking out the explicit aF term& The amochted matrix structure is

Now, the difference betwaa UFU" and oA'oF, which is a armmutator of U-

matrices, is itself a linear combination of U-matricq and is therefore annulled by the yC yp operation. Accordingly,

5-7 Electron mognatic moment 219

and the combination found between bracee in (5-7.162) reducee to the right-hand side of (5-7.163). Inasmuch as OF is everywhere in evidence, the exponential functions of X, and X; can be directly combined and simplified rrs in (5-7.89-91). In particular, the vector k that occunr linearly in (67.160) is effectively replaced by

This g i m

X [(l - u)yn' + (l - u?)ykf]

After the uF terms are separated out, (5-7.160) becoma

which makea use of the combed exponential f o m introduced in Eq& (5-781.86). According to (67.87), without the uF term, we have

and we now exploit the pomi i ty of effectively tramdating X to the extreme left, where it is replaced by u(1- u)equF. This supplie~ another explicit aF term,

Chop. S

The explicit F-term m (5-7.166) umtn'butee

which is

The remainder of (5-7.166) is stated in

1 -0 - - 2 k') e - ~ ~ ] . (6-7.172)

Then, if m e ranova the indicated symmetrizatian with the aid of

perform8 the S-integrale, and adds the 8-integrated explicit spin tenn of (5-7.72), the final contniution to yIp(k') is obtained:

Of the nine sets of terms that constitute y:(k"), only two, thoee labeled as e and i, umtain apmsioas that do not exhiit F explicitly. Thee are subject to checks based on c o m m wi th previous calculations. The simplest of those is

S 7 fleetmm magnetic mommt 221

the consideration of scattering, where the Dirac equation is applicable to simplify the right-hand as well as the left-hand side. Under such circumstan- which include setting F - 0, the distinction between JP and X disappears, symmetrized multiplication is mnecaaq, and

#"-0, y ' + m + O ,

which also incorporatat the photon mawr now required. After these and related rearrangements, such as

(n + H')' 4 2m7' - ik{~~', (6-7.176)

Here ( d t t i n g the prime an K)

in agmement with (4-4.76;), while

which uaee the integrab3 of (4-14.67,68) and the identity (dX2.42). This form-factar result ale0 h c i d e e with the known one displayed in Eqs. (4-4.68,77).

222 Nectrodynamia I1 Chap. S

After this lengthy interlude, it is well to recall what has to be calculated, namely IEq. 6-7.633

in which the prime on k has been dt ted , and we have made explicit the form of M, for a particle field obeying (yII + m)$ - 0. Before engaging in further detailed operations, it is desirable to perfurm the e x o ~ that is implied by the contact term containing c.

This is assisted by rearranging the integrand of (5-7.180) accarding to

We then isolate the f011owing piece of MP:

The integrand here umtains k,y,'(k), which, as given in (6-7.101), already displays OF, permitting us to discard that structure in the denominator of (5-7.182). We shall, furthermore, decompoee k,y:(k) into two par@ the first of which is the contribution of the term - l /m2u2 in (5-7.101).

it &tee the asymptotic form of k,y:(k) for 1k21 , m2. This initial contnion to Mi2) is

We perform the momentum integral in our usual manner,

l - - ic ' /dss&(exp(-&[( l - u)k2 + u(k2 - 2klI)I) ) k2 (l? - + m2

5-7 Woetron magnetic moment 223

If we were! now to combine this s-integral with the very nimillrr expmsion for l' [Eq. (&7.51)], the d t would indeed be finite. There is, however, the danger that the purely mathematical procedure of stopping each S-integral a t a common low- limit is not completely consistent under these circumstances, where the two terms have arisear in quite different ways. For this reason we proceed alternatively to introduce an effective lower limit to S, in a way that has an assured universal meaning, through a modi£ication in the propagation function of the e lec t row n e t i d y neutral photon:

(The preemce of additional factors other than s8 is merely for convenience.) When the calculation that produced f' is repeated with this modi£ication (that of S" remains unchanged in the limit 8 -, 0), we find that

a l q 2 + 8 ) q l - 28) =-- S "(L + f ) , (6-7.187)

2 a 8 r(3-8) 2r 28

where the last form is a sufficient appmximation for 8 a 1. The analogous modification in the calculation of (5-7.185) is

a l nl + 8 ) n l - 28) a I - - SE --(L+ f). (67.188)

4~ 8 r(2 - 8) 2~ 28

md the value thus d g n d to the backet in (S-7.184) is :(a/Zr). Acccdhgly, the^ firat contribution to M{@ is obtained as

with

224 Electrodynamics II Chap. S

The remainder of (5-7.182) is

where

The type of momentum integral met here has been dealt with in (6-7B), from which we recognb that (U in that formula is replaced by y )

and

All the U-integrals then encountered in (S-7.191) are elementary, with the ex- ti- of [d. (5-4.107) ]

and we get

5-7 flutron mqneflc moment 225

with

Now we must waluate

where it will usually be convenient to employ one of the forms

ikAuA, = y,yk + k , = - yk y, - k, . (5-7.199)

There are some parts of (5-7.198) that are similar to the one just considered. They arise h the mtributians to y<(k) that are given by (5-7.128) and the first term of (5-7.166):

Thispiemof c, isfoundtobe

The mamentum integral displayed in (&7.193), for example, might have been computed more simply, in the sense that only a single parametric integral is required. This is a consequence of the similarity of two of the denominators, which differ by a constant. That makw the following partial fraction decomposition advantageous:

Now the individual integrals are evaluated, as illustrated by

226 flutrodynamier II Chap. S

[the redt fm 8 = 0 is already given in (S-7.18S)], and they are combined in

Here is the direct productian of what is realized in (6-7.193) only after the tpintegraticm is pedormed.

The point of this little leer#n beoosnee clearer cm emmhhg the kinds of integrals encountered in the remainder of the calculation, an example of which is (the presence of F as a factor is understood)

A straightforward waluatian would yield a rather complicated double parametric integral. But here we remark that

md, again, that

5-7 Electron mognetic moment 227

Now we evaluate the compent integds, of which the simpler one is

Somewhat more camplicated is

i 2 I)---

2 1 {(l - u ) ( l + 0)lW- - u ( l + v ) l ~ - 1 - v2 l - U l + v U

2 +(l - 0 + 2m) log - ( 2 - u + m ) l o g

l - 0 + 2 w }, (C'7.210)

2 - u + w

and the combination of the two give8

We are not yet ready to proceed on to this final stage of infegration, however, since m e contributions must still be supplied wi th the contact terms that isolate

228 EIoctrodynomicr II Chap. S

the slplidt aF dspcmdenm They are f d in the part. of MP that are implied by yl'(k)l,+i, Eqs. (6-7.156, 174). An euunple of the kind of momentum integral encountered here is

1 (6-7.212)

IY k2 (n- k)' + m2 - e q a ~

The surrounding I&factoxs have been retained in order to emphasize the following property. This object is dimdonl- If m2 were everywhere accompanied by -eqaF, as it is in the eventual substitution, -r -(m2 - eqaF), the outmme would be a pure number (it is multiplied by m in the complete structure), which must be rermwed by a suitable amtact term in order to maintain the normalization condition on M* But the required aF te rm are lacking in

Heace the effective value of (6-7.21 2) is the negative of that multiple o f aF which is needed to repair the deficiency:

it has been simplified by l e g + m2 -r 0, as permitted by the explicit factor uF.

While this argument is quite come&, it may not be convincing to some. And the procedure can fail if the preen= of spin matricee interferes with the simple treatment of m2 - eqaF. For th- recaans, we d d b e a more fonnal prooess. It begins by expanding the last denominator in (S-7.212), now designated as I,

The eawatial o k a t i a n about the fht integral is that both denominators involving contain it in the same cambination, (I2 - k)'. Hence they can be combined in our standard manner, without reference to F. And the &-integration then pmceeds as though F - 0, since the implied aror is - p: The result is a function of P,

which we have expanded with sufficient accuracy to deal with our situation, where + m2 -r equF. The flanking factors of do not interfere with the latter

subetitution, and thus

whese we have omitted the arguments o f f and f'. Then, retaining only the explicit uF terms, we have

1 ZdequF f - m 2 f ' - m 2 -- [ ( i : (h2 - t i n ) '

)l. (5-7218)

To find an eqmdan for f', which is only requhd for F - 0, we differentiate (5-7316) with reepect to D', and then multiply by +W:

That leavet us with

Now we return to f, which is (S-721% evaluated at F - 0, fi + m' - 0, and

230 EIodrodynamicsII Chap. S

ccmsider its depdence upon m. D i m d a a l considerations ahow that it is a numerical multiple of l/m2, or that

On the othea hand, direct differentiation of the inbgd, with n/m treated as invariable, giva

The information supplied by adcling (5-7.221) and (6-722) then simplifia (6-7.220) to

in agmeme~t with (6-7.214). To indicate the caution with which the elementary arjpment should be

applied, coaside the integral

which is also dimdonlee& If the situation were completely analogous to that of (S-7212), the reeult would only refer to the effect of inertkg equF into V, where the asdated coefficient vanishes, ~ c e the vector integral containing kA is proportional to nA, which enters into an effectively null commutation relation with P. But this implicitly aaaumm that the outcame obtained when only m* - cqoF occura is just a umstant to be removed by a contact term. In fact, it neceesarily involvea the combination

owing to the an-cal rather than symmetrical pairing of lI camponenta


The presence of this factar indimtee that the WC integral can be stated as though F - 0:

which yields the non-vanishing reeult

Among the more complicated of the integrals is

Since the integral in the latter form is d imdonlee it might appear that the elementary argument is applicable. This is not quite true, however. The integral is meaninglea8 without its accompanying contact tenn, which d t s in an additional contribution. Let us concentrate on that effect by supposing that m2 has everywhere been supplied with it8 partner -equF. After removing a factor in D' to make the coefficient of k2 unity, we get an expPason of the form

where the symbol p stands for the two additimal parameters, say W and y, which are such that

Concerning the functiana f (p), g(p), we need d y -k that g( p) is always poeitive. As in the reeult of (5-7.150), we have, effectively,

232 Noetrodynomks II Chop. S

The elementary argummt, leading to no aF contniuticm, doee apply to the welldefined integral produced by the second of them terms, where the flanking factors of y and II finally combine as yn -, -m. What remain^ is

where the neceesary amtact term is now explicit. The S-integral that appears here is simply

and thus

where

cancels the aF tenns that have been h s e r t d in W. Concerning the last integral, we note that the farm

leads to

in which


Another type of integral mpirhg some comment is

or, in effect,

We again consider the integral as it would be if only the combination m* - q e F occurred. It has the fonn

where now it is necessary to know that

After performing the K-integration and supplying the contact term, this reads

Chap. S

and

The latter object, which introdues the correct uF dependence, is

l +4 yk- ( (k2 - 2 k ~ ) ~

In both of the integmki that compote Lp we can make the effective subetitu- tia

We also remark that

where it is important not to be misled by the d i m d d e e e nature of the integral an the left. The form of this integral is indicated by


We have now illustrated the various laborgaving devi- that can be used to reduce the remainder of MP to the final integration stage. Since the number of tarms has become fairly large, we give this list as it appears after some algebraic combination has been effectd. It is etated with the aid of the following set of k-integrals (n = 1,2):

whem only the A and B typee appear for n - 1. There is also another set of integrals, which am produced by the mbetitution

lowercam letters designate this type. This final contribution to c, is given, in tenas of the double parametric integration

136 E l d y m o m i c r !I Chap. S

+/$$[-l - 2u + u(2 + u)( l - v)]&

+ /U' [!(l - V ) - U(! + 2(1- 0 ) + *(l - 0)')

+ u2(l - v)(+ + ?(l - v) ) + 3uY1 - 0)(1 - ! ( l - v))] Cs

+/us[-$ +!( l - v ) - ! ( l -v)'+ u(1- v )

X($ + $ ( l - v ) + %(l - v ) ~ ) - u2(1 - v)2(! + $(l - V ) ) ] 4

+ 1 U2-[ - !+u(y+y( l - v ) ) liV +u'(3 - $(l - v ) - +(l -v)2) + u8(l - v)(-3 + +(l - v))]E2

l + + /U(I - .)(l + ~ ) [ 4 q + 2u24 - 3u2d2] + / [ -2u + 6~

( 5 - 7 W

The farm uf the very last team emphahs that the integrand, although here preeented as the sum of a number of different coatributions, should be conaidered as a unit. Indeed, it is a useful check of this rather involved algebra to verify that gingularitiee at d o u s end points of the integration, which occur in individual terms, are cancelled when the whole structure is examined.

Turning to the explicit forms of the functions A,, . . . , &, we recall that the first of the638 is already known [Eq. (6-7.!211)]. We preseat it again, now written as

S 7 Electron maglletic moment 237

The 0 t h ~ ~ functions of the capital class can now be found by combining direct intagration wi th the use of identities such as

and

The d b are

and

238 Hoctrodynamiu I1 Chap. S

The lmercaae functions appear in the single cornbination

The laet two sets of temms in (S-7.255) combine into

Of the lllennaiaing typee, peahape the mast &ble are the integrals of the farm [W = f(1 - v)]

where the last v d a n is produced by expanding the logarithm. The specific ontllmplm encountered here far K, I > 0 are enumerated by

= E 5 = t(3)s n-l

which intdutm the R i m &a function of 8rgumeblt then (K > 0)

5-7 floctron magnetic moment 239

followed by

and finally (k # I > 0)

A related set of numbers is provided by the integrals

The various poaiit iee here, for K, I > 0, are set out as

OnealsDrneetsfamsoftheseintegrdbinwhichI< O,specifidy, l- -1, -2. To deal with them, we v t e out the first, or the first two terms of the seriee in (5-7265) and get

which introducets examplee of the aummatim

240 Elufrodynamia II Chap. S

The latter is specified by (again, the zero value and identity of the indices are handled separately)

and by

In the analogous situations for JkI, we have the simpler relations

For the application of them? d t s , we isolate all terms having L, as a factor, and also retain thaw pieces of


that have inverse *factors. The net contribution obtained in this way is

where we have set aside, for wentual cancellation ekewhere, the singular integral

Next we d d e r the terns having L, as a factor. Rathg untypically, only a relatively small number of them s d v e after heavy cancellation; the explicit structureis

Them integrals are uniformly wahmted by partial integration on v, combined with the introduction of the variable

As a simple example, d d e r

which exploits the pomibility of omitting powem of v and z that are odd unde the reflection of both variable& Another example is

Herre we meet the familiar integral

Chop. S

and the related one,

- 2 f 1 3 w2

II -- 1-0 (2n + l)' 2 6 ' (5 -7rn)

which yields

The outcome obtained in this manner is

The tsnm containing L, that are of greatest difficulty occur in G, whese the following combination appears:

To appreciate its aigdicance, one must know the limiting behviora of

at the boundaries of the vatiablm U and v. Theee are

As a axwqumce, has no singularity at U = 1. But it behaws as ( l - v)-' for v + 1; that aingukity must be isolated. The technical problem in evaluating an

5-7 Elutron magnetic moment 243

integral like lur, is moving the large inverse powers of 1 - U and 1 - v without thereby introducing spurious singularitim specifically, at v = - 1. Here is one procedure for that puqme.

We write

whem the quantity subject to differentiation has been contrived to vanish at both limits of U. We also decompaw the other terms in partial fractions in order to exhibit the powers (1 - v ) - 2 and (1 - v)-'. This yields

& ' l 1 1 1 / " ~ - / ( 1 - v ) 2 ~ h[i-710aG + log -1 l - U

2 U Eloctrodyncwniu II Chap. 5

where the ,last term, which lacke a v-singularity, has already been integrated over v. The U-integral that is the d c i e n t of (1 - v)-2 vanishee, as it should; that of (1 - c)-' turns out to be !. The last tcw is decompowd into a number of integrals like

and

to give the reaulh

with its ccmsequmt v-integral

An example of a compoamt integral evaluated this way is

5-7 Hoetron magnetic moment 245

Numbers of the latter kind, combinatins of fractions and multiplee of r2, are the rule, with two exceptions. They arise from me term:

The first of the novel s t r u m occurs in the integrals

which are m e & d by plvtial integrntim, combined with the mbetitution U -, 1 - U. The introduction of a nerrP variable,

converts the initial infqpd into

For the last term of (6-7.301), we make the subetitution

u - +(l + v ) ,

Chap. S

which, incidentally, also evalua* [cf. Eq. (5-7287)]

and get

The information needed to determine the only new integral appearing here is supplied by

and, employing the substitution

1 ( l ) 0 0 - :R319

and


which then yields

It a h d d also be remarked that the core of the structure which producee 5(3) and r2 log 2 is found m the integral

it is a sommhat more complicated analogue of

The complete list for tbis type of mntn'bution is

together with a singular 0-integral that precisely cancele (5-7281). Finally, we come to the terma involving

which also includes the contribution of the last part of (5-7279). The first thing to observe is this. Under the substitution

we have

which would aeem to indicate that the two sets of tmns should have been united. Unfortunately, they are of sufficiently different structure that no great simplification would thereby d t (which is the general chara-c of this calculation). Neverthelees a few integrals are conveniently evaluated by this tramformation.

248 Etutrodynomicr II Chop. S

The most important of these is

which also c o a s t i m the complete source of thaw! terms in the L, strudumt. For the reek, we proceed mewhat analogously to the L, calculation, but without using the substitution (5-7.319).

We dart with the following combination extracted from the expression for E, I?Eq* (5-7261)B

where

W - $(l + v ) .

On remarking that

we evaluate the zu-integral:

from which we deduce such integrals as

Another set of inbgds is i n f d from the relation

Electron magnetic moment 249

with its cometpence

1 1 1 J1&wrB- --- log -;

~ ~ 1 - ~

an example is

The complete contribution of the r, class is

- 2 l 2 lr2

a(2. 10g2-!3(3)) - ( 3+3+ :++ ) - + 7 + $ - : . (6-7330) 6

The remainder of the integrals contain L, and various powers of U and W.

Examples of these that can be inferred from known results are

As for the others, the following are cited as repremmtative:

The final outcome, combining all terms related to h, is

The artificial nature of the separation of

into the four parts exhi'bitsd in Eqs, (5-7280,209,317, 333) is evident from the relative simplicity of their total:

250 Elutrodynamltr I1 Chap. S

The other piecm of c, listed in Eqa (5-7.54,190,197,201,264) have the wrm

we note separately the vacuum-polarization co~~tributian IEq. (5-7.611

The time has come to add (S-7.336, 336, 337) and, a t long last, infm the numerical coefficient of ( a / 2 ~ ) ~ in the additional electmm magnetic moment. Here it ie:

which uses the number

The implied value of the e l m magnetic moment, bgsed upon the nosninal fine s t n l ~ c o n s t a n t

which is in exceptional agmeme~t with a rece~t meaaurwnent,

To take 881iously the tiny &dual dbmepancy (- 2 X 10-~) would require a knowledge of the ( a / 2 ~ ) ~ uxmction, and more accurate information about the value of a.

The impraved value of the electraa magnetic moment will have implications for various subjects alreatdy extemively discussed, but it seems regsonable not to go further into them mattem now. The length of calculation mquimd for the evaluation of them amall effects has begun to outweigh the instructional value provided by seeing all the details of source theory at work on varied problem& Indeed, as the justumcluded magnetic moment calculation made distregsingly evident, it eventually becomes unfeasible to display the computational details


completely (yet, hopefully, enough clua have been provided to pennit a relatively painless repetition of the labor).

Harold looks appreciative. H.: Let me congratulate you on joining the very small club of people who have

s u c c d y performed this magnetic moment calculation, first correctly carried out by C. Sommerfield [Harvard PhD. Thesis, 1957; Ann. Phys. (N.Y.) 5, 26 (1968)l. How do you feel at this moment?

S.: Exhausted. And somewhat disappointed. While the source calculation, which is rather similar in spirit to that of Sommerfield, is vastly simpler than his (it is quite staggering to find him, at one stage, manipulating as many as seven parameters), the anticipated reduction of the algebraic structure prior to the final integration never quite materialized. As we noted, the eventual numerical form is not particularly complex, in contrast with its component elements, which suggests that some other way of organizing the calculation might be even more effective. It would be pleasant to find it, particularly if one wanted to press on to the ( a / ~ n ) ~ calculation without feeortin& as others have in desperation, to computer tmsb tance. But that is not, for us, an immediate pmqect.

5 - 8 PHOTON PROPAGATION FUNCTION Ill

Although we have just forsworn further largescale electrodynamic computations (deepite some earIier tentative promise^), it may stil l be worthwhile to explore additional aspects of the non-causal calculaticmal methods. It would, for example, be interesting to see how these techniques fare in comparison with the rather extensive causal calculations set out in Section 5-4, where effects of relative order a2 in the photon propagation function were d i s c d . That is the purpoee of this section.

The coupling between two component electromagnetic fields, A, and A,, that is produced by the exchange of a aingle pair of non-interacting spin- $ particlee is conveyed by [compare the action e x p d o n (4-3.9)]

the superscript records the powes of e in the formula. Recalling the expansion IEq. (3-12.23), for example]

Chap. S

we recognize that (6-8.1) ie also part of

f i 'h aly~ , ,~+Ab + cat. (5-84

that is linear in A,, as we might have i n f d directly from (4-8.19). The next dynsmicd level introdu~le~ the exchange of a single virtual photon.

That is indicated in (5-8.3) by replacing the propagation function G + ~ with the modified propagation function g+A. Correct to the relative order a, we have [E~B. (&6.2,3)1

w h q apart from the amtact tmn,

The d d effect of order 3 ie then given by

whem it is undembd that only the tann linear in Ab is mtained. This is made somewhat more explicit by writing (5-8.6) as

whem &, refeaw to the form of M in the h c e of an el-etic potential. We recognize that the firet two of this formula state the effect of

d r y i n g the propagation functions of the two non-intending particleaa This was discuaeed in Section 5-4 as part of the three-particle causal exchange mecha- nirnn. Our mejar concern now is with the last part of (5-8.7). specifically, with the detemhation of the contact tenas that are related to the internal single-photon exchange act. When Ab is made explicit in the third term of (5-8.9, the latter can

5-8 Photon propagotien function 111 253

be written out as

The notation wed here facilitatee the cansideration of an a d d i t i d electmmag- netic field, which, as in the preceding section, we exploit to help clarify the nature of the contact term& An infinitmind variation of that ambient field induces the following change in (5-8.8):

The cyclic prom of the trace, combined with the tramfamation k + -k, has been used in producing the form of the seamd hm. In each of the two contribu- tiane we see the infinitesimal action of an electromagnetic field modifying the prowet3 of single-photon exchange. The appropriate contact terms are known, and are illustrated in a related context in Eqs. (5-7.42,43), for example. The alternative view of (5-89), where gingle-photon exchange is influenced by the differential action of two fields, reqtlires no contact terms. Thus, the contact terrns that supplement (5-8.9) are

As in the preceding section, half of the contact term can be abeorbed in producing the complete structure of M in Eq. (5-8.6), which givee the following detemina- tion of the contact tenn in that fonnula:

In addition to this "internalw contact term, there are, of come, "externalw contact terms which exploit the noa-overlapping arrangement of the fields A, and Ab to satisfy the physical requhments of the theory.

254 Moctrodynomicr II Chop. 5

A useful analysis of the structure of follows from the known decoxnpoei- tion of M into two pnrts [Eq. (5-7.!50)]

w h h& is the gaugecwruiant form of M in the h c e of an el-etic field, and M, depends explicily an field &engthe, We mognize in - G+M,,G+ the m ~ c a t i m in the free-particle propagation function that is given by [cf. Eqa (5-7.55, W1

where

A*(M') - zL'(1- 4 r M' $)'(l ' ) (6-8.11) (M' T m)'

and we have indicated the n d t y for an infra-red m&cation of the kinematical factor near the threshold at M' = m:

The use of the individual ppqptian functions (yn * M')" in (5-8.6) reducee the problem to the known one of two-particle exchange without interaction, but with the mass substitution m -, M' (the algebraic sign of M' is without effect). Thus, we get a first mtrihtion to the modified weight function of the photon propagation functioa, a(M2), by combining the elementary result [Eq. (S-4.186), for example, with m -, Mq and the spectral distribution of the mass M' that is i n f d from (S-8.14),


where we have h included the coafxibution aamdated with [cf. Eq. (5-7.52)], the infra-red-dtive part of the amtact term. That has the advantage of removing all reference to the fictitious mass p at this initial stage of the calculation.

The latter remark is verified by isolating the part of the spectral integd in which M' ranges from m + p to m + 8M, where

a2 ID [(M' - m)2 - J] ID ( l - ( l + 2 $ ) l b M d ( M ' - m ) ( M - m)2

which applies a specidhtiaa of the integral (4-497), or (5-4.79). We thereby replace (5-8.17) with

The spectral integration is easily performed with the aid of the variable v' defined by

it rangee from zero to o - 80, in which

256 Eloctrodynomics I1

The outcame is

Chap. 5

where, as in Eq. (6-4.67').

One can now compute a portion of the integral that is related to atomic energy displacements [Eq. (5-491)]

The use of (5-8223). employing such integrals as

2 "-l 1 E- l-t? 2 n + l k , , 2 k + 1 '

givee the d t

although it is potsi'bly hp ler to prcumd directly from (6-820):

where one utilizee the vahte of this integral that is applicable to the non-interacting system with particlee of effective masa M'. The fractional d a a exhibited in brackets in (k8.27). for example, is lees than half of the known d t given in (6-4.197). In contrast, the high-mass behavior of (5-8.23) overshoots the


mark:

since the logarithmic factor is not prasent in the correct d t of Eq. (5-4201). We now turn to the major problem posed by (5-8.6), which is the evaluation of

For that, we need an appropriate expression for the part of M, that is linear in the field dxengths. The qualifier ''appropriate'' means that the simplification thus far employed in non-causal calculations, the rejection of the factor yn + m on at least one side, is inapplicable here. We must first remove this lacuna in our knowledge (that information is available from the causal calculation of Section 4-6, but our purpose is to illustrate non-causal methods).

Let us begin with the form [equivalent to Eqs. (5-6.4, S)]

where we have the intention of exhibiting M, explicitly in order to facilitate its removal (a different strategy from that followed in the last section, where M,, was simply subtracted). Accordingly, we first commute y, from the right-hand side to produce

The last term, being explicitly linear in the field strengths, is easily evaluated, and we set it aside. Next we decompoee the first tenn of (5-8.32) through the rearrangement

8 Eloctrodynamier II Chop. S

and ccmsidetr the integral cantaining the factor k - u n . Here, we note that

the last statement mcording the result of a partial integration. Accordingly,

which, again, is errplicitly linear in the field strength& That facusee attention an the remaining structure:

Now we make use of the &transformation device of Section 4-14, wbich is such that

+m2u - q u F ( r - &). (6-0s)

Since only linear field tenas are of inter&, we write this as

9 - k2 + u(1- u ) [ - ( ~ ) ~ + m2] + d u 2 + /u&'u'2apk - P ( % - U'€)€ 0

5-8 Photon prapagdion function 111 259

When the explicit fielddrength terms are dt ted , and the k-integration per- f m e d by the usual formula

the d t obtained for (5-8.37) is pdsely K, as presented in Eq. (5-6.54). We therefore confine our attention to the explicit linear F-terms that appear in

an expansion of (e-"X). In doing that we need not include the k tenn of (5-8.39), for, as detailed in (4-14.60), it8 rotational structure annuls the state6 symbolized by ( and ). The ranaining terms are

wheae, as in (4-14m, but without reference to a photon mass,

It is ccmvenient ta amsider a typical Fourier component of the fields, eim, and thereby apply same of the d t s obtained in Section 4-14:

which is the content of Eq. (4-l4.S6), and IEq. (4-14.61)]

In presenting the outcane, it is alao convenient to d d e r a typical particle mat^ element in which l?, standing on the left or the right of F, is replaced by p'

2W Electrodynamics l l

or p", rmps~vdy, whme

pf - p" 'T" p*

Chop. 5

(b8.45)

we find that

w h e the an D es tht U' replam U h the 9 of D. Anoaw way of ~ t a ~ g gaug*htr&mt fam mager?r an a d o p ~ g a hrmtz: gauge, so t b t

5-8 Photon propagation function l11 261

The result of (5-8.48), then appears as

a u(l - do -- 4 r s

I - e q y ~ ( e - * ~ - a-")(ypN + m), (5-8.62) 2

which puts into evidence emamplee of tenns having the factors yp' + n and yp" + m, tenns which must be retained in this calculation.

It behow- us now to retrace our steps and evaluate the integrals that were set aside in Eqa (5-8.32.36). In doing this we meet K-integrals that are the specialization of (4-15.55) to X = 1, aa given by

where cmeaponding primes on X indicate the introduction of p' or p", and of (4-15.6), here written as

We find that

1 cEo

e* d s s 2 m t u - y p ( [ m - ~ ( n - k ) ] ~ - " ~ t ( l + ~ ) s q [ ~ ~ ] e - b ~ ! ( l - ~ ) ) 2 9 Y.

Chap. S

and

In combining these remalts with the evaluation of (6-8.37) that is provided by (&8.48), the foIlowing identity is valuable:

1 4 1 - U)( P" - - u2qi U(1 - U)( p" - p'") - d l - 0. (6-8.67)

D

where [we again rer#rt to the integration sign simplification of (6-7.2S4)]

while

and

S 8 Photon propagation function 111 263

This decompoeitian evidemtly cataloguee the pmmce of yp' + m and yp' + m factors, which are abo implicit in the last term of M'). Only M , survives when bothfactoraaremjected, andtherearewveml (m- teststowhichit can be subjected. One is provided by the form factors displayed in (5-7.178,179). Another refers to the situation considered in Section 4-5, as d d b e d by the coupling stated in Eq. (4-520). Here we deal with a photon field ( p 2 = 0), and d c t one of the pnrticle momenta to be on the IMSS shell (p" + m2 - 0). The d t i n g pure spin coupling inferred h (5-8. is repmmted by the fonn factor

u2(l. - U) G,( p") - 2m21 1 - 0

u(1- u ) ~ ( # & + m2) + m2u2

where

in agreement with the weight function &own in (4-520). The calculation of the preceding section provides a tcst of M, +M',. When the homogeneous field coxmidred there vanishes, the contributions to y:(k') that are labelled e and i d v e to give the functional derivative of M, with respect to -eqA,, with yp' + m and p" + m2 set equal to zero. After some rearrangement, which involvee the use of the identity (S-8.S7), the equivalence of the two v d a n s is indeed realized. This comparison atK, leaves one with the im@on that the forms used in the calculation of Section 5-7 are 80mewhat lacking in felicity.

We are now better prepared for the calculation of (5-8.30). But it might be well to also have before us the analogous non-causal evaluation of the elementary %p compling, which has thus far been diacuased only by the c a d method of Section 4-3. The action @on of (&8.1), without contact terms, is written out a8

where the trace in the charge space has already been &ormed. Let us comment here that a gauge Wormat ion on A&), for example, when combined with the

identity

Chop. S

(5-8.66)

d t s in the following change of (5-8.64):

which we infer from the non-causal cimmstmce of nonsverlap between A,@) and A,(x) [therefore also A(x)]. Thus, if the coupling of (5-8.64) were applied without this space-time d c t i o n , contact tenns would generally be required to maintain gauge invariance. The preferred procedure here is to rearrange the coupling for the non-overlapping arrangement so that gauge invariance is maintained after the space-time extrapolation.

The introduction of the non-causal fonns of the particle propagation functions convert8 (6-8.64) into

p" - p') A:( p' - p")

where Fourier transforms of the fields now appear. We change the momentum integration variablee in two stages, first, by writing

and thean, through the standard o h a t i o n that

5-8

by the translation

The net tmmdormatian,

Photon propagation function 111 265

is combined with the trace evaluation

: hi,) [y,,(m - Y P ? Y , ( ~ - YP'?] =P;# + g& - g,,.( P'P" + m2). (5-8.72)

and the evenneare of the exponential in the find pvariable, to give the coupling

[ ( 1-02 1-02

x ,&p, - g,,, p2 + m2 + -h') 4 - +k,,kV -gpvk2)]

If we now use the integral evaluatim

to provide an effective replacement for 2ppp,, the first (g,,) temvl of (5-8.74) involve the S-integral

266 lodrodynamics II Chop. S

which leaves an explicitly gaugeinvariant coupling* But we must not fail to notice that if the p-integration were performed first, in accordance with the familiar reault

the S factor in the brace of (5-8.76) would become S-', and the indicated null outcame would no longer follow. It is here that the context of non-overhp intervenee, aince any polynomial in k2, specifically that implied by the firat two terms of the exponential expre&on in (5-8.76), can be reinwed fmm e:)(k) without changing the coupling (5-8.73). Thus, we are still able to make $?(k) assume the necessary gauge-covariant form without altering the coupling in the non-overlap situation,

At this stage, we have

which we proceed to rearrange by a partial integratim on v:

The first term on the right side doa~l not contribute to the non-overlapping situation. If it were neverthelas retained in performing the space-time extrapolation, i t would constitute a local coupling of the field strength& It is therefore excluded by the normalization condition that accompanies the initial demiption of the photon. In this way, we come to the final statement that accords with all the physical requkments,

5-8 Pho- propagation function 111 267

the preeence of - it in the denominator is understoad. The addition to the action then implied by (6-8.73) is exactly that given in (4-3.70), with the spectral weight function appearing in the form

as presented in Eq. (5-4.186). The first priority in evaluating (S-8.30) is, clearly, the identification of the

etructure that the S' contact term is intended to excise. A clue is provided by the obeervation that the denominators D and d, which are exhibited in Eqs. (5-8.46, 47), change their character in the limit U -, l, there becoming independent of the particle momenta p' and p". If we inspect the structure of M,, as detailed in M,, MP, Mr, we see that all tenns have a compensating factor 1 - U in the numerator, with one exception. That is the last contribution to M, [Eq. (5-8.59)], which has in its numerator the factor

attention is directed to this piece of M,,

Note, incidmtally, that we have not focused on the factor of u that is common to the whole of D and d, since wery term of M, has a compensating factor of U in its numerator.

A small di&on is now in order, however. As in the preoeding section, the comparison of singular integrals requkt3 that a limiting procees of universal significance be employed; an illustration is the modified photon propagation function of (6-7.186). We proceed analogously here, but find it convenient to alter the convergence factor adopted in the latter equation:

the equivalent of the limit Ct -, 0 is X --, a. Thus, the construction of 3' given in (5-7.187) now reads

a m2u2 + *(l- U) X = -ll&(l - u ) l q

m2 u2 S :(log; + f ) , (68.85)

2 ~ o 2 r

266 EIectrodynomia II Chap. S

where the last form refers to the situation X S m. The advantage of the mbstitu- tion (5-8.84) lilies in the new fonn of the photon propagation function,

in which the additional term mimic# but with the wrong sign, the propagation function of a particle with the mats X. Then, in dhmshg (S-8.831, m e has only to make the substitution

with 4 -D+ (l - u)X2.

Hatold seam d i s t d d H.: Your use of the alternative convergence factars exhiited in (6-884)

reminds one of m e variants of the device known as " w t i o n " , which is widely employed nowadays to attniute mathematical meaning to physically ambiguous theorilies that give divergent integral answers to physical questions. Are you incorporating regdarhtion into the principles of source theory?

S.: Cerhinly not. The contrary should be evident in the present context fiwm the fact that the problem under dimusion has already been solved by c a d methods without reference to such concepts. The non-causal techniquee lead to finite, well-defined expredons which, for convenience of evaluation, are d-- posed in ways that require ccmvergence factors to give meaning to the separate part& Since there is no queetion about the existence of the whole structure, any reasanable convergence factor may be employed. To apply the term " reguMza- tion" to this procedure is to encumber it with a heavy load of misleading associations, which is alwap the danger when m t terminology is transferred to a logically different situation. As K'ung Fu-tzu pointed out some time ago, the neceseary prelude to the solutian of any problem is "the rectification of named'.

We now write out that part of the action @on (5-8.30) which is contributed by M,' [Eq. (5-8.83)], a8 modified by (6-8.87). It is

where the Lomntz gauge hrrs been made explicit, and we have continued the use of the symbol K, rather than p, to denote the photon momentum,

5-8 Photon propagation hnctioo 111 269

The trace appearing here is already known from the elementary calculation [cf. Eq. (5-8.72)]. The principal new feature ia the premnce of the additional denominator, D or 4. For our preeent the simple& procedure for handling the thrm denominators may be the use of the repreeentaticm that is illustrated by

where

1 + 0' l - 0' D b - w-(p"+ m') + w-(p" + m2) + ( l - W )

2 2 u(l - U )

and

is the appropriate replacement when D& occur& The introduction of the variable P, a-g to

p ' - p + i k , p " = p - i k , (5-894)

ccmverts A into

in which we have employed the symbol

5 - too'+ ( l - w)o.

The trmmlatian

p + p - f5k,

270 flwtrodynamics l1 Chop. 5

which is analogous to (5-8.70), nimilarly converts the trace into [cf. Eqs. (5-8.72, 74,7511

which refers specifically to the Lmentz-gauge simplification of the gauge-invariant coupling. Thus, the action expmsicm of Eq. (5-8.89) has become

wi th

In the latter form, the following symbols are used:

The S-integration is now performed to give

5-8 Photon (nopogafiom function 111 271

which anticipatee that 8;' doe^ not contribute in the limit X -r m. Indeed, the cmly role of X is to appear m the integral

where the farm of the right-hand side exploits the fact that 4 is domb&d by A2(1 - w)/u, exoept for values of u that are very clme to unity, when it is n m to retain the term having the factor (l - U)-' . We shall separate this expmsion into two parts:

For the fimt of them, which io independent of W and v', we perform a partial integration on v, taking into account the factor 1 - d in (5-8.103):

The f h t term on the right-hand side here produce8 a local coupling in (S-8.99),

272 Eloctrodynomicr II Chap. S

and is therefore without effect in the non-overlapping arrangement. The second one is evaluated as

which is so written that one recognh the structure af f' aa given in (5-8S). And indeed, according to the prediction of (5-8a) , this contribution to P 4 ) ( k ) is cancelled by - r @ ( k ) , where p ( k ) is the coefficient of g,,, in (S-8.80). The mmainhg part of (5-8.10'7) can be rewritten with the aid of an identity, which is analogous to and inferrable from the identity of (4-12.42):

The reeulting form of P(")(k)ld which now include8 the f' contact term, is

where the required contact term has been incorporated into the last contribution.

5-8 Photon propagation function Ill 273

The cam^ between the meaning of P(k) in the brentz gauge, as illustrated by (5-8B), and the Inrentz-gauge, momentum form of the action e x p d c m of (4-3-70), &m that

The first term on the right side of (5-8.109) thus provides an immediately identifiable piece of M28a(M2), eqnmd in the parametrizatim

We draw attention to the M2 S m2 limit of this part,

for it removes the incorrect logarithmic Mdependence of (5-8.29). Here is further reaseurance that all is going well.

In the following, we shall be primarily concerned wi th the simpler properties of the weight function a(M2). They comprise the computation of the integral of (5-8.25); the evaluation of the constant that, to the present accuracy, states the high-energy limit

lim M2a(M2) - C,; (6-8.114) M 4 oo

and the evaluation of the constant that gives the altered threshold behavior,

All of these are obtainable from limiting aspects of P(k). Thus,

214 Electrodynamics II

and

Chap. S

while

Accordingly, we now proceed to set out the various contributions to P4)(k) . The simplest tam in M, [Eqs. (5-859-6111 is 4. This remark refers to the

waluation of (5-830), where the two free-particle Green's functions are cancelled by the corresponding factors in M,. The definition of (5-8.99), nurtohb wtondis, givee

where

p - p ' - + R - p N + f k .

The d t i n g form of the denominators is

and the pintegral is parametrized as

5-8 Photon propagation fundion 111 215

Two partial integrations on v, supplemented by appropriate contact terms, then yield

and

Utilizing the k2 -, 0 limit, in amordance with (5-8.116), we get

while, for large k2, with only log(k2/m2) retained,

There is no contribution here to G. The calculations associated with M, - M , and MD make use of alI the same

devicee and auxibry computations that we have already amply illustrated.

276 Eluhodymomics II Chop. S

Acmdbgly, we nor direetly state these cantributiam to P4)(k):

8, l - d ~ ( 1 - W ) log - + -k2

8, 4 l - U 8, l - U

a+ l - v2 k2 ~ ( 1 - W ) +m2 -+-- - W ) bog 8+

l - U 4 m2 l-uw

where the three terms are associated with the reepective three terms of (6-8.60), and, similarly,

5-8 Photon propogcrtion hncffom 111 277

In writing theee out we have uaed the symbol 8 of (&8.101), and introduced the related s t r u m

The quantitie~ with + subecripts in (6-8.128) are obtained from these and related symbols by placing v' - + 1, as illustrated by

The asymptotic behavior of the weight function that is expressed by the constant C , [Eqa (6-8.114,llV is easily inferred h the various pi- that we have exhibited. The contributions with labels a - a', p, y, are, respectively,

To t h e we add the &t of combining (6-8.113) with (S-82291, which gives

coinciding with what is &%ed in (6-4201). As for the constant that gives the threehold behavior [ E ~ R (5-8.116, 118)], we find that its only source is the second tenn of the piece labeled a', which is exhibited in Eq. (5-8.109). Its waluatim for -k2 = 4m2 has the following singular part, arising from 8-l:

a? db clo' W 1

2r2 l - u i ~ + z ( l - w ) c ? l - u

where

When modified to incorporate a mall non-zero value of k2 + 4m2, the singular inbgrd factor beamas

278 Electrodynamics I1 Chop. S

while

and we get

in merit with (5-4302). The third computation we have mentioned (5-8.116)J, which is the extmction of the coefficient of (k212 in an expandon of -P4)(k), requim the waluation of a number of elementary integrals of known typeq and we refrain from supplying the details [cf. Eq. (5-4.197) for the result].

Harold has a question. H.: Now that you have done the calculation in both a causal and n o n c a d

manner, which method do you d d e r to be simpler? S.: That is not easy to answer categorically, since each method appears to

have charactddic advantagee and dinndvantagee. The non-causal method seems to come more quickly to a form from which the asymptotic behavim of the weight function can be infe!rred, but the c a d method might be preferable if one wants the detailed structure of the weight function (note that we did not complete this aspect of the non-causal v d m ) . I think that the emp- should be placed on the flexibility of the soptl.ce approach, which pesmita the use of whatever computational method is moet ehfective for the purpose, as seen in the light of expanding experience with that type of problem.

H.: I have another queetian. The photan spectral weight fimctian, as evaluated thus far, seems to have a very simple qslymptotic behavior. And yet, in each calculational method, that emerges only after detailed cancellations between different contribution& Is tham, perhas, yet another way of viewing things which would make this behavior plamile, at least, without detailed calculations?

S.: I'm glad you arPked that, ainde I intend to clue this section with a diemidon of just such an indga t ion of the asymptotic behavior of the photon pmpagatian function, ss it is d d b e d by the spectral weight function.

We begin with the g e n d form of the modified photon propagatian function as given in Eq. (6-4.109), for example, but multiplied by e2 to produce the combhatim that would occur m any dynamical application of this function:

5-8 Photon propamion function 111 279

It t more ccmvenimt to replace the weight function a(M2) by the dimdonleee combination M2a(M2). And we make evident the dynamical nature of this function by exhiiting a factor of eZ, without prejudidng the otherrise arbitrary e2 depeade~ce:

The mass m appearing here either belongs to a specific charged particle, as in the calculatim juet performed, or ia mpmmtative of various particles of not too disparate maseea The qmctral region now of i n M to us refers to magnitudes of R and M that are large in amparison with m. Ita consideration is facilitated by partitioning the spectral integral in

where the last denominator o f (5-8.141), undep the

can be approximated as

The introduction of the symbol

then permits the asymptotic form of (5-8.141) to be written as

2W EIodrodynamkrII Chop. S

in which e~ evidently plays the role of an effective charge for a d d p t i o n in whichall~.111.aaasleesthan h areqlicitlymOV8dhthespsbalintegraL

It is now natural to d t e the dimdonlees weight function s(M2/m2, e2) in a similar manner, by introducing e: in place of e2, and referring M2 to K instead of m2:

This is an identity, in which a new functional f a m acmmpaniee the appearance of a new dimdonle88 variable, m2fi2. It is precisely here that a plausible Simplifi- cation aur be introduced. In a demriptian that operatea at a high level of momentum and maaq all explicit reference to d rrmsses can prenrmably be ignored, thra permitting the neglect of m2/A2 a 1. Indeed, one could raise this to the status of a principle of self-combhcy. The actual dependence on m then beoomee implicit in the structure of e:. Accepting this, we have

where the last f m mmgnks that the arbitrary parameter h can, in particdar, be set equal to M m. Thus, the spectral weight function is asymptotically dependent on the single variable

which relation,

is a functional equation for the weight function. Let us SUM m a b l y enough, that the function a(eM2) can be expanded

2 in a power series when eM a 1,

5-8 Photon propagation h n d h 111 281

One can also expand e$ in an e2 power d e e :

where the asymptotic form of s has been used to infer the logarithmic dependence on M2, but does not fix the additive comtant. The combination of (5-8.151) and (5-8.152) then yields

which, in its two leading terms, exhiits just the simple asymptotic behavior that we set out to understand. The explicit result of (5-4201), for spin-; charged particlea, is conveyed by the parameters

In addition, we have learned the logarithmic M2 dependence of the next power of a, as exhibited in

The functional equation of (5-8.149, 150) merits some further dkussion. Let us introduce the variable

which is such that

282 Eloctrodynonrks II Chop. S

Thea the functional equation reads

which is alao c01lveyed by

where

If m e extmpolatae them definitions down to M' - m2, whm x eamtially reduus to l/e2, the'followhg approximate integrated form is obtained:

The weight function thus depends d y on the above cambinaticm of coupling umsbnt and spectral mass, as given explicitly by

here f ' and f - signifjr the derivative function and the inverse hctian, respectively. The form of f ( x ) that CO-& to the expansion of the preceding paragraph, which refers to X 1, is

With the solution of the functional equation given by (&8.161), m e can verify that (5-8.148) is indeed independent of the arbitrary parameter X. For this, we have only to obsewe that the replacement of M2 by K in the variable x produces l/c,f, according to (5-8.14!5), and then

Photm propqation function 111 283

I t is thus apparent that the replatxment of e and m by ex and X, for arbitrary X, alao maintains the asymptotic functional farm.

In Section 4-3 it has been seen that the use of the asymptotic weight function

leads to a photon propagation function with an inadmbible spacelike singularity (of unphysically large magnitude). This situation would persist if the d tenn were added to (5-8.166). Now, let us tun matters about and ask what is implied by the demand that no such shgularity occur. We first note that the second denominator of (5-8.141) is a monotonically d d g function of k2 ( > O),

Accordingly, if this denominator, which equals unity for k2 - 0, is not to have a zero for aome finite k2 > 0, it must remain non-negative as k2 a, or

The fix& ammquence, which is sparked by the neceeeary existence of the integral, is that s must vaniah as M2 4 m, at least slightly more rapidly than l/log M2. It is this requhment that is violated by the simple form (5-8.166).

Suppoee the equality sign in (68.168) is not realized. Then the function k2e%+(k) approaches a definite limit as k2 -r m:

2 Em k 2 e q + (k) = ew , &=+m

where, indeed, c,' is the limit of e: as A + m. In this situation, which is expreeeea by

284 Elutrodynomicr II Chap. S

one could interpret e, aa the charge appropriate to a more fundamental descrip tion whose MC amcepts refer to an instantaneous characterization of certain irreducible entitie~. None of the practical arguments for source theory would be diminished if this should eventually turn out to be the true dmtion, and it is only for definitenese that we adopt the contrary position (hard-core sourcery), as expressed by the equality sign in (5-6.168):

The variable X of (5-8.1S6) could then be written

and thus approaches rao as M' -. m. The weight function mu& Mniah in the latter limit. The way that it do= this is d c t e d by the implication of (5-8.161) that

which, according to (S-8.160), mquh that u(l/x) approach zero, as x -, 0, at least as rapidly as X. If this behavior is, in fact, linear in X ,

we have

and

Another poesity,

comeqxmds to S vmiddng as X*. Theee examples only illustrate that, without further physical infomation, there are mdl- mathematical pomibilitia of ex- trapolating from the initial asymptotic form of the weight function, given in (5-8.163), to the ultimate asymptotic farm as M2 -+ m. And, whereas the initial

5-9 Photon decoy of the pion 285

behavicw, with the coefficients (S-8.164), refers to pure electrodynamics, the final asymptotic limit involves the totality of physica It is for the latter reason that the statement of (5-8.171), which has the form of an eigenvalue equation for a - e2/4a, cannot be exploited for that purposs in the pmmt or foreseeable state of physical knowledge.

5 - 9 PHOTON DECAY OF THE PION. A CONFRONTATION

We have already made m e reference to the physical proceae r0 -. 27 in Section 4-3. A phenomenological description of that coupling is given in Eq. (4-3.125), and the associated decay rate appears in Eq. (4-3.139). This topic is taken up again in order to discuss a dynamical model of the mechanigm. Neither the model nor our handling of it is realistic; we are not yet ready to conclude the strong- interaction aspects of the procee& Rather, the emphasis stil l remains on electrodynamic& We have a twofold purpoae in discussing this problem. The first is to provide another illustration of non-causal computational methods; the second is to confront views of this situation that have gained widespread credence and popularity in the recent literature. The nature of this confrontation will be indicated as the development p d

The analogy between the pion and an electron-poeitron combination of equivalent quantum numbers has been commented on in Section 4-3, and is implicit in the more extended discusion of Section 5-5. We h a dynamical model on that analogy, in which the pseudoscalar pian field, +(X) , is locally coupled to the appropriate bilinear combination of fields, #(X) , that are associated with charged, spin-; particles, Th- particlee can be thought of as protons, but the only explicit characterization of them that enters our model is a r d c t i o n to large mass, relative to that of the pion,

m m,. . (5-9.1)

The primitive interaction we have d d b e d is exhibited as the Lagrange function tenn

and the initial acticm eqmmion that refers to the charged particlee is therefore

286 ~ ~ y n a m i a 11 Chap. S

Sup- now, that field and source are redefined by the I d transformation

although, despite the appearance of the exponential function, only terms at mast linear in 9 are of intan& in the paent dimusion. The an- of y5 and its anticmnmutativity wi th yO mbine to maintain the form of the source team, while

and

Note &at only the first of thae transformatim has actually involved the reetrictioa to no more than linear +term& We r e q p b in this way that, correct to teams linear in the field +, there is an equivalence between peeudowh and dov do vector ampling:

The question whether this equivalence is indeed r e d i d in explicit calculatio~, qecifically of the mhtive decay, is at the heart of the somewhat c o n t r o v d problems to be studied.

The elementary coupling betwean the field 0 and the fields of two photoas is produced by d d d g the mchange of a pair of charged particle^. Such considerations are entirely analogous to thaw given in Section 4-8 for the dimamion of light by light scattering, and we have only to apply the formula of Eq. (4-8.19), through the subetitution

5-9 Photon duoy of )(re pion 287

to get the following inlmwtion cqmmim for pcmdogcalar coupling:

Owing to the preeence of the matrix y6, the yl? term disappears from the trace, and the desired interaction is produced entirely through the aF tenn in the denominator. Thus, we have

We &all evaluate this eqmmian in an approximation that is based on the mass inequality (5-9.1). Since the photons share equally the total energy m, in the pion rest frame, they cany amall momemta on the scale set by m. Accordingly, it suffice8 to regard their fields as 810wly varying. Then, recalling that [Eq. (4-807611

and employing the kind of 'R evaluation exhibited in (4-8.!56), we get

In stating the final fonn, we have uaed the integral

which ie analogotm to, and derivable from, (4-10.40). As in the dhcumion of the latter, we also remark on the alternative evaluation through transformation to a Euclidean metric (pb ip4) and the use of the surface area of a unit sphere in four dirndona, 2w2:

2 flutrodynomicr II Chap. 5

Than, applying the d t s of Section 4-3 that were cited in the first paragraph of this section, we infer the pian decay rate as

Now let us see what happens when the equivalent peeudovector coupling of (5-9.8) h is used (6-9.10):

g m - y l l - +in-ivy6a+ 2m p n 2 + m 2 - e q a ~ '

In this situation, the pmma of the a d d i t i d matrix v singlss out the ylf tenn in the trace and, as we shall verify in a moment, it sufficee to consider me power of aF in the expadan of the denominator,

Contact with the peeud& form of the coupling should appear on transferring the derivative from the pion field. In doing this, it is important to maintain the appearance of gauge invariance, which is accomplished by writing

iap+ - [ +, n,]

and them

The approximation of slowly varying fields (m, a m) is ueed again to rewrite this as

It is the preearce of the commutator that introduces an additional field-strength fa-, thus juaifying the omission of (OF)'. The use of the basic commutator

5-9 Photon d u a y of the pion 289

in the sufficiently accurate approximate form

The introduction of the Euclideen d c , and an evident symmetry, permits the latter momentum integral to be evaluated as a surface integral extended wer a momentum sphere of large radius:

Its consequence for (5-924) is

in agreement with (6-9.13). Harold lifts an eyebrow. H.: It is very intmeding to see the emergence of a surface integral in momen-

tum space as the imtmment for maintaining the equivalence theorem of (5-9.8). But why have you abandoned your usual exponential representation method for handling such problems? Could it be that it has no counterpart to the comparative subtlety of a surface integral?

S.: Shall we find out? Let us return to the peeudoecalar expmfion (5-9.10) and proceed to run

rapidly through the evaluation

Chap. S

as expectd. It is the peeudmmctor coupling that needs our attention. The latter [&. (5-9.1711 is now written as

which again anticipates that one uF factor wi l l mffice. The we of the commutator form (5-9.19) gives

but this time we evaluate the ammutator directlyt in the sufficient approximation of Eq. (5-923):

The Tr waluatian preeen~ts the interaction as

where the latter form introduces the Euclidean metric, with the related ttansfor-

5-9 Photon d u o y of the pion 291

mation 8 -is, and has employed an obvious four-dimensional 8JfmXIl&Jf0 The question of equivalence thus redu- to whether the final integrations will yield

That the answer m q u h ~ o m e care become6 apparent on inserting the intinite- momentum-epace Euclidean integrals

The left aide of (6-9.33) vanishd This is where we must stop, and think about physics. Implicit in any non-causal

calculation is the nquhment of initial non-overlap between the fields of the emitting and absorbing sources-here, the pion field and the two photon fields, respectively. The complement to this insistence on a M t e spaa+time interval between the two classes of fields is a limitation on the magnitude of the momentum that is exchanged between thean. To convey this restriction in a calculation that has not made it explicit, which is our preeent situation, we must allow the domain of momentum integration to become infinite only at the end of the calculation, corresponding to the final extrapolation to the circumstance of overlapping fields. (A similar remark occurs in Section 4-8, in the context of light-by- light scattering.) With this comment in mind, we return to the test of (5-9.33), and first compute the momentum integrals for a Euclidean sphere of finite radius R

and

Chop. S

The left side of (5-9.33) then reads

where, indeed, if P -r a, inside tbs integratian sign, for any finite 8, the result is zero. But, if we retain a finite but large m2 until the 8-integration is performed, the final integral of (5-9.37) becomes, effectively,

and (6-9.33) is v d e d . In retjroepect, it is evident that the surfaceintegd evaluation aleo refers to a large but finite momentum domain, and that the two camputations are equivalent. To the latter remark we add the specific ohs- tiolrs that first performing the 8-integration in (5-9.33) giv-

and that the four-dimdcmal momeantum integral on the left of (5-9.39) is a b the me encountered in the pmudoecalar calculation [Eq. (S-9.14)J. Now that we have brought to the surface the cimmatmce of initial non-over-

lap and final spacetime extrapolation, it is natural to ask how theee mattem go in a causal calculation, where the extrapolation procedure is quite explicit. Consider, then, a causal arrangement in which an extended pion source emits a pair of charged m c l e 8 that eventually andilate to produce a pair of photon& [The causal diagram can be drawn as an isosceles triangle standing upon its apex, to which the virtual pion line is attached. The opposing, horizoatally drawn side refers to a virtual charged particle,] The primitive pseudacalar coupling defines an effective two-pnrticle d o n source:

while the effective two-particle detection sowce amdated with the two-photon emhian pracess is [cf. Eq. (4-8.3)]

5-9 Photon decay of the pion 293

The vacuum amplitude d m i i n g the two-particle exchange, as inferred from (4-8.4), is then

which also foUm from (5-9.10) on insorting the expansion of G+". The causal situation under consideration is conveyed by the propagation

function fotms

G+ (Xt - rt t ) - i / do, eiHx'-x")(m - YP) ,

and by writing [d* Eq. (4-9.3)J

Ap(x) - A / ( x ) + A t , (5-9.44)

whem a and b deeignate the two photons, with, for example,

A:(%) - i ~ , * ( d ~ ) ' ~ e : * e - ~ * ~ . (S-9.45)

The d t i n g expmdon for the vacuum amplitude is

in which

Owing to the prawn= of matrix y5, the significant stnrctme of the trace, which is

294 Elmctrodymamtcs II Chap. S

illustrated by

reduos to the products of four different components of y', as given in

Them contributions are mmciated, in three diffemmt ways, with the presence of a single factor of m in (6-9.48):

the intermediate atep in this reduction exploits the equality of the vectors p - k, and - ( p f - kb).

The mmequent form of I is the product of (5-9.60) with the invariant momentum-space integral

This i n b g d h easily evaluated in the reet frame of K (K - 0, KO = M), where al l particles and photons have the energy $M, and the integration reduces to one over a scatteakg angle of coeine z:

in which


Having devised the vacuum amplitude (5-9.46) for c a d circumstances, we now make that space-time arrangement explicit by writing

and then proceed to mcomtruct the electromagnetic field, which is already e x p d in terms of gauge-invariant field strengths, After imerting the relation

the vacwm amplitude beam-

and the space-time extrapolation is performed by the substitution

The d t i n g contribution to the action can be presented as

where the form factor has the momentum version

It is normalized at k2 - 0:

296 Electrodynamics II Chap. 5

Under ckcmstmcee of small momentum transfer, )k2) a 4m2, F(x - x3 effeetively becorn= 6(x - X') and (G9.58) reducee to the local coupling of (5-9.13), as erpeeted. In the actual situation, where - k2 = m:, there is a Wall correction factor which, according to (5-9.59), is

If m is takem to be the proton mass, 80 that m J m a 1/6.7, the comction is about 0.2%.

The replacement of peeudmcah by peeudovectm coupling is erpreesed, in the causal vacuum amplitude of (5-9.46,47), by the subetitution

Then, since p and p' are real particle moments, p2 + m2 =pR + m2 = 0, the effect in (6-9.4'7') is such as to immediately recover the paeudoecalar coupling. Why, then, is there any question about the equivalence of the two coupling forms in this pmcas? To provoke that problem one must retain the peeudovector form until after the space-time extrapolation is performed.

The illustrative trace of (5-9.48) now becoma

It can be decompomd into two distinct contri7butiaq of which the firet is the following three terme with the factor m2,

while the remainder is

5-9 Photon decay of the pion 297

The evaluation of the latter trace is assuredly fwsi'ble, but we shall not trouble to do it, since

This means that the term in question does not contribute to the causal process and can only enter as a contact tenn. I t may therefore be put aside until the end of the calculation, wherqany newismy contact term is to be inserted on the basis of the physical requirements that attend the calculation. In the present situation, the only requirement is that gauge invariance be maintained after the space-time extrapolation.

With the alterations we have indicated, the causal vacuum amplitude of (5-9.56) wil l be replaced by

2m2 1 + v do,, eiK(x-x') a'+(xt) - log - l M' 1-0 '

where, as in Eq. (3-8.14),

The issue can now be squarely drawn. Does the paeudwector structure *PPA, have an independent existence that demands for it an explicitly gauge-invariant form, or is i t merely one factor in a complete expression which should be gauge-invariant only in its entirety? The proponents of current algebra take the first view; we champion the second me. With the latter attitude, one has only to remark that the transfer of the derivative in (5-9.68) produces

and we immediaMy recover (5-9.56), which, with no need for a contact term, directly yields the action contribution of (5-9.58). But, if *FPPA, must be made explicitly gaugeinvariant, while maintaining the pseudwector field, it is necessary to make the following substitution

298 Huhodynomics I1 Chap. S

and then the transfer of the first two derivativee produtm the vacuum amplitude

If the d t i n g coupling is applied to the decay proceee where, effectively, an 4 m:, the additional factor of rn:/jU2 < m:/4m2 yields an esseatially null redt for the pion decay constant, And, indeed, this was the conclusion drawn from the initial application of current algebra to the process p. Sutherland, M. Veltman, 1967. Theee and other references, as well as a careful discuseion of the current-algebra viewpoint, can be found in the contribution of R Jackiw to Lectures on Czmmt A & e h and Its Appkntbm, Princeton University Prees, N J, 19723.

The breakdown of current algebra thus revealed could be traced to the neglect of 'anomalous' equal-time commutators of certain current components. This customary language is unfortunate, for although theee additional terms do not appear when formal operator manipulations are employed, their pmtence is demanded by general physical requknents. Here is the reason that no 'anomaly' occurs in the soumetheory discussion-we have utilized the physical nqub ments directly, without reference to operators. Incidentally, we have also seen the podbility of obtaining a null result by purely mathematical manipulations, and then recognized that it originated in insufficient attention to the physical context of the calculation. It may appear to be a trivial aemantic point to deplore the use of the tenn 'anomaly', since the final current-algebraic d d p t i o n of the pion- photon coupling, at the level of dynamic8 now under consideration, is the expected one. Yet, like all inappropriate usages of language, it can and has led to error.

The point at issue refers to higher dynamical levels, where the internal exchange of additional photoas is taken into account. To the current algebraist, the coupling of (5-9.13),

is an anomaly associated wi th the pediaritiee of the 'triangle dhgmm', and no further contributions are crJcpected fram more elaborate mechanisms [S. Adler, 19691. Thus the coupling (5-9.73) is alleged to be valid 'to all orders in a'. And detailed calculations have been carried out by various authors with d t s that

5-9 Photon decay of tha pion 299

are inteqmted as supporting this dictum. But independent source-theoretic calm- lati0118 of the causal type have also been performed by members of the UCLA Sourcery Group [][lester L De Raad, Jr, Kimball A. Milton, Wu-Yang Tsd, Phys. Rev. D 6,1766 (1972); Kirnball A. Milton, Wu-Yang Tsai, W e r L De Raad, Jr., Phys. Rev. D 6, 3491 (1972)l which show that, at the level of one internally exchanged photon, the coupling of (5-9.73) is modified, by the factor

Them calculations have met the sourcetheory requhment of internal comktency by being performed for two different causal arrangements, with concordant d t 8 , But, since they are rather elaborate, one might wish for a more transpar- ent attack on this conceptually important question. We shall fill this need by *g the noncausal approach, and therefore refer the interested reader to the above cited papens for the alternative causal calculations. That we present a simpler method detracts in no way from the significant achievement of the three Sourcemm in pushing their calculations through to a conclusion, and defending it against the firm Establidment ruling that no such effect could exist,

The discussion to follow is quite similar to that of Section 5-8, with one of the photon interactions replaced by the pion coupling, in accordance with (5-9.9) or the alternative of (5-9.8). However, we lack some of the information that was available in the purely electrodynamic discussion, namely, the dynamical modification (to order a) of the primitive interaction (5-9.2) and of its pseudovector equivalent. To that end, let us conaider the appropriate modification of the Eqs. (5-6.2,3) as expmmd, for pseudoscalat coupling, by

Attention now focuses on the part of M that is linear in +:

which we write out as a typical matrix element involving the particle momenta p', p", wi th the pion field supplying the momentum

Chop. S

and

1 1 1 - - - ita 824' u l l ?j h C - ~ X , (&gm) (p' - + m2 ( p M - k)' + d k2 -1

with

which h incoqmratae a finite photon m ~ n a In addition, owing to the appearance of the factors (p' - + m2 arid (p" - k)2 + mZ in (5-9.79), we need the simpler combination

and its analogue with p', X;, D,' + P", X,", D,"-

5-9 Photon doeay of the pion 301

After making substitutions such m

and performing the k-integration, we find that

a & M,- - - i / & ~ l k u ~ e - ~ ~ 2 r

The contact term is now i n f d by impodng the physical normabtion con&- tion that, in the situation of real particle propagation (yp' + m, yp" + m -, 0) and d momentum W e r (9 + 0), the p-ce of M, shan imply no modification in the initial coupling. Hence,

with

and

We illustrate the effect of the final combination, for the situation of real particles and arbitrmy p2, by exhibiting the form factor

302 Elutrodynomitr II Chop. S

whem it has been recognized that the structure of the integral is almoet identical with that .of (4-14.66), lacking only the factor of l - U. Note that the normdb- tion condition, in the fonn

effectively applim to any momentum transfer such that 131 a m2, which includes -3 - m:, .ccading to the Simplifying d c t i o n of (5-9.1). Since it is convenient to work with the separate parts of the maee operator

structure, we made them well defined by introducing the convergence factor

as in (5-8&5), which is only r e q u i d in the eecond of the terms in (S-9.84). and in 5;. The latter beeornet3 (A , m)

The replament of paeud& by peeudwector coupling in (5-9.78) is conveyed by [this is analogous to (5-9.6311

and thedore

where

for example, is the familiar mass operator (without contact term) for a free particle. Accordingly, the contact term associated with M,,

is given by

Photon decay of the pion 363

(S-9.96) mc

S,.- 1,- m. in which [Eq. (5-6S)J

m, a ds - -1- &(l + u)e-umauyl - e - ~ A a ( l - u ) m 2w s 1

a -I1 du ( l + U) log 2~ 0 m2u2

If we also decompoee f, into high- and low-msssaensitive parts,

(p".- (b.+ l;, (5-g#)

we have

m, a X (5-9.99)

and

a m l" = {G*= - - log -. P". w C

Let us also remark on the relationship with the dectromagnetic contact-tenn parameters m. (5-7.62), (5-8.85)], namely

whence

Armed with the requid information, we begin the discuseion of peeudwector coupling by simply following the path that starts at Eq. (5-8.6), where now

304 Elutrodynomics I1 Chap. 5

When one has reached Eq. (&83), which refers qecifically to the third term of (5-8.7). the two photans that are of intereet in the praent problem have been introduced, one explicitly as Ab and the other implicitly in the field variation. We then translate the statement of contact terms given in (5.8.10) as

or, reuniting all photan fields into the field A,

But we shail also find it dosirable to deviate from the earlier treatment by returning to the first two terms of (5-8.7) and proceeding to make explicit the contact tpams that are already incorporated in the mam operator there called MO (while removing that subscript, evocative of a null field, since it is now neceeeary to repreeent a second photon). The contact terms associated with M are [Eq. (5-6.5511

-mc - I,(Y~ + m). (6-9.106)

Accudngly, the following contact teame are contained in the analogue of the firat two parts of (6-8.7):

or, with all photon fields united in A,

The complete list of contact terns that is appended to

5-9 Photon duay of the pion 305

where M is devoid of contact terms, is therefore

One quick approach to the first of the two contact terms in (5-9.110) is through the recognition that

aince the computation of Eqs. (5-9.17-26) givs the latter Tr a value that is independent of m. It is more instructive, however, to repeat the calculation in the same spirit than to merely apply the known result. Accordingly, we use (5-9.19) to get

and the Euclidean momentum integral that now appears [cf. Eq. (5-g.%)] is

owing to the additional factor of p' in the denominator. Having seen this, it is immediately clear that (5-9.109) also vanishd The only information that is required concePning M is the remark that, through its exponential dependence on PS, the final momentum integration over the surface of an arbitrary large sphere will enforce the limit s 4 0, where the convergence factor (5-9.90) vanishes. In 0 t h words, the presence of the mass operator in (5-9.109) cannot reverse the situation already encountered in (&9.111), and both contributions are zero. What remains is the second term of (5-9.110), a multiple of the initial coupling (5-9.17), which multiple, according to (S-9.102), is a/2r. Here is our version of the source-theoretic d t that the initial coupling is modified by the factor

906 IEloctrodynamicr II Chap. 5

A reacling of the operator field-theory papers will show that we are not merely quading about the evaluation of inte2p.als. Erm has entered theee papem just at the point where r e n e t i o n is introduced, for, with the customary emphasis on the removal of divergences, it is taken for granted that two renormalization constants that have the same singular behavior are, in fact, 'equal. The analogue in our procedure would be to reanark that

impliee the equality of l,,,,. and f,, thereby ignoring the finite difference, of a/2r. In short, the sin of the current-algebraista has been to define a cigdirrmt parameter -y, rather than by examining its physical meaning. And that is precisely what source theory is all about,

Our dbmsion is completed by showing, in a ratha different way, that the same conclusion follows from the consideration of pseudoscalar mpling, which is the verification of the equivalence theorem at the next dynamical level. The p6eudu6calar counterpart of Eqs. (5-9.109,llO) is

Again, we begin with the m, term and first remark that

since the effective evaluation of this Tr given in Eqe. (5-9.10-13) shows that it is propodicmal to m". But, alternatively, we have

whare a means that the yxa+ team that is ale0 produced in anticommuting y n with y6+ has bear d W d , as justified by the n d surface integral to which it


would lead. Now note that, through the relation (5-9.96), the last two tenns of (6-9.116) combine into

Hence, the equivaleblce between peudecalar and ptwudovector coupling wil l indeed be maintained at this dynamical level if the first term of (5-9.116) v d e a :

We now verify this by explicit calculation. The procedure of Eq. (5-9.118) convert8 this statement into

The structure of MA for an arbitrarily strong homogeneous field has been given in Section 6-6. But very little of that detail is needed here. We refer tb the c o m c t i m of (5-6-43), sans et., and remark that the odd y-matrix tenn can be omitted, as can field-strength combinations of the fonn F'%,, since only %'cp ie of interest to us. Accordingly, all that survives of (5-6.43) is

and we have been careful to include the convergence factor. In addition, the yp yp structure annihilattee uF and we have, effectively,

308 Noctrodynomicr It Chop. S

whem A2, correctly for them has been treated as a very large quantity in the first of them terms, but it has been noted that the situation of u -+ 1 requires more careful treatment in the second tmn. After the U-integrations are performed, and with the factor of (a/n)m omitted, (5-9.124) reads

in which

The Euclideem momentum integrals thsi finally exprea~ the two different contributions to (5-9.121), with E H produced by expansion of the denominator and by MA, mqectively, m proportional to

The first of theee integrals is evaluated as

and the second one predsely cancels it. All is well. a&mthm The evaluation of the decisive combination of

(&9.102),

was rather indirect. It may be instructive to see a quite direct and elementary computation of just this combination, rather than the separate consideration of the two contact terms. We first observe that the contact terms are designed to mmuve from M CEq. (69.76). with peeudovector coupling] the appropriate linear field interaction, under the physical conditions of real particle propagation [yp' + m, yp" + m -. 01 and negligible momentum transfer [(p' -pt')' -+ 01. It

5-9 Photon duoy of the pion 309

is quite sufficient here to simply place p' = p", and thus the two cantact terms are to be obtained from

and

where the left- and right-hand factor reduction, yp' + m 4 0, is understood. The two exprdons differ only through the presence of m additional factor of iy, in one of them. Accordingly, whenever the central matrix, y' or iy'y,, appears multiplied by a number, or lacks a matrix factor on one side, the resulting contributions to the regpective contact terms are identical and cancel out from the difference. To exploit this property, we write

y'[m - d p ' - h)] - [ m + y(p' - k) ]yP + 2(p' - k)',

and then conclude that we have only to consider the respective structure^

which have already received yp' + m + 0 simplification. At this point, we invoke the mpmmtation [inferred from (4-14.10, ll), for

example, by differentiating with reepect to m2, after which one seta pn, replacing P, equal to -m2]

The k-integration is then performed with the aid of the substitution

310 Eloctrodynsmics II Chap. S

The first of the tana on the right bearmar a multiple of 9 yv ypn and yields identical contributions to the two contact terms, since 7, cornmutee with an even product of y-mattice& Note that this part of the K-integral requim a convergence factor. But the identity of the respective contri'butions, and their exact cancellation in f, - c,, is obviously independent of the choice of that mathematical function. Accordingly, (5-9.133) effectively reducee to

which uset3 the integral (6-9.14), and indeed,

One last renuvk seems to be called for. An additional piece of evidence adduced by operator field t h e an behalf of the claim that the coupling (5-9.73) is exact, refers to the fictitious situation of massle~~ electrodynamics, m - 0. And, indeed, the preeeace of the factor m2 in the first line of (5-9.137) might seem to indicate a null result for m2 -, 0. The erroneous nature of that conclusion is evident in the second line of the same equation; the momentum integral is singular in the limit rn2 -. 0, and the whole structure is actually independent of m2.

Index

h l u t e simultaneity, 2 Action. See a&o Action principle.

associated with single photon exchange, l45

coupling two elecbromagnetic fields through single fermion pair exchange, W(*), 261

for neutral pion coupling to photans, 296

in para poeitmnium, 138 W2'

modified by single virtual photon exchange, W4), 252

non-causal evaluation of, WC4)

contact term in, structural analysis of, ZM

Action principle, 25. Sae a380 Action. for two-particle interacting non-relativ-

istic system, 9 &g equal time fields, 12

Adler, S., 298 Amplitudee, of mixed ortho-para poaitro-

nium states, 127 Analogy, with time development, 146 Angular momentum, and spin $ particle%

W Adhihtion coupling, in ortho poaitro-

nium, 126 Anomaly, in pion decay into photons, 298 k a l vector? 67

Beeeel function, of hmghry argument, K,, 180

Bound state, as a oompoeite particle, B Breit interaction energy, 38

C, 114. See (3eo Charge parity. CP, 114 C a d arrangement, for two-particle in-

teraction ddp t i on , 30 C a d control

in twbparticle exchange, 91 and unstable particles, 9f

Causal Diagram, for pion decay into photons, 292

-8 parity of n-photon state, l13

in poaitronium, 113 Charge reflection, eigenvectors of, 1 s

and a selection rule, 112 Completeness, of eigenfunctians, 16 Compoeite particle, and statistics, 24 Contact terms.

fiction associated with, 152 locality of, 191 for M('), 198 for pseudoscalar coupling, 301 for pseudovector

coupling, 302 dynamical modification, NS relationship with electromagnetic

parameter, 305 for strong field calculation, 160

Convergence factor, 302 Coordinates

center of mass, 13 relative, 13

Current, transveate, symbolic construction of, 36

Cumnt algebra, 291,298,806

Decay constant for magnetic dipole radiation, l7l, 174

312 Index

Decay canstant (cont.) in magnetic field, at high energy, 1a of para poeitronium state, 133 of state in a magnetic field, l68

Decay rate, of mixed orthepara poaitronium atate, 127

De R-4 L 299 Derivative, functional, U Determinant, differential property of, 149 a w e t i c term, in operator, in Divergence equation, 77 Divergences, 140 Double commutator, identity for, 216

Effective charge, in esymptotic photon propagation hct ion, 280

Eigenfunctions, 14. See crlso Wave hmc- tion.

including center of maas motion, 16 completenas property of, l6 differential equation for, 16 and Green's function construction, 16 multi-time, 17

differential equations for, 17 and orthonormality, l8 physical interpretation of, 18

orthononnality of, 15 Electrodynamics, narrow sense of, 112 Energy

of ground atate in strong fields, 164 kinetic, non-relativistic decomposition

of, 14 Energy displacement. See cJsa Eaergy

ahift; Mase, displacement of. additional, in ortho poeitronium, 126 apparent discrepancy in, 68 effect of modified vacuum poldzation

oa, 10s high frequency, of order m/M, W,g9 low frequency, of order m/M, 86.74 of order a(m/M), for Coulomb interac-

tion, 6s relative to fine structure, 66

of order m/M, dimate for, n f t ~ 2p 1 e ~ 4 g9 for 2s level, 87

poeitronium, see Orthepara splitting. relative, modified, of ortho and para

levels, 140 of ortho and para levels, m, m of 2s and 2p levels, m, 109

Energy operator, Dirac, transformed, 62 internal, non-relativistic, 47 non-relativistic, of positrmium, 123 of pRra poeitronium, 122 positronium, submatrix of, 127 rest frame, of poeitronium, 122 tweparticle, approximately relativistic,

46 with large mass ratio, 48 in reet frame, 48

Energy ahift. See also Energy displace ment.

for instantaneous coulmb interaction, 63

for non-local intaraction, 61 of order m/M, one-photon exchange,

78 for 2p level, 79 for 2s level, 79 unit for, 79

technique for evaluating, 76 Energy spectrum

in magnetic field, 167 with a/2w magnetic moment, 169

one-particle, approximately relativistic, 61

relativistic, for E@ 0 particle, M single particle, from second order Dirac

equation, 63 total momentum dependence of, 48 two-particle, expectation values for, 49

first relativistic corrections to, 44 with large mats ratio, M)

and reduced mass, 82 Energy splitting, ortho-para, modification

of, 128 Equations of motion

non-relativistic, 68 solution for constant field, 146

Equivalence, between peeudoecalar and pseudovector couplings, 286

Index 313

Euclidean metric, in momentum space, m, 289, m, 291, m, 308

Euler's dilogarithm, 96 applications of, 91

Exponentials, theorem on combination of, 204

Factor, modifying pion-photon coupling, 299.30s

Feynman diagram, see Non-causal dia- l.!rf'm*

Field, difference, 9 multi-time, equal time speciali211:tion of,

2 single particle, non-relativistic, 2 source of, 10 two-particle, differential equation for,

7 two-particle, equal time, 4

diff erential equation for, 4.8 two-particle non-interacting, non-rela-

tivistic, 2 Fine structure, M)

constant value of, 250 in positrcmium, 112 and reduced mass, 52

Form factor and causal control, 9l causal diagraxn for, 195 charge and magnetic, for scattering, as

checks, 220 effective, in ortho-para splitting, 129 in neutral pion coupling to photons, 296 non-relativistic, as wave function, 100 in photon weight function, 111 for p s e u d d a r coupling, 301 spin i , in vacuum polarization calcula-

tion, 105 in three-photon decay of positronium,

l18 Functional derivative, 24

of Green's function, 76 as symbolic photon source, 29

Gauge invariance, as a check of algebra, 206 maintenance of, 288 and tensor structure, 88 and Ward's identity, 207

Generating function, for energy spectnrm, 36

energy spectsum, physical information in, 36

of S-states, 66 Green's function. See a&o Propagation

function. advanced, 42 approximate equal time equation for,

with non-local interaction, 60 change of order m/M, through single

photon exchange, 77 composition property for, 77 effective single particle, 77 eigenfunction construction of, 15 equal time, differential equation for, 3

symbolic construction of, 26 factorized, 14 functional derivative of, 24,27,32, 75 identical particle symmdry of, 28 for immobile particle, and photon

source, 33 multi-time, eigenfunction expansiuns of,

16 integral equations for, 20 symbolic construction of, 24 unified expression for, 20

multi-time and equal time, relation of, S non-relativistic version of, 76 relativistic, with instantaneous interac-

tion, 38 retarded, 40 single particle, relativistic, equal time

limit of, 39 spin i, 145 transformed, boundary condition for, 43

differential equation for, 41 sets of equations for, 42

two-particle, differential equation for, 7 equal time, differential equation for,

8

3 4 Index

Green's function (cont.) non-interacting, relativistic, 30

unitary tranf3formation of, 40 Ground state

energy in strong magnetic fields, l64 in magnetic field, 167

Hard core murky, 384 Harold

on the asymptotic behavim of the photon spectral weight function, 278

on comparison of causal and non-causal methods, 278

on a comparison with experiment, 144 on a conceptual and computational im-

provement, 140 on the existence of an integral, 138 on the expectation value of yO, 168 on magnetic moment calculations, %l on the quantum correction to synchro-

tron radiation, l87 on "mgularhtion", 268 on surface integrals, 289 on the vacuum polarization calculation,

110 on Ward's identity, 207

Heaviside step function, l67 Heavy photons, 140 Hyperfine structure

interaction for, 48 of muonium, 140 in positronium, 112

Infinite momentum frame, and three-particle kinematid integral, 82

Infra-red cutoffs, 140 Infra-red problem, mathematical origin of,

162 Interaction, additional, of order m/M, 70

non-local, Green's function treatment of, 68

phenomenological, for p a ~ a poeitronium decay, 116

Interaction energy, additional, of order m/M, 64

Coulomb, 34

instantaneous, 38 of positronium with magnetic field, 126 transformed, approximation to, 46

submatrices of, 41 Interaction function, non-relativistic,

property of, 6

Jackiw, R, 298

Karplus, R, 140 Klein, A., 140 K'ung Fu-tzu, 268

Lorentz gauge, 260,2488,270, e73

Magnetic field energy spectrum and eigenfunctioas in,

Is6 -g, 144

Magnetic Moment causal diagrams for, 1921; comparison with d e n t , !B0 contact term, contribution to.

(c!+, 199 ( c2 )a, 233 (c2),3, 225 (%lr, 226 (%)$P 238 (cz)'b, 249

induced, 163 labor saving devices for, 2!H modified propagation fimction contrii

utor t o (%)C, 196 non-causal diagram for, 1921; numerical coefficient of ( u / ~ u ) ~ , 260 strong field modification of, l78 three particle exchange process in, 194 vacuum polarization correction to,

( ~ 2 ) " . pd.9 1909 IQ1 Magnetic polarization, 168

strong field modification of, 178 Mesa, displacement of, in ortho

positronium, 126 in para positrdum, lS3,lSS

Index 3 5

reduced, 14 spectral restrictions on, 85

Mass operator M

decompueition into M, + M,, 200 t&s of cantributians to, 263

M1 non-causal calculatim of, 267 pre4rentation of, 262

M,, canetruction of, 201 magnetic field

for energy state, 166 for first excited level in, l7l for ground state, 160 imaginary part, at high energy, 180 imaginary part of, 167 to quadratic terms for, 170

remainder, M,(*), 201 for single photon exchange, M, 197 for strong field calculation, l65 for two photon exchange M(2), 196

Matrix element, in ortho-para poeitronium transition, 127

Mathematical addendum, direct calculation of relation between pseudovector and electromagnetic con- tactterms,3Q8

Milton, K, 299 Momemta, center of mass, 13

relative, 13 Multiparticle exchange, 1 Multi-photon pmmmm, 28 Muonium, 48

hyperfine splitting, agmment with experiment, 144

a contribution to, 143 non-relativistic formula for, 143

two-photon exchange in, 140 and weak interactions, 112

Neutrino, in muonium decay, l12 Newtan, R, 178 Non-causal diagnun for magnetic mo-

ment, 196 Non-overlap, in n o n a d calculatian,

291

Normalization conditions for M,, 151 for pseudoecalat coupling, 301

Orbital quantum number, for poaitronium, 112

Ortho-para splitting, compIete modification of single-photon annihilation contribution to, 130

contributions to, 139 form factor modification of, 180 modification in annihilation contribu-

tion to, 129 Ortho pogitronium, 114 Orthonormality, of multi-time eigenfmc-

tions, 18 Osdlator, one-dimdanal, 157

P, 114. See also Space parity. Para pcxitronium, 114 Parity

charge, 113 in-c, 113 orbital, 113 apace, 113

Particle, compoeite, and phenomenological d d p t i o n , B

ideartical, 28 spin i, angular momentum properties

of, W Photon mass, fiction associated with, 152 Photon propagation function

and inadmissible spacelike singularity, 283

asymptotic form of, 279 convergence factor in, 267 general form of, 278 modification of, 223

Pi-meeon, see Pion. Pion

decay rate of, 286 photon decay of, 286 peeudoecalar coupling to fermi- S S pseudovector coupling to fermions, 286 two-photon coupling of, 287

3 6 Index

Pdtxdum, Bohr d u b of, l16 energy shift in, 139 =m3Yspectnrm;of,= non-relativistic binding enqiee of, 1l2 ortho, decay rate of, 1!U

p u n d level lifetime of, 12a mixed states of, 125 non-ralativiatic field of, 119 phenamenological field of, 119 single photon exchange in, 126 spectnrm of, 123 three-photon decay of, 116 two-photon decay of, in magnetic

field, 118 ortho-para mixing by magnetic field,

l l 9 , lm ortho-para splitting of, l l 9

ground level, l25 para, decay Fate of, 114

field ppertiee of, 1s fine structure of, 123 ground level lifetime of, 116 instability of, 133 non-relativistic field of, 115 phenomenological field of, 116 two-photon andilation mechanism

of, 130 para and ortho, 114 phenamenological d d p t i a n of, l14 photan decay of, l12 88 pute electdflamic m single and double tmmmw photon ex-

change in, 137 two-photon exchange in, lS5

Pwer clerpdd relativjstic expramion fop, 184 classical eynchrotrpn spectrum of, 103 radiated classically, 108 total, in synchroton radiatian, 1tB

Primitive interaction non-relativistic concept of, 5 for peeudoecalar coupling, 985 peeudoecalar, dynamical modification

of, 299 reptition of, and intagml equation, 7

Principal quantum number, for poeitronium, 112

Principal value, \lac of in form factoa, 93 Probability, Coulomb fkdm for, 89 Propagation function. See crlso Gran's

function. change of order m/M, through two-

photan exchange, 78 modiiied, c, l m multi-time, equal time spechbt im of,

2 non-relativistic, phdd pictur% for, 3

multi-time, and equal time, datiun of, 8

non-relativistic, dj&t form of, S photon, and a ElRWr of pug* 29

~ c t i o n of, 36 covariant, 185 improved treatment of, 80 instantaneoue part of, 37 in radiation gauge, 74 gignificance of gauge teaa3a, !XI spatial compcments of, 57 weight function in, 89

photon, modified, altxmative forms of, 110

in d o - p a r a splitting, l28 for two non-interacting padidea, nm-

relativistic, 2 two-ptuficle non-relativistic, differem-

tial equation for, 2 two-particle, relativistic, with akeletoa

interactions, 30 Propagator. See Propqation function. Ptmdoecalar coupling, dynamical modifi-

cation of, 308

Radiutim gauge+ M advantag- of, 56 construction of photon ppagatim

function in, 74 Relativistic two-particle problem, l Fbormalizatim, and divergmcea, m Riemann zeta Function, 288

Index 317

Scale transformation, modified, 49 and the virial theorem, 47

S e ~ P a p e r s o n ~ E t e o t?va'#, 140

Selection rule, in poeitronium decay, l12 Self-consistency, in neglect of small

masees,= Skeletal interaction theory, 28 Soft photon emission, in pair creation, 87 Sommerfield, C., 261 Source, compoeite particle, and con-

-cy test, 23 multi-time construction of, !44 and statistics, 24

effective, 29 for photon, two spin a particle emis-

sion, 101 two-particle non-relativistic, 5 for two-particle, photon d o n , 86 for two-photon, composite particle

emiseion, 70 for two-photon d o n , 135

effective photon, 39 and functional derivatives, 29

two-particle, equal time, 4 time locality condition for, 5

Source theory, what it's all about, 306 Source theory requirement, of internal

-cy, 299 space parity, in p o e i e u m , 113 Spence function, see Euler's dilogarithm. Spin-orbit coupling, in poe&xmium, 123 Spin quantum n u m k , for positronium,

112 States, of a two-particle system, 22 Statistics, of a composite particle, 24

Fd-Dirac , in positmmium, l13 Surface integral, in momentum space,

289 Sutherland, D, 396 Synchrotron radiation, 182 angular distribution of, 186 classical power spectrum of, 183 Dopples effect in, l85 first quantum c o d o n to, 186

Symmetry, crossing, identical particle, of Green's function, 28

Tensor spin-spin coupling, dgenvalue of, 124

in positdum, 124 Three-particle exchange, kinematid in-

tegral for, 81 Thre&old behavior, of modified photon

weight function, 99 Time, relative, and orthonormality, 18 Total angular momentum quantum num-

ber, for poeitronium, 112 Tramverse interaction, approximate

treatment of, 74 'Mangle diagram, for pion decay into

photons, 298 TBai, W. Y., 299 'Pwo-particle equation, machinery of, l40

relativistic, 32 relativistic and non-relativbtic, com-

pared, 32 7bo-photon exchange, non-relativistic, 89

Unitary transformation, on Green's functions, 40

infinitaind, for generalid virial theorem, 47

simplified, 44

Vacuum amplitude for partial Compton scattering, 198 for photon, two spin + particle ex-

change, 101 for photon exchange by a ampoeite

particle, &d for photon proceseea, 29 for pseudwector description of pion de-

cay, 297,298 symbolic factor in, representing multi-

photon exchange, 30 for threparticle exchange, 88

non-relativistic, 70 for three photon decay, 116 for two non-interacting particlee, non-

relativistic, 1 for two non-interacting spin 3 particles,

30 for two-particle exchange in pion decay,

!m

3 8 Index

Vacuum amplitude (cont.) two-particle, in single systean fonn, 6 for two-photon exchange, 136

in para poeitronium, l30 Vacuum polarization fractional change in, spin 0 , s

8pin ;, 108 modified effect of, 96,1(17

Vector potential, radiation gauge, symbolic construction of, 37

Veltman, M,, 298 Vial theorem, 47

dyadic generalization of, 47

Ward's identity, and gauge invariance, 201 Wave function. See also Eigenfunctions.

short distance behavia, calculational rule for, 138

of 2p state, 60 Weak interactions, and muonium,

112 Weight function a(M2), simpler proper tie^ of, 273 asymptotic

functional equation for, 280 leading powers of a for, 281

constant, G, 278 conetant, C,, 277 photon

modified, contn7,ution to, ft64 modified, interpolation formula for,

100,109 for spin 0 particles, 99

for spin ; particles, 109 threshold behavior of, 99

of photon propagation function, 89 two-particle, change in, 92,107

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